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abstract: 'The effect of uniaxial tensile stress and the resultant strain on the structural/magnetic transition in the parent compound of the iron arsenide superconductor, BaFe$_2$As$_2$, is characterized by temperature-dependent electrical resistivity, x-ray diffraction and quantitative polarized light imaging. We show that strain induces a measurable uniaxial structural distortion above the first-order magnetic transition and significantly smears the structural transition. This response is different from that found in another parent compound, SrFe$_2$As$_2$, where the coupled structural and magnetic transitions are strongly first order. This difference in the structural responses explains the in-plain resistivity anisotropy above the transition in BaFe$_2$As$_2$. This conclusion is supported by the Ginzburg-Landau - type phenomenological model for the effect of the uniaxial strain on the resistivity anisotropy.'
author:
- 'E. C. Blomberg'
- 'A. Kreyssig'
- 'M. A. Tanatar'
- 'R. Fernandes'
- 'M. G. Kim'
- 'A. Thaler'
- 'J. Schmalian'
- 'S. L. Bud’ko'
- 'P. C. Canfield'
- 'A. I. Goldman'
- 'R. Prozorov'
date: 31 October 2011
title: 'Effect of tensile stress on the in-plane resistivity anisotropy in BaFe$_2$As$_2$ '
---
Introduction
============
At ambient conditions, the parent compounds of iron-arsenide superconductors, $A$Fe$_2$As$_2$ ($A$ = Ba, Ca or Sr), crystallize in the tetragonal ThCr$_2$Si$_2$ structure [@Rotter; @Ca-phasetransition]. On cooling, they undergo a structural phase transition with the lattice symmetry lowered from tetragonal to orthorhombic at a characteristic temperature $T_{TO}$ (170 K for $A$=Ca [@Rotter], [@Ca-phasetransition], 205 K for $A$=Sr [@Yan] and 135 K for $A$=Ba [@Rotter]. We denote the compounds as A122 in the following). This transition is accompanied or followed by long-range magnetic ordering into an antiferromagnetic (AFM) stripe phase at the Neél temperature, $T_N$ [@Rotterneutrons]. Indeed $T_N = T_{TO}$ in compounds with $A$=Ca [@Caneutrons] and $A$=Sr [@Srneutrons], where the transition is sharp and strongly first order. In BaFe$_2$As$_2$ $T_N \le T_{TO}$, and the structural transition is second order, whereas the AFM transition is first order [@Birgeneau; @Kreyssigsplit].
Since the doping (or pressure) - dependent superconductivity in A122 iron arsenides exhibits the highest $T_c$ close to the point of complete suppression of the structural/magnetic order, understanding the mechanism of these transitions is very important for understanding the nature of superconductivity. The parent compounds of iron arsenides are metals, so it is suggested that their magnetism is of itinerant character due to a spin density wave (SDW) instability of the multi-band Fermi surface [@MazinSDW; @ChubukovSDW]. On the other hand, it has also been suggested that the magnetism can arise in a local moment picture due to magnetic frustration [@Abrahams] and/or orbital ordering [@orbital1; @orbital2; @orbital3; @orbital4]. Therefore, it is important to conduct measurements that characterize the normal state anisotropy of the electronic structure in the vicinity of $T_N$. From first principles calculations, the electronic anisotropy of iron pnictides was predicted to be fairly high in the orthorhombic $\bf{ab}-$plane below $T_N$ [@detwinning1; @1stprinciple1; @1stprinciple2; @1stprinciple3; @1stprinciple4]. On the other hand, ARPES measurements suggest that a notable energy splitting between $d_{xz}$ and $d_{yz}$ orbitals appears below the transition [@Shen; @Korean].
An insight into intrinsic anisotropy became possible after the development of detwinning techniques, using uniaxial tensile [@detwinning1; @detwinning2] or compressive stress [@Fisher1; @Fisher2] (see Ref. for a review). Electrical resistivity measurements in the detwinned state found the in-plane resistivity anisotropy to have an unusual temperature dependence of the ratio $\rho_b/\rho_a$, peaking just below $T_{TO}$ with maximum $\rho_b/\rho_a=1.2, 1.4, 1.5$ for $A$=Ca, Sr, Ba, respectively [@detwinning1; @detwinning2]. Surprisingly, in BaFe$_2$As$_2$ the high temperature “tail” of the anisotropy is found even in the nominally tetragonal phase above $T_{TO}$. The anisotropy above $T_{TO}$ becomes most pronounced in slightly doped Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ (BaCo122), $x<0.06$[@Fisher1], but is not observed in either Ca122 or Sr122. This difference between Ca122 and Sr122 compounds on one hand, and Ba122 and BaCo122 compounds on the other, may be related to the sharpness of the structural phase transition, i.e., a strongly coupled first order [@Caneutrons], [@Srneutrons] vs. split second order transitions [@Birgeneau; @Kreyssigsplit].
In this paper, we study the evolution of the electronic and structural anisotropy of detwinned Ba122 with special attention to the effects of the applied strain required to detwin the samples. With a much better strain control, we find that the effect above the transition arises from an anomalously large structural susceptibility of the crystals to the applied strain. This strain further separates the already split structural and magnetic transitions in BaFe$_2$As$_2$, as found in the detailed analysis of the temperature-dependent x-ray and polarized optical imaging. We model the effect of the applied strain by using a phenomenological Ginzburg-Landau - type model and show that the difference in the response is directly linked to the order of the magnetic/structural transitions.
![(Color Online) The top right panel shows a schematic of the horseshoe with the potential leads used to apply the strain by adjusting the push-screw. The current leads are mounted strain-free. The top left panel shows the photograph of an actual sample with soldered contacts. The area “A” on the left side of the sample represents the unstrained region. Its polarized light image at 5 K (bottom left panel) reveals clear domain pattern by the alternating blue and orange stripes. The region “B” is located between the potential contacts in the strained part of the sample. It is shown in its detwinned state in the bottom right panel.[]{data-label="1-detwin"}](fig1.pdf){width="0.85\linewidth"}
![(Color Online) Left panels show polarized light images of the strained portion “B” of the sample below (130 K, top) and above (160 K, bottom) the structural transition temperature. The intensity in red, green and blue (RGB) channels for each pixel was digitized using 256 intensity bins. The total intensity was found by summing the intensities of all pixels in the selected region. The RGB color was characterized by the percent contribution of each channel to the total intensity, plotted against the value of each bin to produce the histograms in Fig. \[f2-RGB\]. Right panels show the RGB histograms of a small area of the strained portion “B” of the sample, indicated by the red square in the top left panel. Whereas the blue channel remains almost unchanged, the intensities of the green and red channels shift dramatically indicating overall spectral change. The temperature dependence of this change is quantitatively analyzed in Fig. \[3-color\]. []{data-label="f2-RGB"}](fig2.pdf){width="0.85\linewidth"}
Experimental Methods
====================
Single crystals of Ba122 were grown from an FeAs flux as described in Ref. . Crystals were cut into strips with typical dimensions of 0.6 mm wide, 3 to 5 mm long and had a thickness of approximately 0.1 mm. The lengthwise cuts were made parallel to the tetragonal \[110\] direction, which will become the orthorhombic $\bf{a_o}$ or $\bf{b_o}$ axes below the transition temperature. Cutting directions were estimated by eye using polarized optical images of the domain structure and natural facets on the sample.
Polarized light images were taken at temperatures down to 5 K using a *Leica DMLM* polarization microscope equipped with a flow-type $^{4}$He cryostat, as described in detail in Ref. . High-resolution images were recorded with a spatial resolution of about 1 $\mu$m. Measurements were done with the polarizer and analyzer nearly crossed.
The leads for electrical resistivity measurements were formed by Ag wires, 50 $\mu$m in diameter, soldered to the samples with tin [@SUST]. A photograph of the sample with wires is shown in the top left panel in Fig. \[1-detwin\]. Four - probe measurements were conducted in a *Quantum Design PPMS* from 5 K to 300 K. Measurements were first made on a free standing sample, and then the voltage leads were attached to a horseshoe, as schematically shown in the top-right panel of Fig. \[1-detwin\], while the current leads were mounted so as to produce no strain on the sample. The strain was applied by means of a stainless push screw expanding the legs of the horseshoe. The temperature-dependent resistivity was measured after every strain increment. For the evaluation of the tensile stress value we compared our data with the data of T. Liang [*et. al.*]{} [@Liang], who found a roughly +5 K shift of the tetragonal-to-orthorhombic transition feature upon a stress change from 15 to 50 MPa. In our case a total shift of approximately 3 K was achieved in four equivalent stress increments, which suggests that each strain increment is in the 4-5 MPa range for our horseshoe straining device. Therefore the strain at the highest level is estimated to be in the range of 16-20 MPa.
The sample was periodically imaged via polarized microscopy. The bottom panels in Fig. \[1-detwin\] show two areas of the sample: area $A$ (left panel) is located between current and potential leads and remains twinned during measurements; area $B$ (right panel) is located between the potential contacts in the strained part of the sample and becomes nearly twin-free under strain. The application of uniaxial stress makes it energetically favorable to align domains with their long $\bf{a_o}$ axis along the strain, giving rise to an increasing volume fraction of one domain orientation above the rest.
![(Color Online) Normalized temperature variation of the green color channel’s intensity through the magnetic transition in detwinned crystals of BaFe$_2$As$_2$ and CaFe$_2$As$_2$. Arrows indicate the temperatures of the respective magnetic transitions. While there is a clear signature of the transition at about 135 K in BaFe$_2$As$_2$, the curve changes smoothly through the transition with a second order character. In CaFe$_2$As$_2$, the transition is quite sharp around 160 K, and is strongly first order. []{data-label="3-color"}](fig3.pdf){width="0.85\linewidth"}
The detwinned crystals were studied by high-energy x-ray diffraction in the MU-CAT sector (beamline 6ID-6) at the Advanced Photon Source at Argonne National Laboratory. Measurements were made with X-rays at 99.3 keV, giving an absorption length of approximately 1.5 mm and thus allowing full penetration of the typically 0.1 mm thick samples. Entire reciprocal planes were recorded with the $\bf{c}$-axis parallel to the incident beam. During measurements, the samples were rocked through two independent angles perpendicular to the beam (see Ref. ). The direct beam was blocked by a beam stop and diffraction images were recorded by a MAR345 image plate positioned 1680 mm behind the sample. The beam size was reduced to 0.2x0.2mm$^2$ by a slit system.
![ (Color Online) Temperature evolution of the two-dimensional x-ray diffraction pattern near the tetragonal (220) Bragg diffraction peak. Left and right columns of images show diffraction patterns in the unstrained and strained parts of the crystal, respectively. Four spots in the unstrained part at 6 K (top left) represent four domains in the sample with domain populations (proportional to integrated intensity) ranging between 19 and 31%, close to random (see also Ref. ). In the strained portion of the sample (6 K, top right panel), the dominant domain occupies nearly 90 percent of the volume of the sample area probed by the x-ray beam. Between 132 and 134 K, the sample undergoes an orthorhombic to tetragonal structural transition. The second-order nature of the transition is evidenced by the lack of coexistence of orthorhombic and tetragonal domains. This coexistence is clearly observed in Sr122 (see Ref. ), in which the transition is strongly first-order, see Fig. \[5-schematics\] below for schematic elaboration.[]{data-label="4-bragg"}](fig4.pdf){width="0.7\linewidth"}
![ (Color Online) Schematic diagrams of the displacements of atoms in the twinned orthorhombic phase and the resulting Bragg reflections. As demonstrated in the top panel, a perfect crystal with equal populations of each domain orientation results in a square pattern between the (400) and (040) orthorhombic reflections. Conversely, the bottom panel illustrates the result of an unequal distribution of domain orientations. Here the angle between the O4 and the O2 domain orientations is significantly smaller and consequently moves the reflections closer together. Further, the population of each domain is proportional to the intensity of its Bragg reflection. These effects can be seen in the X-ray data of Fig. \[4-bragg\], especially the $T$=132 K panels.[]{data-label="5-schematics"}](fig5.pdf){width="1\linewidth"}
Results
=======
Polarized Microscopy
--------------------
The unit cell in BaFe$_2$As$_2$ doubles in size and rotates by 45 degrees upon cooling through the tetragonal to orthorhombic transition. This leads to the formation of the domain walls at 45 degrees with respect to the sample edges (see bottom left panel of Fig. \[1-detwin\]). The orthorhombic $\bf{a_o}$ and $\bf{b_o}$ axes inside the domains are at 45 degrees to the twin boundaries (see Ref. ). Therefore the highest contrast of domain imaging is achieved when the sample is aligned with a long \[110\] tetragonal direction at 45$^o$ to the polarization direction of the linearly polarized light (parallel and perpendicular to the orthorhombic $\bf{a_o}$ in different domains.)
The optical contrast of the domains is determined by the anisotropy of the bi-reflectance and is proportional to the orthorhombic distortion. It increases with decreasing temperature, and is weaker in Ba122 than in Ca122 [@domains]. Simultaneously, due to dispersion of the bi-reflectance, initially white incident light on reflection acquires color depending on the orientation of the orthorhombic axes with respect to the polarization direction of the incident light. This results in different colors of structural domains as can be seen in the bottom left panel of Fig. \[1-detwin\]. Therefore, the color of the image contains information about the orthorhombic distortion, albeit in arbitrary units, and can be used for the analysis of its temperature dependence even in the detwinned state below the structural transition or in the strained state above the transition.
Figure \[f2-RGB\] shows images from the strained region B of Fig. \[1-detwin\], below (130 K, left top panel) and above (160 K, left bottom) the transition. This region is completely detwinned by strain. The red square shows the small clean area of the sample, where the color of the image was analyzed numerically. The right panels in Fig. \[f2-RGB\] show red-green-blue (RGB) histograms of that area. The images were taken every 5 K from 80 to 260 K. Fig. \[3-color\] shows the difference between the intensities of the blue and green channels, indicating that the structural distortion does not vanish abruptly at the transition but remains notable up to 200 K.
For reference we show the results of the equivalent analysis in the parent compound Ca122. Here the transition is strongly first order, and the data show no tail above the transition.
X-ray diffraction
-----------------
X-ray analysis was done in both the unstrained (area A) and strained (area B) parts of the same crystal, as shown in Fig. \[1-detwin\]. In the unstrained region, the tetragonal (220) Bragg peak splits below $T_{TO}$ into four peaks, each representing a distinct orthorhombic domain. The four orthorhombic reflections merge back into a single tetragonal Bragg peak on warming above $T_{TO}$ (see Fig. 4).
The integrated intensities of the reflections at 6 K allow us to determine the relative population of orthorhombic domains. In the unstrained area, the population of the four domains ranges 19 to 31 % of the total peak intensity, characteristic of a near random distrobution. In the strained region, the domain whose $\bf{a_O}$-axis lies along the direction of the strain accounts for nearly 90 % of the probed sample volume.
These effects are schematically described in Fig. \[5-schematics\]. The separation between the Bragg peaks resulting from the orthorhombic O1 and O2 domains is fixed because the relative angle between their twinning planes is fixed. The same is true for the peaks from the O3 and O4 domains. However there exists no such rule for the separation between the O2 and O4 peaks because the angle between their twinning planes is determined by their relative domain populations. In a sample with perfectly equal domain populations the four Bragg peaks would produce a square with each reflection having equal intensity. As the relative population of one domain orientation grows, the angle between the twinning planes of the O2 and O4 peaks becomes smaller and consequently the separation of the their Bragg peaks diminishes. This behavior is readily seen in the X-ray data in Fig. \[4-bragg\]. The unstrained region of the crystal manifests relatively similar populations of each domain orientation and produces a pattern not quite square but slightly trapezoidal below the transition temperature. By contrast in the strained region of the crystal, where the dominant domain orientation represents nearly 90% of the sample volume, the O2 and O4 peaks are no longer distinguishable as two separate reflections.
![Orthorhombic distortion, $\epsilon= \frac{(a_O-b_O)}{(a_O+b_O)}$, vs. temperature in strained and unstrained parts of the sample. The strain notably increases the orthorhombic distortion below the transition, and induces a “tail” of orthorhombic distortion above the sharp drop at 135 K. []{data-label="5-delta"}](fig6.pdf){width="0.85\linewidth"}
The temperature evolution of the orthorhombic distortion, $\epsilon \equiv \frac{(a_O-b_O)}{(a_O+b_O)}$, can be clearly seen as an increased splitting distance between the orthorhombic reflections at 6 K as compared to the splitting at 132 K (see Fig. \[4-bragg\]), and is shown in Fig. \[5-delta\]. Application of strain notably increases $\epsilon$ below the transition, and most importantly, a “tail” of the orthorhombic distortion can be tracked to at least 150 K, well above 135 K where the order parameter, $\epsilon$, shows a sharp drop.
Resistivity
-----------
Figure \[6-resistivity\] shows the normalized temperature-dependent resistivity in the twinned, $\rho _t$, and strain-detwinned, $\rho _a$, states of the same sample as measured by $x$-ray diffraction (Fig \[4-bragg\], \[5-delta\]). The third curve was calculated assuming that $\rho _t$ represents an equal mixture of $\rho _a$ and $\rho _b$, $\rho^*_b \equiv 2\rho_t-\rho_a$.
![(Color Online) Temperature-dependent normalized resistivity, $\rho_a$(T)/$\rho_a$(300 K) of the BaFe$_2$As$_2$ sample in the free-standing, $\rho _t$ (black curve), and strain-detwinned, $\rho _a$ (red curve) regions of the same sample used for X-ray measurements in Figs \[4-bragg\], \[5-delta\], and \[8-SrBa\]. The third (blue) curve shows $\rho^* _b$, calculated as $\rho^*_b= 2*\rho_t -\rho_a$. The anisotropy can be seen for all temperatures below the transition, and a slight anisotropy can be found above the transition. Inset: Progression of the effect of increasing strain on the resistivity ($\rho_a$ in the detwinned state). The black curve represents a free standing crystal. Tensile stress incrementally increases until reaching approximately 20 MPa for strain 5, see text for details. Strain 2 is sufficient to detwin the sample, revealing a sharp drop in resistivity at the transition. On further strain increase the jump rounds and its onset shifts up in temperature. []{data-label="6-resistivity"}](fig7.pdf){width="1\linewidth"}
After sufficient stress was applied to detwin the crystals (the sample in Fig. 7 was nearly completely detwinned after the second strain increment, Strain 2, as determined by polarized optical imaging), we performed a careful study of the effect of additional stress on the resistivity anisotropy. The stress, whose magnitude at the highest level is estimated to be in the 20 MPa range, increases the onset temperature of the resistivity anisotropy. However the most dramatic effects of the resistivity change are clearly around the point where the strain is sufficient to detwin the crystal.
Discussion
==========
Strain-induced Anisotropy
-------------------------
Figure \[7-comparison\] shows a direct comparison of the temperature-dependent degree of the orthorhombic distortion, $\epsilon (T)$, and of the resistivity anisotropy, $\rho _b / \rho _a$, in the same sample of BaFe$_2$As$_2$ under identical strain conditions. The two quantities reveal a clear correlation. Both show a rapid rise below approximately 135 K with decreasing temperature. In addition both $\epsilon$ and $\rho_b/\rho_a$ show a clear “tail” above 135 K, in agreement with the color analysis discussed above. In strain-free samples of Ba122, the tetragonal-to-orthorhombic structural transition at $T_{TO}$ is of the second order and precedes a strongly first-order magnetic transition at $T_N \le T_{TO}$ [@Kreyssigsplit]. It is then natural to assign the rapid increase of the anisotropy below 135 K to a magnetic transition, while the “tail” above 135 K correlates with the orthorhombic distortion. The exact meaning of the structural transition in the presence of the strain field becomes unclear, as the order parameter varies smoothly with temperature. Therefore from this direct comparison we conclude that externally applied strain is the cause of the structural and transport anisotropy above $T_{TO}$.
![(Color Online) Comparison of the temperature-dependent resistivity anisotropy, $\frac{\rho^* _b}{\rho_a}$, and the orthorhombic distortion, $\epsilon= \frac{(a_O-b_O)}{(a_O+b_O)}$ in the temperature range close to $T_{TO}$. Both quantities show a pronounced “tail” above a sharp drop in the order parameter at 135 K, revealing that the anisotropy is directly related to strain.[]{data-label="7-comparison"}](fig8.pdf){width="0.85\linewidth"}
Comparison of the effect of strain on first and second order transition: BaFe$_2$As$_2$ vs SrFe$_2$As$_2$
---------------------------------------------------------------------------------------------------------
In Fig. \[8-SrBa\] we compare the temperature dependent orthorhombic distortions, $\epsilon$, for two strained A122 compounds each with a very different character of the transition: strongly first-order in Sr122 and second order in Ba122. The data are plotted on a normalized temperature scale, $T/T_N$. As is clear from the Figure \[8-SrBa\], the “tail” of the anisotropy above the transition is virtually absent in Sr122, whereas it is quite noticeable in Ba122. In the next section we apply Ginzburg-Landau - type theory to model the effect of strain on the resistivity anisotropy considering first and second order transitions in the strain field.
![(Color Online) Comparison of the temperature-dependent orthorhombic distortions, $\epsilon=\frac{a_O-b_O}{a_O+b_O}$, in strain detwinned areas of SrFe$_2$As$_2$, Ref. , and BaFe$_2$As$_2$. The data are presented vs. normalized temperature $T/T_N$. A pronounced “tail” above $T_N$ in BaFe$_2$As$_2$ is caused by an anomalously strong susceptibility of the lattice to strain. []{data-label="8-SrBa"}](fig9.pdf){width="0.85\linewidth"}
Phenomenological model of the effect of the uniaxial strain
-------------------------------------------------------------
Regardless of which electronic degree of freedom $\varphi$ is responsible for the electronic anisotropy, it should be proportional to the orthorhombic distortion, since both break the tetragonal symmetry of the lattice close to $T_s$. For example, $\varphi$ can be associated with magnetic fluctuations [@Rafael1; @Rafael1]. By symmetry, $\phi$ and $\epsilon$ are bilinearly coupled in the free energy expansion, ie. they give rise to the term $\phi$$\epsilon$. Since the external strain $\sigma$ also couples bilinearly to the orthorhombic distortion $\epsilon=(a_O-b_O)/(a_O+b_O)$, it has an effect on $\phi$ similar to that of a magnetic field $h$ on Ising ferromagnets.
In order to compare the effect of a finite $h$ on the second-order and the first-order structural phase transitions, we consider the phenomenological free energy:
$$F=\frac{r}{2}\varphi^{2}+\frac{u}{4}\varphi^{4}+\frac{w}{6}\varphi^{6}-h\:\varphi\label{free_energy}$$
with temperature parameter $r\propto T-T_{s}^{0}$, where $T_{s}^{0}$ is the mean-field structural transition temperature. Here $u$ and $w$ are phenomenological parameters of Ginzburg-Landau theory describing the phase transition. To ensure the stability of the free-energy expansion, $w$ has to be positive. If $u>0$ as well, we have a second-order transition. If $u<0$, we have a first-order transition. In this case, the ratio $-u/w$ determines how strong the first-order transition is, i.e. what is the magnitude of the jump of the order parameter. For $u>0$, we have a second-order phase transition at $r=0$ for $h=0$. The effect of a small but finite $h$ is to extend the region of finite $\varphi$ asymptotically to $r\rightarrow\infty$, giving rise to a “tail” in the plot of $\varphi$ as function of temperature (see Fig. \[Rafael1\]). Formally, there is no strict $T_{s}$, although experimentally there will be a temperature above which the distortion anisotropy is too small to be detected. Notice that, at $T_{s}^{0}$, the value of $\varphi$ scales with the applied field according to $\varphi\sim h^{1/\delta}$, where $\delta=3$ is the mean-field critical exponent.
Let us consider $u<0$, which gives rise to a first-order phase transition. As usual for first-order phase transitions, there is a coexistence region where the states with $\varphi=0$ and $\varphi\neq0$ are both local minima of the free energy. If we consider an adiabatic change of temperature, such that the system always chooses the global minimum, from the minimization of Eq. \[free\_energy\] it is straightforward to find that the transition takes place above $T_{s}^{0}$, at $\frac{r}{w}=\frac{9}{48}\left(\frac{u}{w}\right)^{2}$. We also find that the jump in the order parameter is $\Delta\varphi=\sqrt{\frac{-3u}{4w}}$.
Therefore, the ratio $\left|u\right|/w$ controls the strength of the first-order transition. The effect of a finite field on the jump $\Delta\varphi$ will depend on the value of the ratio $h/\left|u\right|$. In Figure \[Rafael1\], we plot the temperature evolution of $\varphi$ for different values of $\left|u\right|/w=\left\{ 1,\:0.5,\:0.1\right\} $, keeping $h/w=0.01$ constant. The dashed line shows the magnitude of the jump $\Delta\varphi$ for $h=0$. Notice that when the first-order transition is stronger, the jump is barely affected by the finite field. In particular, above the temperature where the jump takes place, the order parameter is never zero but is always very small, giving rise to a rather small “tail”. On the other hand, when the first-order transition is weaker, the same field can completely smear out the jump. This gives rise to a noticeable and continuous “tail”, and therefore to a second-order transition [@Millis].
![(Color Online) Evolution of the anisotropy parameter $\varphi\propto\rho_{b}-\rho_{a}$ vs. temperature parameter $r$, $r\propto T-T_{s}^{0}$. Left column of panels is for $h=0$, right column is for $h=0.01w$. The bottom pair of panels shows a second-order transition for $u$=$w$=1, the other pairs of the panels show first-order transitions for u=-w, u=-0.5w and u=-0.1w (top to bottom). The dashed lines show the size of the jump $\Delta\varphi$ in the absence of an external field.[]{data-label="Rafael1"}](fig10.pdf){width="0.85\linewidth"}
This analysis suggests that the anisotropy above the second order transition originates from the fact that the orthorhombic transition is actually not strictly defined in the strained (and thus orthorhombically distorted) tetragonal phase under uniaxial stress. On the other hand it suggests that the susceptibility to stress is notably enhanced in case of a weak second order transition character.
Conclusions
===========
Systematic characterization of the effect of permanently applied stress on the properties of BaFe$_2$As$_2$ using $x$-ray, polarized optics and electrical resistivity measurements suggest that the applied stress is actually the cause of the resistivity anisotropy in the nominally tetragonal phase. Thus the resistivity anisotropy “tail” above the temperature of the structural transition is solely due to the effect of the uniaxial strain applied to detwin the samples. The difference between $A$Fe$_2$As$_2$ compounds with various alkali earth metals is determined by the character and the strength (order parameter jump) of the structural transition. These conclusions are supported by a phenomenological model of the effect of the uniaxial strain on the structural transition, similar to the effect of a magnetic field on Ising ferromagnets. It is interesting to notice that this strong susceptibility to the effect of strain is also found in the orbital splitting in ARPES measurements [@Shen; @Korean], and in the temperature dependence of the shear modulus in ultrasonic experiments [@Rafael2].
Acknowledgement
---------------
We thank D. Robinson for the excellent support of the high–energy x-ray scattering study. Use of the Advanced Photon Source was supported by the U. S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357. Work at the Ames Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under contract No. DE-AC02-07CH11358.
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|
---
abstract: |
Binary black-hole systems with spins aligned with the orbital angular momentum are of special interest as they may be the preferred end-state of the inspiral of generic supermassive binary black-hole systems. In view of this, we have computed the inspiral and merger of a large set of binary systems of equal-mass black holes with spins aligned with the orbital angular momentum but otherwise arbitrary. By least-square fitting the results of these simulations we have constructed two “spin diagrams” which provide straightforward information about the recoil velocity $|v_{\rm kick}|$ and the final black-hole spin $a_{\rm fin}$ in terms of the dimensionless spins $a_1$ and $a_2$ of the two initial black holes. Overall they suggest a maximum recoil velocity of $|v_{\rm
kick}| = 441.94\,{\rm km/s} $, and minimum and maximum final spins $a_{\rm fin} = 0.3471$ and $a_{\rm fin} = 0.9591$, respectively.
author:
- 'Luciano Rezzolla, Ernst Nils Dorband, Christian Reisswig, Peter Diener, Denis Pollney, Erik Schnetter, Béla Szilágyi'
title: 'Spin Diagrams for equal-mass black-hole binaries with aligned spins'
---
INTRODUCTION {#intro}
============
A number of recent developments in numerical relativity have allowed for stable evolution of binary black holes and opened the door to extended and systematic studies of these systems. Of particular interest to astrophysics are the calculations of the recoil velocity and of the spin of the final black hole produced by the merger. It is well known that a binary with unequal masses or spins will radiate gravitational energy asymmetrically. This results in an uneven flux of momentum, providing a net linear velocity to the final black hole. The knowledge of both the “kick” velocity and of the final spin could have a direct impact on studies of the evolution of supermassive black holes and on statistical studies on the dynamics of compact objects in dense stellar systems.
Over the past year, a number of simulations have been carried out to determine the recoil velocities for a variety of binary black-hole systems. Non-spinning but unequal-mass binaries were the first systems to be studied and several works have now provided an accurate mapping of the unequal-mass space of parameters [@Herrmann:2006ks; @Baker:2006vn; @Gonzalez:2006md]. More recently, the recoils from binaries with spinning black holes have also been considered by investigating equal-mass binaries in which the spins of the black holes are either aligned with the orbital angular momentum [@Herrmann:2007ac; @Koppitz-etal-2007aa], or not. In the first case, a systematic investigation has shown that the largest recoil possible from such systems is of the order of $450\,\mathrm{km/s}$ [@Pollney:2007ss]. In the second case, instead, specific configurations with spins orthogonal to the orbital one have been shown to lead to recoils as high as $2500\,\mathrm{km/s}$ [@Campanelli:2007ew; @Gonzalez:2007hi], suggesting a maximum kick of about $4000\,\mathrm{km/s}$ for maximally-spinning black holes [@Campanelli:2007cg]. Recoil velocities of this magnitude could lead to the ejection of massive black holes from the hosting galaxies, with important consequences on their cosmological evolution.
Here, we extend the analysis carried out in @Pollney:2007ss of binary black hole systems with equal-mass and spins aligned with the orbital one. Our interest in this type of binaries stems from the fact that systems of this type may represent a preferred end-state of the binary evolution. Post-Newtonian studies have shown that in vacuum the gravitational spin-orbit coupling has a tendency to align the spins when they are initially close to the orbital one [@Schnittman:2004vq]. Furthermore, if the binary evolves in a disc, as expected for supermassive black holes, the matter can exert a torque tending to align the spins [@Bogdanovic:2007hp]. Finally, a recoiling supermassive black hole could retain the inner part of its accretion disc and thus the fuel for a continuing QSO phase lasting millions of years as it moves away from the galactic nucleus [@Loeb:2007]. Yet, the analysis of QSOs from the Sloan Digital Sky Survey shows no evidence for black holes carrying an accretion disc and hence for very large recoiling velocities [@Bonning:2007].
-------- --------------- --------------- ------------- ------------- ----------- ----------- ----------------------------------- ---------------------------------------- ------------------------------ ---------------------------------------- ----------------- --------------------------- ------------------------------------- -----------------
[$\pm x/M$]{} [$\pm p/M$]{} [$m_1/M$]{} [$m_2/M$]{} [$a_1$]{} [$a_2$]{} [${\widetilde M_{_{\rm ADM}}}$]{} [${\widetilde J_{_{\mathrm{ADM}}}}$]{} [$|v_{_{\mathrm{kick}}}|$]{} [$|v_{_{\mathrm{kick}}}^{\rm fit}|$]{} [err. ($\%$)]{} [$a_{_{\mathrm{fin}}}$]{} [$a_{_{\mathrm{fin}}}^{\rm fit}$]{} [err. ($\%$)]{}
$r0$ 3.0205 0.1366 0.4011 0.4009 -0.584 0.584 0.9856 0.825 261.75 258.09 1.40 0.6891 0.6883 0.12
$r1$ 3.1264 0.1319 0.4380 0.4016 -0.438 0.584 0.9855 0.861 221.38 219.04 1.06 0.7109 0.7105 0.06
$r2$ 3.2198 0.1281 0.4615 0.4022 -0.292 0.584 0.9856 0.898 186.18 181.93 2.28 0.7314 0.7322 0.11
$r3$ 3.3190 0.1243 0.4749 0.4028 -0.146 0.584 0.9857 0.935 144.02 146.75 1.90 0.7516 0.7536 0.27
$r4$ 3.4100 0.1210 0.4796 0.4034 0.000 0.584 0.9859 0.971 106.11 113.52 6.98 0.7740 0.7747 0.08
$r5$ 3.5063 0.1176 0.4761 0.4040 0.146 0.584 0.9862 1.007 81.42 82.23 1.00 0.7948 0.7953 0.06
$r6$ 3.5988 0.1146 0.4638 0.4044 0.292 0.584 0.9864 1.044 45.90 52.88 15.21 0.8150 0.8156 0.07
$r7$ 3.6841 0.1120 0.4412 0.4048 0.438 0.584 0.9867 1.081 20.59 25.47 23.70 0.8364 0.8355 0.11
$r8$ 3.7705 0.1094 0.4052 0.4052 0.584 0.584 0.9872 1.117 0.00 0.00 0.00 0.8550 0.855 0.00
$ra0$ 2.9654 0.1391 0.4585 0.4584 -0.300 0.300 0.9845 0.8250 131.34 132.58 0.95 0.6894 0.6883 0.16
$ra1$ 3.0046 0.1373 0.4645 0.4587 -0.250 0.300 0.9846 0.8376 118.10 120.28 1.85 0.6971 0.6959 0.17
$ra2$ 3.0438 0.1355 0.4692 0.4591 -0.200 0.300 0.9847 0.8499 106.33 108.21 1.77 0.7047 0.7035 0.17
$ra3$ 3.0816 0.1339 0.4730 0.4594 -0.150 0.300 0.9848 0.8628 94.98 96.36 1.46 0.7120 0.7111 0.13
$ra4$ 3.1215 0.1321 0.4757 0.4597 -0.100 0.300 0.9849 0.8747 84.74 84.75 0.01 0.7192 0.7185 0.09
$ra6$ 3.1988 0.1290 0.4782 0.4602 0.000 0.300 0.9850 0.9003 63.43 62.19 1.95 0.7331 0.7334 0.04
$ra8$ 3.2705 0.1261 0.4768 0.4608 0.100 0.300 0.9852 0.9248 41.29 40.55 1.79 0.7471 0.7481 0.13
$ra10$ 3.3434 0.1234 0.4714 0.4612 0.200 0.300 0.9853 0.9502 19.11 19.82 3.72 0.7618 0.7626 0.11
$ra12$ 3.4120 0.1209 0.4617 0.4617 0.300 0.300 0.9855 0.9750 0.00 0.00 0.00 0.7772 0.7769 0.03
$s0$ 2.9447 0.1401 0.4761 0.4761 0.000 0.000 0.9844 0.8251 0.00 0.00 0.00 0.6892 0.6883 0.13
$s1$ 3.1106 0.1326 0.4756 0.4756 0.100 0.100 0.9848 0.8749 0.00 0.00 0.00 0.7192 0.7185 0.09
$s2$ 3.2718 0.1261 0.4709 0.4709 0.200 0.200 0.9851 0.9251 0.00 0.00 0.00 0.7471 0.7481 0.13
$s3$ 3.4098 0.1210 0.4617 0.4617 0.300 0.300 0.9855 0.9751 0.00 0.00 0.00 0.7772 0.7769 0.03
$s4$ 3.5521 0.1161 0.4476 0.4476 0.400 0.400 0.9859 1.0250 0.00 0.00 0.00 0.8077 0.8051 0.33
$s5$ 3.6721 0.1123 0.4276 0.4276 0.500 0.500 0.9865 1.0748 0.00 0.00 0.00 0.8340 0.8325 0.18
$s6$ 3.7896 0.1088 0.4002 0.4002 0.600 0.600 0.9874 1.1246 0.00 0.00 0.00 0.8583 0.8592 0.11
$t0$ 4.1910 0.1074 0.4066 0.4064 -0.584 0.584 0.9889 0.9002 259.49 258.09 0.54 0.6868 0.6883 0.22
$t1$ 4.0812 0.1103 0.4062 0.4426 -0.584 0.438 0.9884 0.8638 238.37 232.62 2.41 0.6640 0.6658 0.27
$t2$ 3.9767 0.1131 0.4057 0.4652 -0.584 0.292 0.9881 0.8265 200.25 205.21 2.48 0.6400 0.6429 0.45
$t3$ 3.8632 0.1165 0.4053 0.4775 -0.584 0.146 0.9879 0.7906 174.58 175.86 0.73 0.6180 0.6196 0.26
$t4$ 3.7387 0.1204 0.4047 0.4810 -0.584 0.000 0.9878 0.7543 142.62 144.57 1.37 0.5965 0.5959 0.09
$t5$ 3.6102 0.1246 0.4041 0.4761 -0.584 -0.146 0.9876 0.7172 106.36 111.34 4.68 0.5738 0.5719 0.33
$t6$ 3.4765 0.1294 0.4033 0.4625 -0.584 -0.292 0.9874 0.6807 71.35 76.17 6.75 0.5493 0.5475 0.32
$t7$ 3.3391 0.1348 0.4025 0.4387 -0.584 -0.438 0.9873 0.6447 35.36 39.05 10.45 0.5233 0.5227 0.11
$t8$ 3.1712 0.1419 0.4015 0.4015 -0.584 -0.584 0.9875 0.6080 0.00 0.00 0.00 0.4955 0.4976 0.42
$u1$ 2.9500 0.1398 0.4683 0.4685 -0.200 0.200 0.9845 0.8248 87.34 88.39 1.20 0.6893 0.6883 0.15
$u2$ 2.9800 0.1384 0.4436 0.4438 -0.400 0.400 0.9846 0.8249 175.39 176.78 0.79 0.6895 0.6883 0.17
$u3$ 3.0500 0.1355 0.3951 0.3953 -0.600 0.600 0.9847 0.8266 266.39 265.16 0.46 0.6884 0.6883 0.01
$u4$ 3.1500 0.1310 0.2968 0.2970 -0.800 0.800 0.9850 0.8253 356.87 353.55 0.93 0.6884 0.6883 0.01
-------- --------------- --------------- ------------- ------------- ----------- ----------- ----------------------------------- ---------------------------------------- ------------------------------ ---------------------------------------- ----------------- --------------------------- ------------------------------------- -----------------
\
NUMERICAL SETUP AND INITIAL DATA {#numerical simulations}
================================
The numerical simulations have been carried out using the CCATIE code, a three-dimensional finite-differences code using the Cactus Computational Toolkit [@cactusweb] and Carpet mesh refinement infrastructure [@schnetter_etal:2004]. The main features of the code have been recently reviewed in @Pollney:2007ss, where the code has been employed using the so-called “moving-punctures” technique [@Baker:2006yw; @Campanelli:2005dd]. The initial data consists of five sequences with constant orbital angular momentum, which is however different from sequence to sequence. In the $r$ and $ra$-sequences, the initial spin of one of the black holes ${\mathbf
S}_2$ is held fixed along the $z$-axis and the spin of the other black hole is varied so that the spin ratio $a_1/a_2$ takes the values between $-1$ and $+1$, with $a_i\equiv{\mathbf S}_i/M_i^2$. In the $t$-sequence, instead, the spin with a negative $z$-component is held fixed, while in the $s$ and $u$-sequences $a_1/a_2=1$ and $-1$, respectively. In all cases, the masses are $M_i = M/2=1/2$. For the orbital initial data parameters we use the effective-potential method, which allows one to choose the initial data parameters such that the resulting physical parameters (*e.g.,* masses and spins) describe a binary black-hole system on a quasi-circular orbit. The free parameters are: the coordinate locations ${\mathbf C}_i$, the mass parameters $m_i$, the linear momenta ${\mathbf p}_i$, and the spins ${\mathbf S}_i$. Quasi-circular orbits are then selected by setting ${\mathbf p}_1 = -{\mathbf p}_2$ to be orthogonal to ${\mathbf C}_2 - {\mathbf C}_1$, so that ${\mathbf L}
\equiv {\mathbf C}_1 \times {\mathbf p}_1 + {\mathbf C}_2 \times {\mathbf
p}_2$ is the orbital angular momentum. The initial parameters are collected in the left part of Table \[tableone\], while the right part reports the results of simulations. For all of them we have employed 8 levels of refinement and a minimum resolution $0.024\,M$, which has been reduced to $0.018\,M$ for binaries $r5,\,r6$. Note that our results for the $u$-sequence differ slightly from those reported by @Herrmann:2007ac, probably because of our accounting of the integration constant in $|v_{\rm kick}|$ [@Pollney:2007ss].
SPIN DIAGRAMS AND FITS {#spin_diagrams}
======================
Clearly, the recoil velocity and the spin of the final black hole are among the most important pieces of information to be extracted from the inspiral and coalescence of binary black holes. For binaries with equal masses and aligned but otherwise arbitrary spins, this information depends uniquely on the dimensionless spins of the two black holes $a_1,\,a_2$ and can therefore be summarized in the portion of the $(a_1,\,a_2)$ plane in which the two spins vary. It is therefore convenient to think in terms of “spin diagrams”, which summarize in a simple way all of the relevant information. In addition, since the labelling “$1$” and “$2$” is arbitrary, the line $a_1=a_2$ in the spin diagram has important symmetries: the recoil velocity vector undergoes a $\pi$-rotation, *i.e.*, ${\vec v}_{\rm kick}
(a_1,\,a_2) = - {\vec v}_{\rm kick} (a_2,\,a_1)$ but $|v_{\rm kick}
(a_1,\,a_2)| = |v_{\rm kick} (a_2,\,a_1)|$, while no change is expected for the final spin, *i.e.*, $a_{\rm fin}(a_1,\,a_2) = a_{\rm
fin}(a_2,\,a_1)$. These symmetries not only allow us to consider only one portion of the $(a_1,\,a_2)$ space (*cf.* Fig. \[plotone\]), thus halving the computational costs (or doubling the statistical sample), but they will also be exploited later on to improve our fits. The position of the five sequences within the $(a_1,\,a_2)$ space is shown in Fig. \[plotone\].
Overall, the data sample computed numerically consists of 38 values for $|v_{\rm kick}|$ and for $a_{\rm fin}$ which, for simplicity, we have considered to have constant error-bars of $8\ {\rm km/s}$ and $0.01$, which represent, respectively, the largest errors reported in @Pollney:2007ss. In both cases we have modelled the data with generic quadratic functions in $a_1$ and $a_2$ so that, in the case of the recoil velocity, the fitting function is $$|v_{\rm kick}| = | c_0 + c_1 a_1 + c_2 a^2_1 +
d_0 a_1 a_2 + d_1 a_2 + d_2 a^2_2 |\,.
\label{vk_1}$$ Note that the fitting function on the right-hand-side of (\[vk\_1\]) is smooth everywhere but that its absolute value is not smooth along the diagonal $a_1=a_2$. Using (\[vk\_1\]) and a blind least-square fit of the data, we obtained the coefficients (in $\mathrm{km/s}$) $$\begin{aligned}
& c_0 = 0.67 \pm 1.12 \,, \quad
& d_0 = -18.56 \pm 5.34 \,,
\nonumber \\
& c_1 = -212.85 \pm 2.96 \,, \quad
& d_1 = 213.69 \pm 3.57 \,,
\nonumber \\
%
& c_2 = 50.85 \pm 3.48 \,,
& d_2 = -40.99 \pm 4.25 \,,
\label{vk_fit_1}\end{aligned}$$ with a reduced-$\chi^2 = 0.09$. Clearly, the errors in the coefficients can be extremely large and this is simply the result of small-number statistics. However, the fit can be improved by exploiting some knowledge about the physics of the process to simplify the fitting expressions. In particular, we can use the constraint that no recoil velocity should be produced for binaries having the same spin, *i.e.*, that $|v_{\rm kick}|=0$ for $a_1=a_2$, or the symmetry condition across the line $a_1=a_2$. Enforcing both constraints yields $$c_0 = 0\,, \quad
c_1 = -d_1\,, \quad
c_2 = -d_2\,, \quad
d_0 = 0\,,
\label{vk_coeff_2}$$ thus reducing the fitting function (\[vk\_1\]) to the simpler expression $$|v_{\rm kick}| = |c_1 (a_1 - a_2) + c_2 (a_1^{\,2} - a_2^{\,2})|\,.
\label{vk_3}$$
Performing a least-square fit using (\[vk\_3\]) we then obtain $$\begin{aligned}
& c_1 = -220.97 \pm 0.78\,, \quad
& c_2 = 45.52 \pm 2.99\,, \quad
\label{vk_fit_2}\end{aligned}$$ with a comparable reduced-$\chi^2 = 0.14$, but with error-bars that are much smaller on average. Because of this, we consider expression (\[vk\_3\]) as the best description of the data at second-order in the spin parameters. Using (\[vk\_3\]) and (\[vk\_fit\_2\]), we have built the contour plots shown in Fig. \[plottwo\].
A few remarks are worth making. Firstly, we recall that post-Newtonian calculations have so far derived only the linear contribution in the spin to the recoil velocity (see @Favata_etal:2004 and references therein). However, the size of the quadratic coefficient (\[vk\_fit\_2\]) is not small when compared to the linear one and it can lead to rather sizeable corrections. These are maximized when $a_1=0$ and $a_2=\pm 1$, or when $a_1=\pm 1$ and $a_2=0$, and can be as large as $\sim 20\%$; while these corrections are smaller than those induced by asymmetries in the mass, they are instructive in pointing out the relative importance of spin-spin and spin-orbit effects during the merger and can be used as a guide in further refinements of the post-Newtonian treatments. Secondly, expression (\[vk\_3\]) clearly suggests that the maximum recoil velocity should be found when the asymmetry is the largest and the spins are antiparallel, *i.e.*, $a_1=-a_2$. Thirdly, when $a_2={\rm
const.}$, expression (\[vk\_3\]) confirms the quadratic scaling proposed in @Pollney:2007ss with a smaller data set \[*cf.,* eq. (42) there\]. Fourthly, for $a_1=-a_2$, expression (\[vk\_3\]) is only linear and reproduces the scaling suggested by @Herrmann:2007ac. Finally, using (\[vk\_3\]) the maximum recoil velocity is found to be $|v_{\rm kick}| = 441.94 \pm
1.56\ {\rm km/s}$, in very good agreement with the results of @Herrmann:2007ac and @Pollney:2007ss.
In the same way we have first fitted the data for $a_{\rm fin}$, with a function $$a_{\rm fin} = p_0 + p_1 a_1 + p_2 a^2_1 +
q_0 a_1 a_2 + q_1 a_2 + q_2 a^2_2\,,
\label{af_1}$$ and found coefficients with very large error-bars. As a result, also for $a_{\rm fin}$ we resort to physical considerations to constrain the coefficients $p_0 \ldots q_2$. More specifically, we expect that, at least at lowest order, binaries with equal and opposite spins will not contribute to the final spin and thus behave essentially as nonspinning binaries. Stated differently, we assume that $a_{\rm fin}
= p_0$ for binaries with $a_1=-a_2$. In addition, enforcing the symmetry condition across the line $a_1=a_2$ we obtain $$p_1 = q_1 \,, \quad
p_2 = q_2 = q_0/2\,,
\label{af_coeff}$$ so that the fitting function (\[af\_1\]) effectively reduces to $$a_{\rm fin} = p_0 + p_1 (a_1 + a_2) + p_2 (a_1 + a_2)^2\,.
\label{af_2}$$ Performing a least-square fit using (\[af\_2\]) we then obtain $$\begin{aligned}
& p_0 = 0.6883 \pm 0.0003 \,,\quad
& p_1 = 0.1530 \pm 0.0004 \,,
\nonumber \\
& p_2 = -0.0088 \pm 0.0005 \,,
\label{af_fit_2}\end{aligned}$$ with a reduced-$\chi^2=0.02$.
It should be noted that the coefficient of the quadratic term in (\[af\_fit\_2\]) is much smaller then the linear one and with much larger error-bars. Given the small statistics it is hard to assess whether a quadratic dependence is necessary or if a linear one is the correct one (however, see also the comment below on a possible interpretation of expression (\[af\_2\])). In view of this, we have repeated the least-square fit of the data enforcing the conditions (\[af\_coeff\]) together with $p_2=0$ (*i.e.*, adopting a linear fitting function) and obtained $p_0
= 0.6855 \pm 0.0007$ and $p_1 = 0.1518 \pm 0.0012$, with a worse reduced-$\chi^2=0.16$. Because the coefficients of the lowest-order terms are so similar, both the linear and the quadratic fits are well within the error-bars of the numerical simulations. Nevertheless, since a quadratic scaling yields smaller residuals, we consider it to be the best representation of the data and have therefore computed the contour plots in Fig. \[plotthree\] using (\[af\_2\]) and (\[af\_fit\_2\]).
Here too, a few remarks are worth making: Firstly, the fitted value for the coefficient $p_0$ agrees very well with the values reported by several groups [@Gonzalez:2006md; @berti_etal:2007] when studying the inspiral of unequal-mass nonspinning binaries. Secondly, expression (\[af\_2\]) has maximum values for $a_1=a_2$, suggesting that the maximum and minimum spins are $a_{\rm fin} = 0.9591 \pm
0.0022$ and $a_{\rm fin} = 0.3471 \pm 0.0224$, respectively. Thirdly, the quadratic scaling for $a_{\rm fin}$ substantially confirms the suggestions of @Campanelli:2006gf but provides more accurate coefficients. Finally, although very simple, expression (\[af\_fit\_2\]) lends itself to an interesting interpretation. Being effectively a power series in terms of the initial spins of the two black holes, its zeroth-order term can be seen as the orbital angular momentum not radiated in gravitational waves and which amounts, at most, to $\sim 70\%$ of the final spin. The first-order term, on the other hand, can be seen as the contribution to the final spin coming from the initial spins of the two black holes and this contribution, together with the one coming from the spin-orbit coupling, amounts at most to $\sim 30\%$ of the final spin. Finally, the second-order term, which is natural to expect as nonzero in this view, can then be related to the spin-spin coupling, with a contribution to the final spin which is of $\sim 4\%$ at most.
As a side remark we also note that the monotonic behaviour expressed by (\[af\_fit\_2\]) does not show the presence of a local maximum of $a_{\rm fin} \simeq 0.87$ for $a_1 = a_2 \sim 0.34$ as suggested by @Damour:2001tu in the effective one-body (EOB) approximation. Because the latter has been shown to be in good agreement with numerical-relativity simulations of nonspinning black holes [@Damour_Nagar:2007; @Damour_etal:2007], additional simulations will be necessary to refute these results or to improve the EOB approximation for spinning black holes.
Reported in the right part of Table \[tableone\] are also the fitted values for $a_{\rm fin}$ and $|v_{\rm kick}|$ obtained through the fitting functions (\[vk\_3\]) and (\[af\_2\]), and the corresponding errors. The latter are of few percent for most of the cases and increase up to $\sim 20\%$ only for those binaries with very small kicks and which are intrinsically more difficult to calculate. As a concluding remark we note that the fitting coefficients computed here have been constructed using overall moderate values of the initial spin; the only exception is the binary $u4$ which has the largest spin and which is nevertheless fitted with very small errors (*cf.* Table \[tableone\]). In addition, since the submission of this work, another group has reported results from equal-mass binaries with spins as high as $a_1=a_2=\pm
0.9$ [@Marronetti_etal:2007]. Although also for these very high-spin binaries the error in the predicted values is of $1\%$ at most, a larger sample of high-spin binaries is necessary to validate that the fitting expressions (\[vk\_3\]) and (\[af\_2\]) are robust also at very large spins.
CONCLUSIONS {#CONCLUSIONS}
===========
We have performed least-square fits to a large set of numerical-relativity data. These fits, combined with symmetry arguments, yield analytic expressions for the recoil velocity and final black hole spin resulting from the inspiral and merger of equal-mass black holes whose spins are parallel or antiparallel to the orbital angular momentum. Such configurations represent a small portion of the space of parameters, but may be the preferred ones if torques are present during the evolution. Using the analytic expressions we have constructed two spin diagrams that summarize simply this information and predict a maximum recoil velocity of $|v_{\rm kick}| = 441.94 \pm 1.56\ {\rm km/s} $ for systems with $a_1=-a_2=1$ and maximum (minimum) final spin $a_{\rm fin} = 0.9591
\pm 0.0022\, (0.3471 \pm 0.0224)$ for systems with $a_1=a_2=1\,
(-1)$.
It is a pleasure to thank Thibault Damour and Alessandro Nagar for interesting discussions. The computations were performed on the supercomputing clusters of the AEI. This work was supported in part by the DFG grant SFB/Transregio 7.
[NOTE ADDED IN PROOF.]{} Since the publication on the preprint archive of this analysis, our work on the modelling of the final spin has progressed rapidly, yielding new results that complement and complete the ones presented here. In particular, the work published in @Rezzolla:2008a complements the analysis carried here to unequal-mass, equal-spin aligned binaries, while the work reported in @Rezzolla:2008b extends it to generic binaries.
[26]{} natexlab\#1[\#1]{}
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|
---
abstract: 'The purpose of the present paper is to prove existence of *super-exponentially* many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $n$-manifolds. An analogous result holds for non-compact orientable hyperbolic $n$-manifolds of finite volume that are geometric boundaries, for $n \geq 2$.'
address:
- 'Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA'
- 'Institut de mathématiques, Université de Neuchâtel, CH-2000 Neuchâtel, Suisse / Switzerland'
author:
- Michelle Chu
- Alexander Kolpakov
title: 'A hyperbolic counterpart to Rokhlin’s cobordism theorem'
---
Introduction {#sec: Intro}
============
A classical result by V. Rokhlin states that every compact orientable $3$-manifold bounds a compact orientable $4$-manifold, and thus the three-dimensional cobordism group is trivial. Rokhlin also proved that a compact orientable $4$-manifold bounds a compact orientable $5$-manifold if and only if its signature is zero, which is true for all closed orientable hyperbolic $4$-manifolds. One can recast the question of bounding in the setting of hyperbolic geometry, which generated plenty of research directions over the past decades.
A hyperbolic manifold is a manifold endowed with a Riemannian metric of constant sectional curvature $-1$. Throughout the paper, hyperbolic manifolds are assumed to be connected, orientable, complete, and of finite volume, unless otherwise stated. We refer to [@MaRe; @VS] for the definition of an arithmetic hyperbolic manifold.
A connected hyperbolic $n$-manifold $\mathcal{M}$ is said to *bound geometrically* if it is isometric to $\partial \mathcal{W}$, for a hyperbolic $(n+1)$-manifold $\mathcal{W}$ with totally geodesic boundary.
Indeed, some interest in hyperbolic manifolds that bound geometrically was kindled by the works of Long, Reid [@LR1; @LR2] and Niemershiem [@N], motivated by a preceding work of Gromov [@G1; @G2] and a question by Farrell and Zdravkovska [@FZ]. This question is also related to hyperbolic instantons, as described in [@RT1; @RT2].
As [@LR1] shows many closed hyperbolic $3$-manifolds do not bound geometrically: a necessary condition is that the $\eta$-invariant of the $3$-manifold must be an integer. The first example of a closed hyperbolic $3$-manifold known to bound geometrically was constructed by Ratcliffe and Tschantz in [@RT1] and has volume of order $200$.
The first examples of knot and link complements that bound geometrically were produced by Slavich in [@S1; @S2]. However, [@KRR] implies that there are plenty of cusped hyperbolic $3$-manifolds that cannot bound geometrically, with the obstruction being the geometry of their cusps.
In [@LR3], by using arithmetic techniques, Long and Reid built infinitely many orientable hyperbolic $n$-manifolds $\mathcal{N}$ that bound geometrically an $(n+1)$-manifold $\mathcal{M}$, in every dimension $n\geqslant 2$. Every such manifold $\mathcal{N}$ is obtained as a cover of some $n$-orbifold $O_\mathcal{N}$ geodesically immersed in a suitable $(n+1)$-orbifold $O_\mathcal{M}$. However, this construction gives no control on the volume of the manifolds.
In [@BGLS], Belolipetsky, Gelander, Lubotzky, and Shalev showed that the growth rate of all orientable arithmetic hyperbolic manifolds, up to isometry, with respect to volume is super-exponential, in all dimensions $n\geq 3$. Their lower bound used a subgroup counting technique due to Lubotzky [@L95]. In the present paper we shall use the ideas of [@LR3] together with the subgroup counting argument due to Lubotzky [@L95] (also used in [@BGLS]), together with the more combinatorial colouring techniques from [@KS] in order to prove the following facts:
\[prop1\] Let $\kappa_n(x) = $ the number of non-isometric non-orientable compact arithmetic hyperbolic $n$-manifolds of volume $\leq x$. Then we have that $\kappa_n(x) \asymp x^x$ for any $n \geq 3$.
\[prop2\] Let $\nu_n(x) = $ the number of non-isometric non-orientable cusped arithmetic hyperbolic $n$-manifolds of volume $\leq x$. Then we have that $\nu_n(x) \asymp x^x$ for any $n \geq 3$.
Above, the notation “$f(x) \asymp x^x$” for a function $f(x)$ is a shorthand for “There exist positive constants $A_1, B_1, A_2, B_2$, and $x_0$, such that $A_1 x^{B_1 x} \leq f(x) \leq A_2 x^{B_2 x}$, for all $x \geq x_0$.”
The techniques of [@BGLS; @L95] provide us with super-exponentially many manifolds of volume $\leq x$ (for $x$ sufficiently large) by employing a retraction of the manifold’s fundamental group into a free group. In our case we need however to take extra care in order to arrange for the kernel of such retraction comprise an orientation-reversing element. Here Coxeter polytopes and reflection groups come into play as natural sources of orientation-reversing isometries ,as well as building blocks for manifolds.
Then, by using the embedding technique from [@KRS] and the techniques for constructing torsion-free subgroups from [@LR3] (see , also below), we obtain the following theorems establishing that the growth rate with respect to volume of arithmetic hyperbolic manifolds bounding geometrically is the same as that over all arithmetic hyperbolic manifolds.
\[thm1\] Let $\beta_n(x) = $ the number of non-isometric orientable compact arithmetic hyperbolic $n$-manifolds of volume $\leq x$ that bound geometrically. Then we have that $\beta_n(x) \asymp x^x$ for $n \geq 3$.
\[thm2\] Let $\gamma_n(x) = $ the number of non-isometric orientable cusped arithmetic hyperbolic $n$-manifolds of volume $\leq x$ that bound geometrically. Then we have that $\gamma_n(x) \asymp x^x$ for $n \geq 3$.
As a by-product, we provide a different proof to a part of the results in [@KR] and construct a few new Coxeter polytopes not otherwise available on the literature. For dimensions $n = 2, \ldots, 6$ in the compact case, and dimensions $n = 2, \ldots, 13$ in the cusped case, we construct explicit examples of retractions onto free groups. More involved computations may be performed in dimensions $n = 14, 15$ (using the polytopes from [@Allcock]) and $n = 18, 19$ (using the polytopes from [@KapVin]). However, the general case follows from the main result of Bergeron, Haglund, Wise [@BHW] on virtually retractions of arithmetic groups of simplest type onto geometrically finite subgroups.
It is also worth mentioning that a linear lower bound with respect to volume for the number of isometry classes of compact orientable bounding hyperbolic $3$-manifolds was obtained previously in [@MZ] by extending the techniques from [@KMT] and comparing the Betti numbers of the resulting manifolds.
Given the present question’s background, one may think of as a “hyperbolic counterpart” to Rokhlin’s theorem. Indeed, not every compact orientable arithmetic hyperbolic $3$-manifold bounds geometrically, but the number of those that do has the same growth rate as the number of all compact orientable arithmetic hyperbolic $3$-manifolds. In the light of Wang’s theorem [@Wang] and the results of [@BGLM], an analogous statement can be formulated for geometrically bounding hyperbolic $4$-manifolds without arithmeticity assumption.
As for the closed hyperbolic surfaces that bound geometrically, it follows from the work of Brooks [@Brooks] that for each genus $g \geq 2$ the ones that bound form a dense subset of the Teichmüller space. Thus, there are infinitely many of them in each genus $g \geq 2$. However, there are only finitely many arithmetic ones, by [@BGLS]. The argument of applies in this case, and we obtain
\[thm3\] Let $\alpha(g) = $ the number of non-isometric orientable closed arithmetic surfaces of genus $\leq g$ that bound geometrically. Then $\lim_{g \to \infty} \frac{\log \alpha(g)}{g} = \frac{1}{2\pi}$.
\[rem:surfaces-nc\] An analogous statement holds for finite-area non-compact surfaces if we substitute “genus” with “area”, and $\frac{1}{2\pi}$ in the right-hand side of the above limit formula with $1$.
This adds many more (albeit not very explicit) examples to the ones obtained by Zimmermann in [@Z1; @Z2].
The manifolds that we construct in abundance in order to prove – all happen to be orientation double covers. An easy observation implies that any closed orientable manifold $M$ that is an orientation cover bounds topologically: consider $W^\prime = M \times [0,1]$ and quotient one of its boundary components by an orientation-reversing fixed point free involution that $M$ necessarily has in this case. The resulting manifold $W$ is an orientable manifold with boundary $\partial W \cong M$. Indeed, these are manifolds that are *not* orientation covers that may make the cobordism group non-trivial.
Concerning geometrically bounding manifolds, we are not aware at the moment of any that does bounds geometrically and which is not an orientation cover, in both compact and finite-volume cases.
Constructing geodesic boundaries by colorings {#sec: color}
=============================================
The right-angled dodecahedron
-----------------------------
Let $\mathcal{D} \subset \mathbb{H}^3$ be a right-angled dodecahedron. By Andreev’s theorem [@Andreev], it is realisable as a regular compact hyperbolic polyhedron. Suppose that the faces of $\mathcal{D}$ are labelled with the numbers $1$, $\dots$, $12$ as shown in . Let $s_i$ be the reflection in the supporting hyperplane of the $i$th facet of $\mathcal{D}$, for $ i = 1, \dots, 12$, and let $\Gamma_{12} = \mathrm{Ref}(\mathcal{D}) = \langle s_1, s_2, \dots, s_{12} \rangle$ be the corresponding reflection group.
Let $P$ be the pentagonal two-dimensional face of $\mathcal{D}$ labelled $5$ and let $\Gamma_4 = \langle s_1, s_3, s_9, s_{11} \rangle$ be an infinite-index subgroup of $\Gamma_{12}$, which we may consider as a reflection group acting on the supporting hyperplane of $P$, which is isometric to $\mathbb{H}^2$. There is a retraction $R$ of $\Gamma_{12}$ onto $\Gamma_4$ given by $$R: s_i \mapsto
\left\{
\begin{array}{cl}
s_i, &\mbox{ if } i \in \{1, 3, 9 ,11\},\\
\mathrm{id}, &\mbox{ otherwise. }
\end{array}
\right.$$
The group $\Gamma_4$ is virtually free: it contains $F_3 \cong \langle x, y, z \rangle$, a free group of rank $3$, as an index $8$ *normal* subgroup. Indeed, with $x = s_1 s_{11}$, $y = (s_1 s_9)^2$, $z = s_1 s_3 s_{11} s_3$, we have $F_3$ realised as a subgroup of $\Gamma_4$, which is the fundamental group of a $2$-sphere with four disjoint closed discs removed, as depicted in .
Let $P$ be a simple $n$-dimensional polytope (not necessarily hyperbolic) with $m$ facets labelled by distinct elements of $\Omega = \{1, 2, \dots, m\}$. A colouring of $P$, according to [@DJ; @GS; @I; @Vesnin87; @Vesnin], is a map $\lambda: \Omega \rightarrow \mathbb{Z}^n_2$. A colouring is called proper if the colours of facets around each vertex of $P$ are linearly independent vectors of $V = \mathbb{Z}^n_2$.
Proper colourings of compact right-angled polytopes $P\subset \mathbb{H}^n$ give rise to interesting families of hyperbolic manifolds [@GS; @KMT; @Vesnin87; @Vesnin]. Such polytopes $P$ are necessarily simple.
In [@KMT] the notion of a colouring is extended to let $V = \mathbb{Z}^s_2$, $s \geq 2$, be a finite-dimensional vector space over $\mathbb{Z}_2$, and in [@KS] the notion of colouring is extended to polytopes that are not necessarily simple, but rather satisfy a milder constraint of being simple at edges.
A polytope $P \subset \mathbb{H}^n$ is called simple at edges if each edge belongs to exactly $(n-1)$ facets. In the case of a finite-volume right-angled polytope $P \subset \mathbb{H}^n$, $P$ is simple if $P$ is compact, and $P$ is simple at edges if it has any ideal vertices.
A colouring of a polytope $P\subset \mathbb{H}^n$ which is simple at edges is a map $\lambda: \Omega \rightarrow V$, where $V = \mathbb{Z}^s_2$, $s\geq n$, is a finite-dimensional vector space over $\mathbb{Z}_2$. A colouring $\lambda$ is proper if the following two conditions are satisfied:
1. *Properness at vertices:* if $v$ is a simple vertex of $P$, then the $n$ colours of facets around it are linearly independent vectors of $V$;
2. *Properness at edges:* if $e$ is an edge of $P$, then the $(n-1)$ colours of facets around $e$ are linearly independent.
Given a fixed labelling $\Omega$ of the facets of a finite-volume right-angled polytope $P \subset \mathbb{H}^n$, we shall write its colouring as a vector $\mathbf{\lambda} = (\lambda_1, \dots, \lambda_m)$, where $\lambda_i = \sum^{\dim V - 1}_{k = 0} \lambda(i)_k\cdot 2^k$ is a binary representation of the vector $\lambda(i) \in V$, for all $i \in \Omega$.
Let $s_i$ be a reflection in the supporting hyperplane of the $i$-th facets of $P$. Then a proper colouring $\lambda: \Omega \rightarrow V$ defines a homomorphism from the reflection group $\Gamma = \mathrm{Ref}(P) = \langle s_1, s_2, \dots, s_m \rangle$ of $P$ to $V$, such that $\ker \lambda$ is a torsion-free subgroup of $\Gamma$ [@KS].
Let us consider one of the colourings of $\mathcal{D}$ defined in [@GS Table 1], that gives rise to a non-orientable manifold cover of the orbifold $\mathbb{H}^3 \diagup \Gamma_{12}$. Namely, choose $\mathbf{\lambda} = (1, 2, 4, 4, 2, 6, 3, 5, 5, 3, 1, 7)$, so that the $i$-th component of $\mathbf{\lambda}$ corresponds to the colour $\lambda_i$ of the $i$-th face of $\mathcal{D}$. As follows from [@KMT Corollary 2.5], this colouring is indeed non-orientable, since $\lambda_1 + \lambda_2 + \lambda_7 = \mathbf{0}$ in $\mathbb{Z}^3_2$. Thus, $M = \mathbb{H}^3 \diagup \Gamma$, with $\Gamma = \ker \mathbf{\lambda}$ a torsion-free subgroup of $\Gamma_{12}$, is a non-orientable compact hyperbolic $3$-manifold.
The reflection group $\Gamma_{12}$ is an index $120$ subgroup in the reflection group $\mathrm{Ref}(T)$ of the orthoscheme $T = [4,3,5]$, which is arithmetic. Thus, $\Gamma_{12}$ is also arithmetic. Moreover, $\mathrm{Ref}(T) = O^+(q, \mathbb{Z}[\omega])$, with $\omega = \frac{1 + \sqrt{5}}{2}$, for the quadratic form $q = -\omega x^2_0 + x^2_1 + x^2_2 + x^2_3$, as described in [@B §7] and, initially, in [@Bugaenko].
Next, let $\rho: \Gamma \rightarrow R(\Gamma)$ be the restriction of $R$. Observe that $R(\Gamma) = \Gamma_4$, and thus $\rho: \Gamma \rightarrow \Gamma_4$ is an epimorphism. Here we use the fact that $s_1 = \rho(s_1 s_2 s_7)$, $s_3 = \rho(s_3 s_4)$, $s_9 = \rho(s_8 s_9)$, and $s_{11} = \rho(s_2 s_{10} s_{11})$, where all the respective products of $s_i$’s belong to $\Gamma = \ker \mathbf{\lambda}$.
For any subgroup $K \leq F_3$ of index $n$, let us consider $\rho^{-1}(K) = R^{-1}(K) \cap \Gamma$. Then $K$ has index $8n$ in $\Gamma_4$, and $H = \rho^{-1}(K)$ has index $8n$ in $\Gamma$.
Moreover, we produce an orientation-reversing element $\delta \in \Gamma$, such that $\delta \in H$ for every such $H$.
Having established these facts, we know that there are $\asymp (8 n)^{(8 n)} \asymp n^n$ non-conjugate in $\mathrm{Isom}(\mathbb{H}^3)$ subgroups of $\Gamma$ by using the argument of [@BGLS §5.2], and thus there are $\asymp x^x$ non-isometric non-orientable compact arithmetic $3$-manifolds $M = \mathbb{H}^3 \diagup H$ of volume $\leq x$ (for $x > 0$ big enough). This proves the three-dimensional case of .
Now, observe that $x = s_1 s_{11}$, and $\lambda(x) = (1,0,0)^t + (1,0,0)^t = \mathbf{0}$ in $\mathbb{Z}^3_2$. Similarly, $\lambda(y) = \lambda(z) = \mathbf{0}$. Also, $R$ maps $x$, $y$, and $z$ respectively to themselves. Thus, $F_3 = \langle x, y, z \rangle \subset \rho(\Gamma)$. Finally, the element $\delta = s_2 s_4 s_6$ is such that $\lambda(\delta) = (0,1,0)^t + (1,0,0)^t + (1,1,0)^t = \mathbf{0}$, and $\rho(\delta) = \mathrm{id}$, so that $\delta \in \Gamma$ and $\delta \in H = \rho^{-1}(K)$, for every $K \leq F_3$.
Given that $H \leq O^+(q, \mathbb{Z}[\omega])$ for an admissible quadratic form $q$, we have that the argument in the proof of [@KRS Corollary 1.5] applies in this case, and thus the non-orientable compact manifold $M = \mathbb{H}^3\diagup H$ embeds into a compact orientable manifold $N = \mathbb{H}^4\diagup G$, for some arithmetic torsion-free $G \leq O^+(Q, \mathbb{Z}[\tau])$, with $Q = q + x^2_4$. Then, cutting $N$ along $M$ produces a manifold $N // M$, which is connected since $N$ is orientable while $M$ is not. Also, since $M$ is a one-sided submanifold of $N$, the boundary $\partial N'$ is isometric to $\widetilde{M}$, the orientation cover of $M$. Thus we obtain a collection of $\asymp n^n$ orientable arithmetic $3$-manifolds $\widetilde{M}$ that bound geometrically. However, some of them can be isometric, since the same manifold $\widetilde{M}$ can be the orientation cover of several distinct non-orientable manifolds $N_1$, $\dots$, $N_m$.
In order to estimate $m$, observe that each $N_i$ is a quotient of $\widetilde{M}$ by a fixed point free orientation-reversing involution. Let the number of such involutions for $\widetilde{M}$ be $I(\widetilde{M})$. Then $m \leq I(\widetilde{M}) \leq | \mathrm{Isom}(\widetilde{M}) | \leq c_1 \cdot \mathrm{Vol}(\widetilde{M}) \leq c_2 \cdot n = c_3 x$. Indeed, the isometry group of $\widetilde{M}$ is finite, and by the Kazhdan-Margulis theorem [@KM] there exists a lower bound for the volume of the orbifold $\widetilde{M}\diagup \mathrm{Isom}(\widetilde{M}) \geq c_0 > 0$, from which the final estimate follows. Thus, we have at least $\asymp n^n/(c_2 n) \asymp n^n \asymp x^x$ non-isometric compact orientable arithmetic hyperbolic $3$-manifolds $\widetilde{M}$ of volume $\leq x$ that bound geometrically. The upper-bound of the same order of growth follows from [@BGLS]. This proves the three-dimensional case of .
The right-angled 120-cell
-------------------------
Let $\mathcal{C} \subset \mathbb{H}^4$ be the regular right-angled $120$-cell. This polytope can be obtained by the Wythoff construction with the orthoscheme $[4,3,3,5]$ that uses the vertex stabiliser subgroup $[3,3,5]$ of order $(120)^2 = 14400$. The polytope $\mathcal{C}$ is compact and each of its $3$-dimensional facets is a regular right-angled dodecahedron isometric to $\mathcal{D}$ defined above.
Let us choose a facet $F$ of $\mathcal{C}$ and label it $120$. Since $F$ is isometric to $\mathcal{D}$, we can label the neighbouring facets of $F$ as follows:
- choose an isometry $\varphi$ between $F$ and $\mathcal{D}$ and transfer the labelling of $2$-dimensional faces of $\mathcal{D}$ depicted in from $\mathcal{D}$ to $F$ via $\varphi$,
- if $F'$, a facet of $\mathcal{C}$, shares a $2$-face labelled $i \in \{ 1, 2, \dots, 12\}$ with $F$, label $F'$ with $i$.
The remaining facets of $\mathcal{C}$ can be labelled with the numbers in $\{13, \dots, 119\}$ in an arbitrary way. Let $s_i$ denote the reflection on the supporting hyperplane of the $i$-th facet of $\mathcal{C}$, and let $\Gamma_{120} = \mathrm{Ref}(\mathcal{C}) = \langle s_1, s_2, \dots, s_{120} \rangle$.
Now define a colouring $\Lambda$ of $\mathcal{C}$ by using the colouring $\mathbf{\lambda}$ of $\mathcal{D}$ defined above. Namely, we set $$\Lambda(s_i) = \left\{
\begin{array}{cl}
\lambda_i, &\mbox{ for } 1\leq i \leq 12,\\
2^{i-10}, &\mbox{ for } 13\leq i \leq 120.
\end{array}
\right.$$
Observe that $\Lambda$ is a proper colouring of $\mathcal{C}$, as defined in [@KMT], and thus $\Gamma = \ker \Lambda$ is torsion-free. Also, $\Lambda$ is a non-orientable colouring. As in the case of $\mathcal{D}$, we use the retraction $R$ in order to map $\Gamma_{120}$ onto $\Gamma_4$, that contains $F_3$ as a finite-index subgroup. By taking preimages $H = \rho^{-1}(K)$ in $\Gamma$ of index $n$ subgroups $K\leq F_3$ and applying our argument from the previous section, we complete the proof of in the $4$-dimensional case and obtain $\asymp n^n$ non-isometric non-orientable compact arithmetic hyperbolic $4$-manifolds $M = \mathbb{H}^4\diagup H$. The rest of the argument follows from [@KRS Theorem 1.4]. Thus, the $4$-dimensional case of is also proven.
Non-compact right-angled polytopes
----------------------------------
Let $\mathcal{R}_3$ be a right-angled bi-pyramid depicted in , which is the first polytope in the series described by L. Potyagaĭlo and È. Vinberg in [@PV]. The construction in [@PV] produces a series of polytopes $\mathcal{R}_n \subset \mathbb{H}^n$, for $n=3, \dots, 8$, of finite volume, with both finite and ideal vertices, such that each facet of $\mathcal{R}_n$ is isometric to $\mathcal{R}_{n-1}$. Each $\mathcal{R}_n$ is produced by Whythoff’s construction from the quotient of $\mathbb{H}^n$ by the reflective part of $O^+(f_n, \mathbb{Z})$, with $f_n = -x^2_0 + \sum^n_{k=1} x^2_k$, for $n=3, \dots, 8$.
If we provide a non-orientable proper colouring of $\mathcal{R}_3$, as defined in [@KS], we can apply our previous reasoning in order to prove and , as consequence. Let us label the faces of $\mathcal{R}_3$ as shown in , and let the colouring be $\mathbf{\lambda} = (1, 1, 4, 7, 5, 2)$. It is easy to check that $\mathbf{\lambda}$ is indeed proper, since we need to check only the colours around the finite vertices and edges of $\mathcal{R}_3$. Also, $\mathbf{\lambda}$ is non-orientable, since $\lambda_4 + \lambda_5 + \lambda_6 = \mathbf{0}$ in $\mathbb{Z}^3_2$. Let $\Gamma = \ker \mathbf{\lambda}$.
Let $s_i$ be the reflection in the $i$-th facet of $\mathcal{R}_3$, and $\Gamma_6 = \langle s_1, \dots, s_6 \rangle$, and $\Delta = \langle s_1, s_2, s_3 \rangle$. Observe that $\Delta$ contains a free group of rank $2$ as a *normal* subgroup of index $4$. Indeed, $F_2 = \langle x, y \rangle$, with $x = s_1 s_2$, $y = s_3 s_1 s_2 s_3$ is such a subgroup.
Let $R$ be a retraction $\Gamma_6 \rightarrow \Delta$ given by $$R: s_i \mapsto
\left\{
\begin{array}{cl}
s_i, &\mbox{ if } i \in \{1, 2, 3\},\\
\mathrm{id}, &\mbox{ otherwise. }
\end{array}
\right.$$
Since $R$ maps $x$ and $y$ respectively to themselves, and $\lambda(s_1 s_2) = (1,0,0)^t + (1,0,0)^t = \mathbf{0}$ in $\mathbb{Z}^3_2$, we have that $F_3 \subset R(\Gamma)$. Moreover, for $\delta = s_4 s_5 s_6$ it holds that $\lambda(\delta) = \mathbf{0}$, as already verified above, and $R(\delta) = \mathrm{id}$. Then the argument from the previous case of the right-angled dodecahedron applies verbatim.
For the induction step from $\mathcal{R}_{n-1}$ to $\mathcal{R}_n$ we just need to enhance the colouring in the way completely analogous to the extension of a non-orientable colouring of the dodecahedron $\mathcal{D}$ to a non-orientable colouring of the $120$-cell $\mathcal{C}$. Again, the rest of the argument proceeds verbatim in complete analogy to the previous cases.
Surfaces that bound geometrically
---------------------------------
Let $\mathcal{P} \subset \mathbb{H}^2$ be a compact regular right-angled octagon, with sides labelled anti-clockwise $1$, $6$, $2$, $7$, $3$, $8$, $4$, $5$. Let $s_i$ be the reflection in the $i$-th side of $\mathcal{P}$, and $\Gamma_{8} = \mathrm{Ref}(\mathcal{P}) = \langle s_1, s_2, \dots, s_{8} \rangle$ be its reflection group. Also, let $\Gamma_4 = \langle s_1, s_2, s_3, s_4 \rangle$ and $F_3 = \langle x, y, z \rangle$, with $x = s_1 s_2$, $y = s_1 s_3$, $z = s_1 s_4$, be a free subgroup of $\Gamma_4$ of index $2$. The retraction of $\Gamma_8$ onto $\Gamma_4$ is given by $$R: s_i \mapsto
\left\{
\begin{array}{cl}
s_i, &\mbox{ if } i \in \{1, 2, 3, 4\},\\
\mathrm{id}, &\mbox{ otherwise. }
\end{array}
\right.$$
Let us choose a colouring $\mathbf{\lambda} = (1, 1, 1, 1, 2, 3, 5, 6)$ for $\mathcal{P}$, which is a proper and non-orientable one, since $\lambda_1 + \lambda_5 + \lambda_6 = \mathbf{0} \in \mathbb{Z}^3_2$. Let $\Gamma = \ker \lambda$. An easy check ensures that $F_3 \subset R(\Gamma)$, as well as that $R(\delta) = \mathrm{id}$ for an orientation-reversing element $\delta = s_6 s_7 s_8 \in \Gamma$. Then the lower bound $\alpha(x) \geq A_1 x^{B_1 x}$, for some constants $A_1, B_1 > 0$, for the number of geometrically bounding surfaces of area $\leq x$ (for $x$ large enough) follows immediately. The upper bound $\alpha(x) \leq A_2 x^{B_2 x}$, for some constants $A_2, B_2 > 0$, follows from [@BGLS]. Since $\text{area} = 2 \pi (g-1)$, for an orientable genus $g\geq 2$ surface, this proves . The case of finite-area non-compact surfaces mentioned in Remark \[rem:surfaces-nc\] easily follows by analogy.
Constructing geodesic boundaries by arithmetic reductions {#sec: reductions}
=========================================================
We start by recalling the following lemma of Long and Reid [@LR3 Lemma 2.2] (c.f. also the remark after its proof).
\[lem:TF-subgroup\] Let $\Gamma < O^+(n, 1)$ be a subgroup of hyperbolic isometries defined over a number field $K$, and $\delta$ an element of $\Gamma$. Let $\theta_1, \theta_2: \Gamma \rightarrow F_i$ be two homomorphisms of $\Gamma$ onto a group $F_i$, with torsion-free kernels. Let $\Theta(g) = \left( \theta_1(g), \theta_2(g) \right): \Gamma \rightarrow F_1 \times F_2$. Suppose that $\theta_i(\delta)$ has order $k_i < \infty$, $i = 1, 2$, and any prime dividing $gcd(k_1, k_2)$ appears with distinct exponents in $k_1$ and $k_2$. Then $\Theta^{-1} \langle (\theta_1(\delta), \theta_2(\delta)) \rangle$ is a torsion-free subgroup in $\Gamma$ of finite index that contains $\delta$.
The following lemma is used in order to show that the maps that we choose in the sequel as $\theta_i$, $i = 1, 2$, in the Subgroup Lemma above have torsion-free kernels. Its proof is very similar to that of [@LR3 Lemma 2.4].
\[lem:No-torsion\] Let $\Gamma < O^+(n, 1)$ be a finite subgroup defined over the ring of integers $\mathcal{O}_K$ a number field $K$, and let $p \in \mathcal{O}_K$ be an odd rational prime that does not divide the order of $\Gamma$. Then the reduction of $\Gamma$ modulo the ideal $\mathcal{J} = (p)$ is isomorphic to $\Gamma$.
A non-trivial element $g$ of the kernel of the reduction map $\Gamma(K) \rightarrow \Gamma(K/\mathcal{J})$ can be written in the form $g = \mathrm{id} + p^r h$, where $h$ is a matrix not all of whose entries are divisible by $p^r$, with $r$ some positive integer. Let $q < \infty$ be the order of an element $g \in \Gamma$. Then we get $$\mathrm{id} = g^q = \mathrm{id} + q p^r h + \sum^q_{t=2} \binom{q}{t} p^{rt} h^{t},$$ and thus $$q h = \mathbf{0} \mod p^r.$$ The latter implies $p^r$ divides $q$, since $h \neq \mathbf{0} \mod p^r$. Thus, $p$ divides $q$, and $q$ divides the order of $\Gamma$, since $q$ is the order of an element of $\Gamma$. The latter is a contradiction, and thus the reduction map has trivial kernel.
As shown by Vinberg in [@Vinberg], in some cases for an admissible quadratic form $q$ of signature $(n, 1)$ defined over a totally real number field $K$ with ring of integers $O_K$ it holds that $O^+(q, O_K) = \mathrm{Ref}(P) \rtimes \mathrm{Sym}(P)$, where $P \subset \mathbb{H}^n$ is a finite-volume polytope. Here, $\mathrm{Ref}(P)$ denotes the associated reflection group, and $\mathrm{Sym}(P)$ is the group of symmetries of $P$. Also, we assume that $O_K$ is a principal ideal domain in order to keep our account simpler. We refer the reader to [@Guglielmetti-1] for more details.
If the above presentation of $O^+(q, O_K)$ takes place for some finite-volume polytope $P \subset \mathbb{H}^n$, the form $q$ is called *reflective*, and the polytope $P$ is called its *associated polytope*.
An algorithm introduced by Vinberg in [@Vinberg] and implemented in [@Guglielmetti-2] by Guglielmetti allows us to find the associated polytope $P\subset\mathbb{H}^n$ in finite time, for any reflective admissible quadratic form of signature $(n, 1)$.
Compact polytopes in dimensions 5 and 6 {#compact56}
---------------------------------------
Let $\omega=\frac{1+\sqrt{5}}{2}$ and let $P_n \subset \mathbb{H}^n$ be the polytopes associated to the quadratic forms $$q_5=-(-1+2\omega)x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2,$$ $$q_6=-2\omega x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2.$$ The polytopes $P_5$ and $P_6$ are apparently new, and were found by using AlVin and CoxIter software [@Guglielmetti-2; @Guglielmetti-3]. They differ substantially from the polytopes that appear in [@Bugaenko; @Bugaenko2], and have fewer facets.
Let $\Gamma_n = \mathrm{Ref}(P_n)$, be the reflection group of $P_n$, with generators $s_i$, $i \in I_n$, where $I_n$ is the set of outer normals to the facets of $P_n$ or, equivalently, the set of nodes of the Coxeter diagram of $P_n$[^1]. With standard basis $\{v_0, v_1, \dots, v_n\}$, the Vinberg algorithm determines the outer normals for $n=5, 6$ which are given in the Appendix.
The associated reflection group $\Gamma_n$ is arithmetic, and contains a virtually free parabolic subgroup $$\Delta = \begin{cases}
\langle s_5, s_6, s_9 \rangle, &\text{for } n = 5, \\
\langle s_6, s_9, s_{17} \rangle, &\text{for } n = 6;
\end{cases}$$ Indeed, $\Delta$ is isomorphic to the $(2,\infty,\infty)$-triangle group[^2], which contains $F_2$ as a subgroup of index $4$.
The retraction $R: \Gamma_n \rightarrow \Delta$ is defined by sending all but three generators of $\Gamma_n$ to $\mathrm{id}$, with the only generators mapped identically being those of $\Delta < \Gamma_n$.
In order for $R$ being well-defined, we essentially need that the generators of $\Delta$ be connected to the rest of the diagram by edges with even labels only, since any two generators connected by a path of odd-labelled edges are conjugate. This folds, for instance, if the facets corresponding to the reflections generating $\Delta$ are *redoubleable* in terms of [@Allcock].
The element $$\delta = \begin{cases}
s_1 s_2 s_3 s_4 s_7, & \text{for } n=5, \\
s_7 s_{13} s_{18}, & \text{for } n=6.
\end{cases}$$ is orientation-reversing, as it is a product of an odd number of reflections in $\mathbb{H}^n$. Moreover, $\delta \in \ker R$.
[|c|c|c|]{} Polytope & Diagram & LCM\
$P_5$ &
------------------------------
------------------------------
: Polytopes $P_n$, $n=5, 6$, their Coxeter diagrams, and the least common multiple (LCM) of the orders of their parabolic finite subgroups[]{data-label="tab2"}
& $57600 = 2^8 \cdot 3^2 \cdot 5^2$\
$P_6$ &
------------------------------
------------------------------
: Polytopes $P_n$, $n=5, 6$, their Coxeter diagrams, and the least common multiple (LCM) of the orders of their parabolic finite subgroups[]{data-label="tab2"}
& $230400 = 2^{10}\cdot 3^2\cdot 5^2$\
We shall set the map $\Theta$ from to be a pair of reductions modulo various rational primes, and then use in order to ensure that their kernels are torsion-free. Indeed, we need to choose such an odd prime $p \in \mathbb{Z}$ that it does not divide the order of any finite parabolic subgroup in the diagram of $P_n$, $n = 5, 6$. The least common multiples of orders of finite parabolic subgroups for $P_n$, $n = 5, 6$, are given in [^3].
For $\Gamma\in {\mathrm{GL}}(n+1,\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of a number field $K$, let $\phi_p$ denote the homomorphism $\Gamma\rightarrow {\mathrm{GL}}(n+1,\mathcal{O}_K/\mathcal{J})$ induced by reduction modulo $\mathcal{J} = (p)$, the principal ideal generated by a rational integer $p$.
Let us consider the reductions $\phi_{7}$ and $\phi_{11}$ as defined above, and let $\Theta = (\phi_{7}, \phi_{11})$. For $n=5$, the order of $\phi_{7}(\delta)$ equals $800 = 2^5 \cdot 5^2$, while the order of $\phi_{11}(\delta)$ equals $8052 = 2^2\cdot 3^1\cdot 11^1\cdot 61^1$, as follows by straightforward computations, c.f. [@CK-code1; @CK-code2]. For $n=6$, the order of $\phi_{7}(\delta)$ equals $8 = 2^3$, while the order of $\phi_{11}(\delta)$ equals $44 = 2^2\cdot 11^1$.
Then and apply, and $\Gamma = \Theta^{-1}\langle (\phi_{7}(\delta), \phi_{11}(\delta)) \rangle$ is a torsion-free subgroup of finite index in $\Gamma_n$ that contains the orientation-reversing element $\delta$ and retracts onto the free group $\Gamma\cap\Delta$. Then the argument analogous to that of Section \[sec: color\] applies.
Right-angled cusped polytopes in dimensions 4 to 8 {#cusped45678}
--------------------------------------------------
Let $P_n \subset \mathbb{H}^n$ be the right-angled polytopes associated to the principal congruence subgroups of level $2$ for the quadratic forms $$f_n = -x_0^2 + x_1^2 + x_2^2 + \cdots + x_n^2, \text{ for } n=4, \dots, 8.$$
Let $\Gamma_n = \mathrm{Ref}(P_n)$, be the associated reflection group, with generators $s_i$, $i \in I_n$, where $I_n$ is the set of outer normals to the facets of $P_n$. . With standard basis $\{v_0, v_1, \dots, v_n\}$, the Vinberg algorithm starts with the first $n$ outer normals being $$e_i = - v_i, \text{ for } 1 \leq i \leq n,$$ and continues with the next $\binom{n}{2}$ outer normals $$e_{i, j} = v_0 + v_i + v_j, \text{ for } 1 \leq i < j \leq n,$$ all of them being $1$-roots, as is necessary for determining the reflective part of the principle congruence level $2$ subgroup, rather than that of the whole group of units for $f_n$. Let us set $e_{n+1} = e_{1, 2} = v_0 + v_1 + v_2$ and $e_{n+2} = e_{3, 4} = v_0 + v_3 + v_4$, as a more convenient notation.
Such $\Gamma_n$ is arithmetic, and it contains a virtually free parabolic subgroup $\Delta = \langle s_3, s_{4}, s_{n+2} \rangle$. Indeed, $\Delta$ is isomorphic to the $(2, \infty, \infty)$-triangle group which contains $F_2$ as a subgroup of index $4$. Consider the retraction $$R: s_i \mapsto
\left\{
\begin{array}{cl}
s_i, &\mbox{ if } i \in \{3, 4, n+2\},\\
\mathrm{id}, &\mbox{ otherwise. }
\end{array}
\right.$$
The element $\delta= s_1 s_2 s_{n+1}$ is orientation-reversing, as it is a product of $3$ reflections in $\mathbb{H}^n$. Moreover, $\delta \in \ker R$.
For $\Gamma\in {\mathrm{GL}}(n+1,{\mathbb{Z}})$, let $\phi_m$ denote the homomorphism $\Gamma\rightarrow {\mathrm{GL}}(n+1,{\mathbb{Z}}/m{\mathbb{Z}})$ induced by reduction modulo a positive integer $m$. By [@Newman Theorem IX.7] we know that the kernel of $\phi_m$ is torsion-free for $m > 2$. The reduction of $\delta$ modulo $3$ has order $4=2^2$, while its reductions modulo $4$ has order $2$, c.f. [@CK-code1; @CK-code2]. Letting $\Theta = (\phi_3, \phi_4)$, applies, and $\Gamma = \Theta^{-1}\langle(\phi_3(\delta), \phi_4(\delta))\rangle$ is a torsion-free subgroup of finite index in $\Gamma_n$ that contains the orientation-reversing element $\delta$ and retracts onto the free group $\Gamma\cap\Delta$.
Cusped polytopes in dimensions 9 to 13 {#cusped9to13}
--------------------------------------
Let $P_n \subset \mathbb{H}^n$ be the polytopes from Table 7 in [@Vinberg] associated to the quadratic forms $$f_n = -2 x_0^2 + x_1^2 + x_2^2 + \cdots + x_n^2 \text{ for } n=9, 10 \text{ and } 13,$$ while $P_n$, for $n=11, 12$, be the polytopes with Coxeter diagrams given in Figure \[fig:polytopes-cusped-11-12\], respectively. The latter ones appear to be new, and were found by using `AlVin` [@Guglielmetti-3].
Let $\Gamma_n = \mathrm{Ref}(P_n)$, be the reflection group of $P_n$, with generators $s_i$, $i \in I_n$, where $I_n$ is the set of nodes in the Coxeter diagram of $P_n$. Such $\Gamma_n$ is arithmetic, and it contains a virtually free parabolic subgroup $\Delta$ indicated in Table \[tab0\].
Since $\Delta$ is generated by reflections in redoubleable facets we can define a retraction $R: \Gamma_n \rightarrow \Delta$, as before, that send all the generators of $\Gamma_n$ to $\mathrm{id}$, except of those of $\Delta$.
$n$ generators of $\Delta$ triangle group $\cong \Delta$
------ ------------------------------ -------------------------------
$9$ $s_9$, $s_{10}$, $s_{12}$ $(2, \infty, \infty)$
$10$ $s_{10}$, $s_{11}$, $s_{13}$ $(2, 4, \infty)$
$11$ $s_{11}$, $s_{12}$, $s_{18}$ $(4, 4, \infty)$
$12$ $s_{12}$, $s_{13}$, $s_{20}$ $(4, 4, \infty)$
$13$ $s_{13}$, $s_{14}$, $s_{19}$ $(2, \infty, \infty)$
: A virtually free parabolic subgroup $\Delta < \Gamma_n$, for $n = 9, \ldots, 13$[]{data-label="tab0"}
The orientation-reversing element $\delta_n \in \ker R$ is defined by $$\delta_n = \begin{cases}
s_1 s_2 s_3 \ldots s_8 s_{n+2}, & \text{for } n = 9, 10, 13, \\
s_7 s_8 s_{16}, & \text{for } n = 11, \\
s_2 s_{11} s_{18}, & \text{for } n = 12.
\end{cases}$$
Letting $\Theta = (\phi_{m_1}, \phi_{m_2})$, applies with $m_1$ and $m_2$ as in . Here, we also notice that $\delta_n$ for $n= 10, 13$ is an extension of $\delta_9$ by the identity map, which simplifies the computations.
Then $\Gamma = \Theta^{-1}\langle(\phi_{m_1}(\delta), \phi_{m_2}(\delta))\rangle$ is a torsion-free subgroup of finite index in $\Gamma_n$ that contains the orientation-reversing element $\delta$ and retracts onto the free group $\Gamma\cap\Delta$.
$n$ $m_1$ $k_1 = \mbox{order of } \phi_{m_1}(\delta_n)$ $m_2$ $k_2 = \mbox{order of } \phi_{m_2}(\delta_n)$
----------- ------- ----------------------------------------------- ------- -----------------------------------------------
9, 10, 13 $3$ $84 = 2^2\cdot 3^1\cdot 7^1$ $4$ $34 = 2^1\cdot 17^1$
11, 12 $3$ $6 = 2^1 \cdot 3^1$ $4$ $4 = 2^2$
: Orders of the reductions of $\delta$ and their prime factorisations, c.f. [@CK-code1; @CK-code2][]{data-label="tab1"}
Constructing geometric boundaries from virtual retracts
=======================================================
Let $q_n$ be an admissible quadratic form of signature $(n, 1)$ defined over a totally real number field $K$ with ring of integers $O_K$, and let $q_{n+1} = q_n + x^2_{n+1}$. Suppose also that $\Gamma_n < O^+(q_n, O_K)$ is a torsion-free subgroup, either of finite co-volume or co-compact.
Now assume that there exists a retraction $R_n: \Gamma_n \rightarrow \Delta$ of $\Gamma_n$ onto a virtually free subgroup $\Delta$ such that $\ker R_n$ contains an orientation-reversing element $\delta$.
By [@KRS Proposition 7.1], there exists a torsion-free finite-index subgroup $\Gamma'_{n+1} \leq O^+(q_{n+1}, O_K)$, such that $\Gamma_n < \Gamma'_{n+1}$. Moreover, we may assume that $\mathbb{H}^n / \Gamma_n$ is a properly embedded totally geodesic submanifold of $\mathbb{H}^{n+1} / \Gamma_{n+1}$. Thus, the group $\Gamma_n$ is a geometrically finite subgroup of $\Gamma'_{n+1}$. By [@BHW Theorem 1.4], there is a finite index subgroup $G<\Gamma'_{n+1}$, such that $G$ virtually retracts to its geometrically finite subgroups. In particular, $G$ virtually retracts to $G\cap\Gamma_n$. However, since $\Gamma'_{n+1}$ is linear, the arguments of [@LR4 Theorem 2.10] apply to give a virtual retraction from $\Gamma'_{n+1}$ onto $\Gamma_n$. Let $\Gamma_{n+1}$ be the finite index subgroup of $\Gamma'_{n+1}$ which retracts onto $\Gamma_n$. Then the composition $R_{n+1}: \Gamma_{n+1} \rightarrow \Gamma_n \rightarrow \Delta$ is a retraction of $\Gamma_{n+1}$ onto $\Delta$, such that $\delta \in \ker R_{n+1}$.
All the previous arguments from Section \[sec: reductions\] apply, and we obtain Theorems \[thm2\] – \[thm3\] for all $n \geq 2$, since we can use any of our examples worked out in Sections \[sec: color\]–\[sec: reductions\] as a basis for the above inductive procedure.
Acknowledgements {#acknowledgements .unnumbered}
================
[[ The first author thanks Darren Long for many helpful discussions. Both authors would like to thank Mikhail Belolipetsky (IMPA), Vincent Emery (Universität Bern), and Ruth Kellerhals (Université de Fribourg), as well as the Mathematisches Forschungsinstitut Oberwolfach (MFO) administration, for organising the mini-workshop “Reflection Groups in Negative Curvature” (1915b) in April 2019, during which the results of this paper were presented and discussed. They also thank the organisers of the Borel Seminar (Les Diablerets, Switzerland) in December 2018, where this work was started, for stimulating research atmosphere, and the Swiss Doctoral Program – CUSO, Swiss Mathematical Society, and ETH Zürich for the financial support of the event. A word of special gratitude is addressed to the creators and maintainers of SAGE Math, and to Rafael Guglielmetti, the author of `CoxIter` and `AlVin`. The aforementioned software made most of the calculations in the present work possible and provided means for additional verification of those previously done. We also thank the anonymous referees for their suggestions and criticism that helped us improving this paper.]{}]{}
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, *A note on surfaces bounding hyperbolic 3-manifolds*, Monats. Math. **142** (2004), 267–273.
{#app}
Outer normals for compact P5
----------------------------
()
&e\_i = -v\_i+v\_[i+1]{} 1 i4, &\
&e\_5 =-v\_5, &\
&e\_6 =v\_0 + (2 + ) v\_1, &\
&e\_7 =(v\_0 + v\_1 + v\_2 + v\_3), &\
&e\_8 =(1 + ) (v\_0 + v\_1) + (v\_2 + v\_3 + v\_4 + v\_5). &
Outer normals for compact P6
----------------------------
()
&e\_i = -v\_i+v\_[i+1]{} 1 i5, &\
&e\_6 =-v\_6, &\
&e\_7 = v\_0 + w (v\_1 + v\_2), &\
&e\_8 = (v\_0 + v\_1 + v\_2 + v\_3 + v\_4), &\
&e\_9 = v\_0 + 2 v\_1, &\
&e\_[10]{} = (1 + ) (v\_0 + v\_1 + v\_2) + (v\_3 + v\_4 + v\_5 + v\_6), &\
&e\_[11]{} = (1 + 2 ) v\_0 + (1 + 3 ) v\_1 + (1 + ) (v\_2 + v\_3 + v\_4) + (v\_5 + v\_6), &\
&e\_[12]{} = (1 + 2 ) v\_0 + (2 + 3 ) v\_1 + (v\_2 + v\_3 + v\_4 + v\_5 + v\_6), &\
&e\_[13]{} = (2 + 2 ) v\_0 + (1 + 2 ) (v\_1 + v\_2 + v\_3 + v\_4 + v\_5) + v\_6, &\
&e\_[14]{} = (2 + 3 ) v\_0 + (2 + 4 ) v\_1 + (2 + 2 ) v\_2 + (1 + 2 ) (v\_3 + v\_4 + v\_5) + v\_6, &\
&e\_[15]{} = (2 + 3 ) v\_0 + (3 + 4 ) v\_1 + (1 + 2 ) (v\_2 + v\_3 + v\_4) + 2 v\_5, &\
&e\_[16]{} = (2 + 4 ) v\_0 + (3 + 6 ) v\_1 + (1 + 2 ) (v\_2 + v\_3 + v\_4 + v\_5) + v\_6, &\
&e\_[17]{} = (3 + 4 ) v\_0 + (2 + 5 ) v\_1 + (2 + 3 ) (v\_2 + v\_3 + v\_4 + v\_5) + v\_6, &\
&e\_[18]{} = (4 + 5 ) v\_0 + (4 + 6 ) v\_1 + (2 + 4 ) (v\_2 + v\_3 + v\_4 + v\_5). &
Outer normals for cusped P4
---------------------------
()
& e\_i = -v\_i 1 i4, &\
&e\_i = v\_0+v\_[j\_1]{}+v\_[j\_2]{} 5i10 1 j\_1 < j\_2 4. &
Outer normals for cusped P5
---------------------------
()
& e\_i = -v\_i 1 i5, &\
&e\_i = v\_0+v\_[j\_1]{}+v\_[j\_2]{} 6i15 1 j\_1 < j\_2 5, &\
&e\_[16]{} = 2v\_0+v\_1+v\_2+v\_3+v\_4+v\_5. &
Outer normals for cusped P6
---------------------------
()
& e\_i = -v\_i 1 i6, &\
&e\_i = v\_0+v\_[j\_1]{}+v\_[j\_2]{} 7i21 1 j\_1 < j\_2 6,\
&e\_i = 2v\_0+v\_[j\_1]{}++v\_[j\_5]{} 22i27 1 j\_1 < j\_2 < j\_3 < j\_4 < j\_5 6. &
Outer normals for cusped P7
---------------------------
()
& e\_i = -v\_i 1 i7, &\
&e\_i = v\_0+v\_[j\_1]{}+v\_[j\_2]{} 8i28 1 j\_1 < j\_2 7, &\
&e\_i = 2v\_0+v\_[j\_1]{}++v\_[j\_5]{} 29i49 1 j\_1 < j\_2 < j\_3 < j\_4 < j\_5 7, &\
&e\_i = 3v\_0+2v\_[j\_1]{}+v\_[j\_2]{}++v\_[j\_7]{} 50i56 1 j\_1 < j\_2 < j\_3 < j\_4 < j\_5 < j\_6 < j\_7 7. &
Outer normals for cusped P8
---------------------------
()
&e\_i = -v\_i 1 i8, &\
&e\_i = v\_0+v\_[j\_1]{}+v\_[j\_2]{} 9i36 1 j\_1 < j\_2 8, &\
&e\_i = 2v\_0+v\_[j\_1]{}++v\_[j\_5]{} 37i92 1 j\_1 < j\_2 < j\_3 < j\_4 < j\_5 8, &\
&e\_i = 3v\_0+2v\_[j\_1]{}+v\_[j\_2]{}++v\_[j\_7]{} 93i148, &\
&e\_i = 4v\_0+2(v\_[j\_1]{}++v\_[j\_3]{})+v\_[j\_4]{}++v\_[j\_7]{} 149i204 &\
& 1 j\_1 < j\_2 < j\_3 < j\_4 < j\_5 < j\_6 < j\_7 7, &\
&e\_i = 5v\_0+2(v\_[j\_1]{}++v\_[j\_6]{})+v\_[j\_7]{}+v\_[j\_8]{} 205i232, &\
&e\_i = 6v\_0+3v\_[j\_1]{}+v\_[j\_2]{}++v\_[j\_8]{} 233i240 &\
& 1 j\_1 < j\_2 < j\_3 < j\_4 < j\_5 < j\_6 < j\_7 < j\_8 8. &
Outer normals for cusped P9
---------------------------
()
&e\_i = -v\_[i]{} + v\_[i+1]{}, 1 i8, &\
&e\_9 = -v\_9, &\
&e\_[10]{} = v\_0 + 2 v\_1, &\
&e\_[11]{} = v\_0 + v\_1 + v\_2 + v\_3 + v\_4, &\
&e\_[12]{} = 2 v\_0 + v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9. &
Outer normals for cusped P10
----------------------------
()
&e\_i = -v\_[i]{} + v\_[i+1]{}, 1 i9, &\
&e\_[10]{} = -v\_[10]{}, &\
&e\_[11]{} = v\_0 + v\_1 + v\_2 + v\_3 + v\_4, &\
&e\_[12]{} = v\_0 + 2 v\_1, &\
&e\_[13]{} = 2 v\_0 + v\_1 + v\_2 +v\_3 +v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{}. &
Outer normals for cusped P11
----------------------------
()
&e\_i = -v\_i + v\_[i+1]{}, 1 i7, i = 9, 10, &\
&e\_i = -v\_i, i = 8, 11, &\
&e\_[12]{} = v\_0 + v\_9 + v\_[10]{} + v\_[11]{}, &\
&e\_[13]{} = v\_0 + 2 v\_1 + v\_9, &\
&e\_[14]{} = v\_0 + v\_1 + v\_2 + v\_3 + v\_4, &\
&e\_[15]{} = 2 v\_0 + v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_9 + v\_[10]{}, &\
&e\_[16]{} = 2 v\_0 + v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9, &\
&e\_[17]{} = 3 v\_0 + 2 (v\_1 + v\_2) + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{}, &\
&e\_[18]{} = 4 v\_0 + 2 (v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7) + v\_9 + v\_[10]{} + v\_[11]{}. &
Outer normals for cusped P12
----------------------------
()
&e\_i = -v\_i + v\_[i+1]{}, 1 i10, &\
&e\_[i]{} = -v\_[i]{}, i = 11, 12, &\
&e\_[13]{} = v\_0 + 2 v\_1 + v\_[12]{}, &\
&e\_[14]{} = v\_0 + v\_1 + v\_2 + v\_3 + v\_4, &\
&e\_[15]{} = 2 v\_0 + v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_[12]{}, &\
&e\_[16]{} = 2 v\_0 + v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{}, &\
&e\_[17]{} = 3 v\_0 + v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{} + 2 v\_[12]{}, &\
&e\_[18]{} = 3 v\_0 + 2 (v\_1 + v\_2) + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{} + v\_[12]{}, &\
&e\_[19]{} = 3 (v\_0 + v\_1) + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{}, &\
&e\_[20]{} = 5 v\_0 + 2 (v\_1 + v\_2 + v\_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{} + v\_[12]{}). &
Outer normals for cusped P13
----------------------------
()
&e\_i = -v\_[i]{} + v\_[i+1]{}, 1 i12, &\
&e\_[13]{} = -v\_[13]{}, &\
&e\_[14]{} = v\_0 + v\_1 + v\_2 + v\_3 + v\_4, &\
&e\_[15]{} = v\_0 + 2 v\_1, &\
&e\_[16]{} = 2 v\_0 + v\_1 + v\_2 +v \_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{}, &\
&e\_[17]{} = 3 v\_0 + 3 v\_1 + v\_2 +v \_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{} + v\_[12]{}, &\
&e\_[18]{} = 3 v\_0 + 2 v\_1 + 2 v\_2 + v \_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{} + v\_[12]{} + v\_[13]{}, &\
&e\_[19]{} = 5 v\_0 + 2 (v\_1 + v\_2 + v \_3 + v\_4 + v\_5 + v\_6 + v\_7 + v\_8 + v\_9 + v\_[10]{} + v\_[11]{} + v\_[12]{} + v\_[13]{}). &
[^1]: The notation used is as follows: a dashed edge means two reflection hyperplanes have a common perpendicular, a solid edge means parallel (at the ideal boundary) hyperplanes, a double edge means label $4$, a single edge means label $3$, any other edge has a label on it describing the corresponding dihedral angle. The colours are used for convenience only.
[^2]: Here $\Delta$ is not actually generated by reflections in the sides of a hyperbolic finite-area triangle, but is rather only abstractly isomorphic to such a group. However, we are interested in its algebraic rather than geometric properties, regarding its subgroup growth.
[^3]: The orders of all finite parabolic subgroups associated with $P_n$ can be obtained by using `CoxIter` [@Guglielmetti-2] with the `-debug` option.
|
---
abstract: 'We have experimentally investigated 2-dimensional dense bubble flows underneath inclined planes. Velocity profiles and velocity fluctuations have been measured. A broad second-order phase transition between two dynamical regimes is observed as a function of the tilt angle $\theta$. For low $\theta$ values, a block motion is observed. For high $\theta$ values, the velocity profile becomes curved and a shear velocity gradient appears in the flow.'
author:
- 'N.Vandewalle'
- 'S.Trabelsi'
- 'H.Caps'
date: 'Received: / Revised version: date'
title: 'Block to granular-like transition in dense bubble flows'
---
Introduction
============
The dense granular flow of identical particles is a complex phenomenon which involves multiple processes such as friction, correlated motion and inelastic collisions [@duranbook; @pouliquen; @prlpoiseuille]. Various flow regimes have been identified in different experiments. Three major flow regimes are emphasized in Figure \[profiles\]. Block motion occurs for example when particles form some jammed phases [@jam1; @jam2; @bonamy_gm]. In granular flows on a pile, only the surface grains are in motion. This rolling phase has typically a thickness of 10-20 grains and the velocity $v_h$ as a function of the height $h$ is roughly linear for that surface grains [@rajchenbach; @azanza; @scaling].
![Typical velocity profiles $v_{h}$ encountered in dense flows. From left to right: block motion, surface flow and Poiseuille flow.[]{data-label="profiles"}](profiles2.eps){width="8cm"}
A lot of efforts has been devoted to the understanding of dense flows of particles. Some studies [@contact; @grest1] address the problem of contact force networks, i.e. the presence of arches, which plays an important role [@role]. Fluctuations and correlations of the grain motion as a function of the shear rate have been also studied. Collective behaviors have been put into evidence such as the motion of clusters of grains [@bonamy_gm; @bonamy]. Different theoretical approaches have been developed for describing dense flows. Among them, one can cite the Saint-Venant approach which considers some conservation equations in the rolling phase [@saintvenant]. Pouliquen and coworkers [@pouliquen] have also proposed that the granular flows can be considered as quasi-static flows, the global motion resulting from fractures. Nevertheless, no general theory can predict the occurrence of the different flow regimes and corresponding velocity profiles from the various physical parameters such as the density, the shear rate and the friction.
In 1947, Bragg and coworkers proposed a simple experiment [@bragg] in order to put into evidence some crystallographic ordering. This experiment is also famous because it was quoted in the Feynman’s lectures [@feynman]. The experiment consisted in producing small identical bubbles at the surface of a liquid. Hexagonal packed structures appeared and crystallographic domains separated by grain boundaries were observed. Recent experiments [@tamtam; @dimeglio] were devoted to the study of defects in such bubble rafts.
We propose to modify the Bragg’s experiment in order to study dense flows of identical bubbles. The flow of bubbles is of physical interest since contacting bubbles are characterized by a nearly zero friction [@nicolson] ! Our experimental system corresponds thus to an “ideal” dense flow. We show in this letter that this system of bubbles presents a phase transition between two distinct dense flow regimes depending on the kinetic energy of the incoming bubbles.
Experimental setup
==================
![A transparent inclined plane is immersed into water and tilted by an angle $\theta$. Small air bubbles are injected from below at the bottom of the plane and rise towards the top where they aggregate on a linear rough obstacle. A sketch of the experimental setup (left) and typical bubble packing taken by the CCD camera placed above the setup (right) are illustrated. The angle is $\theta = 1^\circ$. Arrows indicate the motion of bubbles.[]{data-label="setup"}](setup.eps "fig:"){width="6cm"}![A transparent inclined plane is immersed into water and tilted by an angle $\theta$. Small air bubbles are injected from below at the bottom of the plane and rise towards the top where they aggregate on a linear rough obstacle. A sketch of the experimental setup (left) and typical bubble packing taken by the CCD camera placed above the setup (right) are illustrated. The angle is $\theta = 1^\circ$. Arrows indicate the motion of bubbles.[]{data-label="setup"}](packing2.eps "fig:"){width="6cm"}
Our experiment is analogous to the filling of a 2D-silo by continuous feeding of granular material from a central hopper. The experimental setup consists in a transparent inclined plane which is immersed into water \[see Fig. \[setup\]\]. The tilt angle $\theta$ is small and ranges from $0^\circ$ to $3^\circ$ in our experiments. The error on $\theta$ is $0.1^\circ$. Spherical air bubbles are injected from below at the bottom of the tilted plane. The bubble production rate is kept constant at $5$ bubbles by one second. The size of the bubbles can be controlled by the air pump. The bubble radius is typically $R=0.75$ mm in the present study. In order to avoid the coalescence of the bubbles, a small quantity of commercial soap (based on SDS surfactants [@broze]) is added to the water. The mixture has a surface tension $\gamma=0.03$ N/m and a viscosity $\eta=0.001$ kg$\,$m${}^{-1}\,$s${}^{-1}$. Due to buoyancy $B$, the bubbles rise beneath the inclined plane. A rough rigid object has been placed at the top of the plane such that bubbles aggregate there. The obstacle roughness is typically one bubble diameter and avoids purely translational motions of the bubble stack [@grest3]. One should note that in the direction perpendicular to the plane, the bubble stack is only one layer of bubbles thick. We thus have a really 2D bubble pile. A CCD camera captures top views of the packing of bubbles through the transparent tilted plane.
The unique parameter to be investigated is the tilt angle $\theta$ of the plane. Indeed, the angle $\theta$ controls the buoyancy $$B = {4 \pi R^3\Delta \rho g \over 3}\sin \theta,$$ where $\Delta \rho \approx 1000$ kg m$^{-3}$ is the difference between air and liquid densities. One should also note that the bubble motion is mainly limited by the viscous drag force $$f \sim \eta R v_i.$$ The bubble/plane friction is small with respect to the drag force $f$ as we will see below. Force equilibrium $B=f$ gives the terminal incoming velocity $v_i$ of the bubbles which is a few millimeters per second since $\sin \theta \approx 10^{-2}$ in our experiments. We are thus in a laminar regime ($Re\approx 1$).
Experimental results
====================
Bubbles tend to aggregate in an ordered hexagonal packing since the bubble size is monodisperse. The packing is continuously fed by new bubbles. As a consequence, the packing evolves and a dense flow of bubbles is seen in both tails. As for usual granular flow, a thin layer of moving grains is observed [@nasuno]. Moreover, smaller motions are also observed deeper inside the packing, even for low $\theta$ values. Figure \[setup\] presents a typical packing of bubbles for $\theta=1^\circ$. The shape of the pile is clearly rounded. If the incoming bubble flow is interrupted, the bubbles slide on each other under the influence of buoyancy and the angle of the bubble stack tends to zero. The pile collapses completely. This indicates that the friction between contacting bubbles is quite low, as discussed in the introduction. As contacting bubbles are sliding on each other, small shape deformations are also observed at the bottom of the pile. Thus, the bubble pile is not a strict granular medium in the sense that no arch is created.
![Part of the packing. Bubbles tend to aggregate in a hexagonal packing. A dislocation is seen in the center of the picture.[]{data-label="pack"}](packing_zoom.eps){width="6cm"}
In the packing, the bubble motion is mainly due to moving dislocations \[see Fig. \[pack\]\]. Those punctual defects are created at the surface of the packing and propagate along bubble lines at various speeds. One should note that the mean dislocation speed is larger than the bubble speed. This is due to the cooperative motion of the bubbles. For larger values of $\theta$, several dislocations are seen in the different parts of the pile. The majority of the bubbles is in motion.
In order to quantify and to measure the bubble flow, we have tracked the motion of every bubble in the tail of the bubble pile. Movies of the evolution of different parts of the bubble stack have been recorded at a frame rate of $12$ fps. The recorded areas contains typically $30\times 30$ bubbles. A precision of $0.03\,$mm on bubble positions is obtained. We have then measured the bubble paths and deduced velocity profiles using a particle tracking algorithm. Figure \[champ\] presents two typical instantaneous ($\Delta
t=0.083$ s) velocity fields in the tail of the bubble pile, for respectively $\theta=0.5^\circ$ and $\theta=1.5^\circ$. One can observe slow bubble motions for a small $\theta$ value (left) and moving layers of bubbles over a quasi-static phase for a larger $\theta$ value (right). Velocity measurements presented hereafter result from averages over long movies (more than $30\,$s).
![Instantaneous velocity fields of a part of the tail of the bubble pile for (left) $\theta=0.5^\circ < \theta_c$ and (right) $\theta=1.5^\circ > \theta_c$. The bubbles are observed to move from right to left. A block motion is observed for $\theta< \theta_c$, while a velocity profile decreasing with the depth $h$ is found for $\theta> \theta_c$. Sketches of the velocity profiles are also illustrated.[]{data-label="champ"}](field_block.eps "fig:"){width="6.5cm"}![Instantaneous velocity fields of a part of the tail of the bubble pile for (left) $\theta=0.5^\circ < \theta_c$ and (right) $\theta=1.5^\circ > \theta_c$. The bubbles are observed to move from right to left. A block motion is observed for $\theta< \theta_c$, while a velocity profile decreasing with the depth $h$ is found for $\theta> \theta_c$. Sketches of the velocity profiles are also illustrated.[]{data-label="champ"}](field_shear.eps "fig:"){width="6.5cm"}
Figure \[activ\] presents the [*mean velocity*]{} $\langle v\rangle$ in the bubble pile as a function of $\sin(\theta)$. We propose a quadratic behavior of the mean velocity as emphasized by the fit in Figure \[activ\]. The mean velocity $\langle v\rangle$ would then be controled by the tilt angle $\theta$ and would be proportional to the kinetic energy ($\sim v^2_i$) of individual bubbles before they reach the packing.
![Mean velocity $\langle v\rangle$ as a function of $\sin(\theta)$. Each dot represents an average over $1500$ pictures and over $1.3\ 10^6$ bubbles. Error bars are indicated. The curve is a fit emphasizing a $\sin^2(\theta)$ behavior.[]{data-label="activ"}](v_h.eps){width="6cm" height="5cm"}
![Velocity fluctuations $\sigma$ defined by Eq.(\[eqT\]) as a function of $\sin(\theta)$. A broad transition is seen at $\sin(\theta_{c})\approx 0.025$ or $\theta_c \approx 1.4^\circ$. Each dot represents an average over $1500$ pictures and over $1.3\ 10^6$ bubbles. Error bars are indicated.[]{data-label="T"}](tg.eps){height="5cm" width="6cm"}
In order to characterize the bubble flow, we have also performed measurements of the velocity fluctuations, similarly to the granular ‘temperature’ [@azanza]. The velocity fluctuations amplitude $\sigma$ is then defined by $$\label{eqT}
\sigma = \sqrt{ \langle v^2 \rangle- \langle v \rangle^2 } \, ,$$ i.e. the second moment of the velocity distribution in both $x$ and $y$ directions [@azanza]. Those velocity fluctuations $\sigma$ are plotted as a function of $\sin(\theta)$ in Figure \[T\]. One can observe a ‘jump’ of the velocity fluctuations $\sigma$ for tilt angles above $\sin(\theta_c)\approx
0.025$. This suggests the presence of a broad second-order phase transition as a function of the tilt angle $\theta$. The critical angle corresponds to $\theta_c\approx 1.4^\circ$. Around this critical angle, two dense flow regimes should be distinguished. They will be characterized below.
For low angle values ($\theta < \theta_c$), the dislocations are created at the surface of the pile. They propagate in diagonal directions and are reflected at the bottom of the bubble packing. As a consequence, we have observed the motion of giant triangular domains of bubbles. Above the critical angle $\theta_c$, we observe numerous dislocations. The density of dislocations is so high that they are interacting by pairs. Indeed, we have observed the collision, the merging, and the annihilation of dislocation pairs. The cooperative motion of the dislocations lead to the motion of small clusters of bubbles.
The above behaviors imply distinct velocity profiles. Velocity profiles have been measured as a function of the vertical position $h$ reduced by the bubble diameter $2R$ and counted from the obstacle. Three typical velocity profiles are given in Figure \[prof\]. Velocity profiles $v_{h}$ reveal two components: (i) a mainly constant profile which dominates for low $\theta$ values and (ii) a curved profile (a gradient) seen for high $\theta$ values. The curvature above $\theta_{c}$ is similar to the velocity profile of granular systems \[see Fig \[profiles\]\] : a velocity profile decreasing abruptly in the more static phase (e.g. deeper in the pile) and growing in the flowing layer [@bonamy_gm]. The velocity profile penetrates as deeply in the pile as the tilt angle $\theta$ value is large, as in granular systems [@bonamy_gm].
![From left to right: velocity profiles $v_{h}$ for respectively $\sin(\theta)=0.012$, $\sin(\theta)=0.023$, and $\sin(\theta)=0.038$. Below $\theta_{c}$, we observe nearly constant vertical profile as in block motions, while above, we obtain a granular-like profile.[]{data-label="prof"}](pro.eps){width="7cm"}
The ‘vertical’ profiles at low $\theta$ values can be explained in terms of block motions of the giant triangular domains of bubbles. Domains are sliced by fast moving dislocations. As the kinetic energy of the incoming bubbles increases, the size of the bubble domains decreases. The density of dislocations increases and the bubble stack becomes more ‘fluid’. At this stage, interacting dislocations which are created at the surface of the flow are observed to annihilate deep in the dense flow. The study of the interaction/annihilation between dislocations could explain the particular curvature of our velocity profiles. This needs more theoretical work and is outside the scope of this letter.
The ‘jump’ in the velocity fluctuations $\sigma$ suggests that we have a transition between block and granular flows depending on the density of dislocations in the packing. Varying the bubble size $R$ or the bubble production rate does not affect qualitatively the results reported herein. Moreover, the bubble motions and dislocation mechanisms reported hereabove are also observed when changing the soap-water mixture.
In the future, the analogy between granular and bubble flows could be developed by changing the geometry of the bubble flow and considering “chute flow”. By tilting the rough obstacle relatively to the incomming bubble flow, one would get a “chute” flow of bubble on an inclined substrate. Tuning the bubble rate production and the tilt angle, could control the thickness of the flowing layer.
Summary
=======
We have studied bubbles moving and aggregating on inclined planes. We have underlined the similarities between dense bubble flows and granular flows. Dense bubble flows can be considered as ‘ideal’ granular flows with nearly zero friction. We have emphasized a broad second-order transition between two dense flow regimes as a function of the tilt angle $\theta$ of the plane below which the bubbles are injected.
This work is financially supported by the contract ARC number 02/07-293. HC thanks FRIA (Brussels, Belgium) for financial support. NV thanks A.Pekalski, O.Pouliquen, and S. Dorbolo for fruitful discussions. The authors are also grateful to G. Broze for surfactant informations (Colgate-Palmolive Research facility).
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---
abstract: 'Post-starburst galaxies can be identified via the presence of prominent Hydrogen Balmer absorption lines in their spectra. We present a comprehensive study of the origin of strong Balmer lines in a volume-limited sample of 189 galaxies with $0.01<z<0.05$, $\log(\mbox{M}_{\star}/\mbox{M}_{\odot})>9.5$ and projected axis ratio $b/a>0.32$. We explore their structural properties, environments, emission lines and star formation histories, and compare them to control samples of star-forming and quiescent galaxies, and simulated galaxy mergers. Excluding contaminants, in which the strong Balmer lines are most likely caused by dust-star geometry, we find evidence for three different pathways through the post-starburst phase, with most events occurring in intermediate-density environments: (1) a significant disruptive event, such as a gas-rich major merger, causing a starburst and growth of a spheroidal component, followed by quenching of the star formation (70% of post-starburst galaxies at $9.5<\log(\mbox{M}_{\star}/\mbox{M}_{\odot})<10.5$ and 60% at $\log(\mbox{M}_{\star}/\mbox{M}_{\odot})>10.5$); (2) at $9.5<\log(\mbox{M}_{\star}/\mbox{M}_{\odot})<10.5$, stochastic star formation in blue-sequence galaxies, causing a weak burst and subsequent return to the blue sequence (30%); (3) at $\log(\mbox{M}_{\star}/\mbox{M}_{\odot})>10.5$, cyclic evolution of quiescent galaxies which gradually move towards the high-mass end of the red sequence through weak starbursts, possibly as a result of a merger with a smaller gas-rich companion (40%). Our analysis suggests that AGN are ‘on’ for $50\%$ of the duration of the post-starburst phase, meaning that traditional samples of post-starburst galaxies with strict emission line cuts will be at least $50\%$ incomplete due to the exclusion of narrow-line AGN.'
author:
- |
M. M. Pawlik$^{1}$, L. Taj Aldeen$^{1,2}$, V. Wild$^{1}$, J. Mendez-Abreu$^{1,3,4}$, N. Lahén$^{5}$,P. H. Johansson$^{5}$, N. Jimenez$^{1,6}$, W. Lucas$^{1}$, Y. Zheng$^{1}$, C. J. Walcher$^{7}$ and K. Rowlands$^{1,8}$\
$^{1}$School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, U.K. (SUPA)\
$^{2}$Department of Physics, College of Science, University of Babylon, Hillah , Babylon, Iraq\
$^{3}$Instituto de Astrofísica de Canarias C/ Vía Láctea, s/n E38205 - La Laguna (Tenerife), Spain\
$^{4}$Departamento de Astrofísica, Universidad de La Laguna, Calle Astrofísico Francisco Sánchez s/n, E-38205 La Laguna, Tenerife, Spain\
$^{5}$Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, FI-00014 Helsinki, Finland\
$^{6}$Unbound, Unit 18, Waterside, 44-48 Wharf Road, London N1 7UX, UK\
$^{7}$Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany\
$^{8}$Department of Physics $\&$ Astronomy, Johns Hopkins University, Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218, USA
bibliography:
- 'ref.bib'
nocite: '[@Pop+2017subm]'
title: 'The origins of post-starburst galaxies at $z<0.05$.'
---
\[firstpage\]
galaxies:evolution, galaxies:stellar content, galaxies:structure, galaxies:starburst, galaxies:interactions
Introduction
============
About one percent of the local galaxies within the Sloan Digital Sky Survey (SDSS) have optical spectra featuring unusually strong Balmer lines in absorption accompanied by weak emission lines [@Goto+2008; @Wong+2012]. Such features have been interpreted as a signature of a rapid decrease in the star-formation activity, likely following a recent starburst [@DresslerGunn1983; @Nolan+2007; @Balogh+2005; @Wild+2007; @vonderLinden+2010]. In this picture, shortly after the starburst, the balance of the newly born stars in a galaxy varies on short timescales as the most massive stars come to the end of their lives and move off the main sequence. This variation is imprinted on the galaxy spectrum in the form of increasing strength of the Balmer absorption lines, which peak for stars of spectral type A. As these stars have main-sequence lifetimes between 0.1 and 1 Gyr, strong Balmer lines in absorption should indicate a starburst not older than $\sim$ 1 Gyr. A coincidental lack of measurable emission lines indicates no ongoing star-formation on a detectable level. This means that these galaxies, often referred to as post-starburst galaxies, have abruptly quenched their star formation in their recent past and could be caught in transition between the star-forming blue cloud and the quiescent red sequence. As such, they offer a unique insight into galaxy evolution and may provide a means of constraining the origin of the bimodality in the population of massive galaxies: blue star-forming gas-rich systems with prominent disks and ‘red and dead’ gas-poor spheroids (e.g. @Strateva+2001 [@Kauffmann+2003b; @Bell+2004; @Baldry+2004; @Baldry+2006; @Bundy+2005]).
The evolutionary scenario in which galaxies migrate from the blue cloud over to the red sequence is supported by observations which reveal that the stellar mass and number density of galaxies on the red sequence has doubled since $z\sim1$, during which time the mass density of the blue cloud has remained nearly constant (see e.g. @Bell+2004 [@Arnouts+2007; @Faber+2007]). The physical processes governing this transition have not yet been determined and it is unlikely that all star-forming galaxies follow the same pathway to the red sequence. The evolutionary path of a galaxy may be determined by a number of factors, such as its mass and structural properties or its environment. As argued by @Peng+2010, more massive galaxies are more likely to become quiescent regardless of what environment they reside in (‘internal’ or ‘mass quenching’) and galaxies in denser environments are more likely to quench their star formation independent of their stellar mass (‘external’ or ‘environmental quenching’). Numerous observations reveal that the build up of the low-mass end of the red-sequence occurs at later times in the history of the Universe than that of the high-mass end (e.g. @Marchesini+2009 [@Moustakas+2013; @Muzzin+2013]), which may simply be related to the fact that the star formation rate of low-mass galaxies declines more slowly than high-mass galaxies (e.g @Asari+2007), or may indicate that quenching events are occurring at later cosmic times for low mass galaxies.
A popular candidate for the internal quenching mechanism in massive galaxies is feedback from the accreting supermassive black hole (active galactic nucleus, AGN) fueled by, e.g. disk instabilities [@Bournaud+2011] or a central bar [@Knapen+2000], that can halt the star formation activity by modifying the interstellar gas conditions or expelling it in powerful galactic winds. External quenching may be driven by a variety of processes depending on the galaxy’s immediate environment. In dense galaxy clusters these include: ram-pressure stripping of the cold interstellar medium [@GunnGott1972], the removal of the hot gas halo or ‘strangulation’ [@Larson+1980; @BaloghMorris2000] or fast encounters with other galaxies also known as ‘harassment’ [@GallagherOstriker1972; @Moore+1998]. In less dense environments it is more likely caused by galaxy mergers which can destroy disks in star-forming galaxies leading to morphologically and kinematically disturbed remnants that over time relax to a state characteristic of the red-sequence population (e.g @Barnes1992 [@NaabBurkert2003; @Bournaud+2005]). Mergers of gas-rich galaxies can lead to powerful centralised starbursts followed by quenching of the star-formation, possibly also related to the AGN feedback on the interstellar medium (e.g. @Sanders+1988 [@Hopkins+2006; @Johansson+2009a]). Alternatively quenching may occur without the presence of AGN feedback as once a galaxy acquires a spheroid-dominated morphology it can shut off its star-formation and turn red due to disk stabilisation against gas clouds fragmentation (‘morphological quenching’, @Martig+2009). Discriminating between the different mechanisms driving the evolution of galaxies from blue to red is not trivial, especially since their relative importance is unlikely to have been constant over cosmic time. It is clear that a single class of galaxies will not hold the answers to all questions regarding the complex picture of galaxy evolution but building up our knowledge about the galaxies caught in transition between the main evolutionary stages, such as post-starburst galaxies, is a step in the right direction. One of the main challenges in studying post-starburst galaxies, perhaps apart from their scarcity, is the large diversity of selection criteria used in the literature. In what follows, we review the different selection methods, the corresponding post-starburst sample properties, and conclusions about their origin and importance for galaxy evolution.
Quiescent post-starburst galaxies
---------------------------------
The first observation of post-starburst galaxies or ‘K+A’ galaxies (here, *quiescent* post-starburst galaxies) goes back to the early 1980s [@DresslerGunn1983], when they were found in distant galaxy clusters ($0.3<z<0.6$). Further observations showed that at these intermediate redshifts ‘K+A’ galaxies reside preferentially in such high density environments (see e.g. @Poggianti+2009 and the references therein). Morphological analysis of their optical images revealed that they are predominantly disk-dominated systems, some of which are interacting or obviously disturbed (e.g. @Couch+1994 [@Couch+1998; @Dressler+1994; @Oemler+1997; @CaldwellRose1997; @Dressler+1999]). Evidence of disk-like structures was also found in the kinematics of some cluster post-starburst galaxies, e.g. by @Franx1993 or, more recently, in integral-field spectroscopic observations where kinematical configurations characteristic of fast rotators [@Emsellem+2007] were found in over $80\%$ of the studied cluster post-starburst galaxies (@Pracy+2009 [@Swinbank+2012; @Pracy+2013]). The proposed mechanisms for the origin of ‘K+A’ galaxies in dense environments include perturbations due to the cluster tidal field [@ByrdValtonen1990], repeated encounters with other galaxies - ‘harassment’ - [@Moore+1996; @Moore+1998] which could induce disturbance in galaxy morphology, or interactions with the intra-cluster medium of newly infalling galaxies to the cluster [@GunnGott1972], leaving the stellar morphology undisturbed and possibly explaining the high incidence of disks in the galaxy samples. It has also been suggested (see e.g. @Poggianti+1999 [@Tran+2003]) that intermediate-redshift cluster post-starburst galaxies could be the progenitors of S0 galaxies that dominate the cores of present-day clusters, therefore playing a significant role in the evolution of the star-forming galaxy fraction in clusters over cosmic time (Butcher-Oemler effect, @ButcherOemler1984).
Other studies revealed that quiescent post-starburst galaxies are not exclusively related to clusters but can also be found in lower-density environments. In the local Universe, they are generally found in the field and loose groups, where dynamical conditions are more favourable for galaxy interactions and mergers (e.g @Zabludoff+1996 [@Blake+2004; @Hogg+2006; @Yang+2008; @Yan+2009; @Goto2007]). A connection to mergers is also revealed in the morphology and structural properties of many quiescent post-starburst galaxies, although the outcomes of different studies are quantitatively diverse. Morphological disturbance signifying an ongoing or past merger has been found in between $15\%$ and $70\%$ of cases, depending on selection criteria and image quality (e.g. @Zabludoff+1996 [@Goto2005; @Yang+2008; @Pracy+2009; @Trouille+2013]). Many studies have reported a high incidence of bulge-dominated early-type morphologies and steep light profiles with high central concentration in post-starburst samples (e.g. @Goto2005 [@Quintero+2004; @Tran+2004; @Blake+2004; @Poggianti+2009; @Mendel+2013]), characteristics typical of red-sequence galaxies and also consistent with merger remnants seen in numerical simulations (e.g @ToomreToomre1972 [@Barnes1988; @NaabBurkert2003]). Evidence for a merger origin has also been found in spatially-resolved studies of some post-starburst galaxies, which revealed centrally concentrated A/F stellar populations [@Pracy+2012; @Swinbank+2011; @Whitaker+2012], in agreement with expectations of gas inflows to the centre of the merger seen in simulations.
Despite the diversity in the findings regarding the properties of the ‘K+A’ galaxies, many studies agree that they are likely transitioning between the blue cloud and the red sequence, both in clusters and in the field. Many lie in the ‘green valley’ of colour-magnitude diagrams (e.g. @Wong+2012). @Tran+2004 estimated that $\sim25\%$ of passive galaxies in the local field underwent a ‘K+A’ phase at $z<1$ (increasing to $70\%$ if only early morphological types are considered) and @Whitaker+2012 argued that their number density evolution of the ‘K+A’ and red-sequence galaxies is consistent with all quiescent galaxies experiencing a ‘K+A’ phase at $z>1$. However, other studies present a contrasting view of the role of post-starburst galaxies in the red-sequence growth. Reservoirs of both neutral and molecular gas have been found in over half of the investigated post-starburst galaxies [@Chang+2001; @Buyle+2006; @Zwaan+2013; @French+2015], meaning that these galaxies are not yet devoid of fuel for star formation (although such conclusions are still limited to small samples). Furthermore, the low incidence of ‘K+A’ galaxies in two clusters at $z\sim0.5$ found by @DeLucia+2009 seems insufficient to represent a dominant channel for the formation of red sequence galaxies. A similar conclusion was arrived at by @Dressler+2013, who proposed that the majority of the ‘K+A’ galaxies in both clusters and the field at $0.3<z<0.54$ represent a phase in an evolutionary cycle within the red sequence, where an already quiescent galaxy accretes a smaller gas-rich companion and passes through a brief post-starburst phase before returning to the red sequence. This was also supported by a morphological analysis of the sample by @Abramson+2013. At somewhat higher redshifts ($0.47<z<1.2$) @Vergani+2010 concluded that a variety of processes could lead to the post-starburst phase and that this channel provides a non-negligible contribution to the red sequence growth, although not higher than $\sim10\%$.
Transitioning post-starburst galaxies
-------------------------------------
All the studies mentioned above relate to post-starburst galaxies in which the star-formation has effectively been quenched, selected based on the lack of measurable nebular emission, usually the \[OII\] line (e.g., @DresslerGunn1983 [@Zabludoff+1996; @Poggianti+1999]) or $H\alpha$ line (e.g., @Goto+2003 [@Quintero+2004; @Balogh+2005]). However, a starburst is not an instantaneous event and, in fact, gas-rich merger simulations (which do not include significant AGN feedback) point to ongoing star-formation for several hundreds of Myr following the initial starburst (see e.g. @Hopkins+2006 [@Wild+2009]). It is therefore reasonable to expect some levels of star-formation to be visible for a while after the onset of the starburst. In light of this, one problem with the traditional definition of post-starburst galaxies is that *the strict cut on emission lines excludes galaxies in the early transition stage between starburst and quiescence*. This early phase is important because the characteristics of the transitioning galaxies may contain information about the event that triggered the transformation and processes occurring during the transition, and these characteristics may fade by the time the galaxy enters the ‘K+A’ phase. For example, @Tremonti+2006 measured high velocity outflows in very young (75-300Myr) post-starburst galaxies which appear to be caused by extreme starbursts rather than AGN as originally postulated @SellTremonti2014. Moreover, *the strict cut on nebular emission lines in the traditional approach does not allow for selection of galaxies with ionisation mechanisms other than star-formation, e.g. AGN or shocks, leading to incomplete samples of post-starburst galaxies* (see also @Wilkinson+2017). More recent studies turned their focus to an alternative broader definition of post-starburst galaxies, in which the condition of quiescence is relaxed. These studies, (examples discussed below), revealed the existence of galaxies whose optical spectra feature both strong Balmer absorption lines as well as nebular emission lines on a measurable level.
However, it is important to note that the nature of galaxies with both strong Balmer lines in absorption and nebular emission lines is still under debate. One of the popular interpretations is that emission-line spectra with strong Balmer absorption, known as $e(a)$ spectra, indicate an *ongoing* starburst with the youngest stellar populations obscured by dust [@Dressler+1999; @Poggianti+1999]. This is supported by some observations which reveal that such spectral characteristics are more common among dusty starbursts and luminous infrared galaxies compared to normal star-forming galaxies [@LiuKennicutt1995; @Smail+1999; @PoggiantiWu2000]. Some studies have suggested that these ongoing dusty starbursts may be the progenitors of some quiescent post-starburst galaxies [@Poggianti+1999; @Balogh+2005].
A second interpretation is that galaxies with strong Balmer lines in absorption and measurable emission lines are true post-starburst galaxies in which the star formation has not been fully quenched. Such decaying starbursts are on the way to becoming traditional ‘K+A’ galaxies. This evolutionary scenario was explored by @Wild+2010 who used a Principal Component Analysis of galaxy spectra (PCA, @Wild+2007), and quantified the shape of the continuum around 4000[Å]{} and the relative strength of Balmer absorption lines to identify post-starburst galaxies *without placing a cut on their emission-lines*. They selected a sample of 400 local galaxies whose spectral characteristics place them on an evolutionary sequence stretching over 600 Myr following the starburst and, from the decay in the $H\alpha$ emission, they found a characteristic star-formation decline timescale of $\sim300$ Myr in broad agreement with merger simulations. A morphological analysis of the images of these post-starburst galaxies revealed the presence of asymmetric faint tidal features in the outskirts of about half of the youngest subset ($t_{SB}<$100 Myr) and a clear decline in the incidence and asymmetry of such features with the starburst age over the following 500 Myr [@Pawlik+2016]. The same study found that the post-starburst galaxies have generally intermediate structural properties between those characteristic of normal star-forming and quiescent galaxies, with no significant structural evolution detected during the first 600 Myr following the starburst. @Rowlands+2015 studied 11 galaxies at $z\sim0.03$ spanning the age sequence of $\sim1$ Gyr from the onset of the starburst, finding a decrease in the molecular gas surface density and effective dust temperature with increasing starburst age. However, the gas and dust fractions were found to be higher than in red-sequence galaxies even 1 Gyr following the starburst. The monotonic trends in the star-formation rate, gas and dust conditions and visual morphology of the post-starburst galaxies with estimated starburst age speak in favour of an evolutionary link between the ongoing starbursts, transitioning post-starburst galaxies with measurable nebular emission, and quiescent ‘K+A’ galaxies.
A class of transitioning post-starburst galaxies was also studied by @Yesuf+2014 who combined the traditional criterion of strong $H\delta$ absorption with a more relaxed criterion on the $H\alpha$ emission line, as well as galaxy colours and flux density ratios in the NUV-optical-IR regime to bridge the gap between starburst and quiescence at $z<0.1$. They found that at $10.3 < \mbox{log}(\mbox{M}_{\star}/\mbox{M}_{\sun}) < 10.7$ the candidate transitioning post-starburst galaxies (with detectable emission lines) are five times as numerous as quiescent post-starburst galaxies and that their structure and kinematics are intermediate between those of blue cloud and red sequence galaxies.
Transitioning post-starburst galaxies have also been found at higher redshifts where, similarly to ‘K+A’ galaxies, they are more numerous compared with the local Universe. Using the above described PCA-based selection method, @Wild+2009 reported an increase in the mass density of the post-starburst galaxies (both transitioning and ‘K+A’) more massive than $\mbox{log}(\mbox{M}_{\star}/\mbox{M}_{\sun})=9.75$ by a factor of 200 between $z\sim0.07$ and $z\sim0.7$. They found that post-starbust galaxies selected with no emission-line cut are found across all environments with no significant difference in the distribution of local densities compared with control samples at $0.5<z<1.0$. More recently, @Wild+2016 used a Principle Component Analysis of the broad-band optical-NIR SED (supercolours, @Wild+2014, see also @Maltby+2016) to study the evolution of post-starburst galaxies from even earlier epochs and found that at $\mbox{log}(\mbox{M}_{\star}/\mbox{M}_{\sun})>10$ their fraction rises from $<1\%$ to $\sim5\%$ of the total galaxy population between $z\sim0.5$ and $z\sim2$. Based on the comparison of the mass functions of the post-starburst and red-sequence galaxies, they argue that rapid quenching of star formation can account for all of the quiescent galaxy population, in the case where the timescale for visibility of the post-starburst spectral features in broad band photometry is not longer than $\sim250$ Myr. A similar analysis by @Rowlands+2017 using spectroscopic surveys at lower redshifts found that the importance of post-starburst galaxies (defined using spectral PCA, as in @Wild+2007) in the build up of the quiescent galaxy population declines rapidly with decreasing redshift and may be insignificant by $z=0$.
AGN and shocks in post-starburst galaxies
-----------------------------------------
Observationally, post-starburst galaxies with emission lines have also been linked with the presence of an AGN and shocks. This was not seen in the early works on post-starburst galaxies which employed emission-line cuts in their selection, as their samples were biased against objects with any kind of ionising sources, including AGN (particularly if the \[OII\]-line was used as the star formation indicator - see @Yan+2006). However, the connection was observed in numerous studies of AGN hosts. For example @Kauffmann+2003a argued that strong $H\delta$ lines in absorption are more common in luminous narrow-line AGN than in star-forming galaxies at $0.02<z<0.3$, and @CidFernandes+2004a found high-order Balmer absorption lines in the nuclear SED of nearly a third of their local low-luminosity AGN sample. Using a conservative $H\alpha$ emission-line cut to select quiescent post-starburst galaxies at $z<0.1$, @Yan+2006 showed that most of them have AGN-like emission-line ratios. @Goto2006 selected a sample of over 800 $H\delta$-strong AGN hosts and used resolved spectroscopy for three such objects at $z<0.1$ to reveal a spatial connection between the post-starburst region and the AGN. @SellTremonti2014 find evidence for AGN activity in 50% of their extreme post-starburst galaxies at $z\sim0.6$.
@Wild+2007 used their PCA selection method to show that, at low redshift, AGN reside in over a half of the studied post-starburst galaxies and that, on average, they are the most luminous AGN within their samples. AGN hosts with post-starburst characteristics may be essential to understanding the causal connection between starbursts and AGN activity, and consequently that between star-formation and black hole growth in galactic centres, as well as give a unique insight into the process of star formation quenching. To that end @Wild+2010 measured a delay between the starburst and AGN activity of about 250 Myr at $0.01<z<0.07$. A similar time delay was found following a different selection technique by @Yesuf+2014 who concluded that AGN are not the primary source of quenching of starbursts, but may be responsible for quenching during the post-starburst phase (see also @Goto2006 and @Davies+2007 for time delays between peaks of starburst and AGN activity found in small AGN samples).
Finally, it is important to note that, aside from star formation and AGN activity, the emission-line features in galaxy spectra may indicate other underlying processes, such as shocks. These are expected to be seen in transitioning galaxies, where the transition is attributed to violent dynamical mechanisms, likely to induce turbulence in the interstellar medium. @Alatalo+2016 built a catalogue of ‘shocked’ post-starburst galaxies or SPOGs, with emission line ratios indicative of the presence of shocks. Such shocks could be related to a number of physical mechanisms, including AGN-driven outflows, mergers or proximity to a cluster (for details see @Alatalo+2016 and references therein).
Summary and goals of this work
------------------------------
Many papers interpret post-starburst galaxies as a transition phase between the star-forming blue cloud and the quiescent red sequence blue cloud- the two major stages of galaxy evolution. However, their true importance for red sequence growth remains a matter of debate. The aim of this paper is to investigate the star-formation histories, visual morphologies, structural properties and environments of galaxies with strong Balmer absorption lines, and a range of emission line properties in order to determine their origins. We also aim to ascertain whether the different classes of Balmer strong galaxies are evolutionarily connected or following separate paths entirely.
This paper is organised as follows: Section 2 describes the samples and their selection criteria; Section 3 - the methodology used to obtain star formation histories, morphology, structure and environment; Section 4 - the results; Section 5 - a discussion, including the analysis of galaxy merger simulations, and focusing on the likely origin of the different post-starburst families and evolutionary pathways through the post-starburst phase; Section 6 - the summary of conclusions. We adopt a cosmology with $\Omega_{m}$ = 0.30, $\Omega_{\Lambda}$ = 0.70 and H$_{0}$ =70kms$^{-1}$Mpc$^{-1}$ and magnitudes are on the AB system.
Data and sample selection
=========================
The spectroscopic catalogue of the SDSS (7th Data Release, SDSS DR7, @Abazajian+2009), containing the optical spectral energy distributions (SED) of $\sim90,000$ galaxies, is a natural choice for selection of objects as rare as low redshift post-starburst galaxies. In our study, we made use of both spectroscopic and imaging data provided by the survey. Additionally, we utilised the information regarding several spectral lines available in the SDSS-MPA/JHU value added catalogue[^1]. The measurements of the Petrosian magnitudes and redshifts of the galaxies were taken directly from the SDSS catalogue and the stellar masses of the galaxies were measured from the five-band SDSS photometry (J. Brinchmann, SDSS-MPA/JHU) using a Bayesian analysis similar to that described in @Kauffmann+2003a. Importantly, this method allows for bursty star formation, varying metallicity and 2-component dust attenuation.
The selection of the post-starburst galaxies as well as control galaxies with ordinary star formation histories was done based on their spectral characteristics. We note the relevance of the widely known aperture bias issue. At the low redshifts considered in this work the fixed 3 aperture diameter of the SDSS fibers probes only the central $\sim 0.6-3$ kpc of massive galaxies. This means that the resulting spectra and all derived quantities may be limited to the central regions of our galaxies and further investigation using spatially resolved spectroscopy is required to investigate the spectral characteristics of these galaxies on a global scale. However, the investigation of such “central" post-starburst galaxies is still of significant interest, not least because in merger simulations the funnelling of gas to the central regions of the merger remnants leads to central starbursts which may be exactly the objects we are identifying in the observations. In what follows, we first introduce the basics of the technique adopted for the classification of the galaxy SED and then describe the criteria used to select the galaxy samples.
Spectral analysis {#sec:spectralanalysis}
-----------------
To distinguish post-starburst galaxies from those with other star-formation histories we used the Principal Component Analysis introduced by @Wild+2007[^2] - a multivariate analysis technique that combines features that vary together in a data set, in this case the optical spectra of galaxies. Regarding a single spectrum - traditionally a 1D array of $n$ flux values - as a single point in an $n$-dimensional space, we can visualise a collection of spectra as a cloud of points in $n$-dimensions. The principal components are the orthogonal vectors corresponding to the lines of greatest variance in the cloud of points, and they constitute the new basis onto which the galaxy spectra are projected upon. The components were calculated using a set of mock spectra created using the @BruzualCharlot2003 [BC03] spectral synthesis code, and therefore contain only stellar light. As this work is focused on post-starburst galaxies, the spectral analysis was limited to the Balmer line region of the galaxy spectra, specifically 3175-4150[Å]{}. Within that region the first two principal components contain information about: PC1 - the 4000[Å]{}-break strength, anti-correlated with Balmer absorption-line strength, which gradually increases with increasing mean stellar age; PC2 - the *excess* Balmer absorption above that expected based on the measured 4000[Å]{}-break strength, which identifies unusual ‘bursty’ star formation histories.
The position in the PC1-PC2 parameter space depends on the stellar content and therefore the current and past star-formation activity of a galaxy. Galaxies with the highest specific star-formation rate are located on the left side of the distribution. Moving towards higher values of PC1 we find galaxies dominated by subsequently older stellar populations and lower specific star-formation rates. PC2 traces the recent star-formation history of the galaxies. Due to the short lifetimes of the most massive O/B stars, the stellar content of galaxies changes rapidly after a starburst, and after about 1 Gyr the galaxy enters an evolutionary sweet spot where A/F stars dominate its energy output. This evolution is imprinted on the galaxy SED as the A/F stars are characterised by the strongest Balmer lines among all stellar types. Therefore we can select robust samples of $\sim$ 1Gyr-old post-starburst galaxies from the ‘bump’ at the top of the distribution in PC1-PC2.
This selection method does not require any emission-line cut and therefore is suitable for the selection of both the traditional post-starburst galaxies in which the star-formation has been quenched, as well as those with detectable emission caused by either ongoing star-formation or other ionisation mechanisms, such as AGN or shocks.
-------- ---------------------------------------------------- ---------------------------------------------------- --------------- ----------- -------------------- ----------------
Sample Counts Counts Balmer-strong H$\alpha$ Dusty AGN
$\mbox{M}_{\star}/\mbox{M}_{\sun}< 3\times10^{10}$ $\mbox{M}_{\star}/\mbox{M}_{\sun}> 3\times10^{10}$ (PC1-PC2) emission (Balmer decrement) (@Kewley+2001)
qPSB 36 (24) 5 (5) Yes No N/A No
agnPSB 33 (26) 5 (5) Yes N/A N/A Yes
ePSB 57(43) 10 (6) Yes Yes No No
dPSB 31(23) 12 (10) Yes Yes Yes No
-------- ---------------------------------------------------- ---------------------------------------------------- --------------- ----------- -------------------- ----------------
 
Sample selection criteria {#sec:sampleselection}
-------------------------
We began our selection with a sample containing 83634 spectroscopically confirmed galaxies with Petrosian $r$-band magnitudes $14.5<\mbox{m}_{r}<17.7$ at $0.01<\mbox{z}<0.05$. In this redshift range selecting galaxies with stellar masses above $\mbox{M}_{\star}/\mbox{M}_{\odot}=10^{9.5}$ yields samples that are statistically complete in red-sequence galaxies, which are defined as galaxies for which the spectral indices PC1 and PC2 (see Section \[sec:spectralanalysis\]) satisfy the relation PC2 $\leq$ PC1 + 0.5. Applying this mass criterion reduced the number of galaxies in the sample to 49148. We then applied a cut on spectral per-pixel signal-to-noise ratio: SNR $>$ 8 in the $g$-band, to ensure accurate measurements of the spectral indices and emission line properties. This removed further $11\%$ of the sample, leaving us with 43811 objects.
Motivated by the bimodal nature of several galaxy properties in the local Universe separating the majority of local galaxies into two distinct families at $\mbox{M}_{\star}/\mbox{M}_{\odot}=3\times10^{10}$ (e.g. @Kauffmann+2003a), we split our sample into these two different mass regimes and refer to the resulting subsets as *low-mass* and *high-mass* galaxies. The low- and high-mass parent samples contain 33438 and 10373 objects, respectively. Before selection of the Balmer-strong galaxies and control samples we applied a further restriction by removing from the parent samples all galaxies observed ‘edge-on’, with projected axis ratio[^3] greater than 0.32. The purpose of this cut was to minimise potential biases in our measurements due to strong attenuation of the stellar light by dust in inclined galactic disks and it resulted in the reduction of the low- and high-mass samples by removing $17\%$ and $13\%$ of the galaxies, respectively. Our final samples from which the Balmer-strong and control samples were drawn contain 27901 and 9001 galaxies in the low- and high-mass regimes, respectively.
### Balmer-strong/post-starburst galaxies {#sec:psbselection}
We found that placing a cut at $\mbox{PC2}=0.0$ at 1$\sigma$ works well for selecting galaxies with prominent Balmer absorption lines, yielding 157 and 32 galaxies in the low- and high mass regimes, respectively. Based on the models of top-hat starbursts superimposed on an old stellar population, investigated by @Wild+2007, the galaxies selected from this extremum of the PC1-PC2 parameter space are expected to have starburst ages greater than $\sim0.6$ Gyr (see also the measured starburst ages in @Wild+2010). We then classified these Balmer-strong galaxies based on their emission-line measurements. We used a cut on the $H\alpha$ emission line equivalent width (EQW) to determine whether the galaxies have ongoing star formation at a measurable level (indicated by $\mbox{EQW}>3$[Å]{} with $\mbox{SNR}>3$). We further used the BPT diagnostics [@Baldwin+1981] to identify potential AGN-host candidates. For this purpose we used the condition introduced by @Kewley+2001, again, requiring that the emission lines have $\mbox{SNR}>3$. We chose this criterion over that introduced by @Kauffmann+2003a to ensure the selection of galaxies with AGN-dominated emission only, excluding those in which the contributions from the AGN and star formation are comparable.
A non-negligible number of the emission-line Balmer-strong galaxies were found to have high values of the Balmer decrement, i.e. the flux ratio of $H\alpha$ to $H\beta$ emission lines measured with respect to the intrinsic ratio of 2.87, indicating considerable dust content. We separate out the ‘dustiest’ galaxies in both mass regimes, in order to test whether they are a separate class of dust-obscured starburst galaxies (@Dressler+1999 [@Poggianti+1999]). A cut on the Balmer decrement was placed to identify the top 10% of dusty galaxies in the parent samples, corresponding to $H\alpha$/$H\beta>5.2$ and $H\alpha$/$H\beta>6.6$ in the low- and high-mass regime, respectively. Although arbitrary, this provides a good base for determining whether the dust-obscured Balmer-strong galaxies are truly different from the dust-unobscured ones. We find $35\%$ and $42\%$ of the Balmer-strong galaxies with emission lines have Balmer decrements above these cuts in the low- and high-mass samples, respectively. Considering the above criteria, we distinguish between four types of Balmer-strong galaxies:
- [**‘Quiescent’ Balmer-strong galaxies (qPSB)**]{} - with no measurable $H\alpha$ emission, equivalent to the traditional definition of post-starburst (or ‘K+A’) galaxies.
- [**Balmer-strong AGN host galaxies (agnPSB)**]{} - located above the @Kewley+2001 demarcation line in the BPT diagram.
- [**Emission-line Balmer-strong galaxies (ePSB)**]{} - galaxies with measurable $H\alpha$ emission line, not classified as dusty or AGN-host candidates. The normal dust content suggests that these are unlikely to be contaminating dust-obscured starbursts. This will be assessed in the paper.
- [**Dusty Balmer-strong galaxies (dPSB)**]{} - with measurable $H\alpha$ emission, classified as dusty but not as AGN host candidates; the subset of ePSB with the highest dust content as indicated by the Balmer decrement. These may be dust obscured starbursts, and are the most likely contaminants of post-starburst samples defined without an emission line cut.
In all cases, we use the label ‘PSB’ for conciseness.
### Control galaxies {#sec:controlselection}
Within both mass regimes, we selected control samples of quiescent and star-forming galaxies pair-matched with the post-starburst galaxies in stellar mass, within $\Delta\mbox{M}_{\star}/\mbox{M}\odot=10^{0.1}$. We randomly selected 5 star-forming and 5 quiescent control galaxies per Balmer-strong galaxy, from the highest-density regions of PC1-PC2 space coinciding with the blue cloud and the red sequence, respectively. As shown in the top left panel of Figure \[fig:pc12\] the star-forming galaxies were selected from regions defined by $-1.0<\mbox{PC2}<-0.5$ with the PC1 criterion depending on the mass regime: $-4.5<\mbox{PC1}<-3.4$ (low-mass), $-3.5<\mbox{PC1}<-2.4$ (high-mass), and the quiescent galaxies from regions defined by $\mbox{PC2}>0.8\times\mbox{PC1}-0.2$ and $\mbox{PC2}<0.8\times\mbox{PC1}-0.6$, and by $-1.0<\mbox{PC1}<-0.2$ (low-mass) and $-0.8<\mbox{PC1}<0.0$ (high-mass). As there are no clear boundaries between the different classes of galaxies in the PC1-PC2 space, we chose to select samples from the regions of highest number density within the blue cloud and the red sequence in order to create clean samples of control galaxies with “typical” properties, and avoid selecting objects with either extreme or intermediate properties.
Additionally, we build a dusty star-forming control sample using the same regions of PC1-PC2 space as the star-forming control samples, with an additional constraint on the Balmer decrement to match the limits used to select the dPSB galaxies.
Summary of sample properties
----------------------------
The galaxy counts of all the Balmer-strong samples along with their selection criteria are summarised in Table \[tab:galcounts\] and, in Figure \[fig:mass\_hist\], we show the stellar mass distributions of the Balmer-strong samples and compare them with the mass distribution of the control samples. In Figure \[fig:pc12\] we present the key spectral indices and line ratios used during the sample selection, measured for all galaxies in Balmer-strong galaxies as well as control samples. Finally, in Figures \[fig:SED\_stack1\] and \[fig:SED\_stack2\] we show the stacked spectra of the different samples, both across the full optical wavelength range and focusing on the individual regions: 1) 3750-4150Å over which the PCA indices are calculated; 2) 4700-5100Å containing the $H\beta$ and OIII lines; 3) 6500-6800Å where the $H\alpha$ and NII spectral lines are located.
     
 
\[fig:SED\_stack1\]
 
\[fig:SED\_stack2\]
Methodology
===========
Star formation histories
------------------------
The galaxy SED in the UV-to-IR regime is generally dominated by the light emitted by its stellar components, reprocessed by the surrounding reservoirs of the interstellar medium (ISM). Therefore, it contains information about the galaxy’s star-formation rate and its star formation history, as well as its dust content. The star formation history can be extracted from the SED through the process of spectral synthesis which essentially breaks down the galaxy SED into its base components. In practice, this is done by fitting the SED with a range of models - here we use an unparameterised approach, meaning that the star formation history is not constrained to be a particular form. We fit a linear combination of starbursts, called simple stellar populations (SSP), spanning a wide range of ages and metallicities.
### Spectral synthesis {#sec:meth_specsynth}
In this work we utilised the SED fitting code STARLIGHT [@CidFernandes+2005], to fit an updated version of the BC03 evolutionary synthesis models, with dust attenuation modelled as a single foreground dust screen. The fitting procedure in STARLIGHT is carried out using a sophisticated multi-stage $\chi^{2}$-minimisation algorithm combining annealing, Metropolis and Markov Chain Monte Carlo techniques. Prior to the fitting, the galaxy spectra were sky-residual subtracted longward of $6700$[Å]{}, in order to correct the effects of the incomplete subtraction of the OH emission lines by the SDSS spectroscopic reduction pipeline [@Wild+2006][^4]. All fluxes were corrected for Galactic extinction using the extinction values provided in the SDSS catalogue which are based upon the @Schlegel98 dust emission maps and the Milky Way extinction curve of @CCM98. The spectra were moved onto air wavelengths to match the models and nebular emission lines of the star-forming, ePSB and dPSB galaxies were masked using a bespoke mask defined from the stacked star-forming galaxy spectrum.
The processed spectra were fit with a linear combination of 300 SSPs spanning 60 stellar ages, that range from 1Myr to 14Gyr, and 5 metallicities: Z/Z$_\odot$ = 0.02, 0.2, 0.4, 1 and 2.5, where we have assumed Z$_\odot=0.02$. The ages were chosen to cover the whole of cosmic time, roughly linearly spaced in log age between $10^8$ and $10^{10}$ years, and with slightly sparser sampling for models younger than $10^8$ years. Although 60 ages bins are far more than can be constrained from a single optical spectrum, sufficient coverage is required across the main sequence lifetime of A/F stars (between 100Myr and 2Gyr) where the strength of the Balmer lines and distinctive shape of post-starburst galaxy spectra change rapidly with time, in principle allowing accurate age dating of the population. We used our star-forming galaxy control sample to test both the standard BC03 population synthesis models which use the Stelib stellar spectral library [@LeBorgne2003], and a new set of models which combine both the Stelib and MILES libraries [@miles2006] to give a total wavelength coverage of 3540.5-8750[Å]{}[^5]. Redwards of 8750[Å]{} both models are based on the theoretical BaSeL 3.1 spectral library [@Westera2002]. Additional changes to the model atmospheres used to model the UV are not relevant to this work. Both models are based on the *Padova 1994* set of stellar evolution tracks.
The traditional BC03 models caused an artificial peak in the star formation history (SFH) of star-forming control galaxies between 1.6-1.9Gyr, as well as a smaller peak at 1.3Gyr; these peaks in star formation histories are a known problem with the models and visible in previous analyses where SFHs from unparameterised fits are presented in a relatively unsmoothed format (e.g. mass-assembly in Fig. 8 of @Asari+2007, @Delgado2017). The spikes are particularly visible in our Balmer-strong samples, which have a significant fraction of their mass formed in this time interval. The updated models provide a more continuous stacked star formation history for the control star-forming galaxies around the critical $0.5-2\times10^9$year timescale which is relevant for quantifying the burst strength and age in post-starburst galaxies. However, a smaller artificial bump is still evident at $\sim$1.3Gyr. Unfortunately this feature in the stellar population models limits the accuracy with which we can age-date the starburst in older post-starburst galaxies. Further investigation as to how these problems can be mitigated when fitting post-starburst galaxy spectra is on-going, but for the purposes of this paper we limit our analysis to parameters that are robust to changes in the library i.e. the total fraction of stars formed in the last 1 and 1.5Gyr, and the star formation history prior to 2Gyr. Additionally, we use our control sample of star-forming galaxies to ensure that bursts are detected above any artificial signals in the star-forming sample.
During fitting, we adopted the attenuation law for starburst galaxies of @Calzetti+2001. Repeating the analysis with the Milky Way extinction law of @Cardelli+1989 produced quantitatively slightly different SFHs, but did not alter our conclusions when comparing SFHs between samples.
Morphology and structure {#sec:method_morph}
------------------------
To characterise the morphology and structure of our galaxies we applied a range of automated measures to their sky-subtracted images in the $g$, $r$ and $i$ bands of the SDSS. The analysis was performed on 1 arc minute cutouts of the SDSS images, centred on the galaxy of interest as defined by the coordinates provided in the SDSS database. Prior to the analysis all images were inspected visually to identify bright light sources in close proximity to the galaxies of interest. Bright nearby sources have the potential to severely contaminate the measures of morphology and structure and, therefore, all images in which such sources had been identified were excluded from the image analysis. This resulted in the reduction of the sample sizes as summarised in Table \[tab:galcounts\].
Our methods follow those presented in @Pawlik+2016; here we outline only the most important details. First, we created a binary pixel map which identifies pixels associated with the galaxy, as opposed to the surrounding sky. The algorithm loops around the SDSS position pixel, searching for connected pixels above a given threshold (1$\sigma$ above the median sky background level). Combined with a running average smoothing filter, the algorithm picks out contiguous features in an image down to a low surface brightness ($\sim$24.7 mag/arcsec$^{2}$).
The binary pixel maps were used to estimate the galaxy radius, $R_{max}$, as the distance between the centre (brightest galaxy pixel) and the most distant pixel from that centre within the map. Generally, this definition of galaxy radius agrees well with the commonly used Petrosian radius [@Petrosian1976; @Blanton+2001; @Yasuda+2001]; however, it provides an advantage in the case of galaxies with extended faint outskirts, like tidal tails.
We then followed standard procedures to measure the S[é]{}rsic index ($n$)[^6], the concentration index ($C$)[^7], the light-weighted asymmetry ($A$), the Gini index ($G$) and the moment of light ($M_{20}$). We additionally measured the new shape asymmetry ($A_{S}$), presented in @Pawlik+2016, which quantifies the disturbance in the faint galaxy outskirts. The shape asymmetry is computed using the same expression as the light-weighted asymmetry parameter, under a 180-degree rotation, but with the measurement performed using the binary pixel maps rather than the galaxy images. This approach allows for equal weighting of all galaxy parts during the measurement, regardless of their relative brightness. Finally, we computed $A_{S90}$ to quantify the shape asymmetry under a 90-degree rotation. This can be used in conjunction with $A_{S}$ to indicate whether the features in galaxy outskirts are elongated (e.g. tidal tails) or circular (e.g. shells). Further details of the methods used to measure each of these parameters are given in @Pawlik+2016 and references therein.
Environment {#sec:method_env}
-----------
We adopted the projected number density of galaxies in the vicinity of the target galaxy as a measure of the environment. The number density was calculated following the method described in @Aguerri+2009, using the projected comoving distance of the target galaxy, $d_{N}$, to its $N$th nearest neighbour: $$\Sigma_{N}=\frac{N}{\pi d_{N}^2}.$$
The nearest neighbours were defined in two ways. The first definition includes all galaxies with spectroscopic redshifts, $z_{s}$, within $\pm$1000km/s of the target galaxy, and with an absolute magnitude difference of not more than $\pm$2. These criteria are similar to those used by @Balogh+2004a and are designed to limit the contamination by background/foreground galaxies even when using projected distances. The second definition uses the SDSS photometric redshift measurements and selects galaxies within $\Delta z_{p}$=0.1 from the target galaxy (see @Baldry+2006 for a similar approach). This does not suffer the same incompleteness of the spectroscopic samples, but has higher contamination due to the less accurate photometric redshifts.
It is important to realise that the values of $\Sigma_{5}$ are approximate estimates of the local number density, with the spectroscopic and photometric measurements representing the lower and upper boundaries[^8], and the uncertainties associated with $\Sigma_{5}$ are expected to be large. To examine the accuracy of the density measurements we also considered the 3rd, 8th and 10th nearest neighbours and found a good agreement with $\Sigma_{5}$. Furthermore, we flagged all galaxies with $d_{N}$ greater than the distance to the edge of the survey, as for such locations the density measurements may be unreliable.
To provide one single estimate of local environment, we took the mean $\Sigma_{5}$ of the spectroscopic and photometric measurements. We stress that the purpose of the mean is merely to provide a single value which is likely to be closer to the true value than the individual measurements, rather than to serve as any statistical measure. We find that the qualitative results and conclusions remain generally unchanged whether we use the mean $\Sigma_{5}$ or the individual spectroscopic/photometric measurements, except for the ePSB sample. For this case we comment on the discrepancies when discussing the results in Section \[sec:results\_env\].
To relate the values of $\Sigma_{5}$ to the different types of environment, we follow the definitions used by previous studies (e.g. @Aguerri+2009, @Walcher+2014):
- $\Sigma_{5}<1\mbox{Mpc}^{-2}$ - very low-density environments,
- $1 \mbox{Mpc}^{-2}<\Sigma_{5}<10 \mbox{Mpc}^{-2}$ - loose groups,
- $\Sigma_{5}>10 \mbox{Mpc}^{-2}$ - compact groups and clusters.
Results
=======
In Table \[tab:results\] we present the recently formed mass fractions estimated by STARLIGHT ($f_{M1}$, $f_{M15}$ - within the last 1Gyr and 1.5 Gyr, respectively), the projected galaxy number density ($\Sigma_{5}$) and the $r$-band measurements of several structural and morphological parameters ($n$, $C$, $A$, $A_{S}$, $A_{S90}$, $G$, $M_{20}$) for all Balmer-strong galaxies studied in this work. Here we present only the top ten rows (galaxies ordered by their *specobjid* number); the full table is available online.
\[tab:results\]
ID SDSS specobjid PSB type $log(M)$ $PC1$ $PC2$ $log([NII]/H\alpha)$ $log([OIII]/H\beta)$ $H\alpha\,EQW$ $H\alpha/H\beta$
----- -------------------- ---------- ---------- -------- -------- ---------------------- ---------------------- ---------------- ------------------
1 75657056748568576 agnPSB 10.06 -4.348 -0.399 0.049 0.268 4.58 2.17
2 78754789168513024 qPSB 9.64 -2.641 -0.408 – – – –
3 78754790712016896 ePSB 10.13 -3.048 -0.903 0.081 – 3.97 –
4 78754790720405504 qPSB 9.59 -3.311 -0.759 – – 0.54 –
5 79034666610327552 qPSB 10.11 -1.892 -0.166 – – 0.66 –
6 82695481800523776 qPSB 10.27 -2.200 -0.068 0.046 – 1.49 –
7 83539864024252416 dPSB 10.67 -2.082 -0.825 -0.099 -0.326 23.78 4.07
8 86071546359054336 agnPSB 10.15 -3.908 -0.317 0.165 0.366 4.91 1.37
9 94798927357804544 ePSB 10.76 -3.490 -0.309 -0.385 -0.515 15.80 1.81
10 100146046004363264 ePSB 9.62 -4.596 -0.364 -0.543 0.061 59.87 1.72
... ... ... ... ... ... ... ... ... ...
ID $f_{M1}$ $f_{M15}$ $n$ $C$ $A$ $A_{S}$ $A_{S90}$ $G$ $M_{20}$ IMG FLAG $\Sigma_{5p}$ $\Sigma_{5s}$ $\Sigma_{5p}$-flag $\Sigma_{5s}$-flag
----- ---------- ----------- ------ ------ ------- --------- ----------- ------- ---------- ---------- --------------- --------------- -------------------- --------------------
1 6.39 8.64 2.58 3.12 0.076 0.121 0.228 0.720 -1.76 0 2.87 1.28 0 0
2 12.63 48.52 2.82 2.88 0.077 0.076 0.331 0.606 -1.80 0 31.91 0.36 0 0
3 1.84 19.35 1.97 3.08 0.044 0.132 0.189 0.638 -2.06 1 10.41 2.20 0 0
4 22.10 63.59 5.48 6.09 0.283 – 0.555 0.657 -3.24 1 25.25 25.25 0 0
5 3.81 44.30 3.26 3.22 0.073 0.146 0.433 0.741 -2.05 0 3.11 0.39 0 0
6 4.16 19.01 – 2.77 0.015 0.158 0.248 0.761 -1.67 0 5.33 0.61 0 0
7 0.02 0.04 2.96 3.21 0.034 0.157 0.175 0.605 -1.78 0 6.25 0.47 0 0
8 14.48 18.40 5.53 3.73 0.198 0.222 0.173 0.748 -0.37 0 0.66 0.20 0 0
9 4.09 7.54 3.49 3.67 0.059 0.170 0.442 0.652 -2.39 0 0.26 0.18 0 0
10 6.01 10.26 3.07 3.29 0.067 0.134 0.446 0.579 -2.07 0 17.77 2.81 0 0
... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
Star formation history {#sec:res_sfh}
----------------------
Using the output of the STARLIGHT code, we explored both recent and earlier (pre-burst) star-formation histories of the Balmer-strong galaxies and compared them with the control samples. The star-formation histories of the dPSB galaxies and associated dusty star-forming control sample output by the code showed almost $100\%$ of their mass assembled at very early cosmic times - inconsistent with their current star forming properties as evident from their nebular emission lines. We therefore believe that the spectral fitting is likely impacted by the high dust contents of these galaxies, and the fact that STARLIGHT can only fit a single-component dust screen, so we exclude these galaxies from this part of the analysis. Further details on the colours and spectral fits of this sample are given in Appendix \[app:dpsb\].
### Recent star formation
The histograms in Figures \[fig:massfrac\_lowM\] and \[fig:massfrac\_highM\] show the distributions of the fractions of recently formed mass for the low-mass and high-mass samples, respectively. The quantities $f_{M1}$ and $f_{M15}$ correspond to the fractions of mass formed in the last 1Gyr and 1.5Gyr, respectively. Each panel shows one Balmer-strong sample (qPSB, agnPSB, ePSB), compared to the control samples of quiescent and star-forming galaxies. Note the change in $x$-axis range between each row. In both samples, we can immediately see that the distributions of $f_{M1}$ and $f_{M15}$ for the Balmer-strong galaxies are skewed towards higher values compared with the control distributions. For the low-mass samples, we show K-S test results comparing the different distributions. The small numbers of objects in the high-mass samples make such tests less useful so are not shown. Table \[tab:massfrac\] presents the percentage of galaxies with very high and very low $f_{M1}$ and $f_{M15}$ in each PSB sample and the control star-forming sample.
In both mass bins, the values of $f_{M1}$ and $f_{M15}$ for the quiescent control samples are consistent with zero and the K-S results clearly show that none of the low-mass Balmer-strong samples are consistent with matching the quiescent control sample. For the low-mass star-forming control galaxies we find the majority $(\sim80\%)$ have $f_{M1}<0.05$ and $f_{M15}<0.12$, and a very small fraction (less than $1\%$) have $f_{M1}>0.10$ and $f_{M15}>0.20$. Looking at Table \[tab:massfrac\], compared to the star-forming galaxies, (1) a much lower fraction of low-mass Balmer-strong galaxies ($14\%-54\%$) formed less than 5$\%$ and $12\%$ of their stellar mass in the last 1Gyr and 1.5Gyr, respectively; 2) a considerably higher fraction of low-mass Balmer-strong galaxies ($21\%-48\%$) have more than $10\%$ and $20\%$ of there stellar mass formed in the last 1Gyr and 1.5Gyr, respectively. This effect is particularly pronounced in the qPSB and agnPSB samples, pointing to a stronger starburst compared with the ePSB galaxies. The K-S tests additionally show that the distributions of $f_{M1}$ and $f_{M15}$ for the agnPSB and qPSB samples are statistically identical, with all other distributions being different from one another.
In the high-mass samples the values of $f_{M1}$ and $f_{M15}$ are generally lower than in the low-mass samples, in agreement with previous studies which found that at low redshifts low-mass galaxies tend to have younger stellar ages and higher specific star-formation rates than the high-mass galaxies (e.g. @Kauffmann+2003b [@Asari+2007]). Unfortunately the small sample sizes do not allow for a meaningful statistical analysis, however both Figure \[fig:massfrac\_highM\] and the results in Table \[tab:massfrac\] suggest a similar picture to the low-mass sample. About 20-50$\%$ of high-mass Balmer-strong galaxies have $f_{M1}>0.03$ and 20-30$\%$ have $f_{M15}>0.2$, compared with $\sim3\%$ and $\sim2\%$ of the star-forming galaxies. In contrast with the low-mass bin, the fractions of recently formed mass are highest for the ePSB sample.
It is clear that a notable number of the Balmer-strong galaxies, particularly in the low-mass ePSB sample, have a fraction of recently formed mass that is consistent with that found in star-forming galaxies. It is of course entirely possible that these are true post-starburst galaxies with weaker bursts than the others, as it is actually the rapid decline in star formation that leads to the distinctive spectral shape of post-starburst galaxies picked up by the PCA selection method, and this is not exactly what is measured by $f_{M1}$ and $f_{M15}$. We may expect weaker bursts to fail to use up the entirety of the available gas, thereby accounting for the ongoing star formation. However, it does raise the question of whether they are true post-starburst galaxies or interlopers with strong Balmer absorption lines caused by something other than their star formation history. A careful investigation showed that the values of $f_{M1}$ and $f_{M15}$ are independent of stellar mass, structure and the environment of the galaxies; however, we found some dependence on the signal-to-noise ratio ($SNR$) and the dust content. Out of the ePSB galaxies with $SNR<15$ (20/57), $70\%$ and $55\%$ have the lowest measured fractions of recently formed stellar mass ($f_{M1}<0.05$ and $f_{M15}<0.1$), compared with $46\%$ and $24\%$, respectively, of those with $SNR\geq15$. This suggests that higher SNR spectra than than the typical in SDSS-DR7 are required to reliably identify recent bursts weaker than $\sim$10% by mass. Furthermore, $76\%$ and $53\%$ of ePSB galaxies with the largest dust content ($H_{\alpha}/H_{\beta} > 4.6$) coincide with $f_{M1}<0.05$ and $f_{M15}<0.1$, respectively, compared with $39\%$ and $25\%$ of the less-dusty ePSB galaxies. This could imply either: (a) the assumption of a single dust screen prevents STARLIGHT from recovering a recent burst in the dustier ePSB galaxies, but they fundamentally do not differ from the rest of the ePSB sample other than by their dust content, or (b) the stronger than average Balmer absorption lines do not reflect a decaying starburst but actually arise from a dust-star geometry such that the O/B stars being obscured behind more dust than average, i.e. these galaxies are not simply more dusty ePSB galaxies but less extreme versions of the “dusty starburst" galaxies [@PoggiantiWu2000]. Untangling these two possibilities is very tricky, pushing us to the limits of spectral fitting techniques, and will be the subject of a future study. We conclude that, while a significant fraction of the ePSB galaxies have had a recent burst of star formation in the past in which typically $\gtrsim10\%$ of the stellar mass was formed, the effects of noise and dust on the galaxy spectra may cause some level of contamination of post-starburst samples in which only weak bursts are identified by spectral fitting. Higher SNR spectra will be needed in order to understand the cause of strong Balmer absorption lines in the majority of ePSB galaxies from their spectra alone. In the following subsections we turn to other properties to further constrain their origins.
Low-mass galaxies qPSB agnPSB ePSB SF
-------------------- -------- -------- -------- --------
$f_{M1}<0.05$ 44$\%$ $45\%$ $54\%$ $78\%$
$f_{M1}>0.1$ $28\%$ $30\%$ $21\%$ $3\%$
$f_{M15}<0.12$ $14\%$ $18\%$ $44\%$ $82\%$
$f_{M15}>0.2$ $46\%$ $48\%$ $23\%$ $3\%$
High-mass galaxies qPSB agnPSB ePSB SF
$f_{M1}<0.015$ 40$\%$ 20$\%$ 20$\%$ 77$\%$
$f_{M1}>0.03$ $20\%$ $40\%$ $50\%$ 3$\%$
$f_{M15}<0.1$ $80\%$ $60\%$ $40\%$ $81\%$
$f_{M15}>0.2$ $20\%$ $20\%$ $30\%$ $2\%$
: The percentage of galaxies in the Balmer-strong samples and control star-forming galaxies that formed a given portion of their stellar mass in the last 1Gyr ($f_{M1}$) and 1.5Gyr ($f_{M15}$), as estimated by STARLIGHT. []{data-label="tab:massfrac"}
 
 
 
### Star formation prior to the starburst
Figure \[fig:sfh\] shows the stacked time evolution of the cumulative fraction of the total stellar mass of the galaxies, with the total mass calculated at 2Gyr in lookback time. This allows us to investigate the star formation history of the galaxies prior to the starburst. For each sample, the solid lines represent the mean values and the shaded regions illustrate the spread of values within the sample, quantified by the standard deviation from the mean. As for the recent star formation history, the difference between the quiescent and star-forming galaxies is clear and in agreement with expectations: the quiescent galaxies clearly build a higher fraction of their stellar mass at earlier times.
In the low-mass regime, the pre-burst star formation histories of all the three Balmer-strong samples, qPSB, agnPSB and ePSB, are almost indistinguishable from the histories of the star-forming galaxies and clearly distinct from the quiescent galaxies. This is consistent with the low-mass Balmer-strong galaxies originating from gas-rich star-forming, rather than quiescent, progenitors. Interestingly, the same is not true at high-mass. In particular, the pre-burst star-formation histories of the massive qPSB galaxies are distinct from the star-forming control, and overlap with those of the quiescent galaxies. This points to red-sequence progenitors, perhaps rejuvenating through minor mergers with gas-rich dwarfs. For the agnPSB and ePSB samples, the stellar mass build-up prior to the burst falls between the two control samples. More detailed inspection revealed that the star-formation histories of both ePSB and agnPSB split roughly equally between those that resemble the quiescent population and those that look more like the star-forming galaxies.
Morphology and structural properties {#sec:res_morph}
------------------------------------
Using the output from the image analysis code described in Section \[sec:method\_morph\] we investigated the morphology and structural properties of all galaxies without nearby stars or other image contaminants (‘clean’ samples, Table \[tab:galcounts\]). In Appendix \[appendix:AGN\] we investigated whether emission from narrow-line AGN affects the measurements of galaxy structure and morphology in our samples. We found no significant effect on any of the light-weighted parameters ($n$, $C$, $G$, $M_{20}$, $A$) measured in the $r$-band and conclude that we can use these measurements to meaningfully compare between galaxies with and without narrow-line AGN. Additionally, in Appendix \[appendix:struct\] we present relations between selected parameters that may be of interest to some readers. These include $A-C$, $G-M_{20}$ and $n-log(\Sigma_{5})$.
Here we present the results of the analysis of the $r$-band images (we found that the analysis of the $g$- and $i$-band images led to the same conclusions). We additionally visually inspected the 3-colour images of the galaxies for signs of past mergers, which can be difficult to identify with automated measurements. These include tidal features that do not form an asymmetric pattern when observed from a given direction and are therefore not detectable with the shape asymmetry ($A_{S}$). The images were inspected by only one reviewer as the aim of the visual classification was merely to provide subsidiary information to that inferred from the automated proxies - the main component of our analysis.
 
qPSB agnPSB ePSB dPSB Q SF
---------------------------------------- ------------- ------------- ------------- ------------- -------------- --------------
Low-mass galaxies with $A_{S}\geq0.2$ $8\%$ (24) $4\%$ (26) $12\%$ (43) $17\%$ (23) $6\%$ (602) $6\%$ (592)
High-mass galaxies with $A_{S}\geq0.2$ $40\%$ (5) $0\%$ (5) $0\%$ (6) $60\%$ (10) $3\%$ (101) $4\%$ (108)
Low-mass post-mergers $11\%$ (36) $3\%$ (33) $21\%$ (57) $16\%$ (31) $2\%$ (783) $8\%$ (785)
High-mass post-mergers $80\%$ (5) $80\%$ (5) $40\%$ (10) $67\%$ (12) $0\%$ (155) $0\%$ (154)
Low-mass galaxies with $n\geq2.0$ $82\%$ (22) $84\%$ (25) $53\%$ (40) $22\%$ (23) $92\%$ (537) $26\%$ (549)
High-mass galaxies with $n\geq2.0$ $100\%$ (5) $80\%$ (5) $100\%$ (5) $63\%$ (8) $98\%$ (94) $55\%$ (103)
### Asymmetries and signs of interaction {#sec:results_morph_A}
Both the visual inspection and automated measurements agree that the Balmer-strong galaxies in our samples are not ongoing mergers. In both low- and high-mass samples, the majority have low light-weighted asymmetry values $A<0.2$, characteristic of normal galaxy types and none have $A>0.35$ commonly found in ongoing mergers. Furthermore, they occupy a similar region of the $G-M_{20}$ parameter space as the control galaxies, with only a few ‘outliers’ in the merger region. The light-weighted asymmetry vs. concentration index and Gini index vs. $M_{20}$ are presented in Figures \[fig:morph\_CA\] and \[fig:morph\_GM20\] respectively. Given the short visibility timescales for merger signatures (0.2-0.4Gyr), peaking before coalescence [@Lotz+2008], and the estimated ages of the starburst ($>0.6$Gyr), it is not surprising to see few ongoing mergers and this does not rule out a merger origin for the Balmer-strong galaxies.
As a merger-induced starburst is believed to occur at coalescence of the progenitor galaxies (except bulgeless galaxies, in which case it may occur earlier, see e.g. @MihosHernquist1996), it is more likely to observe *post-merger* signatures, such as tidal features, in post-starburst galaxy samples (see @Pawlik+2016). In Figure \[fig:morph\_AsAs90\] we show the shape asymmetry measured under 90$^{o}$ and 180$^{o}$ rotation; generally the values of $A_{S}$ fall below 0.2, meaning that the galaxies do not have visible asymmetric post-merger signatures, such as tidal tails. This is consistent with the results of @Pawlik+2016 who found that by 600Myr following the starburst, the shape asymmetry had largely returned to levels similar to control samples. In the top rows of Table \[tab:morph\] we present the fraction of galaxies in each sample with $A_{S}>0.2$. At low-mass, the ePSB and dPSB samples contain the highest fractions of objects with $A_{S}\geq0.2$ ($12\%$ and $17\%$, respectively), which is a little higher than found in the control samples ($6\%$ for both quiescent and star-forming galaxies). The proportions of low-mass qPSB and agnPSB galaxies with $A_{S}\geq0.2$ are low, consistent with those found in the control samples. At high mass, the qPSB and dPSB galaxies have much higher fractions of post-merger candidates than the control samples ($40\%$ \[2/5\] and $60\%$ \[6/10\], respectively), but the other two samples have no positive detections (0/5).
In the middle rows of Table \[tab:morph\] we present the fraction of galaxies in each sample identified as post-merger candidates by visual inspection. We stress that the two post-merger definitions are not equivalent, as the visual classification does not rely on a high degree of asymmetry in the morphological disturbance and is therefore more inclusive. At low-mass, the fraction of ePSB galaxies visually classified as post-mergers is significantly higher than measured with $A_{S}$, and at high mass the same is true for qPSB, agnPSB and ePSB samples. At high mass, the measured fractions reach $80\%$ (4/5) in both qPSB and agnPSB. This increase in post-merger fractions is due to features which are not asymmetric enough to be detected by $A_{S}$. Interestingly, those Balmer-strong galaxies with $A_{S} \geq 0.2$ tend to have low/moderate values of $A_{S90}$, which also points to tidal features with little azimuthal asymmetry. Given that some simulations have shown that symmetric tidal feature patterns such as shells may be formed not only through satellite accretion but also in late stages of major mergers (see e.g. @HernquistSpergel1992, Pop et al. submitted), it is interesting to speculate that symmetric tidal features are more common in more evolved systems, consistent with the starburst ages of $\sim1-1.5$Gyr in these samples. However, as minor mergers may lead to similar signatures, further analysis of simulations would be required to confirm this.
### The profile and central concentration of light
 
 
qPSB agnPSB ePSB dPSB Q SF
------------------ ------------------------- ------------- ------------- ------------- ------------- -------------- --------------
Low-mass sample $\Sigma_{5}<0.0$ $9\%$ (35) $36\%$ (33) $4\%$ (55) $7\%$ (30) $3\%$ (775) $19\%$ (770)
$0.0\leq\Sigma_{5}<1.0$ $74\%$ (35) $58\%$ (33) $82\%$ (55) $87\%$ (30) $49\%$ (775) $74\%$ (770)
$\Sigma_{5}\geq1.0$ $17\%$ (35) $6\%$ (33) $15\%$ (55) $7\%$ (30) $48\%$ (775) $7\%$ (770)
High-mass sample $\Sigma_{5}<0.0$ $100\%$ (5) $20\%$ (5) $70\%$ (10) $58\%$ (12) $21\%$ (153) $52\%$ (151)
$0.0\leq\Sigma_{5}<1.0$ $0\%$ (5) $80\%$ (5) $30\%$ (10) $33\%$ (12) $64\%$ (153) $45\%$ (151)
$\Sigma_{5}\geq1.0$ $0\%$ (5) $0\%$ (5) $0\%$ (10) $8\%$ (12) $15\%$ (153) $3\%$ (151)
In Figure \[fig:Sersic\] we show the distribution of S[é]{}rsic indices, with K-S test results comparing distributions to the control samples and each other. The lower rows of Table \[tab:morph\] show the fraction of each sample with steep light profiles, characterised by $n\geq2.0$.
The low-mass Balmer-strong galaxies span the whole dynamic range in S[é]{}rsic index, with $0.5\le n \le 5.5$, pointing to a range of structural properties, from highly concentrated single component spheroids to disk-dominated systems. The distributions found for qPSB and agnPSB are comparable to the quiescent control sample ($D\sim0.2$, $p=0.28$ and $D\sim0.3$, $p=0.0065$), indicating high central concentration characteristic of massive spheroids ($82\%$ and $84\%$ with $n\geq2$, respectively, compared with $92\%$ of the quiescent galaxies). The ePSB sample has typically lower values of $n$ than both the qPSB and agnPSB samples, with K-S statistics showing the distribution is distinct from the quiescent control sample. The dPSB sample has the lowest values of $n$ for all Balmer-strong samples, with the distribution comparable with the control sample of star-forming galaxies ($D\sim0.3$, $p=0.04$ and $22\%$ with $n\geq0.2$, compared with $26\%$ of the star-forming galaxies). The slight shift towards higher values of $n$ could be due to the high dust content of the dPSB galaxies as a similar shift is observed when comparing the distribution of $n$ between the star-forming galaxies with the control sample of dusty star-forming galaxies. The distributions of $n$ for the dPSB and dusty star-forming control are consistent with being drawn from the same underlying distributions ($D\sim0.2$, $p=0.5$).
The picture is very different at high-mass, where we found high values for the S[é]{}rsic index comparable with the quiescent control sample for the qPSB, agnPSB and ePSB samples ($100\%$, $80\%$ and $100\%$ with $n>2.0$). As for the low-mass sample, the distribution of $n$ for the dPSB galaxies resembles the dusty star-forming control sample.
Similar conclusions are reached from the concentration and Gini indices, however, the separation between samples is less apparent than in the values of $n$ (see Figures \[fig:morph\_CA\] and \[fig:morph\_GM20\]).
Environment {#sec:results_env}
-----------
In Figure \[fig:env\] we present the distributions of the projected number density ($\Sigma_{5}$), and the results are summarised in Table \[tab:env\]. For the low-mass galaxies we show the K-S test results comparing the different distributions. We tested two different cuts on the samples, firstly just removing those galaxies that fell near the edge of the survey, and secondly also removing objects for which the difference between the spectroscopic and photometric value for $\log(\Sigma_{5})$ was greater than 0.4dex [@Baldry+2006]. The results using the additional cuts were not significantly different.
Overall, we see that both low-mass and high-mass Balmer-strong galaxies tend to occupy the low/medium-density environments log($\Sigma_{5})<1.0$, similar to the star-forming control samples. This is evident in the K-S test statistics for the low-mass sample. The only possible difference is for the low-mass agnPSB, where the distribution shifts towards lower values ($36\%$ have log($\Sigma_{5})<0.0$ compared to $19\%$ of the star-forming control), although the K-S test shows that any difference is not formally significant ($D=0.32$, $p=0.0023$). At high-mass there is a possible indication that the qPSB and ePSB galaxies are preferentially found in lower density environments than the star-forming control sample. However, the small numbers prevent any firm conclusions to be drawn.
Qualitatively the above results are generally unaffected by the choice of measurement of the number density, i.e. the distributions of the mean $\Sigma_{5}$ shown in Figure \[fig:env\] point to the same local environments of the post-starburst galaxies relative to the control samples, as the distributions of the individual photometric/spectroscopic measurements. The one exception is the ePSB sample, where $\Sigma_{5}$ derived solely from the photometric redshifts suggests slightly higher-density environments relative to the control samples than the mean values.
We verified that there were no trends in the structural properties of the low-mass Balmer-strong galaxies as a function of their local environment (Figure \[fig:morph\_nE5\]), although the few morphologically disturbed qPSB and agnPSB do tend to reside in under-dense environments ($\Sigma_{5}\lesssim0.0$). The fact that the environments of the low-mass Balmer-strong galaxies are broadly consistent with the star-forming control sample is in agreement with the results presented in Section \[sec:res\_sfh\] showing that the pre-burst star formation histories are also characteristic of star-forming galaxies. The possible tendency of the high-mass Balmer-strong galaxies to be found in lower density environments than the control samples, while their pre-burst star formation histories suggest they are originate from a mix of star-forming and quiescent galaxies, might indicate that environment is the most important factor driving the occurrence of starbursts at high mass. However, larger samples would be needed to verify this conclusion.
Discussion
==========
The wide range of properties of the Balmer-strong galaxies revealed during our analysis implies that there is no unique pathway that leads to their formation, and that the different conclusions drawn about their origins in the literature may all be valid in certain circumstances. Previous observations of Balmer-strong or post-starburst galaxies have suggested they are present in a range of different environments, with varying incidence depending on the cosmic epoch, stellar mass and environment (see e.g. @Poggianti+2009). As such, post-starburst galaxies may represent a phase that is common to a variety of different mechanisms driving galaxy evolution.
A commonly suggested mechanism for the formation of post-starburst galaxies is through mergers of gas-rich galaxies, which can leave faint visual signatures in the morphology of the merger remnant. We therefore begin our discussion by investigating the timescale of visibility of morphological signatures of a past merger using mock galaxy images from hydrodynamical merger simulations (Section \[disc:mergers\]). We then bring together the results of our analysis of the star-formation histories, morphologies, structural properties and the environments of the local Balmer-strong galaxies to discuss which of our samples are true post-starburst galaxies and which are more likely to be interlopers (Section \[disc:psb\]), and to investigate the evolutionary pathways that lead to their formation (Section \[disc:pathways\]).
Timing the visibility of post-merger features {#disc:mergers}
---------------------------------------------
Although the structural evolution of galaxies in merger simulations has already been studied, using various measures of galaxy structure and morphology, the previous studies have not included the new merger-remnant sensitive shape asymmetry. Using hydrodynamical simulations we created mock images of galaxies undergoing a merger, recorded at 20-Myr time intervals, in order to study the evolution of the merger morphology as measured by $A_{S}$.
The simulations, described in detail in Appendix \[sec:mergers\], were performed using the entropy conserving smoothed particle hydrodynamics code [<span style="font-variant:small-caps;">Gadget-3</span>]{} [@Springel2005], with improved SPH implementation - SPHGal [@Hu+2014; @Eisenreich+2017] and include radiative cooling, star formation and feedback from stars and supernovae. In this work we focus mainly on equal-mass mergers of gas-rich galaxies with three different initial morphologies (Sa, Sc and Sd), and in three different dynamical configurations (prograde-prograde 00, prograde-retrograde 07 and retrograde-retrograde 13; see @NaabBurkert2003). We also analyse three simulations with a mass ratio of 1:3, involving galaxies with the same morphologies (Sc) but different orbital configurations (prograde-prograde 00, prograde-retrograde 07 and retrograde-retrograde 13). We limit ourselves to a small number of simulations only to illustrate how the timescale of visibility of the tidal features varies with the conditions of the interaction. A more detailed analysis of a full suite of simulations is left for future work. The mock images were created with the noise properties of the SDSS imaging data (see Appendix \[appendix:mergers\_img\], and analysed with the same code as the real data (see Section \[sec:method\_morph\]), to ensure a truly meaningful comparison with the results presented in this paper.
![The time-evolution of the shape asymmetry (see Section \[sec:method\_morph\]) as measured in the mock images of the simulated galaxy mergers. The spread in the data represents the minimum and maximum values calculated per given simulation, in images synthesised using six different viewing angles. Equal-mass mergers are shown in the top and middle panels and those with mass ratio 1:3, can be found in the bottom panel. In the top panel we vary the progenitor morphology, while in the middle and bottom panels the initial orbital parameters, as indicated in the legend and described in Appendix \[sec:mergers\]. []{data-label="fig:mergers"}](simn_2xSaScSd_07_morphevol_As_r_rerun.jpg "fig:"){width="\columnwidth"} ![The time-evolution of the shape asymmetry (see Section \[sec:method\_morph\]) as measured in the mock images of the simulated galaxy mergers. The spread in the data represents the minimum and maximum values calculated per given simulation, in images synthesised using six different viewing angles. Equal-mass mergers are shown in the top and middle panels and those with mass ratio 1:3, can be found in the bottom panel. In the top panel we vary the progenitor morphology, while in the middle and bottom panels the initial orbital parameters, as indicated in the legend and described in Appendix \[sec:mergers\]. []{data-label="fig:mergers"}](simn_2xSc_00_07_13_morphevol_As_r_rerun.jpg "fig:"){width="\columnwidth"} ![The time-evolution of the shape asymmetry (see Section \[sec:method\_morph\]) as measured in the mock images of the simulated galaxy mergers. The spread in the data represents the minimum and maximum values calculated per given simulation, in images synthesised using six different viewing angles. Equal-mass mergers are shown in the top and middle panels and those with mass ratio 1:3, can be found in the bottom panel. In the top panel we vary the progenitor morphology, while in the middle and bottom panels the initial orbital parameters, as indicated in the legend and described in Appendix \[sec:mergers\]. []{data-label="fig:mergers"}](simn_ScScp3_00_07_13_morphevol_As_r_rerun.jpg "fig:"){width="\columnwidth"}
In Figure \[fig:mergers\] we show the evolution of the shape asymmetry, $A_{S}$, with time since the coalescence of the two galaxies. As described in Section \[sec:method\_morph\], $A_{S}$ can be used to identify asymmetric tidal features in galaxies, signifying a recent morphologically disruptive event, such as a merger. Using SDSS images, @Pawlik+2016 concluded that galaxies displaying asymmetric features are characterised by $A_{S}\geq0.2$, while values of $A_{S}\sim 0.1$ correspond to regular, undisturbed morphologies. This is in good agreement with the end point of the simulations. In the top and middle panel we show equal-mass mergers, and in the bottom panel we show 1:3 mass ratio mergers. The different data sets correspond to different simulations, with varying progenitor morphology (upper panel), and initial orbital configuration of the progenitors (middle and bottom panels).
The timescale of visibility of the tidal features, measurable by $A_{S}\geq0.2$ varies with morphology, orbital configuration and the mass ratio of the progenitors. For the equal-mass mergers, the shortest timescales, measured from the time of coalescence, are observed for the early-type Sa morphology ($<200$ Myr) and the coplanar prograde-prograde orbital configuration (00), in which case the tidal features vanish immediately after coalescence. In the cases where the galaxies have smaller bulge components (Sc and Sd morphologies) and where they collide in more asymmetric dynamical configurations (07 and 13) the asymmetric tidal features induced by the interaction prevail for longer. This is consistent with expectations, as in major merger simulations retrograde configurations produce the most violent effects due to the anti-alignment of the galactic versus orbital angular momenta. In the case of unequal-mass mergers the asymmetry of the tidal features tends to be lower and vanish more rapidly following the coalescence of the progenitors, compared with the 1:1 mergers with the same morphology and orbital parameters.
Depending on the progenitor morphology and initial configuration of orbits, a post-merger with the merger age of $\sim100$ Myr can have a a wide range of values of the shape asymmetry ($\sim0.1$ – $\sim0.4$). Regardless of the initial conditions, the tidal features generally fade away after $\sim500$Myr from the coalescence in 1:1 mergers, and after $\sim400$Myr in those with a mass ratio of 1:3.
The above results could explain the lack of visible post-merger features among our Balmer-strong samples (Section \[sec:results\_morph\_A\]), given that their estimated starburst ages are greater than $\sim0.6$Gyr. The lack of such features is therefore not sufficient to rule out a merger origin of the Balmer-strong galaxies.
The different families of Balmer-strong galaxies {#disc:psb}
------------------------------------------------
As described in Section \[sec:psbselection\] we separated the Balmer-strong galaxies at $0.01<z<0.05$ based on their emission line properties into quiescent ‘K+A’ galaxies (qPSB) with no/weak emission lines, those with a measurable level of nebular emission from ongoing star-formation (ePSB and ‘dusty’ dPSB) or AGN/shock activity (agnPSB). We obtained samples of 36, 33, 57 and 31 qPSB, agnPSN, ePSB and dPSB, respectively, in the low-mass regime ($10^{9.5}<\mbox{M}_{\star}/\mbox{M}_{\sun}<3\times10^{10}$) and 5, 5, 10 and 12 at higher masses ($\mbox{M}_{\star}/\mbox{M}_{\sun}>3\times10^{10}$). The lower mass limit was set to ensure that all samples were complete, including the quiescent control sample. In this section we look at the similarities and differences between the different samples, assessing the likelihood of them being true “post-starburst" galaxies, before progressing onto their likely origins and fate in the following subsection.
### The quiescent Balmer-strong galaxies and AGN/shocks hosts (qPSB and agnPSB)
As the origin of post-starburst galaxies is often linked with violent dynamical processes, we might expect their spectra to show evidence of AGN and shocks. Previous results have shown that strong Balmer absorption lines are common in the spectra of narrow-line AGN samples (e.g. @Kauffmann+2003c [@CidFernandes+2004a]) and narrow-line AGN are common in samples of galaxies with strong Balmer absorption lines (e.g. @Yan+2006 [@Wild+2007]). This motivated @Wild+2009 to discard the emission line cut when selecting post-starburst galaxies. @Tremonti+2006 and @Alatalo+2016 also found evidence of galactic winds and shocks in post-starburst galaxies, respectively. Furthermore, AGN have often been invoked to aid the quenching of star-formation following simulated galaxy mergers, causing the galaxies to become quiescent (e.g. @Hopkins+2006). In this section we compare the properties of the “classical" quiescent Balmer-strong galaxies (qPSB), with the sample that have emission line ratios that lie above the @Kewley+2001 demarcation line (agnPSB). While we have focussed on an AGN as the most likely origin of the high ionisation emission lines, we remind the reader that our selection does not entirely rule out shocks as an alternative origin. However, the requirement for the equivalent width of H$\alpha$ to be larger than 3Å does rule out weak shocks, as well as the class of “retired" galaxies where the high ionisation lines are caused by evolved stellar populations [@CidFernandes2011].
At both high and low masses, the close similarities between the structural properties, star-formation histories and environments of the qPSB and agnPSB suggest that they are of the same physical origin. The distributions of their structural and morphological parameters are indistinguishable from one another, and closely resemble those of the quiescent control sample (Figures \[fig:morph\_AsAs90\], \[fig:Sersic\], \[fig:morph\_CA\], \[fig:morph\_GM20\]) apart from a possible enhancement in visually identified post-merger features (Table \[tab:morph\]). Both are found in intermediate-density environments, matching the star-forming control sample (Figure \[fig:env\]), although at low-mass there is tentative evidence that the agnPSB galaxies are found in slightly lower density environments than the qPSB galaxies on average. In the low mass sample, the mean and distribution of pre-burst star formation histories are barely distinguishable, matching the star-forming control sample (Figure \[fig:sfh\]), and the distributions of recently formed stellar mass are again very similar with $\sim50\%$ indicating a recent starburst in which $>$20% of the stellar mass was formed (Figure \[fig:massfrac\_lowM\] and Table \[tab:massfrac\]). At high mass the pre-burst star formation histories of the qPSB and agnPSB galaxies are mixed, although the majority (4/5 and 3/5) closely resemble the quiescent control sample. The STARLIGHT spectral analysis does not identify a high fraction of mass formed in the last 1.5Gyr in 14% (18%) of the low-mass qPSB (agnPSB) galaxies, and in 80% (60%) of the high-mass qPSB (agnPSB) galaxies. Either the fraction of mass formed in the starbursts was $\lesssim10\%$, and therefore difficult to identify given the limited SNR of the data and limitations of the models, or some Balmer-strong galaxies with no ongoing star-formation are not actually post-starburst. While the latter seems unlikely, further detailed spectral analysis would be required to confirm that these Balmer-strong galaxies are post-starburst, including two-component dust, covariances between fitted parameters, and an assessment of the limitations (and means of improvement) of the stellar population models used.
*From the samples and observations available at the present time, there is no evidence that low-redshift galaxies with strong Balmer absorption lines and high ionisation emission lines do not have the same origin as traditional “K+A" galaxies with strong Balmer absorption lines and weak emission lines.* It would appear that the AGN activity switches on and off on timescales shorter than the post-starburst phase. Because our samples are complete in stellar mass, we can use the relative number of qPSB and agnPSB galaxies to constrain the duty cycle of the AGN during the post-starburst phase (ignoring the possible contribution of shocks to the agnPSB samples). In our low (high) mass sample we find 36 (5) qPSB and 33 (5) agnPSB i.e. the AGN must be “on" for $\sim$50% of the time. This indicates that previous work that selected post-starburst galaxies with stellar masses $>10^{9.5}\mbox{M}_\odot$ using cuts on the emission lines will have underestimated their number density by a factor of two. A similarly high fraction of AGN ($\sim40\%$) was recently found in a sample of bright compact star-forming galaxies at $z=2$, which are likely progenitors of compact quiescent galaxies found at that epoch [@Kocevski+2017]. While the masses and effective radii of our post-starburst galaxies do not imply significant compactness, it is encouraging to find similarly high incidence of AGN among potential progenitors of quiescent galaxies at both low and high redshift.
### The star-forming and dusty Balmer-strong galaxies (ePSB and dPSB) {#sec:discussion_edpsb}
As the transition between the starburst and quiescent post-starburst phases is not instantaneous, declining levels of star-formation should be observed following the starburst. Balmer-strong galaxies with emission indicative of ongoing star-formation are possible candidates for the transitioning phase (e.g. @Wild+2010 [@Pawlik+2016; @Rowlands+2017]); however their true nature is still under debate as some studies regard them as ongoing dust-obscured starbursts (e.g. @Dressler+1999 [@Poggianti+1999]), and they may also be caused by an event that was not sufficiently disruptive to permanently halt the star-formation [@Rowlands+2015]. In this paper we deliberately included all Balmer-strong objects, regardless of their emission line characteristics, in order to make an objective assessment of their likely origins. They account for $\sim60\%$ of the total number of Balmer-strong galaxies (Table \[tab:galcounts\]). Here we compare the properties of the Balmer-strong galaxies with emission-line galaxies indicating star-formation (ePSB) with those in which the Balmer decrement (ratio of $H\alpha$ and $H\beta$ emission lines) points to significant dust contents (dPSB).
The difference in structure between the ePSB and dPSB samples clearly suggests that the two classes represent objects of different physical nature. The ePSB are split roughly equally between low and high values of $n$ and at low-mass the distribution resembles neither the star-forming nor quiescent control samples (Figure \[fig:Sersic\]). On the other hand, the distribution of $n$ measured for the dPSB class is unimodal and resembles that found for the star-forming control sample, in particular those that are highly dust-obscured. In fact, there is no measurement that we were able to make in which the dPSB sample differs significantly from the star-forming control sample, except in the fraction of galaxies with post-merger signatures. Given that their dust content prevents us from analysing their star formation histories, *we are unable to find any evidence that the dPSB galaxies are truly “post-starburst" and we concur with previous literature that they may be dust-enshrouded star-forming galaxies, in which the strong Balmer absorption lines are caused by the preferential obscuration of O/B stars.* It remains possible that the dPSB are progenitors of (some of) the post-starburst galaxies [@Yesuf+2014], however, we are unable to make such a link with the data currently available.
The ePSB galaxies are the most difficult class in which to understand the origin of the strong Balmer lines. In @WildGroves2011 we obtained Spitzer IRS spectra for 11 dusty Balmer-strong galaxies and found that dust-correcting the optical emission lines using the measured $H\alpha/H\beta$ Balmer decrement and a standard attenuation law led to emission line strengths that were entirely consistent with those measured in the mid-IR. We concluded that the Balmer decrement is a reliable estimate of the dust attenuation in Balmer-strong galaxy spectra when it is measurable. The “normal" Balmer decrement distribution matching the star-forming control sample, and normal continuum colours of the ePSB class compared to the very red continuum of the dPSB class both suggest that only a minority of the ePSB galaxies can be dust-enshrouded star-forming galaxies. This can be seen in the stacked spectra presented in Figure \[fig:SED\_stack1\], but to better illustrate the difference in the continuum colors of the ePSB and dPSB galaxies we present their optical and mid-IR colours in Figure \[fig:gi\_w1w2\] as well as some individual examples of spectra in Figure \[fig:sedfits\_dpsb\]. The mid-infrared colours of the dPSB galaxies clearly indicate that they have higher dust content compared with the other samples. However, the dPSB sample is just an arbitrary cut to identify the extreme dustiest of the ePSB class (with the adopted cut dPSB make up $35\%$ and $55\%$ of the ePSB galaxies, respectively). It is possible that this cut should be placed lower, and some of the ePSB class are dusty star-forming galaxies rather than true post-starburst galaxies. Indeed, we find that over half of the dustiest ePSB galaxies coincide with the lowest measured fractions of recently formed stellar mass in our spectral analysis, although this may be indicative of the limitations of spectral fitting models.
As well as their very broad distribution of $n$, covering both star-forming and quiescent control samples, the ePSB galaxies also show on average a higher fraction of stellar mass formed in the recent past than the star-forming control (Figures \[fig:massfrac\_lowM\] and \[fig:massfrac\_highM\], Table \[tab:massfrac\]), and a higher fraction have post-merger features (Figure \[fig:morph\_AsAs90\] and Table \[tab:morph\]). Again, this suggests that at least a fraction are truly post-starburst, but exactly what fraction is difficult to determine. Conservatively taking only the tail of objects for which our spectral analysis identifies a high fraction of stellar mass formed in the last 1-1.5Gyr compared to the star-forming control, we conclude that $\gtrsim20\%$ of the low-mass ePSBs and $\gtrsim50\%$ of the high-mass PSBs are truly post-starburst galaxies. This will exclude objects with smaller starbursts that we are unable to constrain with the spectral analysis. To obtain a less conservative estimate, we can combine the spectral and structural information. While almost all of the ePSBs have pre-burst star formation histories that are consistent with the star-forming control sample, $53\%$ of the low mass and $100\%$ of the high-mass ePSBs have $n>2$ indicative of a spheroid. We can speculate that these galaxies have undergone a violent event leading to a burst of star formation and increase in light concentration through growth of a bulge, or the dominant progenitor was already spheroidal. This is also supported by the higher fraction with post-merger features than seen in the star-forming control sample.*Overall, we find only $20\%$ of the low-mass ePSB sample and none of the high-mass ePSB sample where there is no evidence of either a structural change, recent significant enhancement in star-formation or asymmetric tidal features (i.e. $f_{M1}<0.1$ and $f_{M15}<0.2$ and $n<2$ and $A_{S}<0.2$), compared with $61\%$ and $28\%$ of the low- and high-mass star-forming control samples, respectively.*
Finally, using the merger simulations described in Section \[disc:mergers\] and Appendix \[sec:mergers\], we find further evidence supporting the notion that the ePSB class are truly transitioning post-starburst galaxies with declining levels of star-formation. We created mock spectra of the merging galaxies and combined this with the ongoing SFR to estimate the equivalent width of the $H\alpha$ emission line. We found that the timescale of visibility of the ePSB phase reaches at least $0.7-1.3$Gyr following the starburst[^9], which is roughly the characteristic age of our qPSB galaxies. Naturally, this is a very crude estimate and investigating a larger suite of simulations run for a longer time, that include a wide range of progenitor characteristics and orbital configurations is required to constrain this quantity further. Nevertheless, it is encouraging that all simulations considered in this work show that a merger-induced starburst is followed by the ePSB transition phase during which the star-formation rate in the merger remnant declines over $\sim1$Gyr - a timescale long enough to be observed at the characteristic age of the traditional quiescent post-starburst galaxies. Determining how many ePSB galaxies are truly transitioning to the red-sequence is also difficult. One possibility is that the ePSB galaxies have experienced a more recent burst than the qPSB/agnPSB galaxies, and will subsequently become qPSBs before entering the red-sequence. The morphologies of 53% of the low-mass and 100% of the high mass ePSB galaxies are consistent with this scenario. The STARLIGHT results do indicate that the ePSBs have younger bursts than the qPSB/agnPSB galaxies on average (Table \[tab:massfrac\]): $f_{M1}$ and $f_{M15}$ are high in $\sim20\%$ of low-mass ePSBs, compared to $f_{M15}$ being high in $\sim50\%$ of qPSB/agnPSBs and $f_{M1}$ high in only $\sim30\%$. However, the aliasing seen in the star formation histories of the star-forming control close to the expected age of the starburst prevents strong conclusions from being drawn. Probably a more profitable approach would be through the measurement of the gas contents of the different classes of Balmer-strong galaxies, to investigate the amount of fuel remaining for future star formation (e.g. @Rowlands+2015 [@French+2015]).
Pathways through the post-starburst phase {#disc:pathways}
-----------------------------------------
With an insight to the physical characteristics of the different spectral classes of Balmer-strong galaxies we can build a picture of the different evolutionary pathways of galaxies that can lead to a post-starburst phase. We summarise our conclusions in Figure \[fig:psbevol\]. In this section we do not discuss the dPSB class further, as we concluded in the previous sub-section that the strong Balmer lines were likely due to dust/star geometry rather than recent star formation history. As we are unable to separate the ePSB galaxies that are most likely post-starburst from those that could be interlopers due to dust distribution, we include them here together, but note that there may be some contaminants in this class.
### Low-mass regime {#sec:disc_pathways_lowM}
[*The pre-burst star formation histories and environments of the low-mass qPSB, agnPSB and ePSB galaxies are entirely consistent with those of the star-forming control galaxies, implying that they all originated from star-forming progenitor galaxies.*]{} The structure of $83\%$ qPSB/agnPSB and $53\%$ of the ePSB galaxies is consistent with the quiescent control sample, supporting the scenario that $\sim70$% of low-mass post-starburst galaxies are a transition phase between star-forming spirals and quiescent spheroids. We cannot directly conclude from this study that the starburst and morphological transition were concurrent, although it does not seem plausible that only galaxies that are already spheroidal undergo starbursts strong enough to produce strong Balmer lines. A more plausible scenario is that the same event that triggers the starburst causes the morphological transition. Overall, a higher fraction of the post-starburst galaxies show post-merger signatures than the control samples, although our merger simulations concur that the merger likely occurred too far in the past for signatures to be visible in the majority of the SDSS images. Most of the Balmer-strong galaxies occupy intermediate-density environments (i.e. loose groups), where the dynamical conditions are favourable for galaxy mergers to occur. The remaining ePSB galaxies have a low central concentration of light, characteristic of disks, which suggests they may be returning onto the blue-cloud to continue forming stars in an ordinary fashion.
Based on the above characteristics of the different samples, we propose the following two evolutionary scenarios that lead to the post-starburst phase in the low-mass regime:
- Due to some violent triggering event, a gas-rich star-forming disk galaxy experiences a strong starburst and morphological transition, likely followed by strong AGN activity as the galaxy evolves to become a quiescent spheroid. Whether a galaxy passes through an ePSB phase before reaching the qPSB/agnPSB phase may be determined by the conditions of the triggering event (e.g. orbital configurations in the case of a merger), as well as the properties of the progenitors, such as their structure or their gas content. The limited set of merger simulations considered in this work suggest that the ePSB phase should be visible for $\gtrsim$0.7Gyr following the initial starburst, irrespective of the progenitor morphology and orbital configurations, but a larger variety of simulations is required to better constrain this value.
- A less violent mechanism leads to a starburst that fades more gradually through the ePSB phase, and may or may not result in morphological transition. Depending on the remaining gas reservoir, the galaxy will either return to the star-forming blue cloud, or become a qPSB/agnPSB and subsequently join the red sequence.
If the triggering of the starburst occurs via galaxy mergers, then the strength of the starburst and the associated post-starburst evolutionary pathway may plausibly be determined by the conditions of the interaction (e.g. more major mergers leading to stronger starburst, more rapid shut down in star formation and a more significant morphological transformation).
### High-mass regime {#sec:disc_pathways_highM}
Due to the limited sample sizes, the origins of the high-mass post-starburst galaxies are harder to constrain; however, as our analysis points to some contrasting characteristics compared with their low-mass counterparts, it is worth speculating on their inferred evolutionary pathways.
Contrary to the low-mass counterparts, all qPSB, agnPSB and ePSB tend to have high central concentration, meaning that they could represent different stages of the same evolutionary pathway to the red sequence. All high-mass post-starburst samples have high fractions of morphologically disturbed galaxies, reaching much higher values in the samples of qPSB and agnPSB, again in contrast with what was observed in the low-mass regime. A striking difference between the low- and high-mass qPSB lies in their pre-burst star-formation histories, which very much resemble those characteristic of the quiescent control sample. In the case of the high-mass agnPSB and ePSB, there is a roughly equal split among their star-formation histories into those that resemble the quiescent population and those that look more like the star-forming galaxies. Further investigation shows that the ePSB with likely star-forming progenitors tend to have higher fractions of recently formed mass and more often disturbed morphologies, compared with those whose star-formation histories imply a quiescent progenitor. The pre-burst star formation histories lead to the suggestion that 40% of high-mass post-starburst galaxies are rejuvenation events of an already quiescent elliptical galaxy. Another difference between the low and high-mass samples is in the fractions of stellar mass formed in the recent past, which are typically much lower for the high-mass post-starburst galaxies. This is again consistent with minor-mergers or small rejuvenation events rather than major gas-rich mergers. This makes sense, as it is statistically less likely for two high-mass gas-rich galaxies to merge in the local Universe than two low-mass galaxies or a low and high-mass galaxy.
Based on the above findings we suggest that high-mass galaxies can reach the post-starburst phase through the following two pathways:
- An already quiescent galaxy experiences a relatively weak starburst through a minor merger with a lower mass gas-rich galaxy, after which the galaxy goes through a brief post-starburst phase only to return back onto the red sequence. Depending on the merger conditions, the ePSB phase may be visible before the galaxy attains the qPSB/agnPSB characteristics.
- Due to a violent triggering event, a star-forming spiral galaxy experiences a morphological transition and starburst which fades as the galaxy evolves to the qPSB/agnPSB phase, possibly through an ePSB phase, to reach the red sequence.
Summary
=======
The evolution of galaxies is a complex multistage process, with post-starburst galaxies playing various roles throughout its different stages and channels. In this work we show that the majority of post-starburst galaxies are consistent with representing a transition phase between the blue cloud and the red sequence, following a merger-induced starburst. The quenching of star-formation may be relatively slow, and is likely to be accompanied by flickering AGN activity. However, we find that this evolutionary pathway is not the only one that can lead to the post-starburst phase. In the low-mass regime, post-starburst galaxies also represent a cyclic evolution of galaxies within the blue cloud, in which the stochasticity of star formation leads to a weaker burst, with no morphological transformation or complete quenching. On the other hand, at higher masses, post-starburst galaxies also occur in the cyclic evolution of galaxies within the red sequence, when an already quiescent galaxy gradually builds up its stellar mass through small starbursts. Such episodes could perhaps be resulting from mergers with small gas-rich companions, moving the galaxy towards the high-mass end of the red sequence.
Although in this paper we tried to be exhaustive with the currently available data, it is clear that upcoming and future datasets will have a dramatic impact on our understanding of galaxies transitioning between the blue cloud and red sequence. Of particular importance to the next stages of this work are data from IFU surveys, which will help ascertain whether the galaxies in our sample are nuclear or global post-starbursts, (see e.g. @Pracy+2014) and whether global post-starburst galaxies are more likely to be transitioning, whereas nuclear post-starburst galaxies are more likely to be cyclical events [@Rowlands+2015]. Resolved spectroscopy may also reveal whether our class of post-starburst galaxies with high ionisation lines are predominantly due to AGN or shocks. Finally, a detailed investigation of merger simulations will help place further constraints on the visibility of post-merger features in post-starburst galaxies, including its dependence on the mass, gas content and structural properties of the progenitor galaxies as well as dynamical configuration of the merger.
Acknowledgements
================
We would like to thank Natalia Vale Asari, Gustavo Bruzual and Stephane Charlot for providing invaluable help with interpreting the STARLIGHT spectral fitting results and providing updated models. We also thank the referee for a thorough review of the manuscript and useful suggestions, which helped us clarify some important points and improve the presentation of our results.
MMP, VW, JM-A, NJ and KR acknowledge support of the European Research Council via the award of a starting grant (SEDMorph; P.I. V. Wild). LTA acknowledges support from the Iraqi Ministry of Higher Education and Scientific Research. NL acknowledges the support of the Jenny $\&$ Antti Wihuri Foundation. NL and PHJ acknowledge the support of the Academy of Finland project 274931. YZ acknowledges support of a China Scholarship Council – University of St Andrews Scholarship. WL acknowledges support from the ECOGAL project, grant agreement 291227, funded by the European Research Council under ERC-2011-ADG.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
This publication also makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
 
 
 
Colours and star-formation histories of the dusty Balmer-strong galaxies {#app:dpsb}
========================================================================
The $g-i$ and $W1-W2$ colours {#app:dpsb_colours}
-----------------------------
In Section \[sec:discussion\_edpsb\] we discussed the differences between the ePSB and dPSB classes of the Balmer-strong galaxies. We argued that their discrepant natures are revealed in particular when considering their structural properties, such as the distributions of their S[é]{}rsic indices, as well as their optical colours. The redder colours of the dPSB compared with the ePSB can be inferred from their stacked spectra shown in Figure \[fig:SED\_stack1\]. Additionally, in Figure \[fig:sedfits\_dpsb\] we present examples of individual spectra of some dPSB and ePSB galaxies. These show the differences in the shapes of the continuum of the two classes more clearly, in part due to the linear scaling on the flux axis.
In Figure \[fig:gi\_w1w2\] we present the photometric colours of the dPSB and ePSB galaxies. In addition to SDSS $g-i$ colour we extend the analysis into the mid-infrared regime, using data from the WISE survey [@Wright+2010]. To obtain the W1 and W2 magnitudes (measured in regions centred on 3.4 and 4.6 $\mu$m, respectively) for our Balmer-strong galaxies we used the matched catalogue of 400 million SDSS sources with WISE forced photometry [@Lang+2016]. The W1-W2 colour can be used to probe the dust emission in galaxies, and therefore distinguish between galaxies with different dust-content (see eg. @Nikutta+2014). As expected, the SDSS $g-i$ colour places the Balmer-strong galaxies in between the star-forming and quiescent control-samples, in the so-called “green-valley” with the dPSB galaxies showing a clear tendency towards redder colours, compared with the ePSB galaxies. Moreover, the dPSB galaxies show a strong tendency to have redder $W1-W2$ colour compared with the other samples, which further supports their dusty nature. We note the impact of aperture bias here, which will lead to a difference between the photometric colours and spectroscopic shape and therefore spectral class. For example, the presence of a dust lane through the centre of the galaxy may cause a very red SDSS spectrum, and a “dPSB” classification, while the colours obtained from photometry may be more typical of the control samples.
The star-formation histories {#app:dpsb_sfh}
----------------------------
While discussing the properties of the dPSB galaxies we pointed out a difficulty with interpreting the star-formation histories, obtained from the spectral fitting for this class of galaxies (Section \[sec:res\_sfh\]). Consequently, we excluded these histories from our analysis in the main body of this paper and present them in Figure \[fig:sfh\_dpsb\]. It appears that the dPSB galaxies follow the history of star-formation characteristic of quiescent galaxies, in complete disagreement with their current star-formation activity, inferred based on their spectra featuring prominent emission-lines and Balmer absorption lines. This is likely a consequence of inaccurate spectral fits rather than a true property of the dPSB galaxies. The high dust content and/or its geometry in these objects might prevent the STARLIGHT code from obtaining reliable fits, leading to unrealistic results. Figure \[fig:sedfits\_dpsb\] shows examples of fits to the spectra of the dPSB and ePSB galaxies. Overall, we found that the residuals associated with the fits to the dPSB spectra are about twice as large as those from the fits to the other Balmer-strong galaxy classes.
The impact of the AGN on the structural parameters {#appendix:AGN}
==================================================
A bright centrally located source within a galaxy, such as an AGN, may significantly affect the measurements of some structural parameters. The extent of this effect will depend on the type of the AGN, the morphology of the host galaxy, the wavelength at which the measurement is performed and the nature of the measurement itself. @Pierce+2010 investigated the most extreme case and found that the measurements of the concentration index, $C$ and the $M_{20}$ parameter for quasar-host galaxies are almost certainly unreliable if the quasar contributes at least $20\%$ of the total galaxy light in the B-band. They found that the S[é]{}rsic and Gini indices ($n$, $G$) can also be affected by the emission from the quasar but to a lesser extent, and that the asymmetry parameter, $A$, remains reasonably unaffected, unless the quasar is offset from the galaxy centre. In our study we consider only narrow-line AGN, in which the continuum emission is not visible. For such galaxies, the effects on the structural parameters can only be due to strong narrow emission lines, and are therefore much less significant. @Kauffmann+2007 showed that nearby LINERS and Type II Seyfert galaxies are less likely to suffer significant contamination from the nuclear regions, although their study focused on quantifying the UV-optical colours rather than galaxy structure. Moreover, in our analysis focus primarily on the SDSS $r$-band images, in which we the continuum emission from the AGN should be less dominant, compared with the B-band images considered by @Pierce+2010. We therefore limit ourselves to the following simple check.
In Figure \[fig:AGN\_testmorph\] we compare the values of $n$, $C$, $A$, $G$ and $M_{20}$ measured in the $r$-band for a sample of quiescent galaxies hosting an AGN, selected using the same emission-line ratio cuts as for the agnPSB sample (Section \[sec:psbselection\]), with our control sample of quiescent galaxies. We also show the star-forming galaxy sample to better visualize the dynamic range of the measured parameters.
There is no significant difference between the quiescent AGN and non-AGN samples in terms of their $r$-band measured values of $C$, $G$, and $M_{20}$ but both are significantly different from the star-forming sample. The KS test results indicate the distributions of $n$ and $A$ are different, however in both cases there is no obvious tendency for the AGN sample to display either enhanced or diminished values of either of the measures, compared with the quiescent non-AGN sample. We conclude that the $r$-band measured structural parameters are not significantly affected by the emission from narrow-line AGN and can therefore be used for our purpose of comparison between galaxies which display such nuclear activity and those that do not.
    
Additional results from the image analysis of post-starburst galaxies {#appendix:struct}
=====================================================================
In Figures \[fig:morph\_CA\] and \[fig:morph\_GM20\] we show two further relations between the different morphological measures that we studied, ($C-A$ and $G-M_{20}$, respectively). These relations are often used in quantitative analysis of galaxy morphology and structure, and can help relate between the properties inferred for the post-starburst galaxies and those characteristic of the control star-forming and quiescent samples.
In Figure \[fig:morph\_nE5\] we show that there is no trend between morphology, as measured by the S[é]{}rsic index, and environment for the PSB samples.
 
 
 
Simulated post-mergers {#sec:mergers}
======================
The merger simulations used in this work are based on the N-body smoothed particle hydrodynamics (SPH) code [<span style="font-variant:small-caps;">Gadget-3</span>]{} [@Springel2005], with improved SPH implementation - SPHGal [@Hu+2014; @Eisenreich+2017]. The improved model includes the use of a pressure-entropy formulation of SPH, a Wendland $C^4$ kernel with $100$ neighbours along with an updated velocity gradient estimator, a modified viscosity switch with a strong limiter, artificial conduction of thermal energy and finally a time step limiter (see @Lahen+2017 for details). The cooling of the gas is metallicity-dependent [@Wiersma+2009] and takes into account an UV/X-ray background [@HaardtMadau2001]. In addition, the sub-resolution astrophysics models include models for star formation, stellar feedback with accompanying metal production, and metal diffusion [@Scannapieco+2005; @Scannapieco+2006; @Aumer+2013]. The stellar metal yields are adopted from @Iwamoto+1999 for SNIa, @WoosleyWeaver1995 for SNII and @Karakas2010 for AGB stars, with both energy and metals being released at rates dependent on the age of the stellar particles and the distance to the neighbouring gas particles (see e.g. @Nunez+2017).
Galaxy models {#sec:mergers_models}
-------------
The galaxies were modelled according with the $\Lambda$CDM cosmology ($\Omega_{m}$ = 0.30, $\Omega_{\Lambda}$ = 0.71 and H$_{0}$ =70kms$^{-1}$Mpc$^{-1}$), to resemble the galaxies observed in the local Universe. The physical properties of the galaxies were derived from the parameters describing their host dark matter halos, including the virial mass and velocity ($v_{vir}=160$ km/s, $ M_{vir}=v_{vir}^3/10GH_{0} = 1.34 \times10^{12}M\sun$), with each galaxy consisting of a Hernquist dark matter (DM) halo with mass a $M_\mathrm{DM}=1.286\times 10^{12} \ M_{\sun}$ and a concentration parameter of $c=9$ [@Hernquist1990]. The baryonic mass fraction was set at $m_\mathrm{b}=0.041$ and each galaxy was set to have a two-component structure, including a bulge and a disk, characterised by an exponential light profile. The scale lengths of the components were determined from the conservation of the angular momentum of the system (with the spin-parameter set to $\lambda=0.033$; for details see @Springel+2005 [@Johansson+2009a; @Johansson+2009b]).
In order to produce systems representative of the local star-forming population we chose three types of morphologies, including Sa, Sc and Sd galaxies, characterised by the following parameters:
- Sa: $B/T=0.5$, $r_{disc}=3.75$ kpc, $f_{gas}=0.17$
- Sc: $B/T=0.3$, $r_{disc}=3.79$ kpc, $f_{gas}=0.22$
- Sd: $B/T=0.1$, $r_{disc}=3.85$ kpc, $f_{gas}=0.31$
where $B/T$ is the stellar bulge-to-total mass ratio, $r_{disk}$ is the stellar disk scale length and $f_{gas}$ is the disc gas fraction. In each case the bulge scale length and the stellar disk scale height are to equal to $0.2\times r_{disc}$. The gaseous discs were set to be in hydrostatic equilibrium [@Springel+2005] with gaseous disc scale lengths that equal the stellar disc scale lengths.
Each galaxy consists of $4\times 10^5$ DM particles with a mass resolution per particle of $m_\mathrm{DM}\approx 3.2\times 10^6 \ M_{\sun}$ and $4\times 10^5$ baryonic particles divided according to the mass fraction of each component, as to obtain a mass resolution of $m_\mathrm{b}\approx 1.4 \times 10^5 \ M_{\sun}$ per baryonic particle.
To be fully consistent with the employed sub-resolution models, we have adopted initial metallicity and age distributions for the gaseous and stellar particles in a fashion similar to the one presented in @Lahen+2017. Here we adopted a Milky Way-like metallicity gradient of $0.0585$ dex$/$kpc [@Zatitsky+1994] and abundances as in [@Adelman+1993; @Kilian-Montenbruck+1994], yielding roughly solar total metallicities for all galaxies. The initial star formation rates were set iteratively as SFR$_\mathrm{Sa}=1 \ M_{\sun}/$yr, SFR$_\mathrm{Sc}=2 \ M_{\sun}/$yr and SFR$_\mathrm{Sd}=5 \ M_{\sun}/$yr which define, together with the simulation start time, the initial ages of the stellar particles to be used in stellar feedback within [<span style="font-variant:small-caps;">Gadget-3</span>]{}. Additionally, as the galaxies include up to $\sim30\%$ of the disc mass in gas, the galaxies were relaxed by running each galaxy in isolation for $500$ Myr before setting up the actual merger.
Major merger simulations {#sec:mergers_sims}
------------------------
The merger simulations were set up set up on collisional trajectories with the following initial orbital configurations, with $i_{1}$ and $i_{2}$ being the angles of inclination of the galactic disks with respect to the orbital planes and $\omega_{1}$ and $\omega_{2}$, the arguments of the orbits’ pericentres[^10] (see @NaabBurkert2003):
- G00 - a symmetric configuration - in-plane prograde-prograde orbits with the angular momenta of the galaxies aligned with the orbital angular momentum ($i_{1}=0^{o}$,$i_{2}=0^{o}$,$\omega_{1}=0^{o}$,$\omega_{2}=0^{o}$);
- G07 - retrograde-prograde orbits, with both galactic disks inclined with respect to the orbital planes ($i_{1}=-109^{o}$,$i_{2}=71^{o}$,$\omega_{1}=-60^{o}$,$\omega_{2}=-30^{o}$)
- G13 - retrograde-retrograde orbits with one galaxy inclined and other coplanar with respect to the orbital plane ($i_{1}=-109^{o}$,$i_{2}=180^{o}$,$\omega_{1}=60^{o}$,$\omega_{2}=0^{o}$).
Each simulation started at an initial separation given by the mean of the virial radii of the progenitors (160 kpc/h $\sim$225 kpc) and the pericentric separation being the sum of the disc scale lengths ($\sim$2$\times$2.7 kpc/h $\sim$2$\times$3.8 kpc). The galaxies approached each other following nearly parabolic orbits and, in each case, the evolution of the interacting system was modelled along the the whole merger sequence including the pre-merger stages and the post-coalescence evolution of the remnant (total time of 3 Gyr).
The mock image and spectral synthesis {#appendix:mergers_img}
-------------------------------------
From the positions, ages and star formation histories of the gas and star particles recorded every 0.02 Gyr we created mock images and integrated spectra of the galaxies that could be compared with the real SDSS imaging and spectroscopic data. Using stellar population synthesis models [@BruzualCharlot2003] we assigned a spectral energy distribution (SED) to each particle, based on the information about its age and past star formation history. For the images, these were convolved with the SDSS filter response functions. A simple two-component dust-screen model was used to mimic the attenuation effects from the interstellar medium, assuming an effective optical depth $\tau_V=1.0$ and power law slopes of 0.7 and 1.3 for the diffuse and birth cloud dust respectively (see @daCuhna+2008 and @Wild+2007 for details).
For each time step, we projected particle positions onto a two-dimensional grid, matching the SDSS image resolution (with 0.396 arc seconds per pixel) with a distance to the interacting galaxies corresponding to $z=0.04$, as viewed from six different directions characterised by angles of rotation in a three-dimensional coordinate system: $\alpha$ - around the $x$-axis, $\beta$ - around the $y$-axis and $\gamma$ - around the $z$-axis. They include a face-on orientation ($0$,$0$,$0$), a randomly chosen orientation ($\alpha$,$\beta$,$\gamma$) with each angle between 0 and 2$\pi$, and further five following orientations: ($\alpha$+$\pi$/2,$\beta$,$\gamma$), ($\alpha$+$\pi$,$\beta$,$\gamma$), ($\alpha$+$3\pi$/2,$\beta$,$\gamma$), ($\alpha$,$\beta$+$\pi$/2,$\gamma$) and ($\alpha$,$\beta$+3$\pi$/2,$\gamma$).
An image in a given passband and orientation was created by summing up the luminosities of the particles within each pixel, integrated from the SED convolved with a given filter function. The luminosities were converted to the flux at a distance corresponding to $z=0.04$, calculated within the same cosmological framework as that assumed in the simulations. The flux, $f$, was calibrated to the AB magnitude system, where the zero-point flux density in a given passband is equal to $f_{0}=3631$ Jansky: $$m_{AB} = -2.5\times \mbox{log}_{10}(f/f_{0})$$ The magnitudes were then converted to SDSS counts using the *asinh* magnitude definition [@Lupton+1999]: $$\mbox{counts} = t_{exp}\times f/f_{0}\times 10^{(-0.4(m_{ZP} + kA))},$$ $$f/f_{0} = 2b \times \mbox{sinh}\left(-m_{AB}\left(\mbox{ln}(10)/2.5\right)-\mbox{ln}(b)\right),$$ where $b$ is the ‘softening parameter’ set to $\sim1\sigma$ of the sky noise, $t_{exp}$ is the exposure time, $m_{ZP}$ is the photometric zero-point in the given passband, $k$ is the extinction coefficient and $A$, the airmass at the given position. More information about the photometric calibration can be found at <http://classic.sdss.org/dr7/algorithms/fluxcal.html>.
The obtained idealised images were then convolved with a PSF, built by summing two Gaussian functions with widths and relative maxima matching those characteristic for SDSS images, to mimic the effects of astronomical seeing and camera response on the images. The image synthesis was finalised by addition of Gaussian random noise, with the standard deviation matching the typical error in the photoelectron counts in the SDSS images.
The instantaneous star formation rate output by Gadget was used to estimate the $H\alpha$ line luminosity using the conversion of @K98, and the integrated spectra provided an estimate of the stellar continuum in order to calculate the equivalent width of the $H\alpha$ emission line. To account for the selective extinction by dust of the lines over the stellar continua, the equivalent widths were reduced by $\sim2$.
\[lastpage\]
[^1]: <http://www.mpa-garching.mpg.de/SDSS>
[^2]: <http://www-star.st-and.ac.uk/~vw8/downloads/DR7PCA.html>
[^3]: We defined the projected axis ratio using two SDSS parameters: *expAB* and *devAB* (axis ratios from exponential and deVaucouleurs fits, respecively), measured in the $r$-band. Through careful visual inspection of the galaxy images, we found that the value of $0.32$ works well for isolating the ‘edge-on’ objects and, therefore, we required both parameters to have values above that limit. Assuming that the galaxies have a characteristic intrinsic axis ratio of 0.2, the measured value 0.32 corresponds to 75$^{o}$ inclination.
[^4]: <http://www.sdss.jhu.edu/skypca/spSpec/>
[^5]: XMILESS 2016, available from G. Bruzual on request
[^6]: Computed using the 1D surface-brightness profiles defined by circular apertures.
[^7]: Here we use the growth curve radii enclosing 20$\%$ and 80$\%$ of the total light.
[^8]: The spectroscopic measure is taken as the lower limit because it is expected to underestimate the galaxy number density due to the incompleteness in the spectroscopic redshift data, while the photometric measure will overestimate the true value due to projection effects and is therefore taken as the upper limiting value.
[^9]: In 8 out of 9 cases this estimate is limited by the length of the simulation and it is likely that the ePSB phase will prevail for longer. In the single case where the ePSB phase clearly ends before the end of the simulation, the timescale of visibility is $\sim$0.8Gyr.
[^10]: The angular distance between the pericentre and the ascending node, defined as the point of intersection of the plane of the disk with that of the orbit, where the rotation proceeds from ‘south’ to ‘north’ with respect to the orbital plane.
|
---
abstract: 'A class of stochastic optimal control problems containing optimal stopping of controlled diffusion process is considered. The numerical solutions of the corresponding normalized Bellman equations are investigated. Methods of [@krylov:rate:lipschitz] are adapted. The rate of convergence of appropriate finite difference difference schemes is estimated.'
author:
- 'István Gyöngy, David Šiška'
bibliography:
- 'bibliography.bib'
title: 'On the rate of convergence of finite-difference approximations for normalized Bellman equations with Lipschitz coefficients'
---
Introduction
============
Stochastic optimal control and optimal stopping problems have many applications in mathematical finance, portfolio optimization, economics and statistics (sequential analysis). Optimal stopping problems can be in some cases solved analytically [@shiryaev:statistical]. With most problems, one must resort to numerical approximations of the solutions. One approach is to use controlled Markov chains as an approximation to the controlled diffusion process, see e.g. [@menaldi:some:estimates]. A thorough account of this approach is available in [@kushner:dupuis:numerical].
We are interested in the rate of convergence of finite difference approximations to the payoff function of optimal control problems when the reward and discounting functions may be unbounded in the control parameter. This allows us to treat numerically the optimal stopping of controlled diffusion processes by randomized stopping, i.e. by transforming the optimal stopping into a control problem, see [@krylov:controlled]. This leads us to approximating a normalized degenerate Bellman equation.
Until quite recently, there were no results on the rate of convergence of finite difference schemes for degenerate Bellman equations. A major breakthrough is achieved in Krylov [@krylov:rate:equations] for Bellman equations with constant coefficients, followed by rate of convergence estimates for Bellman equations with variable coefficients in [@krylov:approximating:value] and [@krylov:rate:variable]. The estimate from [@krylov:rate:variable] is improved in [@barles:jakobsen:error:bounds] and [@barles:jakobsen:rate:hamilton]. Finally, Krylov [@krylov:rate:lipschitz] (published in [@krylov:rate:lipschitz:published]) establishes the rate of convergence $\tau^{1/4}+ h^{1/2}$ of finite difference schemes to degenerate Bellman equations with Lipschitz coefficients, where $\tau$ and $h$ are the mesh sizes in time and space respectively.
In the present paper we extend this estimate to a class of normalized degenerate Bellman equations which contains those for optimal stopping of controlled diffusion processes with variable coefficients. Adapting ideas and techniques of [@krylov:rate:lipschitz] we obtain the rate of convergence $\tau^{1/4} + h^{1/2}$, as in [@krylov:rate:lipschitz]. The main ingredient of the proof is a gradient estimate for the solution to the discrete normalized Bellman PDE. This is an extension of the gradient estimate from [@krylov:rate:lipschitz] to our case. After the first version of this article was sent to arxiv, we received [@krylov:apriori] where an essentially more general gradient estimate is proved. This opens the way to proving the same rate of convergence result for normalized Bellman PDEs in more general setting then we present below.
Rate of converge results for optimal stopping are proved for general consistent approximation schemes in [@jakobsen:rate:optimal:stopping]. However, the rate $\tau^{1/4}+ h^{1/2}$ is obtained only when the diffusion coefficients are independent of the time and space variables. For further results on numerical approximations for Bellman equations we refer to [@jakobsen:karlsen:convergence:source:terms], [@jakobsen:karlsen:chioma:error] and [@biswas:jakobsen:karlsen:error].
The paper is organized as follows. The main result is formulated in the next section. In section \[section-existence-of-soln-to-disc-prob\] the existence and uniqueness of the solution to finite difference schemes is proved together with a comparison result. The main technical result, the gradient estimate of solutions to finite difference schemes, is obtained in section \[section-gradient-estimate\]. Some useful analytic properties of payoff functions are presented in section \[section-analytic-ppties\]. Finally, the main result is proved in section \[section-shaking\].
The Main Result {#section-main-result}
===============
Fix $T\in[0,\infty)$. Let $(\Omega, {\mathcal{F}}, {\mathbb{P}}, ({\mathcal{F}}_t)_{t\geq0})$ be a probability space with a right-continuous filtration, such that ${\mathcal{F}}_0$ contains all ${\mathbb{P}}$ null sets. Let $(w_t,{\mathcal{F}}_t)$ be a $d'$ dimensional Wiener martingale. Let $A$ be a separable metric space. For every $t \in [0,T]$, $x \in \R^d$ and $\alpha
\in A$ we are given a $d\times d'$ dimensional matrix $\sigma^\alpha(t,x)$, a $d$ dimensional vector $\beta^\alpha(t,x)$ and real numbers $c^\alpha(t,x)$, $f^\alpha(t,x)$ and $g(x)$.
\[ass-for-optimal-control-problem\] $\sigma, \beta, c, f$ are Borel functions of $(\alpha, t, x)$. The function $g$ is continuous in $x$. There exist an increasing sequence of subsets $A_n$ of $A$, and positive real constants $K, K_n$, and $m, m_n$, such that $\bigcup_{n\in {\mathbb{N}}} A_n=A$ and for each $n \in {\mathbb{N}}$, $\alpha \in A_n$, $$\label{eqn-estimate-of-ass-for-optimal-control-problem}
\begin{split}
|\sigma^\alpha(t,x)
- \sigma^\alpha(t,y)|
+ |\beta^\alpha(t,x) - \beta^\alpha(t,y)| & \leq
K_n|x-y|,\\ |\sigma^\alpha(t,x)| + |\beta^\alpha(t,x)| & \leq
K_n(1+|x|),\\ |c^\alpha(t,x)| + |f^\alpha(t,x)| \leq K_n(1+|x|)^{m_n},
\quad |g(x)| & \leq K(1+|x|)^m
\end{split}$$ for all $x \in \R^d$ and $t\in [0,T]$.
We say that $\alpha \in \mathfrak{A}_n$ if $\alpha = (\alpha_t)_{t\geq 0}$ is a progressively measurable process with values in $A_n$. Let $\mathfrak{A} = \bigcup_{n\in {\mathbb{N}}} \mathfrak{A}_n$. Then under Assumption \[ass-for-optimal-control-problem\] it is well known that for each $s
\in [0,T]$, $x\in \R^d$ and $\alpha \in {\mathfrak{A}}$ there is a unique solution $\{x_t:t\in[0,T-s]\}$ of $$\label{eqn-sde}
x_t = x + \int_0^t \sigma^{\alpha_u}(s+u,x_u)dw_u
+ \int_0^t \beta^{\alpha_u}(s+u,x_u)du,$$ denoted by $x^{\alpha,s,x}_t$. For $s\in[0,T]$ we use the notation ${\mathfrak{T}}(T-s)$ for the set of stopping times $\tau\leq T-s$. Define the payoff function to the optimal stopping and control problem as $$\label{eq:stop-payofffn}
w(s,x)=\sup_{\alpha\in\mathfrak{A}}\sup_{\tau\in{\mathfrak{T}}(T-s)}
v^{\alpha,\tau}(s,x),$$ where $$v^{\alpha,\tau}(s,x)=\E^\alpha_{s,x}
\left[\int_0^{\tau}f^{\alpha_t}(s+t,x_t)
e^{-\varphi_t}dt + g(x_{\tau})e^{-\varphi_{\tau}}
\right],$$ $$\varphi_t
= \int_0^t c^{\alpha_r}(s+r,x_r)dr,$$ and $\E^\alpha_{s,x}$ means expectation of the expression behind it, with $x^{\alpha,s,x}_t$ in place of $x_t$ everywhere. It is worth noticing that for $$w_n(s,x):=\sup_{\alpha\in\mathfrak{A}_n}\sup_{\tau\in{\mathfrak{T}}(T-s)}
v^{\alpha,\tau}(s,x)$$ we have $w_n(s,x)\uparrow w(s,x)$ as $n \to \infty.$ By Theorem 3.1.8 in [@krylov:controlled], $w_n(s,x)$ is bounded from above and below. Hence $w(s,x)$ is bounded from below. However it can be equal to $+\infty$.
Let ${\mathfrak{R}}_n$ contain all progressively measurable, locally integrable processes $r = (r_t)_{(t \geq 0)}$ taking values in $[0,n]$ such that $\int_0^\infty r_t dt
= \infty$. Let ${\mathfrak{R}}= \bigcup_{n
\in {\mathbb{N}}}{\mathfrak{R}}_n$.
We will state without proof a result about randomized stopping. The proof (taking into account the possibility that $g$ is a function of $(t,x)$, continuous in $(t,x)$) can be found in [@siska:gyongy:on:randomized]. If $A=A_n$, $K=K_n$, $m=m_n$ for $n\geq1$ then the following theorem is known from [@krylov:controlled] (see Exercise 3.4.12, via Lemma 3.4.3(b) and Lemma 3.4.5(c)).
\[thm-opt-stop-and-cont-equiv\] Let Assumption \[ass-for-optimal-control-problem\] hold. Then for all $(s,x) \in [0,T] \times \R^d$, either both $w(s,x)$ and $$\begin{split}\label{eq:cont-w-rnd-st}
\sup_{\alpha \in {\mathfrak{A}}}&\sup_{r \in {\mathfrak{R}}}
\E_{s,x}^\alpha \bigg\{\int_0^{T-s}
\big[f^{\alpha_t}(s+t,x_t)e^{-\varphi_t} \\
&+ r_tg(x_t)e^{-\varphi_t}\big] e^{- \int_0^t r_u du} dt
+ g(x_{T-s})e^{-\varphi_{T-s} -
\int_0^{T-s} r_u du} \bigg\}
\end{split}$$ are finite and equal, or they are both infinite.
This theorem is the main tool which allows us to treat the problem of optimal stopping of a controlled process as just an optimal control problem. The aim of this paper is to find the rate of convergence for numerical approximations to the following payoff function: $$\begin{aligned}
\label{eq:payofffn}
v(s,x) & = \sup_{\alpha\in\mathfrak{A}} v^\alpha(s,x), \\
\label{eq:payofffnb}
v^{\alpha}(s,x) & = \E^\alpha_{s,x}
\left[\int_0^{T-s}f^{\alpha_t}(s+t,x_t)
e^{-\varphi_t}dt + g(x_{T-s})e^{-\varphi_{T-s}} \right],\end{aligned}$$ where $x_t$ is the solution to (\[eqn-sde\]). Notice that (\[eq:cont-w-rnd-st\]) can be written in this form such that Assumption \[ass-for-optimal-control-problem\] is satisfied.
The immediate aim now is to find an approximation scheme for the payoff function. From [@krylov:controlled] we know that under some assumptions (stricter then Assumptions \[ass-for-optimal-control-problem\]) the payoff function (\[eq:payofffn\]) satisfies the normalized Bellman equation $$\label{eqn-nbpde}
\begin{split}
\sup_{\alpha \in A} m^\alpha\left(u_t + {{\operatorname{L}}^\alpha}u + f^\alpha \right) & = 0 {\quad \textrm{on} \quad}[0,T)\times \R^d\\
u(T,x) & = g(x) {\quad \text{for} \quad}x \in \R^d,
\end{split}$$ where $${{\operatorname{L}}^\alpha}u = \sum_{i,j} a_{i,j}^\alpha u_{x_i, x_j} + \sum_i \beta_i^\alpha u_{x_i} - c^\alpha u,$$ $a_{ij}^\alpha = 1/2 (\sigma^\alpha \sigma^{\alpha^*})_{ij}$ and $m^\alpha$ is a non-negative function of the control parameter called the normalizing factor. A crucial property of $m^\alpha$ is that $m^\alpha \sigma^\alpha_{ij}$, $m^\alpha b^\alpha_i$, $m^\alpha c^\alpha$ and $m^\alpha f^\alpha$ are bounded as functions $\alpha$. Hence it is natural to approximate $v$ by approximating the solution of the normalized Bellman equation. In this paper we show that the finite difference approximation to the solution of the normalized Bellman PDE converges to $v$. Moreover we find that the rate of convergence is the same as in [@krylov:rate:lipschitz]. It is worth noticing that to show this we don’t need any result about the solvability of the Bellman equation. We now describe the approximation scheme. From now on let $K \geq 1$ be a fixed constant.
\[ass-on-the-scheme\] There exist a natural number $d_1$, vectors $\ell_k \in \R^d$ and functions $$\sigma^\alpha_k:[0,T)\times \R^d \to \R, \quad b^\alpha_k:[0,T)\times \R^d \to \R, \quad \forall k = \pm 1, \ldots, d_1$$ such that $\ell_k = \ell_{-k}$, $|\ell_k| \leq K$, $\sigma^\alpha_k = \sigma^\alpha_{-k}, b^\alpha_k \geq 0$ for $k = \pm 1, \ldots, d_1$ and $$\sigma^\alpha_{ij}(t,x) = \sigma^\alpha_k(t,x) \ell_k^i \ell_k^j, \quad \beta_{i}^\alpha(t,x) = b_k^\alpha(t,x) \ell_k^i$$ for any $\alpha \in A$ and $(t,x) \in [0,T)\times \R^d$.
This might appear to be a restrictive assumption. Actually, all operators ${{\operatorname{L}}^\alpha}$ which admit monotone finite difference approximations satisfy this. See [@dong:krylov:rate:constant]. Fix $\tau > 0$, $h > 0$ and define the grid $$\begin{split}
\bar{\mathcal{M}}_T := \{(t,x) \in [0,T]\times \R^d : (t,x) = ((j\tau) \wedge T, h(i_1\ell_1 + \ldots + i_{d_1} \ell_{d_1})), \\ j \in \{0\} \cup {\mathbb{N}}, i_k \in {\mathbb{Z}}, k = \pm 1, \ldots, \pm d_1\}.
\end{split}$$ Let $\tau_T(t) := \tau$ for $t \leq T-\tau$ and $\tau_T(t) := T-t$ for $t > T-\tau$. So $t+\tau_T(t) = (t+\tau) \wedge T$. Let $Q$ be a non-empty subset of $${\mathcal{M}_T}:= {\bar{{\mathcal{M}_T}}}\cap \left([0,T) \times \R^d\right).$$ Let $\operatorname{T}_{\tau} u (t,x) = u(t + \tau_T(t),x)$, ${\operatorname{T}_{h,\ell_k}}u(t,x) = u(t, x + h_k \ell_k)$, $${\operatorname{\delta}_\tau}u(t,x) = \frac{u(t+\tau_T(t),x) - u(t,x)}{\tau_T(t)},$$ $${\operatorname{\delta}_{h_k,\ell_k}}u(t,x) = \frac{u(t,x+h_k\ell_k) - u(t,x)}{h_k} {\quad \textrm{and} \quad}{\operatorname{\Delta}_{h_k,\ell_k}}u = - {\operatorname{\delta}_{h_k,\ell_k}}\operatorname{\delta}_{h_k,-\ell_k} u.$$ Let $\operatorname{\delta}_{0,\ell} := 0$. Let $a_k^\alpha := (1/2)(\sigma_k^\alpha)^2$. Consider the following finite difference problem: $$\sup_{\alpha \in A} m^\alpha \left({\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u + f^\alpha \right) = 0 \quad \textrm{on}\quad Q, \label{eqn-f-d-problem-a}$$ $$u = g {\quad \textrm{on} \quad}\bar{\mathcal{M}}_T\setminus Q, \label{eqn-f-d-problem-b}$$ where $${{\operatorname{L}}_h^\alpha}u = \sum_k a_k^\alpha {\operatorname{\Delta}_{h_k,\ell_k}}u + \sum_k b_k^\alpha {\operatorname{\delta}_{h_k,\ell_k}}u - c^\alpha u$$ and $m^\alpha$ is a positive function of $\alpha \in A$ such that the following conditions hold.
\[ass-for-the-existence-of-soln-to-discrete-problem\] Let $0 \leq \lambda < \infty$. Functions $\sigma^\alpha_k$, $b^\alpha_k$, $f^\alpha$, $c^\alpha \geq \lambda$, $g$ are Borel in $t$ and continuous in $\alpha \in A$ for each $k = \pm 1, \ldots, d_1$. For any $t \in [0,T]$, $x,y \in \R^d$ and $\alpha \in A$: $$\label{eqn-boundedness-of-diff-and-drift}
\begin{split}
|\sigma^\alpha_k(t,x) - \sigma^\alpha_k(t,y)| + |b^\alpha_k(t,x) - b^\alpha_k(t,y
)| + |c^\alpha(t,x) - c^\alpha(t,y)| \\ + m^\alpha|f^\alpha(t,x) - f^\alpha(t,y)| + |g(x) - g(y)| \leq K|x-y|,\\
|\sigma^\alpha_k| + b^{\alpha}_k + m^\alpha |f^\alpha| + m^\alpha c^\alpha + |g| \leq K
\end{split}$$ and $$\label{eqn-ass-m-c}
\frac{1}{K} \leq m^\alpha\left[1 + c^\alpha(t,x)\right].$$
Notice that if Assumptions \[ass-on-the-scheme\] and \[ass-for-the-existence-of-soln-to-discrete-problem\] are satisfied then (\[eqn-estimate-of-ass-for-optimal-control-problem\]) is also satisfied. On top of Assumptions \[ass-for-optimal-control-problem\], \[ass-on-the-scheme\] and \[ass-for-the-existence-of-soln-to-discrete-problem\], the following is required to obtain the rate of convergence.
\[ass-for-main-result\] For all $\alpha \in A$, $x \in \R^d$ and $t,s \in [0,T]$ $$\begin{split}
|\sigma^\alpha(s,x) - \sigma^\alpha(t,x)| + |b^\alpha(s,x) - b^\alpha(t,x)| \\ + |c^\alpha(s,x) - c^\alpha(t,x)| & \leq K|t-s|^{1/2}, \\
m^\alpha|f^\alpha(s,x) - f^\alpha(t,x)| & \leq K|t-s|^{1/2}.
\end{split}$$
The following theorem is the main result of the paper.
\[theorem-the-main-result\] Let Assumptions \[ass-for-optimal-control-problem\], \[ass-on-the-scheme\], \[ass-for-the-existence-of-soln-to-discrete-problem\] and \[ass-for-main-result\] be satisfied. Let $v$ be the payoff function (\[eq:payofffn\]). Then (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) with $Q = {\mathcal{M}_T}$ has a unique bounded solution ${v_{\tau,h}}$ and $$|v - {v_{\tau,h}}| \leq N_T(\tau^{1/4} + h^{1/2}) {\quad \textrm{on} \quad}{\mathcal{M}_T},$$ where $N_T$ is a constant depending on $K,d,d_1,T, \lambda$.
We briefly outline how this is proved in Section \[section-shaking\]. Following [@krylov:rate:lipschitz] we “shake” the finite difference scheme, we smooth the corresponding solution to get a supersolution of the normalized Bellman PDE . Hence we obtain the estimate $v \leq {v_{\tau,h}}+ N(\tau^{1/4} + h^{1/2})$ by using a comparison of $v$ with supersolutions to the normalized Bellman PDE. To get ${v_{\tau,h}}\leq v + N(\tau^{1/4} + h^{1/2})$ we use the same approach with the roles of $v$ and ${v_{\tau,h}}$ interchanged. We “shake” the optimal control problem (\[eq:payofffn\])-(\[eq:payofffnb\]), again following [@krylov:rate:lipschitz]. We smooth the resulting payoff function to obtain a supersolution of the finite difference scheme. We get the estimate by using a comparison theorem established for the finite difference scheme.
An important consequence of Theorem \[theorem-the-main-result\] is that one gets the same rate of convergence for the optimal stopping and control problem.
Let $w$ be the payoff function for the optimal stopping and control problem (\[eq:stop-payofffn\]). If Assumptions \[ass-for-optimal-control-problem\], \[ass-on-the-scheme\], \[ass-for-the-existence-of-soln-to-discrete-problem\] and \[ass-for-main-result\] are satisfied with some positve function $m^\alpha$, then $$\label{eqn-discrete-bellman-pde-for-opt-stop}
\begin{split}
\sup_{\alpha \in A, r \geq 0} \left(m^\alpha\wedge\frac{1}{1+r}\right) \left[{\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u - r u + f^\alpha +r g \right] = 0 \quad \textrm{on}\quad \mathcal{M}_T, \\
u = g {\quad \textrm{on} \quad}\bar{\mathcal{M}}_T\setminus \mathcal{M}_T,
\end{split}$$ has a unique bounded solution ${w_{\tau,h}}$ and $$|w - {w_{\tau,h}}| \leq N_T(\tau^{1/4} + h^{1/2}) {\quad \textrm{on} \quad}{\mathcal{M}_T},$$ where $N_T$ is a constant depending on $K,d,d_1,T, \lambda$.
First we use Theorem \[thm-opt-stop-and-cont-equiv\] to see that $w(s,x)$ given by (\[eq:stop-payofffn\]) is equal to (\[eq:cont-w-rnd-st\]). Consider a metric space $\bar{A} := A \times [0,\infty)$, with any metric defining the product topology. Let $\bar{A}_n = A_n \times [0,n]$. Then $\bar{A} = \bigcup_{n \in {\mathbb{N}}}\bar{A}_n$. Let $\bar{{\mathfrak{A}}}_n$ denote the set of progressively measurable processes taking values in $\bar{A}_n$. Clearly $$w(s,x) = \sup_{\bar{\alpha} \in \mathfrak{\bar{A}}}\E^{\bar{\alpha}}_{s,x}
\left[\int_0^{T-s}\bar{f}^{\bar{\alpha}_t}(s+t,x_t)
e^{-\bar{\varphi}_t}dt + g(x_{T-s})e^{-\bar{\varphi}_{T-s}} \right],$$ where $$\bar{{\mathfrak{A}}} = \bigcup_{n \in {\mathbb{N}}} \bar{{\mathfrak{A}}}_n, \quad \bar{f}^{\bar{\alpha}} := f^\alpha + rg,\quad \bar{c}^{\bar{\alpha}} = r + c^\alpha, \quad \bar{\varphi}_t = \int_0^t \bar{c}^{\bar{\alpha}_u}(u,x_u)du$$ and $x_t$ is the solution to (\[eqn-sde\]). Notice that we are still using the same $\sigma^\alpha$, $\beta^\alpha$. We can check that Assumptions \[ass-for-optimal-control-problem\], \[ass-on-the-scheme\], \[ass-for-the-existence-of-soln-to-discrete-problem\] and \[ass-for-main-result\] hold with $\bar{A}$, $$\bar{m}^{\bar{\alpha}} = \bar{m}^{(\alpha,r)} := m^\alpha \wedge \frac{1}{1+r},$$ $\bar{c}^\alpha$ and $\bar{f}^\alpha$ in place of $A$, $m^\alpha$, $f^\alpha$ and $c^\alpha$ respectively. Finally we apply Theorem \[theorem-the-main-result\] to $w$ and the scheme .
The system (\[eqn-discrete-bellman-pde-for-opt-stop\]) is equivalent to: $$\label{pde-ineq-for-opt-stop}
\begin{split}
\sup_{\alpha \in A}m^\alpha \left[{\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u + f^\alpha \right] & < 0, \quad g - u = 0 {\quad \textrm{on} \quad}{\mathcal{M}_T},\\
\sup_{\alpha \in A}m^\alpha \left[{\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u + f^\alpha \right] & = 0, \quad g - u \leq 0 {\quad \textrm{on} \quad}{\mathcal{M}_T}\\
{\quad \textrm{and} \quad}u & = g {\quad \textrm{on} \quad}\bar{\mathcal{M}}_T\setminus {\mathcal{M}_T}.
\end{split}$$
Let ${\varepsilon}= \tfrac{1}{1+r}$ in (\[eqn-discrete-bellman-pde-for-opt-stop\]) and take the supremum over ${\varepsilon}\in [0,1]$. Hence (\[eqn-discrete-bellman-pde-for-opt-stop\]) can be rewritten as $$\begin{split}
\sup_{ {\varepsilon}\in [0,1]}\left[ {\varepsilon}\sup_{\alpha \in A} m^\alpha \left( {\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u + f^\alpha \right) + (1-{\varepsilon})(g-u) \right] & = 0 {\quad \textrm{on} \quad}{\mathcal{M}_T},\\
u & = g {\quad \textrm{on} \quad}{\bar{{\mathcal{M}_T}}}\setminus {\mathcal{M}_T}.
\end{split}$$ Notice that the supremum over ${\varepsilon}$ is achieved by either taking ${\varepsilon}= 0$ or ${\varepsilon}= 1$. Hence it can be seen that this is equivalent to (\[pde-ineq-for-opt-stop\]).
On the solutions of the finite difference scheme {#section-existence-of-soln-to-disc-prob}
================================================
First we give two simple examples which justify the condition (\[eqn-ass-m-c\]).
Consider $A=[0,\infty)$, $m^\alpha = (1+\alpha)^{-1}$ and the equation $$\sup_{\alpha \in A} m^\alpha \left({\operatorname{\delta}_\tau}u \right) = 0 {\quad \textrm{on} \quad}{\mathcal{M}_T}$$ with the terminal condition $u = 1$ on ${\bar{{\mathcal{M}_T}}}\setminus {\mathcal{M}_T}$. If $u:{\mathcal{M}_T}\to \R$ is any non-increasing function in $t$, then $m^\alpha \operatorname{\delta}_\tau u \leq 0$. Hence, letting $\alpha \to \infty$, we see that $u$ satisfies the equation. Consequently the solution to the above problem is not unique.
Let $f^\alpha = 1 + \alpha$. Consider now the equation $$\sup_{\alpha \in A}m^\alpha(\operatorname{\delta}_\tau u + f^\alpha) = \sup_{\alpha \in A}m^\alpha\operatorname{\delta}_\tau u + 1 = 0 {\quad \textrm{on} \quad}{\mathcal{M}_T}.$$ If $u$ is a solution then we have $m^\alpha \operatorname{\delta}_\tau u \leq 0$. Hence $\sup_{\alpha \in A}m^\alpha \operatorname{\delta}_\tau u = 0$, which contradicts the equation. Thus the above equation has no solution.
Let Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\] be satisfied throughout the remainder of this section.
\[lemma-existence-of-soln-of-discrete-problem\] There is a unique bounded solution of the finite difference problem (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) on ${\mathcal{M}_T}$.
Let $\gamma = (0,1)$ and define $\xi$ recursively as follows: $\xi(T) = 1$, $\xi(t) = \gamma^{-1} \xi(t + {\tau_T(t)})$ for $t < T$. Then for any function $v$ $${\operatorname{\delta}_\tau}(\xi v) = \gamma \xi {\operatorname{\delta}_\tau}v - \nu \xi v, {\quad \text{where} \quad}\nu = \frac{1-\gamma}{\tau}.$$ To solve (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) for $u$, we could equivalently solve the following for $v$, with $u = \xi v$: $$\label{eqn-proxy-for-eqn-f-d-problem}
v = {\mathbf{1}}_{\bar{\mathcal{M}}_T\setminus Q}\frac{1}{\xi}g + {\mathbf{1}}_{Q}G[v],$$ where for any $\varepsilon > 0$, $$G[v] := v + \varepsilon \xi^{-1} \sup_{\alpha}
m^\alpha \left( {\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u + f^\alpha \right).$$ Then, using the convention that repeated indices indicate summation and always summing up before taking the supremum, $$G[v] = \sup_\alpha \left[ p^\alpha_\tau \operatorname{T}_\tau v + p^\alpha_k \operatorname{T}_{h,l_k} v +
p^\alpha v + \varepsilon m^\alpha (\operatorname{T}_\tau \xi^{-1})f^\alpha \right],$$ with $$p^\alpha_\tau = \varepsilon \gamma \tau^{-1} m^\alpha \geq 0, \quad p^\alpha_k = \varepsilon(2 h^{-2}a^\alpha_k + h^{-1}b^\alpha_k)m^\alpha \geq 0,$$ $$p^\alpha = 1 - p^\alpha_\tau - \sum_k p^\alpha_k - \varepsilon \nu m^\alpha - \varepsilon m^\alpha c^\alpha.$$ Notice that $$p^\alpha_k \leq K \varepsilon \left(\frac{2K}{h^2} + \frac{K}{h} \right)$$ and $$\varepsilon \nu m^\alpha + \varepsilon m^\alpha c^\alpha \leq \varepsilon \tau^{-1}K + \varepsilon K, \quad p_\tau \leq \varepsilon \tau^{-1} K,$$ so for all $\varepsilon$ smaller than some $\varepsilon_0$, $p^\alpha \geq 0$. Also $$\begin{aligned}
0 \leq \sum_kp^\alpha_k + p^\alpha + p_\tau & = & 1 - \varepsilon m^\alpha (\nu + c^\alpha)
\leq 1- (1 \vee \nu)\varepsilon m^\alpha(1 + c^\alpha) \\
& \leq & 1 - (1 \vee \nu) \varepsilon K^{-1} =: \delta < 1,\end{aligned}$$ for some sufficiently small $\varepsilon > 0$. Since the difference of supremums is less than the supremum of a difference $$|G[v](t,x) - G[w](t,x)| \leq \delta \sup_{\bar{\mathcal{M}}_T}|v-w|.$$ Thus the operator $G$ is a contraction on the space of bounded functions on $\bar{\mathcal{M}}_T$. By Banach’s fixed point theorem (\[eqn-proxy-for-eqn-f-d-problem\]) has a unique bounded solution.
\[lemma-discrete-comparison-principle\] Consider $f_1^\alpha, f_2^\alpha$ defined on $A \times \mathcal{M}_T$ and assume that in Q $$\sup_{\alpha} m^\alpha f_2^\alpha < \infty, \quad f_1^\alpha \leq f_2^\alpha.$$ Let there be functions $u_1,u_2$ defined on $\bar{\mathcal{M}}_T$ and a constant $\mu \geq 0$ such that $u_1e^{-\mu|x|}$, $u_2 e^{-\mu|x|}$ are bounded and for some $C \geq 0$ $$\label{eqn-comp-princ-ass1}
\begin{split}
\sup_{\alpha} m^\alpha \left({\operatorname{\delta}_\tau}u_1 + {{\operatorname{L}}_h^\alpha}u_1 + f_1^\alpha + C \right) \qquad \qquad \qquad \qquad \\
\geq \sup_{\alpha} m^\alpha \left({\operatorname{\delta}_\tau}u_2 + {{\operatorname{L}}_h^\alpha}u_2 + f_2^\alpha \right)\quad \textrm{on} \quad Q
\end{split}$$ and $$u_1 \leq u_2 \quad \textrm{on} \quad \bar{\mathcal{M}}_T \setminus Q.$$ Then there exists a constant $\tau^*$ depending only on $K, d_1, \mu$ such for $\tau \in (0, \tau^*)$ $$\label{eqn-comparison-princ-conclusion}
u_1 \leq u_2 + TC {\quad \textrm{on} \quad}\bar{\mathcal{M}}_T.$$ If $u_1$, $u_2$ are bounded on $\mathcal{M}_T$ then (\[eqn-comparison-princ-conclusion\]) holds for any $\tau$.
To prove this lemma we use the following observation.
\[remark-taylors-thm\] Let ${\operatorname{D}_{\ell}}u := u_{x^i} \ell^i$, ${\operatorname{D}_{\ell}}^2 u := u_{x^i x^j} \ell^i \ell^j$, $a^\alpha_k = \frac{1}{2} (\sigma^\alpha_k)^2$. Let $\operatorname{D}_x^n$ denote the collection of all n-th order derivatives in $x$. Consider $${{\operatorname{L}}^\alpha}(t,x)u(t,x) = \sum_k a^\alpha_k(t,x)\operatorname{D}^2_{\ell_k} u(t,x) +
b^\alpha_k(t,x)\operatorname{D}_{\ell_k}u(t,x).$$ For any sufficiently smooth function $\eta(x)$, by Taylor’s theorem, $$m^\alpha|{{\operatorname{L}}^\alpha}\eta(x) - {{\operatorname{L}}_h^\alpha}\eta(x)| \leq N(h^2 \sup_{B_K(x)}|\operatorname{D}^4_x\eta| + h \sup_{B_K(x)} |\operatorname{D}^2_x \eta|).$$
Let $w = u_1 - u_2 -C(T-t)$. From (\[eqn-comp-princ-ass1\]) $$\sup_\alpha m^\alpha \left({\operatorname{\delta}_\tau}w + {{\operatorname{L}}_h^\alpha}w \right) \geq 0$$ hence for any $\varepsilon > 0$ $$w + \varepsilon \sup_\alpha m^\alpha \left({\operatorname{\delta}_\tau}w + {{\operatorname{L}}_h^\alpha}w \right) \geq w.$$ In the proof of Lemma \[lemma-existence-of-soln-of-discrete-problem\], choose $\varepsilon$ such that $p^\alpha$, $p_k^\alpha$ are nonnegative. So the operator $G$ is a monotone operator. Hence for any $\psi \geq w$, $$\label{eqn-w-greater-Gpsi}
\psi + \varepsilon \sup_\alpha m^\alpha \left({\operatorname{\delta}_\tau}\psi + {{\operatorname{L}}_h^\alpha}\psi \right) \geq w.$$ Let $\gamma \in (0,1)$. Use $\xi$ from the proof of Lemma \[lemma-existence-of-soln-of-discrete-problem\]. Fix $\gamma \in (0,1)$ later. Then $${\operatorname{\delta}_\tau}\xi = \xi \frac{1}{\tau}(\gamma - 1).$$ Let $$N_0 = \sup_{\bar{\mathcal{M}}_T} \frac{w_+}{\zeta} < \infty \quad \textrm{and}\quad \psi := N_0\zeta \geq \zeta \frac{w_+}{\zeta} \geq w,$$ where $\eta(x) = \cosh(\mu|x|)$ and $\zeta = \eta \xi$. Then by Remark \[remark-taylors-thm\] and since for $r\geq 0$ $$\frac{\cosh(\mu r)}{r^2} - \frac{\sinh(\mu r)}{r^3} \leq \cosh(\mu r)$$ we get $$m^\alpha {{\operatorname{L}}_h^\alpha}\eta(x) \leq m^\alpha L^\alpha \eta(x) + N_1(h^2 + h)\cosh(\mu|x|+\mu K)
\leq N_2 \cosh(\mu|x| +\mu K),$$ with $N_1, N_2$ depending only on $\mu, d_1, K$. Hence $$m^\alpha \left( {\operatorname{\delta}_\tau}\zeta + {{\operatorname{L}}_h^\alpha}\zeta \right) \leq \zeta[\tau^{-1}(\gamma-1)+N_3], \quad
N_3 := N_2 \frac{\cosh(\mu |x| + \mu K)}{\cosh(\mu|x|)} < \infty.$$ Then $$\begin{split}
w & \leq \psi + \varepsilon \sup_\alpha m^\alpha( {\operatorname{\delta}_\tau}\psi + {{\operatorname{L}}_h^\alpha}\psi ) \\
& \leq N_0\zeta\left[ 1 + \varepsilon
(\tau^{-1}(\gamma - 1) + N_3)\right].
\end{split}$$ Define $\kappa(\gamma) = \tau^{-1}(\gamma - 1) + N_3$. Then for $\tau < \tau^* := N_3^{-1}$ one has $\kappa(0) < 0$ and $\kappa(1) > 0$. So one can have $\gamma$ such that $\kappa < 0$ and $1+\varepsilon \kappa > 0$. By (\[eqn-w-greater-Gpsi\]) $$\label{eqn-proof-of-comp-princ-w-leq}
w \leq \psi + \varepsilon \sup_\alpha m^\alpha \left({\operatorname{\delta}_\tau}\psi + {{\operatorname{L}}_h^\alpha}\psi \right) \leq N_0 \xi \left(1 + \varepsilon \kappa \right) {\quad \textrm{on} \quad}Q.$$ If $w \geq 0$ then $w_+ = w$. Hence $$\label{eqn-proof-of-comp-princ-N0-is-0}
N_0 = \sup_{\bar{\mathcal{M}}_T} \frac{w_+}{\xi} \leq \sup_{\bar{\mathcal{M}}_T} \frac{N_0\xi(1+\varepsilon\kappa)}{\xi}.$$ By the hypothesis $w\leq 0$ on $\bar{\mathcal{M}}_T \setminus Q$. But $\kappa < 0$, hence $N_0$ must be $0$ and so $w\leq 0$ on $\bar{\mathcal{M}}_T$. Finally observe that if $\mu = 0$, $N_2 = N_3 = 0$.
\[corollary-soln-to-disc-bellman-pde-is-bdd\] If $v$ is the solution of (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) then $$v \leq 2K^2(T+1)+\sup_x|g(x)|.$$
Apply Lemma \[lemma-discrete-comparison-principle\] to $v$ and $$\xi(t) = K^2\left[(T-t)+1 \right] + N_1,$$ where $N_1 := \sup_x|g(x)|$. Then on $Q$ $$\begin{aligned}
\sup_\alpha m^\alpha({\operatorname{\delta}_\tau}\xi + {{\operatorname{L}}_h^\alpha}\xi + f^\alpha) & & \\
= \sup_\alpha\left[-m^\alpha( K^2 + c^\alpha(K^2\left[(T-t)+1 \right] + N_1) \right] & \leq & 0,\end{aligned}$$ while on $\bar{\mathcal{M}}_T \setminus Q$ $$\xi(T) \geq \sup_x|g(x)| \geq g(x).$$
Assume that $u_1, u_2$ are functions both satisfying (\[eqn-f-d-problem-a\]) on $Q$ and $u_1 = g_1$, $u_2 = g_2$ on $\bar{\mathcal{M}}_T\setminus Q$. Then $$u_1 \leq u_2 + \sup_x(g_1 - g_2)_+.$$
Apply the comparison principle to $u_1$ and $\bar{u}_2 := \sup_x(g_1 - g_2)_+$.
\[corollary-discrete-case-conditional-holder-continuity\] Fix $(s_0,x_0) \in \bar{\mathcal{M}}_T$. Let $v$ be a solution of (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]). Define $$\nu = \sup_{(s_0,x)\in\bar{\mathcal{M}}_T} \frac{|v(s_0,x) - v(s_0,x_0)|}{|x-x_0|}.$$ Then for all $(t_0,x_0) \in \bar{\mathcal{M}}_T$ such that $t_0 \in [s_0 -1, s_0]$, $$|v(s_0,x_0) - v(t_0,x_0)| \leq N(\nu + 1)|s_0 - t_0|^\frac{1}{2}.$$
Assume, without loss of generality, that $\nu > 1$. Indeed if $\nu < 1$, then in the proof of the theorem use $\bar{\nu} = \nu + 1$. If the result holds for $\bar{\nu}$, then $$N(\bar{\nu}+1) \leq N(\nu + 1 + 1) \leq 2N(\nu + 1).$$ Assume that $s_0 > 0$. Shifting the time axis, so that $t_0 = 0$. Then $s_0 \leq 1$. Let $$\gamma = s_0^{-1/2} {\quad \textrm{and} \quad}\xi(t) = \left\{
\begin{array}{lcr}
e^{s_0 - t} & \text{for} & t < s_0, \\
1 \quad & \text{for} & t \geq s_0.
\end{array}\right.$$ Notice that $\gamma \geq 1/T$. Define $$\psi = \gamma \nu \left[\zeta + \kappa(s_0 - t)\right] + \frac{1}{\gamma}\nu \kappa + K(s_0 - t) + v(s_0,x_0),$$ where $\zeta = \eta \xi$, $\eta(x) = |x-x_0|^2$ and $\kappa$ is a (large) constant to be chosen later, depending on $K, d_1$ and $T$ only. The aim now is to apply Lemma \[lemma-discrete-comparison-principle\] to $v$ and $\psi$ on $\mathcal{M}_{s_0}$. $${\operatorname{\delta}_\tau}\xi(t) = -\theta \xi, {\quad \text{where} \quad}\theta = \tau^{-1}(1-e^{-\tau}) \geq K^{-1}(1-e^{-K}).$$ With $(.,.)$ denoting the inner product in $\R^d$ $${{\operatorname{L}}_h^\alpha}\eta = 2a_k^\alpha|l_k|^2 + b_k^\alpha(l_k,2(x-x_0) + hl_k) - c^\alpha \eta \leq N_1 (1+ |x-x_0|) - c^\alpha \eta.$$ Hence $$\begin{split}
{\operatorname{\delta}_\tau}\zeta + {{\operatorname{L}}_h^\alpha}\zeta & \leq \xi(t)\left[N_1 (1+|x-x_0|) - \theta|x-x_0|^2 \right] - c^\alpha \zeta\\
& \leq N_2(1+|x-x_0|) - \theta |x-x_0|^2 - c^\alpha \zeta.
\end{split}$$ Since $v(s_0,x) \geq -K - \sup_x|g| \geq -N_3$ and hence $$m^\alpha c^\alpha \psi \geq -m^\alpha c^\alpha \frac{\kappa}{\gamma} \nu + m^\alpha c^\alpha N_3.$$ Therefore $$\label{eqn-est-on-psi-as-subsoln}
\begin{split}
& m^\alpha\left({\operatorname{\delta}_\tau}\psi + {{\operatorname{L}}_h^\alpha}\psi + f^\alpha\right) \\
& \leq m^\alpha \left(\gamma \nu \left[N_2(1+|x-x_0|) - \theta|x-x_0|^2-\kappa \right] + c^\alpha N_3 -c^\alpha\frac{\kappa}{\gamma}\nu \right).
\end{split}$$ The expression on the right hand side is a quadratic in $|x-x_0|$ with negative leading coefficient. It achieves its maximum of $$m^\alpha\left\{\gamma \nu \left[N_2 + \frac{1}{2 \theta} - \frac{1}{4\theta N_2^2} -\kappa \right] -K \right\} + K + m^\alpha \left(c^\alpha N_3 -c^\alpha\frac{\kappa}{\gamma}\nu \right)$$ when $|x-x_0| = \frac{1}{2\theta N_2}$. Also $s_0 \leq 1$ and so $\gamma \geq 1$. Since $m^\alpha(1+c^\alpha) \geq K^{-1}$, $m^\alpha(\gamma^2+c^\alpha) \geq (1 \wedge \gamma^2) K^{-1}$. Hence $$\kappa\nu\gamma^{-1} m^\alpha(\gamma^2 + c^\alpha) \geq \kappa \nu \gamma^{-1} (\gamma^2 \wedge 1) K^{-1} \geq K^{-1}\nu \kappa.$$ Finally, $m^\alpha c^\alpha N_T \leq K$. If $\kappa$ is chosen very large (depending on $K,T,d$ only) then (\[eqn-est-on-psi-as-subsoln\]) is non-positive. Hence $$m^\alpha\left(\operatorname{\delta}\psi + {{\operatorname{L}}_h^\alpha}\psi + f^\alpha\right) \leq 0.$$ To apply Lemma \[lemma-discrete-comparison-principle\], $\psi \geq v$ on $\bar{\mathcal{M}}_{s_0} \setminus \mathcal{M}_{s_0}$ must be satisfied. Consider the two cases when either $\gamma|x-x_0| \leq 1$ or $\gamma|x-x_0| > 1$. Then $$\begin{aligned}
\psi(s_0,x) & = & \nu \left[\gamma|x-x_0|^2 + \kappa \gamma^{-1}\right] + v(s_0,x_0) \\
& \geq & \nu |x-x_0| + v(s_0,x_0) \geq v(s_0,x).\end{aligned}$$ By Lemma \[lemma-discrete-comparison-principle\] $$v(t_0,x_0) - v(s_0,x_0) \leq \nu s_0^{-1/2}\kappa s_0 + s_0^{1/2} \nu \kappa + Ks_0
\leq N(\nu + 1)s_0^{1/2}.$$ To get the estimate from the other side, one would consider $$\psi = - \gamma \nu \left[\zeta + \kappa(s_0 - t)\right] - \frac{1}{\gamma}\nu \kappa - K(s_0 - t) + v(s_0,x_0)$$ and then show that for some choice of $\kappa$ $$\sup_\alpha m^\alpha\left(\operatorname{\delta}\psi + {{\operatorname{L}}_h^\alpha}\psi + f^\alpha\right) \geq 0 {\quad \textrm{on} \quad}Q.$$ and that $\psi \leq v$ on $\bar{\mathcal{M}}_{s_0}$.
Gradient Estimate for the Finite Difference Problem {#section-gradient-estimate}
===================================================
The following lemma states some simple properties of the operators $\operatorname{\delta}_{h,l}$ and $\operatorname{\Delta}_{h,l}$, which will be used in this section. The statements of the following lemma are all proved in section 4 of [@krylov:rate:lipschitz].
Let $u$ and $v$ be functions on $\R^d$, $l$ and $x_0$ vectors in $\R^d$ and $h > 0$. Then $$\label{eqn-4.3}
\operatorname{\delta}_{h,l}(uv) = (\operatorname{\delta}_{h,l} u) v + (\operatorname{T}_{h,l} u)\operatorname{\delta}_{h,l}v = v \operatorname{\delta}_{h,l} u + u \operatorname{\delta}_{h,l} v + h(\operatorname{\delta}_{h,l} u)(\operatorname{\delta}_{h,l} v),$$ which can be thought of as the discrete Leibnitz rule. Also $$\label{eqn-4.5}
\begin{split}
\operatorname{\Delta}_{h,l}(uv) & = v\operatorname{\Delta}_{h,l}u + (\operatorname{\delta}_{h,l} u)\operatorname{\delta}_{h,l} v + (\operatorname{\delta}_{h,-l}u)\operatorname{\delta}_{h,-l} v + u \operatorname{\Delta}_{h,l} v,\\
\operatorname{\Delta}_{h,l}(u^2) & = 2u\operatorname{\Delta}_{h,l} u + (\operatorname{\delta}_{h,l}u)^2 + (\operatorname{\delta}_{h,-l} u)^2.
\end{split}$$ Furthermore if $v(x_0) \leq 0$ then at $x_0$ it holds that $$\label{eqn-4.6}
-\operatorname{\delta}_{h,l} v \leq \operatorname{\delta}_{h,l}(v_-), \quad -\operatorname{\Delta}_{h,l} v \leq \operatorname{\Delta}_{h,l} (v_-),$$ $$\label{eqn-4.7}
-\operatorname{\delta}_{h,l}(u_-) \leq \left[{\operatorname{\delta}_{h,\ell}}((u+v)_-)\right]_- + \left[{\operatorname{\delta}_{h,\ell}}(v_-)\right]_+.$$ $$\label{eqn-4.8}
\begin{split}
({\operatorname{\Delta}_{h,\ell}}u)_- \leq [{\operatorname{\delta}_{h,-\ell}}(({\operatorname{\delta}_{h,\ell}}u + v)_-)]_- + [{\operatorname{\delta}_{h,\ell}}(({\operatorname{\delta}_{h,-\ell}}u+w)_-)]_- \\ + [{\operatorname{\delta}_{h,-\ell}}(v_-)]_+ + [{\operatorname{\delta}_{h,\ell}}(w_-)]_+.
\end{split}$$
$$\label{eqn-4.9}
|\operatorname{\Delta}_{h,l} u| \leq |\operatorname{\delta}_{h,-l}((\operatorname{\delta}_{h,l} u)_-)| + |\operatorname{\delta}_{h,l}((\operatorname{\delta}_{h,-l} u)_-)|,$$
$$\label{eqn-4.10}
|\operatorname{\Delta}_{h,l} u| \leq |\operatorname{\delta}_{h,-l}((\operatorname{\delta}_{h,l} u)_+)| + |\operatorname{\delta}_{h,l}((\operatorname{\delta}_{h,-l} u)_+)|,$$
Let Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\] hold. Let $T'$ be the smallest integer multiple of $\tau$ which is greater than or equal to $T$. Choose an arbitrary $\varepsilon \in (0,Kh]$ and $l \in R^d$. Let $h_r = h$ for $r = \pm 1, \ldots, \pm d_1$ and $h_r = {\varepsilon}$ for $r = \pm (d_1 + 1)$. Let $$\bar{\mathcal{M}}_T(\varepsilon) := \{(t,x+i\varepsilon l):(t,x) \in \bar{\mathcal{M}}_T, i = 0, \pm 1, \ldots \}.$$ For a general $Q \subset \bar{\mathcal{M}}_T(\varepsilon)$ let $$\begin{gathered}
Q^0_\varepsilon = \{(t,x): (t+{\tau_T(t)}, x) \in Q, (t,x+h_rl_r) \in Q, \forall r = \pm
1, \ldots , \pm (d_1 + 1)\}, \\
\partial_\varepsilon Q = Q \setminus Q^0_\varepsilon.\end{gathered}$$
\[ass-for-thm-5.2\] Assume that for any $(t,x) \in Q^0_\varepsilon$ $$\label{eqn-first-ass-for-thm-5.2}
|\operatorname{\delta}_{h_r,l_r} b^\alpha_k| \leq K, \quad m^\alpha|\operatorname{\delta}_{h_r,l_r} f^\alpha| \leq K, \quad |\operatorname{\delta}_{h_r,l_r} c^\alpha| \leq K,$$ $$\label{eqn-second-ass-for-thm-5.2}
|\operatorname{\delta}_{h_r,l_r} a^\alpha_k|\leq K\sqrt{a^\alpha_k} + h.$$
Define $a^{\pm} = a_{\pm} = (1/2)(|a| \pm a)$.
\[thm-estimate-on-discrete-derivative\] Let $u$ be a function defined on $\bar{\mathcal{M}}_T(\varepsilon)$ satisfying (\[eqn-f-d-problem-a\]) on $Q$. If Assumption \[ass-for-thm-5.2\] is satisfied, then there is a constant $N > 1$, such that, if there exists some $c_0 \geq 0$ satisfying $$\label{assumption-on-c_0}
\lambda + \tfrac{1}{\tau}(1-e^{-c_0 \tau}) > N,$$ then on $Q$, $$\label{eqn-estimate-on-discrete-derivative}
|\delta_{\varepsilon,\pm l}u| \leq Ne^{c_o}T'\big(1+|u|_{0,Q} + \sup_{k, \partial_\varepsilon Q}\left(|{\operatorname{\delta}_{h_k,\ell_k}}u| + |\delta_{\varepsilon,l}u| + |\delta_{\varepsilon,-l}u| \right) \big).$$
Let $$v_r = \operatorname{\delta}_{h_r,l_r}v, \quad v = \xi u, \quad \xi(t) = \left\{
\begin{array}{ll}
e^{c_0 t} & t < T, \\
e^{c_0 T'} & t = T.
\end{array} \right.$$ Let $(t_0,x_0) \in Q$ be the point where $$V = \sum_r (v_r^-)^2$$ is maximized. By definition, for any $(t,x) \in Q_\varepsilon^o$ we know that $$(t,x+h_rl_r) \in Q.$$ Then either $$v_r(t,x) \leq 0 \quad \textrm{or} \quad -v_r (t,x) = v_{-r}(t,x+h_rl_r) \leq 0.$$ In either case, for any $(t,x) \in Q^0_\varepsilon$ $$|v_r(t,x)| \leq V^{1/2}.$$ Hence $$M_1 := \sup_{Q,r}|v_r| \leq \sup_{\partial_\varepsilon Q,r}|v_r| + V^{1/2},$$ $$|\delta_{\varepsilon, \pm l}u| \leq e^{c_0 T'} \sup_{\delta_\varepsilon Q,r}|\delta_{h_r,l_r}u| + V^{1/2}$$ on $Q$. So we only have to estimate $V$ on $Q$. If $(t_0,x_0)$ belongs to $\partial_{\varepsilon} Q$, then the conclusion of the theorem is trivially true. Thus, we may assume that $(t_0,x_0) \in Q^0_\varepsilon$. Then for any $\bar{\varepsilon}_0 > 0$ there exists ${{\alpha_0}}\in A$ such that at $(t_0,x_0)$, $$m^{{\alpha_0}}\left({\operatorname{\delta}_\tau}u + a^{{{\alpha_0}}}_k {\operatorname{\Delta}_{h_k,\ell_k}}u + b^{{{\alpha_0}}}_k
{\operatorname{\delta}_{h_k,\ell_k}}u - c^{{\alpha_0}}u + f^{{{\alpha_0}}} \right) + \bar{\varepsilon}_0 \geq 0.$$ and so for some $\bar{\varepsilon} \in (0,\bar{\varepsilon}_0]$ $$\label{5.8}
m^{{\alpha_0}}\left({\operatorname{\delta}_\tau}u + a^{{{\alpha_0}}}_k {\operatorname{\Delta}_{h_k,\ell_k}}u + b^{{{\alpha_0}}}_k
{\operatorname{\delta}_{h_k,\ell_k}}u - c^{{\alpha_0}}u + f^{{{\alpha_0}}} \right) + \bar{\varepsilon} = 0.$$ Furthermore (thanks to the fact that $(t_0,x_0) \in Q^0_\varepsilon$) $$\label{5.9}
{\operatorname{T}_{h_r,\ell_r}}\left[m^{{\alpha_0}}\left({\operatorname{\delta}_\tau}u + a^{{{\alpha_0}}}_k {\operatorname{\Delta}_{h_k,\ell_k}}u + b^{{{\alpha_0}}}_k {\operatorname{\delta}_{h_k,\ell_k}}u - c^{{\alpha_0}}u + f^{{{\alpha_0}}} \right) \right] \leq 0.$$ We subtract (\[5.8\]) from (\[5.9\]) and divide by $h_r$ to obtain that for each $r$ $$m^{{{\alpha_0}}} \operatorname{\delta}_{h_r,l_r}\left({\operatorname{\delta}_\tau}u + a^{{{\alpha_0}}}_k {\operatorname{\Delta}_{h_k,\ell_k}}u + b^{{{\alpha_0}}}_k {\operatorname{\delta}_{h_k,\ell_k}}u + f^{{{\alpha_0}}} - c^{{\alpha_0}}u\right) - \tfrac{\bar{\varepsilon}}{h_r} \leq 0.$$ For each $r$, by the discrete Leibnitz rule (\[eqn-4.3\]) $$\label{5.13}
\begin{split}
m^{{\alpha_0}}\left(\delta_\tau (\xi^{-1} v_r) + \xi^{-1}
\left[a_k^{{{\alpha_0}}} {\operatorname{\Delta}_{h_k,\ell_k}}v_r +I_{1r} + I_{2r} + I_{3r}
\right] + \operatorname{\delta}_{h_r,l_r} f^{{\alpha_0}}\right) \\ - \xi^{-1}\operatorname{\delta}_{h_r,l_r}(m^{{\alpha_0}}c^{{\alpha_0}}v) - \tfrac{\bar{\varepsilon}}{h_r} \leq 0,
\end{split}$$ where $$\begin{aligned}
I_{1r} & = & ( \operatorname{\delta}_{h_r,l_r} a^{{{\alpha_0}}}_k ) {\operatorname{\Delta}_{h_k,\ell_k}}v, \\
I_{2r} & = & h_r ( \operatorname{\delta}_{h_r,l_r} a^{{{\alpha_0}}}_k) {\operatorname{\Delta}_{h_k,\ell_k}}v_r, \\
I_{3r} & = & ({\operatorname{T}_{h_r,\ell_r}}b^{{{\alpha_0}}}_k) {\operatorname{\delta}_{h_k,\ell_k}}v_r + (
\operatorname{\delta}_{h_r,l_r} b^{{{\alpha_0}}}_k ) {\operatorname{\delta}_{h_k,\ell_k}}v.\end{aligned}$$ Notice (first equality by (\[eqn-4.5\]), second inequality by (\[eqn-4.6\]), also note that there is a tacit summation in $k$) that $$\begin{aligned}
0 & \geq & {\operatorname{\Delta}_{h_k,\ell_k}}\sum_r (v_r^-)^2 = 2v_r^- {\operatorname{\Delta}_{h_k,\ell_k}}v_r^- + \sum_r
\left[ {\operatorname{\delta}_{h_k,\ell_k}}(v_r^-)^2 + \operatorname{\delta}_{h_k,l_{-k}} (v_r^-)^2\right]
\\
& \geq & - 2v_r^- {\operatorname{\Delta}_{h_k,\ell_k}}v_r + 2\sum_r ({\operatorname{\delta}_{h_k,\ell_k}}v_r^-)^2.\end{aligned}$$ Since $a^{\alpha}_k$ are nonnegative, $$\label{5.10}
0 \leq v_r^- {\operatorname{\Delta}_{h_k,\ell_k}}v_r$$ and $$I := \sum_r a^{{{\alpha_0}}}_k({\operatorname{\delta}_{h_k,\ell_k}}v_r^-)^2 \leq v_r^-
a^{{{\alpha_0}}}_k{\operatorname{\Delta}_{h_k,\ell_k}}v_r.$$ Go back to (\[5.13\]), multiply by $\xi v_r^-$ and sum up in $r$ to get $$\label{5.13.1}
\begin{split}
m^{{\alpha_0}}\Big(\xi v_r^- \delta_\tau (\xi^{-1} v_r) + \tfrac{1}{2}
a^{{{\alpha_0}}}_k v_r^- {\operatorname{\Delta}_{h_k,\ell_k}}v_r + \tfrac{1}{2}I + v_r^-
[I_{1r} + I_{2r} \\ + I_{3r} + \xi \operatorname{\delta}_{h_r,l_r} f^{{\alpha_0}}] \Big) - v_r^- \operatorname{\delta}_{h_r,l_r}(m^{{\alpha_0}}c^{{\alpha_0}}v) - \sum_r \xi (v_r^-) \tfrac{\bar{\varepsilon}}{h_r} \leq 0.
\end{split}$$ By (\[eqn-first-ass-for-thm-5.2\]), $m^\alpha \operatorname{\delta}_{h_r,l_r} f^\alpha_k \geq -K$. If $v_r > 0$, then $v_r^- = 0$, hence $-v_r^- v_r = 0 =
(v_r^-)^2$. On the other hand if $v_r \leq 0$, then $v_r = -v_r^-$ and so $-v_r^-v_r = (v_r^-)^2$. Hence $$\begin{gathered}
m^{{\alpha_0}}v_r^- \xi \operatorname{\delta}_{h_r,l_r} f^{{\alpha_0}}- v_r^- \operatorname{\delta}_{h_r,l_r}(m^{{\alpha_0}}c^{{\alpha_0}}v) \\
= m^{{\alpha_0}}v_r^- \xi \operatorname{\delta}_{h_r,l_r} f^{{\alpha_0}}- m^{{\alpha_0}}{v_r^-}(\operatorname{\delta}_{h_r,l_r} c^{{{\alpha_0}}}) v - m^{{\alpha_0}}{v_r^-}({\operatorname{T}_{h_r,\ell_r}}c^{{{\alpha_0}}} )v_r \\
\geq - e^{c_oT'} K {v_r^-}-m^{{\alpha_0}}{v_r^-}|\operatorname{\delta}_{h_r,l_r} c^{{{\alpha_0}}}| |v| + m^{{\alpha_0}}({\operatorname{T}_{h_r,\ell_r}}c^{{{\alpha_0}}} )({v_r^-})^2 \\
\geq - e^{c_o T'}KM_1 - m^{{\alpha_0}}K M_1 M_o + ({\operatorname{T}_{h_r,\ell_r}}m^{{\alpha_0}}c^{{\alpha_0}}) V.\end{gathered}$$ Use the fact that $V$ attains its maximum at $(t_o,x_o)$, discrete Leibnitz rule and (\[eqn-4.6\]) in order to get a lower bound for the term containing $I_{3r}$. First, $$\begin{aligned}
0 & \geq & \sum_r {\operatorname{\delta}_{h_k,\ell_k}}({v_r^-})^2 = 2{v_r^-}{\operatorname{\delta}_{h_k,\ell_k}}{v_r^-}+ \sum_r h_k ({\operatorname{\delta}_{h_k,\ell_k}}{v_r^-})^2 \\
& \geq & 2 {v_r^-}{\operatorname{\delta}_{h_k,\ell_k}}{v_r^-}\geq -2 {v_r^-}{\operatorname{\delta}_{h_k,\ell_k}}v_r. \end{aligned}$$ Next recall that $b_k^\alpha \geq 0$ and $|\operatorname{\delta}_{h_r,l_r}b^\alpha_k| \leq K$, and so $$-{v_r^-}({\operatorname{T}_{h_r,\ell_r}}b^{{{\alpha_0}}}_k) {\operatorname{\delta}_{h_k,\ell_k}}v_r\leq 0,$$ hence $$v_r^- I_{3r} \geq -{v_r^-}|\operatorname{\delta}_{h_r,l_r}
b^{{{\alpha_0}}}_k||{\operatorname{\delta}_{h_k,\ell_k}}v| \geq -KM_1^2.$$ Apply discrete Leibnitz rule to the very first term of (\[5.13.1\]). Then $$\begin{aligned}
\xi {v_r^-}{\operatorname{\delta}_\tau}(\xi^{-1} v_r ) & = & \xi {v_r^-}\left[\xi^{-1}(t_o
+\tau ){\operatorname{\delta}_\tau}v_r + v_r {\operatorname{\delta}_\tau}\xi^{-1} \right] \\
& = & e^{-c_o\tau} {v_r^-}{\operatorname{\delta}_\tau}v_r - V \xi {\operatorname{\delta}_\tau}\xi^{-1} \geq
- V\xi {\operatorname{\delta}_\tau}\xi^{-1} \\
& = & V\tfrac{1}{\tau}(1-e^{-c_o\tau}).\end{aligned}$$ Using the above estimates we see $$\begin{split}
m^{{\alpha_0}}V\tfrac{1}{\tau}(1-e^{-c_o\tau}) + ({\operatorname{T}_{h_r,\ell_r}}m^{{\alpha_0}}c^{{\alpha_0}}) V + \tfrac{1}{2} m^{{\alpha_0}}a^{{{\alpha_0}}}_k {v_r^-}{\operatorname{\Delta}_{h_k,\ell_k}}v_ r +
\tfrac{1}{2} m^{{\alpha_0}}I \\ + {v_r^-}m^{{\alpha_0}}[I_{1r} + I_{2r}] - \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r} \leq KM_1(e^{c_oT'} + M_0 + m^{{\alpha_0}}M_1).
\end{split}$$ Hence $$\begin{split}
m^{{\alpha_0}}V\tfrac{1}{\tau}(1-e^{-c_o\tau}) + ({\operatorname{T}_{h_r,\ell_r}}m^{{\alpha_0}}c^{{\alpha_0}}) V \leq KM_1(e^{c_oT'} + M_0 + m^{{\alpha_0}}M_1) \\ +m^{{\alpha_0}}{v_r^-}|I_{1r}| +
m^{{\alpha_0}}{v_r^-}|I_{2r}| - \tfrac{1}{2}m^{{\alpha_0}}a^{{{\alpha_0}}}_k {v_r^-}{\operatorname{\Delta}_{h_k,\ell_k}}v_r -
\tfrac{1}{2}m^{{\alpha_0}}I + \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r}.
\end{split}$$ Define $$\begin{aligned}
J_1 & := {v_r^-}|(\operatorname{\delta}_{h_r,l_r} a^{\alpha_o}_k) {\operatorname{\Delta}_{h_k,\ell_k}}v| -
\tfrac{1}{4}\sum_r a^{{{\alpha_0}}}_k ({\operatorname{\delta}_{h_k,\ell_k}}{v_r^-})^2,\\
J_2 & := J_3 - \tfrac{1}{2} a^{{{\alpha_0}}}_k {v_r^-}{\operatorname{\Delta}_{h_k,\ell_k}}v_r -
\tfrac{1}{4}\sum_r a^{{{\alpha_0}}}_k ({\operatorname{\delta}_{h_k,\ell_k}}{v_r^-})^2\\
J_3 & := h_r {v_r^-}|(\operatorname{\delta}_{h_r,l_r} a^{{{\alpha_0}}}_k) {\operatorname{\Delta}_{h_k,\ell_k}}v_r|.\end{aligned}$$ One can rewrite the above inequality as $$\begin{split}
m^{{\alpha_0}}V\tfrac{1}{\tau}(1-e^{-c_o\tau}) + ({\operatorname{T}_{h_r,\ell_r}}m^{{\alpha_0}}c^{{\alpha_0}}) V \qquad \qquad \\
\leq KM_1(e^{c_oT'} + M_0 + m^{{\alpha_0}}M_1) + m^{{\alpha_0}}(J_1 + J_2) + \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r} .
\end{split}$$ So we need to estimate $J_1, J_2$. By (\[eqn-4.9\]) $$|{\operatorname{\Delta}_{h_k,\ell_k}}v| \leq \sum_r |{\operatorname{\delta}_{h_k,\ell_k}}{v_r^-}| +\sum_r |\operatorname{\delta}_{h_k,l_{-k}} {v_r^-}|$$ then because there’s tacit summation in $k = \pm 1, \ldots, \pm d_1$ and because in the second sum we can reverse the order of summation in $k$ $$| {\operatorname{\Delta}_{h_k,\ell_k}}v| \leq 2 \sum_r |{\operatorname{\delta}_{h_k,\ell_k}}{v_r^-}|.$$ We turn our attention to $J_1$. $$\begin{aligned}
{v_r^-}|(\operatorname{\delta}_{h_r,l_r} a_k^{{\alpha_0}}) {\operatorname{\Delta}_{h_k,\ell_k}}v| & \leq & M_1 K|(\sqrt{a_k^{{\alpha_0}}} + h) {\operatorname{\Delta}_{h_k,\ell_k}}v| \\
& \leq & KM_1\sqrt{a_k^{{\alpha_0}}} |{\operatorname{\Delta}_{h_k,\ell_k}}v| + KM_1^2 \\
& \leq & KM_1^2 + 2KM_1 \sqrt{a_k^{{\alpha_0}}} \sum_r {\operatorname{\delta}_{h_k,\ell_k}}{v_r^-}\\
& \leq & KM_1^2 + \sum_r 8KM_1^2 + \tfrac{1}{4} \sum_r a^{{\alpha_0}}_k ({\operatorname{\delta}_{h_k,\ell_k}}{v_r^-})^2, \end{aligned}$$ having used Young’s inequality. So $J_1 \leq NM_1^2$. Since $h_r = h$ or $\varepsilon \in (0, Kh]$, $$J_3 \leq Khv_r^- \sqrt{a_k^{{\alpha_0}}} | {\operatorname{\Delta}_{h_k,\ell_k}}v_r| + Kh^2 {v_r^-}|{\operatorname{\Delta}_{h_k,\ell_k}}v_r|.$$ Also $h^2 |{\operatorname{\Delta}_{h_k,\ell_k}}v_r| \leq 4M_1$ and in general $|a| = 2a^- + a$ and so $$J_3 \leq 2Kh {v_r^-}\sqrt{a_k^{{{\alpha_0}}}} ({\operatorname{\Delta}_{h_k,\ell_k}}v_r)_- + Kh {v_r^-}\sqrt{\alpha_k^{{\alpha_0}}} {\operatorname{\Delta}_{h_k,\ell_k}}v_r + NM_1^2.$$ By (\[eqn-4.8\]) (with summation in $r$ everywhere) $$h({\operatorname{\Delta}_{h_k,\ell_k}}v_r)_- \leq h|{\operatorname{\Delta}_{h_k,\ell_k}}({v_r^-})| \leq |{\operatorname{\delta}_{h_k,\ell_k}}({v_r^-})| + |\operatorname{\delta}_{h_k,l_{-k}} ({v_r^-})|.$$ Therefore $$\begin{aligned}
J_3 & \leq 2K{v_r^-}h \sqrt{a_k^{{\alpha_0}}} {\operatorname{\Delta}_{h_k,\ell_k}}v_r + 4KM_1 \left(\sum_r a_k^{{\alpha_0}}({\operatorname{\delta}_{h_k,\ell_k}}{v_r^-})^2 \right)^{1/2} + NM_1^2 \\
& \leq NM_1^2 + 2K{v_r^-}h \sqrt{a_k^{{\alpha_0}}} {\operatorname{\Delta}_{h_k,\ell_k}}v_r + \tfrac{1}{4}\sum_r a_k^{{\alpha_0}}({\operatorname{\delta}_{h_k,\ell_k}}{v_r^-})^2, \\
J_2 & \leq NM_1^2 - \tfrac{1}{2}(a_k^{{\alpha_0}}- 4Kh\sqrt{a_k^{{\alpha_0}}}) {v_r^-}{\operatorname{\Delta}_{h_k,\ell_k}}v_r.\end{aligned}$$ Consider $\mathcal{K} := \left\{k : a_k^{{\alpha_0}}- 4Kh \sqrt{a_k^{{\alpha_0}}} \geq 0\right\}$. For all $k$ not in $\mathcal{K}$ we have $a_k^{{\alpha_0}}< Nh^2$. Also $h^2|{\operatorname{\Delta}_{h_k,\ell_k}}v_r| \leq 4M_1$. Hence $$\begin{split}
J_2 & \leq NM_1^2 {\quad \textrm{and} \quad}\\
m^{{\alpha_0}}V\tfrac{1}{\tau}(1-e^{-c_o\tau}) + & ({\operatorname{T}_{h_r,\ell_r}}m^{{\alpha_0}}c^{{\alpha_0}}) V \\
& \leq NM_1\left(e^{c_oT'} + M_0 + m^{{\alpha_0}}M_1\right) + \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r} .
\end{split}$$ We know from (5.5), that $M_1 \leq \mu + V^{1/2}$, where $$\mu := \sup_{\partial_\varepsilon Q,r} |v_r| \leq e^{c_o T'} \sup_{\partial_\varepsilon Q,r} |\operatorname{\delta}_{h_r,l_r} u| =: \bar{\mu}.$$ One also defines $$\bar{M_o} = |u|_{0,Q} \geq e^{-c_0 T'}M_0, \quad \bar{V} = e^{-2c_0 T'}V$$ and so, using Young’s inequality $$\begin{gathered}
\label{eqn-5.15}
m^{{\alpha_0}}V\tfrac{1}{\tau}(1-e^{-c_o\tau}) + ({\operatorname{T}_{h_r,\ell_r}}m^{{\alpha_0}}c^{{\alpha_0}}) V \qquad \qquad \\
\leq N(\bar{\mu}+\bar{V}^{1/2})\left(1 + \bar{M_0} + \bar{\mu} + m^{{\alpha_0}}\bar{V}^{1/2}\right) + \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r} \\
\leq N^*\left(1 + \bar{M_0}^2 + \bar{\mu}^2 + m^{{\alpha_0}}\bar{V}\right) + \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r}.\end{gathered}$$ By assumption (5.3), we can find $c_0$ such that $$\tfrac{1}{\tau}(1-e^{-c_o\tau}) > 1 + N^*.$$ Using this and $\bar{V} \leq V$, (\[eqn-5.15\]) becomes $$m^{{\alpha_0}}V \left(1+{\operatorname{T}_{h_r,\ell_r}}c^{{\alpha_0}}\right) \leq N^*\left(1 + \bar{M_0}^2 + \bar{\mu}^2\right) + \sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r}.$$ By our assumptions on $m^\alpha$ and $c^\alpha$, $m^{{\alpha_0}}(1+{\operatorname{T}_{h_r,\ell_r}}c^\alpha) \geq K^{-1}$ and so $$V \leq KN^*\left(1 + \bar{M_0}^2 + \bar{\mu}^2\right) + K\sum_r \xi {v_r^-}\tfrac{\bar{\varepsilon}}{h_r}.$$ Noticing that $\xi {v_r^-}\leq e^{c_oT'} 2M_0 h_r^{-1}$ $$V \leq N^*\left(1 + \bar{M_0}^2 + \bar{\mu}^2\right) + e^{c_oT'} NM_0 \tfrac{\bar{\varepsilon}}{h_r^2}.$$ Neither $N$, nor $M_0$ depend on choice of $\alpha_0$ and so we can let $\bar{\varepsilon} \to 0$. Then $$V \leq N^*\left(1 + \bar{M_0}^2 + \bar{\mu}^2\right).$$
Theorem \[thm-estimate-on-discrete-derivative\] can now be used to estimate the difference between two solutions to the finite difference problem, based on how much the respective drift and diffusion coefficients, discounting factor and reward functions differ on the domain $Q$.
\[remark-on-c\_0\] For small $\tau > 0$, (\[assumption-on-c\_0\]) is satisfied. Indeed, for $\tau^{-1} > 2N$, where $N$ is the constant arising in (\[eqn-estimate-on-discrete-derivative\]) let $c_0 \geq 0$ satisfy $e^{-c_0 \tau} \leq 1/2$. Then $c_0$ depends on $K,d_1$ only.
\[thm-discrete-case-continuous-dependence-on-data\] Let $$\varepsilon = \sup_{\mathcal{M}_T, A, k}\left(|\sigma^\alpha_k - \hat{\sigma}^\alpha_k| + |b^\alpha_k - \hat{b}^\alpha_k| + |c^\alpha - \hat{c}^\alpha| + m^\alpha|f^\alpha - \hat{f}^\alpha| \right).$$ Assume that the functions $u, \hat{u}$ satisfy (\[eqn-f-d-problem-a\]) on $Q = \mathcal{M}_T$, with the coefficients $\sigma, b, c, f$ and $\hat{\sigma}, \hat{b}, \hat{c}, \hat{f}$ respectively. Then $$|u - \hat{u}| \leq Ne^{c_0 T'}T' \varepsilon I,$$ where $$I := \sup_{(T,x) \in \bar{\mathcal{M}}_T} \left(1 + \max_k(|{\operatorname{\delta}_{h_k,\ell_k}}u| + |{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}| + \tfrac{1}{\varepsilon}|u(T,x)-\hat{u}(T,x)|\right).$$
To prove the theorem we need the following lemma. Recall that $a^\alpha_k = (1/2)(\sigma^\alpha_k)^2$.
\[lemma-discrete-case-continuous-dependence-on-data\] Let $u$ and $\hat{u}$ be functions satisfying (\[eqn-f-d-problem-a\]) on some $Q \subset \mathcal{M}_T$ with the coefficients $\sigma, b, c, f$ and $\hat{\sigma}, \hat{b}, \hat{c}, \hat{f}$ respectively. Assume that ${\varepsilon}\in (0,h]$ and for all $\alpha \in A$ $$\begin{split}
|b^\alpha_k - \hat{b}^\alpha_k| + m^\alpha|f^\alpha - \hat{f^\alpha}| + |c^\alpha - \hat{c}^\alpha| & \leq \varepsilon, \\
|a^\alpha_k - \hat{a}^\alpha_k| & \leq 2\varepsilon\sqrt{a^\alpha_k \land \hat{a}^\alpha_k} + \varepsilon h {\quad \textrm{on} \quad}Q.
\end{split}$$ Then on $Q$ $$\begin{split}
& |u - \hat{u}| \\ & \leq {\varepsilon}Ne^{c_0 T'} \left[1 + |u|_{0,Q} + |\hat{u}|_{0,Q} + \sup_{k, \partial Q}\left(|{\operatorname{\delta}_{h_k,\ell_k}}u| + |{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}| + {\varepsilon}^{-1}|u - \hat{u}| \right) \right].
\end{split}$$
We can deduce this from the gradient estimate by introducing an extra coordinate to the grid and by defining a suitable equation on the extended grid.
We will work with $(t,x) = (t,x',x^{d+1}) \in [0,T] \times \R^d \times \R$. Let $$\begin{split}
\tilde{Q} & := Q \times \{0, \pm {\varepsilon}, \ldots, \pm m {\varepsilon}\}, \\
\tilde{Q}^0_{\varepsilon}& := \{(t,x',x^{d+1}): (t+{\tau_T(t)},x',x^{d+1}) \in \tilde{Q}, (t,x'+h\ell_k,x^{d+1}\pm {\varepsilon}) \in \tilde{Q}, \\
& \qquad \qquad \qquad \qquad \forall k = 0, \pm 1, \ldots, \pm d_1\}\\
\partial_\varepsilon \tilde{Q} & := \tilde{Q} \setminus \tilde{Q}^0_{\varepsilon}= \partial Q \times \{0, \pm {\varepsilon}, \ldots, \pm m{\varepsilon}\} \cup Q^0\times\{m{\varepsilon},-m{\varepsilon}\}.
\end{split}$$ where $m \geq \varepsilon^{-1}$ is a fixed integer. Also let $$\tilde{a}^\alpha_k(t,x',x^{d+1}) = \left\{
\begin{array}{lcl}
a^\alpha_k(t,x') & \text{if} & x^{d+1} > 0, \\
\hat{a}^\alpha_k(t,x') & \text{if} & x^{d+1} \leq 0
\end{array} \right.$$ and define $\tilde{b}^\alpha_k, \tilde{c}^\alpha_k$ in a similar way. Let $$\tilde{f}^\alpha_k(t,x',x^{d+1}) = \left\{
\begin{array}{lcl}
f^\alpha_k(t,x')\left[1 - \tfrac{x^{d+1} - \varepsilon}{\varepsilon m}\right] & \text{if} & x^{d+1} > 0, \\
\hat{f}^\alpha_k(t,x')\left[1+\tfrac{x^{d+1}}{\varepsilon m} \right] & \text{if} & x^{d+1} \leq 0
\end{array} \right.$$ and define $\tilde{u}$ analogously. Then $\tilde{u}$ satisfies (\[eqn-f-d-problem-a\]) with $\tilde{a}^\alpha_k, \tilde{b}^\alpha_k, \tilde{c}^\alpha, \tilde{f}^\alpha$ in place of $a^\alpha_k$, $b^\alpha_k$, $c^\alpha$, $f^\alpha$ respectively and with the normalizing factor $m^\alpha$, on the domain $\tilde{Q}$. To apply Theorem \[thm-estimate-on-discrete-derivative\] to $\tilde{u}$ one needs to ascertain that Assumption \[ass-for-thm-5.2\] is satisfied. For $r = \pm 1, \ldots, \pm d_1$ (\[eqn-first-ass-for-thm-5.2\]) follows from Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\]. Furthermore, for $r = \pm (d_1 + 1)$ notice that $$\operatorname{\delta}_{{\varepsilon}, \ell}\tilde{a}^\alpha_k(t,x) = \left\{
\begin{array}{lcl}
0 & \text{if} & x^{d+1} > 0, \\
\varepsilon^{-1}(a^\alpha_k(t,x') - \hat{a}^\alpha_k(t,x')) & \text{if} & x^{d + 1} = 0, \\
0 & \text{if} & x^{d+1} < 0
\end{array} \right.$$ and similarly for $\tilde{b}^\alpha_k, \tilde{c}^\alpha$. So for $\tilde{b}^\alpha_k$ and $\tilde{c^\alpha}$ (\[eqn-first-ass-for-thm-5.2\]) holds, while (\[eqn-second-ass-for-thm-5.2\]) follows from: $$\varepsilon^{-1}|a^\alpha_k(t,x') - \hat{a}^\alpha_k(t,x')| \leq K\sqrt{\tilde{a}^\alpha_k(t,x',0)} + Kh.$$ Finally $$m^\alpha \operatorname{\delta}_{{\varepsilon}, \ell}\tilde{f}^\alpha(t,x) = \left\{
\begin{array}{lcl}
-m^\alpha f^\alpha(t,x')/(\varepsilon m) & \text{if} & x^{d+1} > 0, \\
\varepsilon^{-1} m^\alpha (f^\alpha(t,x') - \hat{f}^\alpha_k(t,x')) & \text{if} & x^{d + 1} = 0, \\
m^\alpha \hat{f}^\alpha(t,x')/(\varepsilon m) & \text{if} & x^{d+1} < 0
\end{array} \right.$$ shows (\[eqn-first-ass-for-thm-5.2\]) for $\tilde{f}^\alpha$. By Theorem \[thm-estimate-on-discrete-derivative\], for all $(t,x') \in \mathcal{M}_T$ $$\label{eqn-3321}
\begin{split}
{\varepsilon}^{-1} |u(t,x') - \hat{u}(t,x')| = |\operatorname{\delta}_{{\varepsilon}, \ell}\tilde{u}(t,x',0)|
\leq NTe^{c_0(T+\tau)}\\
\times \left[1 + |\tilde{u}|_{0, \tilde{Q}} + \sup_{\partial_{\varepsilon} \tilde{Q}}\left(\max_k|{\operatorname{\delta}_{h_k,\ell_k}}\tilde{u}| + |\operatorname{\delta}_{{\varepsilon},\ell}\tilde{u}| + |\operatorname{\delta}_{{\varepsilon},-\ell}\tilde{u}| \right)\right],
\end{split}$$ where $N$ is, in particular, independent of $m$. To complete the proof consider $\partial_{{\varepsilon}}\tilde{Q}$. Either $(t,x',x^{d+1}) \in \partial Q \times {0,\pm {\varepsilon}, \ldots, \pm m {\varepsilon}}$. Then $${\operatorname{\delta}_{h_k,\ell_k}}\tilde{u}(t,x',x^{d+1}) = \left\{
\begin{array}{lcl}
{\operatorname{\delta}_{h_k,\ell_k}}u(t,x')\left[1 - \tfrac{x^{d+1} - \varepsilon}{\varepsilon m}\right] & \text{if} & x^{d+1} > 0, \\
{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}(t,x')\left[1+\tfrac{x^{d+1}}{\varepsilon m} \right] & \text{if} & x^{d+1} \leq 0.
\end{array} \right.$$ And so $|{\operatorname{\delta}_{h_k,\ell_k}}\tilde{u}(t,x',x^{d+1})| \leq |{\operatorname{\delta}_{h_k,\ell_k}}u(t,x')| + |{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}(t,x')|$. Also notice $$\operatorname{\delta}_{{\varepsilon},\ell} \tilde{u}(t,x',x^{d+1}) = \left\{
\begin{array}{lcl}
0 & \text{if} & x^{d+1} \neq 0, \\
{\varepsilon}^{-1}(u(t,x')-\hat{u}(t,x')) & \text{if} & x^{d+1} = 0.
\end{array} \right.$$ $$\operatorname{\delta}_{{\varepsilon},-\ell} \tilde{u}(t,x',x^{d+1}) = \left\{
\begin{array}{lcl}
0 & \text{if} & x^{d+1} \neq 0, \\
{\varepsilon}^{-1}(\hat{u}(t,x')-u(t,x')) & \text{if} & x^{d+1} = 0.
\end{array} \right.$$ Or $(t,x',x^{d+1}) \in Q^0\times \{m{\varepsilon},-m{\varepsilon}\}$. Then $${\operatorname{\delta}_{h_k,\ell_k}}\tilde{u}(t,x',x^{d+1}) = \left\{
\begin{array}{lcl}
{\operatorname{\delta}_{h_k,\ell_k}}u(t,x')\left[1 - \tfrac{{\varepsilon}m - \varepsilon}{\varepsilon m}\right] & \text{if} & x^{d+1} = m{\varepsilon}, \\
{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}(t,x')\left[1+\tfrac{-{\varepsilon}m}{\varepsilon m} \right] & \text{if} & x^{d+1} =-m{\varepsilon}.
\end{array} \right.$$ Notice that $[1 - \tfrac{{\varepsilon}m - {\varepsilon}}{{\varepsilon}m}] = 1/m$ and $[1+\tfrac{-{\varepsilon}m}{{\varepsilon}m}] = 0$. Since $m \geq {\varepsilon}^{-1}$, $m{\varepsilon}\geq 1$ and so for $(t,x',x^{d+1}) \in Q^0\times \{m{\varepsilon},-m{\varepsilon}\}$ $$\operatorname{\delta}_{{\varepsilon},\ell}\tilde{u} = \operatorname{\delta}_{{\varepsilon},-\ell}\tilde{u} = 0.$$ Hence $$\begin{split}
& \sup_{\partial_{\varepsilon} \tilde{Q}}\left(\max_k|{\operatorname{\delta}_{h_k,\ell_k}}\tilde{u}| + |\operatorname{\delta}_{{\varepsilon},\ell}\tilde{u}| + |\operatorname{\delta}_{{\varepsilon},-\ell}\tilde{u}| \right)\\
& \leq \sup_{k,\partial Q}\Big(|{\operatorname{\delta}_{h_k,\ell_k}}u| + 2|{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}| + 2{\varepsilon}^{-1}|u - \hat{u}| + |{\operatorname{\delta}_{h_k,\ell_k}}u|\tfrac{1}{m} \Big).
\end{split}$$ Since the constant $N$ in (\[eqn-3321\]) is independent of $m$, we can let $m \to \infty$ and the conclusion of the lemma follows.
Let $l = (0,\ldots,0,1) \in \R^{d+1}$. From our definition of $\varepsilon$ we clearly have $$\begin{aligned}
|b^\alpha_k - \hat{b}^\alpha_k| + m^\alpha|f^\alpha - \hat{f^\alpha}| + |c^\alpha - \hat{c}^\alpha| \leq \varepsilon.\end{aligned}$$ Assume initially that $\varepsilon \in (0,h]$. Then, using that for any $a,b \geq 0$, $$|a^2 - b^2| = (a+b)|a-b| = 2(a \land b)|a-b| + |a-b|^2$$ we get $$|a^\alpha_k - \hat{a}^\alpha_k| \leq (|\sigma^\alpha_k| \land |\hat{\sigma}^\alpha_k|)|\sigma^\alpha_k - \hat{\sigma}^\alpha_k| + |\sigma^\alpha_k - \hat{\sigma}^\alpha_k|^2 \leq 2\varepsilon\sqrt{a^\alpha_k \land \hat{a}^\alpha_k} + \varepsilon h.$$ So we can use \[lemma-discrete-case-continuous-dependence-on-data\] for $Q=\mathcal{M}_T$ to get $$\begin{split}
& |u - \hat{u}| \\
& \leq {\varepsilon}Ne^{c_0 T'} \left[1 + |u|_{0,Q} + |\hat{u}|_{0,Q} + \sup_{k, \partial Q}\left(|{\operatorname{\delta}_{h_k,\ell_k}}u| + |{\operatorname{\delta}_{h_k,\ell_k}}\hat{u}| + {\varepsilon}^{-1}|u - \hat{u}| \right) \right],
\end{split}$$ on $\mathcal{M}_T$. Use Corollary \[corollary-soln-to-disc-bellman-pde-is-bdd\] to estimate the term $|u|_{0,\mathcal{M}_T} + |\hat{u}|_{0,\mathcal{M}_T}$ to obtain $$|u - \hat{u}| \leq \varepsilon NTe^{c_0(T+\tau)} I.$$ Drop the initial assumption that $\varepsilon \leq h$. For $\theta \in [0,1]$, let $u^\theta$ be the solution of $$\begin{aligned}
\sup_{\alpha}m^\alpha({\operatorname{\delta}_\tau}u^\theta + a^{\theta\alpha}_k {\operatorname{\Delta}_{h_k,\ell_k}}u^\theta + b^{\theta\alpha}_k {\operatorname{\delta}_{h_k,\ell_k}}u^\theta - c^{\theta\alpha}u^\theta + f^{\theta\alpha}) = 0 {\quad \textrm{on} \quad}\mathcal{M}_T\\
(1-\theta)u + \theta \hat{u} = u^\theta {\quad \textrm{on} \quad}\{(T,x) \in \bar{\mathcal{M}}_T\},\end{aligned}$$ where $$(\sigma^{\theta\alpha}_k, b^{\theta\alpha}_k, c^{\theta\alpha}, f^{\theta\alpha}) = (1-\theta)(\sigma^\alpha_k, b^\alpha_k, c^\alpha, f^\alpha) + \theta(\hat{\sigma}^\alpha_k, \hat{b}^\alpha_k, \hat{c}^\alpha, \hat{f}^\alpha)$$ and $a^{\theta\alpha}_k = (1/2)|\sigma^{\theta\alpha}_k|^2.$ For any $\theta_1, \theta_2 \in [0,1]$, $$|\sigma^{\theta_1\alpha}_k - \sigma^{\theta_2 \alpha}_k| + |b^{\theta_1\alpha}_k - b^{\theta_2 \alpha}_k| + |c^{\theta_1\alpha} - c^{\theta_2 \alpha}| + m^\alpha|f^{\theta_1\alpha} - f^{\theta_2 \alpha}| \leq |\theta_1 - \theta_2|\varepsilon.$$ Hence if $\theta_1, \theta_2$ satisfy $|\theta_1 - \theta_2|\varepsilon \leq h$, then, thanks to the first part of the proof (where $u^{\theta_1}$ would play the part of $u$ while $u^{\theta_2}$ would play the part of $\hat{u}$), $$|u^{\theta_1} - u^{\theta_2}| \leq N |\theta_1 - \theta_2|\varepsilon T' e^{c_0(T+\tau)}I(\theta_1, \theta_2),$$ with $$\begin{split}
I(\theta_1, \theta_2) := \sup_{\partial_{\varepsilon} Q}\bigg(1 + \max_k (|{\operatorname{\delta}_{h_k,\ell_k}}u^{\theta_1}| + |{\operatorname{\delta}_{h_k,\ell_k}}u^{\theta_2}|) \\
(|\theta_1 - \theta_2|\varepsilon)^{-1}|u^{\theta_1}(T,x)-u^{\theta_2}(T,x)| \bigg) \leq 4I,
\end{split}$$ since $u^\theta = (1-\theta)u + \theta \hat{u}$ on $\partial_{\varepsilon}Q = \{(T,x) \in \bar{\mathcal{M}_T}\}$. Subdivide the interval $[0,1]$ into intervals of appropriate length to complete the proof for any $\varepsilon > 0$.
Some Properties of Payoff Functions {#section-analytic-ppties}
===================================
We assume in the whole section that Assumption \[ass-for-optimal-control-problem\] and the following assumption hold.
\[assumptions-for-properties-of-payoff-function\] There is a function $m:A \to (0,1]$, such that for all $\alpha \in A$, $t\in[0,T)$, $x\in \R^d$ and $y \in \R^d$ $$\begin{gathered}
m^\alpha c^\alpha(t,x) + m^\alpha \geq K^{-1},\quad |m^\alpha f^\alpha(t,x)| \leq K,\\
m^\alpha|f^\alpha(t,x) - f^\alpha(t,y)| \leq K|x-y|.\end{gathered}$$
\[lemma-lipschitz-continuity\] There exists a constant $N$ such that:
1. For any $(s,x) \in [0,T] \times \R^d$, $$v(s,x) \leq N(1 + (T-s)).$$
2. For any $(s,x), (s,y) \in [0,T] \times \R^d$, $$|v(s,x) - v(s,y)| \leq N(T-s)^2 e^{N(T-s)}|x-y|.$$
Since $m^\alpha c^\alpha(t,x) + m^\alpha \geq K^{-1}$ and $|m^\alpha f^\alpha(s,x)| \leq K$, $$|f^\alpha(s,x)| \leq K^2(1+c^\alpha(s,x)).$$ This and the assumption on the polynomial growth of $g$ together with the estimates of moments for solutions to SDEs gives $$\begin{split}
|v^\alpha (s,x)| \leq & \E_{s,x}^\alpha \int_0^{T-s}|f^{\alpha_t}(s+t,x_t)|e^{-\varphi_t}dt + \E_{s,x}^\alpha|g(x_{T-s})|e^{-\varphi_{T-s}} \\
\leq & K^2\E_{s,x}^\alpha \int_0^{T-s} (1+c^{\alpha_t}(s+t,x_t))e^{-\int_0^tc^{\alpha_u} (s+u,x_u)du} dt + N \\
\leq & N(1+(T-s)) + K^2 \E\int_0^{T-s} c^{\alpha_t}(s+t,x_t)e^{-\int_0^tc^{\alpha_u} (s+u,x_u)du} dt \\
\leq & N(1+(T-s)).
\end{split}$$ The right hand side of the estimate is independent of $\alpha$, hence taking the supremum completes the first part of the proof. If $x^{s,x}_t$ is a solution to a stochastic differential equation $$x^{s,x}_t = x + \int_0^t \sigma(s+u,x_u)dw_u + \int_0^t \beta(s+u,x_u)du$$ where $$|\sigma(t,x) - \sigma(t,y)| \leq L|x-y| {\quad \textrm{and} \quad}|\beta(t,x) - \beta(t,y)| \leq L|x-y|,$$ then, by Theorem 2.5.9 from [@krylov:controlled], with $N$ depending on $L$ only, $$\label{eqn-lip-ppty-of-sde}
\E \sup_{t \leq T-s} |x^{s,x}_t - x^{s,y}_t| \leq Ne^{N(T-s)}|x-y|.$$ First get an estimate for a fixed $\alpha \in A$. $$\begin{aligned}
|v^{\alpha}(s,x) - v^{\alpha}(s,y)| & \leq & \E \int_0^{T-s}|f^{\alpha_t}(s+t,x^{s,x}_t)||e^{-\varphi_t^{\alpha,s,x}}-e^{-\varphi_t^{\alpha,s,y}}|dt \\
& + &\E \int_0^{T-s} |f^{\alpha_t}(s+t,x^{s,x}_t) - f^{\alpha_t}(s+t,x^{s,y}_t)|e^{-\varphi_t^{\alpha,s,y}} dt \\
& + & \E |g(x^{s,x}_{T-s}) - g(x^{s,y}_{T-s})| =: I_1 + I_2 + I_3.\end{aligned}$$ Estimating the above integrals separately: $$I_1 \leq \E \int_0^{T-s}|f^{\alpha_t}(s+t,x^{s,x}_t)||\varphi_t^{\alpha,s,x} - \varphi_t^{\alpha,s,y}|e^{-\min(\varphi_t^{\alpha,s,x},\varphi_t^{\alpha,s,y})}dt.$$ Since $m^\alpha c^\alpha(t,x) + m^\alpha \geq K^{-1}$ and $|m^\alpha f^\alpha(s,x)| \leq K$, $$|f^\alpha(s,x)| \leq \frac{K}{m^\alpha} {\quad \textrm{and} \quad}c^\alpha(s,x) \geq \frac{1}{Km^\alpha} - 1.$$ Therefore, using Lipschitz continuity of $c$ when estimating $|\varphi_t^{\alpha,s,x} - \varphi_t^{\alpha,s,y}|$, $$\begin{aligned}
I_1 & \leq & (T-s)e^{T-s}K^3 \E \int_0^{T-s}\frac{1}{Km^{\alpha_t}}\sup_{r\leq t}|x^{\alpha,s,x}_r-x^{\alpha,s,y}_r|e^{-\int_0^t \frac{1}{Km^{\alpha_u}} du} dt \\
& \leq & (T-s)e^{T-s}K^3 \E\sup_{t\leq T-s}|x^{\alpha,s,x}_t-x^{\alpha,s,y}_t| \int_0^{T-s} -de^{-\int_0^t \frac{1}{Km^{\alpha_u}} du}.\end{aligned}$$ Now $$\begin{split}
I_2 & \leq \int_0^{T-s}(m^{\alpha_t})^{-1}|x^{\alpha,s,x}_t - x^{\alpha,s,y}_t| e^{-\varphi^{\alpha,s,y}} dt \\
&\leq \E \left(\sup_{t\leq T-s}|x^{\alpha,s,x}_t-x^{\alpha,s,y}_t| \int_0^{T-s} K(1+c^{\alpha_t}(s+t,x^{\alpha,s,y}_t)e^{-\varphi^{\alpha,s,y}} dt \right) \\
& \leq \left[K(t-s) + K\right] \E \sup_{t\leq T-s}|x^{\alpha,s,x}_t-x^{\alpha,s,y}_t|.
\end{split}$$ and $$I_3 \leq K (T-s)\E \sup_{t\leq T-s}|x^{\alpha,s,x}_t-x^{\alpha,s,y}_t|.$$ Hence $$|v^{\alpha}(s,x) - v^{\alpha}(s,y)| \leq K^3 (T-s)^2 e^{(T-s)} \E \sup_{t\leq T-s}|x^{\alpha,s,x}_t-x^{\alpha,s,y}_t|.$$ By (\[eqn-lip-ppty-of-sde\]) the right hand side is independent of $\alpha$.
Let $H_T := [0,T) \times R^d$ and $\partial H_T := \{T\}\times \R^d$.
\[lemma-smooth-supersolution-and-payoff-relation\] Let $\psi$ be a smooth function on $H_T$ such that its first order partial derivatives in $x$ grow at most polynomially and for all $\alpha \in A$ it satisfies $$\frac{\partial}{\partial t}\psi + L^{\alpha} \psi - c^\alpha \psi + f^{\alpha} \leq 0 {\quad \textrm{on} \quad}H_T.$$ Let $v$ be the payoff function of the stochastic control problem (\[eq:payofffn\]). Then $$\label{eq:supersoln}
v \leq \psi + \sup_{\partial H_T} [v - \psi]_+ {\quad \textrm{on} \quad}H_T.$$ Hence, clearly, if $v \leq \psi$ on $\partial H_T$, then $v \leq \psi$ on $H_T$.
For any $\varepsilon > 0$ there is a control process $\alpha_t \in {\mathfrak{A}}$ such that $$\begin{split}
& v(s,x) \leq \varepsilon + \E_{s,x} \left(\int_0^{T-s} f^{\alpha_t}(s+t,x_t)e^{-\varphi_t} dt + g(x_{T-s})e^{-\varphi_{T-s}}\right) \\
& \leq \varepsilon - \E_{s,x}\int_0^{T-s}e^{-\varphi_t}\left[\frac{\partial \psi}{\partial t}(s+t,x_t) + L^{\alpha_t}\psi(s+t,x_t) - c^{\alpha_t}\psi(s+t,x_t) \right] dt \\
& + \E_{s,x} g(x_{T-s})e^{-\varphi_{T-s}}.
\end{split}$$ Applying Itô’s formula to $\psi(s+r,x_r)e^{-\varphi_r}$ on the interval $[0,T-s]$ and taking expectation we get $$\begin{split}
\psi(s,x) = & \E_{s,x} \psi(T,x_{T-s})e^{-\varphi_{T-s}} \\
& -\E_{s,x} \int_0^{T-s} e^{-\varphi_t}
\left[{\frac{\partial}{\partial t}}+ L^{\alpha} - c^{\alpha}\right]\psi(s+t,x_t) dt,
\end{split}$$ by noting that the Itô integral has zero expectation, due to moment estimates for $x_r$, since $\psi$ has polynomially growing first order partial derivatives in $x$. Hence $$v(s,x) \leq \varepsilon + \psi(t,x) + \sup_{x \in \R^d} \left[v(T,x) - \psi(T,x)\right]_+,$$ for any ${\varepsilon}> 0$, which yields .
\[lemma-”subsoln”-ppty\] Let $\psi$ be a smooth function on $H_T$ such that its first order partial derivatives in $x$ grow at most polynomially and there exists $\alpha_0 \in A$ such that $$\frac{\partial}{\partial t}\psi + L^{\alpha_0} \psi - c^{\alpha_0} \psi + f^{\alpha_0} \geq 0 {\quad \textrm{on} \quad}H_T.$$ Let $v$ be the payoff function of the stochastic control problem (\[eq:payofffn\]). Then $$\psi \leq v + \sup_{\partial H_T} [\psi - v]_+ {\quad \textrm{on} \quad}H_T.$$ Hence, clearly, if $\psi \leq v$ on $\partial H_T$, then $\psi \leq v$ on $H_T$.
For the constant strategy $\alpha_t := \alpha_0$ for all $t \in [0,T-s]$ $$\begin{split}
& v(s,x) \geq \E_{s,x}^{\alpha} \left(\int_0^{T-s} f^{\alpha_0}(s+t,x_t)e^{-\varphi_t} dt + g(x_{T-s})e^{\varphi_{T-s}} \right) \\
& \geq \E^{\alpha}_{s,x}\bigg(\int_0^{T-s} - \left[\frac{\partial}{\partial t} + L^{\alpha_0} - c^{\alpha_0} \right]\psi(s+t,x_t) e^{-\varphi_t}dt
+ g(x_{T-s})e^{-\varphi_{T-s}} \bigg).
\end{split}$$ By Itô’s formula for $\psi(s+r,x_r)e^{-\varphi_r}$ on the interval $[0,T-s]$ $$\psi(s,x) \leq v(s,x) + \sup_{x\in \R^d} [\psi(T,x) - v(T,x)]_+.$$
\[corollary-holder-continuity\] Assume (\[eqn-boundedness-of-diff-and-drift\]) holds and for all $\alpha \in A$ and $(t,x) \in H_T$ $$\label{ass-for-corollary-holder-continuity}
m^\alpha|f^\alpha| \leq K {\quad \textrm{and} \quad}m^\alpha c^\alpha \leq K.$$ Then there exists a constant $N$ such that for $(s_0,x_0) \in H_T$ and $(t_0,x_0) \in H_T$ satisfying $s_0 - 1 \leq t_0 \leq s_0$ one has $$|v(s_0,x_0) - v(t_0,x_0)| \leq N(\nu + 1)|s_0 - t_0|^\frac{1}{2},$$ where $\nu$ is the Lipschitz constant from part 2 of Lemma \[lemma-lipschitz-continuity\].
Shift the time axis so that $t_0 =0$, $s_0 \leq 1$. Let $(t,x) \in H_{s_0} := [0,s_0)\times \R^d$. Define $\xi(t) = e^{s_0 - t}$. Let $\zeta = \xi \eta$, $\eta(x) = |x-x_0|^2$ and $\gamma := s_0^{-1/2}$. Let $$\psi(t,x) = \gamma \nu \left[\zeta(t,x) + \kappa(s_0 - t)\right] + \frac{1}{\gamma}\nu \kappa + v(s_0,x_0),$$ where $\kappa > 1$ is a large positive constant to be chosen later. To apply Lemma \[lemma-smooth-supersolution-and-payoff-relation\] we need to show that for all $\alpha \in A$ $$m^\alpha \bigg({\frac{\partial}{\partial t}}\psi + L^\alpha \psi -c^\alpha \psi + f^\alpha \bigg) \leq 0 {\quad \textrm{on} \quad}H_{s_0}$$ and $\psi(s_0,x) \geq v(s_0,x)$ for any $x \in \R^d$. Notice $$\begin{gathered}
{\frac{\partial}{\partial t}}\psi = -\gamma \nu \eta \xi - \gamma \nu \kappa, \quad
\frac{\partial}{\partial x_i} \eta(x) = 2(x^i - x_0^i), \quad \frac{\partial^2}{\partial x_i \partial x_j} \eta = 2 \delta_{i,j},\end{gathered}$$ where $\delta_{i,j}$ is the Kronecker delta. Using (\[eqn-boundedness-of-diff-and-drift\]) and $\xi \leq e^T$, $$L^\alpha \psi \leq \gamma \nu 2Kd(1+|x-x_0|) \xi \leq \gamma \nu 2Kd(1+|x-x_0|) e^T.$$ Since $v$ is bounded by $N_T$, for any $\alpha \in A$ and $(t,x) \in H_{s_0}$ $$-c^\alpha(t,x)v(s_0,x_0) \leq c^\alpha(t,x) N_T.$$ Hence for any $\alpha \in A$ $$-c^\alpha \psi \leq - c^\alpha \frac{1}{\gamma}\nu \kappa + c^\alpha N_T {\quad \textrm{on} \quad}H_{s_0}.$$ Then for any $\alpha \in A$ and $(t,x) \in H_{s_0}$, with $N_2 := 2Ke^Td$ and by (\[ass-for-corollary-holder-continuity\]), $$\begin{gathered}
m^\alpha \bigg({\frac{\partial}{\partial t}}\psi + L^\alpha \psi -c^\alpha \psi + f^\alpha \bigg) \\
\leq m^\alpha \left\{\gamma \nu \left[-|x-x_0|^2 + N_2(1+|x-x_0|) - \kappa \right] - c^\alpha \frac{\kappa}{\gamma}\nu + c^\alpha N_T \right\} + K.\end{gathered}$$ The right hand side is a quadratic in $|x-x_0|$ with a negative leading coefficient and so it achieves its maximum when $|x-x_0| = N_2/2$. The maximum is $$\label{eqn-minimum-in-proof-of-holder-ctns-case}
\begin{split}
m^\alpha\left\{\gamma \nu \left[-\frac{N_2}{4} +N_2 + \frac{N_2^2}{2} - \kappa \right] + c^\alpha \kappa \gamma^{-1}\nu + c^\alpha N_T \right\} + K \leq M - L\kappa,
\end{split}$$ where $L := K^{-1}\nu$ and $M$ depends only on $N$, $d$ and $T$. Indeed, $s_0 \leq 1$ so $\gamma \geq 1$. Since $m^\alpha(1+c^\alpha) \geq K^{-1}$ on $H_T$, $m^\alpha(\gamma^2+c^\alpha) \geq (1 \wedge \gamma^2) K^{-1}$ on $H_T$. Hence $$\kappa\nu\gamma^{-1} m^\alpha(\gamma^2 + c^\alpha) \geq \kappa \nu \gamma^{-1} (\gamma^2 \wedge 1) K^{-1} \geq K^{-1}\nu \kappa = L\kappa.$$ If $\kappa$ is large (depending on $K,T,d$ only) then the right hand side of (\[eqn-minimum-in-proof-of-holder-ctns-case\]) is non-positive. Hence $$m^\alpha \bigg({\frac{\partial}{\partial t}}\psi + L^\alpha \psi -c^\alpha \psi + f^\alpha \bigg) \leq 0.$$ By Lemma \[lemma-lipschitz-continuity\] $$\begin{split}
\psi(s_0,x) & = \gamma \nu |x-x_0|^2 + \nu \kappa \gamma^{-1} + v(s_0,x_0) \\
& \geq \gamma \nu |x-x_0|^2 + \nu \kappa \gamma^{-1} - \nu|x-x_0| + v(s_0,x) .
\end{split}$$ The reader can check that $$\gamma \nu |x-x_0|^2 + \nu \kappa \gamma^{-1} - \nu|x-x_0| \geq 0,$$ since it is a quadratic in $|x-x_0|$, its leading coefficient is positive and hence its minimum is $ 4^{-1} \nu \gamma^{-1} + \nu \kappa\gamma^{-1} - 2^{-1}\nu \gamma^{-1}$. Hence for all $x \in \R^d$, $$\psi(s_0,x) \geq v(s_0,x).$$ By Lemma \[lemma-smooth-supersolution-and-payoff-relation\], $\psi(t_0,x_0) \geq v(t_0,x_0)$. But $$v(t_0,x_0) \leq \psi(t_0,x_0) = \gamma \nu \kappa s_0 + \gamma^{-1} \nu \kappa + v(s_0,x_0).$$ Hence, recalling that $\gamma = s_0^{-1/2}$, $$v(t_0,x_0) \leq v(s_0,x_0) + s_0^{-1/2} \nu \kappa s_0 + s_0^{1/2} \nu \kappa \leq v(s_0,x_0) + N_T(\nu+1)s_0^{1/2}.$$ What remains to be shown is that $$v(s_0,x_0) \leq v(t_0,x_0) + N_T(\nu+1)s_0^{1/2}.$$ But that can be achieved by considering $$\psi = - \gamma \nu \left[\zeta + \kappa(s_0 - t)\right] - \frac{1}{\gamma}\nu \kappa + v(s_0,x_0)$$ and applying Lemma \[lemma-”subsoln”-ppty\] to $\psi$.
Shaking the Coefficients {#section-shaking}
========================
The method of shaking the coefficients first introduced in [@krylov:approximating:value] and [@krylov:rate:variable] will be used. Recall that ${v_{\tau,h}}$ is the solution of (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) with $Q = {\mathcal{M}_T}$. Recall that $v$ is the payoff function (\[eq:payofffn\]).
\[remark-on-extending-v\_tau\_h\]
For $(t_0,x_0)$ not in $\bar{\mathcal{M}}_T$ define ${v_{\tau,h}}(t_0,x_0)$ as the solution to (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) on $$\begin{split}
{\mathcal{M}_T}(t_0,x_0) = \big\{\left((t_0 + j\tau) \wedge T,x_0 + h(i_1\ell_1+\ldots + i_{d_1}\ell_{d_1} \right): \\
j\in {\mathbb{N}}\cup \{0\}, i_k \in {\mathbb{Z}}, k = 1,\ldots, d_1 \big\}.
\end{split}$$
Let $B_1$ denote the unit ball centered at the origin, in $R^d$. Consider $y \in S \subset B_1$ and $r \in \Lambda \subset (-1,0)$. Fix $\varepsilon > 0$. Let ${v_{\tau,h}}^\varepsilon$ be the unique solution of $$\label{eqn-shaked-discrete-pde}
\begin{split}
\sup_{\alpha \in A, y \in S, r \in \Lambda} m^\alpha \Big[{\operatorname{\delta}_\tau}u(t,x) + {{\operatorname{L}}_h^\alpha}(t+\varepsilon^2r,x+\varepsilon y) u(t,x)& \\
+ f^\alpha(t+\varepsilon^2r,x+\varepsilon y) \Big] & = 0 \quad \textrm{on}\quad \mathcal{M}_T,
\\
u(T,x) & = \sup_{y\in S}g(x+\varepsilon y).
\end{split}$$ Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\] is satisfied by (\[eqn-shaked-discrete-pde\]) and so the solution exists, is unique and has all the other properties proved in section \[section-existence-of-soln-to-disc-prob\]. Assume, from now on, that $\tau$ is small enough, so that by Remark \[remark-on-c\_0\], (\[assumption-on-c\_0\]) holds.
\[lemma-difference-between-soln-to-disc-and-shaked-soln\] Let Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\] hold. Then there is a constant $N$, such that $$|v^\varepsilon_{\tau,h} - {v_{\tau,h}}| \leq Ne^{N T}T \varepsilon.$$
Let the space of controls be $A \times S \times \Lambda$. Let $$(\hat{\sigma}_k^\alpha, \hat{b}_k^\alpha, \hat{c}^\alpha, \hat{f}^\alpha)(t,x) := (\sigma_k^\alpha, b_k^\alpha, c^\alpha, f^\alpha)(t+\varepsilon^2 r,x+\varepsilon y).$$ Then $$\sup_{\mathcal{M}_T, A \times S \times \lambda, k}\left(|\sigma^\alpha_k - \hat{\sigma}^\alpha_k| + |b^\alpha_k - \hat{b}^\alpha_k| + |c^\alpha - \hat{c}^\alpha| + m^\alpha|f^\alpha - \hat{f}^\alpha| \right) \leq 8K\varepsilon.$$ Apply Theorem \[thm-discrete-case-continuous-dependence-on-data\] to get the conclusion.
The Lipschitz continuity of the solutions to (\[eqn-f-d-problem-a\])-(\[eqn-f-d-problem-b\]) follows. Consider a particular case of (\[eqn-shaked-discrete-pde\]). Let $\Lambda = \{0\}$. Let $\varepsilon = |x-y|$. Fix $x,y \in \R^d$ and let $S = \left\{\varepsilon^{-1}(x-y)\right\}$. Say $z \in S$. Then by uniqueness $$v^\varepsilon_{\tau,h}(t,x) = {v_{\tau,h}}(x-\varepsilon z) = {v_{\tau,h}}(t,y).$$ Hence one obtains:
\[corollary-lipschitz-of-soln-to-f-d-problem\] Let Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\] hold. Then there is a constant $N$ such that for any $t \in [0,T]$ and $x,y \in \R^d$, $$|{v_{\tau,h}}(t,x) - {v_{\tau,h}}(t,y)| \leq NTe^{NT}|x-y|.$$
Let Assumptions \[ass-for-the-existence-of-soln-to-discrete-problem\] and \[ass-for-main-result\] hold. Then there is a constant $N$ such that for any $s,t \in [0,T]$ and $x \in \R^d$, $$|{v_{\tau,h}}(t,x) - {v_{\tau,h}}(s,x)| \leq NTe^{NT} |t-s|^{1/2}.$$
The proof of this corollary is identical to the proof of (6.8) of Lemma 6.2 in [@krylov:rate:lipschitz], if one uses Lemma \[corollary-discrete-case-conditional-holder-continuity\], Corollary \[corollary-lipschitz-of-soln-to-f-d-problem\] and Theorem \[thm-discrete-case-continuous-dependence-on-data\] instead of Corollary 3.7, Lemma 6.1 and Theorem 5.6 from [@krylov:rate:lipschitz], respectively.
In the continuous case, the shaking will be introduced as follows. Let Assumptions \[ass-for-optimal-control-problem\] and \[assumptions-for-properties-of-payoff-function\] hold. Consider the separable metric space $$C = A \times \{(\tau,\xi) \in (-1,0) \times B_1\},$$ with the metric which comes from taking the sum of the metric for $A$ and the metrics which are induced by natural norms on $(-1,0)$ and $B_1$. Extend all the functions $\sigma, \beta, f, c$ for negative $t$ by $\sigma^\gamma(t,x)= \sigma^\gamma(0,x)$ etc. For a fixed $\varepsilon \in (0,1)$, for $\gamma = (\alpha, \tau, \xi)$ let $$\sigma^\gamma(t,x) = \sigma^\alpha(t+\varepsilon^2 \tau, x+\varepsilon \xi)$$ and similarly for $\beta, c$ and $f$. Let $x^{\gamma,s,x}_t$ be the solution of $$\label{eqn-shaked-sde}
x^{s,x}_t = x + \int_0^t \sigma^\gamma(s+u,x_u)dw_u + \int_0^t \beta^\gamma(s+u,x_u)du.$$ Let $C_n := A_n \times \{(\tau,\xi) \in (-1,0) \times B_1\}$, $C = \bigcup_{n \in {\mathbb{N}}}C_n$. Let ${\mathfrak{C}}_n$ be the spaces of admissible control processes defined analogously to ${\mathfrak{A}}_n$ and ${\mathfrak{C}}$ defined analogously to ${\mathfrak{A}}$. Let $$\begin{aligned}
w^\gamma(s,x) & = & \E^\gamma_{s,x}\left[\int_0^{T-s}f^{\gamma_t}(t,x_t)e^{-\varphi_t}dt + g(x_{T -s})e^{-\varphi_{T-s}} \right],\label{eqn-shaked-value-funcion-fixed-control}\\
w_n(s,x) & = & \sup_{\gamma \in {\mathfrak{C}}_n} w^\gamma(s,x), \label{eqn-shaked-value-function-bdd}\\
w(s,x) & = & \sup_{\gamma \in {\mathfrak{C}}} w^\gamma(s,x) \label{eqn-shaked-value-function}.\end{aligned}$$
If $w^\gamma$ and $w$ are defined by (\[eqn-shaked-value-funcion-fixed-control\]) and (\[eqn-shaked-value-function\]), respectively, then $$|w^\gamma - v^\alpha| \leq Ne^{NT}\varepsilon {\quad \textrm{and} \quad}|w-v| \leq Ne^{NT}\varepsilon {\quad \textrm{on} \quad}H_T.$$
Let $x^{\gamma,s,x}$ and $x^{\alpha,s,x}$ denote the solutions to (\[eqn-shaked-sde\]) and (\[eqn-sde\]) respectively. By Theorem 2.5.9 of [@krylov:controlled] $$\begin{split}
\E \sup_{t\leq T-s} |x^{\gamma,s,x}_t - x^{\alpha, s, x}_t|^2 \leq & Ne^{NT} \sup \big[ || \sigma^\alpha(t+\varepsilon^2 \tau,x+\varepsilon \xi) - \sigma^\alpha(t,x)||^2 \\
& + |\beta^\alpha(t+\varepsilon^2 \tau,x+\varepsilon \xi) - \beta^\alpha(t,x)|^2 \big].
\end{split}$$ Hence, using Hölder inequality $$\E \sup_{t\leq T-s} |x^{\gamma,s,x}_t - x^{\alpha, s, x}_t| \leq Ne^{NT}\varepsilon.$$ With this in mind the reader could use the same technique used in proving the second part of Lemma \[lemma-lipschitz-continuity\] in order to get the desired estimate.
\[thm-existence-smooth-supersolution\] For any $\varepsilon \in (0,1]$ there exists $u$ in $C^\infty([0,T]\times \R^d)$ such that $$\sup_{\alpha \in A} \left(u_t + {{\operatorname{L}}^\alpha}u + f^{\alpha}\right) \leq 0 {\quad \textrm{on} \quad}H_{T-\varepsilon^2}\label{eqn-smooth-supersoln-ppty}$$ $$|u-v| \leq Ne^{NT}\varepsilon {\quad \textrm{on} \quad}H_T, \label{eqn-smooth-supersoln-estimate}$$ $$\begin{split}
|\operatorname{D}^2_tu|_{0,[0,T]\times \R^d} + |\operatorname{D}^4_xu|_{0,[0,T]\times \R^d} & \leq Ne^{NT} \varepsilon^{-3} {\quad \textrm{on} \quad}H_T, \\
|\operatorname{D}^2_xu|_{0,[0,T]\times \R^d} & \leq Ne^{NT} \varepsilon^{-1} {\quad \textrm{on} \quad}H_T. \label{eqn-estimates-on-derivatives-of-smooth-sup-soln}
\end{split}$$
Recall that $A = \bigcup_{n\in {\mathbb{N}}} A_n$. By Theorem 2.1 of [@krylov:rate:variable], there exist a smooth function $u_n$ defined on $H_T$ such that for all $\alpha \in A_n$, $$\label{eqn-supersoln-ppty-in-the-bdd-case}
\frac{\partial}{\partial t} u_n + {{\operatorname{L}}^\alpha}u_n + f^\alpha \leq 0 {\quad \textrm{on} \quad}H_{T-\varepsilon^2}.$$ Let $w_n$ be defined by (\[eqn-shaked-value-function-bdd\]). Notice that $f^\gamma$, $c^\gamma$, $\sigma^\gamma$ and $b^\gamma$ in the definition all depend on $\varepsilon$ and $x_t$ is defined by (\[eqn-shaked-sde\]). Let $\zeta \in C_0^\infty((-1,0)\times B_1)$ be non-negative with unit integral. Let $\zeta_\varepsilon(t,x) := \varepsilon^{-d-2}\zeta(t/\varepsilon^2, x/\varepsilon)$. By the proof of Theorem 2.1 of [@krylov:rate:variable], $u_n = w_n * \zeta_\varepsilon$. Let $w$ be defined by (\[eqn-shaked-value-function\]) and let $u := w * \zeta_\varepsilon$. By Lemma (\[lemma-lipschitz-continuity\]) the functions $w, w_n$ are bounded in absolute value by a constant independent of $n$. Then by Lebesgue Dominated Convergence Theorem $$\begin{split}
\lim_{n \to \infty} \frac{\partial}{\partial t} u_n(t,x) & = \lim_{n \to \infty}\frac{\partial}{\partial t} \int_{(-1,0)\times B_1}\zeta_\varepsilon (t-s,x-y) w_n(s,y) ds dy \\
& = \int_{(-1,0)\times B_1} \lim_{n \to \infty} w_n(s,y) \frac{\partial}{\partial t} \zeta_\varepsilon (t-s,x-y) ds dy \\
& = \frac{\partial}{\partial t} u(t,x).
\end{split}$$ Hence $\lim_{n \to \infty}\tfrac{\partial}{\partial t} u_n = \tfrac{ \partial}{\partial t} u$. Similarly $\lim_{n \to \infty} {{\operatorname{L}}^\alpha}u_n = {{\operatorname{L}}^\alpha}u$ on $H_T$, for any $\alpha \in A$. For each fixed $\alpha \in A$, let $n \to \infty$ in (\[eqn-supersoln-ppty-in-the-bdd-case\]). Then for any $\alpha \in A$ $$\frac{\partial}{\partial t} u + {{\operatorname{L}}^\alpha}u + f^\alpha \leq 0 {\quad \textrm{on} \quad}H_{T-\varepsilon^2},$$ which shows (\[eqn-smooth-supersoln-ppty\]). By Lemma \[lemma-lipschitz-continuity\] and Corollary \[corollary-holder-continuity\] $$|w(t,x) - w(s,y)| \leq Ne^{NT}(|t-s|^{1/2}+|x-y|).$$ Then (\[eqn-estimates-on-derivatives-of-smooth-sup-soln\]) follows from known properties of convolutions. Now we want to show (\[eqn-smooth-supersoln-estimate\]). Since $u = w * \zeta_\varepsilon$, $|u(s,x) - w(s,x)|$ is estimated by the right hand side of (\[eqn-smooth-supersoln-estimate\]). So we need only estimate $|w(s,x) - v(s,x)|$, for $(s,x) \in H_T$. To this end we use the fact that difference of supremums is less than the supremum of a difference to see that we need only estimate $$\begin{split}
\E_{s,x}\int_0^{T-s} |f^{\gamma_t}(s+t,x_t^\gamma)e^{-\varphi_t^\gamma} - f^{\alpha_t}(s+t,x_t^\alpha)e^{-\varphi_t^\alpha}|dt,\\
+ \E_{s,x}|g(x^\gamma_{T-s})e^{-\varphi^\gamma_{T-s}} - g(x^\alpha_{T-s})e^{-\varphi^{\alpha}_{T-s}}| =: I + J.
\end{split}$$ Clearly $$\begin{split}
J & \leq K\left[T2\varepsilon + 2\E_{s,x}\sup_{t\leq T-s}|x^\gamma_t - x^\alpha_t|\right], \\
I & \leq \E_{s,x}\int_0^{T-s}|f^{\gamma_t}(s+t,x^\gamma_t)||e^{-\varphi^\gamma_t} - e^{-\varphi^\alpha_t}|dt \\
& + \E_{s,x}\int_0^{T-s}|f^{\gamma_t}(s+t,x^{\gamma}_t) - f^{\alpha_t}(s+t,x^\alpha_t)|e^{-\varphi^\alpha_t}dt =: I_1 + I_2.
\end{split}$$ Like in the proof of Lemma \[lemma-lipschitz-continuity\] (using Assumptions \[ass-for-the-existence-of-soln-to-discrete-problem\], \[ass-for-main-result\] and the definition of $c^\gamma$), we get $$\begin{split}
I_1 & \leq \E_{s,x} \int_0^{T-s}K^2(m^{\alpha_t})^{-1} t \left[2\varepsilon + \sup_{u \leq t}|x^\gamma_u - x^\alpha_u|\right]e^{-\min(\varphi^\gamma_t,\varphi^\alpha_t)}dt \\
& \leq K^3e^T\left[ 2\varepsilon + \E_{s,x}\sup_{t\leq T-s}|x^\gamma_t -x^\alpha_t|\right].
\end{split}$$ By Assumption \[ass-for-the-existence-of-soln-to-discrete-problem\] and the definition of $f^\gamma$ $$\begin{split}
I_2 & = \E_{s,x} \int_0^{T-s} |f^{\alpha_t}(s+t+\varepsilon^2\tau, x^\gamma_t + \varepsilon \xi) - f^{\alpha_t}(s+t, x^{\alpha_t}_t)|e^{-\varphi^\alpha_t} dt \\
& \leq \E_{s,x} \int_0^{T-s} |K(m^{\alpha_t})^{-1}|\left[2\varepsilon + |x^\gamma_t -x^\alpha_t|\right]e^{-\varphi_t^\alpha} dt \\ &
\leq 2e^TK^2\left[\varepsilon + \E_{s,x}\sup_{t\leq T-s}|x^\gamma_t - x^\alpha_t| \right].
\end{split}$$ By Theorem 2.5.9 of [@krylov:controlled] $$\E_{s,x}\sup_{u \leq T-s}|x^\gamma_u - x^\alpha_u| \leq Ne^{NT}\varepsilon,$$ where $x^\alpha$ is a solution of (\[eqn-sde\]), while $x^\gamma$ is a solution of (\[eqn-shaked-sde\]). Hence, noting that the estimate for $I+J$ is independent of $\alpha$, $\tau$ and $\xi$, we get (\[eqn-smooth-supersoln-estimate\]).
\[lemma-main-estimate-from-above\] Let Assumptions \[ass-for-optimal-control-problem\], \[ass-on-the-scheme\], \[ass-for-the-existence-of-soln-to-discrete-problem\] and \[ass-for-main-result\] hold. Then $$v \leq {v_{\tau,h}}+ N_T(\tau^{1/4} + h^{1/2}).$$
Recall that for $\varepsilon > 0$, ${v_{\tau,h}}^\varepsilon$ is defined as the unique bounded solution to (\[eqn-shaked-discrete-pde\]) and by Lemma \[lemma-difference-between-soln-to-disc-and-shaked-soln\] $${v_{\tau,h}}^\varepsilon \leq {v_{\tau,h}}+ N \varepsilon.$$ Assume, without loss of generality, that $h \leq 1$. Let $\xi \geq 0$ be a $C_0^{\infty}([0,T]\times \R^d)$ function with a unit integral and support in $(-1,0) \times B_1$. For any function $w$ defined on $(-\infty, T)\times \R^d$, for which it makes sense, let $$w^{(\varepsilon)}(t,x) = \int_{H_T} w(t-\varepsilon^2r,x-\varepsilon y) \xi(r,y) dr dy.$$ If the function $w(t,x)$ is not defined for negative $t$ then extend it for $t < 0$ by defining $w(t,x) = w(0,x)$. For any $\alpha \in A$ and for all $r \in \Lambda$, $y \in S$ $$\begin{split}
{\operatorname{\delta}_\tau}{v_{\tau,h}}^\varepsilon(t-\varepsilon^2r,x-\varepsilon y) + {{\operatorname{L}}_h^\alpha}(t,x) {v_{\tau,h}}^\varepsilon(t-\varepsilon^2r,x-\varepsilon y) \\
+ f^\alpha(t,x) \leq 0 \quad \textrm{on}\quad H_T.
\end{split}$$ Multiply this by $\xi(r,y)$ and integrate over $H_T$ with respect to $r$ and $y$. Then $${\operatorname{\delta}_\tau}{v_{\tau,h}}^{\varepsilon(\varepsilon)} + {{\operatorname{L}}_h^\alpha}{v_{\tau,h}}^{\varepsilon(\varepsilon)} + f^\alpha \leq 0 {\quad \textrm{on} \quad}H_T.$$ Use Taylor’s Theorem, to get that on $H_{T-2\varepsilon^2}$: $$\begin{gathered}
|{\operatorname{\delta}_\tau}{v_{\tau,h}}^{\varepsilon(\varepsilon)} - \operatorname{D}_t {v_{\tau,h}}^{\varepsilon(\varepsilon)}|
\leq N\tau|\operatorname{D}^2_t {v_{\tau,h}}^{\varepsilon(\varepsilon)}|_{0,H_{T-2\varepsilon^2}} =: M_1\\
m^\alpha|{{\operatorname{L}}_h^\alpha}{v_{\tau,h}}^{\varepsilon(\varepsilon)} - {{\operatorname{L}}^\alpha}{v_{\tau,h}}^{\varepsilon(\varepsilon)}| \leq Nh^2|\operatorname{D}^4_x {v_{\tau,h}}^{\varepsilon(\varepsilon)}|_{0,H_{T-2\varepsilon^2}}\\
+ h|\operatorname{D}^2_x {v_{\tau,h}}^{\varepsilon(\varepsilon)}|_{0,H_{T-2\varepsilon^2}} =: M_2.\end{gathered}$$ With $M:= M_1 + M_2$, for any $\alpha \in A$ on $H_{T-2\varepsilon^2}$ $$\left[\operatorname{D}_t + {{\operatorname{L}}^\alpha}\right]({v_{\tau,h}}^{\varepsilon(\varepsilon)} + (T-t)M) + f^\alpha \leq 0.$$ By Lemma \[lemma-smooth-supersolution-and-payoff-relation\] $$v \leq {v_{\tau,h}}^{\varepsilon(\varepsilon)} + 2(T-t)M + \sup_{\partial H_{T-2\varepsilon^2}} (v - {v_{\tau,h}}^{\varepsilon(\varepsilon)})_+.$$ By Hölder continuity in time of $v$ and also ${v_{\tau,h}}^{\epsilon (\epsilon)}$: $$\begin{split}
\sup_{\partial H_{T-2\varepsilon^2}} (v - {v_{\tau,h}}^{\varepsilon(\varepsilon)})_+ & \leq \sup_{x\in \R^d} ( |v(T-\varepsilon^2,x) - g(x)| + | {v_{\tau,h}}^{\varepsilon(\varepsilon)}(T-\varepsilon^2,x) - g(x)|) \\
& \leq 4N_T\varepsilon.
\end{split}$$ By standard properties of convolutions: $$\begin{split}
|\operatorname{D}^2_t {v_{\tau,h}}^{\varepsilon(\varepsilon)}|_{0,H_{T-2\varepsilon^2}} + |\operatorname{D}^4_x {v_{\tau,h}}^{\varepsilon(\varepsilon)}|_{0,H_{T-2\varepsilon^2}} + |\operatorname{D}^2_x {v_{\tau,h}}^{\varepsilon(\varepsilon)}|_{0,H_{T-2\varepsilon^2}}\\
\leq N\varepsilon^{-3} + N\varepsilon^{-3} + N\varepsilon^{-1}.
\end{split}$$ Since $\varepsilon = (\tau + h^2)^{1/4}$, $$\begin{split}
v \leq {v_{\tau,h}}^{\varepsilon(\varepsilon)} + 4N\varepsilon + N\tau \varepsilon^{-3} + N h \varepsilon^{-1} + N h^2 \varepsilon^{-3}
\leq {v_{\tau,h}}^{\varepsilon(\varepsilon)} + N(\tau +h^2)^{1/4}.
\end{split}$$ Then $$v \leq {v_{\tau,h}}+ N (\tau^{1/4} + h^2) {\quad \textrm{on} \quad}H_T,$$ since $|{v_{\tau,h}}^{\varepsilon} - {v_{\tau,h}}| \leq N\varepsilon$ and $|{v_{\tau,h}}^{\varepsilon(\varepsilon)} - {v_{\tau,h}}^{\varepsilon}| \leq N\varepsilon$.
\[lemma-main-estimate-from-below\] Let Assumptions \[ass-for-optimal-control-problem\], \[ass-on-the-scheme\], \[ass-for-the-existence-of-soln-to-discrete-problem\] and \[ass-for-main-result\] hold. Then $${v_{\tau,h}}\leq v + N_T(\tau^{1/4} + h^{1/2}) {\quad \textrm{on} \quad}H_T.$$
Let $\varepsilon = (\tau + h^2)^{1/4}$. On $(T -\varepsilon^2, T]$ the estimate is trivial consequence of the Hölder continuity in time of both $v$ and ${v_{\tau,h}}$. Let $S:= T - \varepsilon^2$. It remains to prove the estimate on $H_S$. By Theorem \[thm-existence-smooth-supersolution\], there is a smooth function $u$ defined on $[0,T]\times \R^d$ satisfying $$\label{eqn-u-is-a-supsoln}
\sup_{\alpha \in A} \left(u_t + {{\operatorname{L}}_h^\alpha}u + f^{\alpha}\right) \leq 0 {\quad \textrm{on} \quad}H_S.$$ Apply Taylor’s Theorem to see that in $H_{T-\tau}$ $$\label{eqn-consequence-of-taylor-on-u}
\begin{split}
|{\operatorname{\delta}_\tau}u - \operatorname{D}_t u| \leq N\tau|\operatorname{D}^2_t u|_{0,H_{T}} =: M_1\\
m^\alpha|{{\operatorname{L}}_h^\alpha}u - {{\operatorname{L}}^\alpha}u| \leq Nh^2|\operatorname{D}^4_x u|_{0,H_{T}} + h|\operatorname{D}^2_x u|_{0,H_T} =: M_2
\end{split}$$ Let $M=M_1 + M_2$. By Theorem \[thm-existence-smooth-supersolution\]: $$M_1 + M_2 \leq N_T \tau \varepsilon^{-3} + N_T h^2 \varepsilon^{-3} + N_T h \varepsilon^{-1} \leq N_T(\tau + h^2)^{1/4}.$$ Since $\tau < 1$, $\varepsilon^2 > \tau$. By (\[eqn-u-is-a-supsoln\]) and (\[eqn-consequence-of-taylor-on-u\]) $$\begin{split}
& \sup_{\alpha} m^\alpha ({\operatorname{\delta}_\tau}{v_{\tau,h}}+ {{\operatorname{L}}_h^\alpha}{v_{\tau,h}}+ f^\alpha) = 0 \\
\geq & \sup_{\alpha} ({\operatorname{\delta}_\tau}u + {{\operatorname{L}}_h^\alpha}u + f^\alpha - M) {\quad \textrm{on} \quad}H_S.
\end{split}$$ Let $u' = \sup_{H_T \setminus H_S} ({v_{\tau,h}}- u)_+ + u$. The aim now is to apply Lemma \[lemma-discrete-comparison-principle\] to ${v_{\tau,h}}$ and $u'$. On the discrete boundary $H_T \setminus H_{S}$, $u' \geq {v_{\tau,h}}$. By Lemma \[lemma-discrete-comparison-principle\] $${v_{\tau,h}}\leq u + TM + \sup_{H_T \setminus H_S} ({v_{\tau,h}}- u)_+.$$ By Hölder continuity in time of ${v_{\tau,h}}$ and $v$ and by Theorem \[thm-existence-smooth-supersolution\] $$\begin{split}
\sup_{H_T \setminus H_S} ({v_{\tau,h}}- u)_+ \leq \sup_{H_T \setminus H_S} |{v_{\tau,h}}- g| + \sup_{H_T \setminus H_S} |g - v| + \sup_{H_T \setminus H_S} |v - u| \leq N_T\varepsilon.
\end{split}$$ By Theorem \[thm-existence-smooth-supersolution\], $|v-u| \leq N_T\varepsilon$. Hence $$\begin{split}
{v_{\tau,h}}& \leq v + N_T(\varepsilon + \tau^{1/2} + (\tau+h^2)^{1/4}) \leq v + N_T(\tau + h^2)^{1/4} \\
& \leq v + N_T(\tau^{1/4} + h^{1/2}) {\quad \textrm{on} \quad}H_S.
\end{split}$$
[**Acknowledgment.**]{} The authors are grateful to Nicolai Krylov in Minnesota for useful information on the subject of this paper.
|
---
abstract: 'This paper addresses the four enabling technologies, namely multi-user sparse code multiple access (SCMA), content caching, energy harvesting, and physical layer security for proposing an energy and spectral efficient resource allocation algorithm for the access and backhaul links in heterogeneous cellular networks. Although each of the above mentioned issues could be a topic of research, in a real situation, we would face a complicated scenario where they should be considered jointly, and hence, our target is to consider these technologies jointly in a unified framework. Moreover, we propose two novel content delivery scenarios: 1) single frame content delivery (SFCD), and 2) multiple frames content delivery (MFCD), where the time duration of serving user requests is divided into several frames. In the first scenario, the requested content by each user is served over one frame. However, in the second scenario, the requested content by each user can be delivered over several frames. We formulate the resource allocation for the proposed scenarios as optimization problems where our main aim is to maximize the energy efficiency of access links subject to the transmit power and rate constraints of access and backhaul links, caching and energy harvesting constraints, and SCMA codebook allocation limitations. Due to the practical limitations, we assume that the channel state information values between eavesdroppers and base stations are uncertain and design the network for the worst case scenario. Since the corresponding optimization problems are mixed integer non-linear and nonconvex programming, NP-hard, and intractable, we propose an iterative algorithm based on the well-known alternate and successive convex approximation methods. In addition, the proposed algorithms are studied from the computational complexity, convergence, and performance perspectives. Moreover, the proposed caching scheme outperforms the existing traditional caching schemes like random caching and most popular caching. We also study the effect of joint and disjoint considerations of enabling technologies for the performance of next-generation networks. We also show that the proposed caching strategy, MFCD and joint solutions have 43%, 9.4% and %51.3 performance gain compared to no cahcing, SFCD and disjoint solutions, respectively. ***Index Terms–*** Heterogeneous cellular networks, Content caching, Physical layer security, Energy harvesting, Imperfect CSI.'
author:
- 'Mohammad R. Abedi, Nader Mokari, Mohammad R. Javan, and Eduard . A. Jorswieck [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'Bibliography.bib'
title: 'Single or Multiple Frames Content Delivery for Next-Generation Networks?'
---
Introduction
============
Background and Motivation
-------------------------
Over recent years, the growth of high data rate of mobile traffic, energy, content storing, security, and limited knowledge of channels over mobile networks are the major challenges of network design and implementation. To tackle these issues and cope with the users’ requirements, the next-generation of wireless communications is introduced which uses multiple advanced techniques such as energy harvesting (EH), physical layer (PHY) security, new multiple access techniques, and content caching. Hence, all of these issues must be considered together and efficient joint radio resource allocation and content placement algorithms must be applied to provide high performance for the designed networks. However, devising efficient algorithms to handle all these issues is a challenging task, and to the best our knowledge, no research exists addressing all these issues together in a unified framework. Although each of the mentioned issues could be an interesting research topic, our main contribution is to study the joint effect of security, EH, content caching, and imperfect and limited channel knowledge in a unified joint access and backhaul links framework. In this regards, we develop a comprehensive model and mathematical representation, and design a robust resource allocation algorithm. Although the resulting optimization problem is complicated, effective optimization methods are used to achieve the solution. The outline of each issue, applicable solutions, and related works are explained in the sequel.
### Growth of High Data Rate Mobile Traffic
Incredible growth in high data rate mobile applications requires high capacity in radio access and backhaul wireless links. However, the centralized nature of mobile network architectures can not provide enough capacity on the wireless access and backhaul links to satisfy high demand for rich multimedia content. Heterogeneous network consisting of multiple low power radio access nodes and the traditional macrocell nodes, is a promising solution to improve coverage and to provide high capacity [@mokari2016limited].
### Content Caching
Multimedia services can be provided using recent advanced mobile communication technologies by new types of mobile devices such as smart phones and tablets. However, transferring the same content several times in a short period imposes capacity pressures on the network. To overcome this, content caching at the network edge has recently been emerged as a promising technique in next-generation networks. Caching in next-generation mobile networks also reduces the mobile traffic by eliminating the redundant traffic of duplicate transmissions of the same content from servers. The deployment of content cashing relevant to evolved packet core and radio access network (RAN) are studied in [@Woo13]. By caching, contents can be closer to the end-users, and backhaul traffic can be offloaded [@Bastug14; @rezvani2017fairness] to the edge of the network. The authors in [@ahlehagh2014video; @shanmugam2013femtocaching] investigate caching the contents in RAN with the aim to store contents closer to users. The content caching in small-cell base stations is studied in [@shanmugam2013femtocaching; @golrezaei2013femtocaching]. In [@gregori2015joint], the authors reduce both the load and energy consumption of the backhaul links by caching the most popular contents at small base stations (SBSs). In [@zhang2017cost], the authors consider two-tier heterogeneous wireless networks (HetNets) with hierarchical caching, where the most popular files are cached at SBSs while the less popular ones are cached at macro base stations (MBSs). The goal of [@zhang2017cost] is to maximize network capacity with respect to the file transmission rate requirements by optimizing the cache sizes for MBSs and SBSs.
### Energy Harvesting
The offer of high-rate services increases the energy consumption at receivers which degrades the battery life. Therefore, the trade-off between high-rate requirement and long battery life is required to achieve good performance. Energy harvesting has emerged as a promising approach to provide sustainable networks with the long-term sustainable operation of power supplies. In EH communication networks, nodes acquire energy from environmental energy sources including random motion and mechanical vibrations, light, acoustic, airflow, heat, RF radio waves [@Yeatman04; @Paradiso05]. The design of novel transmission policies due to highly random and unpredictable nature of harvestable profile of the harvested energy is required.
### New Multiple Access Techniques
Sparse code multiple access (SCMA) with near optimal spectral efficiency is a promising technique to improve capacity of wireless radio access [@Hosein13]. This multiple-access technique that is based on non-orthogonal codebook assignment provides massive connectivity and improves spectral efficiency [@Hosein13; @Au14; @moltafet2017comparison]. By performing an appropriate codebook assignment, a subcarrier in SCMA networks can be shared among multiple users. Joint codebook assignment and power allocation for SCMA is studied in [@li2016joint]. The codebook assignment and power allocation is also investigated in [@li2016cost]. The authors formulate energy-efficient transmission problem to maximize the network energy efficiency (EE) subject to system constraints.
### Imperfect Channel State Information
In most previous works, the authors assume perfect channel state information (CSI) of all links for BSs. However, in practice, knowing of perfect CSI in BSs requires a huge amount of bandwidth for signalling through the feedback links which is not possible. Moreover, due to time varying channel, feedback delay, quantization error, and estimation errors, perfect CSI may not be available at transmitters. In this regard, some works aim to tackle the performance degradation caused by the limited and imperfect CSI [@ahmed2006outage; @mokari2016limited; @javan2017resource]. In [@ahmed2006outage], the authors investigate the power and subcarrier allocation by the quantized CSI. It is assumed that the perfect CSI does not exist at transmitters and imperfect CSI can be achieved via limited rate feedback channels. In [@mokari2016limited], joint power and subcarrier allocation is studied for the uplink of an orthogonal frequency-division multiple access (OFDMA) HetNet assuming imperfect CSI. In [@javan2017resource], a limited rate feedback scheme is considered to maximize the average achievable rate for decode-and-forward relay cooperative networks.
### Security
The broadcast nature of wireless transmission makes security against eavesdropping a major challenge for the next generation wireless networks [@csiszar2013secrecy]. In this regards, physical-layer security is a promising method to provide security in wireless networks [@wyner1975wire; @alavi2017limited]. This technique explores the characteristics of the wireless channel to provide security for wireless transmission. In [@abedi2016limited], the authors consider physical layer security for relay assisted networks with multiple eavesdroppers. They maximize the sum secrecy rate of network with respect to transmission power constraint for each transmitter via imperfect CSI. In [@wang2017new], the authors investigate the benefits of three promising technologies, i.e., physical layer security, content caching, and EH in heterogeneous wireless networks.
### Joint Backhaul and Access Resource Allocation
Joint resource allocation at backhaul and access links is investigated in [@sharma2017joint] for heterogeneous networks. In [@sharma2017joint], the full duplex self-backhauling capacity is used to simultaneously communicate over the backhaul and access links. In [@dhifallah2015joint], joint access and backhaul links optimization is considered to minimize the total network power consumption. In [@hua2016wireless], the authors study joint wireless backhaul and the access links resource allocation optimization. The goal is to maximize the sum rate subject to the backhaul and access constraints. Joint backhaul and access links optimization is considered in [@shariat2015joint] for dense small cell networks. Joint resource allocation in access and backhaul links is considered for ultra dense networks in [@zhuang2017joint] where the goal is to maximize the throughput of the network under system constraints. In [@mirahsanjoint], the authors consider joint access and backhaul resource allocation for the admission control of service requests in wireless virtual network. The access and backhaul links optimizations are considered for small cells in the mmW frequency in [@niu2015exploiting].
Our Contributions
-----------------
This paper addresses the above joint provisioning of resources between the wireless backhauls and access links by using multi-user SCMA (MU-SCMA) to improve the network energy efficiency. We consider secure communications in EH enabled SCMA downlink communications with imperfect channel knowledge. In our work, we combine and extend several techniques to improve performance of network and formulate an optimization problem with the aim of maximizing EE with respect to system constraints. There are several works which consider each of these topics separately. However, in a real situation, these issues should be considered jointly. To the best of our knowledge, none of the existing works considered the above issues in a unified framework. The main contributions of this work are as follows:
- We provide a unified framework in which physical layer security, content caching, EH, and imperfect knowledge of channel information is considered jointly in the design of wireless communication networks
- We consider SCMA as a non-orthogonal multiple access technology where the codebooks are allowed to be used several times among users which increases the spectral efficiency.
- We propose two novel scenarios for content delivery, namely single frame content delivery (SFCD), and multiple frames content delivery (MFCD). We compare the performance of the proposed delivery scenarios with each other for different system parameters. Due to the random energy arrivals in the EH based communication, there may not be enough energy to send the entire file within the desired frames. Therefore, the first scenario may interrupt sending the file. To overcome this difficulty, we can use the second scenario. There, due to the file transfer in multiple frames, the probability of interrupting will be very low. It should be noted that the second scenario can be suitable for applications with large file sizes.
- We consider the access and backhaul links jointly and formulate the resource allocation for the proposed scenarios as optimization problems whose objectives are to maximize the energy efficiency of the network while transmit power and rate constraints, EH constraints, codebook assignment constraints, as well as caching constraints should be satisfied.
- We provide mathematical frameworks for our proposed resource allocation problems where fractional programming, alternative optimization, and successive convex approximation methods are successfully applied to achieve solutions for the resource allocation optimization problems. We further study the convergence and the computational complexity of the proposed resource allocation algorithms.
- We evaluate and assess the performance of the proposed scheme for different values of the network parameters using numerical experiments.
The following notations is used in the paper: $[x]^+=\max\{0,x\}$. $|\mathcal{S}|$ denotes the cardinality of a set $\mathcal{S}$. $[.]^{\dagger}$ represents the conjugate transpose. $\|.\|$ denotes the Euclidean norm of a matrix/vector.
The rest of the paper is organized as follows. Section \[System-Model\] defines the system model. Section \[THE OPTIMIZATION FRAMEWORK\] is dedicated to the optimization frameworks where the objectives and the constraints are explained. Section \[Resource Allocation based on worst case CSI\] describes the details of scheduling, power allocation algorithm, content placement, EH, codebook assignment, and subcarrier allocation. In Section \[simulationsresults\], we provide the numerical analysis, and Section \[Conclusion\] concludes the paper.
System Model {#System-Model}
============
Consider the downlink SCMA transmission of a wireless heterogeneous cellular network comprising of $O$ MBSs and $J$ SBSs in a two dimensional Euclidean plane $\mathbb{R}^2$, as shown in Fig. \[fig-System-Model\]. Let us denote by $\mathcal{O}=\{1,2,\dots,O\}$ the set of the MBSs and by $\mathcal{J}=\{1,2,\dots,J\}$ the set of the SBSs. Each cache-capable BS, i.e., $b\in\mathcal{B}=\{1,\dots,B\}=\mathcal{O}\bigcup\mathcal{J}$ with size $B=|\mathcal{B}|$, is connected to the core network via backhaul[^2] links which are wireless links. The paper assumes that there is no interference between the wireless backhaul and access links and these links are out-of-band. A set of total number of users, $\mathcal{U}_b=\{1,2,\dots,U_b\}$ is served by BS $b$ with size $U_b=|\mathcal{U}_b|$. The set of network users is $\mathcal{U}=\bigcup_{b=1}^{B}\mathcal{U}_b$. The system consists of $Q$ eavesdroppers which are indexed by $q\in\mathcal{Q}=\{1,2,\dots,Q\}$ with size $Q=|\mathcal{Q}|$. The total transmit bandwidth of the access, i.e., BW, is divided into $N$ subcarriers where the bandwidth of each subcarrier is $BW_n$ ($BW=N\times
BW_n$). $K$ social media $\omega_{k},k\in\mathcal{K}=\{1,\dots,K\}$, as the main traffic of internet contents, are requested by the users in the network. We assume that during the runtime of the network optimization process, user-BS association is fixed.
{width="6"}
The message passing algorithm (MPA) can be used to detect multiplexed signals on the same subcarriers [@Hoshyar08]. In our resource allocation framework, we consider two tasks: content caching and delivery resource allocations. The content caching task deals with determining which content should be cached in which storage. However, the delivery task deals with performing resource allocation such that the contents are delivered to the requesting users within serving time. We assume that the time is split into several super frames. We further assume that each super frame is divided into $F$ frames of duration $T$ seconds. throughout each supper frame, the arriving users requests, which should be served over the next supper frame, are gathered by the network control system. We emphasize that our proposed content caching and resource allocation algorithms are run for each super frame. Throughout the network run time, the network monitors the file requests and estimate the content distribution (content popularity). At the beginning of each super frame, if a change in the statistics of the contents popularity is detected, joint content caching and radio resource allocation is performed, and otherwise, only radio resource allocation is performed. Note that the proposed resource allocation problem is solved at the beginning of each supper frame, and hence, the information about the CSIs and energy harvesting profile over all $F$ frames of the considered supper frame are required and should be known in advanced. With such assumption, we rely on the off-line approach which is common in the context of energy harvesting[^3] [@luo2013optimal; @minasian2014energy]. The proposed transmission structure is shown in Fig. \[Frame-structure\].
{width="6.4"}
Let $\mathbf{s}=\{s^{mt}_{bu}\}$ denote the codebook assignment at BS $b$ at frame $t$ where $s^{mt}_{bu}$ is an indicator variable that is 1 if codebook $m$ is assigned to user $u$ at BS $b$ at frame $t$ and 0 otherwise. Furthermore, let $\mathbf{p}=\{p^{mt}_{bu}\}$ denote the allocated transmit power vector with $p^{mt}_{bu}$ representing the transmit power for user $u$ at BS $b$ at frame $t$ on codebook $m$. Thus, the total transmit power of BS $b$ at frame $t$ is $\sum_{u\in
\mathcal{U}_b}\sum_{m\in
\mathcal{M}}s^{mt}_{bu}p^{mt}_{bu},\forall b\in \mathcal{B},t\in
\mathcal{F}$. To transmit the codewords to the designated users, the transmit power $\mathbf{p}$ is finally allocated on the corresponding subcarriers. However, different from OFDMA based networks, the transmit power $p^{mt}_{bu}$ is allocated on subcarrier $n$ according to a given proportion $\eta_{nm}$, which is determined by the codebook design ($0<\eta_{nm}<1$ when $c_{nm}=1$ and $\eta_{nm}=0$ when $c_{nm}=0$ [@Hosein13]). Therefore, the signal-to-interference-plus-noise ratio (SINR) of user $u$ in BS $b$ when using codebook $m$ can be expressed as follows: $$\gamma^{mt}_{bu}=\frac{\sum_{n\in
\mathcal{N}}\eta_{nm}s^{mt}_{bu}p^{mt}_{bu}g^{nt}_{bu}}{I^{mt}_{bu}+(\sigma^{n}_u)^2},$$ where $I^{mt}_{bu}=\sum_{\acute{b}\in
\mathcal{B}\setminus\{b\}}\sum_{\acute{u}\in
\mathcal{U}_{\acute{b}}}\sum_{n\in
\mathcal{N}}\eta_{nm}s^{mt}_{\acute{b}\acute{u}}
p^{mt}_{\acute{b}\acute{u}}g^{nt}_{\acute{b}u}$ and $g^{nt}_{bu}$ denotes the channel power gain between BS $b$ and user $u$ on subcarrier $n$ at time $t$. $(\sigma^{n}_u)^2$ is the noise power on subcarrier $n$ at user $u$. Each of the subcarriers can be assumed to undergo a block-fading, and hence, the channel coefficients are kept constant within each frame. The achievable rate for the $u^{\text{th}}$ user in BS $b$ at frame $t$ on codebook $m$ is given by $R^{\text{D},mt}_{bu}= \log_2\left(1+\gamma^{mt}_{bu}\right).$
We assume that the eavesdroppers only wiretap the access link[^4]. Therefore, the SINR of eavesdropper $q$ in BS $b$ when using codebook $m$ can be expressed as: $$\hat{\gamma}^{mt}_{buq}=\frac{\sum_{n\in
\mathcal{N}}\eta_{nm}s^{mt}_{bu}p^{mt}_{bu}
h^{nt}_{bq}}{\hat{I}^{mt}_{buq}+(\sigma^{n}_q)^2},$$ where $\hat{I}^{mt}_{buq}=\sum_{\acute{b}\in
\mathcal{B}\setminus\{b\}}\sum_{\acute{u}\in
\mathcal{U}_{\acute{b}}}\sum_{n\in
\mathcal{N}}\eta_{nm}s^{mt}_{\acute{b}\acute{u}}p^{mt}_{\acute{b}\acute{u}}h^{nt}_{\acute{b}q}$ and $h^{mt}_{bq}$ denotes the channel power gain between BS $b$ and eavesdropper $q$ on subcarrier $n$. $(\sigma^{n}_q)^2$ is the noise power on subcarrier $n$ at eavesdropper $q$. The achievable rate for the $q^{\text{th}}$ eavesdropper in BS $b$ at frame $t$ is evaluated by $R^{\text{E},mt}_{buq}=\log_2\left(1+\hat{\gamma}^{mt}_{buq}\right).$ The achievable secrecy access rate for non-colluding eavesdroppers and the $u^{\text{th}}$ user in BS $b$ at frame $t$ on codebook $m$ is expressed as [@pinto2012secure], $$\label{eq-R-S}
R^{\text{S},mt}_{bu}=\left[R^{\text{D},mt}_{bu}-\max_{q\in
\mathcal{Q}}R^{\text{E},mt}_{buq}\right]^+.$$
THE OPTIMIZATION FRAMEWORK {#THE OPTIMIZATION FRAMEWORK}
==========================
In this section, we provide the design objective and a characterization of the constraints that must be satisfied by content caching, EH, codebook assignment, and power allocations.
System Constraints
------------------
### Content Caching Constraints
Let the finite size of cache memory at the $b^{\text{th}}$ BS is denoted by $V_b$. If the requested file $k$ by user $u$ exists in the cache, then the file is sent to the user immediately. This event is referred as a cache hit. However, if file $k$ does not exist in the cache, then the request is forwarded to the core network via backhaul, then downloaded file $k$ from the core network via backhaul is forwarded to the user. The size of the social media, $\alpha_k,k\in\mathcal{K}$ is assumed to be Log-Normal distributed with parameters $\mu$ and $\kappa$ [@sobkowicz2013lognormal]. As the total cached media should not exceed the finite size of cache memory at BS $b$, we have $$\label{eq-V}
\sum_{k\in\mathcal{K}}\theta_{bk}{\alpha}_k \leq V_b,\forall
b\in\mathcal{B},$$ where $\theta_{bk}$ is a binary indicator declaring whether social media $\omega_k$ is cached at BS $b$.
### Content Delivery
The content delivery consists of two phases: 1) a cache placement phase, and 2) a content delivery phase. In the cache placement phase, the cache content is determined at each BS, and in the content delivery phase, the requested files are delivered to users over wireless channels. In this paper, two new delivery scenarios are considered for content delivery phase. In the first scenario, the user’s requested file $k$ with size $\alpha_k$ is sent in a single frame, while in the second scenario the user’s requested file $k$ is divided into several parts with sizes $\{\beta^t_k\}, \forall t, k$, which are sent over several frames. The scenarios are shown in Fig. \[Frame-structure\]. To ensure that all parts of each file are transmitted to user, the following constraint should be satisfied $$\label{eq--1}
\sum_{t\in F}\beta^t_k=\alpha_k, \forall k.$$
### Access and Backhaul Links Constraints
Let $\upsilon_{ku}$ denote whether user $u$ needs $\omega_k$. The backhaul traffic constraint for BS $b$ for the SFCD scenario is written as follows $$\begin{aligned}
\label{eq-Upsilon}
&\sum_{k\in\mathcal{K}}\sum_{u\in\mathcal{U}_b}\sum_{m\in\mathcal{M}}s^{mt}_{bu}
(1-\theta_{bk}).\min\left\{\sum_{u\in\mathcal{U}_b}\upsilon_{ku},1\right\}\alpha_k\\\nonumber&\leq
T\sum_{n\in\mathcal{N}}\zeta_{bn}\tilde{R}^{nt}_b,\forall
t\in\mathcal{F}, b\in\mathcal{B},\end{aligned}$$ where the left hand side term of (\[eq-Upsilon\]) is the backhual traffic for BS $b$ and the right hand side term of (\[eq-Upsilon\]) is backhaul traffic capacity, which must be greater than the backhaul traffic for each BS. The backhaul link is a simple P2M link with OFDMA technology. $\zeta_{bn}\in\{0,1\}$ denotes whether BS $b$ uses subcarrier $n$. For the MFCD scenario, $\alpha_k$ in (\[eq-Upsilon\]) is replaced by $\beta^t_k$ as follows $$\begin{aligned}
\label{eq-Upsilon-1}
&\sum_{k\in\mathcal{K}}\sum_{u\in\mathcal{U}_b}\sum_{m\in\mathcal{M}}s^{mt}_{bu}
(1-\theta_{bk}).\min\left\{\sum_{u\in\mathcal{U}_b}
\upsilon_{ku},1\right\}\beta^t_k\leq\\\nonumber&
T\sum_{n\in\mathcal{N}}\zeta_{bn}
\tilde{R}^{nt}_b,\forall
t\in\mathcal{F}, b\in\mathcal{B}.\end{aligned}$$
Note that for all requests of social media $\omega_k$ from BS $b$, if social media $\omega_k$ is not stored at BS $b$, the requested social media $\omega_k$ is disseminated to BS $b$ from the core network just once. $\tilde{R}^{nt}_b$ is the rate of backhaul link for BS $b$ on subcarrier $n$ which is calculated by $$\tilde{R}^{nt}_b=\log_2\left(1+\tilde{\gamma}^{nt}_b\right),\forall
t\in\mathcal{F},b\in\mathcal{B},n\in\mathcal{N}.$$
When BS receives the data from core network, it transmits the file back on the downlink. Let $\tilde{\gamma}^{nt}_b$ denotes the received SNR at BS $b$ from the core network when the backhaul is used to fetch the files from the core network for BS $b$. $\tilde{\gamma}^{nt}_b$ can be written as $\tilde{\gamma}^{nt}_b=\frac{\tilde{p}^{nt}_b\tilde{h}^{nt}_b}{(\sigma^n_b)^2},$ where ${\tilde{p}^{nt}}_b$ is the transmit power of each wireless backhaul link connected to BS $b$ on subcarrier $n$ and $\tilde{h}^{nt}_b$ denotes the channel power gain between the $b^{\text{th}}$ BS on subcarrier $n$ and the core network and $(\sigma^n_b)^2$ is the noise power at the $b^{\text{th}}$ BS on subcarrier $n$. Also the downlink traffic should not exceed the traffic capacity of each downlink. This yields for the first delivery scenario $$\label{eq-R-S-constraint}
\sum_{k\in\mathcal{K}}\sum_{m\in\mathcal{M}}s^{mt}_{bu}
\upsilon_{ku}\alpha_k\leq
T\sum_{m\in\mathcal{M}}R^{\text{S},mt}_{bu}, \forall
t\in\mathcal{F}, b\in\mathcal{B}, u\in\mathcal{U}_b.$$
Note that for the MFCD scenario, $\alpha_k$ in (\[eq-R-S-constraint\]) is replaced by $\beta^t_k$ as follows: $$\label{eq-R-S-constraint-1}
\sum_{k\in\mathcal{K}}\sum_{m\in\mathcal{M}}s^{mt}_{bu}
\upsilon_{ku}\beta^t_k\leq T
\sum_{m\in\mathcal{M}}R^{\text{S},mt}_{bu}, \forall
t\in\mathcal{F}, b\in\mathcal{B}, u\in\mathcal{U}_b.$$
### Power Allocation Constraints
To determine the constraints that must be satisfied by any feasible power allocation, let $p^{mt}_{bu}$ and $\tilde{p}^{nt}_{b}$ denote the power allocated to link the $b^{\text{th}}$ BS-the $u^{\text{th}}$ user at time frame $t$ on codebook $m$ and to link core network-the $b^{\text{th}}$ BS at frame $t$. The elements of $p^{mt}_{bu}$ and $\tilde{p}^{nt}_{b}$ must satisfy the followings: $$\label{eq-P-constraint-1}
p^{mt}_{bu}\geq0,\forall
b\in\mathcal{B},u\in\mathcal{U}_b,m\in\mathcal{M},t\in\mathcal{F},$$ $$\label{eq-tilde-P-constraint-1}
\tilde{p}^{nt}_{b}\geq0,\forall
b\in\mathcal{B},n\in\mathcal{N},t\in\mathcal{F}.$$
In a practical network, core network has a power budget, ${P}^{\text{Total},t}$, which bounds the total power allocated by core network on the core network-$b$ BS links and subcarriers at frame $t$. This constraint can be written as: $$\label{eq-tilde-P-constraint-2}
\sum_{b\in\mathcal{B}}\sum_{n\in\mathcal{N}}\zeta_{bn}\tilde{p}^{nt}_{b}\leq
{P}^{\text{Total},t},\forall t\in\mathcal{F}.$$
### EH Constraints
We assume that the $b^{\text{th}}$ BS is connected to a rechargeable battery with capacity $E^{\text{max}}_b$, and obtains its power supply through an EH renewable sources such as solar. The renewable sources are used to charge batteries during the day. $E^t_b\in[0,E^{\text{max}}_b]$ is defined as the energy remaining in the battery at the start of the $t^{\text{th}}$ frame. Then $E^t_b$ can be written in recursive form as: $$\begin{aligned}
\label{eq-energy-23456}\nonumber
E^{t+1}_b=&\min\left(E^t_b-T\sum_{m\in\mathcal{M}}
\sum_{u\in\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}+
\tilde{E}^t_b,E^{\text{max}}_b\right),\\& \forall b\in\mathcal{B}.
t\in\mathcal{F},\end{aligned}$$ where $\tilde{E}^t_b$ denotes the amount of energy is harvested during the $t^{\text{th}}$ frame at the $b^{\text{th}}$ BS. The energy arrival takes place as a Poisson arrival process with mean $\Gamma_b$ [@ozel2011transmission; @dhillon2014fundamentals]. The unit amount of energy harvested at each arrival at each BS is denoted by $\rho^t_b$, which depends on the EH capabilities of renewable energy source at each BS. Therefore, $\tilde{E}^t_b=\varpi^t_b \rho^t_b$, where $\varpi^t_b$ is the number of arrivals within $T$ with a mean value of $\Gamma_b T$. In designing of optimal transmission policies for EH communication systems, there are main constraints referred to as energy consumption causality constraints, which state that the energy packets which do not arrive yet, cannot be used by a source. These constraints can be expressed as: $$\label{eq-energy-2}
\sum_{t=1}^{f}\sum_{m\in\mathcal{M}}\sum_{u\in\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}
\leq \frac{1}{T}\sum_{t=1}^{f}E^t_b, \forall b\in\mathcal{B},
f\in\mathcal{F}.$$
If battery capacity is not enough to store the newly arrived energy packet, the energy will be wasted at the beginning of a transmission interval. By considering the following energy overflow constraint on our problem, we avoid this battery overflow by enfrocing the following constraint: $$\label{eq-energy-3}
\sum_{t=1}^{f+1}E^t_b-T\sum_{t=1}^{f}\sum_{m\in\mathcal{M}}\sum_{u\in\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}
\leq E^{\text{max}}_b,\forall b\in\mathcal{B}, f\in\mathcal{F}.$$
### Scheduling Constraints
In order to improve the detection performance, we should use the codebooks which have less subcarriers in common. This means that, it must be guaranteed that each subcarrier cannot be reused more than a certain value $D$, i.e., the maximum number of differentiable constellations generated by the codebook-specific constellation function, as follows [@li2016cost] $$\label{eq-S}
\sum_{b\in\mathcal{B}}\sum_{u\in\mathcal{U}_b}\sum_{m\in\mathcal{M}}c_{nm}s^{mt}_{bu}\leq
D,\forall n\in\mathcal{N},t\in\mathcal{F}.$$
In addition, (\[eq-S-1\]), (\[eq-S-3\]), and (\[eq-S-2\]) together denote that codebooks are exclusively allocated among users of each BS. For the SFCD scenario, we have $$\label{eq-S-1}
\sum_{m\in\mathcal{M}}
\sum_{t\in\mathcal{F}}
\sum_{u\in\mathcal{U}_b}
s^{mt}_{bu}\leq 1,\forall
b\in\mathcal{B},$$ and for the MFCD scenario, we have $$\label{eq-S-3}
\sum_{m\in\mathcal{M}}
\sum_{u\in\mathcal{U}_b}s^{mt}_{bu}\leq 1,\forall b\in\mathcal{B},t\in\mathcal{F},$$ $$\label{eq-S-2}
s^{mt}_{bu}\in\{0,1\},\forall
b\in\mathcal{B},u\in\mathcal{U}_b,m\in\mathcal{M},t\in\mathcal{F}.$$
### Worst Case Channel Uncertainty Model
For the channels between the $b^{\text{th}}$ BS and the $q^{\text{th}}$ eavesdropper, only the estimated value $\tilde{h}^{nt}_{bq}$ is available at the $b^{\text{th}}$ BS. We define the channel error as $e_{h^{nt}_{bq}}=|h^{nt}_{bq}-\tilde{h}^{nt}_{bq}|$, and we assume that the channels mismatches are bounded as follows: $$\label{eq-e-3}
e_{h^{nt}_{bq}}\leq \varepsilon_{h^{nt}_{bq}},\forall b\in\mathcal{B},q\in\mathcal{Q},n\in\mathcal{N},t\in\mathcal{F},$$ where $\varepsilon_{h^{nt}_{bq}}$ is known constant. Hence the actual channel power gain value lies in the region $h^{nt}_{bq}\in\mathcal{H}^{nt}_{bq}=[\tilde{h}^{nt}_{bq}-\varepsilon_{h^{nt}_{bq}}~\tilde{h}^{nt}_{bq}+\varepsilon_{h^{nt}_{bq}}]$ [@liang2009compound].
The Optimization Problem {#The Optimization
Problem}
------------------------
We formulate the utility maximization problem with power allocation, codebook assignment, and content caching subject to energy causality and power budget constraints at each BS for the SFCD scenario as: $$\begin{aligned}
\label{eq--2}
\max_{\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta}}
\min_{\textbf{h}\in\boldsymbol{\mathcal{H}}}&~
\Xi_{\text{EE}}(\mathbf{p},\mathbf{s}),\\\nonumber
\text{s.t.}~&(\ref{eq-V}),(\ref{eq-Upsilon}),(\ref{eq-R-S-constraint}),(\ref{eq-P-constraint-1})-(\ref{eq-S-1}),(\ref{eq-S-2}),(\ref{eq-e-3}),\end{aligned}$$ where $\Xi_{\text{EE}}(\mathbf{p},\mathbf{s})=\frac{\sum_{m\in
\mathcal{M}}\sum_{t\in \mathcal{F}}\sum_{b\in
\mathcal{B}}\sum_{u\in \mathcal{U}_b}R^{\text{S},mt}_{bu}}
{\sum_{m\in \mathcal{M}}\sum_{t\in \mathcal{F}}\sum_{b\in
\mathcal{B}}\sum_{u\in \mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}}$, ${\textbf{h}}=[{\textbf{h}}^1,\dots,{\textbf{h}}^t,\dots,{\textbf{h}}^F],{\textbf{h}}^t=[h^{1t}_{11},\dots,h^{Nt}_{11},h^{1t}_{12},...,h^{Nt}_{1Q},...,h^{Nt}_{BQ}],\boldsymbol{\mathcal{H}}=\mathcal{H}^{11}_{11}\times\dots\times\mathcal{H}^{nt}_{bq}\times\dots\times\mathcal{H}^{NF}_{BQ}$. Note that for the MFCD scenario, constraint (\[eq–1\]) is added to the optimization problem (\[eq–2\]). We also replace (\[eq-Upsilon\]), (\[eq-R-S-constraint\]) and (\[eq-S-1\]) by (\[eq-Upsilon-1\]), (\[eq-R-S-constraint-1\]) and (\[eq-S-3\]), respectively. It should also be noted that in the second scenario, $\boldsymbol{\beta}$ is itself an optimization variable that must be obtained in the optimization problem. The optimization problem (\[eq–2\]) consisting of non-convex objective function and both integer and continuous variables. Hence, it is mixed-integer nonlinear programming (MINLP), non-convex, intractable and NP-hard problem [@murty1987some].
The optimization problem (\[eq–2\]) is NP-hard.
Please see Appendix \[appendix-A\].
It is very difficult to find the global optimal solution within polynomial time. Hence, the available methods to solve convex optimization problem can not be applied directly. To solve this problem, an iterative algorithm based on the well-known and well-proven alternating, Dinkelbach and successive convex approximation methods is proposed where in each iteration, the main problem is decoupled into several sub-problems subject to some optimization variables.
PROPOSED SOLUTION {#Resource Allocation based on worst case
CSI}
=================
The difficulty of solving the problem (\[eq–2\]) arises from the nonconvexity of both the objective function and feasible domain. As far as we know, there is no standard method to solve such a nonconvex optimization problem. In this section, some optimization methods such as alternative optimization, fractional programming, and difference-of-two-concave-functions (DC) programming, are jointly applied to solve the primal problem by transforming it into simple subproblems step by step. To facilitate solving (\[eq–2\]), an alternate optimization method is adopted to solve a multi-level hierarchical problem which consists of the several subproblem. The core idea of the alternate optimization is that only one of the optimization parameters is optimized in each step while others are fixed. When each parameter is given, the resulting subproblem can be reformulated as the form of DC problem and solved by DC programming. Moreover, a sequential convex program is finally solved by convex optimization methods at each iteration of the DC programming. In this section, we propose a solution for the SFCD scenario which is suitable for MFCD, too. The transformation process for solving this problem mainly consists of the following steps: I. *Transformation of the primal problem:* By using the epigraph method, the inner maximization in the objection function in (\[eq-worstcase-p-123\]) can be simplified and the secondary problem can be naturally derived. II. *Alternate optimization over some variables:* In this step, the alternate optimization method is adopted to cope with the non-convexity of the resulting parametrized secondary problems which is further rewritten as five sub-problems, namely, access power allocation, access code allocation, backhaul power allocation, backhaul subcarrier allocation, content placement, and channel uncertainty. III. *DC programming for the nonconvex constraint elimination:* In this step, we reformulate the nonconvex constraint (\[eq-4321\]) as a canonical DC programming which can be settled by iteratively solving a series of sequential convex constraints. Finally, these convex constraints can be solved by convex programming. IV. *Fractional programming:* Applying fractional programming, the parameterized secondary subproblem is solved with a given parameter in each iteration.
Transformation of the primal problem
------------------------------------
For simplifying (\[eq–2\]), we herein introduce auxiliary variables $\boldsymbol{\varphi}=\{\varphi^{mt}_{bu}\in\mathbb{R}\}$. Additionally, we can rewrite (\[eq–2\]) equivalently as
\[eq-worstcase-p-123\] $$\begin{aligned}
&\max_{\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\boldsymbol{\varphi}}
\min_{\textbf{h}\in\boldsymbol{\mathcal{H}}}~A,\\\nonumber
&\text{s.t.}~\sum_{k\in\mathcal{K}}\sum_{m\in\mathcal{M}}s^{mt}_{bu}
\upsilon_{ku}\alpha_k\leq
\sum_{m\in
\mathcal{M}}\max\left\{R^{\text{D},mt}_{bu}-\varphi^{mt}_{bu},0\right\},\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\forall t\in\mathcal{F}, b\in\mathcal{B},
u\in\mathcal{U}_b,\\\label{eq-4321}&R^{\text{E},mt}_{buq}\leq\varphi^{mt}_{bu},\forall
m\in\mathcal{M},
t\in\mathcal{F},b\in\mathcal{B},u\in\mathcal{U}_b,q\in\mathcal{Q},
\\\nonumber&(\ref{eq-V}),(\ref{eq-Upsilon}),(\ref{eq-P-constraint-1}),(\ref{eq-tilde-P-constraint-1}),(\ref{eq-S})
-(\ref{eq-e-3}).\end{aligned}$$
where $A=\frac{\sum_{m\in \mathcal{M}}\sum_{t\in
F}\sum_{b\in \mathcal{B}}\sum_{u\in
\mathcal{U}_b}\max\left\{R^{\text{D},mt}_{bu}-\varphi^{mt}_{bu},0\right\}}
{\sum_{t\in \mathcal{F}}\sum_{b\in \mathcal{B}}\sum_{m\in
\mathcal{M}}\sum_{u\in
\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}}.$ To solve the optimization problem (\[eq-worstcase-p-123\]), we should further transform it. We first rewrite $\max\left\{R^{\text{D},mt}_{bu}-\varphi^{mt}_{bu},0\right\}$ as [@van2013solution]: $\max\left\{R^{\text{D},mt}_{bu}-\varphi^{mt}_{bu},0\right\}=
\max\left\{-R^{\text{D},mt}_{bu2}-\varphi^{mt}_{bu},-R^{\text{D},mt}_{bu1}\right\}
+R^{\text{D},mt}_{bu1}$ where $$\begin{aligned}
&R^{\text{D},mt}_{bu1}=\log_2\Big(\sum_{b\in
\mathcal{B}}\sum_{u\in \mathcal{U}_b}\sum_{n\in \mathcal{N}}
\Big(\eta_{nm}s^{mt}_{bu}p^{mt}_{bu}g^{nt}_{bu}+(\sigma^{n}_u)^2\Big)\Big),
\\&R^{\text{D},mt}_{bu2}=\log_2\left(\sum_{\acute{b}\in
\mathcal{B}\setminus\{b\}}\sum_{\acute{u}\in
\mathcal{U}_{\acute{b}}}\sum_{n\in
\mathcal{N}}\left(\eta_{nm}s^{mt}_{\acute{b}\acute{u}}
p^{mt}_{\acute{b}\acute{u}}g^{nt}_{\acute{b}u}+(\sigma^{n}_u)^2\right)\right).\end{aligned}$$
By introducing auxiliary variables $\boldsymbol{\delta}=
\{\delta^{mt}_{bu}\in\mathbb{R}\}$, (\[eq-worstcase-p-123\]) is equivalently reformulated as [@van2013solution]
\[eq-worstcase-p-1234\] $$\begin{aligned}
&\max_{\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\boldsymbol{\varphi}
,\boldsymbol{\delta}}\min_{\textbf{h}\in\boldsymbol{\mathcal{H}}}
~\Theta(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\boldsymbol{\varphi},\mathbf{e}_h),\\\label{eq-EEE-1}
&\text{s.t.}~\sum_{k\in\mathcal{K}}\sum_{m\in\mathcal{M}}s^{mt}_{bu}
\upsilon_{ku}\alpha_k\leq
\sum_{m\in \mathcal{M}}\left\{\delta^{mt}_{bu}
+R^{\text{D},mt}_{bu1}\right\},\\\nonumber& \forall t\in\mathcal{F},
b\in\mathcal{B},
u\in\mathcal{U}_b,\\\label{eq-EEE}&R^{\text{E},mt}_{buq}\leq\varphi^{mt}_{bu},\forall
m\in\mathcal{M},
t\in\mathcal{F},b\in\mathcal{B},u\in\mathcal{U}_b,q\in\mathcal{Q},\\\label{eq-EEE-2}
&-R^{\text{D},mt}_{bu2}-\varphi^{mt}_{bu}
\leq\delta^{mt}_{bu},\forall m\in\mathcal{M},
t\in\mathcal{F},b\in\mathcal{B},u\in\mathcal{U}_b,
\\\label{eq-EEE-3}&-R^{\text{D},mt}_{bu1}\leq\delta^{mt}_{bu},\forall m\in\mathcal{M},
t\in\mathcal{F},b\in\mathcal{B},u\in\mathcal{U}_b,
\\\nonumber&(\ref{eq-V}),(\ref{eq-Upsilon}),(\ref{eq-P-constraint-1})
-(\ref{eq-e-3}).\end{aligned}$$
where $\Theta(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\boldsymbol{\varphi},\mathbf{e}_h)=\frac{\sum_{m\in \mathcal{M}}\sum_{t\in
F}\sum_{b\in \mathcal{B}}\sum_{u\in
\mathcal{U}_b}\left\{\delta^{mt}_{bu}
+R^{\text{D},mt}_{bu1}\right\}} {\sum_{t\in \mathcal{F}}\sum_{b\in
\mathcal{B}}\sum_{m\in \mathcal{M}}\sum_{u\in
\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}}$.
Alternate optimization over optimization variables
--------------------------------------------------
Due to the combined non-convexity of both objective function and the constraint with respect to optimization parameters, the optimization problem (\[eq–2\]) is difficult to solve. According to alternate optimization method, we can always optimize a function by first optimizing over some of the variables, and then optimizing over the remaining ones. For convenience, the feasible domain of (\[eq–2\]) is denoted by $\mathbb{D}$ as $\mathbb{D}\triangleq
\left\{(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h):
(\ref{eq-V}),(\ref{eq-Upsilon}),(\ref{eq-R-S-constraint})-(\ref{eq-e-3})\right\}$. For fixed $\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta}$, $\mathbf{e}_h$-section of the feasible domain of $\mathbb{D}$, i.e., $\mathbb{D}_{\mathbf{e}_h}$, is defined as $\mathbb{D}_{\mathbf{e}_h}\triangleq \left\{\mathbf{e}_h:
(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h)
\in\mathbb{D}\right\}$. Likewise, for fixed $\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h$, $\mathbf{p}$-section of the feasible domain of $\mathbb{D}$, i.e., $\mathbb{D}_{\mathbf{p}}$, is defined as $\mathbb{D}_{\mathbf{p}}\triangleq \left\{\mathbf{p}:
(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h)
\in\mathbb{D}\right\}$. Similarly, for fixed $\mathbf{p},\tilde{\mathbf{p}},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h$, $\mathbf{s}$-section of the feasible domain of $\mathbb{D}$, i.e., $\mathbb{D}_{\mathbf{s}}$, is defined as $\mathbb{D}_{\mathbf{s}}\triangleq \left\{\mathbf{s}:
(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h)
\in\mathbb{D}\right\}$. In the same way, for fixed $\mathbf{p},\mathbf{s},\boldsymbol{\theta},\mathbf{e}_h$, $\tilde{\mathbf{p}}\times \boldsymbol{\zeta}$-section of the feasible domain of $\mathbb{D}$, i.e., $\mathbb{D}_{\tilde{\mathbf{p}}\times \boldsymbol{\zeta}}$, is defined as $\mathbb{D}_{\tilde{\mathbf{p}}\times \boldsymbol{\zeta}}\triangleq
\left\{\tilde{\mathbf{p}},
\boldsymbol{\zeta}:
(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},
\boldsymbol{\zeta},\mathbf{e}_h)\in\mathbb{D}\right\}$. Correspondingly, for fixed $\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h$, $\boldsymbol{\theta}$-section of the feasible domain of $\mathbb{D}$, i.e., $\mathbb{D}_{\boldsymbol{\theta}}$, is defined as $\mathbb{D}_{\boldsymbol{\theta}}\triangleq
\left\{\boldsymbol{\theta}:
(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\mathbf{e}_h)
\in\mathbb{D}\right\}$. Finally, the alternate optimization is used to solve the following hierarchical five-level optimization subproblem: $$\max_{\substack{\mathbf{p}\in\mathbb{D}_{\mathbf{p}}\\\boldsymbol{\varphi}\in\mathbb{R}\\\boldsymbol{\delta}\in\mathbb{R}}}
\left[\max_{\mathbf{s}\in\mathbb{D}_{\mathbf{s}}}
\left[\max_{\substack{\tilde{\mathbf{p}},\boldsymbol{\zeta}\\\in\mathbb{D}_{\tilde{\mathbf{p}}\times \boldsymbol{\zeta}}}}
\left[\max_{\boldsymbol{\theta}\in\mathbb{D}_{\boldsymbol{\theta}}}
\left[\min_{\mathbf{e}_h\in\mathbb{D}_{\mathbf{e}_h}}
\Theta(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},\boldsymbol{\zeta},\boldsymbol{\varphi},\mathbf{e}_h)\right]\right]\right]\right].$$
In conclusion, the subproblems can be solved sequentially at each iteration of alternate optimization. In the first optimization subproblem, we find $\mathbf{e}_h$ for a given $\mathbf{p}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\varphi}_{\varrho}$, and $\boldsymbol{\delta}_{\varrho}$: $$\begin{aligned}
\label{AS-1}
\min_{\mathbf{e}_h\in\mathbb{D}_{\mathbf{e}_h}}
\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho}),\end{aligned}$$ where $\varrho$ is the iteration number of alternate optimization algorithm. By defining the solution of (\[AS-1\]) as ${\mathbf{e}_h}_{\varrho+1}$, the second level subproblem is solved to find $\boldsymbol{\theta}$ with a given $\mathbf{p}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\varphi}_{\varrho}$, and $\boldsymbol{\delta}_{\varrho}$: $$\begin{aligned}
\label{AS-2}
\max_{\boldsymbol{\theta}\in\mathbb{D}_{\boldsymbol{\theta}}}
\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$ Similarly, by defining the solution of (\[AS-2\]) as $\boldsymbol{\theta}_{\varrho+1}$, the third level subproblem is solved to find $\boldsymbol{\zeta}$ and $\tilde{\mathbf{p}}$ with a given $\mathbf{p}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\varphi}_{\varrho}$, and $\boldsymbol{\delta}_{\varrho}$: $$\begin{aligned}
\label{AS-3}
\max_{\tilde{\mathbf{p}},\boldsymbol{\zeta}\in\mathbb{D}_{\tilde{\mathbf{p}}\times \boldsymbol{\zeta}}}
\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$
Correspondingly, by defining the solution of (\[AS-3\]) as $\tilde{\mathbf{p}}_{\varrho+1}$ and $\boldsymbol{\zeta}_{\varrho+1}$, the fifth level subproblem is solved to find $\mathbf{s}$ with a given $\mathbf{p}_{\varrho},\boldsymbol{\varphi}_{\varrho}$, and $\boldsymbol{\delta}_{\varrho}$: $$\begin{aligned}
\label{AS-4}
\max_{\mathbf{s}\in\mathbb{D}_{\mathbf{s}}}
\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$ Finally, by defining the solution of (\[AS-4\]) as $\mathbf{s}_{\varrho+1}$, the fifth level subproblem is solved to find $\mathbf{p},\boldsymbol{\varphi},\boldsymbol{\delta}$ with a given $\mathbf{s}_{\varrho}$: $$\begin{aligned}
\label{AS-6}
\max_{\mathbf{p}\in\mathbb{D}_{\mathbf{p}},\boldsymbol{\varphi}
\boldsymbol{\delta}}\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$
Let $(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho}
,\boldsymbol{\zeta}_{\varrho},{\mathbf{e}_h}_{\varrho})$ denote the obtained solution at the $\varrho$-th iteration, which should be used for the $\varrho+1$-th iteration. With a convergence threshold $\epsilon_1$, the stop condition of alternate optimization algorithm is then given by $$\begin{aligned}
&|\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho})-\\\nonumber&
\Theta(\mathbf{p}_{\varrho+1},\tilde{\mathbf{p}}_{\varrho+1},\mathbf{s}_{\varrho+1},\boldsymbol{\theta}_{\varrho+1},\boldsymbol{\zeta}_{\varrho+1},\boldsymbol{\varphi}_{\varrho+1},{\mathbf{e}_h}_{\varrho+1})|\leq\epsilon_1.\end{aligned}$$
We can also present a maximum allowed number $\Psi_1$ for $\varrho_1$. Alternate optimization algorithm is illustrated in Table. \[Alternate Optimization Algorithm\]. Furthermore, the following Theorem. \[Theorem 2\] can verify the convergence of the alternate optimization algorithm.
\[Theorem 2\] If (\[AS-1\]) -(\[AS-6\]) are solvable, in each iteration, the sequence of each solution, i.e., $\{\Theta(\mathbf{p}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\varphi}_{\varrho}
,\boldsymbol{\delta}_{\varrho})\}$, is monotonically decreasing.
Please see Appendix \[appendix-B\].
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**Algorithm 1**: Alternate Optimization Algorithm
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
**Step1**: Select a starting point $(\mathbf{p}_0,\tilde{\mathbf{p}}_0,
\mathbf{s}_0,\boldsymbol{\theta}_0,
\boldsymbol{\zeta}_0,\boldsymbol{\varphi}_0
,\boldsymbol{\delta}_0)\in\mathbb{D}$, and Set iteration number $\varrho=0$;
**Step2**: Compute $\Theta(\mathbf{p}_0,
\mathbf{s}_0,\boldsymbol{\varphi}_0
,\boldsymbol{\delta}_0)$;
**Repeat**
**Step3**: For fixed $\mathbf{p}_{\varrho}$, solve (\[eq-worstcase-e\]) to obtained the ${\mathbf{e}_h}_{\varrho+1}$ (Linear programming);
**Step4**: For the obtained ${\mathbf{e}_h}_{\varrho+1}$, and fixed $\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},
\mathbf{s}_{\varrho},
\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho}
,\boldsymbol{\delta}_{\varrho}$, solve (\[AS-2\]) to find $\boldsymbol{\theta}_{\varrho+1}$ (Linear programming);
**Step5**: For the obtained ${\mathbf{e}_h}_{\varrho+1}$, $\boldsymbol{\theta}_{\varrho+1}$, and fixed $\mathbf{p}_{\varrho},
\mathbf{s}_{\varrho},\boldsymbol{\varphi}_{\varrho} ,\boldsymbol{\delta}_{\varrho}$, solve (\[AS-3\]) to find $\tilde{\mathbf{p}}_{\varrho+1}$ and $\boldsymbol{\zeta}_{\varrho+1}$ (Convex programming);
**Step6**: For the obtained ${\mathbf{e}_h}_{\varrho+1}$, $\boldsymbol{\theta}_{\varrho+1}$, $\boldsymbol{\zeta}_{\varrho+1}$ and $\tilde{\mathbf{p}}_{\varrho+1}$, and fixed $\mathbf{p}_{\varrho},\boldsymbol{\varphi}_{\varrho}
,\boldsymbol{\delta}_{\varrho}$, solve (\[AS-4\]) to find $\mathbf{s}_{\varrho+1}$ (DC programming);
**Step7**: For the obtained ${\mathbf{e}_h}_{\varrho+1}$, $\boldsymbol{\theta}_{\varrho+1}$, $\boldsymbol{\zeta}_{\varrho+1}$,$\tilde{\mathbf{p}}_{\varrho+1}$ and $\mathbf{s}_{\varrho+1}$, solve (\[AS-6\]) to find $\mathbf{p}_{\varrho+1},\boldsymbol{\varphi}_{\varrho+1}$ and $\boldsymbol{\delta}_{\varrho+1}$ (DC programming);
**Step8**:Compute $\Theta(\mathbf{p}_{\varrho+1},\mathbf{s}_{\varrho+1},\boldsymbol{\varphi}_{\varrho+1}
,\boldsymbol{\delta}_{\varrho+1})$;
**Step9**: $\varrho=\varrho+1$;
**Step10**: **If** $|\Theta(\mathbf{p}_{\varrho},\mathbf{s}_{\varrho},
\boldsymbol{\varphi}_{\varrho} ,\boldsymbol{\delta}_{\varrho})-
\Theta(\mathbf{p}_{\varrho+1},\mathbf{s}_{\varrho+1},
\boldsymbol{\varphi}_{\varrho+1}
,\boldsymbol{\delta}_{\varrho+1})|\leq\epsilon_1$ or $\varrho>\Psi_1$ goto Step12, **else** goto Step3;
**End**
**Step11**: Return $(\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta},
\boldsymbol{\zeta},\boldsymbol{\varphi}
,\boldsymbol{\delta})=(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},
\boldsymbol{\theta}_{\varrho},
\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho}
,\boldsymbol{\delta}_{\varrho})$.
\[Alternate Optimization Algorithm\]
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
### Channel Uncertainty Problem
For minimizing the worst-case problem over $\boldsymbol{\mathcal{H}}$ in (\[AS-1\]), we solve the following problem for each $t$, $b$, $u$, $m$, and $q$: $$\begin{aligned}
\label{eq-worstcase-e}
\max_{h^{nt}_{bq}\in\mathcal{H}^{nt}_{bq}}~&
R^{\text{E},mt}_{buq}\equiv\max_{h^{nt}_{bq}\in\mathcal{H}^{nt}_{bq}}~\frac{\sum_{n\in
\mathcal{N}}\eta_{nm}s^{mt}_{bu}p^{mt}_{bu}
h^{nt}_{bq}}{\hat{I}^{mt}_{buq}+(\sigma^{n}_q)^2}$$ We can rewrite (\[eq-worstcase-e\]) as follows:
\[eq-worstcase-e2\] $$\begin{aligned}
\max_{\textbf{h}^t}~&\frac{(\bar{\textbf{c}}^{mt}_{buq})^\text{T}{\textbf{h}}^t}{(\hat{\textbf{c}}^{mt}_{buq})^\text{T}{\textbf{h}}^t+(\sigma^{n}_q)^2},\\\text{s.t.}~&h^{nt}_{\acute{b}\acute{q}}\leq\tilde{h}^{nt}_{\acute{b}\acute{q}}+\varepsilon_{h^{nt}_{\acute{b}\acute{q}}},~\forall \acute{b},\acute{q},\acute{u},n\\
&\tilde{h}^{nt}_{\acute{b}\acute{q}}-\varepsilon_{h^{nt}_{\acute{b}\acute{q}}}\leq h^{nt}_{\acute{b}\acute{q}}, ~ \forall \acute{b},\acute{q},\acute{u},n,\end{aligned}$$
where $\bar{\textbf{c}}^{mt}_{buq}$ is a vector of the same dimension as ${\textbf{h}}^t$ with all zero enry expect for $[\bar{\textbf{c}}^{mt}_{buq}]_{\acute{b}=b,\acute{q}=q,n}=\eta_{nm}s^{mt}_{bu}p^{mt}_{bu}, \forall n\in\mathcal{N}$, and $\hat{\textbf{c}}^{mt}_{buq}$ is a vector of the same dimension as ${\textbf{h}}^t$ with $[\hat{\textbf{c}}^{mt}_{buq}]_{\acute{b},\acute{q},n}=\sum_{\acute{u}\in
\mathcal{U}_{\acute{b}}}\eta_{nm}s^{mt}_{\acute{b}\acute{u}}p^{mt}_{\acute{b}\acute{u}}, \forall \acute{b}\neq b,\acute{q}\neq q, n$, and $[\hat{\textbf{c}}^{mt}_{buq}]_{\acute{b},\acute{q},n}=0$ for all other entries. This problem has a linear fractional objective function, for which, Charnes-Cooper transformation can be used to reformulate it into the following linear programming optimization problem [@parsaeefard2017robust]:
\[eq-worstcase-e3\] $$\begin{aligned}
\max_{\bar{\textbf{h}}^t,\mu}~&(\bar{\textbf{c}}^{mt}_{buq})^\text{T}\bar{\textbf{h}}^t,\\\text{s.t.}~~&(\hat{\textbf{c}}^{mt}_{buq})^\text{T}\bar{\textbf{h}}^t+\mu(\sigma^{n}_q)^2=1,\\&\bar{h}^{nt}_{\acute{b}\acute{q}}\leq\mu\tilde{h}^{nt}_{\acute{b}\acute{q}}+\mu\varepsilon_{h^{nt}_{\acute{b}\acute{q}}},~\forall \acute{b},\acute{q},\acute{u},n\\
&\mu\tilde{h}^{nt}_{\acute{b}\acute{q}}-\mu\varepsilon_{h^{nt}_{\acute{b}\acute{q}}}\leq \bar{h}^{nt}_{\acute{b}\acute{q}}, ~ \forall \acute{b},\acute{q},\acute{u},n,\end{aligned}$$
where $\bar{\textbf{h}}^t={\textbf{h}}^t/\mu$, $\bar{\textbf{h}}^t\succeq \textbf{0}$ and $\mu> 0$. Problem (\[eq-worstcase-e3\]) can now be efficiently solved using interior-point based methods by some off-the-shelf convex optimization toolboxes, e.g., CVX.
### Content Placement
A linear programming (LP) with respect to $\boldsymbol{\theta}$ for the content placement problem can be obtained. This problem can be easily solved by existing LP available standard optimization softwares such as CVX with the internal solver MOSEK [@mokari2016limited; @michael2012matlab].
### Backhaul Power and Subcarrier Allocation
The optimization problem is still a mixed-integer non-convex programming with respect to $\boldsymbol{\zeta}$ and $\tilde{\boldsymbol{p}}$, which is difficult to tackle. To make this problem tractable, we first relax each $\boldsymbol{\zeta}$ to a continuous interval, i.e., $\boldsymbol{\zeta}\in$\[0,1\]. Further, new variables $\textbf{x}=\boldsymbol{\zeta}\tilde{\boldsymbol{p}}$ is defined to replace $\tilde{\boldsymbol{p}}$. Then, we can transform the nonconvex optimization problem into the convex one. This problem can be easily solved by available standard optimization softwares such as CVX with the internal solver MOSEK [@mokari2016limited]. Note that this relaxation is called time sharing which shows the time percentage that each subcarrier should be used [@tao2008resource].
### Access Power and Codebook Allocation
The optimization problem is still non-convex with respect to $\textbf{p}$ and $\textbf{s}$. The difficulty of solution comes from the non-convexity of both objective function and secrecy rate constraint. There is no standard approach to solve such a non-convex problem. Therefore, we exploit DC and fractional programming in the next sections to transform it into a tractable problem. In the following, we develop a solution for power allocation optimization problem and we remark that this solution can be developed for code assignment in the same way.
Difference-of-Two-Concave-Functions (D.C.) Approximation
--------------------------------------------------------
Due to the non-convexity of (\[eq-EEE\]), the optimization problem (\[eq-worstcase-p-1234\]) is still difficult to solve. The standard D.C. optimization problem can be written as $\min_{\mathbf{x}}\{F(\mathbf{x})=F_1(\mathbf{x})-F_2(\mathbf{x})\}$ where $F_1$ and $F_2$ are two convex components with convex feasible domain. This problem can be solved iteratively by solving a sequential convex program as follows: $$\min_{\mathbf{x}}\{F_1(\mathbf{x})-F_2(\mathbf{x}_{\varrho})-\langle
\nabla
F_2(\mathbf{x}_{\varrho}),\mathbf{x}-\mathbf{x}_{\varrho}\rangle\},$$ at each iteration, where $\mathbf{x}_{\varrho}$ is the optimal solution of the $\varrho^{\text{th}}$ iteration used for the $(\varrho+1)^{\text{th}}$ iteration and $\nabla F_2(\mathbf{x})$ is the gradient of $F_2(\mathbf{x})$ evaluated at $\mathbf{x}_{\varrho}$. By the sequential convex approximation, DC subproblems are equivalently reformulated as:
\[eq-worstcase-p-1234-1\] $$\begin{aligned}
\max_{\mathbf{p},\boldsymbol{\varphi} ,\boldsymbol{\delta}}
&~\Theta(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho},\boldsymbol{\zeta}_{\varrho},\boldsymbol{\varphi}_{\varrho},{\mathbf{e}_h}_{\varrho}),\\\nonumber
\text{s.t.}~&-\left(R^{\text{E},mt}_{buq2}-R^{\text{E},mt}_{buq1}-\left\langle\nabla
R^{\text{E},t}_{bu1},
p^{mt}_{bu}-p^{mt}_{bu}(\varrho)\right\rangle
\right)\leq\\&\varphi^{mt}_{bu},\forall m\in\mathcal{M},
t\in\mathcal{F},b\in\mathcal{B},u\in\mathcal{U}_b,q\in\mathcal{Q},
\\\nonumber&(\ref{eq-P-constraint-1}),(\ref{eq-energy-2}),(\ref{eq-energy-3}),(\ref{eq-EEE-1}),(\ref{eq-EEE-2}),(\ref{eq-EEE-3}),\end{aligned}$$
where $$\begin{aligned}
& R^{\text{E},mt}_{buq1}=\log_2\left(\sum_{b\in
\mathcal{B}}\sum_{u\in
\mathcal{U}_b}\sum_{n\in
\mathcal{N}}\left(\eta_{nm}s^{mt}_{bu}
p^{mt}_{bu}h^{nt}_{bq}+(\sigma^{n}_q)^2\right)\right)\\
&R^{\text{E},mt}_{bu2}=\log_2\left(\sum_{\acute{b}\in
\mathcal{B}\setminus\{b\}}\sum_{\acute{u}\in
\mathcal{U}_{\acute{b}}}\sum_{n\in
\mathcal{N}}\left(\eta_{nm}s^{mt}_{\acute{b}\acute{u}}
p^{mt}_{\acute{b}\acute{u}}
h^{nt}_{\acute{b}q}+(\sigma^{n}_q)^2\right)
\right).\end{aligned}$$
We first express $R^{\text{E},mt}_{buq}$ in a D.C. form as: $$\label{E}
R^{\text{E},mt}_{buq}=-(R^{\text{E},mt}_{buq2}-R^{\text{E},mt}_{buq1}).$$ Based on (\[E\]), the gradient $\nabla R^{\text{E},mt}_{bu1}$ with respect to $\mathbf{p}$ is given by $$\begin{aligned}
\label{eq-53}
&\nabla R^{\text{E},mt}_{bu1}=\frac{\partial
R^{\text{E},mt}_{bu1}}{\partial p^{mt}_{bu}}=
\frac{\partial R^{\text{E},mt}_{buq1}}{\partial
p^{mt}_{bu}}=\\\nonumber&\frac{s^{mt}_{bu}}{\ln2} \frac{\sum_{n\in
\mathcal{N}}\left(\eta_{nm}h^{nt}_{bq}\right)} {\sum_{b\in
\mathcal{B}}\sum_{u\in \mathcal{U}_b}\sum_{n\in
\mathcal{N}}\left(\eta_{nm}s^{mt}_{bu}p^{mt}_{bu}h^{nt}_{bq}+(\sigma^{n}_q)^2\right)}.\end{aligned}$$
The sequence $R^{\text{E},mt}_{buq}$ derived from the D.C. programming algorithm is monotonically decreasing.
Please see Appendix \[appendix-C\].
Fractional Programming
----------------------
The objective function in (\[eq-worstcase-p-1234\]) is non-convex. The form of (\[eq-worstcase-p-1234\]) can be classified into the nonlinear fractional programming [@Dinkelbach67]. Therefore, after replacing nonconvex constraints by convex constraints using the D.C. method in the previous section, the Dinkelbach’s algorithm use to solve convex fractional programming. We define the maximum objective functions $(\chi^{\mathbf{p}})^*$ of the considered system as: $$\begin{aligned}
\label{eq-zeta}\nonumber
&(\chi^{\mathbf{p}})^*=\max_{\mathbf{p},\boldsymbol{\varphi},\boldsymbol{\delta}}\frac{\sum_{m\in
\mathcal{M}}\sum_{t\in F}\sum_{b\in \mathcal{B}}\sum_{u\in
\mathcal{U}_b}\left\{\delta^{mt}_{bu}
+R^{\text{D},mt}_{bu1}\right\}} {\sum_{t\in \mathcal{F}}\sum_{b\in
\mathcal{B}}\sum_{m\in \mathcal{M}}\sum_{u\in
\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}}\\&=\frac{\Xi_{\text{Num}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}
{\Xi_{\text{Den}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}.\end{aligned}$$
We are now ready to introduce the following theorem for $\chi^{\mathbf{p}}$.
The maximum value of $(\chi^{\mathbf{p}})^*$ is achieved if and only if $$\begin{aligned}
\label{eq-zeta-2}
\max_{\mathbf{p},\boldsymbol{\varphi}
,\boldsymbol{\delta}}~&
\left\{{\Xi_{\text{Num}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}-(\chi^{\mathbf{p}})^*
{\Xi_{\text{Dem}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}\right\}\\\nonumber&
={\Xi_{\text{Num}}(\mathbf{p}^*,\mathbf{s},\boldsymbol{\varphi}^*,\boldsymbol{\delta}^*)}-(\chi^{\mathbf{p}})^*
{\Xi_{\text{Num}}(\mathbf{p}^*,\mathbf{s},\boldsymbol{\varphi}^*,\boldsymbol{\delta}^*)}=0.\end{aligned}$$ For $\Xi_{\text{Num}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})\geq0$ and $\Xi_{\text{Dem}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})>0$, where $$\label{eq-20}
\max_{\mathbf{p},\boldsymbol{\varphi}
,\boldsymbol{\delta}}~\left\{{\Xi_{\text{Num}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}-\chi^{\mathbf{p}}
{\Xi_{\text{Dem}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}\right\},$$ is defined as a parametric program with parameter $\chi^{\mathbf{p}}$.
Please refer to [@Dinkelbach67; @Schaible76].
By the Dinkelbach’s method \[23\] with a initial value $\chi^{\mathbf{p}}_0$ of $\chi^{\mathbf{p}}$, (\[eq-20\]) can be solved iteratively by solving the following problem: $$\label{eq-21}
\max_{\mathbf{p},\boldsymbol{\varphi}
,\boldsymbol{\delta}}~{\Xi_{\text{Num}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}-\chi^{\mathbf{p}}_{\varrho}
{\Xi_{\text{Dem}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})},$$ with a given $\chi^{\mathbf{p}}_{\varrho}$ at the $\varrho^{\text{th}}$ iteration, where $\varrho$ is the iteration index. $\chi^{\mathbf{p}}_{\varrho}$ can be explained as the secure EE obtained at the previous iteration. In (\[eq-21\]), the maximization problem is equivalent to $$\label{eq-22}
\min_{\mathbf{p},\boldsymbol{\varphi}
,\boldsymbol{\delta}}~{\chi^{\mathbf{p}}_{\varrho}
{\Xi_{\text{Dem}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})-\Xi_{\text{Num}}(\mathbf{p},\mathbf{s},\boldsymbol{\varphi},\boldsymbol{\delta})}}.$$
Let $\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}), \boldsymbol{\varphi}(\chi^{\mathbf{p}}_{\varrho})$ and $\boldsymbol{\delta}(\chi^{\mathbf{p}}_{\varrho})$ denote the solution of (\[eq-23\]) for a given $\chi^{\mathbf{p}}_{\varrho}$. After each iteration, $\chi^{\mathbf{p}}_{\varrho}$ should be updated by $$\label{eq-23}
\chi^{\mathbf{p}}_{\varrho+1}=\frac
{\Xi_{\text{Dem}}(\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\varphi}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\delta}(\chi^{\mathbf{p}}_{\varrho}),\mathbf{s})}
{\Xi_{\text{Num}}(\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\varphi}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\delta}(\chi^{\mathbf{p}}_{\varrho}),\mathbf{s})}.$$
The iteration process will be stopped when (\[eq-zeta-2\]) is satisfied. In practice, we define the terminated condition of the iterative process as: $$\begin{aligned}
&\Big|\chi_{\varrho}\Xi_{\text{Dem}}(\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\varphi}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\delta}(\chi^{\mathbf{p}}_{\varrho}),\mathbf{s})-\\\nonumber&
\Xi_{\text{Num}}(\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\varphi}(\chi^{\mathbf{p}}_{\varrho}),\boldsymbol{\delta}(\chi^{\mathbf{p}}_{\varrho})),\mathbf{s})\Big|\leq\epsilon_3,\end{aligned}$$ with a small convergence tolerance $\epsilon_3>0$. The algorithm of fractional programming is clarified in Algorithm 2, where $\Psi_3$ is the maximum allowed number of iterations considering the computational time. We use the fractional programming Dinkelbach’s algorithm for the convexified problem (\[eq-worstcase-p-1234-1\]).
If problems (\[eq-worstcase-p-1234-11\]) are solvable, the sequence $\{\chi_{\varrho}\}$ obtained by Algorithm 1 has the following properties: 1) $\chi^{\mathbf{p}}_{\varrho+1}>\chi^{\mathbf{p}}_{\varrho}$; 2) $\lim_{\varrho\rightarrow\infty}\chi^{\mathbf{p}}_{\varrho}=(\chi^{\mathbf{p}})^*$.
Please refer to [@Dinkelbach67].
Based on the fractional programming, subproblems (\[eq-worstcase-p-1234-11\]) are associated with a parametric program problem stated as follows:
\[eq-worstcase-p-1234-11\] $$\begin{aligned}
\max_{\mathbf{p},\boldsymbol{\varphi} ,\boldsymbol{\delta}}
&~\chi^{\mathbf{p}}\sum_{t\in \mathcal{F}}\sum_{b\in \mathcal{B}}\sum_{m\in
\mathcal{M}}\sum_{u\in
\mathcal{U}_b}s^{mt}_{bu}p^{mt}_{bu}-\\\nonumber&\sum_{m\in
\mathcal{M}}\sum_{t\in F}\sum_{b\in \mathcal{B}}\sum_{u\in
\mathcal{U}_b}\left\{\delta^{mt}_{bu}
+R^{\text{D},mt}_{bu1}\right\},\\\nonumber
\text{s.t.}~&-\left(R^{\text{E},mt}_{buq2}-R^{\text{E},mt}_{buq1}-\left\langle\nabla
R^{\text{E},t}_{bu1},
p^{mt}_{bu}-p^{mt}_{bu}(\varrho)\right\rangle
\right)\leq\varphi^{mt}_{bu},\\&\forall m\in\mathcal{M},
t\in\mathcal{F},b\in\mathcal{B},u\in\mathcal{U}_b,q\in\mathcal{Q},
\\\nonumber&(\ref{eq-P-constraint-1}),(\ref{eq-energy-2}),(\ref{eq-energy-3}),
(\ref{eq-EEE-1}),(\ref{eq-EEE-2}),(\ref{eq-EEE-3}).\end{aligned}$$
We propose an iterative algorithm (known as the Dinkelbach method [@Dinkelbach67]) for solving (\[eq-worstcase-p-1234-11\]) with an equivalent objective function. The proposed algorithm to obtain power allocation policy $\textbf{p}$ is summarized in Table. \[Fractional Programming Algorithm\]. The convergence to the appropriate energy efficiency is guaranteed. Note that similar algorithm can be used to obtain code allocation policy $\textbf{s}$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**Algorithm 3**: Fractional Programming Algorithm
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
**Step1**: Initialize the maximum number of iterations $\Psi_3$ and the maximum tolerance $\epsilon_3$;
**Step2**: Choose an initial value $\chi^{\mathbf{p}}_0$ and set iteration index $\varrho=0$;
**Repeat**
**Step3**: Solve problem (\[eq-worstcase-p-1234-11\]) for a given $\chi^{\mathbf{p}}_{\varrho}$ and obtain power allocation policy $\mathbf{p}(\chi^{\mathbf{p}}_{\varrho})$ (Convex programming);
**Step5**: Update $\chi^{\mathbf{p}}_{\varrho}$ by (\[eq-23\]) to obtain $\chi^{\mathbf{p}}_{\varrho+1}$;
**Step6**:$\varrho=\varrho+1$;
**Step7**: If $|\chi^{\mathbf{p}}_{\varrho}\Xi_{\text{Den}}(\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}),\mathbf{s})-
\Xi_{\text{Num}}(\mathbf{p}(\chi^{\mathbf{p}}_{\varrho}),\mathbf{s})|<\epsilon_3$ or $\varrho>\Psi_3$ goto Step7, **else** goto Step3;
**Step8**: Return $\mathbf{p}^*=\mathbf{p}(\chi^{\mathbf{p}}_{\varrho-1})$, $(\chi^{\mathbf{p}})^*=\chi^{\mathbf{p}}_{\varrho}$.
**End**
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
To solve the primary optimization problem, the main optimization problem is decomposed into several subproblems, with each subproblem being in a hierarchical order of the main problem. Depending on different methods to solve each subproblem, the computational complexity of the proposed algorithm is analyzed in Section \[computationalanalysis\].
Analysis of Computational Complexity of Proposed Algorithm {#computationalanalysis}
==========================================================
To solve the primary optimization problem, the main optimization problem is decomposed into several subproblems, with each subproblem being in a hierarchical order of the main problem. The fast gradient algorithm can be used to solve the inner convex subproblems [@richter2012computational]. Then, the convergence of fast gradient algorithm can be written as follows [@richter2012computational]: $$\varrho^{\iota}_{\tau}=\mathcal{O}(1)
\min\left\{\sqrt{\frac{l_{\iota}}{\xi_{\iota}}}\ln\left(\frac{l_{\iota}}{\tau}\right),
\sqrt{\frac{l_{\iota}}{\xi_{\iota}}}\right\},\iota=1,2,$$ where $\tau$ is convergence tolerance. $l_1$ and $l_2$ are defined as the Lipschitz constants by which the Lipschitz conditions are satisfied with the gradients $\nabla
R^{\text{E},t}_{buq,1}(\mathbf{p},\mathbf{s})$ in subproblem (\[AS-2\]) and (\[AS-3\]), respectively. Moreover, $\xi_1$ and $\xi_2$ denote the convexity parameters to satisfy strong convexity of $f_1$ and $f_2$, respectively. In other words, the fast gradient method is used to solve optimization problem (\[AS-2\]) and (\[AS-3\]) in $\varrho^{1}_{\tau}$ and $\varrho^{2}_{\tau}$ iterations, respectively. The iteration numbers $\varrho_1, \varrho_2$, and $\varrho_3$ are corresponding to the convergence tolerance $\epsilon_1, \epsilon_2$, and $\epsilon_3$ related to subalgorithms. Then, the overall computational complexity associated with the proposed algorithm is dominated by $\varrho_1\varrho_3(\varrho^1_2\varrho^{1}_{\tau}+\varrho^2_2\varrho^{2}_{\tau})$, where $\varrho^1_2$ and $\varrho^2_2$ are the iteration numbers of DC programming algorithm. The computational complexity of the proposed solution of the content placement problem is equal to $\mathcal{O}(I_{\theta}B^{3.5}K^{3.5})$ [@ben2001lectures] where $I_{\theta}$ is the expected iteration numbers. The computational complexity of the our proposed MFCD problem is equal to $\mathcal{O}(I_{\beta}F^{3.5}K^{3.5})$ [@ben2001lectures] where $I_{\beta}$ is the expected iteration numbers. The computational complexity of the proposed backhaul power and subcarrier allocation problems are equal to $\mathcal{O}(\hat{I}_{\tilde{p}}NFB
\log_2(1/\epsilon_{\tilde{p}}))$ and $\mathcal{O}(\hat{I}_{\zeta}NFB
\log_2(1/\epsilon_{\zeta}))$ where $\epsilon_{\tilde{p}}$, $\hat{I}_{\tilde{p}}$, $\epsilon_{\zeta}$ and $\hat{I}_{\zeta}$ are the maximum error tolerance and expected iteration numbers, respectively.
By the proposed algorithm, the main problem, which is NP-hard, can be solved in polynomial time. Therefore, the complexity of the proposed algorithm compared to the original problem is far less and manageable. To manage the complexity, Graphics Processing Unit (GPU) can be utilized. By exploiting GPU instead of central processing unit (CPU), new processing methods to accelerate the processing time can be used. In [@naderazmi2018arxiv], a new framework to accelerate the iterative-based resource allocation by using ASM and SCA has been devised. By using GPU-based resource allocation, the processing time speed-up of about 1500 times compared to CPU-based methods can be achieved. Due to the space limitation, we omit the study of GPU-based version of the proposed algorithms and leave it as future work.
Simulation Results {#simulationsresults}
==================
For simulations, we consider a multi-cell downlink SCMA system where $U$ users are randomly distributed in an area of circle with the radius of 1 km for each BS as the center. The number of users in circle area of $b^{\text{th}}$ BS is set to $U_b=4,\forall b$, and the total number of subcarriers and codebooks are set to 8 and 28, respectively. The bandwidth of each subcarrier is 180 kHz [@etsi2011136]. The channels between the MBS and its users and SBS and its users are generated with a normalized Rayleigh fading component and a distance-dependent path loss in urban and suburban areas, modeled as $PL(dB)=128.1+37.6\log_{10}(d)+X$ and $PL(dB)=38+30\log_{10}(d)+X$, respectively [@etsi2011136], where $d$ is the distance from user to BS in kilometers and $X $ is 8 dB log-normal shadowing. We set the frame duration to $T=0.01$ s [@castiglione2014energy; @zhang2015delay]. The noise power, $(\sigma^n_u)^2=(\sigma^n_b)^2=\sigma^2,\forall u,b,n$ is set to $-125$ dBm. We set $D=2$ and $\eta_{nm}=0.5,\forall n,m$ for SCMA [@nikopour2013sparse]. We set the amount of harvested energy per arrival to $\rho^t_b=\rho=0.8, \forall t, b$ J, $\Gamma_b=\Gamma=0.1,\forall b$ and users request contents by normal random generator. In the most popular caching case, the most popular contents is cached at each BS until its storage is full. In this case, the content popularity is modeled as the Zipf distribution with Zipf parameter equals to 0.8. Simulation results are obtained by averaging over 1000 simulation runs.
Effect of Maximum Allowable Backhaul Transmission Power
-------------------------------------------------------
In this part, we obtain the backhaul rate for different values of backhaul transmission power with different values of $\alpha$. The simulation results are compared for different caching scenarios such as no caching, random caching, most popular caching and the proposed caching methods. In no caching case, no contents are stored by any BS. Hence, all the requested contents are served by the core network over the backhaul links [@park2016joint; @stephen2016green; @hsu2016resource]. In the random caching strategy, the contents are randomly cached by BSs until storage of BSs is full. Content popularity does not matter in this strategy. In the most popular caching strategy, each BS caches the most popular contents until its storage is full [@stephen2016green; @park2016joint]. The results are reported in Fig. \[Fig\_R\_backhual\_vs\_p\_diff\_alpha\]. As can be seen, for a fixed transmit power, when $\alpha$ is increased, the resulting backhaul rate increases. As can be seen from this figure, utilizing the caching strategies can reduce backhaul traffic compared to the no caching scheme. Our proposed caching strategy has nearly 43%, 23.4% and 18.5% performance gain in terms of backhaul rate reduction compared to the no caching scheme for different values of $\alpha=$1, 2, and 3, respectively. It is also notable that the most popular caching strategy causes more reduction in the backhaul traffic, compared to the random caching scheme. However, when all the caching placement are done jointly with the allocation of other network resources, the network performance improves dramatically. This improvement is due to the fact that content placement is done according to network conditions and resources. Besides, as shown in Fig. \[Fig\_R\_backhual\_vs\_p\_diff\_alpha\], our caching scheme reduces the total backhaul rate close to almost 11% compared to the most popular caching strategy.
Effect of Energy Harvesting
---------------------------
Fig. \[Fig\_Energy\_efficiency\_vs\_E\_b\_diff\_U\] shows the EE as a function of harvested energy per arrival for the SFCD scenario. We compare different EH strategy in terms of EE. In general, by increasing the EH value, the EE is also increased. For larger number of users, the EE is increased. In other words, for the small number of users, there is sufficient power resources, therefore by increasing users, the EE is also increased as shown in Fig. \[Fig\_Energy\_efficiency\_vs\_E\_b\_diff\_U\]. However, for too more users, the power resource will be exhausted and thus some users can not access to network. Even so, due to multiuser diversity, the EE will still increase. For limited battery, due to overflow conditions (\[eq-energy-3\]), the stored energy must be used such that there is enough capacity in the battery for newly arrived energy. In this regard, increasing the value of $\rho$ will increase the energy efficiency at first, but with further increasing $\rho$, the energy efficiency decreases. This is because, from the energy efficiency point of view, the energy consumption would be limited to the amount which maximizes the bit-per-joule quantity. However, for unlimited battery, with increasing $\rho$, the energy efficiency increases at first, and by further increasing $\rho$, the energy efficiency becomes constant since no more energy would be consumed as all the arriving energy could be stored in the battery.
Effect of File Spliting
-----------------------
Fig. \[fig\_EE\_vs\_Packet\_Spliting\] shows EE as a function of the harvested energy per arrival, $\rho$ for the SFCD and MFCD scenarios As can be seen from Fig. \[fig\_EE\_vs\_Packet\_Spliting\], the MFCD scheme outperforms the SFCD scenario. In the EE communication networks, due to random energy arrivals, there may be not enough energy to transmit file that has big size in the SFCD scheme. In contrast, in the MFCD schemes, file is splitted into the several small size files which can be transmitted in the suitable frames to increase EE. In the uniform file splitting, the file is uniformly splitted into several smaller files with the same size. This scheme has better performance than the SFCD scheme, However, we can improve the network performance by using the our proposed method. In the our proposed MFCD scheme, the best size of each splitted file is obtained to enhance the network performance. This figure also shows that by reducing the size of file, the distance between the graphs for the three scenarios decreases. As seen, the MFCD-proposed file splitting and MFCD-uniform file splitting have closed to almost 9.4% and 6% performance gain in terms of EE compared to SFCD scheme, respectively.
Transmission Inutility
----------------------
In this section, we investigate the transmission inutility for the SFCD and MFCD schemes. The transmission inutility is defined by multiplying the outage probability in the transmission delay. The outage probability is defined as probability that there is not enough battery to send content files and the transmission delay is defined as number of frames to send files. Fig. \[Fig\_Outage\_probability\_vs\_F\_diff\_alpha\] demonstrates the transmission inutilities of our proposed schemes for different content file size. As can be seen, by increasing the size of file, the transmission inutility is increased for both schemes. This is due to the fact that there may be not enough harvested energy to send file, and the energy deficiency probability can be increased. Therefore, the outage probability approaches to one in sufficiently big size of files. This deterioration in the MFCD schemes are less than the SFCD scheme. Because in the MFCD schemes, the deficiency probability of energy can be reduced by dividing the content file into several parts and sending each part in different frames. In the proposed splitting scheme, we find the best fractional of content file for each frame which reduces the outage probability more than before. As shown in Fig. \[Fig\_Outage\_probability\_vs\_F\_diff\_alpha\], for larger content file sizes, the MFCD scheme has a higher efficiency in reducing the outage probability. As can be seen, for the size of content files less than 3 MBits, the SFCD scheme is better, while for the size of large files, the MFCD scheme is better.
Effect of Channel Uncertainity
------------------------------
Fig. \[Fig3\] shows the access secrecy rate versus channel uncertainty for the SFCD scenario. We see that at bigger channel uncertainty, the secrecy access rate clearly has low value. This is due to the fact that when the uncertainty increases, for the worst case scenario, we must guarantee the security for the worst (biggest) channel value of eavesdroppers which leads to low values of secrecy rate. As can be seen, as the number of eavesdroppers increases, the secrecy access rate decreases due to the multiuser diversity gain for eavesdroppers.
Comparison Between Joint backhaul and access optimization and Disjoint Optimization Problem Solution
----------------------------------------------------------------------------------------------------
Fig. \[Fig\_compare\_joint\_disjoint\] illustrates the comparison between joint optimization and disjoint optimization problem solutions versus different backhaul transmission power for the SFCD scenario. In solving the main problem disjointly, one must solve two optimization problems, one for the access part and the other for the backhaul part. Here, we first solve the backhaul problem since in most cases the backhaul capacity is the limiting factor. For the backhaul problem, the objective is maximization of the backhaul rate, and the constraint is the total transmit power of the backhaul. Solving this problem, the supported transmission rate of the backhaul is obtained which will be used in the access problem. The access problem is similar to the problem . The differences between the access problem and problem are that the optimization variables of the backhaul and the constraints relating to the backhaul are removed, and one additional constraint, which states the the access secrecy rate should be above the backhaul transmission rate (obtained in the backhaul problem), is included in the access problem. It can be noticed that the joint backhaul and access optimization approach has better solution than the disjoint backhaul and access optimization approach. This is mainly because that in the joint scenario, the feasibility set of the optimization problem is bigger than the disjoint one. Indeed, in the access problem, we have the constraint which enforces that the access rate should above the backhaul transmission rate (obtained in the backhaul problem) which makes the feasibility set of the disjoint problems smaller than that of the joint one. From Fig. \[Fig\_compare\_joint\_disjoint\], it can be observed that there exists nearly 51.3% and 50% performance gap between joint and disjoint approachs in terms of backhaul and access rates, respectively.
Effect of super frame size
--------------------------
Fig. \[Fig\_Energy\_efficiency\_backhual\_rate\_access\_rate\_vs\_F\] shows the variation of the EE with the number of frames, $F$ for the SFCD scenario. It is seen that by increasing super frame size, the value of EE increases. In other words, by increasing super frame size, the transmitter can transmit data stream over different frames, then the secrecy access rate and EE will increase. For limited battery storage, with increasing super frame size, at first the EE increases. However, with further increasing super frame size, due to energy overflow constraints, (\[eq-energy-3\]), which enforce the transmitters to spend energy, the energy efficiency decreases. Note that, as the value of $\rho$ becomes larger, this decrease in EE happens in lower super frame sizes. For unlimited battery storage, since the overflow constrains, (\[eq-energy-3\]), are absent, all the harvested energy is stored in the battery. In this case, increasing super frame size will increase the diversity gain, and hence, the energy efficiency increases.
The Convergence of the Proposed Algorithm
-----------------------------------------
In this part, we investigate the performance of the proposed resource allocation algorithm. In Fig. \[Fig\_EE\_vs\_iteration\_number\], we show EE after each iteration at the proposed alternate optimization algorithm. As can be seen, the convergence of the proposed algorithm can averagely be achieved within 700 iterations.
Conclusion {#Conclusion}
==========
In this paper, we provided a unified framework for radio resources allocation and content placement considering the physical layer security and the channel uncertainty to provide higher energy efficiency. To do so, we considered downlink SCMA scenarios, and we aimed at maximizing the worst case energy efficiency subject to system constraints which determines the radio resources allocation and content placement parameter. Moreover, we proposed two novel content delivery scenarios: 1) single frame content delivery, and 2) multiple frames content delivery. In the first scenario, the requested content by each user is served over one frame. However, in the second scenario, the requested content by each user can be delivered over several frames. Since the optimization problems are noncovex and NP-hard, we provided an iterative method converging to a local solution. Finally, we showed the resulting secrecy access rate, backhaul rate, and energy efficiency for different values of maximum backhaul transmit power as well as different number of users and various content size. In addition, we compared the performance of the proposed caching scheme with the existing traditional caching schemes. Based on simulation results, via our proposed caching scheme, the performance is approximately improved by 14% and 21% compared to the most popular and random caching schemes, respectively. Moreover, it can be seen that the MFCD scheme can approximately enhance the system performance by 5.2% and 11.1% for small and large files, respectively.
Proof of Lemma 1 {#appendix-A}
================
We jointly find the optimization variables $\mathbf{p},\tilde{\mathbf{p}},\mathbf{s},\boldsymbol{\theta}$, and $\boldsymbol{\zeta}$ such that the EE of proposed system is maximized. Hence, (\[eq–2\]) is MINLP and non-convex. We assume that the optimization variables $\tilde{\mathbf{p}},\boldsymbol{\theta}$, and $\boldsymbol{\zeta}$ are constant. In access link, we also assume that one subcarrier is exclusively assigned to at most $D$ users within the cell. We consider the downlink of an SCMA-based access link consisting of $N$ subcarriers, $U$ users and $D=2$. By assuming that the special $u^{\text{th}}$ user’s channel gain on all subcarriers is the largest among all users, the optimal power assigned to user $u$ is equal to $p^{mt}_{bu}/N$, where $p^{mt}_{bu}$ is the transmit power assigned to user $u$ at BS $b$ at frame $t$ on codebook $m$. Then, the challenge is how to allocate the remaining power resource $E^t_b/T-p^{mt}_{bu}$ to $U-1$ users over all subcarriers. Thus, no subcarrier can be assigned to more than one user. Therefore, a special case of power and codebook optimization problem with $D>1$ is equivalent to the NP-hard problem considered in [@lei2015joint], and the result follows. Finally, it can be concluded that the main problem (\[eq–2\]) is also NP-hard.
Proof of Theorem 1 {#appendix-B}
==================
In accordance to the foregoing discussions, for (\[AS-1\]) with a given ${\mathbf{e}_h}_{\varrho}$, $(\mathbf{p}_{\varrho+1},\tilde{\mathbf{p}}_{\varrho+1},\mathbf{s}_{\varrho+1},\boldsymbol{\theta}_{\varrho+1}
,\boldsymbol{\zeta}_{\varrho+1})$ is its optimal solution, while $(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho}
,\boldsymbol{\zeta}_{\varrho})$ is only its feasible solution. We get that $$\begin{aligned}
&\Xi_{EE}(\mathbf{p}_{\varrho+1},\tilde{\mathbf{p}}_{\varrho+1},\mathbf{s}_{\varrho+1},
\boldsymbol{\theta}_{\varrho+1}
,\boldsymbol{\zeta}_{\varrho+1},{\mathbf{e}_h}_{\varrho})\leq\\\nonumber&
\Xi_{EE}(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho}
,\boldsymbol{\zeta}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$ Likewise, for (\[AS-2\]) with a given $\mathbf{p}_{\varrho}$, $(\tilde{\mathbf{p}}_{\varrho+1},\mathbf{s}_{\varrho+1},\boldsymbol{\theta}_{\varrho+1}
,\boldsymbol{\zeta}_{\varrho+1},{\mathbf{e}_h}_{\varrho+1})$ is its optimal solution, while $(\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho}
,\boldsymbol{\zeta}_{\varrho},{\mathbf{e}_h}_{\varrho})$ is only its feasible solution. It follows that $$\begin{aligned}
&\Xi_{EE}(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho+1},\mathbf{s}_{\varrho+1},
\boldsymbol{\theta}_{\varrho+1}
,\boldsymbol{\zeta}_{\varrho+1},{\mathbf{e}_h}_{\varrho+1})\leq\\\nonumber&
\Xi_{EE}(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho}
,\boldsymbol{\zeta}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$ For relations (\[AS-3\]), (\[AS-4\]), and (\[AS-6\]), this trend is similar. It is naturally concluded that $$\begin{aligned}
&\Xi_{EE}(\mathbf{p}_{\varrho+1},\tilde{\mathbf{p}}_{\varrho+1},\mathbf{s}_{\varrho+1},\boldsymbol{\theta}_{\varrho+1}
,\boldsymbol{\zeta}_{\varrho+1},{\mathbf{e}_h}_{\varrho+1})\leq\\\nonumber&
\Xi_{EE}(\mathbf{p}_{\varrho},\tilde{\mathbf{p}}_{\varrho},\mathbf{s}_{\varrho},\boldsymbol{\theta}_{\varrho}
,\boldsymbol{\zeta}_{\varrho},{\mathbf{e}_h}_{\varrho}).\end{aligned}$$
Proof of Theorem 2 {#appendix-C}
==================
Because of the convexity of $R^{\text{E},mt}_{buq1}$, it follows that $$\begin{aligned}
\label{eq-001}
&R^{\text{E},mt}_{buq1}({\varrho+1})\geq\\\nonumber&
R^{\text{E},mt}_{buq1}({\varrho})-\left\langle\nabla
R^{\text{E},t}_{bu1}({\varrho}),
p^{mt}_{bu}({\varrho+1})-p^{mt}_{bu}(\varrho)\right\rangle,\end{aligned}$$ for $p^{mt}_{bu}(\varrho)$ and $p^{mt}_{bu}(\varrho+1)$ in the feasible domain. We can deduce that $$\begin{aligned}
\label{eq-002}
&-\left\langle\nabla
R^{\text{E},t}_{bu1}({\varrho}),
p^{mt}_{bu}({\varrho+1})-p^{mt}_{bu}(\varrho)\right\rangle \\\nonumber&-(R^{\text{E},mt}_{buq2}({\varrho+1})-R^{\text{E},mt}_{buq1}({\varrho}))\leq
-(R^{\text{E},mt}_{buq2}({\varrho})-R^{\text{E},mt}_{buq1}({\varrho})),\end{aligned}$$
Combined with (\[eq-001\]) and (\[eq-002\]), we conclude that $$\label{eq-003}
-(R^{\text{E},mt}_{buq2}({\varrho+1})-R^{\text{E},mt}_{buq1}({\varrho+1}))
\leq
-(R^{\text{E},mt}_{buq2}({\varrho})-R^{\text{E},mt}_{buq1}({\varrho})).$$
Obviously, the current value $R^{\text{E},mt}_{buq}({\varrho+1})$ is smaller than the previous value $R^{\text{E},mt}_{buq}({\varrho})$ while the current solution $p^{mt}_{bu}(\varrho+1)$ is better than the previous solution $p^{mt}_{bu}(\varrho)$. As a result, the theorem is proved.
[^1]: Mohammad R. Abedi and Nader Mokari are with ECE Department, Tarbiat Modares University, Tehran, Iran.
Mohammad R. Javan is with the Department of Electrical and Robotics Engineering, Shahrood University, Shahrood, Iran.
E . A. Jorswieck is with the Department of Systems and Computer Engineering, Dresden University of Technology (TUD), Germany.
[^2]: Point-to-multipoint (P2M) technologies are considered as backhaul networks for small cell which is an effective way of sharing the backhaul resource between several BSs. PMP backhaul has high spectral efficiency, and speed and flexibility of deployment, and have been successfully deployed in the Middle East, Africa and in Europe by major operators [@BACKHAUL-10].
[^3]: In the context of energy harvesting, there is another approach which is called on-line approach. This approach assumes that the information is available only causally and use the Markov decision process method for resource allocation over $F$ frames [@minasian2014energy]. Although the availability of noncausal information is no practical, the off-line approach would provide a benchmark for energy harvesting networks. We leave the on-line approach as a future research direction
[^4]: Due to the high computing power at the BSs, stronger cryptography is used for the links between core and BSs, hence eavesdropping of these links is hard and difficult. Therefore, we assume that only access link can be wiretapped by eavesdroppers.
|
---
abstract: |
A new geometric argument is introduced to exclude binary collisions with order constraints. Two applications are given in this paper.
The first application is to show the existence of a new set of periodic orbits in the planar three-body problem with mass $M=[1, \, m, \, m]$, where we study the action minimizer under topological constraints in a two-point free boundary value problem. The main difficulty is to exclude possible binary collisions under order constraints, which is solved by our geometric argument.
The second application is to study the set of retrograde orbits in the planar three-body problem with mass $M=[1, \, m, \, m]$. We can show the existence for any $m>0$ and any rotation angle $\theta \in (0, \pi/2)$. Specially, in the case when $\theta=\pi/2$, the action minimizer coincide with either the Schubart orbit or the Broucke-Hénon orbit, which partially answers the open problem proposed by Venturelli.
author:
- |
Wentian Kuang\
Chern Institute of Mathematics, Nankai University\
Tianjin 300071, China\
Emails: [email protected]
- |
Rongchang Liu, Duokui Yan\
School of Mathematics and System Sciences, Beihang University\
Beijing 100191, China\
Email: [email protected], [email protected]
title: A new way to exclude collisions with order constraints
---
[**Key word:**]{} Three-body problem, variation method, topological constraint, geometric argument.\
[**AMS classification number:**]{} 37N05, 70F10, 70F15, 37N30, 70H05, 70F17\
Introduction
============
After the pioneering work of the figure-eight orbit [@CM], many new orbits have been shown to exist by variational method. One of the main difficulties is to exclude possible collisions in the action minimizer. In the last two decades, two powerful methods have been introduced to study the isolated collisions. One is the local deformation method [@CV; @CH3; @FT; @FU; @Ven; @Yu; @Yu1; @Yu2; @Zh], and the other is the level estimate method [@CH; @CH1; @CH2; @Zhang; @Yan3]. However, when topological constraints are imposed to the N-body problem, it is difficult to eliminate collisions by applying the two methods in general. New progress has been made recently in [@Yu1; @Yu2], where the author imposed strong topological constraints and successfully applied his local deformation method to show the existence of many choreographic and double choreographic solutions.
In this paper, we introduce a geometric argument to exclude collisions and apply it to two sets of periodic orbits in the planar three-body problem. Let $M=[m_1, \, m_2, \, m_3]=[1, \, m, \, m]$ with $m>0$. For simplicity, we denote $\mathsf{Q}_i$ as the $i$-th $(i=1, 2, 3, 4)$ quadrant in the Cartesian $xy$ coordinate system and $\overline{\mathsf{Q}_i}$ as its closure. For example, $\mathsf{Q}_1=\{ (x, \, y) \, |\, x>0,\, y>0 \}$ and $\overline{\mathsf{Q}_1}=\{(x, \, y) \, | \, x\geq 0, \, y \geq 0\}$. Assume the center of mass to be at the origin. That is, $q \in \chi$, where $$\label{chi}
\chi=\left \{ q=\begin{bmatrix}
q_{1} \\
q_{2}\\
q_{3}
\end{bmatrix}=\begin{bmatrix}
q_{1x} & q_{1y} \\
q_{2x} & q_{2y} \\
q_{3x} & q_{3y}
\end{bmatrix} \in \mathbb{R}_{3 \times 2} \, \bigg{|} \, \sum_{i=1}^3 m_iq_i =0 \right \}.$$Let $$Z_1=q_2-q_3, \qquad Z_2=q_1-\frac{q_2+q_3}{2}=(1+\frac{1}{2m})q_1.$$ Our geometric result is as follows.
\[maingeometricresult\] Assume the two boundaries $(Z_1(0), \, Z_2(0))$ and $(Z_1(1), \, Z_2(1))$ are fixed and let $(Z_1, \, Z_2)\in H^1([0,1],\mathbf{R}^4)$ be the action minimizer of a fixed boundary value problem $$\mathcal{A}(Z_1, \, Z_2)= \inf_{\mathcal{P}(\overline{Z}_1, \, \overline{Z}_2)}\mathcal{A}= \inf_{\mathcal{P}(\overline{Z}_1, \, \overline{Z}_2)} \int_0^1 (K+U)\, dt,$$ where $K$ in and $U$ in are the standard kinetic energy and the potential energy respectively, and $$\mathcal{P}(\overline{Z}_1, \, \overline{Z}_2)= \left\{(\overline{Z}_1, \, \overline{Z}_2) \in H^1([0,1],\mathbf{R}^4) \, \big| \, \overline{Z}_i(0)=Z_i(0), \, \, \overline{Z}_i(1)=Z_i(1) \, (i=1,2) \right\}.$$ Let $Z_1(0), \, Z_1(1)\in \overline{\mathsf{Q}_i} \, $ and $ \, Z_2(0), \, Z_2(1)\in \overline{\mathsf{Q}_j}$, while $\overline{\mathsf{Q}_i}$ and $\overline{\mathsf{Q}_j}$ are two adjacent closed quadrants. Then $Z_1(t)$ and $Z_2(t)$ are always in two adjacent closed quadrants for all $t \in [0,\, 1]$ and $(Z_1, \, Z_2)$ must satisfy one of the following three cases:
1. $Z_1(t)$ and $Z_2(t)$ can not touch the coordinate axes for all $t\in(0, \, 1)$;
2. $Z_1(t)$ and $Z_2(t)$ are on the coordinate axes for all $t \in [0,\, 1]$;
3. the motion is a part of the Euler solution with $Z_2(t) \equiv 0$ for all $t \in [0,\, 1]$.
As its applications, we consider the following two-point free boundary value problems with topological constraints. The first application is as follows. Similar to [@Yan3], we set $$\label{boundarysetting}
Q_s=\begin{bmatrix}
m(a_1-a_2) & 0\\
-(m+1)a_1-ma_2 & 0\\
ma_1+(m+1)a_2 & 0
\end{bmatrix}, \qquad Q_e=\begin{bmatrix}
2mb_2 & 0\\
-b_2 & b_1 \\
-b_2 & -b_1
\end{bmatrix} R(\theta),$$ where $a_1\geq 0$, $a_2 \geq 0$, $b_1 \in \mathbb{R}$, $b_2 \in \mathbb{R}$, and $R(\theta)=\begin{bmatrix}
\cos(\theta)& \sin(\theta)\\
-\sin(\theta)& \cos(\theta)
\end{bmatrix}$. The two configuration sets are defined as follows: $$\label{QSQE}
Q_S=\left\{ Q_s \, \bigg|\, a_1 \geq 0, \, a_2 \geq 0\right\},\quad Q_E=\left\{ Q_e \, \bigg|\, b_1 \in \mathbb{R}, \, b_2 \in \mathbb{R} \right\}.$$ Geometrically, the configuration $Q_s$ is on a horizontal line with order constraints $q_{2x}(0) \leq q_{1x}(0) \leq q_{3x}(0)$. The configuration $Q_e$ is an isosceles triangle with $q_1$ as its vertex, and the symmetry axis of each $Q_e$ in is a counterclockwise $\theta$ rotation of the $x-$axis. Pictures of the two configurations $Q_s$ and $Q_e$ are shown in Fig. \[picQsQe\] respectively.
(-1.5, -1.5)(1.75, 1.2) (-1.5, 0)(-0.15, 0) (-1.45, 0.25)[$q_2$]{} (-1.05, 0.25)[$q_1$]{} (-0.35, 0.25)[$q_3$]{} (-1.4, 0) (-1, 0) (-0.3, 0) (-1.5, 1.05)[$Q_s$:]{} (0.3, 0)(1.5, 0) (0.7, -0.3)(1.5, 0.5)
(1, 0)[0.6]{}[0]{}[22]{}
(0.65, 0.15)(0.78, -0.73) (1.5, 0.5)(0.78, -0.73) (1.5, 0.5)(0.65, 0.15)
(1.5, 0.5) (0.65, 0.15) (0.78, -0.73)
(1.23, 0.12)(1.68, 0.15)
(0.7, 0.4)[$q_2$]{} (1.6, 0.55)[$q_1$]{} (0.88, -0.8)[$q_3$]{} (1.75, 0.15)[$\theta$]{}
(0.3, 1.05)[$Q_e$:]{}
For each $\theta \in [0, \pi/2)$, standard results [@CH3; @FT; @Yan2; @Yan5] imply that there exists an action minimizer $\mathcal{P}_{m, \, \theta} \in H^1([0,1], \chi)$, such that $$\label{actionminsetting}
\mathcal{A}(\mathcal{P}_{m, \, \theta})= \inf_{q \in P(Q_S,\, Q_E)} \mathcal{A} = \inf_{q \in P(Q_S, \, Q_E)} \int_0^1 (K+U) \, dt,$$ where $K=\frac{1}{2}\sum_{i=1}^3 m_i |\dot{q}_i|^2$, $U=\sum_{1 \le i<j\le 3} \frac{m_i m_j}{|q_i-q_j|}$ and $$\mathrm P(Q_S,\, Q_E)=\left \{q\in H^1([0,1],\, \chi) \, \big{|} \, q(0) \in Q_S, \, q(1) \in Q_E \right \}.$$ By the celebrated results of Marchal [@Mar] and Chenciner [@CA], the action minimizer $\mathcal{P}_{m, \, \theta}$ is free of collision in $(0,1)$. We are only left to exclude possible boundary collisions. Our main results are as follows.
\[mainthm1\] For each given $\theta \in [0, \pi/2)$ and mass set $M=[m_1,\, m_2, \, m_3]=[1, \, m, \, m]$ with $m>0$, the minimizer $\mathcal{P}_{m, \, \theta}$ is collision-free and it can extended to a periodic or quasi-periodic orbit.
The new idea in the proof is to apply our geometric result (Theorem \[maingeometricresult\]) to exclude binary collisions under order constraints. It was first introduced in [@Yan4] to study retrograde double-double orbit in the planar equal-mass four-body problem, and it is also used to study the Schubart orbit(Fig. \[twoorbits\] (a)) and the Broucke-Hénon orbit (Fig. \[twoorbits\] (b)) in the equal mass case [@Yan5].
Besides excluding possible collisions, we need to show that $\mathcal{P}_{m, \, \theta}$ is nontrivial, which means that it does not coincide with a relative equilibrium.
In the case when $m=1$, we can show that $\mathcal{P}_{m, \, \theta}$ is nontrivial by introducing test paths for each $\theta$, which extends the result in [@Yan3].
\[mainthm2\] When $m=1$ and $\theta \in [0, \, 0.183\pi]$, the minimizer $\mathcal{P}_{m, \, \theta}$ is nontrivial.
In [@Yan3], we apply Chen’s level estimate method [@CH; @CH2] and show that for each $\theta \in [0.084 \pi, 0.183\pi]$, $\mathcal{P}_{m, \, \theta}$ is collision-free and can be extended to a nontrivial periodic or quasi-periodic orbit. It is clear that the results (Theorem \[mainthm1\] and Theorem \[mainthm2\]) in this paper are stronger than the results in [@Yan3].
When $m=1$ and $\theta=0$, the orbit is shown in Fig. \[orbitatt0\] and it looks very simple. However, to the authors’ knowledge, there is no existence proof for $\mathcal{P}_{1, \, 0}$. Actually, it is difficult to apply the local deformation method [@FT] to $\mathcal{P}_{1, \, 0}$, while the level estimate method [@CH; @CH2; @Yan3] can only exclude collisions in $\mathcal{P}_{m, \, \theta}$ with $\theta \in [0.084 \pi, \, 0.183\pi]$. In this sense, our geometric argument has its own advantage in eliminating collisions with topological constraints.
![ \[orbitatt0\]At $t=0$, the three masses (in dots) form a collinear configuration with body 1 in the middle. At $t=1$, they (in crosses) form an isosceles configuration with body 1 as the vertex. The minimizing path $\mathcal{P}_{1, \, 0}$ connects a collinear configuration with order constraints $q_{2x}(0)\leq q_{1x}(0) \leq q_{3x}(0)$ and an isosceles triangle configuration, while its periodic extension has a simple and symmetric shape.](henon2piece2.eps){width="2.7in" height="2.1in"}
The second application is to show the existence of the retrograde orbits [@CH; @CH2]. By introducing the level estimate method, Chen [@CH; @CH2] can show the existence of retrograde orbit for most of the rotation angles and most of the masses. Specially, in the case when $M=[m_1,\, m_2, \, m_3]=[1, \, m, \, m]$, Chen’s result [@CH2] requires two inequalities between the rotation angle $\theta$ and the mass $m>0$. In fact, when $\theta>0.4 \pi$, the two inequalities in [@CH2] fail for all $m>0$. Fortunately, our geometric result can be applied successfully to the existence of retrograde orbits for all $\theta \in (0, \, \pi/2)$ and all $m>0$. Furthermore, when $\theta=\pi/2$, we can show that the action minimizer coincide with either the Schubart orbit or the Broucke-Hénon orbit, which partially answers the open problem proposed by Venturelli (Problem 6 in [@VEOP]).
Before stating our result, we first introduce the variational setting of the retrograde orbits. Let $$\label{boundarysetting2}
Q_{s_1}=\begin{bmatrix}
-2ma_1-ma_2 & 0\\
a_1-ma_2 & 0\\
a_1+(m+1)a_2 & 0
\end{bmatrix}, \qquad Q_{e_1}=\begin{bmatrix}
2mb_2 & 0\\
-b_2 & b_1 \\
-b_2 & -b_1
\end{bmatrix} R(\theta),$$ where $M=[1,\, m,\, m]$, $a_1\geq 0$, $a_2 \geq 0$, $b_1 \in \mathbb{R}$, $b_2 \in \mathbb{R}$, and $R(\theta)=\begin{bmatrix}
\cos(\theta)& \sin(\theta)\\
-\sin(\theta)& \cos(\theta)
\end{bmatrix}$. The two configuration sets are defined as follows: $$\label{QSQE2}
Q_{S_1}=\left\{ Q_{s_1} \, \bigg|\, a_1 \geq 0, \, a_2 \geq 0\right\},\qquad Q_{E_1}=\left\{ Q_{e_1} \, \bigg|\, b_1 \in \mathbb{R}, \, b_2 \in \mathbb{R} \right\}.$$ Note that in the two boundary settings $\{Q_{S}, \, Q_{E} \}$ in and $\{Q_{S_1}, \, Q_{E_1} \}$ in , $Q_{E_1}=Q_E$ at $t=1$. At $t=0$, $Q_{S_1}$ in is on the $x$-axis with order constraints $$q_{1x}(0) \leq q_{2x}(0) \leq q_{3x}(0),$$ while $Q_{S}$ in is on the $x$-axis with different order constraints $$q_{2x}(0) \leq q_{1x}(0) \leq q_{3x}(0).$$
For each $\theta \in (0, \pi/2]$, standard results [@CH3; @FT; @Yan2; @Yan5] imply that there exists an action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}} \in H^1([0,1], \chi)$, such that $$\label{actionminsetting2}
\mathcal{A}(\widetilde{\mathcal{P}_{m, \, \theta}})= \inf_{q \in P(Q_{S_1}, \, Q_{E_1})} \mathcal{A} = \inf_{q \in P(Q_{S_1}, \, Q_{E_1})} \int_0^1 (K+U) \, dt,$$ where $$\mathrm P(Q_{S_1},\, Q_{E_1})=\left \{q\in H^1([0,1],\chi) \, \big{|} \, q(0) \in Q_{S_1}, \, q(1) \in Q_{E_1} \right \}.$$
By applying Theorem \[maingeometricresult\], we can show that $\widetilde{\mathcal{P}_{m, \, \theta}}$ is collision-free when $\theta \in (0, \, \pi/2)$, which implies the existence of the retrograde orbits. Furthermore, in the case when $\theta=\pi/2$, $\widetilde{\mathcal{P}_{m, \, \theta}}$ is either a part of the Schubart orbit or a part of the Broucke-Hénon orbit.
\[retrograderesult\] For each given $\theta \in (0, \, \pi/2)$ and mass set $M=[m_1,\, m_2, \, m_3]=[1, \, m, \, m]$ with $m>0$, the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ in is collision-free. And it can be extended to a periodic or quasi-periodic orbit.\
When $\theta=\pi/2$, the action minimizer $\widetilde{\mathcal{P}_{m, \, \pi/2}}$ in coincides with either the Schubart orbit or the Broucke-Hénon orbit.
The paper is organized as follows. In Section \[exclusiontotalcollision\], we show that $\mathcal{P}_{m, \, \theta}$ is collision free at $t=1$ and it is also free of total collision. In Section \[geoarg\], a general geometric result is introduced and it can be applied to $\mathcal{P}_{m, \, \theta}$ to show that it is collision-free. In Section \[compareEuleratm1\], we show that $\mathcal{P}_{m, \, \theta}$ is nontrivial when $m=1$ and $\theta \in [0, \, 0.183\pi]$. In the last section (Section \[applytoretrogradeorbit\]), we apply our geometric result (Theorem \[maingeometricresult\]) to the retrograde orbits and the Broucke-Hénon orbit.
Exclusion of total collision {#exclusiontotalcollision}
============================
In this section, we exclude possible total collisions in the action minimizer $\mathcal{P}_{m, \, \theta}$ for all $\theta \in [0, \pi/2)$ and $m>0$. Also, we exclude possible binary collision at $t=1$ by the standard deformation method in the end.
For any $\theta \in [0, \pi/2)$ and $m>0$, the action minimizer $\mathcal{P}_{m, \, \theta}$ has no total collision.
We first obtain a lower bound of action when $\mathcal{P}_{m, \, \theta}$ has a total collision in $[0,1]$. We denote the action by $\mathcal{A}_{total}$ if $\mathcal{P}_{m, \, \theta}$ has a total collision. By Chen’s level estimate [@CH], $$\mathcal{A}_{total} \geq \frac{m^2+2m}{1+2m} \cdot \frac{3}{2} (1+2m)^{2/3} \pi^{2/3} = \frac{3m}{2} \pi^{2/3} \frac{m+2}{(1+2m)^{1/3}}.$$ By the setting of $Q_S$ and $Q_E$ in , we can choose a piece of Euler orbit as the test path. In this Euler orbit, at $t=0$, it is on the $x$-axis with body 1 at the origin. While at $t=1$, $Q_e$ degenerates to a straight line. The corresponding action $\mathcal{A}_{test}$ is $$\begin{split}
\mathcal{A}_{test}&= \frac{3}{2} \left(\pi/2 -\theta\right)^{2/3} \left[2m^3(2+m/2)^2\right]^{1/3} \\
&\leq \frac{3}{2} \pi^{2/3} \left(\frac{m^3(2+m/2)^2}{2}\right)^{1/3}= \frac{3m}{2} \pi^{2/3} \frac{(4+m)^{2/3}}{2}.
\end{split}$$ To show that $\frac{3m}{2} \pi^{2/3} \frac{m+2}{(1+2m)^{1/3}}>\frac{3m}{2} \pi^{2/3} \frac{(4+m)^{2/3}}{2}$, it is equivalent to show that $$\label{totaltestcomp}
\frac{m+2}{(1+2m)^{1/3}} > \frac{(4+m)^{2/3}}{2}.$$ In order to prove , we consider the difference $g(m)= 8(m+2)^3-(4+m)^2(1+2m)$. Then to prove is equivalent to show that $g(m)>0$ for any $m>0$.
In fact, the derivative $\frac{d g(m)}{d m}= 18 m^2+62 m+56>0$ for any $m>0$. And $g(0)= 60>0$. It follows that $g(m)>0$ for any $m>0$. That is, $\mathcal{A}_{total} > \mathcal{A}_{test}$.
Therefore, $\mathcal{P}_{m, \, \theta}$ has no total collision. The proof is complete.
Note that by the definition of $Q_E$ in , the only possible binary collision at $t=1$ is between bodies 2 and 3. Standard local deformation result can imply this fact.
The following blow-up results are needed in proving that $\mathcal{P}_{m, \, \theta}$ is free of binary collisions at $t=1$. It is known that the bodies involved in a partial collision or a total collision will approach a set of central configurations. More information can be known if the solution under concern is an action minimizer:
\[Vench\] If a minimizer $q$ of the fixed-ends problem on time interval $[\tau_1, \tau_2]$ has an isolated collision of $k\leq N$ bodies, then there is a parabolic homethetic collision-ejection solution $\hat{q}$ of the $k-$body problem which is also a minimizer of the fixed-ends problem on $[\tau_1, \tau_2]$.
\[blowup\] Let $X$ be a proper linear subspace of $\mathbb{R}^d$. Suppose a local minimizer $x$ of $\mathcal{A}_{t_0, \, t_1}$ on $$B_{t_0, \, t_1}(x(t_0), X):= \{ x \in H^1([t_0, t_1], (\mathbb{R}^d)^N) \, \big|\, x(t_0) \, \text{is fixed, and} \, \, x_i(t_1) \in X, \, i=1,2, \dots, N \}$$ has an isolated collision of $k \leq N$ bodies at $t=t_1$. Then there is a homothetic parobolic solution $\bar{y}$ of the $k$-body problem with $\bar{y}(t_1)=0$ such that $\bar{y}$ is a minimizer of $\mathcal{A}^{*}_{\tau, \, t_1}$ on $B_{\tau, \, t_1}(\bar{y}(\tau), X)$ for any $\tau<t_1$. Here $\mathcal{A}^{*}_{\tau, \, t_1}$ denotes the action of this $k$-body subsystem.
By applying Lemma \[Vench\] and Lemma \[blowup\], the following result holds.
\[nobinarycollisionat1\] For each given $\theta \in [0, \pi/2)$ and mass set $M=[m_1,\, m_2, \, m_3]=[1, \, m, \, m]$ with $m>0$, there is no binary collision at $t=1$ in $\mathcal{P}_{m, \, \theta}$.
Note that $Q_E= \left\{ Q_e \, \bigg|\, b_1 \in \mathbb{R}, \, b_2 \in \mathbb{R} \right\}$ where $Q_e=\begin{bmatrix}
2b_2/m & 0\\
-b_2 & b_1 \\
-b_2 & -b_1
\end{bmatrix} R(\theta)$. It is clear that $Q_E$ is a two-dimensional vector space and the only possible binary collision in $Q_E$ is between bodies 2 and 3.
Assume that $q_2$ and $q_3$ collide at $t=1$ in $\mathcal{P}_{m, \, \theta}$. By the analysis of blow up in Lemma \[Vench\] and Lemma \[blowup\], there exists a parobolic homothetic solution $q_i(t)=\xi_i t^{\frac23}, \, (i=2,\, 3)$, which is also a minimizer of the $2$-body problem on $[1-\tau, 1]$ for any $\tau>0$. In fact, $(\xi_2,\, \xi_3)$ forms a central configuration with $m\xi_2=-m\xi_3$, and the two vectors $\xi_2$, $\xi_3$ satisfy the energy constraint: $$\sum_{i=2,3} \frac{1}{2} |\frac{2}{3} \xi_i|^2 - \frac{1}{|\xi_2-\xi_3|}=0.$$ For a given $\epsilon>0$ small enough, we fix $q_i(\epsilon) \, (i=2,3)$. Next, we perturb $q_i$ to $\bar{q}_i$ (i=2,3) such that $\bar{q}_i(1-\epsilon)= q_i(1-\epsilon) , (i=2,3)$, $\bar{q}_2(1) \neq \bar{q}_3(1)$ and $\bar{q}_i(1) \, (i=2,3)$ satisfy the boundary condition $Q_E$. Let $$\overrightarrow{\bar{q}_3\bar{q}_2(1)}= \frac{\bar{q}_2(1)- \bar{q}_3(1)}{|\bar{q}_2(1)- \bar{q}_3(1)|},$$ where $\bar{q}_2(1)$ and $\bar{q_3}(1)$ are the perturbed vectors of $q_1$ and $q_2$ at $t=1$. Since the boundary set $Q_E$ is a vector space, it follows that one can always choose the local deformation $\bar{q}_i$ such that $\bar{q}_i(1) \, (i=2,3)$ satisfies $$\langle \overrightarrow{\bar{q}_3\bar{q}_2(1)}, \frac{\xi_2}{|\xi_2|} \rangle \neq -1.$$ By [@CH4; @FU; @Yu], there exist $\bar{q}_2$ and $\bar{q}_3$, such that the action of $\bar{q}_2$ and $\bar{q}_3$ in $[1-\tau, \, 1]$ is strictly smaller than the action of the parabolic ejection solution: $q_2$ and $q_3$. Contradiction!
Therefore, there is no binary collision at $t=1$ in $\mathcal{P}_{m, \, \theta}$. The proof is complete.
Geometric result {#geoarg}
================
In this section, we study the geometric property of the action minimizer connecting two given boundaries. The main result is Theorem \[geometricthm\], which can be applied to exclude the possible binary collisions with order constraints.
Let $M=[m_1, \, m_2, \, m_3]=[1, \, m, \, m]$. Let $$Z_1=q_2-q_3, \qquad Z_2=q_1-\frac{q_2+q_3}{2}=(1+\frac{1}{2m})q_1.$$ The kinetic energy $K$ and the potential energy $U$ can be rewritten as $$\label{Kformulajacob}
K=\sum_{i=1}^3 \frac{1}{2} m_i |\dot{q}_i|^2= \frac{m}{4} |\dot{Z}_1|^2+ \frac{m}{2m+1} |\dot{Z}_2|^2,$$
$$\label{Uformulajacob}
U= \sum_{1 \le i<j\le 3} \frac{m_i m_j}{|q_i-q_j|}=\frac{m^2}{|Z_1|}+ \frac{m}{|\frac{1}{2}Z_1+ Z_2|}+ \frac{m}{|\frac{1}{2}Z_1-Z_2|},$$
while the action functional is $$\label{Aformulajacob}
\mathcal{A}=\int_0^1{(K+U)}dt.$$
For convenience, in the Cartesian $xy$ coordinate system, the $i$-th quadrant is denoted by $ \mathsf{Q}_i \, (i=1, 2, 3, 4)$, while its closure is denoted by $\overline{\mathsf{Q}_i}$. For example, $\mathsf{Q}_1=\{ (x, \, y) \, |\, x>0,\, y>0 \}$ and $\overline{\mathsf{Q}_1}=\{(x, \, y) \, | \, x\geq 0, \, y \geq 0\}$. The main result in this section is as follows.
\[geometricthm\] Assume the two boundaries $(Z_1(0), \, Z_2(0))$ and $(Z_1(1), \, Z_2(1))$ are fixed and let $(Z_1, \, Z_2)\in H^1([0,1],\mathbf{R}^4)$ be the action minimizer of a fixed boundary value problem $$\mathcal{A}(Z_1, \, Z_2)= \inf_{\mathcal{P}(\overline{Z}_1, \, \overline{Z}_2)}\mathcal{A},$$ where $$\mathcal{P}(\overline{Z}_1, \, \overline{Z}_2)= \left\{(\overline{Z}_1, \, \overline{Z}_2) \in H^1([0,1],\mathbf{R}^4) \, \big| \, \overline{Z}_i(0)=Z_i(0), \, \, \overline{Z}_i(1)=Z_i(1) \, (i=1,2) \right\}.$$ Let $Z_1(0), \, Z_1(1)\in \overline{\mathsf{Q}_i} \, $ and $ \, Z_2(0), \, Z_2(1)\in \overline{\mathsf{Q}_j}$, while $\overline{\mathsf{Q}_i}$ and $\overline{\mathsf{Q}_j}$ are two adjacent closed quadrants. Then $Z_1(t)$ and $Z_2(t)$ are always in two adjacent closed quadrants for all $t \in [0,\, 1]$ and $(Z_1, \, Z_2)$ must satisfy one of the following three cases:
1. $Z_1(t)$ and $Z_2(t)$ can not touch the coordinate axes for all $t\in(0, \, 1)$;
2. $Z_1(t)$ and $Z_2(t)$ are on the coordinate axes for all $t \in [0,\, 1]$;
3. the motion is a part of the Euler solution with $Z_2(t) \equiv 0$ for all $t \in [0,\, 1]$.
Before proving Theorem \[geometricthm\], we introduce several preliminary results. If both $Z_1(t)$ and $Z_2(t)$ are nonzero vectors, we define an angle $\Delta \equiv \Delta(Z_1(t), \, Z_2(t))$ by $$\label{Deltadef}
\Delta\equiv \Delta(Z_1(t), \, Z_2(t))= \left\{\begin{aligned}
& \beta(Z_1(t), \, Z_2(t)),\ \ \text{if} \ \ \beta(Z_1(t), \, Z_2(t)) \le\frac{\pi}{2}; \\
& \pi-\beta(Z_1(t), \, Z_2(t)),\ \ \text{if}\ \ \beta(Z_1(t),\, Z_2(t))>\frac{\pi}{2},\\
\end{aligned}
\right.$$ where $\beta(Z_1(t), \, Z_2(t))= \arccos{\frac{\langle Z_1(t), \, Z_2(t) \rangle}{|Z_1(t)|||Z_2(t)|}}$ is the angle between the two nonzero vectors $Z_1(t)$ and $Z_2(t)$. By , it follows that $\Delta \in [0, \pi/2]$. Intuitively, $\Delta= \Delta(Z_1, \, Z_2)$ is the angle between the two straight lines spanned by $Z_1$ and $Z_2$.
A new formula of $U \equiv U(Z_1, \, Z_2)$ can then be derived by the law of cosines: $$\begin{aligned}
\label{formulaofUinz1z2theta}
U(Z_1, \, Z_2) &=& \frac{m^2}{|Z_1|}+ \frac{m}{|\frac{1}{2}Z_1+ Z_2|}+ \frac{m}{|\frac{1}{2}Z_1-Z_2|} \nonumber\\
&=& \frac{m^2}{|Z_1|}+ \frac{m}{\sqrt{\frac{1}{4} |Z_1|^2+ |Z_2|^2 + |Z_1||Z_2| \cos (\Delta) }} \nonumber\\
& & \, \, + \frac{m}{\sqrt{\frac{1}{4} |Z_1|^2+ |Z_2|^2 - |Z_1||Z_2| \cos (\Delta) }}.\end{aligned}$$
By , $U(Z_1,Z_2)$ is a function of three variables: $|Z_1|$, $|Z_2|$ and $\Delta$ when both $Z_1 \not= 0$ and $Z_2 \not= 0$ hold. Let $U(|Z_1|, \, |Z_2|, \, \Delta) \equiv U(Z_1, \, Z_2)$. Indeed, $U(|Z_1|, \, |Z_2|, \, \Delta)$ satisfies the following property.
\[Uintheta\] Fix $|Z_1| \neq 0$ and $|Z_2| \neq 0$ and assume that the potential energy $U(|Z_1|,|Z_2|, \Delta)=U(Z_1, \, Z_2)$ in is finite. Then $U(|Z_1|, \, |Z_2|, \, \Delta)$ is a strictly decreasing function with respect to $\Delta$.
Fixing $|Z_1|$ and $|Z_2|$ and taking the derivative of $U(|Z_1|, \, |Z_2|, \, \Delta)$ in with respect to $\Delta$, it follows that $$\begin{aligned}
& &\frac{\partial U(|Z_1|,\, |Z_2|, \, \Delta)}{\partial \Delta}\\
&=& \frac{\frac{1}{2} |Z_1||Z_2| \sin (\Delta) }{ \left[\frac{1}{4} |Z_1|^2+ |Z_2|^2 + |Z_1||Z_2| \cos (\Delta) \right]^{3/2}} - \frac{\frac{1}{2} |Z_1||Z_2| \sin (\Delta) }{\left[\frac{1}{4} |Z_1|^2+ |Z_2|^2 - |Z_1||Z_2| \cos (\Delta) \right]^{3/2}}.\end{aligned}$$ Note that $\Delta \in [0,\frac{\pi}{2}]$. It implies that $$\frac{\partial U(|Z_1|,\, |Z_2|, \, \Delta )}{\partial \Delta } \leq 0,$$ and $\frac{\partial U(|Z_1|,\, |Z_2|,\, \Delta )}{\partial \Delta }=0$ if and only if $\Delta = 0$ or $\Delta = \pi/2$. The proof is complete.
Given a nonzero point $Z_k=(Z_{kx}, \, Z_{ky}) \, (k=1,2)$, we consider its four reflection points: $(\pm|Z_{kx}|, \, \pm|Z_{ky}|)$. For each $t \in [0, \, 1]$, we choose $Z_{1i}(t)$ to be one of the four reflection points such that $Z_{1i}(t) \in \overline{\mathsf{Q}_i}$. For example, $Z_{11}(t)= (|Z_{1x}(t)|, \, |Z_{1y}(t)|)$. Similarly, we can choose $Z_{2j}(t) \in \overline{\mathsf{Q}_j}$ for all $t \in [0,\, 1]$. Then the following result holds.
\[Ucomparetilde\] For each $t \in [0,1]$ and any $i$, $j$ $\in$ $\{1, \, 2, \, 3, \, 4\}$, the potential function $U(Z_{1i}(t),\, Z_{2j}(t))$ must be one of the following two values: $U_1(t)$ and $U_2(t)$. It satisfies
$$\label{Uinij}
U(Z_{1i}(t),\, Z_{2j}(t))=\left\{\begin{aligned}
&U_1(t), \, \, \text{if} \ Z_{1i}(t), \, Z_{2j}(t) \ \text{are in two adjacent closed quadrants};\\\\
&U_2(t), \,\, \text{otherwise.}\\
\end{aligned}\right.$$
Moreover, $U_1(t)<U_2(t)$.
Note that if two nonzero vectors $Z_1(t)$ and $Z_2(t)$ are in two adjacent closed quadrants, the angle $\Delta(Z_1(t), \, Z_2(t))=\Delta(Z_{1i}(t), \, Z_{2j}(t))$ if their reflection points $Z_{1i}(t)$ and $Z_{2j}(t)$ are also in two adjacent closed quadrants.
If $Z_1(t)$ or $Z_2(t)$ belongs to the coordinate axes (including the case when $Z_1(t)=0$ or $Z_2(t)=0$ for some $t$), it is easy to check that $Z_{1i}(t)$ and $Z_{2j}(t)$ are always in two adjacent closed quadrants. Hence, $U(Z_{1i}(t), \, Z_{2j}(t))=U_1(t)$. Thus we only consider the case when both $Z_1(t)$ and $Z_2(t)$ are away from the coordinate axes.
In fact, similar to the definition of $\Delta$, we define two angles $\alpha_1$ and $\alpha_2$ as follows $$\label{alpha1alpha2}
\begin{split}
\alpha_1&=\min \left\{ \arccos \frac{\langle Z_1(t), \, \vec{s}_1 \rangle}{|Z_1(t)|}, \, \, \pi- \arccos \frac{\langle Z_1(t), \, \vec{s}_1 \rangle}{|Z_1(t)|} \right\}, \\
\alpha_2&=\min \left\{ \arccos \frac{\langle Z_2(t), \, \vec{s}_1 \rangle}{|Z_2(t)|}, \, \, \pi- \arccos \frac{\langle Z_2(t), \, \vec{s}_1 \rangle}{|Z_2(t)|} \right\},
\end{split}$$ where $\vec{s}_1=(1,0)$. It is clear that $\alpha_1, \alpha_2 \in (0, \pi/2)$.
If $Z_{1i}$ and $Z_{2j}$ are in two adjacent quadrants respectively, then $$\label{Z1Z2Deltaangle1}
\Delta_1\equiv \Delta (Z_{1i}(t), \, Z_{2j}(t))= \min \left\{ \alpha_1+\alpha_2, \, \, \pi-\alpha_1-\alpha_2 \right\}.$$
If $Z_{1i}$ and $Z_{2j}$ are not in two adjacent quadrants, we have $$\label{Z1Z2Deltaangle2}
\Delta_2\equiv\Delta (Z_{1i}(t), \, Z_{2j}(t))= |\alpha_1-\alpha_2|.$$
Note that both $Z_1(t)$ and $Z_2(t)$ are away from the coordinate axes, it implies that $\alpha_1, \alpha_2 \in (0, \, \pi/2)$. If $\alpha_1 +\alpha_2 \le\frac{\pi}{2}$, then $$\min\{\alpha_1+\alpha_2, \, \pi-\alpha_1-\alpha_2 \}=\alpha_1+\alpha_2 > |\alpha_1-\alpha_2|.$$ If $\alpha_1+\alpha_2>\frac{\pi}{2}$, then $$\begin{aligned}
\min\{\alpha_1+\alpha_2, \, \pi-\alpha_1-\alpha_2\}&=\pi-\alpha_1-\alpha_2 \\
&> \frac{\pi}{2}- \min \{\alpha_1, \alpha_2 \} > |\alpha_1-\alpha_2|.\end{aligned}$$ Hence, $\Delta_1 < \Delta_2$. Recall that $U(Z_{1i}(t), \, Z_{2j}(t))=U(|Z_1|, \, |Z_2|, \, \Delta(Z_{1i}(t), \, Z_{2j}(t)))$. Let $$U_1(t)=U(|Z_1|, \, |Z_2|,\, \Delta_1), \quad U_2(t)=U(|Z_1|,\, |Z_2|,\, \Delta_2).$$ It is clear that for each given $t \in [0, 1]$, the values of $U(Z_{1i}(t), \, Z_{2j}(t))\, (i, j=1,2,3,4)$ can only be either $U_1(t)$ or $U_2(t)$. By proposition \[Uintheta\], it follows that $$U_1(t) < U_2(t).$$ The proof is complete.
\[z1z2sametimetangent\] For the minimizing path $(Z_1,\, Z_2)$ in Theorem \[geometricthm\], if there exists some $t_0 \in (0, 1)$, such that both $Z_1(t_0)$ and $Z_2(t_0)$ are tangent to the axes, then both $Z_1$ and $Z_2$ must stay on the corresponding axes for all $t \in [0, 1]$.
The proof basically follows by the uniqueness of solution of the initial value problem of an ODE system. Note that for $t \in (0, 1)$, $q_i \, (i=1,2,3)$ are the solutions of the Newtonian equations. Without loss of generality, we assume $Z_1(t_0)$ is tangent to the $x$-axis with $0< t_0<1$. Note that $Z_1(t_0) \neq 0$ and $Z_1=q_2-q_3$. It follows that $$\label{q1ya2yt0equal}
q_{2y}(t_0)=q_{3y}(t_0), \qquad \dot{q}_{2y}(t_0)= \dot{q}_{3y}(t_0).$$
If $Z_2(t_0)$ is also on the $x$-axis and tangent to it, it implies that $$q_{1y}(t_0)=0, \qquad \dot{q}_{1y}(t_0)= 0.$$ Note that the center of mass is fixed at $0$, it follows that $$\label{q1q2q3yall0}
q_{1y}(t_0)=q_{2y}(t_0)=q_{3y}(t_0)=0, \qquad \dot{q}_{1y}(t_0)= \dot{q}_{2y}(t_0)=\dot{q}_{3y}(t_0)= 0.$$ The Newtonian equations and imply that $$\label{qidouble0}
\ddot{q}_{1y}(t_0)= \ddot{q}_{2y}(t_0)=\ddot{q}_{3y}(t_0)= 0.$$ Since the set $$\{(q_1, \, q_2, \, q_3) \,|\, q_{1y}=q_{2y}=q_{3y}=0, \, \, \, \dot{q}_{1y}= \dot{q}_{2y}=\dot{q}_{3y}= 0\}$$ is invariant, it imply that $$q_{1y}(t)=q_{2y}(t)=q_{3y}(t)=0, \qquad \forall \, t \in [0, 1].$$ It follows that both $Z_1(t)$ and $Z_2(t)$ stay on the $x$-axis for all $t \in [0, 1]$.
If $Z_2(t_0)$ is tangent to the $y$-axis, we have $$\label{q3xequal0}
q_{1x}(t_0)=0, \quad q_{2x}(t_0)=-q_{3x}(t_0), \quad \dot{q}_{1x}(t_0)= 0, \quad \dot{q}_{2x}(t_0)=-\dot{q}_{3x}(t_0).$$ Note that in , $$q_{2y}(t_0)=q_{3y}(t_0), \qquad \dot{q}_{2y}(t_0)= \dot{q}_{3y}(t_0).$$ By the Newtonian equations, and imply that $$\ddot{q}_{1x}(t_0)= 0, \qquad \ddot{q}_{2y}(t_0)= \ddot{q}_{3y}(t_0).$$ Note that the set $$\{(q_1, \, q_2, \, q_3) \,|\, q_{1x}=0,\, \, q_{2x}=-q_{3x}, \, \, q_{2y}=q_{3y}, \,\, \dot{q}_{1x}= 0, \, \, \dot{q}_{2x}=-\dot{q}_{3x}, \, \, \dot{q}_{2y}= \dot{q}_{3y}\}$$ is invariant, it follows hat $$q_{1x}(t)=0, \qquad q_{2y}(t)=q_{3y}(t), \quad \forall \, t \in [0, 1].$$ It implies that $Z_1$ stays on the $x$-axis and $Z_2$ stays on the $y$-axis for all $t \in [0, 1]$. The proof is complete.
**Proof of Theorem \[geometricthm\]:** The key point in the proof is the observation that for all $t\in [0,1]$, $Z_1(t)$ and $Z_2(t)$ must belong to two adjacent quadrants.
Without loss of generality, we assume $Z_1(0), \, Z_1(1)\in \overline{\mathsf{Q}_2}$ and $Z_2(0), \, Z_2(1)\in \overline{\mathsf{Q}_1}$. We define a path $(\widetilde{Z}_1, \, \widetilde{Z}_2)=(\widetilde{Z}_1(t), \, \widetilde{Z}_2(t))$ by $$\label{tildeZ1Z2def}
\widetilde{Z}_1(t)= (-|Z_{1x}(t)|, \, |Z_{1y}(t)|), \ \ \widetilde{Z}_2(t)= (|Z_{2x}(t)|, \, |Z_{2y}(t)|), \ \ \, \, \forall \, t \in [0,1].$$ It is clear that $\widetilde{Z}_1(t) \in \overline{\mathsf{Q}_2}$, $\widetilde{Z}_2(t) \in \overline{\mathsf{Q}_1}$ and $|\widetilde{Z}_i(t)|=|Z_i(t)|$ for all $t \in [0,1]$. Also, $\displaystyle \int_0^1 K(\widetilde{Z}_1, \, \widetilde{Z}_2) \, dt=\int_0^1 K(Z_1, \, Z_2) \, dt$. By lemma \[Uinij\], we have $$\label{Uineq}
U(Z_1(t),Z_2(t))\ge U(\widetilde{Z}_1(t),\widetilde{Z}_2(t)),\ \ \forall t\in[0,1] .$$ Thus $$\mathcal{A}(Z_1,Z_2)\ge\mathcal{A}(\widetilde{Z}_1,\widetilde{Z}_2).$$ On the other hand, since $(Z_1, \, Z_2)$ is a minimizing path connecting the two fixed boundaries, and $(\widetilde{Z}_1, \, \widetilde{Z}_2)$ connects the same boundaries, it implies that $$\mathcal{A}(Z_1,Z_2)\le\mathcal{A}(\widetilde{Z}_1,\widetilde{Z}_2).$$ Hence $$\label{AZ1Z2equalAtilde}
\mathcal{A}(Z_1,Z_2)=\mathcal{A}(\widetilde{Z}_1,\widetilde{Z}_2).$$ By the smoothness of $U(Z_1(t), \, Z_2(t))$ in $(0,1)$, it follows that $$U(Z_1(t),Z_2(t))= U(\widetilde{Z}_1(t),\widetilde{Z}_2(t)).$$ Then by lemma \[Uinij\], $Z_1(t)$ and $Z_2(t)$ are in two adjacent closed quadrants for all $t\in [0,1]$.
Note that by the celebrated results of Marchal [@Mar] and Chenciner [@CA], an action minimizer $(Z_1, \, Z_2)$ is collision-free in $(0,\, 1)$ and it is a solution of the Newtonian equations in $(0,\, 1)$.
By , it implies that both $(Z_1,Z_2)$ and $(\widetilde{Z}_1,\widetilde{Z}_2)$ are smooth paths in $(0, \, 1)$. Thus if $Z_1$ or $Z_2$ touches the coordinate axes in $(0,1)$, it must be tangent to the coordinate axes.
Note that $Z_1=q_2-q_3$, it follows that $Z_1(t) \not=0$ for all $t \in (0,\, 1)$. If $Z_2(t_0)=0$, by the smoothness of $Z_2$ and $\widetilde{Z}_2$, it follows that $\dot{Z}_2(t_0)=0$. Since $\{(q_1,q_2,q_3)\, |\, q_1=0, \, \dot{q}_1=0\}$ is an invariant set, it implies that $Z_2\equiv 0$ for all $t \in [0,\,1]$. In this case, it is one part of the Euler solution, which is case $($c) in Theorem \[geometricthm\].
If $Z_2\not \equiv 0$, by the Newtonian equations, $Z_1$ and $Z_2$ satisfy $$\label{Z1Z2equation}
\begin{split}
&\ddot{Z}_1=\frac{Z_2-Z_1/2}{|Z_2-Z_1/2|^3} - \frac{Z_2+Z_1/2}{|Z_2+Z_1/2|^3} - \frac{2mZ_1}{|Z_1|^3},\\
&\ddot{Z}_2=-\frac{1+2m}{2}\left[\frac{Z_2-Z_1/2}{|Z_2-Z_1/2|^3}+\frac{Z_2+Z_1/2}{|Z_2+Z_1/2|^3}\right].
\end{split}$$
Next, we show that if there exists some $t_0 \in (0, \,1)$, such that $Z_1(t_0)$ or $Z_2(t_0)$ is on the axes, then $Z_1(t)$ and $Z_2(t)$ must stay on the axes for all $t \in [0,1]$.\
In fact, we first assume that $Z_1(t_0)$ is on the positive $x$-axis for some $t_0\in (0,1)$ and $Z_2(t_0)\in \mathsf{Q}_1$. By , the acceleration in the $y$-direction $\ddot{Z}_{1y}(t_0)$ satisfies $$\ddot{Z}_{1y}(t_0)>0.$$ The smoothness of $Z_1$ and $\widetilde{Z}_1$ implies that the velocity satisfies $\dot{Z}_{1y}(t_0)=0$. Consequently, for small enough $\epsilon>0$, $Z_1(t_0+\epsilon)\in \mathsf{Q}_1, \, Z_2(t_0+\epsilon)\in \mathsf{Q}_1$. Contradiction to the property that $Z_1(t)$ and $Z_2(t)$ belong to two adjacent closed quadrants for all $t\in [0,1]$! Hence $Z_2(t_0)\notin \mathsf{Q}_1$. Similarly, $Z_2(t_0)\notin \mathsf{Q}_i$ for $i=1,2,3,4$, i.e $Z_2(t_0)$ is on the coordinate axes.
We then discuss the case when $Z_1(t_0)$ is on the negative $x$-axis. If $Z_2(t_0) \in \mathsf{Q}_1$, it follows that $\dot{Z}_{1y}(t_0)=0$ and $\ddot{Z}_{1y}(t_0)<0$. Hence, there exists small enough $\epsilon>0$, such that $Z_1(t_0+\epsilon) \in \mathsf{Q}_3$ and $Z_2(t_0+\epsilon)\in \mathsf{Q}_1$. Contradiction to the result that $Z_1(t)$ and $Z_2(t)$ must belong to two adjacent quadrants! Similarly, $Z_2(t_0)\notin \mathsf{Q}_i$ for $i=1,2,3,4$. Therefore, whenever $Z_1(t_0)\not =0$ is on the $x$-axis for some $t_0 \in (0, \, 1)$, $Z_2(t_0)$ must be on the axes. By Lemma \[z1z2sametimetangent\], $Z_1$ and $Z_2$ must stay on the corresponding axes for all $t\in [0,1]$.
The same argument works for $Z_1(t_0)$ on the $y$-axis or $Z_2(t_0)$ on one of the axes. Therefore, whenever there is some $t_0 \in (0, \, 1)$ such that $Z_1(t_0)$ or $Z_2(t_0)$ is on the axes, both $Z_1(t)$ and $Z_2(t)$ must stay on the corresponding axes for all $t \in [0, \, 1]$. That is, case (b) in Theorem \[geometricthm\] holds whenever there exists some $t_0 \in (0, \, 1)$ such that $Z_1(t_0)$ or $Z_2(t_0)$ is on the axes.
In the end, if both case (b) and $($c) fail, we have both $Z_1(t)$ and $Z_2(t)$ are away from the axes for all $t \in (0,\, 1)$. Then case (a) holds. The proof is complete.
Now we can apply Theorem \[geometricthm\] to exclude the possible collisions in the action minimizer $\mathcal{P}_{m, \, \theta}$.
\[nocollision\] For each given $\theta \in [0, \pi/2)$ and mass set $M=[m_1,\, m_2, \, m_3]=[1, \, m, \, m]$ with $m>0$, the minimizer $\mathcal{P}_{m, \, \theta}$ is collision-free.
By the celebrated work of Marchal [@Mar] and Chenciner [@CA], $\mathcal{P}_{m, \, \theta}$ is collision-free in $(0, 1)$. By Section \[exclusiontotalcollision\], $\mathcal{P}_{m, \, \theta}$ has no total collision. Furthermore, it has no binary collision at $t=1$.
Recall that at $t=0$, the three bodies are on the $x$-axis and satisfy the order $q_{2x}(0) \le q_{1x}(0) \le q_{3x}(0)$. It implies that he only possible collisions are $q_1(0)=q_2(0)$ and $q_1(0)=q_3(0)$. Note $\mathcal{P}_{m, \, \theta}$ is free of total collision, it implies that $q_{2x}(0)<0$ and $q_{3x}(0)>0$.
If $Z_{2x}(0) = (1+\frac{1}{2m})q_{1x}(0)=0$, it is an Euler configuration. It is clear that it can not have any binary collision at $t=0$. If $Z_{2x}(0) > 0$, the only possible binary collision is $q_1(0)=q_3(0)$. Similarly, if $Z_{2x}(0) = (1+\frac{1}{2m})q_{1x}(0) < 0$, the only possible binary collision is $q_1(0)=q_2(0)$. We only discuss the case when $Z_{2x}(0) = (1+\frac{1}{2m})q_{1x}(0) > 0$. The other case $Z_{2x}(0)< 0$ is exactly the same.
If $Z_{2x}(0) > 0$, we are left to eliminate the possible binary collision $q_1(0)=q_3(0)$. In what follows, we exclude the binary collision between bodies 1 and 3 by contradiction.
When $\theta\in (0,\pi/2)$, we claim that the collision minimizer $(Z_1,Z_2)$ satisfies $$\label{Z1Z2quadrant}
Z_1(t)\in \overline{\mathsf{Q}_2},\ \ \ Z_2(t)\in \overline{\mathsf{Q}_1},\ \ \ \forall t\in [0,1].$$ In fact, by the definition of $Q_{S}$ and $Q_{E}$ in , $Z_1(0)< 0$ is on the negative $x$-axis, while $Z_2(0)>0$ is on the positive $x$-axis. $Z_1(1)\in \mathsf{Q}_2 \, \, \text{or} \, \, \mathsf{Q}_4$ and $Z_2(1) \in \mathsf{Q}_1 \, \, \text{or} \, \, \mathsf{Q}_3$.\
Assume $Z_1(1)\in \mathsf{Q}_4$. A new path $(\widetilde{Z}_1, \, \widetilde{Z}_2)$ can be defined by , such that $$\widetilde{Z}_1(t)= (-|Z_{1x}(t)|, \, |Z_{1y}(t)|)\in \overline{\mathsf{Q}_2},\qquad \widetilde{Z}_2(t)= (|Z_{2x}(t)|, \, |Z_{2y}(t)|)\in \overline{\mathsf{Q}_1},\quad\forall \, t\in [0,1],$$ and Lemma \[Ucomparetilde\] implies that $$\mathcal{A}(Z_1,Z_2)\ge\mathcal{A}(\widetilde{Z}_1,\widetilde{Z}_2).$$ Note that $(\widetilde{Z}_1, \, \widetilde{Z}_2)$ also satisfies the boundary condition , it implies that $ \mathcal{A}(Z_1,Z_2)\le\mathcal{A}(\widetilde{Z}_1,\widetilde{Z}_2)$. Hence, $$\mathcal{A}(Z_1,Z_2)= \mathcal{A}(\widetilde{Z}_1,\widetilde{Z}_2).$$ Since $\widetilde{Z}_1(1)\in \mathsf{Q}_2$ and $\widetilde{Z}_2(1)\in \overline{\mathsf{Q}_1}$, by Theorem \[geometricthm\] we have $\widetilde{Z}_1(t)\in \mathsf{Q}_2$ for all $t\in (0,1)$. On the other hand, $Z_1(0)$ is on the negative $x$-axis and $Z_1(1)\in \mathsf{Q}_4$. Thus there is some $t_0\in (0,1)$ such that $Z_1(t_0)$ is on the $y$-axis. Then $\widetilde{Z}_1(t_0)$ is also on the $y$-axis. Contradict to $\widetilde{Z}_1(t)\in \mathsf{Q}_2$ for all $t\in (0,1)$! Hence, $Z_1(1)\in \mathsf{Q}_2$. A similar argument shows that $Z_2(1)\in \overline{\mathsf{Q}_1}$. By Theorem \[geometricthm\], holds.
When $\theta =0$, we have $Z_1(1)$ is on the $y$-axis and $Z_2(1)$ is on the $x$-axis. By the same argument as above, we actually have $$Z_1(t)\in \overline{\mathsf{Q}_2},\ \ \ Z_2(t)\in \overline{\mathsf{Q}_1},\ \ \ \forall t\in [0,1],$$ or $$Z_1(t)\in \overline{\mathsf{Q}_3},\ \ \ Z_2(t)\in \overline{\mathsf{Q}_4},\ \ \ \forall t\in [0,1].$$ Here we only consider the case $Z_1(t)\in \overline{\mathsf{Q}_2}, \, Z_2(t)\in \overline{\mathsf{Q}_1}$, while the other case follows similarly.
For $\theta \in [0,\pi/2)$, the minimizer must satisfies that $Z_1(t)$ and $Z_2(t)$ belongs to two adjacent closed quadrants for all $t\in [0,1]$. This is enough to exclude the binary collision at $t=0$.
Assume at $t=0$, $q_1(0)=q_3(0)$. Since $q_{2x}\le q_{1x} \le q_{3x}$, it follows that $q_1(0)=q_3(0)>0$ and $q_2(0)<0$. In fact, by [@CH4; @FU; @Yu3], the following limits hold: $$\label{velocityat0limit}
\lim_{t \to 0^{+}} \frac{q_1-q_3}{|q_1-q_3|}=(1, \, 0), \quad \lim_{t \to 0^{+}} \frac{d}{dt}\left(\frac{q_1-q_3}{|q_1-q_3|}\right)=(0, \, 0).$$
It is known that an isolated binary collision in the three-body problem can be regularized. By , it implies that for small enough $t>0$, there exists $\alpha_0>2/3$, such that $$q_1-q_3 = ( a t^{2/3}+o(t^{2/3}), \, \, b t^{\alpha_0}+o(t^{\alpha_0}) ).$$ Hence, if $b \neq 0$, $$\begin{aligned}
&&\frac{d}{dt} \left( \frac{q_1-q_3}{|q_1-q_3|} \right) \\
&=& \left( \frac{-ab^2 t^{2 \alpha_0 - 7/3}(3\alpha_0 -2) }{3 (a^2+ b^2 t^{2 \alpha_0 - 4/3})^{3/2}} +o(t^{2 \alpha_0 - 7/3}), \, \, \frac{-ab^2 t^{ \alpha_0 - 5/3}(3\alpha_0 -2) }{3 (a^2+ b^2 t^{2 \alpha_0 - 4/3})^{3/2}} +o(t^{\alpha_0 - 5/3})\right).\end{aligned}$$ By , it follows that $$2 \alpha_0 - 7/3 >0, \quad \alpha_0 - 5/3 >0.$$ Therefore, $$\alpha_0 > 5/3.$$ It follows that $$\label{velocity12ycol}
\lim_{t \to 0^{+}} \left(\dot{q}_{1y}-\dot{q}_{3y} \right)=0.$$ If $b=0$, it is clear that holds. Hence, at $t=0$, $q_1$ and $q_3$ has the same $y$-velocity. Thus $$\label{yvelocity}
\dot{q}_{1y}(0)=\dot{q}_{3y}(0) =-\frac{m}{m+1}\dot{q}_{2y}(0).$$ By the above discussion, for $t\in [0,1]$ there always holds $Z_1(t)\in \overline{\mathsf{Q}_2}, \, Z_2(t)\in \overline{\mathsf{Q}_1}$. Together with , we have $$\dot{q}_{2y}\ge 0,\ \ \ \dot{q}_{1y}=\dot{q}_{3y}\le 0.$$ When $\dot{q}_{2y}= 0$, by Lemma 6.2 in \[20\], the path $(Z_1, Z_2) = (Z_1(t), Z_2(t))(t\in [0,1])$ must stay on the $x$-axis. However, by the definition of $Q_E$ in , the collision-free isosceles configuration at t = 1 can never become a collinear configuration on the x-axis. Contradiction!\
When $\dot{q}_{2y}(0)> 0$, By , it follows that $\dot{q}_{1y}(0)< 0$. Then for $t\in (0,\epsilon]$ with $\epsilon>0$ small enough we have $$\dot{q}_{1y}(t)<0, \qquad \forall t \in (0, \epsilon].$$ Hence for $t\in (0,\epsilon)$, $Z_2(t)\notin \mathsf{Q}_1$. But $Z_1(t)\in \mathsf{Q}_2, \, Z_2(t) \in \mathsf{Q}_1$ holds for all $t\in (0,1)$. Contradiction!
The above argument implies that body 1 and 3 can not collide at $t=0$ in $\mathcal{P}_{m, \, \theta}$ in the case when $Z_{2x}>0$. For the case $Z_{2x}<0$, the argument is similar. Therefore, $\mathcal{P}_{m, \, \theta}$ is free of collision. The proof is complete.
By Theorem \[nocollision\], for each given $\theta \in [0, \, \pi/2)$ and each $m>0$, the action minimizer $\mathcal{P}_{m, \, \theta}$ is actually a solution of the Newtonian equations. By applying the formulas of first variation as in Section 5 of [@Yan3], one can easily show that $\mathcal{P}_{m, \, \theta}$ can be extended to a periodic or quasi-periodic orbit.
Compare $\mathcal{P}_{m, \, \theta}$ and the Euler orbit for $m=1$ {#compareEuleratm1}
==================================================================
In previous sections, we have shown that $\mathcal{P}_{m, \, \theta}$ is collision-free. In this section, we show that $\mathcal{P}_{m, \, \theta}$ is nontrivial when $m=1$ and $\theta \in [0, 0.183 \pi]$.
\[nontrivialatm1\] When $m=1$ and $\theta \in [0, \, 0.183\pi]$, the minimizer $\mathcal{P}_{m, \, \theta}$ is nontrivial.
The case when $\theta \in [0.084\pi, 0.183 \pi]$ has been shown in [@Yan3]. In what follows, we only show the case when $\theta \in [0, 0.084 \pi]$.
Let $\mathcal{A}_{Euler}$ be the action of an Euler orbit connecting $Q_S$ and $Q_E$ in . Then $$\mathcal{A}_{Euler} = 3 \left(\frac{5}{4}\right)^{2/3} \left(\frac{\pi}{2}-\theta \right)^{2/3}.$$ We need to define a test path $\mathcal{P}_{test}$ connecting $Q_S$ and $Q_E$, such that its action $\mathcal{A}_{test}=\mathcal{A}\left( \mathcal{P}_{test} \right)$ satisfies $ \mathcal{A}_{test} < \mathcal{A}_{Euler}$ for each $\theta \in [0, 0.084\pi]$. The test path $\mathcal{P}_{test}$ is defined as follows. We choose $\theta_0=0.053\pi$. Let $q= \begin{bmatrix}
q_1\\
q_2\\
q_3
\end{bmatrix}$ be the position matrix path of the minimizer $\mathcal{P}_{1, \theta_0}$, and $\tilde{q}=\begin{bmatrix} \tilde{q}_{1} \\
\tilde{q}_{2}\\
\tilde{q}_{3} \end{bmatrix}$ be the position matrix path of $\mathcal{P}_{test}=\mathcal{P}_{test, \theta}$. We can then define a test path $\mathcal{P}_{test, \theta}$ by connecting the following 11 points: $$\tilde{q}\left(\frac{i}{10}\right)= q\left(\frac{i}{10}\right), \,\, (i=0,1,\dots, 9), \quad \tilde{q}\left(1\right)=q(1)R(-\theta_0)R(\theta).$$ In fact, $\tilde{q}(t) $ satisfies $$\label{defptest}
\tilde{q}(t) = \tilde{q}\left(\frac{i}{10}\right)+ 10\left(t- \frac{i}{10} \right) \left[ \tilde{q}\left(\frac{i+1}{10}\right)- \tilde{q}\left(\frac{i}{10}\right) \right], \quad t \in \left[ \frac{i}{10}, \frac{i+1}{10} \right],$$ where $i=0,1,\dots, 9$. It is easy to check that $\tilde{q}(0) \in Q_S$ and $\tilde{q}(1) \in Q_E$, where $Q_S$ and $Q_E$ are the boundary sets defined in . Once the values of $q\left(\frac{i}{10}\right) \, (i=0, 1, \dots, 10)$ in $\mathcal{P}_{1, \theta_0}$ are given, the action of the test path $\mathcal{A}_{test}= \mathcal{A}\left( \mathcal{P}_{test, \theta} \right)$ can be calculated accurately as in Lemma 3.1 of [@Yan3]. The data of the test path and the corresponding figures of action values are shown in Table \[table1\] and Fig. \[fig3\].
[ |c|c|c|]{}\
$t$ & $\tilde{q}_1$ &$\tilde{q}_2$\
$ 0$ & $ ( 0.3067 , \, 0 ) $ & $ ( -0.9504 , \, 0 ) $\
$ 0.1 $ & $ ( 0.34241260 , \, 0.11622575 ) $ & $ ( -0.94529039 , \, 0.06856631 ) $\
$ 0.2 $ & $ ( 0.41572598 , \, 0.18977437 ) $ & $ ( -0.93016235 , \, 0.13667709 ) $\
$ 0.3 $ & $ ( 0.49233787 , \, 0.22736882 ) $ & $ ( -0.90538226 , \, 0.20366579 ) $\
$ 0.4 $ & $ ( 0.56164714 , \, 0.24183944 ) $ & $ ( -0.87127416 , \, 0.26870901 ) $\
$ 0.5 $ & $ ( 0.62118018 , \, 0.24108669 ) $ & $ ( -0.82813029 , \, 0.33090644 ) $\
$ 0.6 $ & $ ( 0.67056091 , \, 0.22979410 ) $ & $ ( -0.77626334 , \, 0.38932578 ) $\
$ 0.7 $ & $ ( 0.70992537 , \, 0.21093951 ) $ & $ ( -0.71604331 , \, 0.44303606 ) $\
$ 0.8 $ & $ ( 0.73947888 , \, 0.18657625 ) $ & $ ( -0.64791266 , \, 0.49113901 ) $\
$ 0.9 $ & $ ( 0.75936088 , \, 0.15823937 ) $ & $ ( -0.57237992 , \, 0.53279730 ) $\
$ 1 $ & $ ( 0.77840337 , \, 0 )R(\theta) $ & $ ( -0.38920168 , \, 0.64061834 )R(\theta) $\
![\[fig3\] In the figure, the horizontal axis is $\theta/\pi$, and the vertical axis is the action value $\mathcal{A}$. The black curve is the graph of the test path’s action $\mathcal{A}_{test}$ and the blue curve is the graph of $\mathcal{A}_{Euler}$.](3b0to0084.eps){width="3.1in"}
For the given set of 11 interpolation points, it is shown in [@Yan3] that the action of the test path $\mathcal{A}_{test}$ is a smooth function with respect to $\theta$. In the last step, we compare the value of the two smooth functions: $\mathcal{A}_{test}$ and $\mathcal{A}_{Euler}$ when $\theta\in[0,0.084\pi]$ in Fig. \[fig3\]. To do so, we calculate the value of the two functions with a step $\pi\times10^{-3}$.
The error of the linear interpolation method used to compare the two function is $\frac{1}{8} (3.14 \times 10^{-3})^2 \tilde{\Delta}$, where $\tilde{\Delta}$ is the maximum of the second derivative of the corresponding function. For $\theta \in [0, 0.084 \pi]$, it turns out that $\tilde{\Delta} \leq \frac{40}{\pi}$ for both functions. It implies that the error is bounded by $$\frac{1}{8} (3.14 \times 10^{-3})^2 \Delta_1 \leq \frac{1}{8} (3.14 \times 10^{-3})^2 \frac{40}{\pi} \approx 1.57 \times 10^{-5}.$$ Numerically, for $\theta \in [0, 0.084 \pi]$, the minimum value of $\mathcal{A}_{Euler}-\mathcal{A}_{test}$ in Fig. \[fig3\] is $ 9.49 \times 10^{-3} > 1.57 \times 10^{-5}$.
Therefore, for each given $\theta \in [0, \, 0.084 \pi]$, the action of the test path $\mathcal{A}_{test}$ satisfies $$\mathcal{A}_{test} < \mathcal{A}_{Euler}.$$ It follows that the action minimizer $\mathcal{P}_{1, \theta}$ is nontrivial and collision-free when $\theta \in [0, \, 0.084 \pi]$. The proof is complete.
Application to the retrograde orbit {#applytoretrogradeorbit}
===================================
In this section, we apply our geometric result (Theorem \[geometricthm\]) to the retrograde orbit. Note that the retrograde orbit can be characterized as an action minimizer connecting two free boundaries. Let $$\label{boundarysetting02}
Q_{s_1}=\begin{bmatrix}
-2ma_1-ma_2 & 0\\
a_1-ma_2 & 0\\
a_1+(m+1)a_2 & 0
\end{bmatrix}, \qquad Q_{e_1}=\begin{bmatrix}
2mb_2 & 0\\
-b_2 & b_1 \\
-b_2 & -b_1
\end{bmatrix} R(\theta),$$ where $M=[1,\, m,\, m]$, $a_1\geq 0$, $a_2 \geq 0$, $b_1 \in \mathbb{R}$, $b_2 \in \mathbb{R}$, and $R(\theta)=\begin{bmatrix}
\cos(\theta)& \sin(\theta)\\
-\sin(\theta)& \cos(\theta)
\end{bmatrix}$. The two configuration sets are defined as follows: $$\label{QSQE02}
Q_{S_1}=\left\{ Q_{s_1} \, \bigg|\, a_1 \geq 0, \, a_2 \geq 0\right\},\quad Q_{E_1}=\left\{ Q_{e_1} \, \bigg|\, b_1 \in \mathbb{R}, \, b_2 \in \mathbb{R} \right\},$$ For each $\theta \in (0, \pi/2]$, standard results [@CH3; @FT; @Yan2; @Yan5] imply that there exists an action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}} \in H^1([0,1], \chi)$, such that $$\label{actionminsetting02}
\mathcal{A}(\widetilde{\mathcal{P}_{m, \, \theta}})= \inf_{q \in P(Q_{S_1}, \, Q_{E_1})} \mathcal{A} = \inf_{q \in P(Q_{S_1},Q_{E_1})} \int_0^1 (K+U) \, dt,$$ where $$\mathrm P(Q_{S_1}, \, Q_{E_1})=\left \{q\in H^1([0,1],\chi) \, \big{|} \, q(0) \in Q_{S_1}, \, q(1) \in Q_{E_1} \right \}.$$
We first show that the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ is free of total collision by applying the level estimate method [@CH; @CH2]. The test path here is a modified version of the test path in [@CH2].
\[nototalretro\] $\widetilde{\mathcal{P}_{m, \, \theta}}$ is free of total collision for all $m>0$ and $\theta\in(0,\, \frac{\pi}{2}]$.
The proof is based on Theorem 2 in [@CH2]. To be consistent, we use the same notation as in [@CH2]. Let the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ is for $t \in [0, T]$ with $T=1/4$.
We first obtain a lower bound of action when the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ has a total collision. We denote the action by $\mathcal{A}_{total}$ if $\widetilde{\mathcal{P}_{m, \, \theta}}$ has a total collision. By Chen’s level estimate [@CH], if $T=1/4$, the lower action bound of total collision path is $$\mathcal{A}_{total} \geq \frac{m^2+2m}{1+2m} \cdot \frac{3}{2} (1+2m)^{2/3} \pi^{2/3} \Bigl(\frac14\Bigr)^{\frac13}= \frac{3m}{2} \pi^{2/3} \frac{m+2}{(1+2m)^{1/3}}\Bigl(\frac14\Bigr)^{\frac13}.$$ By the setting of $Q_{S_1}$ and $Q_{E_1}$ in , we consider an artificial path $$x(t)=(x_1(t), \, x_2(t), \, x_3(t))$$ on $[0,1]$ with $x_i(t) \in \mathbb{C}$.
Let $M=2m+1$, $\phi = 4\theta\in(0, \, 2\pi]$, $\alpha=\Bigl(\frac{\phi^2}{M}\Bigr)^{\frac13}\frac{1}{(2m)^{\frac23}(2\pi)^{\frac23}}$ and $$J(s)= \displaystyle \int_0^1\frac{1}{|1-se^{2\pi ti}|}dt \quad \quad \text{for} \quad 0\leq s<1.$$ We set $$Q(t)=\frac{1}{(M\phi)^{\frac23}}e^{\phi ti}, \qquad R(t)=\frac{1}{(2m)^{\frac23}(2\pi)^{\frac23}}e^{(\phi-2\pi)ti}$$ and consider the path $$x(t)=(x_1(t), \, x_2(t), \, x_3(t))=\left(-2mQ(t),\, \, \, \, Q(t)-mR(t), \, \, \, \, Q(t)+mR(t) \right).$$ Note that $[0,\, 1/4]$ is a fundamental domain of this path and $x(0)\in Q_{S_1}, \, x(1/4)\in Q_{E_1}$. A direct calculation shows that $$|\dot{x_1}|^2 = (2m)^2\Bigl(\frac{\phi}{M^2}\Bigr)^{\frac23},$$ $$|\dot{x_2}|^2 = \Bigl(\frac{\phi}{M^2}\Bigr)^{\frac23}+\frac{m^2(\phi-2\pi)^2}{(2m)^{\frac43}(2\pi)^{\frac43}}
-\Bigl(\frac{\phi}{M^2}\Bigr)^{\frac13}\frac{2m(\phi-2\pi)}{(2m)^{\frac23}(2\pi)^{\frac23}}\cos(2\pi t),$$ $$|\dot{x_3}|^2 = \Bigl(\frac{\phi}{M^2}\Bigr)^{\frac23}+\frac{m^2(\phi-2\pi)^2}{(2m)^{\frac43}(2\pi)^{\frac43}}
+\Bigl(\frac{\phi}{M^2}\Bigr)^{\frac13}\frac{2m(\phi-2\pi)}{(2m)^{\frac23}(2\pi)^{\frac23}}\cos(2\pi t),$$ $$\int_0^1 K(\dot{x})dt =m \left[\Bigl(\frac{\phi}{M^2}\Bigr)^{\frac23}+\frac{m^2(\phi-2\pi)^2}{(2m)^{\frac43}(2\pi)^{\frac43}}\right]
+\frac12(2m)^2\Bigl(\frac{\phi}{M^2}\Bigr)^{\frac23}.$$ And the potential energy is $$\begin{aligned}
&\int_0^1 U(x)dt \\
&= \int_0^1 \frac{m}{|x_1-x_2|}+\frac{m}{|x_1-x_3|}+\frac{m^2}{|x_2-x_3|}dt\\
&= \int_0^1 \Bigl(\frac{\phi^2}{M}\Bigr)^{\frac13}\frac{m}{|1-m\alpha e^{-2\pi ti}|}+\Bigl(\frac{\phi^2}{M}\Bigr)^{\frac13}\frac{m}{|1+m\alpha e^{-2\pi ti}|}+2^{\frac13}m^{\frac53}\pi^{\frac23}dt\\
&= 2^{\frac13}m^{\frac53}\pi^{\frac23}+2m\Bigl(\frac{\phi^2}{M}\Bigr)^{\frac13}J(m\alpha).\end{aligned}$$ Note that $m\alpha= m \Bigl(\frac{\phi^2}{M}\Bigr)^{\frac13}\frac{1}{(2m)^{\frac23}(2\pi)^{\frac23}} \leq \frac{m}{(2m+1)^{\frac13} (2m)^{\frac23}} < \frac{1}{2}$ and $J(s)$ is increasing in $[0,1)$ ([@CH]). Therefore the action of the path over the fundamental domain $[0, \, 1/4]$ is $$\begin{aligned}
\mathcal{A}_{test}&=\frac14\int_0^1 K(\dot{x})+ U(x)dt\\
&< \frac{m}{4} \Bigl( \frac{\pi^2}{M}\Bigr)^{\frac13} \left[ 3 \times 2^{-\frac23}m^{\frac23}M^{\frac13}+
\left( 2J \left(\frac12 \right)+1 \right) \times 2^{\frac23} \right].\end{aligned}$$ Now we are left to prove $\mathcal{A}_{test}<\mathcal{A}_{total}$ for all $m>0$.
If $0<m\le 1$, we have $$\begin{aligned}
\label{m01}
\frac{\mathcal{A}_{test}}{\mathcal{A}_{total}} &< \frac{3 \times 2^{-\frac23} m^{\frac23}M^{\frac13}+
\left(2J\left(\frac{1}{2} \right)+1 \right)2^{\frac23}}{6\times\Bigl(\frac14\Bigr)^{\frac13}(m+2)} \nonumber\\
&\leq \frac{1}{2 \times 3^{\frac{2}{3}}}+ \frac{\left[ 2J\left(\frac{1}{2} \right)+1 \right] 2^{\frac23}}{6 \times 2^{\frac13}} \approx 0.9011 <1.\end{aligned}$$ If $m \ge 1$, let $f(m)= f_1(m) -f_2(m)$, where $$f_1(m)=\frac{m}{4} \Bigl( \frac{\pi^2}{M}\Bigr)^{\frac13} \left[ 3 \times 2^{-\frac23}m^{\frac23}M^{\frac13}+
\left( 2J \left(\frac12 \right)+1 \right) \times 2^{\frac23} \right]$$ and $$f_2(m)= \frac{3m}{2} \pi^{2/3} \frac{m+2}{(1+2m)^{1/3}}\Bigl(\frac14\Bigr)^{\frac13}.$$ Since $\mathcal{A}_{test} < f_1(m)$ and $\mathcal{A}_{total} \geq f_2(m)$. In order to show $\mathcal{A}_{test}<\mathcal{A}_{total}$, it is sufficient to show that $f(m)= f_1(m) -f_2(m)<0$.
In fact, $$\begin{aligned}
f(m) & := f_1(m) -f_2(m) \\
& = \frac{m}{4}\Bigl(\frac{\pi^2}{M}\Bigr)^{\frac13} \left[ 3 \times 2^{-\frac23} m^{\frac23}M^{\frac13}+
\left(2J \left(\frac12 \right)+1\right)2^{\frac23}-3\times 2^{\frac13}(m+2)\right].\end{aligned}$$ Let $g(m)=3 \times 2^{-\frac23} m^{\frac23}M^{\frac13}+
\left(2J \left(\frac12 \right)+1\right)2^{\frac23}-3\times 2^{\frac13}(m+2)$, then $$f(m)= \frac{m}{4}\Bigl(\frac{\pi^2}{M}\Bigr)^{\frac13} g(m).$$ A direct calculation shows that $$\begin{aligned}
\frac{d g(m)}{dm}= 2^{\frac{1}{3}}
\left[ \Bigl(\frac{2m+1}{m}\Bigr)^{\frac13}+\Bigl(\frac{m}{2m+1} \Bigr)^{\frac23}\right]-3\times 2^{\frac13}
<0\end{aligned}$$ for every $m\ge 1$. It follows that $g(m) < g(1)$ for any $m>1$. However, when $m=1$, it is shown that $f(1) <0$. It implies $g(1)<0$ and $g(m)<0$ for all $m>1$. So, $$\label{mge1}
\mathcal{A}_{test}-\mathcal{A}_{total}< f(m) <0, \quad \quad \forall \quad m\geq 1.$$ Therefore, by and , the inequality $\mathcal{A}_{test}<\mathcal{A}_{total}$ holds for any $m>0$. The proof is complete.
By the same argument of Lemma \[nobinarycollisionat1\] in Section \[exclusiontotalcollision\], the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}= \widetilde{\mathcal{P}_{m, \, \theta}}([0,1])$ in is collision-free at $t=1$. We are then left to exclude the possible binary collisions at $t=0$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$.
Let $$Z_1=q_2-q_3, \quad \text{and} \quad Z_2=q_1-\frac{q_2+q_3}{2}=(1+\frac{1}{2m})q_1.$$ Let $(\hat{Z}_1(t), \, \hat{Z}_2(t)) \, (t \in [0,\, 1])$ be the minimizing path corresponding to the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$. Next, we apply Theorem \[geometricthm\] to exclude possible binary collisions at $t=0$.
\[nocollision2an3at0\] For any $m>0$ and $\theta \in (0, \pi/2)$, $q_2(0) \not= q_{3}(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$.
We show it by contradiction! Note that by Lemma \[nototalretro\], there is no total collision at $t=0$. If $q_2(0) = q_{3}(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$, it follows that $$\hat{Z}_1(0)=0, \quad \hat{Z}_{2x}(0)<0, \quad \text{and} \quad \hat{Z}_{2y}(0)=0,$$ while $$\hat{Z}_{1}(1) \in \mathsf{Q}_2 \, \, \text{or} \, \, \mathsf{Q}_4, \qquad \hat{Z}_{2}(1) \in \mathsf{Q}_1 \, \, \text{or} \, \, \mathsf{Q}_3.$$
Lemma \[Ucomparetilde\] and Theorem \[geometricthm\] imply that for any $m>0$ and $\theta \in (0, \, \pi/2)$, the minimizing path $(\hat{Z}_1(t), \, \hat{Z}_2(t)) \, (t \in [0,\, 1])$ must satisfy that the two curves $\hat{Z}_1$ and $\hat{Z}_2$ stay in two adjacent closed quadrants and they don’t touch the axes in $(0,1)$. It implies that $\hat{Z}_2(t) \in \overline{\mathsf{Q}_3}$.
On the other hand, note that by [@Yu3], when $q_2(0) = q_{3}(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$, the motion of the collision pair satisfies $$\lim_{t \to 0^{+}} \frac{q_2-q_3}{|q_2-q_3|}=(1, \, 0), \quad \lim_{t \to 0^{+}} \left(\dot{q}_{2y}-\dot{q}_{3y} \right)=0.$$ By Theorem \[geometricthm\], $\hat{Z}_1(t) \in \overline{\mathsf{Q}_4}$. Since $\hat{Z}_2(t) \in \overline{\mathsf{Q}_3}$, it follows that $\dot{q}_{1y}(0)\leq 0, \, \displaystyle \dot{q}_{2y}(0)= \lim_{t \to 0^{+}} \dot{q}_{2y} \geq 0$.
When $\epsilon>0$ small enough, we consider the following identity: $$\label{doubledotqident}
q_{2y}(\epsilon)- q_{3y}(\epsilon)= \int_0^{\epsilon} \int_0^t \left[\ddot{q}_{2y}- \ddot{q}_{3y} \right] \, ds \,dt.$$ Theorem \[geometricthm\] implies that we can choose small enough $\epsilon>0$, such that $\hat{Z}_1(\epsilon) \in \mathsf{Q}_4$. That is $\hat{Z}_{1y}(\epsilon)= q_{2y}(\epsilon)- q_{3y}(\epsilon)<0$.
However, the Newtonian equations imply that $$\ddot{q}_{2y}- \ddot{q}_{3y} = \frac{2(q_{3y}-q_{2y})}{|q_2-q_3|^3} + \frac{q_{1y}-q_{2y}}{|q_1-q_2|^3}-\frac{q_{1y}-q_{3y}}{|q_1-q_3|^3}.$$ Since $\hat{Z}_1(t) \in \overline{\mathsf{Q}_4}$ and $ \hat{Z}_2(t) \in \overline{\mathsf{Q}_3}$ for all $t \in [0, \epsilon]$ with $\epsilon>0$ small enough, it follows that for all $t \in [0, \epsilon]$, $$\frac{2(q_{3y}(t)-q_{2y}(t))}{|q_2(t)-q_3(t)|^3}\geq 0,$$ $$|q_1(t)-q_3(t)|\leq |q_1(t)-q_2(t)|, \quad q_{1y}(t)-q_{3y}(t) \leq q_{1y}(t)-q_{2y}(t)\leq 0.$$ Hence, $$\frac{2(q_{1y}(t)-q_{2y}(t))}{|q_1(t)-q_2(t)|^3}\geq 0, \qquad \frac{q_{1y}(t)-q_{2y}(t)}{|q_1(t)-q_2(t)|^3}-\frac{q_{1y}(t)-q_{3y}(t)}{|q_1(t)-q_3(t)|^3}\geq 0, \quad \forall \, t \in [0, \epsilon].$$ It implies that $$q_{2y}(\epsilon)- q_{3y}(\epsilon)= \int_0^{\epsilon} \int_0^t \left[\ddot{q}_{2y}- \ddot{q}_{3y} \right] \, ds \,dt \geq 0.$$ Contradict to $q_{2y}(\epsilon)- q_{3y}(\epsilon)<0$!
Therefore, $\widetilde{\mathcal{P}_{m, \, \theta}}$ has no binary collision between bodies 2 and 3 at $t=0$. The proof is complete.
\[nocollision1an2at0\] For any $m>0$ and $\theta \in (0, \pi/2)$, $q_1(0) \not= q_{2}(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$.
We show it by contradiction! Note that by Lemma \[nototalretro\], there is no total collision at $t=0$. If $q_1(0) = q_{2}(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$, it follows that $$\hat{Z}_{1x}(0)<0, \quad \hat{Z}_{2x}(0)<0, \quad \text{and} \quad \hat{Z}_{1y}(0)=\hat{Z}_{2y}(0)=0,$$ while $$\hat{Z}_{1}(1) \in \mathsf{Q}_2 \, \, \text{or} \, \, \mathsf{Q}_4, \qquad \hat{Z}_{2}(1) \in \mathsf{Q}_1 \, \, \text{or} \, \, \mathsf{Q}_3.$$ Lemma \[Ucomparetilde\] and Theorem \[geometricthm\] imply that for any $m>0$ and $\theta \in (0, \pi/2)$, the minimizing path $(\hat{Z}_1(t), \, \hat{Z}_2(t)) \, (t \in [0,\, 1])$ must satisfy $$\label{Q2Q3Z1Z2sec}
\hat{Z}_1(t) \in \overline{\mathsf{Q}_2}, \qquad \hat{Z}_2(t) \in \overline{\mathsf{Q}_3}.$$ By [@Yu3], when $q_1(0) = q_{2}(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$, the motion of the collision pair satisfies $$\lim_{t \to 0^{+}} \frac{q_1-q_2}{|q_1-q_2|}=(1, \, 0), \quad \lim_{t \to 0^{+}} \left(\dot{q}_{1y}-\dot{q}_{2y} \right)=0.$$ Since $\hat{Z}_2(t) \in \overline{\mathsf{Q}_3}$, it follows that $\displaystyle \dot{q}_{1y}(0)= \lim_{t \to 0^{+}} \dot{q}_{1y}=\dot{q}_{2y}(0)<0$ and $\dot{q}_{3y}(0)>0$. Then $\hat{Z}_1(t)= q_2(t)-q_3(t)$ is in $\mathsf{Q}_3$ for all $t \in (0, \, \epsilon]$. Contradict to $\hat{Z}_1(t) \in \overline{\mathsf{Q}_2}$!
Therefore, there is no binary collision between bodies 1 and 2 at $t=0$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$. The proof is complete.
In the case when $\theta=\pi/2$, the only difference is that $Q_{E_1}$ can be degenerated to an Euler configuration on the $x$-axis. We show that the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ is either one part of the Schubart orbit (Fig. \[twoorbits\] (a)) or one part of the Broucke-Hénon orbit (Fig. \[twoorbits\] (b)).
Let $\theta=\pi/2$ and $m>0$. The action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ coincide with either the Schubart orbit or the Broucke-Hénon orbit.
By Theorem \[geometricthm\], the minimizing path $(\hat{Z}_1, \, \hat{Z}_2)$ must satisfy one of the three cases. Similar to the arguments in Lemma \[nocollision2an3at0\] and Lemma \[nocollision1an2at0\], case (a) in Theorem \[geometricthm\] implies that the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ is collision-free.
Next, we consider cases (b) and $($c) of Theorem \[geometricthm\]. Note that by the definition of $Q_{E_1}$ in , it can be degenerated to an Euler configuration with body 1 at the origin. If case ($c$) happens, it implies that $(\hat{Z}_1, \, \hat{Z}_2)$ coincides with the Euler solution and $Z_2 \equiv 0$ for all $t \in [0,1]$. However, by the definition of $Q_{S_1}$ in , it implies that $q_1(0)=q_2(0)=q_3(0)=0$, which is a total collision. Contradiction to Lemma \[nototalretro\]! Therefore, the minimizing path $(\hat{Z}_1, \, \hat{Z}_2)$ must stay on the $x$-axis for all $t \in [0,\, 1]$.
Next, we show that $q_2(0)\not=q_3(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$. If not, note that $\displaystyle \lim_{t \to 0^{+}} \frac{q_2-q_3}{|q_2-q_3|}=(1, \, 0)$ and $\widetilde{\mathcal{P}_{m, \, \theta}}$ is collision free in $(0,\, 1]$, it implies that the three bodies should keep the order at $t=1$: $$\label{orderat1}
q_{1x}(1) < q_{3x}(1) < q_{2x}(1), \quad q_{1y}(1) = q_{3y}(1) = q_{2y}(1)=0.$$ However, by the definition of $Q_{E_1}$, when it becomes a straight line on the $x$-axis, the order of the three bodies can only be $$q_{2x}(1) < q_{1x}(1) < q_{3x}(1), \quad \text{or} \quad q_{3x}(1) < q_{1x}(1) < q_{2x}(1).$$ Contradict to ! Hence, there is no binary collision between bodies 2 and 3 in $\widetilde{\mathcal{P}_{m, \, \theta}}$.
If at $t=0$, $q_1(0)=q_2(0)$ in $\widetilde{\mathcal{P}_{m, \, \theta}}$, i.e. bodies 1 and 2 collide at $t=0$. Since $(\hat{Z}_1, \, \hat{Z}_2)$ stays on the $x$-axis for all $t \in [0,\, 1]$, it implies that $$\dot{q}_{1y}(0)=\dot{q}_{2y}(0)= \dot{q}_{3y}(0)= 0.$$ By Lemma 6.2 in [@Yu3], the motion must be collinear and it must be part of the Schubart orbit.
Therefore, for any given $m>0$ and $\theta=\pi/2$, the action minimizer $\widetilde{\mathcal{P}_{m, \, \theta}}$ coincides with either the Schubart orbit or the Broucke-Hénon orbit. The proof is complete.
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author:
- Matty van Son
bibliography:
- 'biblio\_ALL.bib'
title: ' [Uniqueness conjecture for extended Markov numbers]{} '
---
\[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{}
\[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Conjecture]{}
\[theorem\][Remark]{}
.2 in
Introduction {#introduction .unnumbered}
============
Triples of *regular Markov numbers* are the solutions to the Diophantine equation $$x^2+y^2+z^2=3xyz.$$ These *Markov triples* are the subject of the uniqueness conjecture of regular Markov numbers, introduced by Frobenius in 1913 [@frobenius1913].
Markov triples are uniquely defined by their largest element.
Both $(1,5,2)$ and $(5,29,2)$ are Markov triples. By the theory of Markov numbers the number 5 appears in infinitely many Markov triples. The uniqueness conjecture states that the only Markov triple in which 5 is the largest element is $(1,5,2)$.
This conjecture is well studied, and shows up in many interesting areas. We refer to the book by Aigner [@aigner2015] for a general reference. Although the conjecture is not proven, some cases are known, see for example [@baragar1996; @button2001].
In this note we extend the uniqueness conjecture for graphs of general Markov numbers, and show that for certain graphs the extended uniqueness conjecture fails (Theorem \[theorem: uniq break\]). To define these graphs we first show how regular Markov numbers may be obtained from a graph of sequences.
Generalised uniqueness conjecture
=================================
The author and O. Karpenkov [@karpenkov2018] showed an extension for regular Markov numbers. We define this extension Subsection \[subsection: general Markov numbers\] and introduce the generalised uniqueness conjecture. We develop the first counterexamples to the conjecture in Subsection \[subsection: counterexamples\].
Development of general Markov numbers {#subsection: general Markov numbers}
-------------------------------------
We start with definitions of continued fractions.
Let $\alpha=\big(a_i\big)_{i=1}^n$ and $\beta=\big(b_i\big)_{i=1}^m$ be finite sequences of positive integers. The *concatenation* of $\alpha$ and $\beta$ is $\alpha\oplus\beta=(a_1,\ldots,a_n,b_1,\ldots,b_m)$. We often shorten the notation $\alpha\oplus\beta$ to $\alpha\beta$.
The *continued fraction expansion of $\alpha$* is $$a_1+\cfrac{1}{\ddots+\cfrac{1}{a_n}}.$$ and is denoted by $[a_1;a_2:\ldots:a_n]$.
Next we define an important notion in the study of Markov numbers and sequences.
For a sequence of positive integers $(a_1,\ldots,a_n)$ let $c$ and $d$ be the integers with $\gcd(c,d)=1$ such that $$[a_1;a_2:\ldots:a_{n-1}]=\frac{c}{d}.$$ Define the *integer sine* of $\alpha$ to be $c$. We use the notation $\breve{K}(\alpha)=c$.
The term *integer sine* comes from the study of integer geometry. We recommend the book on this topic by Karpenkov [@karpenkov2013] for interested readers.
To calculate the integer sine of a sequence one may evaluate the continued fraction, or a polynomial of elements of the sequence called the *continuant*. For an explanation of continuants see the book by Graham, Knuth, and Patashnik [@graham1990].
We define a graph structure that is used to study Markov numbers.
Define operations $\mathcal{L}$ and $\mathcal{R}$ on triples of finite sequences of positive integers by $$\begin{aligned}
\mathcal{L}(\alpha,\gamma,\beta)&=(\alpha,\alpha\gamma,\gamma),\\
\mathcal{R}(\alpha,\gamma,\beta)&=(\gamma,\gamma\beta,\beta).
\end{aligned}$$ Define a binary graph ${\mathcal{G}}(\alpha,\beta)$ with root $(\alpha,\alpha\beta,\beta)$, where two vertices $v$ and $w$ are connected by an edge $(v,w)$ if $$w=\mathcal{L}(v)\quad \mbox{or} \quad w=\mathcal{R}(v).$$
We define operations $\chi$ and $X$ to obtain a triple graph of positive integers from a graph of triple sequences.
Let $\chi$ be the map acting on triples of sequences by $$\chi(\alpha,\gamma,\beta)=\big(\breve{K}(\alpha),\breve{K}(\gamma),\breve{K}(\beta)\big).$$ Define a map $X$ taking a triple graph of sequences ${\mathcal{G}}(\alpha,\beta)$ to a triple graph of integers, where vertices $v$ are mapped to $\chi(v)$, and edges $(v,w)$ are mapped to $\big(\chi(v),\chi(w)\big)$. We call the graph $${\mathcal{T}}\big((1,1),(2,2)\big)=X\Big({\mathcal{G}}\big((1,1),(2,2)\big)\Big)$$ the *graph of regular Markov numbers*. The triples appearing as vertices in this graph are the solutions to the Diophantine equation $$x^2+y^2+z^2=3xyz.$$
A more complete treatment of the relation between Markov numbers and triple graphs of positive integers, and also triple graphs of ${\operatorname{SL}(2,{\mathbb{Z}})}$ matrices and binary quadratic forms, may be found in the papers [@karpenkov2018; @vanson2018].
Let us collect some known results about the graph of regular Markov numbers.
- - Every triple at a vertex of the graph ${\mathcal{T}}\big((1,1),(2,2)\big)$ is a Markov triple.
- The graph contains every possible Markov triple.
- The vertices $v=(a_1,M,a_2)$ with $a_1\leq M$ and $a_2\leq M$, and $w=(b_1,Q,b_2)$ with $b_1\leq Q$ and $b_2\leq Q$ are connected by an edge $(v,w)$ if and only if either $$w=(a_1,3Ma_1-a_2,M)\quad \mbox{or} \quad w=(M,3Ma_2-a_1,a_2).$$
- The Markov graph is a tree (no loops or double edges).
This proposition is a collection of classical results in the theory of regular Markov numbers. One may find a proof of each statement in the books by Cusick [@cusick1989] or Aigner [@aigner2015].
One may define triple graphs of integers in the same way as Markov numbers but with different sequences. In this note we consider the graphs ${\mathcal{T}}\big((a,a),(b,b)\big)$ where $a$ and $b$ are positive integers and $a<b$. We call this a *graph of general Markov numbers*. We have the analogue of the uniqueness conjecture for regular Markov numbers.
Let $a$ and $b$ be positive integers with $a<b$. Then each triple of integers at a vertex of the graph of general Markov numbers ${\mathcal{T}}\big((a,a),(b,b)\big)$ is uniquely defined by it’s largest element.
First counterexamples to the general uniqueness conjecture {#subsection: counterexamples}
----------------------------------------------------------
We define certain graphs of general Markov numbers for which this conjecture is false.
For any positive integer $n$ define positive integers $a_n$ and $b_n$ by $$\begin{aligned}
a_n&=n^2+3,\\
b_n&=n^4+5n^2+5.
\end{aligned}$$ Note that $\gcd(a_n,b_n)=1$, and that the ratio $b_n/a_n$ is equal to the continued fraction $$\frac{b_n}{a_n}=\left[ 1+n^2 ; 1 : 2+n^2 \right]$$
Define the sequences $S_n(0)$ and $S_n(1)$ by $$S_n(0)=(na_n,na_n),\ \ S_n(1)=(nb_n,nb_n).$$ We notate the graphs given with these sequences by ${\mathcal{T}}_n=X\Big({\mathcal{G}}\big(S_n(0),S_n(1)\big)\Big)$.
We show the sequences $S_n(0)$ and $S_n(1)$ for $n=1,\ldots,5$, along with the continued fraction of $b_n/a_n$, in Table \[table: sn\].
$n$ $S_n(0)$ $S_n(1)$ $b_n/a_n$
----- ------------ -------------- ---------------
1 (4,4) (11,11) \[2; 1: 3\]
2 (14, 14) (82, 82) \[5; 1: 6\]
3 (36, 36) (393, 393) \[10; 1: 11\]
4 (76, 76) (1364, 1364) \[17; 1: 18\]
5 (140, 140) (3775, 3775) \[26; 1: 27\]
: Sequences $S_n(0)$ and $S_n(1)$ for $n=1,\ldots,5$.[]{data-label="table: sn"}
We present the main result.
\[theorem: uniq break\] The uniqueness conjecture for general Markov numbers does not hold for any graph ${\mathcal{T}}_n$, where $n$ is a positive integer.
In the graph of Markov numbers ${\mathcal{T}}_n$ for any $n>0$ there are triples defined $$\begin{aligned}
&\Big( \breve{K}\big(S_n(0)\big),\quad \breve{K}\big( S_n(0)^{5j+1} S_n(1) \big) ,\quad \breve{K}\big(S_n(0)^{5j}S_n(1)\big) \Big),\\
&\Big( \breve{K}\big(S_n(0) S_n(1)^{3j}\big),\quad \breve{K}\big( S_n(0) S_n(1)^{3j+1} \big) ,\quad \breve{K}\big(S_n(1)\big) \Big),
\end{aligned}$$ for all $j\geq 1$. We show that the largest element of these triples are equal, and hence the uniqueness conjecture for general Markov numbers fails for the graphs ${\mathcal{T}}_n$. More specifically we have the following proposition.
\[proposition: uniq break\] For all positive integers $n$ and $j$ we have that $$\breve{K} \big( S_n(0)^{5j+1} S_n(1) \big) = \breve{K} \big( S_n(0) S_n(1)^{3j+1} \big).$$
The simplest examples are in the graph ${\mathcal{T}}_1$, which contains the triples $$\begin{aligned}
\Big( \breve{K}(4,4),\ &\breve{K}\big( (4,4)^6 \oplus (11,11) \big) ,\ \breve{K}\big((4,4)^5\oplus (11,11)\big) \Big),\\
\Big( \breve{K}\big((4,4)\oplus(11,11)^3 \big),\ &\breve{K}\big( (4,4)\oplus(11,11)^4\big) ,\ \breve{K}(11,11) \Big),
\end{aligned}$$ which, when evaluated, give $$\begin{aligned}
(4,\, &355318099,\, 19801199),\\
(2888956,\, &355318099,\, 11).
\end{aligned}$$
Proof of Theorem \[theorem: uniq break\]
----------------------------------------
Theorem \[theorem: uniq break\] follows from Proposition \[proposition: uniq break\]. To prove this proposition we first define sequences of positive integers $\big(L_n(j)\big)_{j>0}$ and $\big(R_n(j)\big)_{j>0}$ containing the values $$\breve{K} \big( S_n(0)^{5j+1} S_n(1) \big)\quad\mbox{and}\quad \breve{K} \big( S_n(0) S_n(1)^{3j+1} \big).$$ We show in Lemmas \[lemma: R is a subsequence of A\] and \[lemma: L is a subsequence of A\] that both $\big(L_n(j)\big)_{j>0}$ and $\big(R_n(j)\big)_{j>0}$ are subsequences of another sequence $\big(A_n(j)\big)_{j>0}$ for every $n>0$. Then we show that their elements align within $\big(A_n(j)\big)_{j>0}$ in such a way that Proposition \[proposition: uniq break\] holds.
Let $n$ be a positive integer and let $a_n=n^2+3$ and $b_n=n^4+5n^2+5$. Define $$\begin{aligned}
l_n&=(na_n)^2+2, \quad r_n=(nb_n)^2+2,\\
L_n(1)&=\breve{K}(na_n,na_n,nb_n,nb_n), \quad L_n(2)=\breve{K}(na_n,na_n,na_n,na_n,nb_n,nb_n),\\
R_n(1)&=\breve{K}(na_n,na_n,nb_n,nb_n), \quad R_n(2)=\breve{K}(na_n,na_n,nb_n,nb_n,nb_n,nb_n).
\end{aligned}$$ For $j>2$ define $$L_n(j)=l_nL_n(j-1)-L_n(j-2) \quad \mbox{and} \quad R_n(j)=r_nR_n(j-1)-R_n(j-2).$$
We relate the sequences $\big(L_n(j)\big)_{j>0}$ and $\big(R_n(j)\big)_{j>0}$ to the numbers in Theorem \[theorem: uniq break\].
The following statements are equivalent:
- For all positive integers $n$ and $i$ we have that $$\breve{K} \big( S_n(0)^{5i+1}\oplus S_n(1) \big) = \breve{K} \big( S_n(0)\oplus S_n(1)^{3i+1} \big).$$
- For all positive integers $n$ and $i$ we have that $$L_n(5i+1) = R_n(3i+1).$$
The proof of this proposition relies on the recurrence relation for general Markov numbers which may be found in [@karpenkov2018 Theorem 7.15].
The first $6$ elements of the sequences $\big(L_1(j)\big)_{j>0}$ and $\big(R_1(j)\big)_{j>0}$ are $$\begin{aligned}
\big(L_1(j)\big)_{j=1}^{6}&=(191, 3427, 61495, 1103483, 19801199, 355318099),\\
\big(R_1(j)\big)_{j=1}^{6}&=(191, 23489, 2888956, 355318099, 43701237221, 5374896860084).
\end{aligned}$$
Next we define the sequence $\big(A_n(j)\big)_{j>0}$.
Let $A_n(1)=1$ and $A_n(2)=n(n^2+4)$. Then for $j>2$ define $$A_n(j)=nA_n(j-1)+A_n(j-2).$$
We guessed the structure of this sequence from looking at the case for $n=1$, where $\big(A_1(j)\big)_{j>0}$ is the sequence A022095 in [@oeis].
\[lemma: R is a subsequence of A\] For all positive integers $n$ and $j$ we have that $$R_n(j)=A_n(10j).$$
For any $n>0$ we have $na_n=3n+n^3$, $b=n^5+5n^3+5n$, and also that $r_n=(nb_n)^2+2$. Hence $$R_n(1)=\breve{K}(na_n,na_n,nb_n,nb_n)=n^{11}+11n^9+44n^7+76n^5+51n^3+8n.$$ By computation we see that $A_n(10)=R_n(1)$. Also we see that $$\begin{aligned}
R_n(2)
=&{n}^{21}+21\,{n}^{19}+189\,{n}^{17}+951\,{n}^{15}+2926\,{n}^{13}+5655\,{n}^{11}+\\
&6787\,{n}^{9}+4818\,{n}^{7}+1827\,{n}^{5}+301\,{n}^{3}+13\,n
=A_n(20).
\end{aligned}$$ This serves as a base of induction.
Assume that $A_n(10j)=R_n(j)$ for all $j=1,\ldots,k{-}1$, for some $k>2$. Then $$\begin{aligned}
R_n(k)&=r_nR_n(k-1)-R_n(k-2)\\
&=r_nA_n\big(10(k-1)\big)-A_n\big(10(k-2)\big),\\
&=r_nA_n(10k-10)-A_n(10k-20),
\end{aligned}$$ with the first equality following definition, and the second equality following the induction hypothesis. Note that $$\begin{aligned}
A_n(10k-10)&=nA_n(10k-10-1)+A_n(10k-10-2)\\
&=(n^2+1)A_n(10k-10-2)+A_n(10k-10-3)\\
&\vdots\\
&=x_1A_n(10k-10-9)+x_2A_n(10k-10-10),
\end{aligned}$$ where $x_1$ and $x_2$ are given through direct computation (we used Maple software) by $$\begin{aligned}
x_1&={n}^{9}+8\,{n}^{7}+21\,{n}^{5}+20\,{n}^{3}+5\,n,\\
x_2&={n}^{8}+7\,{n}^{6}+15\,{n}^{4}+10\,{n}^{2}+1.
\end{aligned}$$ In the same manner we have $$\begin{aligned}
A_n(10k)&=nA_n(10k-1)+A_n(10k-2)\\
&=(n^2+1)A_n(10k-2)+A_n(10k-3)\\
&\vdots\\
&=y_1A_n(10k-19)+y_2A_n(10k-20),
\end{aligned}$$ where $y_1$ and $y_2$ are given through direct computation by $$\begin{aligned}
y_1=&{n}^{19}+18\,{n}^{17}+136\,{n}^{15}+560\,{n}^{13}+1365\,{n}^{11}+\\
&2002
\,{n}^{9}+1716\,{n}^{7}+792\,{n}^{5}+165\,{n}^{3}+10\,n
,\\
y_2=&{n}^{18}+17\,{n}^{16}+120\,{n}^{14}+455\,{n}^{12}+1001\,{n}^{10}+\\
&1287
\,{n}^{8}+924\,{n}^{6}+330\,{n}^{4}+45\,{n}^{2}+1
.
\end{aligned}$$ Using this information we have that $$\begin{aligned}
R_n(k)&=r_nA_n(10k-10)-A_n(10k-20)\\
&=r_n\big(x_1A_n(10k-10-9)+x_2A_n(10k-10-10)\big)-A_n(10k-20)\\
&=r_nx_1A_n(10k-19)+(r_nx_2-1)A_n(10k-20).
\end{aligned}$$ Through direct computation we see that $y_1=r_nx_1$ and $y_2=r_nx_2-1$. Hence we have that $$R_n(k)=A_n(10k),$$ so the induction holds and the proof is complete.
Lemma \[lemma: R is a subsequence of A\] says that $\big(R_n(j)\big)_{j>0}$ is a subsequence of $\big(A_n(j)\big)_{j>0}$. Now we show an analogous statement for $\big(L_n(j)\big)_{j>0}$.
\[lemma: L is a subsequence of A\] For all positive integers $n$ and $i$ we have that $$L_n(j)=A_n\big(10+6(j-1)\big).$$
We use induction. We have that $A_n(10)=L_n(1)$, as in the proof for Lemma \[lemma: R is a subsequence of A\] . We also have $l_n=(n^3+3n)^2{+}2$, and that $$\begin{aligned}
A_n(16)&={n}^{17}+17{n}^{15}+119{n}^{13}+441{n}^{11}+925{n}^{9}+1086{
n}^{7}+658{n}^{5}+169{n}^{3}+11n\\
&=L_n(2).
\end{aligned}$$ This is our base of induction. Assume that $A_n\big(10+6(j-1)\big)=A_n(6j+4)=L_n(j)$ for all $j=1,\ldots,k{-}1$, for some $k>2$. Then $$\begin{aligned}
L_n(k)&=l_nL_n(k-1)-L_n(k-2)\\
&=l_nA_n(6k-2)-A_n(6k-8),
\end{aligned}$$ with the first equality following definition, and the second equality following the induction hypothesis. In a similar way to the proof for $\big(R_n(j)\big)_{j>0}=\big(A_n(j)\big)_{j>0}$ we have that $$A_n(6k-2)=w_1A_n(6k-7)+w_2A_n(6k-8),$$ where $w_1=n^5+4n^3+3n$ and $w_2=n^4+3n^2+1$. Also we have $$A_n\big(10+6(k-1)\big)=A_n(6k+4)=z_1A_n(6k-7)+z_2A_n(6k-8),$$ where $$\begin{aligned}
z_1&={n}^{11}+10\,{n}^{9}+36\,{n}^{7}+56\,{n}^{5}+35\,{n}^{3}+6\,n,\\
z_2&={n}^{10}+9\,{n}^{8}+28\,{n}^{6}+35\,{n}^{4}+15\,{n}^{2}+1.
\end{aligned}$$ Now we have that $$\begin{aligned}
L_n(k)&=l_nA_n(6k-2)-A_n(6k-8),\\
&=l_nw_1A_n(6k-7)+(l_nw_2-1)A_n(6k-8),
\end{aligned}$$ and by direct computation we see that $z_1=l_nw_1$ and $z_2=l_nw_2-1$.
So $L_n(k)=A_n\big(10+6(k-1)\big)$ and induction holds. This completes the proof.
We prove Proposition \[proposition: uniq break\].
Given Lemmas \[lemma: R is a subsequence of A\] and \[lemma: L is a subsequence of A\] we need only show the values $L_n(5j+1)$ and $R_n(3j+1)$ align within the sequence $\big(A_n(j)\big)_{j>0}$. Indeed we have that $$\begin{aligned}
L_n(5j+1)&=A_n\big(10+6(5j+1-1)\big)=A_n(30j+10),\\
R_n(3j+1)&=A_n\big(10(3j+1)\big)=A_n(30j+10).
\end{aligned}$$ Hence the claim is proved.
Theorem \[theorem: uniq break\] follows as a corollary.
|
---
abstract: |
We present a search for associated production of Higgs and $W$ bosons in $p\bar{p}$ collisions at a center of mass energy of $\sqrt{s}=1.96$ TeV in 5.3 fb$^{-1}$ of integrated luminosity recorded by the D0 experiment. Multivariate analysis techniques are applied to events containing one lepton, an imbalance in transverse energy, and one or two $b$-tagged jets to discriminate a potential $WH$ signal from standard model backgrounds. We observe good agreement between data and background, and set an upper limit of [4.5]{} (at 95% confidence level and for $m_H=115$ GeV) on the ratio of the $WH$ cross section multiplied by the branching fraction of $H \rightarrow b \bar{b}$ to its standard model prediction. A limit of [4.8]{} is expected from simulation.\
date: 'December 3, 2010'
title: ' Search for $\bm{WH}$ associated production in 5.3 fb$\bm{^{-1}}$ of $\bm{p\bar{p}}$ collisions at the Fermilab Tevatron\'
---
author\_list.tex
The only unobserved particle of the standard model (SM) is the Higgs boson ($H$) which emerges from the spontaneous breaking of electroweak symmetry. Its observation would support the hypothesis that the Higgs mechanism generates the masses of the weak gauge bosons and accommodates finite masses of fermions through their Yukawa couplings to the Higgs field. The mass of the Higgs boson ($m_H$) is not predicted by the SM, but the combination of direct searches at the CERN $e^+e^-$ Collider (LEP) [@sm-lep] and precision measurements of other electroweak parameters constrain $m_H$ to $114.4<m_H<185$ GeV at the 95% CL [@elweak]. While the region $158 < m_{H} < 175$ GeV has been excluded at the 95% CL by a combination of searches at CDF and D0 [@ww-cdf; @ww-dzero; @ww-combo; @tev-combo], the remaining mass range continues to be probed at the Fermilab Tevatron Collider. The associated production of a Higgs boson and a leptonically-decaying $W$ boson is among the cleanest Higgs boson search channels at the Tevatron, and provides the largest event yield for the decay $H \rightarrow b \bar{b}$ in the range $m_H < 135$ GeV. Several searches for $WH$ production at a $p\bar{p}$ center-of-mass energy of $\sqrt{s}=1.96$ TeV have been published. Three of these [@hep-ex/0410062; @wh-plb; @wh-prl] use subsamples (0.17 fb$^{-1}$, 0.44 fb$^{-1}$, and 1.1 fb$^{-1}$) of the data analyzed in this paper, while three from the CDF collaboration are based on cumulative samples (0.32 fb$^{-1}$, 0.95 fb$^{-1}$ and 2.7 fb$^{-1}$) of integrated luminosity [@CDF-wh; @CDF-wh-1fb; @CDF-wh-2.7fb].
We present a new search using an improved multivariate technique based on data collected with the D0 detector, corresponding to an integrated luminosity of $5.3$ fb$^{-1}$. The search selects events with one charged lepton ($\ell$ = electron, $e$, or muon, $\mu$), an imbalance in transverse energy () that arises from the unobserved neutrino in the $W\to\ell\nu$ decay, and either two or three jets, with one or two of these selected as candidate $b$-quark jets ($b$-tagged).
The channels are separated into independent categories based on the number of $b$-tagged jets in an event (one or two). Single $b$-tagged events contain three important sources of backgrounds: (i) multijet events, where a jet is misidentified as an isolated lepton, (ii) $W$ boson production in association with $c$-quark or light-quark jets, and (iii) $W$ boson production in association with two heavy-flavor ($b\bar{b},c\bar{c}$) jets. In events with two $b$-tagged jets, the dominant backgrounds are from $Wb\bar{b}$, $t\bar{t}$, and single top-quark production.
The analysis relies on the following components of the D0 detector [@run2det]: (i) a central-tracking system, which consists of a silicon microstrip tracker (SMT) and a central fiber tracker (CFT), both located within a 2 T superconducting solenoidal magnet; (ii) a liquid-argon/uranium calorimeter containing electromagnetic, fine hadronic, and coarse hadronic layers, segmented into a central section (CC), covering pseudorapidity $|\eta|<1.1$ relative to the center of the detector [@defs], and two end calorimeters (EC) extending coverage to $|\eta|\approx 4.0$, all housed in separate cryostats [@run1det], with scintillators between the CC and EC cryostats providing sampling of developing showers for $1.1<|\eta|<1.4$; (iii) a muon system located beyond the calorimetry consisting of layers of tracking detectors and scintillation trigger counters, one before and two after the 1.8 T iron toroids. A 2006 upgrade of the D0 detector added an inner layer of silicon [@layer0] to the SMT and an improved calorimeter trigger [@l1cal2b]. The integrated luminosity is measured using plastic scintillator arrays located in front of the EC cryostats at $2.7 < |\eta| < 4.4$. The trigger and data acquisition systems are designed to accommodate high instantaneous luminosities.
Events in the electron channel are triggered by a logical OR of several triggers that require an electromagnetic (EM) object or an EM object in conjunction with a jet. Trigger efficiencies are taken into account in the Monte Carlo (MC) simulation through a weighting of events based on an efficiency derived from data, and parametrized as a function of electron $\eta$ and azimuth $\phi$, and jet transverse momentum $p_T$.
We accept events for the muon channel from a mixture of single high-$p_T$ muon, jet, and muon plus jet triggers, and expect this inclusive trigger to be fully efficient for our selection criteria. We verify this by comparing events that pass a well-modeled subset of high-$p_T$ muon triggers to those that are selected by the inclusive set of triggers. Good agreement is observed between data and MC for this high-$p_T$ muon subset of triggers. Events not selected by a high-$p_T$ muon trigger tend to be selected by a jet trigger. The efficiency of this complementary set of triggers is modeled as a function of the scalar sum of jet $p_T$ in an event ($H_T$). This model provides a gain in efficiency relative to the high-$p_T$ muon triggers, and produces good agreement between data and MC for the combination of all triggers following its application to the simulation.
The [@pythia] MC generator is used to simulate production of dibosons with inclusive decays ($WW$, $WZ$, and $ZZ$), $WH {\rightarrow}l \nu b \bar{b}$ and $ZH {\rightarrow}l l b \bar{b}$ ($l = e$, $\mu$, or $\tau$). The contribution from $ZH$ events in which one lepton is not identified to the total signal yield is approximately 5%. Background from $W/Z$ ($V$)+jets and $t\bar{t}$ events is generated with [@ALPGEN] interfaced to for parton showering and hadronization. The samples are produced using the MLM parton-jet matching prescription [@ALPGEN]. The $V$+jets samples are divided into $V$+light jets and $V$+heavy-flavor jets. The $V$+light jets samples include $Vjj$, $Vbj$, and $Vcj$ processes, where $j$ is a light-flavor ($u,d,s$ quarks or gluons) jet, while the $V$+heavy-flavor samples for $Vb\bar{b}$ and $Vc\bar{c}$ are generated separately. Production of single top-quark events is generated using [@COMPHEP; @COMPHEP2], with used for parton evolution and hadronization. Simulation of both background and signal processes relies on the CTEQ6L1 [@CTEQ] leading-order parton distribution functions for all MC events. These events are processed through a full D0 detector simulation based on [@GEANT] using the same reconstruction software as used for D0 data. Events from randomly chosen beam crossings are overlaid on the simulated events to reproduce the effect of multiple $p\bar{p}$ interactions and detector noise.
The simulated background processes are normalized to their predicted SM cross sections, except for $W$+jets events, which are normalized to data before applying $b$-tagging, where contamination from the $WH$ signal is expected to be negligible. The signal cross sections and branching fractions are calculated at next-to-next-to-leading order (NNLO) and are taken from Refs. [@signal1; @signal2; @signal3; @signal4; @signal5], while the [$t\overline{t}$]{} single $t$, and diboson cross sections are at next-to-leading order (NLO), and taken from Ref. [@ttbar_xsecs], Ref. [@stop_xsecs], and the program [@mcfm], respectively. As a cross check, we compare data with NLO predictions for $W$+jets based on , and find a relative data/MC normalization factor of $1.0 \pm 0.1$, where the normalization for data is obtained after subtracting all other expected background processes. The normalizations of the $Vb\bar{b}$ and $Vc\bar{c}$ yields in MC relative to data are consistent with the ratio of LO/NLO cross sections predicted by . Therefore we apply these ratios to the corresponding $W$+heavy-flavor and $Z$+heavy-flavor jet processes.
This analysis is based on a preselection of events with an electron of $p_{T}>15$ GeV, with $|\eta|<$ 1.1 or $1.5 < |\eta| < 2.5$, or a muon of $p_{T}>15$ GeV, with $| \eta| < 1.6$. Preselected events are also required to have $>20$ GeV, either two or three jets with $p_{T}>20$ GeV (after correction of the jet energy [@jes]) and $|\eta|<2.5$, and $H_T > 60$ GeV for 2-jet events, or $H_T > 80$ GeV for 3-jet events. The is calculated from the individual calorimeter cells in the EM and fine hadronic layers of the calorimeter, and is corrected for the presence of muons. All energy corrections to electrons and jets (including energy from the coarse hadronic layers associated with jets) are propagated into the . To suppress multijet background, events with GeV are removed, where $M_W^T$ is the transverse mass of the $W$ boson candidate. Events with additional charged leptons isolated from jets that pass the flavor-dependent $p_T$ thresholds $p_{T}^e>15$ GeV, $p_T^{\mu}>10$ GeV, and $p_T^{\tau}>10$ or 15 GeV, depending on $\tau$ decay channel [@tauid], are rejected to decrease dilepton background from $Z$ boson and $t\bar{t}$ events. Events must have a reconstructed [$p\overline{p}$]{} interaction vertex (containing at least three associated tracks) that is located within $\pm40$ cm of the center of the detector in the longitudinal direction.
Lepton candidates are identified in two steps. In the first step, each candidate must pass “loose” identification criteria. For electrons, we require 95% of the energy in a shower to be deposited in the EM section of the calorimeter, isolation from other calorimeter energy deposits, spatial distributions of calorimeter energies consistent with those expected for EM showers, and a reconstructed track matched to the EM shower that is isolated from other tracks. For the “loose” muon, we require hits in each layer of the muon system, scintillator hits in time with a beam crossing (to veto cosmic rays), a spatial match with a track in the central tracker, and isolation from jets within $\Delta\mathcal{R} < 0.5$ [@defs] to reject semileptonic decays of hadrons. In the second step, the loose leptons are subjected to a more restrictive “tight” selection. Tight electrons must satisfy more restrictive calorimeter isolation fractions and EM energy-fraction criteria, and satisfy a likelihood test developed on $Z\to ee$ data based on eight quantities characterizing the EM nature of the particle interactions [@ttbar-prd]. Tight muons must satisfy stricter isolation criteria on energy in the calorimeter and momenta of tracks near the trajectory of the muon candidate. Inefficiencies introduced by lepton-identification and isolation criteria are determined from $Z\to\ell\ell$ data. The final selections rely only on events with tight leptons, with events containing only loose leptons used to determine the multijet background. Jets are reconstructed using a midpoint cone algorithm [@blazey] with radius 0.5. Identification requirements for jets are based on longitudinal and transverse shower profiles, and minimize the possibility that the jets are caused by noise or spurious depositions of energy. For data taken after 2006, and in the corresponding simulation, jets must have at least two associated tracks emanating from the reconstructed [$p\overline{p}$]{} interaction vertex. Any difference in efficiency for jet identification between data and simulation is corrected by adjusting the jet energy and resolution in simulation to match those measured in data. Comparison of with other generators and with data shows small discrepancies in distributions of jet pseudorapidity and dijet angular separations [@alw]. The data are therefore used to correct the $W$+jets and $Z$+jets MC events through polynomial reweighting functions parameterized by the leading and second-leading jet $\eta$, and $\Delta \mathcal{R}$ between the two leading jets, that bring these distributions for the total simulated background and the high statistics sample of events prior to $b$-tagging into agreement.
Instrumental background and that from semileptonic decays of hadrons, referred to jointly as the multijet background, are estimated from data. The instrumental background is significant in the electron channel, where a jet with a high EM fraction can pass electron-identification criteria, or a photon can be misidentified as an electron. In the muon channel, the multijet background is less important and arises mainly from semi-leptonic decay of heavy-flavor quarks, where the muon passes isolation criteria.
To estimate the number of events that contain a jet that passes the “tight” lepton selection, we determine the probability $f_{T|L}$ for a “loose” lepton candidate, originating from a jet, to also pass tight identification. This is done in events that pass preselection requirements without applying the selection on $M_W^T$, i.e., events that contain one loose lepton and two jets, but small ($5-15$ GeV). The total non-multijet background is estimated from MC and subtracted from the data before estimating the contribution from multijet events. For electrons, $f_{T|L}$ is determined as a function of electron $p_T$ in three regions of $|\eta|$ and four of $\Delta\phi($$,e)$, while for muons it is taken as a function of $|\eta|$ for two regions of $\Delta\phi($$,\mu)$. The efficiency for a loose lepton to pass the tight identification ($\varepsilon_{T|L}$) is measured in $Z\to\ell\ell$ events in data, and is modeled as a function of $p_T$ for electrons and muons. The estimation of multijet background described in Ref. [@ttbar-prd] is used to determine the multijet background directly from data, where each event is assigned a weight that contributes to the multijet estimation based on $f_{T|L}$ and $\varepsilon_{T|L}$ as a function of event kinematics. Since $f_{T|L}$ depends on , the scale of this estimate of the multijet background must be adjusted when comparing to data with $ > 20$ GeV. Before applying $b$-tagging, we perform a fit to the $M_W^T$ distribution to set the scales for the multijet and $W$+jets backgrounds simultaneously.
Efficient identification of $b$ jets is central to the search for $WH$ production. The D0 neural network (NN) $b$-tagging algorithm [@NNcert] for identifying heavy-flavored jets is based on a combination of seven variables sensitive to the presence of tracks or secondary vertices displaced significantly from the primary vertex. All tagging efficiencies are determined separately for data and for simulated events. We first use a low threshold on the NN output that corresponds to a misidentification rate of 2.7% for light-flavor jets of $p_T \geq 50$ GeV that are mistakenly tagged as heavy-flavored jets. If two jets in an event pass this $b$-tagging requirement, the event is classified as double-$b$-tagged (DT). Events that are not classified as DT are considered for placement in an independent single-$b$-tag (ST) sample, which requires exactly one jet to satisfy a more restrictive NN operating point corresponding to a misidentification rate of 0.9%. The efficiencies for identifying a jet that contains a $b$ hadron for the two NN operating points are $(63\pm 1)$% and $(53\pm 1)$%, respectively, for a jet with a $p_T$ of 50 GeV. These efficiencies are determined for “taggable” jets, i.e., jets with at least two tracks, each with at least one hit in the SMT. Simulated events are corrected to have the same fraction of jets satisfying the taggability and $b$-tagging requirements as found in preselected data.
The expected event yields following these selection criteria for specific backgrounds and for $m_H=115$ GeV are compared to the observed number of events in Table \[tab:table3\]. Distributions in dijet invariant mass for the two jets of highest $p_T$, in 2-jet and 3-jet events are shown for the ST and DT samples in Fig. \[emu-two-tags\](a–d). The data are well-described by the sum of the simulated SM processes and multijet background. The contributions expected from a Higgs boson with $m_H=115$ GeV, multiplied by a factor of ten, are also shown for comparison.
\
------------- ------ ------- ----- ------ ------- ----- ------ ------- ----- ------ ------- -----
$WZ$ 153 $\pm$ 18 22.5 $\pm$ 3.3 33.9 $\pm$ 4.8 2.6 $\pm$ 1.1
$Wb\bar{b}$ 1601 $\pm$ 383 346 $\pm$ 93 358 $\pm$ 90 48 $\pm$ 13
$W+lf$ 1290 $\pm$ 201 57.5 $\pm$ 9.2 210 $\pm$ 35 12.1 $\pm$ 1.8
$t\bar{t}$ 417 $\pm$ 54 177 $\pm$ 35 633 $\pm$ 96 176 $\pm$ 35
Single $t$ 203 $\pm$ 33 58 $\pm$ 11 53.6 $\pm$ 9.1 13.0 $\pm$ 2.7
MJ 663 $\pm$ 43 56.5 $\pm$ 4.2 186 $\pm$ 13 12.7 $\pm$ 1.0
All Bkg. 4326 $\pm$ 501 718 $\pm$ 120 1474 $\pm$ 160 264 $\pm$ 44
$WH$ 9.7 $\pm$ 0.9 6.5 $\pm$ 1.0 2.1 $\pm$ 0.3 0.8 $\pm$ 0.2
Data 4316 709 1463 301
------------- ------ ------- ----- ------ ------- ----- ------ ------- ----- ------ ------- -----
: \[tab:table3\] [ Summary of event yields for the $\ell$ + $b$-tagged jets + final state. Event yields in data are compared with the expected number of ST and DT events in the samples with $W$ boson candidates plus two or three jets, comprised of contributions from simulated diboson pairs (labeled “$WZ$” in the table), $W/Z$+$b\bar{b}$ or $c\bar{c}$ (“$W b\bar{b}$”), $W/Z$+light-quark jets (“$W+lf$”), and top-quark (“$t\bar{t}$” and “Single $t$”) production, as well as data-derived multijet background (“MJ”). The quoted uncertainties include both statistical and systematic contributions, including correlations between background sources and channels. The expectation for $WH$ signal is given for $m_H=115$ GeV. ]{}
We use a random forest (RF) multivariate technique [@RF1; @RF2] to separate the SM background from signal, and search for an excess, which is expected primarily at large values of RF discriminant. A separate RF discriminant is used for each combination of jet multiplicity (two or three), lepton flavor ($e$ or $\mu$), and number of $b$-tagged jets (one or two). The 2-jet events are divided into data-taking periods, before and after the 2006 detector upgrade, for a total of twelve separately trained RFs for each chosen Higgs boson mass. Each RF consists of a collection of individual decision trees, with each tree considering a random subset of the twenty kinematic and topological input variables listed in Table \[tab:RFlist\]. The final RF output is the average over the individual trees. The input variables $\sqrt{\hat{s}}$ and $\Delta\mathcal{R}$(dijet,$\ell+\nu$) each have two solutions arising from the two possibilities for the neutrino $p_{z}$, assuming the lepton and ($\nu$) constitute the decay products of an on-shell $W$ boson. The angles $\theta^{*}$ and $\chi$ are described in Ref. [@spincorr], and exploit kinematic differences arising from the scalar nature of the Higgs and the spins of objects in the $Wb\bar{b}$ background. The RF outputs from 2-jet ST and DT events are shown in Fig. \[emu-two-tags\](e,f).
Variable Definition
------------------------------------------------------------ -------------------------------------------------
$p_{T}$($j_1$) Leading jet $p_{T}$
$p_{T}$($j_2$) Sub-leading jet $p_{T}$
$E(j_{2})$ Sub-leading jet energy
$\Delta\mathcal{R}$($j_1$,$j_2$) $\Delta\mathcal{R}$ between jets
$\Delta \phi$($j_1$,$j_2$) $\Delta \phi$ between jets
$\Delta \phi$($j_1$, $\ell$) $\Delta \phi$ between lepton and leading jet
$p_{T}$(dijet system) $p_{T}$ of dijet system
$m_{jj}$ Dijet invariant mass
$p_{T}$($\ell$- system) $p_{T}$ of $W$ candidate
Missing transverse energy
aplanarity See Ref. [@aplan]
$\sqrt{\hat{s}}$ Invariant mass of the $\nu$+$\ell$+dijet system
\[0ex\]\[-1.5ex\][$\Delta\mathcal{R}$(dijet,$\ell+\nu$)]{} $\Delta\mathcal{R}$ between the
dijet system and $\ell+\nu$ system
$M_W^T$ Lepton- transverse mass
\[0ex\]\[-1.5ex\][$H_{T}$]{} Scalar sum of the transverse momenta
of all jets in the event
\[0ex\]\[-1.5ex\][$H_{Z}$]{} Scalar sum of the longitudinal momenta
of all jets in the event
\[0ex\]\[-1.5ex\][$\cos \theta^*$]{} Cosine of angle between $W$ candidate
and beam direction in zero-momentum frame
$\cos \chi$ See Ref. [@ref:coschi]
: List of RF input variables, where $j_1$ ($j_2$) refers to the jet with the highest (second highest) $p_T$. \[tab:RFlist\]
The dijet mass distribution is especially sensitive to $WH$ production, and was used previously to set limits on $\sigma(p\bar{p}\to WH) \times \mathcal{B}(H \rightarrow b \bar{b})$ in Ref. [@wh-plb]. However, the gain in sensitivity using the RF output as the final discriminant is about 20% for a Higgs mass of 115 GeV, which, in terms of the expected limit on the $WH$ cross section, is equivalent to a gain of about 40% in integrated luminosity.
The systematic uncertainties that affect the signal and SM backgrounds can be categorized by the nature of their source, i.e., theoretical (e.g., uncertainty on a cross section), MC modeling (e.g., reweighting of samples), or experimental (e.g., uncertainty on integrated luminosity). Some of these uncertainties affect only the normalization of the signal or backgrounds, while others also affect the differential distribution of the RF output.
Theoretical uncertainties include uncertainties on the $t\bar{t}$ and single top-quark production cross sections (10% and 12%, respectively [@ttbar_xsecs; @stop_xsecs]), an uncertainty on the diboson production cross section (6% [@mcfm]), and an uncertainty on $W$+heavy-flavor production (20%, estimated from ). These uncertainties affect only the normalization of the backgrounds.
Uncertainties from modeling that affect the distribution in the RF output include uncertainties on trigger efficiency as derived from data (3–5%), lepton identification and reconstruction efficiency (5–6%), reweighting of MC samples (2%), the MLM matching applied to $W/Z$+light-jet events ($<0.5$%), and the systematic uncertainties associated with choice of renormalization and factorization scales in as well as the uncertainty on the strong coupling constant (2%). Uncertainties on the renormalization and factorization scales are evaluated by adjusting the nominal scale for each, simultaneously, by a factor of 0.5 and 2.0.
Experimental uncertainties that affect only the normalization of the signal and SM backgrounds arise from the uncertainty on integrated luminosity (6.1%) [@luminosity_ID]. Those that also affect the distribution in RF output include jet taggability (3%), $b$-tagging efficiency (2.5–3% per heavy quark-jet), the light-quark jet misidentification rate (10%), acceptance for jet identification (5%); jet-energy calibration and resolution (varies between 15% and 30%, depending on the process and channel). The background-subtracted data points for the RF discriminant for $m_H=115$ GeV, with all channels combined, are shown with their systematic uncertainties in Fig. \[syst\_plot\].
We observe no excess relative to expectation from SM background, and we set upper limits on the production cross section $\sigma(WH)$ using the RF outputs for the different channels. The binning of the RF output is adjusted to assure adequate population of background events in each bin. We calculate all limits at the 95% CL using a modified frequentist approach and a Poisson log-likelihood ratio as test statistic [@junkLim; @readLim]. The likelihood ratio is studied using pseudoexperiments based on randomly drawn Poisson trials of signal and background events. We treat systematic uncertainties as “nuisance parameters” constrained by their priors, and the best fits of these parameters to data are determined at each value of $m_H$ by maximizing the likelihood ratio [@collie]. Independent fits are performed to the background-only and signal-plus-background hypotheses. All appropriate correlations of systematic uncertainties are maintained among channels and between signal and background. The systematic uncertainties before and after fitting are indicated in Fig. \[syst\_plot\]. The log-likelihood ratios for the background-only model and the signal-plus-background model as a function of $m_H$ are shown in Fig. \[limits\_plot\](a).
The upper limit on the cross section for $\sigma( p\bar{p} \rightarrow WH) \times \mathcal{B}(H \rightarrow b \bar{b})$ at the 95% CL is a factor of [4.5]{} larger than the SM expectation for $m_H=115$ GeV. The corresponding upper limit expected from simulation is [4.8]{}. The analysis is repeated for ten other $m_H$ values from 100 to 150 GeV; the corresponding observed and expected 95% CL limits relative to their SM expectations are given in Table \[limits\] and in Fig. \[limits\_plot\](b).
$m_H$ \[GeV\] 100 105 110 115 120 125 130 135 140 145 150
--------------- ----- ----- ----- ----- ----- ----- ----- ------ ------ ------ ------
Exp. ratio 3.3 3.6 4.2 4.8 5.6 6.8 8.5 11.5 16.5 23.6 36.8
Obs. ratio 2.7 4.0 4.3 4.5 5.8 6.6 7.0 7.6 12.2 15.0 30.4
In conclusion, $\ell+$+2 or 3-jet events have been analyzed in a search for $WH$ production in 5.3 fb$^{-1}$ of [$p\overline{p}$]{} collisions at the Fermilab Tevatron. The yield of single and double $b$-tagged jets in these events is in agreement with the expected background. We have applied a Random Forest multivariate analysis technique to further separate signal and background. We have set upper limits on $\sigma(p\bar{p} \rightarrow WH) \times \mathcal{B}(H \rightarrow b \bar{b})$ relative to their SM expectation for Higgs masses between 100 and 150 GeV. For $m_H=115$ GeV, the observed (expected) 95% CL limit is a factor of [4.5]{} ([4.8]{}) larger than the SM expectation.
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|
---
abstract: 'Recent work has constructed economic mechanisms that are both truthful and differentially private. In these mechanisms, privacy is treated separately from the truthfulness; it is not incorporated in players’ utility functions (and doing so has been shown to lead to non-truthfulness in some cases). In this work, we propose a new, general way of modelling privacy in players’ utility functions. Specifically, we only assume that if an outcome $o$ has the property that any report of player $i$ would have led to $o$ with approximately the same probability, then $o$ has small privacy cost to player $i$. We give three mechanisms that are truthful with respect to our modelling of privacy: for an election between two candidates, for a discrete version of the facility location problem, and for a general social choice problem with discrete utilities (via a VCG-like mechanism). As the number $n$ of players increases, the social welfare achieved by our mechanisms approaches optimal (as a fraction of $n$).'
author:
- 'Yiling Chen[^1]'
- 'Stephen Chong[^2]'
- 'Ian A. Kash[^3]'
- 'Tal Moran[^4]'
- 'Salil Vadhan[^5]'
bibliography:
- 'biblio.bib'
title: 'Truthful Mechanisms for Agents that Value Privacy[^6] '
---
**Keywords:** differential privacy, mechanism design, truthfulness, elections, VCG
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was inspired by discussions under the Harvard Center Research for Computation and Society’s “Data Marketplace” project. We are grateful to the other participants in those meetings, including Scott Kominers, David Parkes, Felix Fischer, Ariel Procaccia, Aaron Roth, Latanya Sweeney, and Jon Ullman. We also thank Moshe Babaioff and Dave Xiao for helpful discussions and comments.
[^1]: Center for Research on Computation and Society and School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA. E-mail: `[email protected]`.
[^2]: Center for Research on Computation and Society and School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA. E-mail: `[email protected]`. Supported by NSF Grant No. 1054172.
[^3]: Microsoft Research Cambridge, 7 J J Thomson Ave, Cambridge CB3 0FB, UK. E-mail: `[email protected]`.
[^4]: Efi Arazi School of Computer Science, IDC Herzliya. Email: `[email protected]`.
[^5]: Center for Research on Computation and Society and School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA. E-mail: `[email protected]`
[^6]: Work begun when all the authors were at the Harvard Center for Computation and Society, supported in part by a gift from Google, Inc. and by NSF Grant CCF-0915016.
|
`NORDITA 2020-022`
**Quiver CFT at strong coupling**
*Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden*\
*Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark*\
`[email protected]`
**Abstract**
The circular Wilson loop in the two-node quiver CFT is computed at large-$N$ and strong ’t Hooft coupling by solving the localization matrix model.
Introduction
============
An $SU(N_c)$ gauge theory with $N_f=2N_c$ fundamental hypermultiplets, often called super-QCD, is perhaps the simplest $\mathcal{N}=2$ superconformal theory. Since its conformal anomaly does not satisfy $a=c$, a putative holographic dual must always remain stringy, no matter how large the ’t Hooft coupling is [@Gadde:2009dj], in contradistinction, for instance, to $\mathcal{N}=4$ super-Yang-Mills (SYM). In spite of this striking difference, SQCD and SYM are connected by a family of superconformal theories, all having weakly-coupled duals. It would be interesting to understand how the string description breaks down or becomes strongly-coupled at the SQCD point.
![\[qui\]Two-node quiver.](Quiver.jpg){width="5cm"}
The interpolating theory is obtained by gauging the flavor group of SQCD. The result is an $SU(N)\times SU(N)$ quiver with bi-fundamental matter and two independent couplings (fig. \[qui\]). Once flavor gauge fields decouple at $\lambda _2=0$, the quiver becomes equivalent to SQCD augmented with a free vector multiplet that restores $a=c$. For equal couplings, the symmetry is enhanced by an extra $\mathbbm{Z}_2$. This is not accidental, as at $\lambda _1=\lambda _2$ the quiver is equivalent to the $\mathbbm{Z}_2$ orbifold of $\mathcal{N}=4$ SYM [@Lawrence:1998ja]. The orbifold and the parent SYM theory share the same planar diagrams [@Bershadsky:1998cb] and hence are equivalent at $N\rightarrow \infty $.
The holographic dual of the quiver is string theory on the $AdS_5\times (S^5/\mathbbm{Z}_2)$ orbifold [@Kachru:1998ys], where $\mathbbm{Z}_2$ acts by flipping the four coordinates of $S^5$ in the $\mathbbm{R}^6$ embedding, reflecting the 2+4 split of the $\mathcal{N}=4$ scalars between the vector and hypermultiplet of $\mathcal{N}=2$.
The vastly different strong-coupling behavior of SYM and SQCD manifests itself in the expectation value of the circular Wilson loop, which can be computed from first principles in both cases using localization [@Pestun:2007rz]. The SYM Wilson loop nicely exponentiates [@Erickson:2000af; @Drukker:2000rr]: $$\label{N=4WL}
W_{\rm SYM}=\frac{2}{\sqrt{\lambda }}\,I_1\left(\sqrt{\lambda }\right)
\stackrel{\lambda \rightarrow \infty }{\simeq} \sqrt{\frac{2}{\pi }}\,\lambda ^{-\frac{3}{4}}\,{\rm e}\,^{\sqrt{\lambda }},$$ in agreement with the minimal area law in $AdS_5$. Indeed, the regularized area of the circle is $-2\pi $ [@Berenstein:1998ij; @Drukker:1999zq], the string tension is $$\label{String-Ten}
T=\frac{\sqrt{\lambda }}{2\pi }\,.$$ Together they give $\sqrt{\lambda }$ in the exponent.
The Wilson loop in the quiver CFT also exponentiates, in terms of the effective coupling [@Rey:2010ry]: $$\label{eff-coupling}
\frac{2}{\lambda }=\frac{1}{\lambda _1}+\frac{1}{\lambda _2}\,,$$ in accord with expectations from AdS/CFT, as exactly the same coupling controls the string tension [@Lawrence:1998ja; @Klebanov:1999rd; @Gadde:2010zi], while the minimal surface is unaffected by the orbifold projection. The notion of effective coupling actually applies to a larger class of $\mathcal{N}=2$ superconformal theories and goes beyond the strong-coupling regime [@Mitev:2014yba; @Mitev:2015oty].
On the contrary, in SQCD the Wilson loop does not exponentiate (we denote the SQCD ’t Hooft coupling by $\lambda _1$, keeping in mind its embedding in the quiver) [@Passerini:2011fe]: $$\label{WSQCD}
W_{\rm SQCD}\stackrel{\lambda _1\rightarrow \infty }{\simeq} \,{\rm const}\,\frac{\lambda_1 ^3}{\left(\ln\lambda_1 \right)^{\frac{3}{2}}}\,.$$ Such a power+log behavior is hardly consistent with a semi-classical string interpretation.
To the leading order the Wilson loop only depends on the average of the inverse couplings. The difference does not show up in the exponent. In string theory, the difference defines a theta-angle on the worldsheet [@Lawrence:1998ja; @Klebanov:1999rd; @Gadde:2010zi]: $$\label{theta}
\theta =\pi -\pi \,\frac{\frac{1}{\lambda _1}-\frac{1}{\lambda _2}}{\frac{1}{\lambda _1}+\frac{1}{\lambda _2}}=\frac{2\pi \lambda _1}{\lambda _1+\lambda _2}\,.$$ Proper definition of the corresponding term in the string action requires resolution of the orbifold singularity. Supersymmetry-preserving resolution involves a non-contractable two-cycle collapsing to zero size when regularization is removed. The theta-term measures the wrapping number of the worldsheet around this non-contractable cycle. Interestingly, the symmetric point ($\lambda _1=\lambda _2$) corresponds to the $\pi$-flux ($\theta =\pi $) and not zero as one could possibly expect. The theta-term breaks CP such that interchanging the two gauge groups ($\lambda _1\leftrightarrow\lambda _2$) entails a parity transformation on the worldsheet: $\theta \rightarrow 2\pi -\theta $.
This wonderful picture calls for a quantitative test. A first-principles string calculation would be particularly interesting. This is not what we will do here. Instead we will explore the circular Wilson loop in the stringy regime, but by purely field-theoretic methods, namely by solving the localization matrix model [@Pestun:2007rz] to the first order in the strong-coupling expansion, expending the results in [@Rey:2010ry] beyond the leading exponential. The leading order does not carry any theta-dependence and the Wilson loop expectation value is essentially the same as in SYM. The “one-loop” correction we are going to compute can serve as a testbed for string theory on the orbifold with the B-flux along with the spectral data known in quite a detail at any coupling [@Gadde:2009dj; @Gadde:2010zi; @Gadde:2010ku].
Localization
============
The field content of the $SU(N)\times SU(N)$ quiver consists of two vectors multiplets in the adjoint[^1]: $(A_{a\mu} ,\Phi _a,\Phi '_a)$, $a=1,2$, and bi-fundamental matter: $(X,Y,X^\dagger ,Y^\dagger )$: $D_\mu X=\partial _\mu X+A_{1\mu} X-XA_{2\mu} $. We will be interested in the Wilson loop expectation value $$W_a=\left\langle \frac{1}{N}\,{\rm P}\exp
\left[\oint_C ds\,\left(i\dot{x}^\mu A_{a\mu} +|\dot{x}|\Phi _a\right) \right]
\right\rangle,$$ for the circular contour $C$.
After the theory is placed on the four-sphere the problem reduces to a finite-dimensional matrix integral over zero modes of the vector-multiplet scalars. In the eigenvalue representation, $\Phi _a=\mathop{\mathrm{diag}}(a_{a1}\ldots a_{aN})$, the localization integral is [@Pestun:2007rz]: $$\label{localizationMM}
Z=\int_{}^{}\prod_{a=1}^{2}\prod_{i}^{}da_{ai}\,\,
\frac{\prod\limits_{a}^{}\prod\limits_{i<j}^{}(a_{ai}-a_{aj})^2H^2(a_{ai}-a_{aj})}{\prod_{ij}^{}H^2(a_{1i}-a_{2j})}\,
\,{\rm e}\,^{-\sum_{a}^{}\frac{8\pi ^2N}{\lambda _a}\sum_{i}^{}a_{ai}^2},$$ where $H(x)$ admits a product representation: $$H(x)=\prod_{n=1}^{\infty }\left(1+\frac{x^2}{n^2}\right)^n\,{\rm e}\,^{-\frac{x^2}{n}}.$$
The circular Wilson loops correspond to simple exponentials in the localization matrix model: $$W_a=\left\langle \frac{1}{N}\sum_{i}^{}\,{\rm e}\,^{2\pi a_{ai}}\right\rangle.$$ In contradistinction to $\mathcal{N}=4$ SYM, where the matrix model is Gaussian [@Erickson:2000af; @Drukker:2000rr], the quiver matrix integral is interacting even at the orbifold point $\lambda _1=\lambda _2$. This demonstrates very clearly that the orbifold equivalence is a dynamical phenomenon and only holds in the strict large-$N$ limit. Even at large-$N$ equivalence to the Gaussian model is not immediately obvious. It can be formally established by inspecting the large-$N$ saddle-point equations.
When written in terms of the the eigenvalue densities, $$\rho _a(x)=\left\langle \frac{1}{N}\,\sum_{i}^{}\delta (x-a_{ai})\right\rangle,$$ the saddle-point equations [@Brezin:1977sv] become $$\begin{aligned}
\strokedint_{-\mu _1}^{\mu _1}dy\,\rho _1(y)\left(\frac{1}{x-y}-K(x-y)\right)
+\int_{-\mu _2}^{\mu _2}dy\,\rho _2(y)K(x-y)&=&\frac{8\pi ^2}{\lambda _1}\,x
\label{inteq1}
\\
\strokedint_{-\mu _2}^{\mu _2}dy\,\rho _2(y)\left(\frac{1}{x-y}-K(x-y)\right)
+\int_{-\mu _1}^{\mu _1}dy\,\rho _1(y)K(x-y)&=&\frac{8\pi ^2}{\lambda _2}\,x,
\label{inteq2}\end{aligned}$$ where $$K(x)=-\frac{H'(x)}{H(x)}=x\left(\psi (1+ix)+\psi (1-ix)+2\gamma \right).$$ The Wilson loops are given by $$\label{wils-int}
W_a=\int_{-\mu _a}^{\mu _a}dx\,\rho_a (x)\,{\rm e}\,^{2\pi x}.$$ This setup has been used to study Wilson loops in SQCD and quiver CFT, mostly at weak coupling [@Fiol:2015mrp; @Mitev:2015oty; @Billo:2018oog; @Billo:2019fbi]. The leading-order strong-coupling solution of the saddle-point equations was obtained in [@Rey:2010ry]. We will extend it to the next order in $1/\sqrt{\lambda }$.
When $\lambda _1=\lambda _2=\lambda $, the equations are consistent with the symmetric ansatz $\rho _1=\rho _2$, for which the $K$-terms cancels and one is left with the saddle-point equation of the Gaussian matrix model whose solution is the Wigner semicircle: $$\rho (x)=\frac{2}{\pi \mu ^2}\,\sqrt{\mu ^2-x^2}$$ with $$\mu =\frac{\sqrt{\lambda }}{2\pi }\,.$$ This is how orbifold equivalence operates at large $N$.
As observed in [@Rey:2010ry] the semicircular distribution is a good approximation even for unequal $\lambda _1$, $\lambda _2$, provided that both couplings are large and comparable in magnitude. The argument goes as follows. The saddle-point equations reflect the balance of forces between eigenvalues. The $1/(x-y)$ repulsion smoothens the distribution on short scales but dies out at large distances. The external linear force confines the eigenvalues to a finite interval but at strong coupling is only operative at very large $x$. The bulk of the distribution is thus controlled by the two-body forces mediated by $K(x-y)$. The function $K(x)$ is overall positive and grows as $x\ln x$ at large $x$. As a result, the like eigenvalues attract, while the opposite eigenvalues repel with a force that grows with distance. To balance this force and prevent large terms appearing in the integral equations, the two eigenvalue distributions “lock” making the densities $\rho _{1,2}$ approximately equal. The locking cancels large terms with $K(x-y)$. The cancellation is only approximate in each of the equations (\[inteq1\]) and (\[inteq2\]), but an almost perfect cancellation occurs in their sum [@Rey:2010ry]. Thus $\rho _{1}\approx \rho _{2}$ implies that both densities are given by the Wigner distribution whose width is determined by the effective coupling (\[eff-coupling\]).
![\[densities-fig\]The eigenvalue densities $\rho _1$ (purple line) and $\rho _2$ (blue line) obtained by numerically solving (\[inteq1\]), (\[inteq2\]) for $\lambda _1=5320$, $\lambda _2=2797$. The dashed line is the Wigner distribution with the effective coupling $\lambda =3667$. The density for the gauge group with a larger coupling ($\rho _1$) tends to spread more because the restoring force is weaker, hence $\mu _1>\mu_2$, but in spite of considerable disparity in the coupling strength the difference between $\rho _1$ and $\rho _2$ is very small. This is the locking effect. The difference is most pronounced near the spectral edge.](Densities.jpg){width="8cm"}
This picture agrees very well with numerics (fig. \[densities-fig\]). The two densities are approximately the same and deviate from the Wigner distribution only near the spectral edge. But Wilson loops are controlled precisely by the edge, because of their exponential dependence on the eigenvalues. We thus need to know the edge behavior of the densities in detail.
Since $\mu _{1,2}$ are large the Wilson loop exponentiates at strong coupling, as in the SYM, but with a different prefactor determined by the structure of the eigenvalue density near the endpoint. Exactly the same behavior was found in the $\mathcal{N}=2^*$ theory [@Chen:2014vka], where the leading order solution is approximately Gaussian [@Buchel:2013id], while the first strong-coupling correction is determined by a fairly complicated boundary dynamics. We conjecture that these features are common to all $\mathcal{N}=2$ theories with weakly-coupled holographic duals. Wigner density in the bulk is accompanied by $\mathcal{O}(1)$ deviations at the edge. As in [@Chen:2014vka] we will solve the integral equations in two steps, first in the bulk and then at the boundary, matching the two solutions in their overlapping regime of validity.
Bulk
====
It does not make sense to plug $\rho _1=\rho _2\equiv \rho_{\rm Wigner} (x)$ back into the integral equations (\[inteq1\]), (\[inteq2\]). One gets a non-sensical result if $\lambda _1$, $\lambda _2$ are different. This is a rather disturbing feature of the leading-order solution that only relies on the sum of the two equations. To accommodate the difference, the solution needs to be refined.
Since $\mu _a\gg 1$, the kernels in the integral equations can be approximated by their large-distance asymptotics: $$K(x)\simeq x\ln x^2+2\gamma x+\frac{1}{6x}\equiv K^\infty (x).$$ The Wigner distribution and its cousins have simple convolution with the asymptotic kernel: $$\begin{aligned}
&&\int_{-\mu }^{\mu }dy\,\sqrt{\mu ^2-y^2}\,K^\infty (x-y)
=\frac{\pi }{3}\,x^3+ \left(\pi \mu ^2\ln\frac{\mu \,{\rm e}\,^{\gamma +\frac{1}{2}}}{2}+\frac{\pi }{6}\right)x
\nonumber \\
&&\int_{-\mu }^{\mu }dy\,\,\frac{K^\infty (x-y)}{\sqrt{\mu ^2-y^2}}=2\pi x\ln\frac{\mu \,{\rm e}\,^{\gamma +1}}{2}
\nonumber \\
&&
\int_{-\mu }^{\mu }dy\,\,
\frac{K^\infty (x-y)}{\left(\mu ^2-y^2\right)^{n+\frac{1}{2}}}
=-\frac{2^n(n-1)!\pi }{(2n-1)!!\mu ^{2n}}\,x,\qquad n=1,2,\ldots \end{aligned}$$ This observation suggests the following ansatz: $$\label{ansatz-bulk}
\rho _a(x)=A\sqrt{\mu_a ^2-x^2}+\frac{2\mu _aAB_a}{\sqrt{\mu_a ^2-x^2}}
+\frac{4\mu _a^2AC_a}{\left(\mu _a^2-x^2\right)^{\frac{3}{2}}}+\ldots$$ Each consecutive term adds an extra power of $1/\mu $, and hence of $1/ \sqrt{\lambda }$, so this ansatz naturally represents the strong-coupling expansion of the density. While $\mu _1=\mu _2$ at the leading order, due to the locking effect, the two endpoints split at higher orders. On the contrary, the overall normalization constant $A$ must remain the same to all orders in $1/\sqrt{\lambda }$, as will become clear shortly.
The asymptotic integral operators generate only cubic and linear terms in $x$ at each order in $1/\mu $. Moreover, the cubic terms only arise from the Wigner function. Cancellation of the cubic terms is precisely the condition that the overall constant $A$ is the same for the two densities. But the linear terms do not cancel automatically. Matching them gives two scalar equations: $$\begin{aligned}
\label{long-eq}
&&1-\mu _{1,2}^2\ln\frac{\mu _{1,2}\,{\rm e}\,^{\gamma +\frac{1}{2}}}{2}+\mu _{2,1}^2\ln\frac{\mu _{2,1}\,{\rm e}\,^{\gamma +\frac{1}{2}}}{2}
-4B_{1,2}\mu _{1,2}\ln\frac{\mu _{1,2}\,{\rm e}\,^{\gamma +1}}{2}
\nonumber \\ &&
+4B_{2,1}\mu _{2,1}\ln\frac{\mu _{2,1}\,{\rm e}\,^{\gamma +1}}{2}
+8C_{1,2}-8C_{2,1}=\frac{8\pi }{A\lambda _{1,2}}\,.\end{aligned}$$
The unit normalization of the densities gives another two conditions that can be used to eliminate $B_a$: $$\label{Ba}
B_a=\frac{1}{2\pi A\mu _a}-\frac{\mu _a}{4}\,.$$ When (\[Ba\]) is substituted in (\[long-eq\]) the latter considerably simplifies: $$1+\frac{\mu _{1,2}^2}{2}-\frac{\mu _{2,1}^2}{2}-\frac{2}{\pi A}\,\ln\frac{\mu _{1,2}}{\mu _{2,1}}+8C_{1,2}-8C_{2,1}=\frac{8\pi }{A\lambda _{1,2}}\,.$$ The sum of the two equations determines $A$: $$A=\frac{4\pi }{\lambda _1}+\frac{4\pi }{\lambda _2}=\frac{8\pi }{\lambda }\,,$$ while their difference gives: $$\label{mainconstraint}
\mu _1^2-\mu_2^2-\frac{\lambda }{2\pi ^2}\,\ln\frac{\mu _1}{\mu _2}+
16(C_1-C_2)=\lambda\left(\frac{1}{\lambda _1}-\frac{1}{\lambda _2}\right).$$
![\[edges\]The endpoint structure of the eigenvalue distribution: $\Delta $ is the gap between $\mu _1$ and $\mu _2$, while $\alpha $ is the offset of the midpoint from the Gaussian-model prediction $\mu =\sqrt{\lambda }/2\pi $ (see also fig. \[densities-fig\]).](Edge.jpg){width="10cm"}
The constants $B_a$ should stay finite in the large-$\lambda $ limit, which requires cancellation between the two terms in (\[Ba\]), nominally of order $\mathcal{\mathcal{O}}(\sqrt{\lambda })$ each. This requirement fixes $\mu _a=\sqrt{\lambda }/2\pi +\mathcal{O}(1)$. If we parameterize the endpoints of the eigenvalue distributions as in fig. \[edges\]: $$\label{muzy}
\mu _{1,2}=\frac{\sqrt{\lambda }}{2\pi }+\alpha \pm\frac{\Delta }{2}\,,$$ the normalization condition (\[Ba\]) boils down to $$\label{B12}
B_{1,2}=-\frac{\alpha }{2}\mp\frac{\Delta }{4}\,.$$ All terms of order $\mathcal{O}(\lambda) $ in (\[mainconstraint\]) also neatly cancel leaving behind one more equation: $$\label{C-condition}
\alpha \Delta +4(C_1-C_2)=\frac{1}{2}-\frac{\theta }{2\pi }\,,$$ with the $\theta $-parameter introduced in (\[theta\]).
All in all, the saddle-point equations and normalization conditions fix $A$ and $B_a$ and impose one constraint on the four remaining variables, $\mu _{a}$ and $C_a$, or $\alpha $, $\Delta $ and $C_a$. It seems that the ansatz (\[ansatz-bulk\]) introduces more unknowns than the equations can fix. At the same time, general theorems [@Gakhov] guarantee uniqueness of the solution to (\[inteq1\]), (\[inteq2\]). We found a three-parametric family. Why do general theorems fail? An apparent contradiction is resolved if we recall that the general theorems rely on the boundary conditions at the endpoints in a crucial way [@Gakhov], while the correct boundary behavior breaks down for the ansatz (\[ansatz-bulk\]). The density explodes at the endpoints starting with the second order, allowing the ansatz to evade the uniqueness theorems. This also means that the ansatz is not applicable for $x$ very close to $\pm \mu _a$, and indeed at $x\pm \mu_a \sim \mathcal{O}(1)$ all the terms in the expansion are of the same order signaling the breakdown of the strong-coupling expansion. The equations have to be solved separately near the boundary. It will become clear later that matching to the bulk will eventually fix all the remaining ambiguities.
Boundary
========
The bulk solution suggests the following behavior near the endpoints: $$\label{scaling}
\rho _a(x)\simeq A\sqrt{2\mu _a}\,f_a(\mu _a-x),$$ where $f_{1,2}(\xi )$ are some order-one scaling functions. Their large-distance asymptotics is fixed by matching to the bulk solution (\[ansatz-bulk\]): $$f_a (\xi )\stackrel{\xi \rightarrow \infty }{\simeq }\sqrt{\xi }+\frac{B_a}{\sqrt{\xi }}+\frac{C_a}{\xi ^{\frac{3}{2}}}\equiv f_a^\infty (\xi ).$$
Integral equations for the scaling functions can be derived in two steps. The difficulty lies in the non-locality of the original, exact saddle-point equations. Even if we zoom in onto the spectral edge, the integrals would receive contributions from the whole eigenvalue interval. To isolate the boundary region we can use the following trick [@Chen:2014vka]. Consider exact saddle-point equations, schematically written as $$R_{ab}*\rho _b=\frac{8\pi ^2}{\lambda _a}\,x,$$ where $*$ represents convolution. The perturbative bulk solution satisfies $$R^\infty _{ab}*\rho^\infty _b=\frac{8\pi ^2}{\lambda _a}\,x,$$ where $R^\infty $ is $R$ with $K$ replaced by $K^\infty $. This equation is actually exact, inspite of all approximations made. Hence, $$R*\rho= R^\infty *\rho^\infty.$$ Subtracting $R*\rho ^\infty $ from both sides we get: $$R*(\rho -\rho ^\infty )=(R^\infty -R)*\rho ^\infty .$$
These formal manipulations achieve our goal. Now taking $x=\mu -\xi $ with $\xi =\mathcal{O}(1)$, we find that only $y=\mu -\eta $ with $\eta =\mathcal{O}(1)$ contribute to the convolution integrals. Indeed, $R(\xi-\eta )$ grows as $(\eta -\xi )\ln(\eta -\xi) $, but $\rho -\rho ^\infty $ decays as $\eta ^{-5/2}$ away from the boundary. The convolution integral in $R*(\rho -\rho ^\infty )$ thus converges and can be extended to infinity. Likewise, $\rho ^\infty $ grows as $\eta ^{1/2}$, but $R-R^\infty $ decays as $1/(\eta -\xi) ^2$, so all integrals converge and the upper limit of integration can be safely removed: $$\label{boundaryinteq}
\int_{0}^{\infty }R_{ab}(\xi -\eta )\left(f_b(\eta )-f^\infty _b(\eta )\right)
=\int_{0}^{\infty }\left(R^\infty _{ab}(\xi -\eta )-R_{ab}(\xi -\eta )\right)f_b^\infty (\eta ).$$
The explicit form of the kernel in the last equation is $$R_{ab}(\xi )=\begin{pmatrix}
\frac{1}{\xi }-K(\xi) & K(\xi -\Delta ) \\
K(\xi +\Delta ) & \frac{1}{\xi }-K(\xi) \\
\end{pmatrix},$$ and the same for $R^\infty $ with $K\rightarrow K^\infty $. The shift by $\Delta $ in the off-diagonal terms occurs because of the gap between the endpoints of $\rho _1$ and $\rho _2$ (fig. \[edges\]) and the way we have defined the scaling functions in (\[scaling\]).
The resulting equation is of the Wiener-Hopf type and can be solved by Fourier transform $$f_a(\xi )=\int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }\,\,\,{\rm e}\,^{-i\omega \xi }f_a(\omega ).$$ Since $f_a(\xi )=0$ for $\xi <0$, its Fourier image is analytic in the upper half plane of $\omega $.
The integral equation cannot be straightforwardly Fourier transformed, because it holds only for positive $\xi $. The equation can be extended to the whole real line at the expense of introducing another unknown function, different from zero at negative $\xi $. After that the equation can be integrated and becomes algebraic in the Fourier space: $$\label{WH-bare}
R(f-f^\infty )=(R^\infty -R)f^\infty +X_-.$$ The subscript indicates that $X_-$ vanishes for $\xi >0$ and is therefore negative-half-plane analytic function of $\omega $.
The Wiener-Hopf method is based on the analytic factorization of the kernel: $$\label{Riemann-Hilbert}
G_-R=G_+,$$ where $G_\pm$ are matrix functions analytic in the upper/lower half-planes. Multiplying the two sides of (\[WH-bare\]) by $G_-$, we get: $$\label{WH-dressed}
G_+(f-f^\infty )=(G_-R^\infty -G_+)f^\infty +G_-X_-.$$ This equation contains two unknown functions, $f$ and $X_-$, but they are analytic in different halves of the complex plane and can be disentangled with the help of the projection operators: $$\label{pm-projector}
\mathcal{F}_\pm(\omega )=\pm\int_{-\infty }^{+\infty }\frac{d\nu }{2\pi i}\,\,
\frac{\mathcal{F}(\nu )}{\nu -\omega \mp i\epsilon }\,,$$ that singles out a half-plane analytic part of $\mathcal{F}$.
The $+$ projection of (\[WH-dressed\]) gives: $$G_+(f-f^\infty )=\left[(G_-R^\infty -G_+)f^\infty \right]_+.$$ Linearity of the projection and upper-half-plane analyticity of $f^\infty $ then give: $$f=G_+^{-1}\left[G_-R^\infty f^\infty \right]_+.$$ This equation constitutes a formal solution of the boundary problem. It still remains to analytically factorize the kernel.
The Fourier images of the functions appearing in the construction are $$\begin{aligned}
\label{Rw}
R(\omega )&=&2\pi i \mathop{\mathrm{sign}}\omega \coth\frac{\omega }{2}\begin{bmatrix}
\coth \omega & -\frac{\,{\rm e}\,^{i\Delta \omega }}{\sinh\omega } \\
-\frac{\,{\rm e}\,^{-i\Delta \omega }}{\sinh\omega } & \coth\omega \\
\end{bmatrix}
\\
R^\infty (\omega )&=&\frac{4\pi i\mathop{\mathrm{sign}}\omega }{\omega ^2}\begin{bmatrix}
1+\frac{5\omega ^2}{12} & \left(-1+\frac{\omega ^2}{12}\right)\,{\rm e}\,^{i\Delta \omega } \\
\left(-1+\frac{\omega ^2}{12}\right)\,{\rm e}\,^{-i\Delta \omega } & 1+\frac{5\omega ^2}{12} \\
\end{bmatrix}
\\
\label{fa-inf}
f^\infty_a (\omega )&=&\frac{\sqrt{\pi }\,i^{\frac{3}{2}}}{2(\omega +i\epsilon )^{\frac{3}{2}}}\left(1-2i\omega B_a+4\omega ^2C_a\right).\end{aligned}$$ The analytic form of $\mathop{\mathrm{sign}}\omega $ is implied here: $$\mathop{\mathrm{sign}}\omega =\lim_{\epsilon \rightarrow 0}
\frac{\sqrt{\omega +i\epsilon }}{\sqrt{\omega -i\epsilon }}\,,$$ where the branch cut of $\sqrt{\omega \mp i\epsilon }$ extends into the upper/lower half-plane.
Incidentally, the fractional powers of $\omega +i\epsilon $ cancel in the product $R^\infty f^\infty $, leaving a triple pole $\omega =-i\epsilon $ as the only singularity in the lower half-plane. Closing the contour of the $+$ projection in the lower half-plane picks the residue: $$\label{residue-formula}
f(\omega )=G^{-1}_+(\omega )\mathop{\mathrm{res}}_{\nu =0}
\frac{G_-(\nu )R^\infty (\nu )f^\infty (\nu )}{\omega -\nu }\,.$$ This equation expresses the scaling functions $f_a$ through the Wiener-Hopf factors of the kernel. The problem reduces to analytic factorization of the matrix function (\[Rw\]) according to (\[Riemann-Hilbert\]).
Analytic matrix factorization is known as the Riemann-Hilbert problem and has numerous applications in the theory of solitons [@Faddeev:1987ph] and in algebraic geometry. For a scalar function ($1\times 1$ matrix), the problem can be solved in quadratures by taking the logarithm, applying the projection (\[pm-projector\]) and exponentiating back. This procedure does not work for matrices due to non-commutativity of matrix multiplication. Matrix factorization is a substantially more complicated problem (see [@Its-RH-review] for a review) for which there is no simple plug-in solution. Fortunately, for the particular case of (\[Rw\]) the Riemann-Hilbert factorization has been carried out explicitly [@Antipov]. The Wiener-Hopf factors were found in [@Antipov] by exploiting analytic properties of the hypergeometric functions and linear identities among them. In principle, an explicit formula is all we need, but we would like to present a derivation that highlights connections to the inverse scattering problem. This perspective can be useful in view of possible generalizations and may hint on the links to integrability of the dual string theory [@Bena:2003wd; @Kazakov:2004qf].
Matrix factorization
--------------------
Consider Schrödinger equation with the Pöschl-Teller potential: $$-\frac{d^2\psi }{dx^2}+\frac{1}{4\cosh^2x}\,\psi =k^2\psi .$$ Its scattering theory is conveniently formulated in terms of the Jost functions characterized by purely exponential asymptotics at infinity: $$\psi _L^\pm\simeq \,{\rm e}\,^{\mp ikx}~~(x\rightarrow -\infty ),\qquad
\psi _R^\pm\simeq \,{\rm e}\,^{\pm ikx}~~(x\rightarrow +\infty ).$$ The Jost functions $\psi ^-_{L,R}$ describe in-type scattering states with the incident wave moving left or right and the amplitude of the transmitted wave normalized to one, while $\psi _{L,R}^+$ are the $T$-conjugate out-states. The four Jost functions are related by parity and complex conjugation.
The Jost functions admit analytic continuation into the complex momentum plane. Moreover, $\psi _{L,R}^+$ are analytic in the upper half-plane and $\psi _{L,R}^-$ are analytic in the lower half-plane, after oscillating exponentials are knocked off: $$\chi _{L,R}^+=\,{\rm e}\,^{\pm ikx}\psi^+ _{L,R},\qquad
\chi _{L,R}^-=\,{\rm e}\,^{\mp ikx}\psi^- _{L,R}.$$ These functions are faithfully half-plane analytic in $k$.
For the Pöschl-Teller potential the Jost functions can be found explicitly: $$\begin{aligned}
\psi _R^\pm&=&\,{\rm e}\,^{\pm ikx}\sqrt{1+\,{\rm e}\,^{-2x}}\,{}_2\!F_1\left(\frac{1}{2}\mp ik,\frac{1}{2}\,;1\mp ik;-\,{\rm e}\,^{-2x}\right)
\nonumber \\
\psi _L^\pm&=&\,{\rm e}\,^{\mp ikx}\sqrt{1+\,{\rm e}\,^{2x}}\,{}_2\!F_1\left(\frac{1}{2}\mp ik,\frac{1}{2}\,;1\mp ik;-\,{\rm e}\,^{2x}\right).
\nonumber \end{aligned}$$ The four Jost functions are linearly dependent, because they are solutions of a second-order differential equation, and all of them can be expressed through any two chosen as the basis.
In the case at hand, the linear relations follow from transformation rules of the hypergeometric function under argument inversion. For example, applying the $x\rightarrow -x$ transformation to $\psi _R^\pm$, we get: $$\label{left-right}
\psi _R^\pm=\mp\frac{i}{\sinh\pi k}\,\psi _L^\pm\pm i\coth\pi k\,
\frac{B\left(\frac{1}{2}\pm ik,\frac{1}{2}\right)}{B\left(\frac{1}{2}\mp ik,\frac{1}{2}\right)}\,
\psi _L^\mp.$$
More conventionally, the in-states are chosen as the basis. The out-states are then related to them by the S-matrix. Reshuffling (\[left-right\]) we find: $$\begin{bmatrix}
\psi ^+_L & \psi ^+_R \\
\end{bmatrix}
= i\,\frac{B\left(\frac{1}{2}+ ik,\frac{1}{2}\right)}{B\left(\frac{1}{2}- ik,\frac{1}{2}\right)}\,
\begin{bmatrix}
\psi ^-_R & \psi ^-_L \\
\end{bmatrix}
\begin{bmatrix}
\tanh \pi k & -\frac{i}{\cosh\pi k} \\
-\frac{i}{\cosh\pi k} & \tanh \pi k \\
\end{bmatrix}.$$ The same relation holds for the derivatives of the Jost functions and hence for their Wronskians $$W^+=\begin{bmatrix}
\psi _L^+ & \psi _R^+ \\
\frac{d\psi _L^+}{dx} & \frac{d\psi _R^+}{dx} \\
\end{bmatrix},\qquad
W^-=\begin{bmatrix}
\psi _R^- & \psi _L^- \\
\frac{d\psi _R^-}{dx} & \frac{d\psi _L^-}{dx} \\
\end{bmatrix}.$$ Namely, $$W^+=W^-S.$$ This is already close to what we need. One can say that Wronskians factorize the S-matrix, but Wronskians by themselves are not yet analytic. The oscillating factors in the Jost functions have to be offset by a similarity transformation: $$W^\pm\rightarrow W^\pm\Omega ,\qquad S\rightarrow \Omega^{-1} S\Omega$$ with $$\Omega =\mathop{\mathrm{diag}}(\,{\rm e}\,^{ikx+\frac{x}{2}},\,{\rm e}\,^{-ikx-\frac{x}{2}}).$$
The truly analytic factorization formula is slightly more complicated: $$\begin{aligned}
&& B\left(\frac{1}{2}- ik,\frac{1}{2}\right)
\begin{bmatrix}
\psi _L^+\,{\rm e}\,^{ikx+\frac{x}{2}} & \psi _R^+ \,{\rm e}\,^{-ikx-\frac{x}{2}}
\\
\frac{d\psi _L^+}{dx}\,\,{\rm e}\,^{ikx+\frac{x}{2}} &
\frac{d\psi _R^+}{dx}\,\,{\rm e}\,^{-ikx-\frac{x}{2}} \\
\end{bmatrix}
\nonumber \\
\nonumber
&&=iB\left(\frac{1}{2}+ ik,\frac{1}{2}\right)
\begin{bmatrix}
\psi _R^-\,{\rm e}\,^{ikx+\frac{x}{2}} & \psi _L^- \,{\rm e}\,^{-ikx-\frac{x}{2}}
\\
\frac{d\psi _R^-}{dx}\,\,{\rm e}\,^{ikx+\frac{x}{2}} &
\frac{d\psi _L^-}{dx}\,\,{\rm e}\,^{-ikx-\frac{x}{2}} \\
\end{bmatrix}
\begin{bmatrix}
\tanh \pi k & -\frac{i\,{\rm e}\,^{-2ikx-x}}{\cosh\pi k} \\
-\frac{i\,{\rm e}\,^{2ikx+x}}{\cosh\pi k} & \tanh \pi k \\
\end{bmatrix}.\end{aligned}$$ Remarkably, the similarity transformation not only rendered all wavefunction half-plane analytic, but also brought the S-matrix into the form very similar to (\[Rw\]). In fact, $\Omega^{-1} S\Omega$ coincides with $R(\omega )$ up to an overall scalar factor after the following change of variables: $$k\rightarrow \frac{\omega }{\pi }+\frac{i}{2}\,,\qquad
x\rightarrow -\frac{\pi \Delta }{2}\,.$$ The scalar factor is easily factorizable by itself: $$\frac{1}{2\pi ^2}\,\mathop{\mathrm{sign}}\omega \coth\frac{\omega }{2}=\frac{1}{\sqrt{\omega +i\epsilon }\,B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\frac{1}{2}\right)}\,
\cdot \,
\frac{1}{\sqrt{\omega -i\epsilon }\,B\left(\frac{1}{2}+\frac{i\omega }{2\pi }\,,\frac{1}{2}\right)}\,.$$ The solution of the Riemann-Hilbert problem thus follows from the scattering theory of the Pöschl-Teller potential!
The final result is rather bulky, and is best written in the shorthand notation: $$\label{Q-function}
Q(\alpha ,\beta ;q)=B(\alpha ,\beta ){}_2\!F_1(\alpha ,\beta ;\alpha +\beta ;-q).$$ The salient properties of this function are summarized in the appendix. The solution of the Riemann-Hilbert problem (\[Riemann-Hilbert\]) takes the following form: $$\begin{aligned}
G_+&=&\frac{4\pi ^2}{\sqrt{\omega +i\epsilon }\,\,B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\frac{1}{2}\right)}
\begin{bmatrix}
a_+ & b_+ \\
c_+ & d_+ \\
\end{bmatrix}
\begin{bmatrix}
\,{\rm e}\,^{-\frac{\pi \Delta }{2}} & 0 \\
0 & \,{\rm e}\,^{\frac{\pi \Delta }{2}} \\
\end{bmatrix}
\\
G_-&=&\frac{1}{\pi }\,\sqrt{\omega -i\epsilon }\,B\left(\frac{1}{2}+\frac{i\omega }{2\pi }\,,\frac{1}{2}\right)
\begin{bmatrix}
a_- & b_- \\
c_- & d_- \\
\end{bmatrix}\end{aligned}$$ with $$\begin{aligned}
a_+&=&Q\left(1-\frac{i\omega }{\pi }\,,\frac{1}{2}\,;\,{\rm e}\,^{-\pi \Delta }\right)
\nonumber \\
b_+&=&Q\left(1-\frac{i\omega }{\pi }\,,\frac{1}{2}\,;\,{\rm e}\,^{\pi \Delta }\right)
\nonumber \\
c_+&=&\left(\frac{1}{2}-\frac{i\omega }{\pi }\right)Q\left(1-\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,{\rm e}\,^{-\pi \Delta }\right)+\frac{1}{1+\,{\rm e}\,^{\pi \Delta }}\,Q\left(1-\frac{i\omega }{\pi }\,,\,\frac{3}{2}\,;\,{\rm e}\,^{-\pi \Delta }\right)
\nonumber \\
d_+&=&-\left(\frac{1}{2}-\frac{i\omega }{\pi }\right)Q\left(1-\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,{\rm e}\,^{\pi \Delta }\right)-\frac{1}{1+\,{\rm e}\,^{-\pi \Delta }}\,Q\left(1-\frac{i\omega }{\pi }\,,\,\frac{3}{2}\,;\,{\rm e}\,^{\pi \Delta }\right)
\nonumber \\
a_-&=&Q\left(\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,{\rm e}\,^{\pi \Delta }\right)
\nonumber \\
b_-&=&Q\left(\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,{\rm e}\,^{-\pi \Delta }\right)
\nonumber \\
c_-&=&\left(\frac{1}{2}-\frac{i\omega }{\pi }\right)Q\left(\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,\,{\rm e}\,^{\pi \Delta }\right)-\frac{1}{1+\,{\rm e}\,^{-\pi \Delta }}\,Q\left(\frac{i\omega }{\pi }\,,\,\frac{3}{2}\,;\,{\rm e}\,^{\pi \Delta }\right)
\nonumber \\
d_-&=&-\left(\frac{1}{2}-\frac{i\omega }{\pi }\right)Q\left(\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,\,{\rm e}\,^{-\pi \Delta }\right)+\frac{1}{1+\,{\rm e}\,^{\pi \Delta }}\,Q\left(\frac{i\omega }{\pi }\,,\,\frac{3}{2}\,;\,{\rm e}\,^{-\pi \Delta }\right).\end{aligned}$$
We also need the inverse of $G_+$. The standard Wronskian identity appears useful in that regard: $$\begin{bmatrix}
\psi _L^+ & \psi _R^+ \\
\frac{d\psi _L^+}{dx} & \frac{d\psi _R^+}{dx} \\
\end{bmatrix}^{-1}=\frac{1}{2ik}
\begin{bmatrix}
\frac{d\psi^+ _R}{dx} & -\psi _R^+ \\
-\frac{d\psi _L^+}{dx} & \psi _L^+ \\
\end{bmatrix}.$$ Using this identity we get: $$G^{-1}_+=\frac{\sqrt{\omega +i\epsilon }}{8\pi ^3}\,B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\frac{1}{2}\right)
\begin{bmatrix}
1+\,{\rm e}\,^{\pi \Delta } & 0 \\
0 & 1+\,{\rm e}\,^{-\pi \Delta } \\
\end{bmatrix}
\begin{bmatrix}
-d_+ & b_+ \\
c_+ & -a_+ \\
\end{bmatrix}$$ Checking that $G_+^{-1}G_+=1$ by a direct calculation is a really fun exercise.
Solving the boundary problem
----------------------------
With all the ingredients at hand, we can now find the scaling functions from (\[residue-formula\]). Evaluating the residue with the help of (\[small-alpha-Q\]) we get: $$\label{prelim-sol}
f(\omega )=\frac{2\pi ^{\frac{5}{2}}i^{\frac{3}{2}}}{\omega ^2}\,G_+^{-1}(\omega )\left(u+\frac{i\omega }{\pi }\,v\right),$$ with $$\begin{aligned}
u&=&\begin{bmatrix}
1 \\
-\frac{1}{2}\,\tanh\frac{\pi \Delta }{2} \\
\end{bmatrix}
\nonumber \\
\label{v-vec}
v&=&\left(\pi \alpha-\ln\cosh\frac{\pi \Delta }{2}\right)
\begin{bmatrix}
1 \\
-\frac{1}{2}\,\tanh\frac{\pi \Delta }{2} \\
\end{bmatrix}
+\begin{bmatrix}
0 \\
\tanh\frac{\pi \Delta }{2} \\
\end{bmatrix},\end{aligned}$$ where the explicit form of $B_{1,2}$ from (\[B12\]) has been used.
The densities should vanish as a square root at the boundary and so should the scaling functions $f_a(\xi )$. The right behavior at $\xi =0$ is not at all guaranteed for the solution obtained above and has to be imposed by hand as an extra condition. The endpoint behavior in the coordinate space is determined by the dependence of the Fourier image on large imaginary frequencies. The square root maps to $\omega ^{-3/2}$ in the Fourier space, and the right boundary conditions correspond to $$f_a(i\pi \kappa )\stackrel{\kappa \rightarrow +\infty }{\simeq }
\frac{Z_a}{\kappa ^\frac{3}{2}}$$ with some constant $Z_a$.
The general solution as given above is not consistent with this requirement. An expansion of $G_+^{-1}$ at large imaginary frequencies follows from (\[large-alpha-Q\]), and starts with $\kappa ^{1/2}$: $$G^{-1}_+(i\pi \kappa )\stackrel{\kappa \rightarrow +\infty }{=}
\frac{\sqrt{i\kappa }}{4\sqrt{2}\,\pi ^2}\begin{bmatrix}
\sqrt{1+\,{\rm e}\,^{\pi \Delta }} & 0 \\
\sqrt{1+\,{\rm e}\,^{-\pi \Delta }} & 0 \\
\end{bmatrix}+\mathcal{O}\left(\frac{1}{\sqrt{\kappa }}\right),$$ which means that in general $f(i\pi \kappa )$ will scale as $\kappa ^{-1/2}$ because of the $v$-term in (\[prelim-sol\]). In the coordinate space $1/\sqrt{\kappa }$ translates to $1/\sqrt{\xi }$, an expected asymptotics of a generic solution to the integral equation [@Gakhov]. But we are seeking a special solution where this leading asymptotic cancels leaving behind the desired $\sqrt{\xi }$ behavior. This happens if $$\begin{bmatrix}
\sqrt{1+\,{\rm e}\,^{\pi \Delta }} & 0 \\
\sqrt{1+\,{\rm e}\,^{-\pi \Delta }} & 0 \\
\end{bmatrix}v=0.$$ The next term scales as $\kappa ^{-3/2}$ and if this condition is imposed the solution has the right boundary asymptotics.
One may expect that the boundary conditions impose two constraints for each of the two independent functions, but $G_+^{-1}(i\pi \kappa )$ degenerates as a matrix at $\kappa \rightarrow +\infty $ and, as a result, only one condition survives. The condition is actually very simple, it basically requires the top component of $v$ to vanish. From the explicit formula (\[v-vec\]) we find that this is equivalent to $$\label{alpha-delta}
\alpha =\frac{1}{\pi }\,\ln\cosh\frac{\pi \Delta }{2}\,.$$ We get an extra constraint, invisible in the bulk, that relates two of the remaining four parameters of the solution.
Interestingly, $\alpha $ appears to be always positive. This implies the following inequality: $$\frac{\mu _1+\mu _2}{2}{\geqslant}\mu ,$$ illustrated in fig. \[edges\]. The density for the weaker coupling ($\rho _2$) squeezes compared to the Wigner semicircle, while the density for the larger coupling ($\rho _1$) expands. This is intuitively clear, because the extent of the density is controlled by the overall linear force inversely proportional to the coupling. What is less obvious is that the expansion of $\rho _1$ is always more pronounced than the squeezing of $\rho _2$. It would be interesting to understand this behavior at a qualitative level.
The boundary solution really simplifies once the condition (\[alpha-delta\]) is imposed. The scaling functions (\[prelim-sol\]) become $$\begin{aligned}
f_{1,2}(\omega )&=&\frac{i^{\frac{3}{2}}B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\,\frac{1}{2}\right)}{4\sqrt{\pi }\,(\omega+i\epsilon ) ^{\frac{3}{2}}}
\left[
\left(1-\frac{2i\omega }{\pi }\right)Q\left(1-\frac{i\omega }{\pi }\,,\,\frac{1}{2}\,;\,{\rm e}\,^{\pm \pi \Delta }\right)
\right.
\nonumber \\
&&\vphantom{\frac{i^{\frac{3}{2}}B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\,\frac{1}{2}\right)}{4\sqrt{\pi }\,(\omega+i\epsilon ) ^{\frac{3}{2}}}}
\left.
+\,{\rm e}\,^{\pm \pi \Delta }
Q\left(1-\frac{i\omega }{\pi }\,,\,\frac{3}{2}\,;\,{\rm e}\,^{\pm \pi \Delta }\right)
\right].\end{aligned}$$ They admit an integral representation $$\label{f12-omega}
f_{1,2}(\omega )=\frac{i^{\frac{3}{2}}B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\,\frac{1}{2}\right)}{2\sqrt{\pi }\,(\omega+i\epsilon ) ^{\frac{3}{2}}}
\int_{0}^{1}du\,\left(\frac{1+\,{\rm e}\,^{\pm\pi \Delta }u^2}{1-u^2}\right)^{\frac{i\omega }{\pi }}
\left(1-\frac{2i\omega }{\pi }\,\,\frac{1}{1+\,{\rm e}\,^{\pm\pi \Delta }u^2}\right),$$ that follows from (\[integral-Q\]) upon a change of variables $t=u^2$. This form is particularly convenient for Taylor expansion at small $\omega $.
The scaling functions should match with the bulk solution at large $\xi $. In practice, matching means that the Taylor expansion at small $\omega $ coincides with (\[fa-inf\]). The first three orders can be easily found from the integral representation: $$\begin{aligned}
f_{1,2}(\omega )&\stackrel{\omega \rightarrow 0}{=}&
\frac{\sqrt{\pi }\,i^{\frac{3}{2}}}{2\omega ^{\frac{3}{2}}}\left[
1+\frac{i\omega }{\pi }\ln\frac{1+\,{\rm e}\,^{\pm\pi \Delta }}{2}
\right.
\nonumber \\
&&\left.
+\frac{\omega ^2}{\pi ^2}\left(
\frac{\pi ^2}{8}-2\arctan^2\,{\rm e}\,^{\pm\frac{\pi \Delta }{2}}
-\frac{1}{2}\,\ln^2\frac{1+\,{\rm e}\,^{\pm\pi \Delta }}{2}
\right)+\ldots \right].\end{aligned}$$ Comparing to (\[fa-inf\]) we find that $$B_{1,2}=-\frac{1}{2\pi }\,\ln\cosh\frac{\pi \Delta }{2}\pm\frac{\Delta }{4}\,.$$ Taking into account (\[alpha-delta\]), this gives the same expression (\[B12\]) that was inferred from the bulk normalization condition. We get nothing new, this is not even a consistency check because the first two orders are guaranteed to match by construction.
New data is contained in the next term. Reading off its coefficient and comparing to (\[fa-inf\]) we find: $$\label{C12}
C_{1,2}=\frac{1}{32}-\frac{1}{2\pi ^2}\,\arctan^2\,{\rm e}\,^{\pm\frac{\pi \Delta }{2}}-\frac{1}{8\pi ^2}\left(\ln\cosh\frac{\pi \Delta }{2}\pm\frac{\pi \Delta }{2}\right)^2.$$ This determines the two remaining unknowns and fixes all the parameters of the bulk solution.
Wilson loops
============
We can now complete the circle and use the remaining bulk condition (\[C-condition\]) to find $\Delta $. To this end, we infer from (\[C12\]) that $$C_1-C_2=\frac{1}{8}-\frac{1}{2\pi }\,\arctan\,{\rm e}\,^{\frac{\pi \Delta }{2}}
-\frac{\Delta }{4\pi }\,\ln\cosh\frac{\pi \Delta }{2}\,.$$ Upon substitution of this formula along with (\[alpha-delta\]) into (\[C-condition\]) many terms cancel, the relationship between $\Delta $ and $\theta $ simplifies and can be inverted, and at the end we find a simple analytic expression $$\label{Delta-theta}
\Delta =\frac{2}{\pi }\,\ln\tan\frac{\theta }{4}\,.$$ The other parameter that characterizes the eigenvalue distribution, $\alpha $, can be found from (\[alpha-delta\]): $$\alpha =-\frac{1}{\pi }\,\ln\sin\frac{\theta }{2}\,.$$
![\[pictorial-muz\]The endpoint positions relative to the mean-field value $\mu =\sqrt{\lambda }/2\pi $, as functions of the $\theta $-parameter. The dots are obtained by picking $\lambda _1$, $\lambda _2$ randomly between $0$ and $8000$ and numerically solving the integral equations. Certain scatter in the numerical data is due to unaccounted $1/\sqrt{\lambda }$ corrections which are different for different points.](Muz.jpg){width="8cm"}
The endpoints are determined by the definition (\[muzy\]): $$\begin{aligned}
\mu _1&=&\frac{\sqrt{\lambda }}{2\pi }-\frac{1}{\pi }\ln\left(2\cos^2\frac{\theta }{4}\right)+\mathcal{O}\left(\frac{1}{\sqrt{\lambda }}\right)
\nonumber \\
\mu _2&=&\frac{\sqrt{\lambda }}{2\pi }-\frac{1}{\pi }\ln\left(2\sin^2\frac{\theta }{4}\right)+\mathcal{O}\left(\frac{1}{\sqrt{\lambda }}\right)\end{aligned}$$ This result is plotted in fig. \[pictorial-muz\]. The picture is symmetric under $\theta \rightarrow 2\pi -\theta $, $\mu _1\leftrightarrow \mu _2$, as expected.
The main contribution to the Wilson loop average (\[wils-int\]) comes from the largest eigenvalues located near the edge of the distribution. The density under the integral in (\[wils-int\]) can thus be replaced by its scaling form (\[scaling\]). Since the exponential weight guarantees fast convergence, the integration can be safely extended to infinity: $$W_a\simeq A\sqrt{2\mu _a}\,\,{\rm e}\,^{2\pi \mu _a}\int_{0}^{\infty }
d\xi \,f_a(\xi )\,{\rm e}\,^{-2\pi \xi }.$$ The integral is the Fourier image of the scaling function at pure imaginary frequency: $$W_{1,2}\simeq 8\sqrt{\pi }\lambda ^{-\frac{3}{4}}\,{\rm e}\,^{\sqrt{\lambda }+2\pi \alpha \pm \pi \Delta }f_{1,2}(2\pi i).$$
Using the explicit solution (\[f12-omega\]) and substituting (\[alpha-delta\]) for $\alpha $ we find: $$W_{1,2}=\cosh^2\frac{\pi \Delta }{2}\,
\left(1\pm 2\sin\frac{\pi \Delta }{2}\,\arctan\,{\rm e}\,^{\pm\frac{\pi \Delta }{2}}
\right)\sqrt{\frac{2}{\pi }}\,\lambda ^{-\frac{3}{4}}\,{\rm e}\,^{\sqrt{\lambda }}.$$ Finally, expressing $\Delta $ as a function of $\theta $ with the help of (\[Delta-theta\]), we obtain $$\label{W1W2}
W_1= w(\theta )\sqrt{\frac{2}{\pi }}\,\lambda ^{-\frac{3}{4}}\,{\rm e}\,^{\sqrt{\lambda }},\qquad
W_2= w(2\pi -\theta )\sqrt{\frac{2}{\pi }}\,\lambda ^{-\frac{3}{4}}\,{\rm e}\,^{\sqrt{\lambda }},$$ where $$w(\theta )=\frac{1-\frac{\theta }{2}\,\cot\frac{\theta }{2}}{\sin^2\frac{\theta }{2}}\,.$$ This is the main result of the paper.
![\[wilson-figure\]The circular Wilson loops in the quiver theory normalized by that in the $\mathcal{N}=4$ SYM, plotted as a function of the $\theta $-parameter. The dots represent the same data as in fig. \[pictorial-muz\].](Wil.jpg){width="8cm"}
The function $w(\theta )$, shown in fig. \[wilson-figure\], encodes the difference between the quiver CFT and $\mathcal{N}=4$ SYM. Indeed, the asymptotic strong-coupling expectation value in the SYM is given by (\[N=4WL\]). Comparing to (\[W1W2\]) we see that $w(\theta )$ is an extra factor that arises in the quiver theory: $$\lim_{\lambda \rightarrow \infty }\frac{W_1}{W_{\rm SYM}}=w(\theta ),
\qquad
\lim_{\lambda \rightarrow \infty }\frac{W_2}{W_{\rm SYM}}=w(2\pi -\theta ).$$ The ratio of Wilson loops is much easier to compute in string theory than a separate Wilson loop on its own. The disc amplitude for the circular loop in $AdS_5\times S^5$ has been known for a long time [@Drukker:2000ep] as a formal ratio of potentially divergent determinants. But in the ratio all divergences cancel making the Wilson loop normalized by its SYM counterpart an ideal playground for studying quantum string effects in holography [@Forini:2015bgo; @Faraggi:2016ekd].
Observables better suited for comparison to string theory are the twisted and untwisted loop correlators: $$\label{wpm}
w_\pm=\frac{W_1\pm W_2}{2W_{\rm SYM}}\,.$$ The disc amplitude, normalized by the undeformed $AdS_5\times S^5$ counterpart, maps directly to $w_+$, while $w_-$ describes the disc with the twist operator inserted. Localization gives the following predictions at strong coupling: $$w_+(\theta )=\frac{1+\frac{\pi -\theta }{2}\cot\frac{\theta }{2}}{\sin^2\frac{\theta }{2}}\,,\qquad
w_-(\theta )=- \frac{\pi }{2}\,\,\frac{\cos\frac{\theta }{2}}{\sin^3\frac{\theta }{2}}\,.$$ It would be very interesting to test these predictions by an explicit string-theory calculation.
![\[wilson-figure-untwisted\]The untwisted Wilson loop.](wper.jpg){width="7cm"}
The Wilson loops depend on $\theta $ almost trigonometrically, in accord with expectations that $\theta $ is a periodic variable in the dual string picture. However, the dependence on $\theta $ is not entirely analytic, for instance the untwisted Wilson loop diverges as $1/|\theta |^3$ when $\theta$ approaches zero, or any integer multiple of $2\pi $ (fig. \[wilson-figure-untwisted\]). The singularity signals the breakdown of the string description and happens precisely where the gauge theory becomes weakly coupled.
Decoupling limit
----------------
We can explore the vicinity of the singular point by considering the limiting case of $\lambda _1\gg\lambda _2$, still assuming $\lambda _2\gg 1$. This can be called the supergravity decoupling limit to distinguish it from the true decoupling where $\lambda _2\rightarrow 0$. All the above formulas then apply with $\theta $ approaching $2\pi $. The effective coupling in this limit coincides with the smaller one: $$\lambda \simeq 2\lambda _2,\qquad \theta \simeq 2\pi \left(1-\frac{\lambda _2}{\lambda _1}\right).$$ The Wilson loop of the weaker-coupled gauge group stays finite: $$W_2\simeq \frac{\,{\rm e}\,^{\sqrt{2\lambda _2}}}{3\cdot 2^{\frac{1}{4}}\pi ^\frac{1}{2}\lambda _2^\frac{3}{4}}\,,$$ while the stronger-coupled one diverges as $\lambda _1^3$: $$\label{W1-dec}
W_1\simeq \frac{\,{\rm e}\,^{\sqrt{2\lambda _2}}}{2^{\frac{1}{4}}\pi ^{\frac{5}{2}}\lambda _2^\frac{15}{4}}\,\lambda _1^3.$$
The limiting expression for $W_1$ resembles the SQCD Wilson loop (\[WSQCD\]) but does not coincide with it in all the detail. The cubic scaling with $\lambda _1$ is reproduced, but the log-suppression is missing and the coefficient of proportionality still depends on $\lambda _2$. The limit $\lambda _{1,2}\rightarrow \infty $, $\lambda _2/\lambda _1\rightarrow 0$, accessible from supergravity, is thus different from the true decoupling where $\lambda _1$ is fixed and $\lambda _2\rightarrow 0$ (it is enough to take $\lambda _2\sim 1$).
It is actually easy to understand why the limits do not commute. The endpoints of the eigenvalue distributions in the supergravity limit behave as $$\begin{aligned}
\mu _1&\simeq &\frac{\sqrt{2\lambda _2}}{2\pi }+\frac{2}{\pi }\,\ln\lambda _1
-\frac{2}{\pi }\,\ln\frac{\pi ^2}{2}
\nonumber \\
\mu _2&\simeq &\frac{\sqrt{2\lambda _2}}{2\pi }-\frac{1}{\lambda }\,\ln 2.\end{aligned}$$ Upon true decoupling (in SQCD), one gets [@Passerini:2011fe] $$\mu _{\rm SQCD}\simeq \frac{2}{\pi }\ln\lambda _1-\frac{1}{\pi }\ln\ln\lambda _1+\,{\rm const}\,,$$ again very similar to $\mu _1$, but different in detail.
The logarithmic growth with $\lambda _1$ in the supergravity limit is an endpoint effect, we still assume that the background, bulk density is a Wigner distribution with a parametrically large width of order $\sqrt{\lambda _2}$, and in particular $\sqrt{\lambda _2}\gg \ln\lambda _1$. Likewise, $W_1$ in (\[W1-dec\]) depends on $\lambda _1$ through a prefactor, on the background of the leading exponential behavior controlled by $\sqrt{\lambda _2}$. In SQCD, on the contrary, $\ln\lambda _1$ is the largest scale. As $\lambda _2$ decreases, both $W_1$ and $\mu _1$ decrease and should settle to their SQCD values at $\lambda _2\sim 1$. Large logs, $\ln \lambda _1$ and $\ln\ln\lambda _1$, should arise as a remnant of the transitory regime where $\sqrt{\lambda _2}$ and $\ln \lambda _1$ are equally important.
It is instructive to see what happens to the densities in the decoupling limit. The gap between the endpoints $\mu _1$ and $\mu _2$ grows large when $\lambda _1\gg\lambda _2$. Indeed $\Delta \rightarrow \infty $ as $\theta \rightarrow 2\pi $, which means that $\rho _1$ acquires a long tail extending parametrically far beyond the Wigner distribution. The functional shape of the tail is given by (\[f12-omega\]) with $\Delta \rightarrow \infty $: $$f_1(\omega )\stackrel{\Delta \rightarrow \infty }{\simeq }
\frac{i^{\frac{3}{2}}B\left(\frac{1}{2}-\frac{i\omega }{2\pi }\,,\,\frac{1}{2}\right)
B\left(1-\frac{i\omega }{\pi }\,,\,\frac{1}{2}+\frac{i\omega }{\pi }\right)}{4\sqrt{\pi }\,\omega ^{\frac{3}{2}}}\,\,{\rm e}\,^{i\omega \Delta }.$$ The last factor is the Fourier image of a shift operator, as a result $f_1$ becomes effectively a function of $\Delta -\xi $ extending over large distances $\xi \sim \Delta \sim\ln \lambda _1/\lambda _2$.
In the coordinate representation the tail is exponential: $$f_1(\xi )\simeq \frac{2\Gamma ^2\left(\frac{3}{4}\right)}{\pi ^{\frac{3}{2}}}\,
\,{\rm e}\,^{-\frac{\pi }{2}\left(\Delta -\xi \right)},$$ or, for the original density, $$\rho _1(x)\simeq \frac{2^{\frac{11}{4}}\Gamma ^2\left(\frac{3}{4}\right)}{\pi \lambda _2^\frac{3}{4}}\,\,{\rm e}\,^{\sqrt{\frac{\lambda _2}{8}}-\frac{\pi x}{2}}.$$ This is similar but not identical to the asymptotic eigenvalue distribution in SQCD, which at infinite coupling approaches [@Passerini:2011fe]: $$\rho _{\rm SQCD}(x)\stackrel{\lambda _1=\infty }{=}\frac{1}{2\cosh\frac{\pi x}{2}}
\simeq \,{\rm e}\,^{-\frac{\pi x}{2}}\,.$$ The SQCD eigenvalue density has the same exponential tail but with a different prefactor. Importantly, the behavior at $x\sim 1$ is markedly different: in SQCD the density has a coupling-independent universal shape, while the $\rho _1$ merges with the Wigner distribution at $x\sim \mu _1\sim \sqrt{\lambda _2}$.
Conclusions
===========
We have studied the expectation value of the circular Wilson loop in the superconformal quiver CFT at strong coupling, starting with the localized partition function on $S^4$. The circular loop is not the only observable accessible via localization. Other marked examples are Wilson loops in higher representations [@Fraser:2011qa; @Fraser:2015xha], correlation functions of local operators [@Gerchkovitz:2016gxx; @Rodriguez-Gomez:2016ijh; @Baggio:2016skg; @Pini:2017ouj; @Beccaria:2018xxl], correlators between local operators and a Wilson loop [@Billo:2018oog; @Beccaria:2018owt] and the Bremsstrahlung function [@Fiol:2015spa; @Mitev:2015oty; @Gomez:2018usu], all potentially calculable by similar methods.
The results for the circular loop are qualitatively consistent with the dual string picture. The coupling constant dependence comes out mostly trigonometric, in line with interpretation of $\theta $ as a theta-angle in the string sigma-model, the b-flux through the vanishing cycle of the $AdS_5\times (S^5/\mathbbm{Z}_2)$ orbifold. In view of the recent progress on similar problem in $AdS_5\times S^5$ [@Forini:2015bgo; @Faraggi:2016ekd; @Forini:2017whz; @Cagnazzo:2017sny; @Medina-Rincon:2018wjs], a more precise, quantitative comparison may actually be within reach. We will not attempt to set up the string calculation here, but will make some general remarks on its salient features.
One can envisage expanding around the minimal surface for the circle, which is an $AdS_2$ hemi-sphere embedded in $AdS_5$ and sitting at a single point on $S^5$ exactly on the orbifold locus. Quantum fluctuations of the string explore the tangent plane to $S^5$ which in the quiver theory becomes the $\mathbbm{R}\times \mathbbm{C}^2/\mathbbm{Z}_2$ orbifold. The effective string description of the circular Wilson loop is thus a partially massive theory on $AdS_2$ whose massless sector is the $\mathbbm{R}\times \mathbbm{C}^2/\mathbbm{Z}_2$ orbifold. Massive modes originate from fluctuations in $AdS_5$ and presumably cancel once the Wilson loop is normalized to its $\mathcal{N}=4$ value. In all the likelihood the normalized expectation value (\[wpm\]) is the ratio of the orbifold partition functions on $AdS_2$ at different values of the b-flux: $$w_+(\theta )=
\lim_{\epsilon \rightarrow 0}\frac{Z_{(\mathbbm{C}^2/\mathbbm{Z}_2)_{\epsilon ,\theta }}}{Z_{(\mathbbm{C}^2/\mathbbm{Z}_2)_{\epsilon ,\pi}}}\,,$$ where $\epsilon $ is the blowup parameter that regularized the orbifold geometry.
The orbifold partition function is naturally represented by an instanton sum: $$Z_{(\mathbbm{C}^2/\mathbbm{Z}_2)_{\epsilon ,\theta }}
=\sum_{k}^{}\mathcal{A}_k\,{\rm e}\,^{-\sqrt{\lambda }\,\epsilon |k|+ik\theta }.$$ At finite resolution the instantons are exponentially suppressed but the suppression disappears in the orbifold limit, in accord with our findings. However, an attempt to extract individual instanton amplitudes from (\[wpm\]) runs into problems because of the divergences at $\theta = 0$ and $2\pi $. While we understand the origin of these divergences, it is unclear how to regularize them. The principal value prescription does not work, for example[^2]. The theory at $\theta =2\pi$ has $\lambda _2\sim \mathcal{O}(1)$ and is no longer strongly coupled, even if $\lambda _1\gg 1$. It would be very interesting to make the above arguments more precise and to see how the divergences are resolved (or how they arise) in string theory.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank C. Bachas, R. Klabbers, I. Klebanov, T. McLoughlin, C. Nunez, H. Ouyang, A. Parnachev, E. Pomoni and A. Tseytlin for interesting discussions and D. Medina-Rincon for comments on the manuscript. This work was supported by the grant “Exact Results in Gauge and String Theories” from the Knut and Alice Wallenberg foundation and by RFBR grant 18-01-00460 A.
Function $Q$
============
The function defined in (\[Q-function\]) admits an integral representation: $$\label{integral-Q}
Q(\alpha ,\beta ;q)=\int_{0}^{1}dt\,t^{\beta -1}(1-t)^{\alpha -1}(1+qt)^{-\alpha },$$ The only singularities of $Q$ in the finite part of the complex plane are simple poles at non-positive integer $\alpha $. Analyticity in $\alpha $ for $\mathop{\mathrm{Re}}\alpha >0$ easily follows from the integral representation.
It is also easy to develop asymptotic expansions at small and large $\alpha $. At large positive $\alpha $, $$\label{large-alpha-Q}
Q(\alpha ,\beta ;q)\stackrel{\alpha \rightarrow +\infty }{=}
\frac{\Gamma (\beta )}{\alpha ^\beta (1+q)^\beta }+\mathcal{O}\left(\frac{1}{\alpha ^{\beta +1}}\right).$$ At small $\alpha$, $$\label{small-alpha-Q}
Q(\alpha ,\beta ;q)\stackrel{\alpha \rightarrow 0 }{=}
\frac{1}{\alpha }-\ln(1+q)-\psi (\beta )-\gamma +\mathcal{O}(\alpha ).$$
[10]{}
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[^1]: Only bosonic fields are displayed.
[^2]: It does not work for the untwisted Wilson loop. The twisted Wilson loop is analytic in $\theta $ and the principal-value prescription should work.
|
An Infinite Level Atom coupled to a Heat Bath An Infinite Level Atom Martin Koenenberg [^1] M.Koenenberg
We consider a $W^*$-dynamical system $({\mathfrak{M}_\beta},{\tau})$, which models finitely many particles coupled to an infinitely extended heat bath. The energy of the particles can be described by an unbounded operator, which has infinitely many energy levels. We show existence of the dynamics ${\tau}$ and existence of a $(\beta,{\tau})$ -KMS state under very explicit conditions on the strength of the interaction and on the inverse temperature $\beta$.
Primary 81V10; Secondary 37N20, 47N50.
KMS-state, thermal equilibrium, $W^*$-dynamical system, open quantum system.
Martin Könenberg Fakultät für Mathematik und Informatik FernUniversität Hagen Lützowstra[ß]{}e 125 D-58084 Hagen, Germany. martin.koenenberg@fernuni -hagen.de
Introduction
============
In this paper, we study a $W^*$-dynamical system $({\mathfrak{M}_\beta},{\tau})$ which describes a system of finitely many particles interacting with an infinitely extended bosonic reservoir or heat bath at inverse temperature $\beta$. Here, ${\mathfrak{M}_\beta}$ denotes the $W^*$-algebra of observables and ${\tau}$ is an automorphism-group on ${\mathfrak{M}_\beta}$, which is defined by $${\tau}_t(X):= e^{{i}t {\mathcal{L}_Q}}\,X\, e^{-{i}t {\mathcal{L}_Q}},\ X\in {\mathfrak{M}_\beta},\ t\in {\mathbbm{R}}.$$ In this context, $t$ is the time parameter. ${\mathcal{L}_Q}$ is the Liouvillean of the dynamical system at inverse temperature $\beta$, ${Q}$ describes the interaction between particles and heat bath. On the one hand the choice of ${\mathcal{L}_Q}$ is motivated by heuristic arguments, which allow to derive the Liouvillean ${\mathcal{L}_Q}$ from the Hamiltonian ${H}$ of the joint system of particles and bosons at temperature zero. On the other hand we ensure that ${\mathcal{L}_Q}$ anti-commutes with a certain anti-linear conjugation ${\mathcal{J}}$, that will be introduced later on. The Hamiltonian, which represents the interaction with a bosonic gas at temperature zero, can be the Standard Hamiltonian of the non-relativistic QED, (see or instance [@BachFroehlichSigal1998a]), or the Pauli-Fierz operator, which is defined in [@DerezinskiJaksic2001; @BachFroehlichSigal1998a], or the Hamiltonian of Nelson’s Model. We give the definition of these Hamiltonians in the sequel of Definition \[Def:Ho-Hg\].\
Our first result is the following:
${\mathcal{L}_Q}$, defined in , has a unique self-adjoint realization and ${\tau}_t(X)\in {\mathfrak{M}_\beta}$ for all $t\in {\mathbbm{R}}$ and all $X\in {\mathfrak{M}_\beta}$.
The proof follows from Theorem \[Thm1\] and Lemma \[LemInv\]. The main difficulty in the proof is, that ${\mathcal{L}_Q}$ is not semi-bounded, and that one has to define a suitable auxiliary operator in order to apply Nelson’s commutator theorem.\
Partly, we assume that the isolated system of finitely many particles is confined in space. This is reflected in Hypothesis \[Hyp1\], where we assume that the particle Hamiltonian ${H_{el}}$ possesses a Gibbs state. In the case where ${H_{el}}$ is a Schrödinger-operator, we give in Remark \[Bem:Gibbs\] a sufficient condition on the external potential $V$ to ensure the existence of a Gibbs state for ${H_{el}}$. Our second theorem is
Assume Hypothesis \[Hyp1\] and that ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-(\beta/2)({\mathcal{L}_0}+{Q})})$. Then there exists a $(\beta,{\tau})$-KMS state ${\omega^\beta}$ on ${\mathfrak{M}_\beta}$.
This theorem ensures the existence of an equilibrium state on ${\mathfrak{M}_\beta}$ for the dynamical system $({\mathfrak{M}_\beta},{\tau})$. Its proof is part of Theorem \[Thm3\] below. Here, ${\mathcal{L}_0}$ denotes the Liouvillean for the joint system of particles and bosons, where the interaction part is omitted. ${\Omega_0^\beta}$ is the vector representative of the $(\beta,{\tau})$-KMS state for the system without interaction. In a third theorem we study the condition ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-(\beta/2)({\mathcal{L}_0}+{Q})})$:
\[Thm2\] Assume Hypothesis \[Hyp1\] is fulfilled. Then there are two cases,
1. If $0\,{\leqslant}\, \gamma\, <\,1/2$ and ${\underline}{\eta}_1\,(1+\beta)\,\ll\, 1$, then ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}+{Q})})$.
2. If $\gamma=1/2$ and $(1+\beta)({\underline}{\eta}_1+{\underline}{\eta}_2)\,\ll\, 1$, then ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}+{Q})})$.
Here, $\gamma\in[0,1/2)$ is a parameter of the model, see and ${\underline}{\eta}_1,{\underline}{\eta}_2$ are parameters, which describe the strength of the interaction, see . In a last theorem we consider the case where ${H_{el}}= -\Delta_q + \Theta^2q^2$ and the interaction Hamiltonian is $\lambda\, q\, \Phi(f)$ at temperature zero for $\lambda \not=0$. Then,
\[Thm4a\] ${\Omega_0^\beta}$ is in ${\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}\,+\,{Q})})$ for all $\beta \in (0,\, \infty)$, whenever $$|2\Theta^{-1}\,\lambda|\,\||k|^{-1/2}\,f\|_{{\mathcal{H}_{ph}}}\, <\,1.$$
Furthermore, we show that our strategy can not be improved to obtain a result, which ensures existence for all values of $\lambda$, see .\
In the last decade there appeared a large number of mathematical contributions to the theory of open quantum system. Here we only want to mention some of them [@BachFröhlichSigal2000; @DerezinskiJaksicPillet2003; @DerezinskiJaksic2003; @FroehlichMerkli2004; @FroehlichMerkliSigal2004; @JaksicPillet1996a; @JaksicPillet1996b; @Merkli2001], which consider a related model, in which the particle Hamilton ${H_{el}}$ is represented as a finite symmetric matrix and the interaction part of the Hamiltonian is linear in annihilation and creation operators. In this case one can prove existence of a $\beta$- KMS without any restriction to the strength of the coupling. (In this case we can apply Theorem \[Thm2\] with $\gamma=0$ and ${\underline}{\eta}_1=0$). We can show existence of KMS-states for an infinite level atom coupled to a heat bath. Furthermore, in [@DerezinskiJaksicPillet2003] there is a general theorem, which ensures existence of a $(\beta,{\tau})$-KMS state under the assumption, that ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-(\beta/2) {Q}})$, which implies ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-(\beta/2)({\mathcal{L}_0}+{Q})})$. In Remark \[Bem3d\] we verify that this condition implies the existence of a $(\beta,{\tau})$-KMS state in the case of a harmonic oscillator with dipole interaction $\lambda\, q\, \cdot \,\Phi(f)$, whenever $(1+\beta) \lambda\|(1+|k|^{-1/2})f\| \ll 1$.\
Mathematical Preliminaries
==========================
Fock Space, Field- Operators and Second Quantization
----------------------------------------------------
We start our mathematical introduction with the description of the joint system of particles and bosons at temperature zero. The Hilbert space describing bosons at temperature zero is the *bosonic Fock space* ${\mathcal{F}_b}$, where $$\label{FockSpace}
{\mathcal{F}_b}\,:=\, \mathcal{F}_b[{\mathcal{H}_{ph}}] \,:=\, {\mathbbm{C}}\oplus \bigoplus_{n=1}^\infty {\mathcal{H}_{ph}}^{(n)},
\qquad {\mathcal{H}_{ph}}^{(n)}\,:=\,\bigotimes_{sym}^n {\mathcal{H}_{ph}}.$$ ${\mathcal{H}_{ph}}$ is either a closed subspace of $L^2({\mathbbm{R}}^3)$ or $L^2({\mathbbm{R}}^3 \times \{\pm\})$, being invariant under complex conjugation. If phonons are considered we choose ${\mathcal{H}_{ph}}= L^2({\mathbbm{R}}^3 )$, if photons are considered we choose ${\mathcal{H}_{ph}}=L^2({\mathbbm{R}}^3 \times \{\pm\})$. In the latter case “+” or “-” labels the polarization of the photon. However, we will write $\langle\, f\,|\,g\,\rangle_{{\mathcal{H}_{ph}}} \,:=\, \int {\overline}{f(k)}\, g(k)\,dk$ for the scalar product in both cases. This is an abbreviation for $\sum_{p\,=\,\pm} \int {\overline}{f(k,p)}\, g(k,p)\,dk$ in the case of photons.\
${\mathcal{H}_{ph}}^{(n)}$ is the $n$-fold symmetric tensor product of ${\mathcal{H}_{ph}}$, that is, it contains all square integrable functions $f_n$ being invariant under permutations $\pi$ of the variables, i.e., $f_n(k_1,\ldots,k_n)\,=\,f_n(k_{\pi(1)},\ldots,k_{\pi(n)})$. For phonons we have $k_j\in {\mathbbm{R}}^3$ and $k_j\in {\mathbbm{R}}^3\times \{\pm\}$ for photons. The wave functions in ${\mathcal{H}_{ph}}^n$ are states of $n$ bosons.\
The vector $\Omega \,:=\, (1,\,0,\,\ldots)\,\in {\mathcal{F}_b}$ is called the *vacuum*. Furthermore we denote the subspace ${\mathcal{F}_b}$ of finite sequences with ${{\mathcal{F}_b}}^{fin}$. On ${{\mathcal{F}_b}}^{fin}$ the *creation and annihilation operators*, $a^*(h)$ and $a(h)$, are defined for $h\in {\mathcal{H}_{ph}}$ by $$\begin{aligned}
\label{CreaAnnOps}
(a^*(h)&\,f_{n})(k_1,\ldots,k_{n+1})\\ \nonumber
&\,=\,(n+1)^{-1/2}\sum_{i=1}^{n+1}h(k_i)\,f_{n}(k_1,\ldots,k_{i-1},k_{i+1},\ldots ,k_{n+1}),\\
(a(h)&\,f_{n+1})(k_1,\ldots,k_{n})\\ \nonumber
&\,=\,(n+1)^{1/2}\int h(k_{n+1})\,f_{n+1}(k_1,\ldots,k_{n+1})\,dk_{n+1},\nonumber\end{aligned}$$ and $a^*(h)\,\Omega\,=\,h,\ a(h)\,\Omega\,=\,0 $. Since $a^*(h)\,\subset\,(a(h))^*$ and $a(h)\,\subset\,(a^*(h))^*$, the operators $a^*(h)$ and $a(h)$ are closable. Moreover, the canonical commutation relations (CCR) hold true, i.e., $$\begin{gathered}
\label{CCR}
[a(h)\,,\,a(\widetilde{h})]\,=\,[a^*(h)\,,\,a^*(\widetilde{h})]\,=\,0,
\qquad [a(h)\,,\,a^*(\widetilde{h})] \,=\,\langle\, h \,|\, \widetilde{h}\,\rangle_{{\mathcal{H}_{ph}}}.\end{gathered}$$ Furthermore we define field operator by $$\label{FieldOp}
\Phi(h)\,:=\,2^{-1/2}\,(a(h)\,+\,a^*(h)),\qquad h\in {\mathcal{H}_{ph}}.$$ It is a straightforward calculation to check that the vectors in ${{\mathcal{F}_b}}^{fin}$ are analytic for $\Phi(h)$. Thus, $\Phi(h)$ is essentially self-adjoint on ${{\mathcal{F}_b}}^{fin}$. In the sequel, we will identify $a^*(h),\, a(h)$ and $ \Phi(h)$ with their closures. The Weyl operators ${W}(h)$ are given by ${W}(h)\,=\,\exp({i}\, \Phi(h))$. They fulfill the CCR-relation for the Weyl operators, i.e., $$\label{WeylCCR}
{W}(h)\,{W}(g)\,=\, \exp( {i}/2\, {\textrm{Im}\,}\langle\, h\,|\,g\,\rangle_{{\mathcal{H}_{ph}}} )\,{W}(g+h),$$ which follows from explicit calculations on ${{\mathcal{F}_b}}^{fin}$. The Weyl algebra ${W}({\mathfrak{f}})$ over a subspace ${\mathfrak{f}}$ of ${\mathcal{H}_{ph}}$ is defined by $$\label{WeylAlgebra}
{W}({\mathfrak{f}}) \,:= \,{\operatorname{cl}}{\operatorname{LH}}\{ {W}(g)\in \mathcal{B}({\mathcal{F}_b})\,:\, g\in {\mathfrak{f}}\}.$$ Here, ${\operatorname{cl}}$ denotes the closure with respect to the norm of $\mathcal{B}({\mathcal{F}_b})$, and “${\operatorname{LH}}$” denotes the linear hull.\
Let $\alpha\,:\, {\mathbbm{R}}^3\,\rightarrow\, [0,\,\infty)$ be a locally bounded Borel function and ${\operatorname{dom}}(\alpha )\,:=\, \{ f\in {\mathcal{H}_{ph}}\,:\, \alpha f\in {\mathcal{H}_{ph}}\}$. Note, that $(\alpha f)(k)$ is given by $\alpha(k)\,f(k,p)$ for photons. If ${\operatorname{dom}}(\alpha)$ is dense subspace of ${\mathcal{H}_{ph}}$, $\alpha$ defines a self-adjoint multiplication operator on ${\mathcal{H}_{ph}}$. In this case, the second quantization $d\Gamma(\alpha)$ of $\alpha$ is defined by $$\label{SecQuant}
(d\Gamma(\alpha)\,f_n)(k_1,\ldots,k_n)\,:=\, (\alpha(k_1)+\alpha(k_2)+\ldots+\alpha(k_n))\,f_n(k_1,\ldots,k_n)$$ and $d\Gamma(\alpha)\,\Omega\,=\,0$ on its maximal domain.
Hilbert space and Hamiltonian for the particles
-----------------------------------------------
Let ${\mathcal{H}_{el}}$ be a closed, separable subspace of $L^2(X,\, d\mu)$, that is invariant under complex conjugation. The Hamiltonian ${H_{el}}$ for the particle is a self-adjoint operator on ${\mathcal{H}_{el}}$ being bounded from below. We set ${H_{el,+}}\,:=\,{H_{el}}\,-\,\inf\sigma({H_{el}})\,+\,1$. Partly, we need the assumption
\[Hyp1\] Let $\beta>0$. There exists a small positive constant $\epsilon>0$, and $$\label{GibbsCond}
{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{ e^{-(\beta\,-\,\epsilon)\, {H_{el}}}\}\, <\,\infty .$$ The condition implies the existence of a Gibbs state $$\label{GibbsState}
{\omega^\beta_{el}}(A)\,=\,{\mathcal{Z}}^{-1}\,{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{ e^{-\beta\, {H_{el}}} A \},\quad A\in \mathcal{B}({\mathcal{H}_{el}}),$$ for ${\mathcal{Z}}\, = \,{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{ e^{-\beta\, {H_{el}}} \}$.
\[Bem:Gibbs\] Let ${\mathcal{H}_{el}}=L^2({\mathbbm{R}}^n, d^n\,x)$ and ${H_{el}}\,=\, -\Delta_x \,+\, V_1\,+\,V_2$, where $V_1$ is a $-\Delta_x$-bounded potential with relative bound $a\,<\,1$ and $V_2$ is in $L^2_{loc}({\mathbbm{R}}^n,\, d^nx)$. Thus ${H_{el}}$ is essentially self-adjoint on $\mathcal{C}_c^\infty({\mathbbm{R}}^n)$. Moreover, if additionally $$\int e^{-(\beta\,-\,\epsilon)\, V_2(x)}\,d^n\,x<\,\infty,$$ then one can show, using the Golden-Thompson-inequality, that Hypothesis \[Hyp1\] is satisfied.
Hilbert space and Hamiltonian for the interacting system {#SecGeneralizedFieldop}
--------------------------------------------------------
The Hilbert space for the joint system is ${\mathcal{H}}\,:=\, {\mathcal{H}_{el}}{\otimes}{\mathcal{F}_b}$. The vectors in ${\mathcal{H}}$ are sequences $f= (f_n)_{n\in {\mathbbm{N}}_0}$ of wave functions, $f_n\in {\mathcal{H}_{el}}{\otimes}{\mathcal{H}_{ph}}^{(n)}$, obeying $$\begin{gathered}
{\underline}{k}_n \,\mapsto \,f_n(x,\,{\underline}{k}_n)\in {\mathcal{H}_{ph}}^{(n)}
\qquad \textrm{ for } \mu\textrm{- almost every x}\\
x\, \mapsto \,f_n(x,\,{\underline}{k}_n)\in {\mathcal{H}_{el}}\qquad
\text{ for Lebesgue - almost every }{\underline}{k}_n,\end{gathered}$$ where ${\underline}{k}_n \,=\, (k_1,\ldots,k_n)$. The complex conjugate vector is ${\overline}{f}\,:=\, (\, {\overline}{f}_n\,)_{n\in {\mathbbm{N}}_0}$.\
Let $G^j\,:=\,\{G^j_k\}_{k\in{\mathbbm{R}}^3},\, H^j\,:=\,\{H^j_k\}_{k\in{\mathbbm{R}}^3}$ and $F\,:=\,\{F_k\}_{k\in{\mathbbm{R}}^3}$ be families of closed operators on ${\mathcal{H}_{el}}$ for $j\,=\,1,\ldots,r$. We assume, that ${\operatorname{dom}}(F^\sigma_k)\,\supset\, {\operatorname{dom}}({H_{el,+}}^{1/2})$ and that $$k\,\mapsto\, G_k^j,\ (H_k^j),\ F_k {H_{el,+}}^{-1/2},\ (F_k)^* {H_{el,+}}^{-1/2} \in \mathcal{B}({\mathcal{H}_{el}})$$ are weakly (Lebesgue-)measurable. For $\phi \in {\operatorname{dom}}({H_{el,+}}^{1/2})$ we assume that $$\begin{gathered}
k\,\mapsto\, (G_k^j\,\phi)(x),\ (H_k^j\,\phi)(x),\ (F_k\,\phi)(x) \in {\mathcal{H}_{ph}},\\
k\,\mapsto\, ((G_k^j)^*\,\phi)(x),\ ((H_k^j)^*\,\phi)(x),\ ((F_k)^*\,\phi)(x) \in {\mathcal{H}_{ph}}, \textrm{ for }x\in X.\end{gathered}$$ Moreover we assume for $\vec G\,=\,(G^1,\ldots,\,G^r),\
\vec H \,:=\, (H^1,\ldots,\,H^r)$ and $F$, that $$\|\vec G\|_w\,<\,\infty,\ \|\vec H\|_w \,<\,\infty ,\ \|F\|_{w,1/2}\,<\,\infty,$$ where $$\begin{gathered}
\label{DefNorm}
\|G_j\|^2_w \,:=\,\int (\alpha(k)\,+\,\alpha(k)^{-1})
\,\big(\|(G_k^j)^*\|_{\mathcal{B}({H_{el}})}^2\,+\,\|G_k^j\|_{\mathcal{B}({H_{el}})}^2\big)\,dk\\
\|\vec G\|^2_w\, :=\, \sum_{j=1}^r\|G_j\|^2_w,
\qquad \|F\|^2_{w,1/2}\,:=\, \|F{H_{el,+}}^{-1/2}\|^2_{w}\,+\,\|F^*{H_{el,+}}^{-1/2}\|^2_{w}.\end{gathered}$$ We define for $f\,=\,(f_n)_{n=0}^\infty\in {\operatorname{dom}}({H_{el,+}}^{1/2}){\otimes}{\mathcal{F}_b}^{fin}$ the (generalized) creation operator $$\begin{aligned}
\label{GenCreaOp}
(a^*(F)\,f_{n})&(x,\,k_1,\ldots,\,k_{n+1})\\ \nonumber
&\,:=\,(n+1)^{-1/2}\,\sum_{i=1}^{n+1}(F_{k_{i}}\,f_{n})(x,\,k_1,\ldots,\,k_{i-1},\,k_{i+1},\ldots ,\,k_{n+1})\end{aligned}$$ and $a(F)\,f_0(x)\,=\, 0$. The (generalized) annihilation operator is $$\begin{aligned}
\label{GenAnnOp}
(a(F)\,f_{n+1})&(x,\,k_1,\ldots,k_{n})\\ \nonumber
&\,:=\,(n+1)^{1/2}\,\int(F^*_{k_{n+1}}\,f_{n+1})(x,\,k_1,\ldots,\,k_{n},\,k_{n+1})\, dk_{n+1}.\end{aligned}$$ Moreover, the corresponding (generalized) field operator is $ \Phi(F)\,:=\, 2^{-1/2}\, (a(F)\,+\,a^*(F))$. $\Phi(F)$ is symmetric on ${\operatorname{dom}}({H_{el,+}}^{1/2}){\otimes}{\mathcal{F}_b}^{fin}$. The bounds follow directly from Equations and . $$\begin{aligned}
\label{RelBoundGen}
\|\,a(F){H_{el,+}}^{-1/2}\,f\|^2_{{\mathcal{H}}}
&\,{\leqslant}\,& \int |\alpha(k)|^{-1} \|F^*_k\,{H_{el,+}}^{-1/2}\|^2_{\mathcal{B}({\mathcal{H}_{el}})}\,dk \cdot
\|d\Gamma(|\alpha|)^{1/2}f\|^2_{{\mathcal{H}}}\\ \nonumber
\|a^*(F){H_{el,+}}^{-1/2}\,f\|^2_{{\mathcal{H}}}
&\,{\leqslant}\,& \int |\alpha(k)|^{-1} \|F_k\,{H_{el,+}}^{-1/2}\|^2_{\mathcal{B}({\mathcal{H}_{el}})}\,dk\cdot
\|d\Gamma(|\alpha|)^{1/2}\,f\|^2_{{\mathcal{H}}}\\ \nonumber
&&\, +\,\int \|F_k\,{H_{el,+}}^{-1/2}\|^2_{\mathcal{B}({\mathcal{H}_{el}})}\,dk
\cdot \|f\|^2_{{\mathcal{H}}}.\end{aligned}$$ For $(G_k)^j,\, (H_k)^j\in \mathcal{B}({\mathcal{H}_{el}})$, the factor ${H_{el,+}}^{-1/2}$ can be omitted. The Hamiltonians for the non-interacting, resp. interacting model are
\[Def:Ho-Hg\]On ${\operatorname{dom}}({H_{el}}){\otimes}{\operatorname{dom}}(d\Gamma(\alpha))\cap {\mathcal{F}_b}^{fin}$ we define $$\label{Def:Ho-Hg}
{H_0}\, :=\, {H_{el}}{\otimes}{\mathbf{1}}\,+\, {\mathbf{1}}{\otimes}d\Gamma(\alpha), \qquad
{H}\, :=\, {H_0}\,+\, {W},$$ where ${W}\, :=\, \Phi(\vec G )\,\Phi(\vec H )\,+\, {\operatorname{h.c.}}\,+\,\Phi(F)$ and $\Phi(\vec G )\,\Phi(\vec H )\,:=\, \sum_{j=1}^r\Phi( G^j )\,\Phi(H^j )$. The abbreviation “h.c.” means the formal adjoint operator of $\Phi(\vec G )\,\Phi(\vec H )$.
We give examples for possible configurations:\
Let $\gamma \in {\mathbbm{R}}$ be a small coupling parameter.\
$\blacktriangleright$ The Nelson Model:\
${\mathcal{H}_{el}}\subset L^2({\mathbbm{R}}^{3N})$, ${H_{el}}\,:=\, -\Delta\,+\,V$, ${\mathcal{H}_{ph}}\,=\,L^2({\mathbbm{R}}^3)$ and $\alpha(k)\,=\,|k|$. The form factor is $F_k\,=\,\gamma\, \sum_{\nu\,=\,1}^N\,e^{- {i}\, k x_\nu}\,|k|^{-1/2}\,
{\mathbf{1}}[\,|k|\,{\leqslant}\, \kappa],\ x_\nu\in {\mathbbm{R}}^3$ and $H^j,\, G^j\,=\,0$.\
$\blacktriangleright$ The Standard Model of Nonrelativistic QED:\
${\mathcal{H}_{el}}\subset L^2({\mathbbm{R}}^{3N})$, ${H_{el}}\,:=\, -\Delta\,+\,V$, ${\mathcal{H}_{ph}}\,=\,L^2({\mathbbm{R}}^3\times \{\pm\})$ and $\alpha(k)\,=\,|k|$. The form factors are $$\begin{gathered}
F_\mathbf{k}\,=\, 4\gamma^{3/2} \,\pi^{-1/2}\,\sum_{\nu=1}^N(-{i}\nabla_{x_\nu} \cdot \epsilon(k,p))
e^{- {i}\,\gamma^{1/2} k x_\nu}\,(2|k|)^{-1/2}\,
{\mathbf{1}}[\,|k|\,{\leqslant}\, \kappa]+ {\operatorname{h.c.}},\\
G^{i,\,\nu}_\mathbf{k}\, = \, H^{i,\,\nu}_\mathbf{k}
\,=\,2\gamma^{3/2} \,\pi^{-1/2}\,\epsilon_i(k,\,p)\, e^{- {i}\,\gamma^{1/2}\, k x_\nu}\,(2|k|)^{-1/2}\,
{\mathbf{1}}[\,|k|\,{\leqslant}\, \kappa]\end{gathered}$$ for $i\,=\,1,\,2,\,3,\ \nu\,=\,1,\ldots,\, N,\ x_\nu\in {\mathbbm{R}}^3$ and $\mathbf{k}=(k,p)\in {\mathbbm{R}}^3\times\{\pm\}$. $\epsilon_i(k,\,\pm)\in{\mathbbm{R}}^3 $ are polarization vectors.\
$\blacktriangleright$ The Pauli-Fierz-Model:\
${\mathcal{H}_{el}}\subset L^2({\mathbbm{R}}^{3N})$, ${H_{el}}\,:=\, -\Delta\,+\,V$, ${\mathcal{H}_{ph}}\,=\,L^2({\mathbbm{R}}^3)$ or ${\mathcal{H}_{ph}}\,=\,L^2({\mathbbm{R}}^3\times \{\pm\})$, and $\alpha(k)\,=\,|k|$. The form factor is $F_k\,=\,\gamma \sum_{\nu\,=\,1}^N {\mathbf{1}}[\,|k|\,{\leqslant}\, \kappa] \, k\cdot x_\nu$ and $G^j_k=H^j_k=0$
The Representation ${\pi}$ {#sec:Rep}
==========================
In order to describe the particle system at inverse temperature $\beta$ we introduce the algebraic setting. For ${\mathfrak{f}}\, =\,\{ f\in {\mathcal{H}_{ph}}\,:\, \alpha^{-1/2}f \in {\mathcal{H}_{ph}}\}$ we define the algebra of observables by $$\label{Algebra}
{\mathfrak{A}}\,=\, \mathcal{B}({\mathcal{H}_{el}}){\otimes}{ \mathcal{W}}({\mathfrak{f}}).$$ For elements $A\in {\mathfrak{A}}$ we define $\tilde{\tau}^0_t(A)\,:=\, e^{{i}\,t\, {H_0}}\,A\, e^{-{i}\, t\, {H_0}}$ and\
$\tilde{\tau}^g_t(A)\,:=\, e^{{i}\,t\,{H}}A e^{-\,{i}\, t\, {H}}$. We first discuss the model without interaction.
The Representation ${ \pi_f}$
-----------------------------
The time-evolution for the Weyl operators is given by $$\label{TimeEvField}
e^{{i}\, t \,{ \check{H} }}\,{ \mathcal{W}}(f)\,e^{-{i}\, t\, { \check{H} }}\,=\, { \mathcal{W}}(e^{{i}\, t\, \alpha }\,f).$$ For this time-evolution an equilibrium state ${ \omega^\beta_f}$ is defined by $${ \omega^\beta_f}({ \mathcal{W}}(f))\,=\, \langle\, f \,|\, (1 \,+\, 2\,\varrho_\beta) \,f\,\rangle_{{\mathcal{H}_{ph}}},$$ where $\varrho_\beta(k)\,=\, \big(\exp(\beta \,\alpha(k)) \,-\,1\big)^{-1}$. It describes an infinitely extended gas of bosons with momentum density $\varrho_\beta$ at temperature $\beta$. Since ${ \omega^\beta_f}$ is a quasi-free state on the Weyl algebra, the definition of ${ \omega^\beta_f}$ extends to polynomials of creation and annihilation operators. One has $$\begin{gathered}
\label{EvOnePointTwoPoint}
{ \omega^\beta_f}(a(f))\,=\,{ \omega^\beta_f}(a^*(f))\,=\,{ \omega^\beta_f}(a(f)\,a(g))\,=\,{ \omega^\beta_f}(a^*(f)\,a^*(g))\,=\,0,\\
{ \omega^\beta_f}(a^*(f)\,a(g))\,=\, \langle\, g\,|\, \varrho_\beta \,f \,\rangle_{{\mathcal{H}_{ph}}}.\end{gathered}$$ For polynomials of higher degree one can apply Wick’s theorem for quasi-free states, i.e., $$\label{QuasiFreeExp}
{ \omega^\beta_f}(a^{\sigma_{2m}}(f_{2m})\cdots a^{\sigma_1}(f_1))
\,=\,\sum_{P\in \mathcal{Z}_2}\prod_{\stackrel{\{i,j\}\in P}{{i>j}}}
{ \omega^\beta_f}\big(a^{\sigma_i}(f_i)\,a^{\sigma_j}(f_j)\big),$$ where $a^{\sigma_{k}}\,=\,a^*$ or $a^{\sigma_{k}}\,=\,a$ for $k=1,\ldots,\,2m$. $\mathcal{Z}_2$ are the pairings, that is
$P\in\mathcal{Z}_2,$ iff $P\,=\,\{ Q_1,\ldots,Q_m\},\ \#Q_i=2$ and $\bigcup_{i=1}^m\,Q_i\,=\,\{1,\ldots,\,2m\}$.\
The Araki-Woods isomorphism ${ \pi_f}\,:\, { \mathcal{W}}({\mathfrak{f}})\,\rightarrow\, \mathcal{B}({\mathcal{F}_b}{\otimes}{\mathcal{F}_b})$ is defined by $$\begin{gathered}
\label{ArakiWoodsIso}
{ \pi_f}[{ \mathcal{W}}(f)] \,:=\, { \mathcal{W}}_\beta(f)\,:=\, \exp( {i}\, \Phi_\beta(f)),\\
\Phi_\beta(f)\,:=\, \Phi( (1\,+\,\varrho_\beta)^{1/2}\, f){\otimes}{\mathbf{1}}\,+\, {\mathbf{1}}{\otimes}\Phi( \varrho_\beta^{1/2}\, {\overline}{f}).\end{gathered}$$ The vector ${\Omega^\beta_{f}}\,:=\, \Omega{\otimes}\Omega$ fulfills $$\label{AWRep}
{ \omega^\beta_f}({ \mathcal{W}}(f))\,=\, \langle\, {\Omega^\beta_{f}}\,|\, { \pi_f}[{ \mathcal{W}}(f)]\, {\Omega^\beta_{f}}\,\rangle.$$
The representation ${\pi^{el}}$
-------------------------------
The particle system without interaction has the observables $\mathcal{B}({\mathcal{H}_{el}})$, the states are defined by density operators $\rho$, i.e., $\rho \in \mathcal{B}({\mathcal{H}_{el}}),\ 0{\leqslant}\rho,\ {\operatorname{Tr}}\{\rho\}\,=\,1$. The expectation of $A\in \mathcal{B}({\mathcal{H}_{el}})$ in $\rho$ at time $t$ is $${\operatorname{Tr}}\{\, \rho\, e^{{i}\, t\, {H_{el}}}\,A\, e^{-{i}\, t\, {H_{el}}}\}.$$ Since $\rho$ is a compact, self-adjoint operator, there is an ONB $(\phi_n)_n$ of eigenvectors, with corresponding (positive) eigenvalues $(p_n)_n$. Let $$\label{Vektor-rep}
\sigma(x,\,y) \,=\,\sum_{n=1}^\infty p_n^{1/2}\, \phi_n(x)\, {\overline}{\phi_n(y)}\in {\mathcal{H}_{el}}{\otimes}{\mathcal{H}_{el}}.$$ For $\psi \in {\mathcal{H}_{el}}$ we define $\sigma\, \psi\,:=\, \int \sigma(x,\,y)\,\psi(y)\, d\mu(y)$. Obviously, $\sigma$ is an operator of Hilbert-Schmidt class. Note, ${\overline}{\sigma}\,\psi \,:=\, {\overline}{\sigma\, {\overline}{\psi}}$ has the integral kernel ${\overline}{\sigma(x,\,y)}$. It is a straightforward calculation to verify that $${\operatorname{Tr}}\{ \rho\, e^{{i}\, t\, {H_{el}}}\,A\, e^{-{i}\, t \,{H_{el}}}\}
\,=\, \langle\, e^{-{i}\, t \,{\mathcal{L}_{el}}}\,\sigma \,|
\, (A{\otimes}1)\, e^{-{i}\, t\, {\mathcal{L}_{el}}}\sigma\,\,\rangle_{{\mathcal{H}_{el}}{\otimes}{\mathcal{H}_{el}}},$$ where ${\mathcal{L}_{el}}\,=\, {H_{el}}{\otimes}{\mathbf{1}}- {\mathbf{1}}{\otimes}{\overline}{H}_{el}$. This suggests the definition of the representation $${\pi^{el}}\,:\, \mathcal{B}({\mathcal{H}_{el}})\rightarrow \mathcal{B}({\mathcal{H}_{el}}{\otimes}{\mathcal{H}_{el}}),\quad A\mapsto A{\otimes}{\mathbf{1}}.$$ Now, we define the representation map for the joint system by $$\pi\,:\, {\mathfrak{A}}\, \rightarrow\, \mathcal{B}({\mathcal{K}}),\qquad \pi\,:=\, \pi_{el}{\otimes}\pi_f,$$ where ${\mathcal{K}}\,:=\,{\mathcal{H}_{el}}{\otimes}{\mathcal{H}_{el}}{\otimes}{\mathcal{F}_b}{\otimes}{\mathcal{F}_b}$. Let ${\mathfrak{M}_\beta}\,:=\, \pi[{\mathfrak{A}}]''$ be the *enveloping $W^*$-algebra*, here $\pi[{\mathfrak{A}}]'$ denotes the commutant of $\pi[{\mathfrak{A}}]$, and $\pi[{\mathfrak{A}}]''$ the bicommutant. We set ${\mathcal{D}}:= U_1\otimes \overline{U_1}\otimes \mathcal{C}$, where $\mathcal{C}$ is a subspace of vectors in $\mathcal{F}_{b}^{fin}\otimes\mathcal{F}_{b}^{fin}$, with compact support, and $U_1:=\cup_{n=1}^\infty \operatorname{ran}\,\mathbbm{1}[{\mathcal{H}_{el}}{\leqslant}n]$. On ${\mathcal{D}}$ the operator ${\mathcal{L}_0}$, given by $$\begin{aligned}
{\mathcal{L}_0}&\,:=\,& {\mathcal{L}_{el}}{\otimes}{\mathbf{1}}\,+\, {\mathbf{1}}{\otimes}{ \mathcal{L}_f},\quad \textrm{on } {\mathcal{K}},\\
{ \mathcal{L}_f}&\,:=\,& d\Gamma(\alpha){\otimes}{\mathbf{1}}\,-\, {\mathbf{1}}{\otimes}d\Gamma(\alpha),\quad \textrm{on } {\mathcal{F}_b}{\otimes}{\mathcal{F}_b},\end{aligned}$$ is essentially self-adjoint and we can define $${ \tau^0}_t(X)\,:=\, e^{{i}\,t\, {\mathcal{L}_0}}\,X\, e^{-{i}\,t\, {\mathcal{L}_0}}\in {\mathfrak{M}_\beta},\quad X\in {\mathfrak{M}_\beta},\quad t\in {\mathbbm{R}},$$ It is not hard to see, that $$\pi[\tilde{\tau}^0_t(A)]\,=\,{ \tau^0}_t(\pi[A]),\quad A\in {\mathfrak{A}},\,\quad t\in {\mathbbm{R}}$$ On ${\mathcal{K}}$ a we introduce a conjugation by $${\mathcal{J}}\,( \phi_1 {\otimes}\phi_2 {\otimes}\psi_1 {\otimes}\psi_2)
\,=\,{\overline}{\phi_2} {\otimes}{\overline}{\phi_1} {\otimes}{\overline}{\psi_2} {\otimes}{\overline}{\psi_1 }.$$ It is easily seen, that ${\mathcal{J}}\, {\mathcal{L}_0}\,=\, -{\mathcal{L}_0}\, {\mathcal{J}}$. In this context one has ${\mathfrak{M}_\beta}'\,=\,{\mathcal{J}}\,{\mathfrak{M}_\beta}\,{\mathcal{J}}$, see for example [@BratteliRobinson1987]. In the case, where ${H_{el}}$ fulfills Hypothesis \[Hyp1\], we define the vector representative ${\Omega^\beta_{el}}\in {\mathcal{H}_{el}}{\otimes}{\mathcal{H}_{el}}$ of the Gibbs state ${\omega^\beta_{el}}$ as in for $\rho= e^{-\beta {H_{el}}}\,{\mathcal{Z}}^{-1}$.
\[FreeSystem\] Assume Hypothesis \[Hyp1\] is fulfilled. Then, ${\Omega_0^\beta}\,:=\, {\Omega^\beta_{el}}{\otimes}{\Omega^\beta_{f}}$ is a *cyclic* and *separating* vector for ${\mathfrak{M}_\beta}$. $e^{-\beta/2 {\mathcal{L}_0}}$ is a *modular operator* and ${\mathcal{J}}$ is the *modular conjugation* for ${\Omega_0^\beta}$, that is $$X{\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\, {\mathcal{L}_0}}),\quad {\mathcal{J}}\,X \,\Omega\,=\, e^{-\beta/2 \,{\mathcal{L}_0}}\,X^*\,{\Omega_0^\beta}$$ for all $X\in{\mathfrak{M}_\beta}$ and ${\mathcal{L}_0}\, {\Omega_0^\beta}\,=\,0$. Moreover, $${\omega_0^\beta}(X)\,:=\, \langle\, {\Omega_0^\beta}\,|\, X\,{\Omega_0^\beta}\,\rangle_{{\mathcal{K}}},\qquad X\in {\mathfrak{M}_\beta}$$ is a $({ \tau^0},\,\beta)$-KMS-state for ${\mathfrak{M}_\beta}$, i.e., for all $X,\,Y\in {\mathfrak{M}_\beta}$ exists $F_\beta(X,\,Y,\cdot)$, analytic in the strip $S_\beta\,=\,\{ z\in {\mathbbm{C}}\,:\, 0\,<\,{\textrm{Im}\,}z\, <\beta\}$, continuous on the closure and taking the boundary conditions $$\begin{aligned}
F_\beta(X,\,Y,\,t)&\,=\,&{\omega_0^\beta}(X\,{ \tau^0}_t(Y))\\
F_\beta(X,\,Y,\,t\,+\,{i}\,\beta)&\,=\,&{\omega_0^\beta}({ \tau^0}_t(Y)\,X)\end{aligned}$$
For a proof see [@JaksicPillet1996b].
The Liouvillean ${\mathcal{L}_Q}$
=================================
In this and the next section we will introduce the Standard Liouvillean ${\mathcal{L}_Q}$ for a dynamics ${\tau}$ on ${\mathfrak{M}_\beta}$, describing the interaction between particles and bosons at inverse temperature $\beta$. The label $Q$ denotes the interaction part of the Liouvillean, it can be deduced from the interaction part $W$ of the corresponding Hamiltonian by means of formal arguments, which we will not give here. In a first step we prove self-adjointness of ${\mathcal{L}_Q}$ and of other Liouvilleans. A main difficulty stems from the fact, that ${\mathcal{L}_Q}$ and the other Liouvilleans, mentioned before, are not bounded from below. The proof of self-adjointness is given in Theorem \[Thm1\], it uses Nelson’s commutator theorem and auxiliary operators which are constructed in Lemma \[Lem1.1x\]. The proof, that ${\tau}_t(X)\in {\mathfrak{M}_\beta}$ for $X\in {\mathfrak{M}_\beta}$, is given in Lemma \[LemInv\]. Assuming ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2({\mathcal{L}_0}+{Q})})$ we can ensure existence of a $({\tau},\beta)$-KMS state ${\omega^\beta}(X)=\langle {\Omega^\beta}\,|X\,{\Omega^\beta}\rangle\cdot \|{\Omega^\beta}\|^{-2}$ on ${\mathfrak{M}_\beta}$, where ${\Omega^\beta}=e^{-\beta/2({\mathcal{L}_0}+{Q})}{\Omega_0^\beta}$. Moreover, we can show that $e^{-\beta{\mathcal{L}_Q}}$ is the modular operator for ${\Omega^\beta}$ and conjugation ${\mathcal{J}}$. This is done in Theorem \[Thm3\].\
Our proof of \[Thm3\] is inspired by the proof given in [@DerezinskiJaksicPillet2003]. The main difference is that we do not assume, that ${Q}$ is self-adjoint and that ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta {Q}})$. For this reason we need to introduce an additional approximation ${Q_N}$ of ${Q}$, which is self-adjoint and affiliated with ${\mathfrak{M}_\beta}$, see Lemma \[Lem5d\].
The interaction on the level of Liouvilleans between particles and bosons is given by ${Q}$ , where $${Q}\,:=\, \Phi_\beta(\vec G )\,\Phi_\beta(\vec H )\,+\,{\operatorname{h.c.}}\,+\,\Phi_\beta(F),
\quad \Phi_\beta(\vec G )\,\Phi_\beta(\vec H )\,:=\, \sum_{j=1}^r \Phi_\beta( G^j )\,\Phi_\beta(H^j ).$$ For each family $K=\{K_k\}_{k}$ of closed operators on ${\mathcal{H}_{el}}$ with $\|K\|_{w,1/2}<\infty$ we set $$\Phi_\beta(K)\,:= \,
\big(a^*( (1\,+\,\varrho_\beta)^{1/2}\, K){\otimes}{\mathbf{1}}\,+\, {\mathbf{1}}{\otimes}a^*( \varrho_\beta^{1/2}\, K^*)\big) \,+\, {\operatorname{h.c.}}.$$ Here, $K_k$ acts as $K_k\otimes {\mathbf{1}}$ on ${\mathcal{H}_{el}}\otimes {\mathcal{H}_{el}}$. A Liouvillean, that describes the dynamics of the joint system of particles and bosons is the so-called *Standard Liouvillean* $$\label{Def:Lg}
{\mathcal{L}_Q}\,\phi \,:=\, ({\mathcal{L}_0}\,+\, {Q}\,-\, {Q}^ {\mathcal{J}})\,\phi,\qquad \phi\in {\mathcal{D}},$$ which is distinguished by ${\mathcal{J}}\, {\mathcal{L}_Q}=-{\mathcal{L}_Q}\, {\mathcal{J}}$. For an operator $A$, acting on ${\mathcal{K}}$, the symbol $A^{\mathcal{J}}$ is an abbreviation for ${\mathcal{J}}\,A\,{\mathcal{J}}$. An important observation is, that $[ {Q}\,,\, {Q}^{\mathcal{J}}]\,=\,0$ on ${\mathcal{D}}$. Next, we define four auxiliary operators on ${\mathcal{D}}$ $$\begin{aligned}
\label{eq3.1x}
{\mathcal{L}_{a}}^{(1)}&\,:=\, ({H_{el,+}}{\otimes}{\mathbf{1}}\,+\, {\mathbf{1}}{\otimes}{\overline}{H}_{el,+}\big){\otimes}{\mathbf{1}}+{\mathbf{1}}{\otimes}{ \mathcal{L}_{f,a}}+{\mathbf{1}}\\ \nonumber
{\mathcal{L}_{a}}^{(2)}&\,:=\, {H_{el,+}}^Q+({H_{el,+}}^Q)^{ {\mathcal{J}}} +c_1{\mathbf{1}}{\otimes}{ \mathcal{L}_{f,a}}+c_2\\ \nonumber
{\mathcal{L}_{a}}^{(3)}&\,:=\, {H_{el,+}}^Q+ ({H_{el,+}})^{{\mathcal{J}}}+c_1{\mathbf{1}}{\otimes}{ \mathcal{L}_{f,a}}+c_2\\ \nonumber
{\mathcal{L}_{a}}^{(4)}&\,:=\, {H_{el,+}}{\otimes}{\mathbf{1}}+ ({H_{el,+}}^Q)^{ {\mathcal{J}}}+c_1{\mathbf{1}}{\otimes}{ \mathcal{L}_{f,a}}+c_2,\end{aligned}$$ where ${ \mathcal{L}_{f,a}}$ is an operator on ${\mathcal{F}_b}{\otimes}{\mathcal{F}_b}$ and ${H_{el,+}}^Q$ acts on ${\mathcal{K}}$. Furthermore, $$\begin{aligned}
{ \mathcal{L}_{f,a}}\,&=\, d\Gamma(1+\alpha){\otimes}{\mathbf{1}}+ {\mathbf{1}}{\otimes}d\Gamma(1+\alpha)+{\mathbf{1}},\\
{ \mathcal{L}_{el,a}}\,&=\,{H_{el,+}}{\otimes}{\mathbf{1}}\,+\, {\mathbf{1}}{\otimes}{\overline}{H}_{el,+}
\quad {H_{el,+}}^Q\,:=\, {H_{el,+}}{\otimes}{\mathbf{1}}+{Q}.\end{aligned}$$ Obviously, ${\mathcal{L}_{a}}^{(i)},\ i\,=\,1,\,2,\,3,\,4$ are symmetric operators on ${\mathcal{D}}$.
\[Lem1.1x\] For sufficiently large values of $c_1,\,c_2\,{\geqslant}\,0$ we have that ${\mathcal{L}_{a}}^{(i)},\ i\,=\,1,\,2,\,3,\,4$ are essentially self-adjoint and positive. Moreover, there is a constant $c_3\,>\,0$ such that $$\label{eq3.2}
c_3^{-1}\,\|{\mathcal{L}_{a}}^{(1)}\,\phi\|\,{\leqslant}\,\|{\mathcal{L}_{a}}^{(i)}\,\phi\|
{\leqslant}c_3\,\|{\mathcal{L}_{a}}^{(1)}\,\phi\|,\qquad \phi \in{\operatorname{dom}}({\mathcal{L}_{a}}^{(1)}).$$
Let $a,\,a'\in \{l,\,r\}$ and $K_i,\ i\,=\,1,\,2$ be families of bounded operators with $\|K_i\|_{w}<\infty$. Let $\Phi_l(K_i)\,=\,\Phi(K_i){\otimes}{\mathbf{1}}$ and $\Phi_r(K_i)\,:=\,{\mathbf{1}}{\otimes}\Phi(K_i).$ We have for $\phi \in \mathcal{D}$ $$\begin{aligned}
\|\,\Phi_a(\eta K_1)\,\Phi_{a'}(\eta'K_2)\,\phi\,\|
\,&{\leqslant}\, {\operatorname{const}}\|\,{ \mathcal{L}_{f,a}}\,\phi\,\|\\ \nonumber
\|\,\Phi_a(\eta F)\,\phi\,\| \,&{\leqslant}\, {\operatorname{const}}\| ({ \mathcal{L}_{el,a}})^{1/2}({ \mathcal{L}_{f,a}})^{1/2}\,\phi\,\|,\end{aligned}$$ where $\eta,\,\eta'\in\{ (1+\varrho_\beta)^{1/2},\, \varrho^{1/2}_\beta\}$. Note, that the estimates hold true, if $\Phi_a(\eta K_i)$ or $\Phi_a(\eta F)$ are replaced by ${\Phi_a(\eta K_i)}^{{\mathcal{J}}}$ or ${\Phi_a(\eta F)}^{{\mathcal{J}}}$. Thus, we obtain for sufficiently large $c_1\,\gg\, 1 $, depending on the form-factors, that $$\label{eq3.6}
\|{Q}\,\phi\|\,+\,\|{{Q}}^{{\mathcal{J}}}\,\phi\|\,{\leqslant}\,1/2\,
\big\| \big({ \mathcal{L}_{el,a}}+c_1\,{ \mathcal{L}_{f,a}}\big)\,\phi\big\|.$$ By the Kato-Rellich-Theorem ( [@ReedSimonII1980], Thm. X.12) we deduce that ${\mathcal{L}_{a}}^{(i)}$ is self-adjoint on ${\operatorname{dom}}({ \mathcal{L}_{el,a}}+c_1\,{ \mathcal{L}_{f,a}})$, bounded from below and that ${ \mathcal{L}_{el,a}}+c_1\,{ \mathcal{L}_{f,a}}$ is ${\mathcal{L}_{a}}^{(i)}$-bounded for every $c_2\,{\geqslant}\, 0$ and $i\,=\,2,\,3,\,4$. In particular, ${\mathcal{D}}$ is a core of ${\mathcal{L}_{a}}^{(i)}$. The proof follows now from $\|{\mathcal{L}_{a}}^{(i)}\,\phi\|\,{\leqslant}\, \|({ \mathcal{L}_{el,a}}+c_1\,{ \mathcal{L}_{f,a}})\,\phi\|\,{\leqslant}\,c_1\, \|{\mathcal{L}_{a}}^{(1)}\,\phi\|$ for $\phi\in {\mathcal{D}}$.
\[Thm1\] The operators $$\label{eq3.7}
{\mathcal{L}_0},\quad {\mathcal{L}_Q}\,=\, {\mathcal{L}_0}+{Q}-{{Q}}^{{\mathcal{J}}},
\quad{\mathcal{L}_0}+{Q},\quad {\mathcal{L}_0}-{{Q}}^{{\mathcal{J}}},$$ defined on $\mathcal{D}$, are essentially self-adjoint. Every core of ${\mathcal{L}_{a}}^{(1)}$ is a core of the operators in line .
We restrict ourselves to the case of ${\mathcal{L}_Q}$. We check the assumptions of Nelson’s commutator theorem ([@ReedSimonII1980], Thm. X.37). By Lemma \[Lem1.1\] it suffices to show $\|{\mathcal{L}_Q}\phi\|\,{\leqslant}\,{\operatorname{const}}\|{\mathcal{L}_{a}}^{(1)}\phi\|$ and $|\langle\, {\mathcal{L}_Q}\phi\,|\,{\mathcal{L}_{a}}^{(2)}\phi \rangle
-\langle\, {\mathcal{L}_{a}}^{(2)}\phi|\,{\mathcal{L}_Q}\phi \rangle |
\,{\leqslant}\,{\operatorname{const}}\|({\mathcal{L}_{a}}^{(1)})^{1/2}\phi\|^2$ for $\phi\in \mathcal{D}$. The first inequality follows from Equation . To verify the second inequality we observe $$\begin{aligned}
\label{eq3.9}
\big|&\big\langle\, {\mathcal{L}_Q}\,\phi\,\big| \,{\mathcal{L}_{a}}^{(2)}\,\phi \,\big\rangle
-\big \langle\, {\mathcal{L}_{a}}^{(2)}\,\phi\,\big| \,{\mathcal{L}_Q}\,\phi \,\big\rangle \big|\\ \nonumber
&{\leqslant}c_1 \Big|\big\langle\, {Q}\phi\,\big|\,{ \mathcal{L}_{f,a}}\phi \big\rangle
-\big \langle\, { \mathcal{L}_{f,a}}\phi\, \big|\, {Q}\phi\big\rangle\Big|\\ \nonumber
&\phantom{{\leqslant}}+ c_1 \Big|\big\langle\, {{Q}}^{{\mathcal{J}}} \phi\,\big|\, { \mathcal{L}_{f,a}}\phi\,\big\rangle
- \big\langle\, { \mathcal{L}_{f,a}}\phi\,\big|\,{{Q}}^{{\mathcal{J}}} \phi\,\big\rangle\Big|\\ \nonumber
&\phantom{{\leqslant}} + \Big|\big\langle\, { \mathcal{L}_f}\phi \big|\,{Q}\phi\,\big\rangle
- \langle\, {Q}\phi\,\big|\, { \mathcal{L}_f}\phi\,\big\rangle\Big|
+ \Big|\big\langle\, { \mathcal{L}_f}\phi\,\big|\,{{Q}}^{{\mathcal{J}}} \phi\,\big\rangle
-\big\langle\, {{Q}}^{{\mathcal{J}}} \phi\,\big|\, { \mathcal{L}_f}\phi\,\big\rangle \Big|,\end{aligned}$$ where we used, that $
\big[{H_{el,+}}^Q ,\, ({H_{el,+}}^Q)^{{\mathcal{J}}}\big]\,=\,0.
$ Let $K_i\in \{ G_j,\,H_j\}$ and $\eta,\,\eta' \in \{\varrho^{1/2},\, (1\,+\,\varrho)^{1/2}\}$. We remark, that $$\begin{aligned}
\label{eq3.10}
[\Phi_a(\eta\, K_1)\,\Phi_{a'}(\eta'\, K_2)\,,\,{ \mathcal{L}_{f,a}}]
&\,=\,{i}\, \Phi_a({i}\,(1\,+\,\alpha)\, \eta\, K_1)\,\Phi_{a'}v(\eta'\,K_2)\\ \nonumber
&\,+\,{i}\,\Phi_a(\eta\, K_1)\,\Phi_{a'}({i}\,(1\,+\,\alpha)\, \eta'\,K_2)\\ \nonumber
[\Phi_a(\eta \,F)\,,\ { \mathcal{L}_{f,a}}]&\,=\, {i}\,\Phi_a({i}\,(1\,+\,\alpha) \,\eta\, F).\end{aligned}$$ Hence, for $\phi\in {\operatorname{dom}}({\mathcal{L}_{a}}^{(2)})$, we have by means of that $$\begin{aligned}
\label{eq3.11}
\big|\big \langle\, \phi\,|\, [\Phi_a(\eta K_1)\,\Phi_{a'}(\eta' K_2),\ { \mathcal{L}_{f,a}}]\phi \,\big \rangle \big|
\,&{\leqslant}\, {\operatorname{const}}\|{ \mathcal{L}_{f,a}}^{1/2}\phi \|^2\\ \nonumber
\big|\big \langle \phi\,|\, [\Phi_a(\eta F),\ { \mathcal{L}_{f,a}}]\phi\, \big \rangle \big|
\,&{\leqslant}\, {\operatorname{const}}\|{ \mathcal{L}_{f,a}}^{1/2}\phi \|\,\| ({ \mathcal{L}_{el,a}})^{1/2}\phi\|.\end{aligned}$$ Thus, is bounded by a constant times $ \|({\mathcal{L}_{a}}^{(1)})^{1/2}\phi\|^2$. The essential self-adjointness of ${\mathcal{L}_Q}$ follows now from estimates analog to and , where ${ \mathcal{L}_{f,a}}$ is replaced by ${ \mathcal{L}_f}$ in and in the left side of $\eqref{eq3.11}$. For ${\mathcal{L}_0}+{Q}$ and ${\mathcal{L}_0}-{{Q}}^{{\mathcal{J}}}$ one has to consider the commutator with ${\mathcal{L}_{a}}^{(3)}$ and ${\mathcal{L}_{a}}^{(4)}$, respectively.
\[RemarkSelfadjoint\] In the same way one can show, that ${H}$ is essentially self-adjoint on any core of $H_1:={H_{el}}\,+\, d\Gamma(1\,+\,\alpha)$, even if ${H}$ is not bounded from below.
Regularized Interaction and Standard Form of ${\mathfrak{M}_\beta}$ {#RegVer}
===================================================================
In this subsection a regularized interaction ${Q_N}$ is introduced: $$\label{DefQN}
{Q_N}\,:=\, \Big\{\Phi_\beta(\vec G_{N})\,\Phi_\beta(\vec H_{N})\,+\, {\operatorname{h.c.}}\Big\}
\,+\,\Phi_\beta(F_{N}).$$ The regularized form factors $\vec G_{N},\,\vec H_{N},\,F_{N}$ are obtained by multiplying the finite rank projection $P_N:=\mathbf{1}[{H_{el}}{\leqslant}N]$ from the left and the right. Moreover, an additional ultraviolet cut-off ${\mathbf{1}}[\alpha{\leqslant}N]$, considered as a spectral projection, is added. The regularized form factors are $$\begin{gathered}
\nonumber
\vec G_{N}(k)\,:= \,{\mathbf{1}}[ \alpha \,{\leqslant}\,N]\, P_N \,\vec G(k)\,P_N,\qquad
\vec H_{N}(k)\,:=\, {\mathbf{1}}[ \alpha \,{\leqslant}\, N]\,P_N \,\vec H(k)\,P_N,\\ F_{N}(k)\,:= \,{\mathbf{1}}[ \alpha\, {\leqslant}\, N]\,P_N\,F(k)\,P_N.\end{gathered}$$
\[Lem5d\] i) ${Q_N}$ is essentially self-adjoint on $\mathcal{D} \subset{\operatorname{dom}}({Q_N})$. ${Q_N}$ is affiliated with ${\mathfrak{M}_\beta}$, i.e,. ${Q_N}$ is closed and $$X'\,{Q_N}\subset {Q_N}\,X',\quad \forall\, X'\in {\mathfrak{M}_\beta}'.$$ ii) ${\mathcal{L}_0}+{Q_N}$, ${\mathcal{L}_0}-{\mathcal{J}}{Q_N}{\mathcal{J}}$ and ${\mathcal{L}_0}+{Q_N}-{\mathcal{J}}{Q_N}{\mathcal{J}}$ converges in the strong resolvent sense to ${\mathcal{L}_0}+{Q}$, ${\mathcal{L}_0}-{\mathcal{J}}{Q}{\mathcal{J}}$ and ${\mathcal{L}_0}+{Q}-{\mathcal{J}}{Q}{\mathcal{J}}$, respectively.\
Let ${Q_N}$ be defined on $\mathcal{D}$. With the same arguments as in the proof of Theorem \[Thm1\] we obtain $$\|{Q_N}\phi\|\,{\leqslant}\, C \|{ \mathcal{L}_{f,a}}\phi\|,
\quad \big|\big\langle\, {Q_N}\phi\,\big|\,{ \mathcal{L}_{f,a}}\phi \big\rangle
\,-\,\big\langle\, { \mathcal{L}_{f,a}}\phi\,\big|\,{Q_N}\phi \big\rangle \big|\,
{\leqslant}\,C \big\|({ \mathcal{L}_{f,a}})^{1/2}\phi\big\|^2,$$ for $\phi \in \mathcal{D}$ and some constant $C>0$, where we have used that $\|F_N\|_{w}\,<\,\infty$. Thus, from Theorem \[Thm1\] and Nelson’s commutator theorem we obtain that $\mathcal{D}$ is a common core for ${Q_N},\ {\mathcal{L}_0}+{Q_N},\ {\mathcal{L}_0}\,-\,{Q_N}^{{\mathcal{J}}},\ {\mathcal{L}_0}\,+\,{Q_N}\,-{Q_N}^{{\mathcal{J}}}$ and for the operators in line . A straightforward calculation yields $$\lim_{N\rightarrow\infty}\,{Q_N}\phi
\,=\,{Q}\phi,\quad
\lim_{N\rightarrow\infty}\,{\mathcal{J}}{Q_N}{\mathcal{J}}\phi
\,=\,{\mathcal{J}}{Q}{\mathcal{J}}\phi
\qquad \forall\, \phi\in \mathcal{D}.$$ Thus statement ii) follows.\
Let $N_f\,:=\, d\Gamma(1){\otimes}{\mathbf{1}}+ {\mathbf{1}}{\otimes}d\Gamma({\mathbf{1}})$ be the number-operator. Since ${\operatorname{dom}}(N_f)\supset \mathcal{D}$ and ${{ \mathcal{W}}_\beta(f)}^{{\mathcal{J}}}\,:\,{\operatorname{dom}}(N_f)\,\rightarrow\, {\operatorname{dom}}(N_f)$, see [@BratteliRobinson1987], we obtain $${Q_N}(A{\otimes}{\mathbf{1}}{\otimes}{ \mathcal{W}}_\beta(f))^{{\mathcal{J}}} \phi\,=\, (A{\otimes}{\mathbf{1}}{\otimes}{ \mathcal{W}}_\beta(f))^{{\mathcal{J}}}{Q_N}\phi$$ for $A\in\mathcal{B}({\mathcal{H}_{el}}),\ f\in {\mathfrak{f}}$ and $\phi \in \mathcal{D}$. By closedness of ${Q_N}$ and density arguments the equality holds for $\phi \in {\operatorname{dom}}({Q_N})$ and $X\in{\mathfrak{M}_\beta}$ instead of $A{\otimes}{\mathbf{1}}{\otimes}{ \mathcal{W}}_\beta(f)$. Thus ${Q_N}$ is affiliated with ${\mathfrak{M}_\beta}$ and therefore $e^{{i}\,t {Q_N}}\in {\mathfrak{M}_\beta}$ for $t\in {\mathbbm{R}}$.\
\[LemInv\] We have for $X\in {\mathfrak{M}_\beta}$ and $t\in{\mathbbm{R}}$ $$\begin{aligned}
{\tau}_t(X)= e^{{i}t({\mathcal{L}_0}+{Q})}\,X\,e^{{i}t({\mathcal{L}_0}+{Q})},\quad
{ \tau^0}_t(X)= e^{{i}t({\mathcal{L}_0}-{Q}^{\mathcal{J}})}\,X\,e^{{i}t({\mathcal{L}_0}-{Q}^{\mathcal{J}})}\end{aligned}$$ Moreover, ${\tau}_t(X)\in {\mathfrak{M}_\beta}$ for all $X\in {\mathfrak{M}_\beta}$ and $t\in {\mathbbm{R}}$, such as $$E_{{Q}}(t)\,:= \,e^{{i}\,t \,({\mathcal{L}_0}\,+\,{Q})}\,e^{-{i}\, t\,{\mathcal{L}_0}}=
e^{{i}\,t \,{\mathcal{L}_Q}}\,e^{-{i}\, t\,({\mathcal{L}_0}-{Q}^{{\mathcal{J}}})}
\in {\mathfrak{M}_\beta}.$$
First, we prove the statement for ${Q_N}$, since ${Q_N}$ is affiliated with ${\mathfrak{M}_\beta}$ and therefore $e^{{i}t{Q_N}}\in {\mathfrak{M}_\beta}$. We set $$\begin{gathered}
\hat{\tau}_t^N(X)=e^{{i}\, t\,({\mathcal{L}_0}\,+\,{Q_N})}\,X \,e^{-{i}\,t\,({\mathcal{L}_0}\,+\,{Q_N})},\quad
\hat{\tau}_t(X)=e^{{i}\, t\,({\mathcal{L}_0}\,+\,{Q})}\,X \,e^{-{i}\,t\,({\mathcal{L}_0}\,+\,{Q})}\end{gathered}$$ On account of Lemma \[Lem5d\] and Theorem \[Thm1\] we can apply the Trotter product formula to obtain $$\begin{aligned}
\hat{\tau}_t^N(X)&
= {\operatorname{w-lim}}_{n\,\to \,\infty}
\big(e^{{i}\,\frac{t}{n}{\mathcal{L}_0}}\,e^{{i}\,\frac{t}{n}{Q_N}} \big)^n
X \big(e^{-{i}\, \frac{t}{n}{Q_N}}\,e^{-{i}\,\frac{t}{n}{\mathcal{L}_0}} \big)^n\\ \nonumber
&= {\operatorname{w-lim}}_{n\to \infty}
{ \tau^0}_{\frac{t}{n}}\big(
e^{{i}\,\frac{t}{n}\,{Q_N}}\cdots
{ \tau^0}_{\frac{t}{n}}(e^{{i}\, \frac{t}{n}\,{Q_N}}\,X\, e^{-{i}\,\frac{t}{n}{Q_N}})
\cdots e^{-{i}\,\frac{t}{n}{Q_N}}
\big).\end{aligned}$$ Since $e^{{i}\,\frac{t}{n}{Q_N}}, X\in {\mathfrak{M}_\beta}$ and since ${ \tau^0}$ leaves ${\mathfrak{M}_\beta}$ invariant, $\hat{\tau}^N_t(X)$ is the weak limit of elements of ${\mathfrak{M}_\beta}$, and hence $\hat{\tau}^N_t(X)\in {\mathfrak{M}_\beta}$. Moreover, $$\hat{\tau}_t(X)\,=\,{\operatorname{w-lim}}_{N\,\rightarrow\, \infty} \hat{\tau}^N_t(X)\in {\mathfrak{M}_\beta}.$$ For $E_N(t)\,:=\, e^{{i}\,t\,({\mathcal{L}_0}\,+\,{Q_N})}e^{-{i}\,t \,{\mathcal{L}_0}} \in \mathcal{B}({\mathcal{K}})$ we obtain $$\begin{aligned}
e^{{i}t ({\mathcal{L}_0}\,+\,{Q_N})}e^{-{i}\,t\,{\mathcal{L}_0}}
&= {\operatorname{s-lim}}_{n\rightarrow \infty}\big(e^{{i}\, \frac{t}{n}\, {\mathcal{L}_0}}
e^{{i}\, \frac{t}{n}\, {Q_N}}\big)^n\,e^{-{i}\,t\, {\mathcal{L}_0}}\\ \nonumber
&={\operatorname{s-lim}}_{n\rightarrow \infty}{ \tau^0}_{\frac{t}{n}}(e^{{i}\,\frac{t}{n}\, {Q_N}})
{ \tau^0}_{\frac{2t}{n}}(e^{{i}\, \frac{t}{n}\, {Q_N}}) \cdots
{ \tau^0}_{\frac{nt}{n}}(e^{{i}\,\frac{t}{n}\,{Q_N}})\in {\mathfrak{M}_\beta}.\end{aligned}$$ By virtue of Lemma \[Lem5d\] we get $E_{{Q}}(t)\,:=\, e^{{i}\,t\,({\mathcal{L}_0}\,+\,{Q})}\,e^{-{i}\,t \,{\mathcal{L}_0}}\,
=\,{\operatorname{w-lim}}_{N\,\rightarrow\,\infty}\, E_N(t)\in {\mathfrak{M}_\beta}$. Since ${\mathcal{J}}$ leaves ${\mathcal{D}}$ invariant and thanks to Lemma \[Lem5d\], we deduce, that ${\mathcal{D}}$ is a core of ${\mathcal{J}}{Q_N}{\mathcal{J}}$. Moreover, we have $ e^{-{i}t {Q_N}^{\mathcal{J}}}={\mathcal{J}}e^{{i}t {Q_N}}{\mathcal{J}}\in {\mathfrak{M}_\beta}'$. Since we have shown, that $\hat{\tau}^N$ leaves ${\mathfrak{M}_\beta}$ invariant, we get $$\begin{aligned}
{\tau^N}_t(X)&= {\operatorname{w-lim}}_{n\to \infty}
(e^{{i}\frac{t}{n}({\mathcal{L}_0}+{Q_N})}
e^{{i}\frac{t}{n}(-{Q_N}^{\mathcal{J}})})^n
X\,(e^{-{i}\frac{t}{n}(-{Q_N}^{\mathcal{J}})}\,
e^{-{i}\frac{t}{n}({\mathcal{L}_0}+{Q_N})})^n \\
&={\operatorname{w-lim}}_{n\to \infty}\hat{\tau}^N_{\frac{t}{n}}\big(
e^{-{i}\,\frac{t}{n}\,{Q_N}^{\mathcal{J}}}\cdots
\hat{\tau}^N_{\frac{t}{n}}(e^{-{i}\, \frac{t}{n}\,{Q_N}^{\mathcal{J}}}
\,X\, e^{{i}\,\frac{t}{n}{Q_N}^{\mathcal{J}}})
\cdots e^{{i}\,\frac{t}{n}{Q_N}^{\mathcal{J}}}\big)\\
&= \hat{\tau}^N_t(X).\end{aligned}$$ Thanks to Lemma \[Lem5d\] we also have $${\tau}_t(X)={\operatorname{w-lim}}_{n\to\infty}{\tau^N}_t(X)={\operatorname{w-lim}}_{{\mathbbm{N}}\to\infty}\hat{\tau}^N_t(X)=\hat{\tau}_t(X).$$ The proof of ${ \tau^0}_t(X)= e^{{i}t({\mathcal{L}_0}-{Q}^{\mathcal{J}})}\,X\,e^{{i}t({\mathcal{L}_0}-{Q}^{\mathcal{J}})}$ follows analogously. Using the Trotter product formula we obtain $$\begin{aligned}
e^{{i}t ({\mathcal{L}_0}+{Q_N})}\,e^{-{i}t {\mathcal{L}_0}}
&\,=\,& {\operatorname{s-lim}}_{n\,\to\, \infty}\big(e^{{i}\frac{t}{n}\, {\mathcal{L}_0}}
e^{{i}\frac{t}{n}{Q_N}}\big)^n\,e^{-{i}t {\mathcal{L}_0}}\\ \nonumber
&=&{\operatorname{s-lim}}_{n\,\to\, \infty} { \tau^0}_{\frac{t}{n}}(e^{{i}\frac{t}{n} {Q_N}})
{ \tau^0}_{\frac{2t}{n}}(e^{{i}\frac{t}{n} {Q_N}}) \cdots
{ \tau^0}_{\frac{nt}{n}}(e^{{i}\frac{t}{n} {Q_N}})\\ \nonumber
&=& {\operatorname{s-lim}}_{n\,\to \,\infty}\big(e^{{i}\frac{t}{n}({\mathcal{L}_0}-{{Q_N}}^{{\mathcal{J}}})}
e^{{i}\frac{t}{n} {Q_N}}\big)^n\,e^{-{i}t ({\mathcal{L}_0}-{{Q_N}}^{{\mathcal{J}}})}\\ \nonumber
&=& e^{{i}t ({\mathcal{L}_0}+{Q_N}-{\mathcal{J}}{Q_N}{\mathcal{J}})}\,e^{-{i}t ({\mathcal{L}_0}-{{Q_N}}^{{\mathcal{J}}})}.\end{aligned}$$ By strong resolvent convergence we may deduce $E(t)=e^{{i}t {\mathcal{L}_Q}}\,e^{-{i}t ({\mathcal{L}_0}-{{Q}}^{{\mathcal{J}}})}$ .
Let $\mathcal{C}$ be the natural positive cone associated with ${\mathcal{J}}$ and ${\Omega_0^\beta}$ and let ${\mathfrak{M}_\beta^{ana}}$ be the ${\tau}$-analytic elements of ${\mathfrak{M}_\beta}$, (see [@BratteliRobinson1987]).
\[Thm3\] Assume Hypothesis \[Hyp1\] and ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}\,+\,{Q})})$. Let ${\Omega^\beta}\,:=\, e^{-\beta/2\,({\mathcal{L}_0}\,+\,{Q})}\,{\Omega_0^\beta}$. Then $$\begin{gathered}
{\mathcal{J}}\, {\Omega^\beta}\,=\,{\Omega^\beta},\qquad
{\Omega^\beta}\,=\, e^{\beta/2\,({\mathcal{L}_0}\,-\,{{Q}}^{{\mathcal{J}}})}\,{\Omega_0^\beta},\\ \nonumber
{\mathcal{L}_Q}\,{\Omega^\beta}\,=\,0,\qquad
{\mathcal{J}}\, X^*\,{\Omega^\beta}\,= \,e^{-\beta/2\,{\mathcal{L}_Q}}X\,{\Omega^\beta},\quad \forall\,X\in{\mathfrak{M}_\beta}\end{gathered}$$ Furthermore, ${\Omega^\beta}$ is separating and cyclic for ${\mathfrak{M}_\beta}$, and ${\Omega^\beta}\in\mathcal{C}$. The state ${\omega^\beta}$ is defined by $${\omega^\beta}(X):= \|{\Omega^\beta}\|^{-2}\,\langle {\Omega^\beta}\,|X\,{\Omega^\beta}\rangle,\ X\in {\mathfrak{M}_\beta}$$ is a $({\tau},\, \beta)$-KMS state on ${\mathfrak{M}_\beta}$.
First, we define $\Omega(z)\,=\, e^{-z\,({\mathcal{L}_0}+{Q})}\,{\Omega_0^\beta}$ for $ z\in {\mathbbm{C}}$ with $0{\leqslant}{\textrm{Re}\,}z{\leqslant}\beta/2$. Since ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}+{Q})})$, $\Omega(z)$ is analytic on $\mathcal{S}_{\beta/2}:=
\{ z\in {\mathbbm{C}}\,:\, 0\,<\, {\textrm{Re}\,}(z)\,<\,\alpha\}$ and continuous on the closure of $\mathcal{S}_{\beta/2}$, see Lemma \[Lem2App\] below.\
$\blacktriangleright$ Proof of ${\mathcal{J}}\,\Omega(\beta/2)\,=\,\Omega(\beta/2)$:\
We pick $\phi \in \bigcup_{n\in{\mathbbm{N}}} {\operatorname{ran}}{\mathbf{1}}[|{\mathcal{L}_0}|\,{\leqslant}\,n]$. Let $
f(z)\,:=\,\langle\, \phi\,|\, {\mathcal{J}}\,\Omega(\overline{z})\,\rangle$ and $
g(z)\,:=\,\langle\, e^{-(\beta/2\,-\,\overline{z})\,{\mathcal{L}_0}}\,\phi\,|\, e^{-z\,({\mathcal{L}_0}\,+\,{Q})}\,{\Omega_0^\beta}\,\rangle$. Both $f$ and $g$ are analytic on $\mathcal{S}_{\beta/2}$ and continuous on its closure. Thanks to Lemma \[LemInv\] we have $E_{{Q}}(t)\in {\mathfrak{M}_\beta}$, and hence $$f({i}t)\,=\, \langle\, \phi\,|\, {\mathcal{J}}\, E_{{Q}}(t)\,{\Omega_0^\beta}\,\rangle
\,=\,\langle\, \phi\,|\, e^{-\beta/2 \,{\mathcal{L}_0}}\, E_{{Q}}(t)^*\,{\Omega_0^\beta}\,\rangle
\,=\, g({i}\,t),\ t\in {\mathbbm{R}}.$$ By Lemma \[Lem1App\], $f$ and $g$ are equal, in particular in $z\,=\,\beta/2$. Note that $\phi$ is any element of a dense subspace.\
$\blacktriangleright$ Proof of ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{\beta/2\,({\mathcal{L}_0}- {{Q}}^{{\mathcal{J}}})})$ and $\Omega(\beta/2)\,=\, e^{\beta/2\,({\mathcal{L}_0}-{{Q}}^{{\mathcal{J}}})}\,{\Omega_0^\beta}$:\
Let $\phi \in \bigcup_{n\in{\mathbbm{N}}} {\operatorname{ran}}{\mathbf{1}}[|{\mathcal{L}_0}\,-{{Q}}^{{\mathcal{J}}}|\,{\leqslant}\, n]$. We set $g(z)\, :=\, \langle\, e^{{\overline}{z}({\mathcal{L}_0}\,-{{Q}}^{{\mathcal{J}}})}\,\phi\,|\, e^{-z\,{\mathcal{L}_0}}\,{\Omega_0^\beta}\,\rangle$. Since ${E_{{Q}}(t)}^{{\mathcal{J}}}\,=\, e^{{i}\,t\,({\mathcal{L}_0}-{{Q}}^{{\mathcal{J}}})}\,e^{-{i}\,t\,{\mathcal{L}_0}}$, $g$ coincides for $z\,=\,{i}\,t$ with $f(z)\,:=\,\langle\, \phi\,| \,{\mathcal{J}}\,\Omega({\overline}{z})\,\rangle $. Hence they are equal in $z\,=\,\beta/2$. The rest follows since $e^{\beta/2\,({\mathcal{L}_0}- {{Q}}^{{\mathcal{J}}})}$ is self-adjoint.\
$\blacktriangleright$ Proof of ${\mathcal{L}_Q}\,\Omega(\beta/2)\,=\,0$:\
Choose $\phi \in \bigcup_{n\in{\mathbbm{N}}} {\operatorname{ran}}{\mathbf{1}}[|{\mathcal{L}_Q}|\,{\leqslant}\,n]$. We define $g(z)\,:=\, \langle\, e^{-{\overline}{z}{\mathcal{L}_Q}}\phi \,|\,e^{z\,({\mathcal{L}_0}-{{Q}}^{{\mathcal{J}}})}\,{\Omega_0^\beta}\,\rangle$ and $f(z)\,:= \, \langle\, \phi\,|\, \Omega(z)\,\rangle$ for $z$ in the closure of $ \mathcal{S}_{\beta/2}$. Again both functions are equal on the line $z\,=\,{i}\,t,\ t\in {\mathbbm{R}}$. Hence $f$ and $g$ are identical, and therefore $\Omega(\beta/2)\in {\operatorname{dom}}(e^{-\beta/2\, {\mathcal{L}_Q}})$ and $e^{-\beta/2 {\mathcal{L}_Q}}\,\Omega(\beta/2)\,=\,\Omega(\beta/2)$. We conclude that ${\mathcal{L}_Q}\,\Omega(\beta/2)\,=\,0$.\
$\blacktriangleright$ Proof of ${\mathcal{J}}\, X^*\,\Omega(\beta/2)\,
=\, e^{-\beta/2{\mathcal{L}_Q}}\,X\,\Omega(\beta/2),\ \forall\,X\in {\mathfrak{M}_\beta}$:\
Fore $A\in {\mathfrak{M}_\beta^{ana}}$ we have, that $$\begin{aligned}
\nonumber
{\mathcal{J}}\, A^*\,\Omega(-{i}t)
\,&=\,{\mathcal{J}}\,A^*\,E_{{Q}}(t)\,{\Omega_0^\beta}\,=\,e^{-\beta/2 {\mathcal{L}_0}}\, E_{{Q}}(t)^*A{\Omega_0^\beta}\\
&=\,e^{-(\beta/2\,-\,{i}\,t)\, {\mathcal{L}_0}}\,e^{-{i}t({\mathcal{L}_0}+{Q})} A\,{\Omega_0^\beta}\\
&=e^{-(\beta/2-{i}t) {\mathcal{L}_0}}\,{\tau}_{-t}(A)\,e^{-{i}t({\mathcal{L}_0}+ {Q})} \,{\Omega_0^\beta}.\end{aligned}$$ Let $\phi \in \bigcup_{n\in{\mathbbm{N}}} {\operatorname{ran}}{\mathbf{1}}[|{\mathcal{L}_0}|{\leqslant}n]$. We define $
f(z)\,=\, \langle\, \phi\,|\,{\mathcal{J}}\, A^*\,\Omega({\overline}{z})\,\rangle$ and $g(z)\,=\, \langle\, e^{-(\beta/2-{\overline}{z}) {\mathcal{L}_0}}\,\phi\,|\,
{\tau}_{{i}z}(A)\,\Omega(z)\,\rangle$. Since $f$ and $g$ are analytic and equal for $z\,=\, {i}t$, we have ${\mathcal{J}}A^*\Omega(\beta/2)\,=\,{\tau}_{{i}\, \beta/2}(A)\,\Omega(\beta/2)$. To finish the proof we pick $\phi \in \bigcup_{n\in{\mathbbm{N}}} {\operatorname{ran}}{\mathbf{1}}[|{\mathcal{L}_Q}|\,{\leqslant}\,n]$, and set $f(z)\, :=\, \langle\, \phi\,|\,{\tau}_{{i}z}( A)\,\Omega(\beta/2)\,\rangle$ and $ g(z)\,:=\, \langle\, e^{-\overline{z}{\mathcal{L}_Q}}\phi\,|\,A\Omega(\beta/2)\,\rangle$. For $z\,=\,{i}\,t$ we see $$g({i}t)\,=\,\langle\, \phi\,|
\, e^{-{i}t{\mathcal{L}_Q}}\,A\, e^{{i}t{\mathcal{L}_Q}}\Omega(\beta/2)\,\rangle
\,=\,\langle\, \phi\,|\, {\tau}_{-t}(A)\,\Omega(\beta/2)\,\rangle
\,=\,f({i}\,t).$$ Hence $A\,\Omega(\beta/2)\in {\operatorname{dom}}(e^{-\beta/2\,{\mathcal{L}_Q}})$ and ${\mathcal{J}}A^*\Omega(\beta/2)\,=\, e^{-\beta/2{\mathcal{L}_Q}}A\Omega(\beta/2)$.\
Since ${\mathfrak{M}_\beta^{ana}}$ is dense in the strong topology, the equality holds for all $X\in {\mathfrak{M}_\beta}$.\
$\blacktriangleright$ Proof, that ${\Omega^\beta}$ is separating for ${\mathfrak{M}_\beta}$:\
Let $A\in {\mathfrak{M}_\beta^{ana}}$. We choose $\phi \in \bigcup_{n\in{\mathbbm{N}}} {\operatorname{ran}}{\mathbf{1}}[|({\mathcal{L}_0}+Q)|\,{\leqslant}\, n]$. First, we have $${\mathcal{J}}\, A^*\,\Omega(\beta/2)\,=\,{\tau}_{{i}\beta/2}(A)\,\Omega(\beta/2).$$ Let $f_\phi(z)=\langle \phi| {\tau}_z(A)\Omega(\beta/2)\rangle$ and $g_\phi(z)=\langle e^{{\overline}{z}({\mathcal{L}_0}+{Q})}\phi\,
|\,A e^{-(\beta/2+z)({\mathcal{L}_0}+{Q})}{\Omega_0^\beta}\rangle$\
for $-\beta/2\,{\leqslant}\,{\textrm{Re}\,}z \,{\leqslant}\,0$. Both functions are continuous and analytic if $-\beta/2\,<\, {\textrm{Re}\,}z \,< 0$. Furthermore, $f_\phi({i}\,t)\,=\,g_\phi({i}\,t)$ for $t\in {\mathbbm{R}}$. Hence $f_\phi\,=\,g_\phi$ and for $z\,=\,-\beta/2$ $$\langle\, \phi\,|\,{\mathcal{J}}\,A^*\Omega(\beta/2)\,\rangle
\,=\,\langle\, e^{-\beta/2\,({\mathcal{L}_0}+{Q})}\phi\,|\,A{\Omega_0^\beta}\,\rangle.$$ This equation extends to all $A\in {\mathfrak{M}_\beta}$, we obtain $A\,{\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}+{Q})})$, such as $e^{-\beta/2({\mathcal{L}_0}+{Q})}\,A\,{\Omega_0^\beta}\,=\,{\mathcal{J}}\, A^*\Omega(\beta/2)$ for $A\in {\mathfrak{M}_\beta}$. Assume $A^*\,\Omega(\beta/2)\,=\,0$, then\
$e^{-\beta/2\,({\mathcal{L}_0}+{Q})}A{\Omega_0^\beta}\,=\,0$ and hence $A{\Omega_0^\beta}\,=\,0$. Since ${\Omega_0^\beta}$ is separating, it follows that $A=0$ and therefore $A^*\,=\,0$.\
$\blacktriangleright$ Proof of ${\Omega^\beta}\in\mathcal{C}$, and that ${\Omega^\beta}$ is cyclic for ${\mathfrak{M}_\beta}$:\
To prove that $\phi\in \mathcal{C}$ it is sufficient to check that $\langle\, \phi\,|\, A{\mathcal{J}}A{\Omega_0^\beta}\rangle \,{\geqslant}\,0$ for all $A\in{\mathfrak{M}_\beta}$. We have $$\begin{aligned}
\langle\, \Omega(\beta/2)|\, A {\mathcal{J}}A\,{\Omega_0^\beta}\,\rangle\,
&=\, {\overline}{\langle\, {\mathcal{J}}A^*\Omega(\beta/2)\,| A{\Omega_0^\beta}\,\rangle}\\
\,&=\, {\overline}{\langle\, e^{-\beta/2({\mathcal{L}_0}+{Q})} \,A{\Omega_0^\beta}\,|\, A{\Omega_0^\beta}\,\rangle}\,{\geqslant}\, 0.\end{aligned}$$ The proof follows, since every separating element of $\mathcal{C}$ is cyclic. $\blacktriangleright$ Proof, that ${\omega^\beta}$ is a $({\tau},\,\beta)$-KMS state:\
For $A,\,B\in {\mathfrak{M}_\beta}$ and $z\in S_{\beta}$ we define $$F_\beta(A,\,B,\,z)\,
=\, c\,\langle\, e^{-{i}{\overline}{z}/2{\mathcal{L}_Q}}A^*{\Omega^\beta}\,|\,e^{{i}z/2{\mathcal{L}_Q}}B{\Omega^\beta}\,\rangle,$$ where $c\,:=\, \|{\Omega^\beta}\|^{-2}$. First, we observe $$\begin{aligned}
F_\beta(A,\,B,\,t)&\,=\,c\,\langle\, e^{-{i}t/2{\mathcal{L}_Q}}A^*{\Omega^\beta}\,|\,e^{{i}t/2{\mathcal{L}_Q}}B{\Omega^\beta}\,\rangle
\,=\,c\,\langle\, {\Omega^\beta}\,|A{\tau}_t(B){\Omega^\beta}\,\rangle\\ \nonumber
&\,=\,{\omega^\beta}(A\,{\tau}_t(B))\end{aligned}$$ and $$\begin{aligned}
{\omega^\beta}({\tau}_t(B)A)&\,=\,c\,\langle\, {\tau}_t(B^*){\Omega^\beta}\,|\,A{\Omega^\beta}\,\rangle
\,=\,c\,\langle\, {\mathcal{J}}A{\Omega^\beta}\,|\,{\mathcal{J}}{\tau}_t(B^*){\Omega^\beta}\,\rangle \\ \nonumber
&\,=\,c\,\langle\, e^{-\beta/2 {\mathcal{L}_Q}}A^*{\Omega^\beta}\,|\,e^{-\beta/2\, {\mathcal{L}_Q}}{\tau}_t(B){\Omega^\beta}\,\rangle\\
&\,=\,c\,\langle\, e^{-{i}{\overline}{({i}\beta+t)}/2 {\mathcal{L}_Q}}A^*{\Omega^\beta}\,|
\,e^{{i}\,({i}\beta+t)/2 {\mathcal{L}_Q}}B{\Omega^\beta}\,\rangle\\ \nonumber
&\,=\,F_\beta(A,\,B,t+{i}\beta).\end{aligned}$$ The requirements on the analyticity of $F_\beta(A,\,B,\,\cdot)$ follow from Lemma \[Lem2App\].
Proof of Theorem \[Thm2\] {#Eqstate}
==========================
For ${\underline}{s}_{\,n} \,:=\, (s_n,\ldots,\,s_1)\in {\mathbbm{R}}^n$ we define $$\label{DefQNs}
{Q_N}({\underline}{s}_{\,n})\,:= \,{Q_N}(s_n)\cdots{Q_N}(s_1),\qquad {Q_N}(s)\,:=\, e^{-s {\mathcal{L}_0}}{Q_N}e^{s\,{\mathcal{L}_0}},\ s\in {\mathbbm{R}}$$ At this point, we check that ${Q_N}({\underline}{s}_{\,n}){\Omega_0^\beta}$ is well defined, and that it is an analytic vector of ${\mathcal{L}_0}$, see Equation . The goal of Theorem \[Thm2\] is to give explicit conditions on ${H_{el}}$ and ${W}$, which ensure ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}+{Q})\,}).$ Let $$\begin{aligned}
\label{FormFactorBound}
{\underline}{\eta}_1 &\,:=\,& \int \big(\|\vec G(k)\|^2_{\mathcal{B}({\mathcal{H}_{el}})}
+\|\vec H(k)\|^2_{\mathcal{B}({\mathcal{H}_{el}})}\big)(2+4\alpha(k)^{-1})\,dk\\ \nonumber
{\underline}{\eta}_2 &\,:=\,& \int \big(\| F(k)\,{H_{el,+}}^{-\gamma}\|^2_{\mathcal{B}({\mathcal{H}_{el}})}
+\|F(k)^*\,{H_{el,+}}^{-\gamma} \|_{\mathcal{B}({\mathcal{H}_{el}})}\big)(2 +4\alpha(k)^{-1})\,dk\end{aligned}$$ The idea of the proof is the following. First, we expand $ e^{-\beta /2({\mathcal{L}_0}+{Q_N})}e^{{\mathcal{L}_0}}$ in a Dyson-series, i.e., $$\begin{aligned}
\label{DysonSer}
e&^{-\beta /2({\mathcal{L}_0}+{Q_N})}e^{{\mathcal{L}_0}} \\\nonumber
&= {\mathbf{1}}+\sum_{n=1}^\infty(-1)^n\int_{\Delta_{\beta/2}^{n}}
e^{-s_n {\mathcal{L}_0}}{Q_N}e^{s_n\,{\mathcal{L}_0}}\cdots e^{-s_1 {\mathcal{L}_0}}{Q_N}e^{s_1\,{\mathcal{L}_0}}\, d\underline{s}_{\,n}.\end{aligned}$$ Under the assumptions of Theorem \[Thm2\] we obtain an upper bound, uniform in $N$, for $$\begin{aligned}
\label{DysonExp}
\langle {\Omega_0^\beta}\,|&\,e^{-\beta ({\mathcal{L}_0}+{Q_N})}{\Omega_0^\beta}\rangle\\ \nonumber
&= 1 +\sum_{n=1}^\infty(-1)^n
\int_{\Delta_{\beta}^{n}}\langle {\Omega_0^\beta}\,|\,
e^{-s_n {\mathcal{L}_0}}{Q_N}e^{s_n\,{\mathcal{L}_0}}\cdots e^{-s_1 {\mathcal{L}_0}}{Q_N}e^{s_1\,{\mathcal{L}_0}}{\Omega_0^\beta}\rangle\, d\underline{s}_{\,n}.\end{aligned}$$ This is proven in Lemma \[MainEstimate\] below, which is the most important part of this section. In Lemma \[DysExp\] and Lemma \[Lem0.1\] we deduce from the upper bound for an upper bound for $\|e^{-(\beta/2)({\mathcal{L}_0}+{Q_N})}{\Omega_0^\beta}\|$, which is uniform in $N$. The proof of Theorem \[Thm2\] follows now from Lemma \[Lem1.1\], where we show that ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-(\beta/2)({\mathcal{L}_0}+{Q})})$.
\[DysExp\] Assume $$\limsup_{n\,\rightarrow \,\infty} \sup_{0\,{\leqslant}\,x\, {\leqslant}\, \beta/2}
\Big\| \int_{\Delta_x^n}{Q_N}({\underline}{s}_{\,n})\,d{\underline}{s}_{\,n}\Big\|^{1/n}\,<\,1.$$ for all $N\in {\mathbbm{N}}$. Then ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-x({\mathcal{L}_0}+{Q_N})}),\ 0\,<\, x\,{\leqslant}\, \beta/2$ and $$\label{Lem:DysExp}
e^{-x\,({\mathcal{L}_0}+{Q_N})}{\Omega_0^\beta}\,=\,{\Omega_0^\beta}+\sum_{n=1}^\infty(-1)^n\int_{\Delta_{x}^{n}}
{Q_N}({\underline}{s}_{\,n}){\Omega_0^\beta}\, d\underline{s}_{\,n}.$$ In this context $\Delta_{x}^{n}\,=\, \{(s_1,\ldots,\,s_n)\in {\mathbbm{R}}^n\,
:\, 0\,{\leqslant}\,s_n\,{\leqslant}\ldots {\leqslant}s_1\,{\leqslant}\, x\}$ is a simplex of dimension $n$ and sidelength $x$.
Let $\phi \in {\operatorname{ran}}{\mathbf{1}}[ |{\mathcal{L}_0}\,+\,{Q_N}|\,{\leqslant}\, k]$ and $0\,{\leqslant}\,x\,{\leqslant}\, \beta/2$ be fixed. An $m$-fold application of the fundamental theorem of calculus yields $$\begin{aligned}
\nonumber
\lefteqn{
\langle\, e^{-x\,({\mathcal{L}_0}\,+\,{Q_N})}\,\phi\, | \, e^{x\,{\mathcal{L}_0}}\,{\Omega_0^\beta}\,\rangle
= \Big\langle\, \,\phi \,|\,{\Omega_0^\beta}\,+\, \sum_{n=1}^m(-1)^n \int_{\Delta_{x}^{n}}\,
{Q_N}({\underline}{s}_{\,n})\,{\Omega_0^\beta}\,d\underline{s}_{\, n}\,\Big\rangle }\\ \label{eq3.16}
&&+ (-1)^{m+1}\int_{\Delta_{x}^{m+1}}\, \big\langle\, e^{-s_{m+1}\,({\mathcal{L}_0}\,+\,{Q_N})}\phi
\,|\, e^{s_{m+1}\,{\mathcal{L}_0}}\,{Q_N}({\underline}{s}_{\,m+1})\,{\Omega_0^\beta}\,\big\rangle\,d\underline{s}_{\, m+1}.
$$ Since ${\mathcal{L}_0}\,{\Omega_0^\beta}\,=\,0$ we have for $r({\underline}{s}_{\,m+1})\,:=\,(s_m-s_{m+1},\ldots,\, s_1-s_{m+1})$ that $$e^{s_{m+1}{\mathcal{L}_0}}\,{Q_N}({\underline}{s}_{\,m+1}){\Omega_0^\beta}\,=\,{Q_N}\, {Q_N}(r({\underline}{s}_{\,m+1})){\Omega_0^\beta},$$ We turn now to the second expression on the right side of Equation , after a linear transformation depending on $s_{m+1}$ we get $$(-1)^{m+1}\,\int_0^x \Big\langle\, e^{-s_{m+1}({\mathcal{L}_0}+{Q_N})}\phi \,|\,{Q_N}\,\int_{\Delta^m_{x-s_{m+1}}}
{Q_N}({\underline}{r}_{\,m}){\Omega_0^\beta}\, d\underline{r}_{\, m}\,\Big\rangle\,ds_{m+1}.$$ Since $\| e^{-s_{m+1}\,({\mathcal{L}_0}\,+\,{Q_N})}\,\phi\|\,{\leqslant}\,e^{\beta/2\, k}\,\|\phi\|$, and using that ${Q_N}({\underline}{r}_{\,m})\,{\Omega_0^\beta}$ is a state with at most $2m$ bosons, we obtain the upper bound $${\operatorname{const}}\|\phi\|\, \sqrt{(2m)\,(2m+1)}\,\sup_{0\,{\leqslant}\, x\,{\leqslant}\, \beta/2}
\Big \| \int_{\Delta^m_{x-s_{m+1}}}
{Q_N}({\underline}{r}_{\,m})\,{\Omega_0^\beta}\, d\underline{r}_{\, m}\Big \|.$$ Hence, for $m\,\rightarrow\,\infty$ we get $$\langle\, e^{-x({\mathcal{L}_0}+{Q_N})}\phi\mid {\Omega_0^\beta}\,\rangle
= \Big\langle\, \phi\,|\, {\Omega_0^\beta}\,+\,\sum_{n=1}^\infty(-1)^n \int_{\Delta_{x}^{n}}\,
{Q_N}({\underline}{s}_{\,n})\,{\Omega_0^\beta}\,d\underline{s}_{\, n}\,\Big\rangle .$$ Since $\bigcup_{k=1}^\infty {\operatorname{ran}}{\mathbf{1}}[ |{\mathcal{L}_0}\,+\,{Q_N}|\,{\leqslant}\,k]$ is a core of $e^{-x({\mathcal{L}_0}\,+\,{Q_N})}$, the proof follows from the self-adjointness of $e^{-x({\mathcal{L}_0}\,+\,{Q_N})}$.
\[Lem0.1\] Let $0\,<\,x\,{\leqslant}\,\beta/2$. We have the identity $$\begin{aligned}
\lefteqn{
\int_{\Delta_{x/2}^{n}}\, \int_{\Delta_{x/2}^{m}}
\big \langle\, {Q_N}({\underline}{r}_{\,m}){\Omega_0^\beta}\,|\,{Q_N}({\underline}{s}_{\,n}){\Omega_0^\beta}\,\big\rangle\, d\underline{r}_{\,m}\,d\underline{s}_{\,n}}\\ \nonumber
&=&
\int_{\Delta_{\beta}^{n+m}}\, \mathbf{1}[z_m\,{\geqslant}\,\beta-x\,{\geqslant}\, x\, {\geqslant}\, z_{m+1}]\,
\big \langle\, {\Omega_0^\beta}\,\big|\,{Q_N}({\underline}{z}_{\,n+m}){\Omega_0^\beta}\,\big \rangle\,d\underline{z}_{n+m}.\end{aligned}$$ For $m\,=\,n$ it follows $$\Big\|\int_{\Delta_{x/2}^{n}}{Q_N}({\underline}{s}_{\,n}){\Omega_0^\beta}\, d{\underline}{s}_{\,n}\Big\|^2
{\leqslant}\int_{\Delta_{\beta}^{2n}}\big|\big\langle\, {\Omega_0^\beta}\,|\,{Q_N}({\underline}{s}_{\,2n}){\Omega_0^\beta}\,
\big\rangle \big|\,d{\underline}{s}_{\,2n}.$$
Recall Theorem \[FreeSystem\] and Lemma \[Lem5d\]. Since ${\mathcal{J}}$ is a conjugation we have $\langle\, \phi\,|\, \psi \,\rangle =\langle\, {\mathcal{J}}\,\psi\,|\, {\mathcal{J}}\,\phi \,\rangle $, and for every operator $X$, that is affiliated with ${\mathfrak{M}_\beta}$, we have ${\mathcal{J}}\, X\,{\Omega_0^\beta}\,=\, e^{-\beta/2\,{\mathcal{L}_0}}\,X^*\,{\Omega_0^\beta}$. Thus, $$\begin{aligned}
\label{eq3.17a}
\lefteqn{
\int_{\Delta_{x /2}^{n}}\, \int_{\Delta_{x/2}^{m}}
\big \langle\, {Q_N}({\underline}{r}_{\,m}){\Omega_0^\beta}\,|\,{Q_N}({\underline}{s}_{\,n})\,{\Omega_0^\beta}\, \big \rangle\, d\underline{r}_{\,m}\,d\underline{s}_{\,n}
}\\ \nonumber
&=& \int_{\Delta_{x/2}^{n}} \int_{\Delta_{x/2}^{m}}
\big \langle\, e^{-\beta/2{\mathcal{L}_0}}\,{Q_N}({\underline}{s}_{\,n})^*\,{\Omega_0^\beta}\,\big|\,e^{-\beta/2{\mathcal{L}_0}}\,{Q_N}({\underline}{r}_{\,m})^*\,{\Omega_0^\beta}\,\big \rangle\,
d\underline{r}_{\,m}\,d\underline{s}_{\,n}\end{aligned}$$ Since ${\mathcal{L}_0}\, {\Omega_0^\beta}\,=\,0$ we have $$e^{-\beta{\mathcal{L}_0}}\,{Q_N}({\underline}{r}_{\,m})^*\,{\Omega_0^\beta}\, = \,{Q_N}(\beta-r_1)\cdots {Q_N}(\beta-r_m)\,{\Omega_0^\beta}.$$ Next, we introduce new variables for $\underline{r}$, namely $y_i \,:=\, \beta \,-\,r_{m-i+1}$. Let $D_{x/2}^{m}\,:=\, \{ \underline{y}_{\,m}\in {\mathbbm{R}}^m\,:\
\beta-x \,{\leqslant}\, y_m\,{\leqslant}\ldots {\leqslant}\, y_1\, {\leqslant}\, \beta\}$. Thus the right side of Equation equals $$\begin{aligned}
\lefteqn{
\int_{\Delta_{x/2}^{n}}\,
\int_{D_{x/2}^{m}}\,
\big \langle\, {\Omega_0^\beta}\,\big|\,{Q_N}({\underline}{s}_{\,n})\,{Q_N}({\underline}{y}_{\,m})^*\,{\Omega_0^\beta}\,\big \rangle \,d\underline{s}_{\,n}\,d\underline{y}_{\,m}
}\\
&=&
\int_{\Delta_{\beta}^{n+m}}\, {\mathbf{1}}[z_m\,{\geqslant}\, \beta\,-\,x\,{\geqslant}\, x\, {\geqslant}\, z_{m+1}]\,
\big \langle\, {\Omega_0^\beta}\,\big|\,{Q_N}({\underline}{z}_{\,n+m})\,{\Omega_0^\beta}\,\big \rangle\,d\underline{z}_{n+m}.\end{aligned}$$ The second statement of the Lemma follows by choosing $n\,=\,m$.
\[Lem1.1\] Assume $\sup_{N\in {\mathbbm{N}}}\|e^{-x({\mathcal{L}_0}+{Q_N})}{\Omega_0^\beta}\|\,<\,\infty$ then ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-x({\mathcal{L}_0}+{Q})})$ and $$\|e^{-x({\mathcal{L}_0}+{Q})}{\Omega_0^\beta}\|\,{\leqslant}\,\sup_{N\in {\mathbbm{N}}}\|e^{-x({\mathcal{L}_0}+{Q_N})}\,{\Omega_0^\beta}\|$$
For $f\in \mathcal{C}^\infty_0({\mathbbm{R}})$ and $\phi\in{\mathcal{K}}$ we define $\psi_N\,:=\,f({\mathcal{L}_0}\,+\,{Q_N})\,\phi$. Obviously, for $g(r)\,=\,e^{-x\,r}\,f(r)\in \mathcal{C}^\infty_0({\mathbbm{R}})$ we have $e^{-x\,({\mathcal{L}_0}\,+\,{Q_N})}\,\psi_N\,=\,g({\mathcal{L}_0}\,+\,{Q_N})\,\phi $. Since ${\mathcal{L}_0}\,+\,{Q_N}$ tends to ${\mathcal{L}_0}\,+\,{Q}$ in the strong resolvent sense as $N\to \infty$, we know from [@ReedSimonI1980] that $\lim_{N\,\rightarrow\, \infty}\psi_N\,=\,f({\mathcal{L}_0}\,+\,{Q})\,\phi\,=:\, \psi$ and $$\lim_{N\rightarrow \infty}e^{-x\,({\mathcal{L}_0}\,+\,{Q_N})}\,\psi_{N}
\,=\,\lim_{N\rightarrow \infty}g({\mathcal{L}_0}\,+\,{Q_N})\,\phi
\,=\,g({\mathcal{L}_0}\,+\,{Q})\,\phi\,=\, e^{-x\,({\mathcal{L}_0}\,+\,{Q})}\,\psi.$$ Thus, $$\begin{aligned}
|\langle\, e^{-x({\mathcal{L}_0}+{Q})}\,\psi\,|\,{\Omega_0^\beta}\,\rangle |
\,&=\, \lim_{N\to \infty} |\langle\, e^{-x({\mathcal{L}_0}+{Q_N})}\psi_N \,|\,{\Omega_0^\beta}\,\rangle |\\
\,&{\leqslant}\, \sup_{N\in {\mathbbm{N}}}\|e^{-x\,({\mathcal{L}_0}+{Q_N})}{\Omega_0^\beta}\|\, \|\psi\|,\end{aligned}$$ Since $\{ f({\mathcal{L}_0}+{Q})\,\phi\in{\mathcal{K}}\,:\,\phi \in {\mathcal{K}},\, f\in \mathcal{C}^\infty_0({\mathbbm{R}})\}$ is a core of $e^{-x({\mathcal{L}_0}+{Q})}$, we obtain ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-x({\mathcal{L}_0}+{Q})})$.
\[MainEstimate\] For some $C>0$ we have $$\begin{aligned}
\int_{\Delta_{\beta}^n}
&\Big|\big\langle\, {\Omega_0^\beta}\,|\, {Q_N}({\underline}{s}_{\,n}){\Omega_0^\beta}\,\big\rangle \Big|\,d{\underline}{s}_{\,n}\\ \nonumber
&{\leqslant}{\operatorname{const}}\,(n+1)^2\,(1+\beta)^n\,
\Big(8{\underline}{\eta}_1 +\frac{(8C{\underline}{\eta}_2)^{1/2} }{(n+1)^{(1-2\gamma)/2}} \Big)^{n},\end{aligned}$$ where ${\underline}{\eta}_1$ and ${\underline}{\eta}_2$ are defined in .
First recall the definition of ${Q_N}$ and ${Q_N}({\underline}{s}_{\,n})$ in Equation and Equation , respectively. Let $$\int_{\Delta_{\beta }^{n}}\big|\big\langle\, {\Omega_0^\beta}\,|\, {Q_N}({\underline}{s}_{\,n})\,{\Omega_0^\beta}\,\big\rangle \big|\,d{\underline}{s}_{\,n}
\,=:\, \int_{\Delta_{1}^{n}}\beta^n\,J_n(\beta,{\underline}{s})\,d{\underline}{s}_{\,n},$$ The functions $J_n(\beta,{\underline}{s})$ clearly depends on $N$, but since we want to find an upper bound independent of $N$, we drop this index. Let $W_1=\Phi(\vec G )\,\Phi(\vec H )\,+\,{\operatorname{h.c.}}\,$, $W_2:=\Phi(F)$ and $W:=W_1+W_2$. By definition of ${\omega_0^\beta}$ in , see also , we obtain $$\begin{aligned}
J_{\,n}(\beta,\,{\underline}{s}_{\,n})\,&=\,
{\omega_0^\beta}\Big(\big(e^{-\beta\, s_n\,{H_0}}\,W\,e^{\beta s_n{H_0}}\big)
\cdots\big(e^{-\beta \,s_1\,{H_0}}\,W\,e^{\beta \,s_1\,{H_0}}\big)\Big)\\
&=({\mathcal{Z}})^{-1}\sum_{\kappa\in \{ 1,\,2\}^n}
{ \omega^\beta_f}\Big({\operatorname{Tr}}_{{\mathcal{H}_{el}}}\big\{e^{-\beta\, {H_{el}}}\big(e^{-\beta \,s_n\,{H_0}}\,W_{\kappa(n)}
\,e^{\beta\, s_n\,{H_0}}\big)\cdots\\
&\phantom{({\mathcal{Z}})^{-1}\sum_{\kappa\in \{ 1,\,2\}^n}
{ \omega^\beta_f}\Big({\operatorname{Tr}}_{{\mathcal{H}_{el}}}\big\{e^{-\beta\, {H_{el}}}\ }
\cdots\big(e^{-\beta\, s_1\,{H_0}}\,W_{\kappa(1)}\,e^{\beta\, s_1\,{H_0}}\big)\big\}\Big)\end{aligned}$$ By definition of ${ \omega^\beta_f}$ it suffices to consider expressions with an even number of field operators. In the next step we sum over all expression, where $n_1$ times $W_1$ occurs and $2n_2$ times $W_2$. The sum of $n_1$ and $n_2$ is denoted by $m$. For fixed $n_1$ and $n_2$ the remaining expressions are all expectations in ${ \omega^\beta_f}$ of $2m$ field operators. In this case the expectations in ${ \omega^\beta_f}$ can be expressed by an integral over ${\mathbbm{R}}^{2m}\times \{\pm\}^{2m}$ with respect to $\nu$, which is defined in Lemma \[Lem-N1\] below. To give a precise formula we define $$M(m_1,\,m_2)\,=\, \{ \kappa\in \{1,\,2\}^n\,
:\,\#\kappa^{-1}(\{i\})\,=\,m_i,\quad i=1,\,2\}.$$ Thus we obtain $$\begin{aligned}
\label{eq3.41}
J_n&(\beta,\,{\underline}{s}_{\,n})
\,=\,({\mathcal{Z}})^{-1}\sum_{\stackrel{(n_1,\,n_2)\in {\mathbbm{N}}^2}{n_1\,+\,2n_2\,=\,n}}\qquad
\sum_{\stackrel{\kappa\in M(n_1,\,2n_2)}{m\,:=\,n_1\,+\,n_2}}
\int\,\nu(d{\underline}{k}_{\,2m}{\otimes}d{\underline}{\tau}_{\,2m})\\ \nonumber
&\phantom{=}
{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\big\{ e^{-(\beta\,-\,\beta(s_1\,-\,s_{2m})){H_{el}}}I_{2m}
e^{-\beta\,(s_{2m-1}\,-\,s_{2m})\,{H_{el}}}
\cdots
e^{-\beta\,(s_1\,-\,s_{2})\,{H_{el}}}\,I_{1}\big\}\,,\end{aligned}$$ Of course $I_{j}$ depends on ${\underline}{k}_{\,2m}\times {\underline}{\tau}_{\,2m}$, namely for $\kappa(j)\,=\,1,\,2$ we have $$\begin{aligned}
I_j
&\,=\,&
\begin{cases}
I_j(m,\,\tau,\,m',\,\tau'),& \kappa(j)\,=\,1\\
I_j(m,\,\tau),& \kappa(j)\,=\,2,
\end{cases} \end{aligned}$$ where $(m,\,\tau),\, ( m',\,\tau')\in \{ (k_j,\tau_j)\,:\, j=1,\ldots,m\}$. For $\kappa(j)\,=\,1$ we have that $$\begin{aligned}
I_j(m,\,+,\,m',\,-)&\,=\,& \vec{G}^*(m)\, \vec{H}(m')\,+\,\vec{H}^*(m)\, \vec{G}(m')\\ \nonumber
I_j(m,\,-,\,m',\,+)&\,=\,& \vec{G}(m)\, \vec{H}^*(m')\,+\,\vec{H}(m)\, \vec{G}^*(m')\\ \nonumber
I_j(m,\,+,\,m',\,+)&\,=\,& \vec{G}^*(m)\, \vec{H}^*(m')\,+\,\vec{H}^*(m)\, \vec{G}^*(m')\\ \nonumber
I_j(m,\,-,\,m',\,-)&\,=\,& \vec{G}(m)\, \vec{H}(m')\,+\,\vec{H}(m)\, \vec{G}(m')\end{aligned}$$ and for $\kappa(j)\,=\,2$ we have that $$\begin{aligned}
I_j(m,\,+)&\,=\,& F^*(m)\\ \nonumber
I_j(m,\,-)&\,=\,& F(m).\end{aligned}$$ In the integral we insert for $(m,\,\tau)$ and $(m',\,\tau')$ in the definition of $I_j$ from left to right $k_{2m},\,\tau_{2m},\ldots,\,k_1,\,\tau_1$.\
For fixed $({\underline}{k}_{\,2m},{\underline}{\tau}_{\,2m})$ the integrand of is a trace of a product of $4m$ operators in ${\mathcal{H}_{el}}$. We will apply Hölder’s-inequality for the trace, i.e., $$|{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{A_{2m}\,B_{2m}\cdots A_1\,B_1\}|{\leqslant}\prod_{j=1}^{2m}\|B_{j}\|_{\mathcal{B}({\mathcal{H}_{el}})}
\cdot \prod_{j=1}^{2m}{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{A_i^{p_j}\}^{p_j^{-1}}.$$ In our case $p_i\,:=\,(s_{i-1}\,-\,s_i)^{-1}$ for $i=2,\ldots,\,2m$ and $p_1\,:=\,(1\,-\,s_1\,+\,s_{2m})^{-1}$ and $$(A_j,\, B_j)\,:=\,
\begin{cases}
\big( e^{-\beta\, p_j^{-1}\, {H_{el}}},\, I_j(m,\,\tau,\,m',\,\tau')\big),& \kappa(j)\,=\,1\\
\big( e^{-\beta\, p_j^{-1}\, {H_{el}}}\,{H_{el,+}}^\gamma,\, {H_{el,+}}^{-\gamma}\,I_j(m,\tau,)\big),& \kappa(j)\,=\,2
\end{cases}.$$ We define $$\begin{aligned}
\eta_1(k) \,&=\,\max\big\{ \|\vec G(k)\|_{\mathcal{B}({\mathcal{H}_{el}})^r},\, \|\vec H(k)\|_{\mathcal{B}({\mathcal{H}_{el}})^r} \big\}\\
\eta_2(k) \,&=\,\max\big\{\|F(k)\,{H_{el,+}}^{-\gamma}\|_{\mathcal{B}({\mathcal{H}_{el}})},
\,\|F^*(k)\,{H_{el,+}}^{-\gamma}\|_{\mathcal{B}({\mathcal{H}_{el}})}\big\}.\end{aligned}$$ By definition of $B_j$ we have $$\label{UpperBoundB}
\|B_{j}\|_{\mathcal{B}({\mathcal{H}_{el}})}{\leqslant}\begin{cases}
\eta_1(m)\eta_1(m'),& \kappa(j)=1\\
\eta_2(m),& \kappa(j)=2
\end{cases}.$$ Furthermore, $$\begin{aligned}
{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{A_i^{p_j}\}^{p_j^{-1}}&={\operatorname{Tr}}_{{\mathcal{H}_{el}}}\big\{ e^{-\beta {H_{el}}}\,{H_{el,+}}^{p_j\gamma}\big\}^{p_j^{-1}}\\
&{\leqslant}\| e^{-\epsilon {H_{el}}}\,{H_{el,+}}^{p_j\gamma}\|^{p_j^{-1}}_{{\mathcal{H}_{el}}}\,
{\operatorname{Tr}}_{{\mathcal{H}_{el}}}\big\{ e^{-(\beta-\epsilon)\, {H_{el}}}\big\}^{p_j^{-1}},\quad k(j)=2\end{aligned}$$ Let $E_{gs}\,:=\,\inf\, \sigma({H_{el}})$. The spectral theorem for self-adjoint operators implies $$\| e^{-\epsilon\, {H_{el}}}\,{H_{el,+}}^{p_i\,\gamma}\|^{p_i^{-1}}_{{\mathcal{H}_{el}}}
\,{\leqslant}\, \sup_{r\,{\geqslant}\,E_{gs}}\, e^{-\epsilon\, p_i^{-1} \,r}(r\,-\,E_{gs}\,+1)^{\gamma}
\,{\leqslant}\, \epsilon^{-\gamma}\,p_i^\gamma\, e^{-\epsilon \,p_i^{-1}\,(E_{gs}\,-\,1)}.$$ Inserting this estimates we get $$\begin{aligned}
{\operatorname{Tr}}_{{\mathcal{H}_{el}}}&\big\{ e^{-(\beta\,-\,\beta(s_1\,-\,s_{2m})){H_{el}}}I_{2m}
e^{-\beta\,(s_{2m-1}\,-\,s_{2m})\,{H_{el}}}
\cdots
e^{-\beta\,(s_1\,-\,s_{2})\,{H_{el}}}\,I_{1}\big\}\\
&{\leqslant}C_{\kappa}({\underline}{s}_{n})
\prod_{j=1}^{2m}\|B_j\|_{\mathcal{B}({\mathcal{H}_{el}})}\end{aligned}$$ where $$C_{\kappa}({\underline}{s}_{n}):=(1-\,s_1\,+\,s_{n})^{-\alpha_1}
\prod_{i=1}^{n-1}(s_i\,-\,s_{i+1})^{-\alpha_i}$$ and $$\alpha_i=\begin{cases}
0,&\kappa(i)=1\\
1/2,&\kappa(i)=2
\end{cases}$$ Now, we recall the definition of $\nu$. Roughly speaking, one picks a pair of variables $(k_i,k_j)$ and integrates over $\delta_{k_i,k_j}\coth(\beta/2\alpha(k_i))\,dk_i dk_j$. Subsequently one picks the next pair and so on. At the end one sums up all $\frac{(2m)!}{2^m\, m!}$ pairings and all $4^m$ combinations of ${\underline}{\tau}_{\,2m}$. Inserting Estimate and that $$\begin{aligned}
\int &\eta_\nu(k)\eta_{\nu'}(k)\coth(\beta/2\alpha(k))\,dk {\leqslant}(1+\beta^{-1}){\underline}{\eta}_{\,\nu}^{1/2}{\underline}{\eta}_{\,\nu'}^{1/2},\end{aligned}$$ we obtain $$\begin{aligned}
|J_n(\beta,\,{\underline}{s})|
&{\leqslant}\frac{(1+\beta^{-1})^n}{{\mathcal{Z}}} \sum_{\stackrel{(n_1,\,n_2)\in {\mathbbm{N}}_0^2}{n_1\,+\,2n_2\,=\,n}}
\ \sum_{\stackrel{\kappa\in M(n_1,\,2n_2)}{m\,:=\,n_1\,+\,n_2}}
(\,{\underline}{\eta}_1)^{n_1}\,(C{\underline}{\eta}_2)^{n_2}\,\frac{(2m)!2^m}{ m!}C_\kappa({\underline}{s})\end{aligned}$$ By Lemma \[Lem0.2\] below and since $(2m)!/(m!)^2{\leqslant}4^m$ we have $$\begin{aligned}
\int_{\Delta_{\beta }^{n}}&\big|\big\langle\, {\Omega_0^\beta}\,|\, {Q_N}({\underline}{s}_{\,n})\,{\Omega_0^\beta}\,\big\rangle \big|\,d{\underline}{s}_{\,n}\\
&\,{\leqslant}{\operatorname{const}}(1+\beta)^n \sum_{\stackrel{(n_1,\,n_2)\in {\mathbbm{N}}_0^2}{n_1\,+\,2n_2\,=\,n}}
{ n \choose n_1} \frac{(8{\underline}{\eta}_1)^{n_1}\,(8C'{\underline}{\eta}_2)^{n_2}}{(n+1)^{(1-2\gamma)\,n_2 -2} } \end{aligned}$$ This completes the proof.
The Harmonic Oscillator {#HarmOsc}
=======================
Let $L^2(X,\,d\mu)=L^2({\mathbbm{R}})$ and ${H_{el}}=: {H_{osc}}:= -\Delta_q+\Theta^2 q^2$ be the one dimensional harmonic oscillator and ${\mathcal{H}_{ph}}=L^2({\mathbbm{R}}^3)$. We define $${H}\,=\, {H_{osc}}\,+\, \Phi(F) \,+\, { \check{H} },\qquad { \check{H} }\,:= \, d\Gamma(|k|),$$ where $\Phi(F)\,=\, q\, \cdot \,\Phi(f)$, with $\lambda\,(|k|^{-1/2}\,+\,|k|^{1/2})\,f\in L^2({\mathbbm{R}}^3)$.\
${H_{osc}}$ is the harmonic oscillator, the form-factor $F$ comes from the dipole approximation.\
The Standard Liouvillean for this model is denoted by $\mathcal{L}_{osc}$. Now we prove Theorem \[Thm4a\].
We define the creation and annihilation operators for the electron. $$\begin{gathered}
\label{eq4.1}
{A}^*\,=\,\frac{\Theta^{1/2}\,q\,-\,{i}\, \Theta^{-1/2}\, p}{\sqrt{2}},
\quad {A}\,=\,\frac{\Theta^{1/2}\,q\,+\,{i}\, \Theta^{-1/2}\,p}{\sqrt{2}},\quad p\,=\,-{i}\,\partial_x,\\
\Phi(c)\,=\, c_1\,q\,+\,c_2 \, p,\quad \textrm{for}\ c\,=\,c_1\,+\,{i}\,c_2\in {\mathbbm{C}},\ c_i\in {\mathbbm{R}}.\end{gathered}$$ These operators fulfill the CCR-relations and the harmonic- oscillator is the number-operator up to constants. $$\begin{gathered}
\label{eq4.2}
[{A},\,{A}^*]\,=\,1,\qquad [{A}^*,\,{A}^*]\,=\,[{A},\,{A}]\,=\,0,\qquad {H_{osc}}\,=\,\Theta {A}^*\,{A}\,+\,\Theta/2,\\
[{H_{osc}},\,{A}]\,=\,-\Theta \,{A},\qquad [{H_{osc}},\,{A}^*]\,=\,\Theta \,{A}^*.\end{gathered}$$ The vector $\Omega\,:=\, \left(\frac{\Theta}{\pi}\right)^{1/4} \,e^{-\Theta\, q^2\,/2}$ is called the vacuum vector. Note, that one can identify $\mathcal{F}_b[{\mathbbm{C}}]$ with $L^2({\mathbbm{R}})$, since ${\operatorname{LH}}\{ (A^*)^n\,\Omega\, | \, n\in {\mathbbm{N}}^0\}$ is dense in $L^2({\mathbbm{R}})$. It follows, that ${\omega_\beta^{osc}}$ is quasi-free, as a state over ${W}({\mathbbm{C}})$ and $${\omega_\beta^{osc}}({W}(c))\,=\, ({\mathcal{Z}})^{-1}\, {\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{ e^{-\beta {H_{el}}} \,{W}(c)\}
\,=\, \exp\big( -1/4\, \coth(\beta\, \Theta/2)\,|c|^2\big),$$ where ${\mathcal{Z}}\,=\, {\operatorname{Tr}}_{{\mathcal{H}_{el}}}\{ e^{-\beta\,{\mathcal{H}_{el}}} \}$ is the partition function for ${\mathcal{H}_{el}}$.\
First, we remark, that Equation is defined for this model without regularization by $P_N \,:=\, {\mathbf{1}}[{H_{el}}\,{\leqslant}\, N]$. Moreover we obtain from Lemma \[Lem0.1\], that $$\Big\|\int_{\Delta_{\beta/2}^{n}}{Q}({\underline}{s}_{\,n}){\Omega_0^\beta}\,d{\underline}{s}_{\,2n}\Big\|^2
\,{\leqslant}\,
\int_{\Delta_{\beta}^{2n}}\big|\big\langle\, {\Omega_0^\beta}\,|\,
{Q}({\underline}{s}_{\,2n})\,{\Omega_0^\beta}\,\big\rangle \big|\,d{\underline}{s}_{\,2n}\,=:\,
h_{2n}(\beta,\, \lambda).$$ To show that ${\Omega^\beta}\in {\operatorname{dom}}(e^{-\beta/2 \,({\mathcal{L}_0}\,+\,{Q})})$ is suffices to prove, that $\sum_{n\,=\,0}^\infty h_{2n}(\beta,\,\lambda)^{1/2}\,<\,\infty$. We have $$\begin{gathered}
\label{eq4.3}
h_{2n}(\beta,\, \lambda)\,=\, \frac{(-\beta\,\lambda)^{2n}}{{\mathcal{Z}}} \int_{\Delta_1^{2n}}\,
{\omega_\beta^{osc}}\big( \big( e^{-\beta \,s_{2n}\, {H_{el}}}\,q\,e^{\beta\, s_{2n}\, {H_{el}}}\big) \\ \nonumber
\cdots \big( e^{-\beta\, s_1\,{H_{el}}}\,q\,e^{\beta \,s_1\,{H_{el}}}\big)\big)\\ \nonumber
\cdot { \omega^\beta_f}\big((e^{-\beta\, s_{2n}\, { \check{H} }}\,\Phi(f)\,e^{\beta \,s_{2n} \,{ \check{H} }})
\cdots (e^{-\beta \,s_1\, { \check{H} }}\,\Phi(f)\,e^{\beta \,s_1\, { \check{H} }})\big)\, d\underline{s}_{\,2n}.\end{gathered}$$ Moreover, we have $$\begin{aligned}
\nonumber\label{eq4.4y}
e^{-\beta\, s_i \,{H_{el}}} \,q\, e^{\beta\, s_i \,{H_{el}}}
&= (2\Theta)^{-1/2}\big(e^{-\beta\, \Theta\, s_i}\,A^*\,+\,e^{\beta \,\Theta \,s_i}\,A\,\big)\\
e^{-\beta\, s_i \,{ \check{H} }}\,\Phi(f)\,e^{\beta\, s_i\, { \check{H} }}&= 2^{-1/2}
\Big( a^*(e^{-\beta\, s_i\,|k|}\,f)+ a(e^{\beta\, s_i\,|k|}\,f)\Big).\end{aligned}$$ Inserting the identities of Equation in Equation and applying Wick’s theorem [@BratteliRobinson1996 p. 40] yields $$\begin{aligned}
\nonumber
h_{2n}& (\beta,\,\lambda)
= (\beta\,\lambda)^{2n} \int_{\Delta_1^{2n}}
\sum_{P\in \mathcal{Z}_2}\ \prod_{\{i,\,j\}\in P}
K_{osc}(|s_i\,-\,s_j|,\,\beta)\\ \nonumber
&\phantom{\sum_{P\in \mathcal{Z}_2}\ \prod_{\{i,\,j\}\in P}
K_{osc}(|s_i\,-}
\cdot\sum_{P'\in \mathcal{Z}_2}\ \prod_{\{k,\,l\}\in P'}
K_{f}(|s_k-s_l|,\,\beta)\, d{\underline}{s}_{\,2n}\\ \label{eq4.6}
&=\frac{(\beta\,\lambda)^{2n}}{(2n)!} \int_{[0,\,1]^{2n}}
\sum_{P,\,P'\in \mathcal{Z}_2} \prod_{\stackrel{\{i,\,j\}\in P}{\{k,\,l\}\in P'}}
K_{osc}(|s_i-s_j|,\,\beta)\,K_{f}(|s_k-\,s_l|,\beta)\, d{\underline}{s}_{\,2n},\end{aligned}$$ where for $k\,<\,l$ and $i\,<\,j$, such as $$\begin{aligned}
K_{f}(|s_k\,-\,s_l|,\,\beta)
&\,:= \,&{ \omega^\beta_f}((e^{-\beta\, s_k \,{ \check{H} }}\,\Phi(f)\,e^{\beta \,s_k \,{ \check{H} }})\,
(e^{-\beta \,s_l \,{ \check{H} }}\,\Phi(f)\,e^{\beta \,s_l\, { \check{H} }}))\\
K_{osc}(|s_i\,-\,s_j|,\,\beta)
&\,:=\,&{\omega_\beta^{osc}}(e^{-\beta\, s_i\, {H_{el}}}\,q\,e^{\beta \,s_i \,{H_{el}}}\,
e^{-\beta \,s_j \,{H_{el}}}\,q\,e^{\beta \,s_j\, {H_{el}}}). \end{aligned}$$ The last equality in holds, since the integrand is invariant with respect to a change of the axis of coordinates.\
We interpret two pairings $P$ and $P'\in \mathcal{Z}_2$ as an indirected graph $G\,=\,G(P,\,P')$, where $M_{2n}\,=\,\{1,\ldots,\,2n\}$ is the set of points. Any graph in $G$ has two kinds of lines, namely lines in $L_{osc}(G)$, which belong to elements of $P$ and lines in $L_{f}(G)$, which belong to elements of $P'$.\
Let $\mathcal{G}(A)$ be the set of undirected graphs with points in $A\subset M_{2n}$, such that for each point “i” in $A$, there is exact one line in $L_f(G)$, which begins in “i”, and exact one line in $L_{osc}(G)$, which begins with “i”. $\mathcal{G}_c(A)$ is the set of connected graphs. We do not distinguish, if points are connected by lines in $L_f(G)$ or by lines in $L_{osc}(G)$.\
Let $$\begin{aligned}
\mathcal{P}_k\, :=\, \Big\{ P\,:&\, P=\{A_1,\ldots,\, A_k \},\
\emptyset\not=A_i\subset M_{2n},\\
\,
& A_i\cap A_j \,=\,\emptyset \textrm{ for } i\,\not=\,j,\,
\bigcup_{i\,=\,1}^k\, A_i\,=\, M_{2n}\Big\}\end{aligned}$$ be the family of decompositions of $M_{2n}$ in $k$ disjoint set. It follows $$\begin{aligned}
\nonumber\label{eq4.7}
h_{2n}(\beta,\, \lambda)&\,=\,
\frac{(\beta\,\lambda)^{2n}}{(2n)!}\,\sum_{G\in \mathcal{G}(M_{2n})} \int_{M_{2n}}
\prod_{\stackrel{\{i,j\}\in L_{osc}(G)}{\{k,\,l\}\in L_{f}(G)}}
K_{osc}(|s_i\,-\,s_j|,\,\beta)\\ \nonumber
&\phantom{\,=\,}
K_{f}(|s_k\,-\,s_l|,\,\beta)\, d{\underline}{s}_{\,n}\\ \nonumber
&=\frac{(\beta\lambda)^{2n}}{(2n)!}\ \sum_{k\,=\,1}^{2n}\ \sum_{ \{A_1,\ldots,\, A_k\}\in \mathcal{P}_k }
\ \sum_{\stackrel{(G_1,\ldots,\,G_k)}{G_a\in \mathcal{G}_c(A_a)}}\prod_{a\,=\,1}^k J(G_a,\,A_a,\,\beta)\\
&=\frac{(\beta\lambda)^{2n}}{(2n)!} \ \sum_{k\,=\,1}^{2n}\ \frac{1}{k!} \
\sum_{ \stackrel{A_1,\ldots,\,A_k\subset M_{2n},}{ \{A_1,\ldots,\, A_k\}\in \mathcal{P}_k} }\
\sum_{\stackrel{(G_1,\ldots,\,G_k)}{G_a\in \mathcal{G}_c(A_a)}}\prod_{a\,=\,1}^k \,J(G_a,\,A_a,\,\beta), \end{aligned}$$ where $$\label{eq4.8}
J(G_a,\,A_a,\,\beta)\,:=\,\int_{A_a}\prod_{\stackrel{\{i,\,j\}\in L_{osc}(G_a)}{\{k,l\}\in L_{f}(G_a)}}
K_{osc}(|s_i\,-\,s_j|,\,\beta) K_{f}(|s_k\,-\,s_l|,\,\beta)\,d\underline{s}.$$ $\int_{A_a}\,d{\underline}{s}$ means, $\int_{-1}^1\,ds_{j_1}\int_{-1}^1\,ds_{j_2}\ldots \int_{-1}^1\,ds_{j_m}$, where $A_a\,=\,\{ j_1,\ldots,\,j_m\}$ and $\#A_a\,=\,m$.\
From the first to the second line we summarize terms with graphs, having connected components containing the same set of points. From the second to the third line the order of the components is respected, hence the correction factor $\frac{1}{k!}$ is introduced. Due to Lemma \[Lem1\] the integral depends only on the number of points in the connected graph, i. e. $J(G,\,A,\,\beta)\,=\, J(\#A,\,\beta)$. Moreover, Lemma \[Lem1\] states that $\beta^{\#A}\cdot J(\#A,\,\beta)\,
{\leqslant}\,(2\||k|^{-1/2}\,f\|_2 \,(\Theta\, \beta)^{-1})^{\#A}\,(C\,\beta\,+\,1).$ To ensure that $\mathcal{G}_c(A_a)$ is not empty, $\#A_a$ must be even. For $(m_1,\ldots,\,m_k)\in {\mathbbm{N}}^k$ with $m_1\,+\cdots+\,m_k\,=\,n$ we obtain $$\label{eq4.9}
\sum_{ \stackrel{A_1,\ldots,\,A_k\subset M_{2n}, \#A_i\,=\,2m_i}
{ \{A_1,\ldots,\, A_k\}\in \mathcal{P}_k}} 1
\,=\, \frac{(2n)!}{(2m_1)!\cdots(2m_k)!}.$$ Let now be $A_a\subset M_{2n}$ with $\#A_a\,=\,2m_a\,>\,2$ fixed. In $G_a$ are $\#A_a$ lines in $L_{osc}(G_a)$, since such lines have no points in common, we have $\frac{(2m_a)!}{m_a!\, 2^{m_a}}$ choices. Let now be the lines in $L_{osc}(G_a)$ fixed. We have now $\big((2m_a\,-\,2)(2m_a\,-\,4)\cdots 1\big)$ choices for $m_a$ lines in $L_{f}(G_a)$, which yield a connected graph. Thus $$\label{eq4.10}
\sum_{G_a\in \mathcal{G}_c(A_a),}\, 1
\,=\, \frac{(2m_a)!}{m_a!\, 2^{m_a}}\big((2m_a\,-\,2)\,(2m_a-\,\,4)\cdots 1\big)
\,=\, \frac{(2m_a)!}{2m_a }.$$ For $\#A_a\,=\,2$ exists only one connected graph. We obtain for $h_{2n}$ $$\begin{aligned}
\label{eq4.11}
\lefteqn{
h_{2n}(\beta,\, \lambda)\,=\,(\lambda)^{2n}
\sum_{k\,=\,1}^{2n}\frac{1}{k!} \sum_{ \stackrel{(m_1,\ldots,\,m_k)\in {\mathbbm{N}}^k}{m_1\,+\ldots+\,m_k\,=\,n}}
\prod_{a\,=\,1}^k \frac{J(2m_a,\,\beta)(\beta^2)^{m_a}}{2m_a} }\\ \nonumber
&{\leqslant}&(2\Theta^{-1}\, \||k|^{-1/2}\,f\|\,\lambda)^{2n}
\sum_{k\,=\,1}^{2n}\frac{1}{k!} \sum_{ \stackrel{(m_1,\ldots,\,m_k)\in {\mathbbm{N}}^k}{m_1\,+\ldots+\,m_k\,=\,n}}
\prod_{a\,=\,1}^k \frac{(C\,\beta\,+\,1)}{2m_a}\\ \nonumber
&{\leqslant}&(2\Theta^{-1}\,\||k|^{-1/2}\,f\|\, \lambda)^{2n} \sum_{k\,=\,1}^{2n}
\frac{\big((C\,\beta\,+\,1)/2\sum_{m\,=\,1}^n\frac{1}{m}\big)^k}{k!}.\end{aligned}$$ Since the $\sum_{m=1}^n\frac{1}{m}$ can be considered as a lower Riemann sum for the integral $\int_1^{m\,+\,1} r^{-1}\,dr$, we have $\sum_{m\,=\,1}^n\frac{1}{m}\,{\leqslant}\,\ln(n+1)$. Thus, $$\begin{aligned}
\label{eq4.11a}
h_{2n}(\beta,\, \lambda)
&{\leqslant}& (2\Theta^{-1} \,\||k|^{-1/2}\,f\|\, \lambda)^{2n}
\sum_{k=1}^{2n}\frac{\big((C\,\beta\,+\,1)/2\,\ln(n+1)\big)^k}{k!}\\ \nonumber
&{\leqslant}& (2\Theta^{-1}\, \||k|^{-1/2}\,f\|\, \lambda)^{2n} (n\,+\,1)^{(C\,\beta\,+\,1)/2}.\end{aligned}$$ Since $2 |\lambda|\, \||k|^{1/2}\,f\|\,<\,\Theta$ the series $\sum_{n=0}^\infty h_{2n}(\beta,\, \lambda)^{1/2}$ converges absolutely for all $\beta\,>\,0$. It follows, that $$e^{-\beta/2 \,({\mathcal{L}_0}\,+\,{Q})}\,{\Omega_0^\beta}\,=\, {\Omega_0^\beta}\,+\,
\sum_{n\,=\,1}^\infty \int_{\Delta_{\beta/2}^{n}}{Q}({\underline}{s}_{\,n})\,{\Omega_0^\beta}\,d{\underline}{s}_{\,n}$$ exists.
Conversely, Equation and Lemma \[Lem1\] imply $$\label{eq4.12}
h_{2n}(\beta,\, \lambda)\,{\geqslant}\,(\lambda/2)^{2n}
\frac{J(2n,\,\beta)\,\beta^{2n}}{2n}
\,=\, \frac{\Big(\Theta^{-1}\,\int \frac{\beta^2\,\lambda^2/4\,|f(k)|^2}
{\sinh(|k|\,\beta/2) \sinh(\beta\,\Theta /2)}\,dk\Big)^{n}}{2n}.$$ Hence for every $\beta\,>\,0$ exists a $\lambda\in{\mathbbm{R}}$, such that $h_{2n}(\beta,\, \lambda)\,{\geqslant}\,\frac{1}{2n}$. Thus $\sum_{n=1}^\infty h_{2n}(\beta,\,\lambda)^{1/2}\,=\,\infty$
We can therefore not extended Theorem \[Thm4\] to an existence proof for all $\lambda\,>\,0$.
\[Lem1\] Following statements are true. $$\begin{aligned}
J(G,\,A,\,\beta)&= J(\#A,\beta),\ G\in \mathcal{G}_c(A)\\
J(\#A,\,\beta)&{\leqslant}(2\||k|^{-1/2}\,f\|_2 \,(\Theta\, \beta)^{-1})^{\#A}\cdot(C\,\beta\,+\,1)\\
J(\#A,\,\beta) &{\geqslant}\Big(\Theta^{-1}\,
\int \frac{|f(k)|^2}{\sinh(|k|\,\beta/2) \sinh(\Theta\, \beta/2)}\,dk\Big)^{\#A/2},\end{aligned}$$ where $\#A\,=\,2m$ and $C\,=\, (1/2)\,\frac{ \|f\|^2}{\||k|^{1/2}\,f\|^2}$.
A relabeling of the integration variables yields $$\begin{aligned}
J(G,\,A,\,\beta)
{\leqslant}\,& {\overline}{K}_f\,\int_{[0,1]^{2m}}\,
K_{osc}(|t_1-t_2|,\,\beta)\,K_{f}(|t_2\,-\,t_3|,\,\beta)\cdots\\ \nonumber
&\cdots K_{osc}(|t_{2m-1}\,-\,t_{2m}|,\,\beta)
\,d\underline{t}\end{aligned}$$ for ${\overline}{K}_f\,:=\,\sup_{s\in [0,1]}K_{f}(s,\,\beta)$. We transform due to $s_i \,:=\, t_i\,-\,t_{i\,+\,1},\ i\,{\leqslant}\, 2m-1$ and $s_{2m}\,=\, t_{2m}$, hence $-1\,{\leqslant}\, s_i\,{\leqslant}\,1,\ i\,=\,1,\ldots,\, 2m$, since integrating a positive function we obtain $$\begin{aligned}
J(G,\,A,\,\beta)
&{\leqslant}\Big(\int_{-1}^1\,K_{osc}(|s|,\,\beta)\,ds\Big)^m
\Big(\int_{-1}^1\,K_{f}(|s|,\,\beta)\,ds\Big)^{m\,-\,1}\\ \nonumber
& \cdot\sup_{s\in [0,1]}K_{f}(s,\,\beta).\end{aligned}$$ We recall that $$\int_{-1}^1K_{osc}(|s|,\,\beta)\,ds
\,=\,(2\Theta)^{-1}\int_{-1}^1\frac{\cosh(\beta \,\Theta \,|s|\,-\,\Theta\,\beta/2)}{\sinh(\Theta\,\beta/2)}\,ds
\,=\, 2(\Theta^2\, \beta)^{-1}$$ and $$\begin{aligned}
\int_{-1}^1K_{f}(|s|,\,\beta)\,ds
\,&=\,\int_{-1}^1\int \frac{\cosh(\beta\,|s|\,|k|-\beta|k|/2)\,|f(k)|^2}{2\sinh(\beta\,|k|/2)}\,dk\,ds\\
\,&=\, 2\int \frac{|f(k)|^2}{\beta\,|k|}\,dk.\end{aligned}$$ Using $\coth(x){\leqslant}1+1/x$ and using convexity of $\cosh$, we obtain $$\sup_{s\in [0,\,1]}K_{f}(s,\,\beta)
{\leqslant}(1/2)\int |f(k)|^2\,dk \,+\, \frac{1}{\beta}\,\int \frac{|f(k)|^2}{|k|}\,dk.$$ Due to the fact, that $t\,\mapsto\, K_f(t,\,\beta)$ and $t\,\mapsto \,K_{osc}(t,\,\beta)$ attain their minima at $t\,=\,1/2$, we obtain the lower bound for $J(\#A,\,\beta)$.
\[Bem3d\] In the literature there is one criterion for ${\Omega_0^\beta}\in {\operatorname{dom}}(e^{-\beta/2\,({\mathcal{L}_0}\,+\,{Q})})$, to our knowledge, that can be applied in this situation [@DerezinskiJaksicPillet2003]. One has to show that $\|e^{-\beta/2 \,{Q}}\,{\Omega_0^\beta}\|\,<\,\infty$. If we consider the case, where the criterion holds for $\pm \lambda$, then the expansion in $\lambda$ converges, $$\begin{aligned}
\|e&^{-\beta/2\, {Q}}\,{\Omega_0^\beta}\|^2
\,=\,\sum_{n\,=\,0}^\infty \frac{(\lambda\,\beta)^{2n}}{(2n)!}\,{\omega^\beta_{el}}(q^{2n})\,{ \omega^\beta_f}(\Phi(f)^{2n})\\ \nonumber
&=\sum_{n\,=\,0}^\infty \frac{(\lambda\,\beta)^{2n}}{(2n)!}\Big(\frac{(2n)!}{n!\,2^n}\Big)^2
\,K_{osc}(0,\,\beta)^n\, K_{f}(0,\,\beta)^n\\ \nonumber
&=\sum_{n=0}^\infty (\lambda\,\beta)^{2n}\,\Theta^{-n}\,{2n\choose n}2^{-2n}
\Big(\coth(\Theta\, \beta/2)\int |f(k)|^2 \,\coth(\beta\,|k|/2)\,dk\Big)^n\\ \nonumber
&\,{\geqslant}\, \sum_{n\,=\,0}^\infty (\lambda\,\beta)^{2n}(4\,\Theta)^{-n}\,
\Big(\int |f(k)|^2\, dk \Big)^n.\end{aligned}$$ Obviously, for any value of $\lambda\,\not=\,0$, there is a $\beta\,>\,0$, for which $\|e^{-\beta/2 \,{Q}}\,{\Omega_0^\beta}\|\,<\,\infty$ is not fulfilled.
**Acknowledgments**:\
This paper is part of the author’s PhD requirements. I am grateful to Volker Bach for many useful discussions and helpful advice. The main part of this work was done during the author’s stay at the Institut for Mathematics at the University of Mainz. The work has been partially supported by the DFG (SFB/TR 12).
{#App1}
\[Lem1App\] Let $f,\, g\,:\, \{ z\in {\mathbbm{C}}\,:\, 0\,{\leqslant}\, {\textrm{Re}\,}(z)\,{\leqslant}\,\alpha\}\,\rightarrow \,{\mathbbm{C}}$ continuous and analytic in the interior. Moreover, assume that $f(t)\,=\,g(t)$ for $t\in {\mathbbm{R}}$. Then $f\,=\,g$.
Let $h\,:\, \{ z\in {\mathbbm{C}}\,:\, |\,{\textrm{Im}\,}(z)\,<\,\alpha\}\rightarrow {\mathbbm{C}}$ defined by $$h(z):=
\begin{cases} f(\,z\,)\,-\, g(\,z\,),& \textrm{on } \{ z\in {\mathbbm{C}}\,:\, 0\,{\leqslant}\, {\textrm{Im}\,}(z)\,<\,\alpha\}\\
{\overline}{f(\,{\overline}{z}\,)}- {\overline}{g(\,{\overline}{z}\,)},& \textrm{on } \{ z\in {\mathbbm{C}}\,:\, -\alpha\,<\, {\textrm{Im}\,}(z)\,<\,0\}
\end{cases}$$ Thanks to the Schwarz reflection principle $h$ is analytic. Since $h(t)\,=\,0$ for all $t\in{\mathbbm{R}}$, we get $h\,=\,0$. Hence $f=g$ on $\{ z\in {\mathbbm{C}}\,:\, 0\,{\leqslant}\, {\textrm{Re}\,}(z)\,<\,\alpha\}$. Since both $f$ and $g$ are continuous, we infer that $f=g$ on the whole domain.
\[Lem2App\] Let $H$ be some self-adjoint operator in $\mathcal{H}$, $\alpha>0$ and $\phi\in {\operatorname{dom}}( e^{\alpha\,H} )$. Then $\phi\in {\operatorname{dom}}( e^{z \,H})$ for $z\in \{ z\in {\mathbbm{C}}\,:\, 0\,{\leqslant}\, {\textrm{Re}\,}(z)\,{\leqslant}\,\alpha\}$. $z\mapsto e^{z \,H}\phi$ is continuous on $\{ z\in {\mathbbm{C}}\,:\, 0\,{\leqslant}\, {\textrm{Re}\,}(z)\,{\leqslant}\,\alpha\}$ and analytic in the interior.
Due to the spectral calculus we have $$\int e^{2{\textrm{Re}\,}\, z\, s} d\langle \phi\,|\,\mathbbm{E}_s\,\phi\rangle{\leqslant}\int (1+ e^{2\alpha\, s}) d\langle \phi\,|\,\mathbbm{E}_s\,\phi\rangle=:C_1^2<\infty.$$ Thus $\phi\in {\operatorname{dom}}( e^{z \,H})$. Let $\psi\in \mathcal{H}$ and $f(z)=\langle \psi\,|\,e^{z \,H}\,\phi\rangle$. There is a sequence $\{\psi_n\}$ with $\psi_n\in \bigcup_{m\in {\mathbbm{N}}} \operatorname{ran}\,\mathbbm{1}[ |H|{\leqslant}m]$ and $\lim_{n\to\infty}\psi_n=\psi$. We set $f_n(z)= \langle \psi_n\,|\,e^{z\,H}\,\phi\rangle.$ It is not hard to see that $f_n$ is analytic, since $\psi_n$ is an analytic vector for $H$, and that $|f_n(z)|{\leqslant}C_1\,\|\psi_n\|$ and $\lim_{n\to \infty}f_n(z)=f(z)$. Thus $f$ is analytic and hence $z\mapsto e^{z \,H}\phi$ is analytic. Thanks to the dominated convergence theorem the right side of $$\|e^{z_n H}\phi-e^{z H}\phi\|^2
{\leqslant}\int (e^{2{\textrm{Re}\,}z_n s}+e^{2{\textrm{Re}\,}z s}- e^{\bar{z_n}s+zs}-e^{\bar{z}s+z_n s})
d\langle \phi\,|\,\mathbbm{E}_s\,\phi\rangle$$ tends to zero for $\lim_{n\to\infty}z_n=z$. This implies the continuity of $z\mapsto e^{z \,H}\phi$.
\[Lem0.2\] We have for $n_1+n_2{\geqslant}1$$$\label{eq3.52}
\int_{\Delta_1^{n}} C_\kappa({\underline}{s})\, d{\underline}{s}_n
\,{\leqslant}\, \frac{{\operatorname{const}}\,C^{n_2}}{(n_1+n_2)!\,(n+1)^{(1-2\gamma)\,n_2 -2} }$$
We turn now to the integral $$\label{eq3.54}
\int_{\Delta_1^{n}} C_\kappa({\underline}{s})\, d{\underline}{s}_{\,n}
\,=\,\int_{\Delta_1^{n}} (1\,-\,s_1\,+\,s_{n})^{-\alpha_1}
\prod_{i=1}^{n\,-\,1}(s_i\,-\,s_{i\,+\,1})^{-\alpha_i}\,d\underline{s}_n.$$ We define for $k\,=\,1,\ldots,\,2n$, a change of coordinates by $s_k\,=\, r_1\,-\,\sum_{j\,=\,2}^k r_j$, the integral transforms to $$\begin{aligned}
\label{eq3.55}
\int_{S^{n}} &(1\,-\,(r_2\,+\cdots+\,r_{n}))^{-\alpha_1}
\prod_{i\,=\,2}^{n}r_i^{-\alpha_i}\, d\underline{r}_{\,n}\\ \nonumber
&= \int_{T^{n-1}} (1\,-\,(r_2\,+\cdots+\,r_{n}))^{1\,-\,\alpha_1}
\prod_{i=2}^{n}r_i^{-\alpha_i}\,d\underline{r}_{n-1}\\ \nonumber
\,&=\, \frac{\Gamma(1\,-\,\alpha_1)^{-1}\Gamma\big( 1\,-\,\gamma\big)^{2n_2}}
{\Gamma\big( n_1\,+\, 2n_2\,(1-\gamma)\big)}\end{aligned}$$ where $ S^{2n} \,:=\,\{ \underline{r}\in {\mathbbm{R}}^{2n}\, :\, 0{\leqslant}\, r_i\,{\leqslant}\, 1,\ r_2\,+\cdots +\,r_{2n}\,{\leqslant}\, r_1\}$ and $T^{2n-1} :=\{ \underline{r}\in{\mathbbm{R}}^{2n-1}\, :\, 0\,{\leqslant}\,r_i{\leqslant}\, 1,\ r_2\,+\cdots +\,r_{2n} \,{\leqslant}\, 1\}$. From the first to the second formula we integrate over $dr_1$. The last equality follows from [@GradshteynRyzhik1980 Formula 4.635 (4)], here $\Gamma$ denotes the Gamma-function.\
From Stirling’s formula we obtain $$\label{eq3.56}
(2\pi)^{1/2}\, x^{x\,-\,1/2}\,e^{-x}\,{\leqslant}\, \Gamma(x)\,{\leqslant}\, (2\pi)^{1/2}\, x^{x\,-\,1/2}\,e^{-x\,+\,1},\quad x\,{\geqslant}\, 1.$$ Since $n_1\,+\,n_2\,{\geqslant}\, 1$ get $$\label{eq3.57}
\frac{\Gamma(n_1+n_2+1)}{\Gamma(n_1+2(1-\gamma)\,n_2)}
\,{\leqslant}\, (n+1)^2
\Big(\frac{n_1+2(1-\gamma)n_2}{e}\Big)^{-(1-2\gamma)n_2}.$$ Note that $\Gamma(n_1+n_2+1)=(n_1+n_2)!$.
\[Lem-N1\] Let $(1\,+\,\alpha(k)^{-1/2})\,f_1,\ldots,(1\,
+\,\alpha(k)^{-1/2})\,f_{2m}\in {\mathcal{H}_{ph}}$ and $\sigma\in\{+,-\}^{2m}$. Let $a^+\,=\,a^*$ and $a^-\,=\,a$ $$\begin{aligned}
\label{eq3.39}
\lefteqn{
{ \omega^\beta_f}\big(a^{\sigma_{2m}}(e^{-\sigma_{2m} \,s_{2m}\,\alpha(k)}\,f_{2m})
\cdots a^{\sigma_1}(e^{-\sigma_1\, s_1\,\alpha(k)}\,f_1)\big)}\\ \nonumber
&=& \int f_{2m}^{\sigma_{2m}}(k_{2m},\,\tau_{2m})\cdots f_1^{\sigma_1}(k_{1},\,\tau_1)\,
\nu(d {\underline}{k}_{2m}{\otimes}d{\underline}{\tau}_{2m}),\end{aligned}$$ where $\nu(d {\underline}{k}_{2m}{\otimes}d{\underline}{\tau}_{2m}) $ is a measure on $({\mathbbm{R}}^3)^{2m}\times\{+,\,-\}^{2m}$ for phonons, respectively on $({\mathbbm{R}}^3\times\{\pm\})^{2m}\times\{+,\,-\}^{2m}$ for photons, and $$\label{eq3.43}
\nu(d{\underline}{k}_{2m}{\otimes}d{\underline}{\tau}_{2m})
{\leqslant}\sum_{P\in\mathcal{Z}_{2m}}\sum_{{\underline}{\tau}\in \{+,\,-\}^{2m}} \prod_{\{i\,>\,j\}\in P}
\Big(\delta_{k_i,\,k_j}\coth(\beta\,\alpha(k_i)/2)\Big)\,d{\underline}{k}_{2m}.$$ for $f^+(k,\,\tau)\,:=\, f(k)\,{\mathbf{1}}[\tau\,=\,+]$ and $f^+(k,\,\tau)\,:=\,{\overline}{f(k)}\,{\mathbf{1}}[\tau\,=\,-]$.
Since ${ \omega^\beta_f}$ is quasi-free, we obtain with $a^+\,:=\,a^*$ and $a^-\,:=\,a$ $$\begin{aligned}
\label{eq3.36}
\lefteqn{
{ \omega^\beta_f}\big(a^{\sigma_{2m}}(e^{-\sigma_{2m} \,s_{2m}\,\alpha(k)}\,f_{2m})
\cdots a^{\sigma_1}(e^{-\sigma_1 \,s_1\,\alpha(k)}\,f_1)\big)}\\ \nonumber
&=&\sum_{P\in \mathcal{Z}_2}\prod_{\stackrel{\{i,\,j\}\in P}{{i\,>\,j}}}
{ \omega^\beta_f}\big(a^{\sigma_i}(e^{-\sigma_i\, s_i\, \alpha(k)}\,f_i)\,
a^{\sigma_j}(e^{-\sigma_j \,s_j\,\alpha(k)}\,f_j)\big),\end{aligned}$$ see Equation . For the expectation of the so called two point functions we obtain: $$\label{EvalOneTwoPoinfOmf}
{ \omega^\beta_f}\big(a^+(e^{ s_i\, \alpha(k)}\,f_i)\,a^{+}(e^{ s_j\,\alpha(k)}\,f_j)\big)\,=\,
0\,=\,{ \omega^\beta_f}\big(a(e^{- s_i\,\alpha(k)}\,f_i)\,a(e^{- s_j\,\alpha(k)}\,f_j)\big),$$ such as $$\begin{aligned}
\label{EvalOmf}
{ \omega^\beta_f}\big(a^+(e^{ xs_i\alpha(k)}f_i)a^{-}(e^{- x\,s_j\alpha(k)}f_j)\big)
&\,=\,&\int f_i(k)\,{\overline}{f_j(k)}\
\frac{e^{ x\,(s_i\,-\,s_j)\alpha(k)}}{e^{\beta \,\alpha(k)}-1}\,dk\\ \nonumber
{ \omega^\beta_f}\big(a^-(e^{ xs_i\alpha(k)}f_i)\,a^{+}(e^{- xs_j\alpha(k)}f_j)\big)
&\,=\,&\int f_j(k)\,{\overline}{f_i(k)} \
\frac{e^{ (\beta+ xs_j-xs_i)\alpha(k)}}{e^{\beta\alpha(k)}-1}\,dk\end{aligned}$$ Hence it follows $$\begin{aligned}
\label{QuasiFreeExp2}
\lefteqn{
{ \omega^\beta_f}\big(a^{\sigma_{2m}}(e^{-\sigma_{2m}\, s_{2m}\,\alpha(k)}\,f_{2m})
\cdots a^{\sigma_1}(e^{-\sigma_1 \,s_1\,\alpha(k)}\,f_1)\big)}\\ \nonumber
&=& \int f_{2m}^{\sigma_{2m}}(k_{2m},\,\tau_{2m})\cdots f_1^{\sigma_1}(k_{1},\,\tau_1)\,
\nu(d{\underline}{k}_{2m}{\otimes}d{\underline}{\tau}_{2m}),\end{aligned}$$ where $f^+(k,\,\tau)\,:=\, f(k)\,{\mathbf{1}}[\tau=+]$ and $f^-(k,\,\tau)\,:=\,{\overline}{f(k)}\,{\mathbf{1}}[\tau\,=\,-]$.\
$\nu(d^{3(2m)}k{\otimes}d^{2m}\tau) $ is a measure on $({\mathbbm{R}}^3)^{2m}\times\{+,\,-\}^{2m}$, which is defined by $$\begin{aligned}
\label{DefNu}
\sum_{P\in\mathcal{Z}_{2m}}&\ \sum_{{\underline}{\tau}\in \{+,\,-\}^{2m}}\, \prod_{\{i\,>\,j\}\in P}
\delta_{\tau,\,-\tau}\,\delta_{k_i,\,k_j}\, \\ \nonumber
&
\Big(\delta_{\tau,\,+}
\,\frac{e^{x\,(s_i\,-\,s_j)\,\alpha(k_i)}}{e^{\beta \,\alpha(k_i)}\,-\,1}\,
+\,\delta_{\tau,\,-}\,
\frac{e^{(\beta\,-\,x\,(s_i\,-\,s_j))\,\alpha(k_i)}}{e^{\beta \,\alpha(k_i)}\,-\,1}\Big)\,d{\underline}{k}_{2m}.\end{aligned}$$
[1]{} H. Araki, E. Woods. Representations of the Canonical Commutation Relations describing a non-relativistic infinite free Bose Gas. , 4:637–662, 1963. V. Bach, J. Fr[ö]{}hlich, and I. M. Sigal. Quantum electrodynamics of confined non-relativistic particles. , 137:299–395, 1998. V. Bach, J. Fröhlich, I. M.Sigal. Return to Equilibrium. , 41:3985–4060, 2000. O. Bratteli, D. Robinson. Operator Algebras and Quantum Statistical Mechanics 1, Text and Monographs in Physics, Springer-Verlag, 1987. O. Bratteli, D. Robinson. Operator Algebras and Quantum Statistical Mechanics 2, Text and Monographs in Physics, Springer-Verlag, 1996. Derezinski J., Jaksic V., Pillet C.-A. , 15: 447-489, 2003. J. Derezinski, V. Jaksic. Spectral theory of [P]{}auli-[F]{}ierz operators. , 180 : 243–327, 2001 J. Derezinski, V. Jaksic. Return to Equilibrium for [P]{}auli-[F]{}ierz Operators. , 4 : 739–793, 2003 J. Fr[ö]{}hlich, M. Merkli. Another Return to Equilibrium , 251: 235–262, 2004. J. Fr[ö]{}hlich, M. Merkli, I. M.Sigal. Ionization of Atoms in a Thermal Field. , 116: 311–359, 2004. DOI 10.1023/B:JOSS.0000037226.16493.5e I.S. Gradstein, I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, 1980, 4 R. Haag, N. Hugenholz, M. Winnink. On the Equilibrium States in Quantum Statistical Mechanics. , 5: 215–236, 1967. V. Jakši[ć]{}, C. A. Pillet. On a Model for Quantum Friction. [I]{}[I]{}: [F]{}ermi’s Golden Rule and Dynamics at Positive Temperature. , 176:619–643, 1996 V. Jakši[ć]{}, C. A. Pillet. On a Model for Quantum Friction [I]{}[I]{}[I]{}: Ergodic Properties of the Spin-Boson System. , 178:627–651, 1996 M. Merkli. Positive Commutators in Non-Equilibrium Statistical Quantum Mechanics, , 223: 327–362, 2001. M. Reed,B. Simon. Methods of Modern Mathematical Physics: [I]{}. [F]{}unctional Analysis Academic Press, 1980. M. Reed,B. Simon. Methods of Modern Mathematical Physics: [I]{}[I]{}. [F]{}ourier Analysis and Self-Adjointness Academic Press, 1980.
[^1]: Supported by the DFG (SFB/TR 12)
|
---
author:
- 'E. Bittencourt,'
- 'L. G. Gomes'
- 'and G. B. Santos'
title: Effects of tidal forces and heat flow on the late evolution of the universe
---
Introduction
============
Several attempts to describe our actual clumped universe have been made by adopting fewer symmetries than the ones present in the standard cosmological model, described by the Friedmann-Lemaître-Robertson-Walker (FLRW) class of geometries, which is invariant under a six-parameter group of isometries and whose surfaces of transitivity are three-dimensional spacelike hypersurfaces of constant curvature [@ellis_mac_marteens]. A number of inhomogeneous geometries have been constructed following this former idea in order to give a more realistic view of cosmology [@steph03; @kra]. Special attention is given to the models dealing with the averaging and backreaction issues [@wiltshire; @kolb; @buchert], and the exact Lemaître-Tolman (LT) and Szekeres models [@krasinski].
A particularly interesting technique to generate inhomogeneous cosmologies is the one developed by Szafron and Collins [@coll_79; @coll1_79; @coll2_79] in which exact solutions to the Einstein’s field equations (EFE) are found through the so-called intrinsic symmetries approach, where only submanifolds of the whole space-time admit certain group of isometries. For instance, when the fluid flow is irrotational, one could impose symmetries on the hypersurfaces orthogonal to the flow seen as three-dimensional manifolds and still the full space-time would possess no symmetries whatsoever.
Following this procedure, exact solutions were studied considering simple matter contents, as vacuum [@wolf86], irrotational dust [@bona92; @sop00] and perfect fluids [@arg85; @wolf86a]. In [@wolf86], flatness of three-dimensional hypersurfaces (that could be either timelike or spacelike) was imposed together with the condition of a traceless extrinsic curvature, for simplicity. In [@bona92] irrotational dust metrics were considered by demanding that the hypersurfaces orthogonal to the fluid flow had constant curvature, and the authors found that the resulting solutions are either contained in the Szekeres dust solutions [@sze75] or are homogeneous of certain Bianchi types. Also in [@sop00] an irrotational dust solution was considered and a maximal group of isometries, that is, a six-parameter group of motions, was imposed on the spacelike hypersurfaces orthogonal to the fluid velocity, and the conclusion was that all the irrotational dust solutions of EFE with flat spatial geometry were either Bianchi I or were subfamilies of the Szekeres geometry.
In this work we propose an explicit inhomogeneous solution of the EFE by restricting the hypersurfaces orthogonal to the irrotational fluid flow to be flat and considering the matter content to be general, in principle. We find that we only need to determine the energy density function in order to completely specify the geometry and the matter distribution. As we specify the equations of state for the anisotropic stresses and the pressure, we show how the inhomogeneities in the model yield an accelerated expansion. Compared to the FLRW model, this leads to a behavior usually attributed to the dark energy component.\
We begin by presenting in the next section the inhomogeneous solution in detail. Then we move, in Sec.\[III\], to the analysis of the luminosity distance and the calculation of the deceleration parameter in the approximation of small values of the redshift. We then proceed to put bounds, in Sec. \[IV\], on the parameters of our model based on the observed value for the deceleration parameter. Concluding remarks are presented in Sec. \[V\] and, for completeness, we compute the Newman-Penrose invariants of the geometry in the Appendix. We adopt conventions as in Ref. [@ellis_mac_marteens], except that Greek indices $\alpha, \beta, \gamma \ldots$ run from $0$ to $3$ and Latin indices $i, j, k \ldots$ run from $1$ to $3$ (the three spatial directions). Geometric units are assumed such that $c=\kappa=1$, where c is the speed of light and $\kappa$ is the Einstein constant.
The solution {#II}
============
The properties of the metric we propose here are inspired by the observed homogeneity and isotropy of the universe encoded in the cosmological principle, which is usually put forward by adopting the FLRW model with flat spatial curvature scalar. However, this hypothesis can be relaxed by imposing that just the hypersurfaces at constant time $t$ are maximally symmetric without requiring that their corresponding isometries are also symmetries of the whole space-time. With this in mind, the simplest manner to implement this scenario, in the flat case, is through the infinitesimal line element $$\label{met_f}
ds^2=-e^{f(t,x,y,z)}dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ where $f(t,x,y,z)$ is an arbitrary function (at least of class ${\cal C}^2$) and $a(t)$ is the scale factor. With no further assumptions, this is an algebraically general metric of Petrov type I (see the Appendix for details). Furthermore, from a recent discussion in the literature [@green; @buchert15; @green1]—in which it was elucidated that linear perturbations of homogeneous and isotropic models lead to different results when compared to a linearized inhomegeneous solutions—we can state that the exact solution of EFE derived here is indeed the simplest generalization of the flat standard model in terms of intrinsic symmetries.
In this model, the fluid flow will be described by the normalized, irrotational and shear-free vector field $u^\mu=e^{-f/2}\delta^\mu{}_0$ and the flat 3-hypersurfaces, with metric given by $h_{\mu\nu}=g_{\mu\nu}+ u_{\mu}u_{\nu}$, are orthogonal to it. The energy-momentum tensor can be generally decomposed by this field as
$$T_{\mu\nu}=\rho u_\mu u_\nu+ph_{\mu\nu}+q_{\mu} u_{\nu}+q_{\nu} u_{\mu}+\pi_{\mu\nu},$$
where $\rho$ is the energy density, $p$ is the isotropic pressure, $q_\mu$ is the heat flow, $\pi_{\mu\nu}$ is the traceless symmetric anisotropic pressure tensor and $q_\nu\,u^\nu=0$ and $\pi_{\mu\nu}\, u^\nu =0$.
In terms of this matter content, the $0-0$ component of the EFE $G_{\mu\nu}=T_{\mu\nu}$ gives $$\label{coord_trans1}
e^{f(t, x,y,z)}={\frac{\textstyle{3H^2(t)}}{\textstyle{\rho(t,x,y,z)}}},$$ where $H=\dot{a}/a$ and dot means derivative with respect to the $t$-coordinate. Similarly to [@leandro], we can rewrite the line element using the scale factor as the time coordinate and introduce an explicit dependence of the metric with respect to the energy density through relation (\[coord\_trans1\]), obtaining $$ds^2=-{\frac{\textstyle{3}}{\textstyle{a^2\rho(a,x,y,z)}}}da^2+a^2(dx^2+dy^2+dz^2).$$ In this new coordinate system, the $0-0$ component of Einstein’s equations is an identity. The other components lead to $$\begin{aligned}
{\frac{\textstyle{1}}{\textstyle{\sqrt{3\,\rho}}}} \,\partial_i\rho&=&q_i\label{heat__eisn_eq}\\
\sqrt{\rho}\left[\nabla^2\left(\rho^{-\frac{1}{2}}\right) - \partial_{ij}\left(\rho^{-\frac{1}{2}}\right)\right] - \frac{a^3}{3}\,\frac{\partial\rho}{\partial a} - a^2\rho &=&a^2p+\pi_{ij},\qquad \mbox{for}\,\, i=j,\label{pres__eisn_eq}\\
\frac{1}{4}\left({\frac{\textstyle{2\rho\,\partial_{ij}\rho - 3\partial_i\rho \,\partial_j\rho}}{\textstyle{\rho^2}}}\right)&=&\pi_{ij},\qquad \mbox{for}\,\, i\neq j,\label{apres__eisn_eq}\end{aligned}$$ where $\partial_i\equiv\frac{\partial}{\partial x^i}$, $\partial_{ij}\equiv\frac{\partial^2}{\partial x^i\, \partial x^j}$ and $\nabla^2:= \delta^{ij}\partial_{ij}$ is the 3-dimensional Euclidean Laplacian. Note that the energy density in the equations above comes from the geometry through EFE. Once it is somehow provided, all the other fluid components are determined. Indeed, Eqs. (\[heat\_\_eisn\_eq\])-(\[apres\_\_eisn\_eq\]) can be seen as a set of physically meaningful equations of state for a viscous fluid that is compatible with Einstein’s equations. The physical meaning of this assertion will become clearer afterwards.
Concerning the non-trivial kinematic quantities associated to the fluid four-velocity $u^{\mu}$, we find that the expansion coefficient is given by $$\vartheta=u^\mu{}_{;\mu}=\sqrt{3\rho},$$ while the acceleration vector $a_\mu=u_{\mu;\nu}u^\nu$ is expressed as $$a_i=-\frac{1}{2}\,\partial_i\ln\rho.$$ Note that the expansion is not homogeneous and the acceleration is nonzero due to the spatial dependence of the energy density. Thus, we see that we are not dealing with a geodesic congruence, even though it is irrotational and shear-free. Due to the non-null acceleration, the evolution of the deviation vector along the fluid flow lines, $\eta^{\mu}$, after substituting the EFE, becomes $$\label{geo_devia}
\frac{d^2\eta^{\mu}}{d\lambda^2}=\left[\frac{1}{2}\pi^{\mu}{}_{\alpha} - E^{\mu}{}_{\alpha} - \frac{(\rho+3p)}{6}h^{\mu}{}_{\alpha}\right]\eta^{\alpha} + a^{\mu}{}_{;\alpha}\eta^{\alpha},$$ where $E_{\mu\nu}=C_{\mu\alpha\nu\beta}u^{\alpha}u^{\beta}$ denotes the electric part of the Weyl tensor $C_{\mu\alpha\nu\beta}$.
The phenomenological and nonrelativistic equation of state for the anisotropic pressure ($\pi_{\mu\nu}=-\xi\sigma_{\mu\nu}$) is not suitable here, since the matter content is described in terms of shear-free comoving observers. In this way, if viscosity is still present in the cosmological fluid in the form of an anisotropic pressure, then it could be related to dissipation of the gravitational potential energy, in particular, through tidal forces [@douce]. Thus, we have to appeal to an extension of the phenomenological approach by allowing $\pi_{\mu\nu}$ to be a function of the electric part of the Weyl tensor too. In general cases, the equation of state for the anisotropic pressure should be $$\pi_{\mu\nu}=\pi_{\mu\nu}(\sigma_{\alpha\beta},E_{\alpha\beta}).$$ For the sake of compatibility with the evolution of the kinematical quantities, in particular to the shear evolution (see details in [@ellis_mac_marteens; @bit_review]), there is a natural and unique equation of state given by[^1] $$\label{epi}
\pi_{\mu\nu}=-2E_{\mu\nu},$$ indicating that the viscosity of the fluid and the tidal forces acting on it are indeed related. Remarkably, the terms involving the acceleration vector and its covariant derivatives in the evolution equation for the shear tensor are proportional to the electric part of the Weyl tensor, showing the consistency of the equation of state aforementioned. It should also be noticed that such relation is not the one derived in the existing literature (see [@mimoso; @coley94; @bss14; @bss15]). In our case, the spatial components of $E_{\mu\nu}$ can be written down as $$\label{comp_elec}
E_{ij}=\sqrt{\rho}\left[\frac{1}{2}\,\partial_{ij}\left(\rho^{-\frac{1}{2}}\right)-\frac{1}{6}\,\delta_{ij}\,\nabla^2\left(\rho^{-\frac{1}{2}}\right)\right].$$
Comparison of Eq. (\[comp\_elec\]) with Eq. (\[apres\_\_eisn\_eq\]), using the equation of state (\[epi\]), gives that the off-diagonal spatial components of Einstein’s equations are identically satisfied. Thus, the remaining diagonal spatial components of the Einstein tensor are all equal to $$\frac{\nabla^2(\rho^{-\frac{1}{2}})}{\rho^{-\frac{1}{2}}} =\frac{a^2}{2}\left(a\,\frac{\partial\rho}{\partial a} + 3\gamma\rho\right),$$ where we have assumed a barotropic equation of state of the form $p=(\gamma-1)\rho$ in which $\gamma$ is considered a function of the scale factor.
In the sequence we consider the energy density as a product of a function depending only on $a$ and a function depending only on the spatial coordinates in the convenient following manner $$\label{definicao_Xi}
\rho(a,x,y,z)=\frac{\epsilon(a)}{[\chi(x,y,z)]^2},$$ resulting, for the time dependent part, $$\label{eq_rho_a}
a^3\,\frac{d\epsilon}{da}+3\gamma a^2 \epsilon=-2\kappa,$$ and for the spatial dependent part $$\label{laplacian}
\chi\nabla^2\chi=- \kappa,$$ where $\kappa$ is a constant of separability and the minus sign was chosen for later convenience. First, let us analyze the dependence of the energy density with respect to the scale factor. Eq. (\[eq\_rho\_a\]) can be directly integrated in the case of $\gamma$ constant, yielding
$$\label{sol_eq_rho_a}
\gamma=const. \quad \Rightarrow \quad
\epsilon(a)=\epsilon_1 \left(\frac{ a_0}{a}\right)^{3\gamma}-\frac{2\kappa}{(3\gamma-2)\, a^2},$$
where $\epsilon_1 = \epsilon_0 +\frac{2\kappa}{(3\gamma-2)\,a_0^2}$ and $\epsilon_0 = \epsilon(a=a_0)$. The case $\gamma=2/3$ has to be solved separately, resulting in $$\label{sol_eq_rho_a_gama23}
\gamma=2/3 \quad \Rightarrow \quad
\epsilon(a)=\epsilon_0 \left(\frac{ a_0}{a}\right)^{2}-\frac{2\kappa}{a^2}\,\ln\left(\frac{a}{a_0}\right).$$
The first term on the right hand side of Eq. (\[sol\_eq\_rho\_a\]) is the same as in the flat FLRW models and the second term (proportional to $a^{-2}$) mimics a spatial curvature term for $\kappa\neq0$ and $\gamma\neq2/3$ which plays an important role at late times in the evolution of the universe, as we shall see. It should also be noticed that this solution is invariant under the transformation $a\rightarrow \lambda a$ and $\kappa\rightarrow\lambda^2\kappa$. In an expanding universe model there is a combination of the constants such that the energy density goes to zero at a finite value of the scale factor given by
$$\frac{a}{a_0}=\, \left[\frac{2\kappa}{ (3\gamma -2) \,\epsilon_1\,a_0^2}\right]^{\frac{1}{2-3\gamma}},\quad\mbox{for}\quad \gamma\neq\frac{2}{3}\qquad\mbox{and}\qquad
\frac{a}{a_0}= \, \exp \left(\frac{\epsilon_0 \, a_0^2}{2 \kappa}\right),\quad\mbox{for}\quad \gamma=\frac{2}{3}.$$
When this happens the metric becomes singular in the same manner as the closed FLRW case.
Therefore, by assuming that the relation in Eq. (\[epi\]) is valid and assuming a separation of variables of the form (\[definicao\_Xi\]) with a linear equation of state for the isotropic pressure, the geometry is then completely determined by solutions of Eq. (\[laplacian\]) with appropriate boundary conditions. In the literature, this is well-known and corresponds to the static version of a nonlinear Klein-Gordon equation whose potential is inversely proportional to $\chi$ [@polyanin].
Solutions of the elliptic partial differential equation (\[laplacian\]) can be found in the implicit form $$c_1x + c_2y + c_3z + c_4 +c_5- \sqrt{c_1^2 + c_2^2 + c_3^2}\int\frac{d\chi}{\sqrt{-2\kappa\ln\chi- (c_1^2 + c_2^2 + c_3^2) c_4}}=0.$$ However, this general expression is not needed for our purposes here, as we shall see in the next section.
Redshift and luminosity distance {#III}
================================
We perform now an analysis of the luminosity distance in this geometry, restricting ourselves afterwards in the regime of small values of the redshift. With this in mind, we start by computing the geodesic equations for the metric (\[met\_f\]), which are given by $$\begin{aligned}
&&\frac{d k^0}{d\lambda} + H (k^0)^2 + 2\, \frac{\chi'}{\chi}\,k^0 =0,\label{geo1}\\[1ex]
&&\frac{d\vec k}{d\lambda} + \frac{\chi\nabla\chi}{a^2}(k^0)^2 + 2 H\,k^0\,\vec k=0,\label{geo2}$$ where $k^0=dt/d\lambda$, $\vec k=\left(\frac{dx}{d\lambda},\frac{dy}{d\lambda},\frac{dz}{d\lambda}\right)$ and $\chi'=\vec k\cdot \nabla\chi$ with $\lambda$ as the affine parameter of the geodesics and where dot ($\cdot$) stands for the usual Euclidean scalar product. The line element provides the constraint $$\label{first_line_el}
b=-\chi^2(k^0)^{2}+a^2(t) (\vec k\cdot\vec k),$$ where $b=-1,0,+1$ if the geodesic is time-like, light-like or space-like, respectively. In particular, to find the corrected expression for the redshift, we only need to study the null geodesics. Thus, substitution of Eq. (\[first\_line\_el\]) into Eq. (\[geo1\]) yields $$\label{first_int_geo1}
k^0=\frac{E}{a\chi^2},$$ where $E$ is an integration constant.
According to [@ellis_mac_marteens], the redshift definition is $$1+z=\frac{(-u_{\mu}k^{\mu})|_{e}}{(-u_{\mu}k^{\mu})|_{o}},$$ in which the subscript $e$ indicates the spacetime event where the photon was emitted and the subscript $o$ indicates the spacetime event where the photon was observed. The spatial dependence of $z$ through $\chi$ makes the redshift expression completely different from the one in FLRW models. In this way, the equation for $z$ becomes
$$\label{z_lambda}
1+z(\lambda) =\frac {a_{0}\chi_0}{a(\lambda)\chi(\lambda)},$$
where $a_0$ is the scale factor today and $\chi_0$ is the inverse square root of the spatial energy density at the location in which the photon is observed. In virtue of the freedom in the parametrization of the null geodesics, we can choose $(-u_{\mu}k^{\mu})|_{o}=1$. It implies here that $E=a_0\chi_0$ .
Now we proceed to calculate the luminosity distance. First, we consider the Sachs equations [@ferreira; @bentivegna] $$\begin{aligned}
&&\frac {d^2D_A}{d\lambda^2} + \left( {\Sigma}^{2}+\frac{1}{2}R_{\mu\nu}k^{\mu}k^{\nu} \right)\,D_A=0,\\
&&\frac {d\Sigma}{d\lambda}+2\left( \frac {d\ln D_A}{d\lambda}\right)\Sigma=C_{\alpha\beta\mu\nu}m^{\alpha}k^\beta m^{\mu}k^\nu\end{aligned}$$ for the angular diameter distance $D_A$ and the shear $\Sigma$ of the null congruence, where $m^\mu$ is a space-like vector orthogonal to $k^\mu$. As we set the initial conditions for the observer $$D_A(0)=0 \;{\rm and}\;\frac{dD_A}{dz}{\bigg|}_0=\frac{1}{H_0} ,$$ where $H_0$ is the Hubble parameter at the instant of time in which the photon is observed, we see that for $z=0$ we have $\frac{d^2D_A}{d\lambda^2}=0$, which implies, $$\label{DerivadaSegundaDA}
\left(\frac{dz}{d\lambda}\right)^{2} \frac{d^2D_A}{dz^{2}} = - \left(\frac{d^{2}z}{d\lambda^{2}}\right)\frac{1}{H_0} \qquad \text{as $z=0$}.$$ In a certain way, the analysis of $D_A(z)$ up to terms of order $z^2$ around $z=0$ ressembles the Dyer-Roeder approach for studying light propagation in inhomogeneous backgrounds [@dyer73], where the optical shear $\Sigma$ and $R_{\mu\nu}k^{\mu}k^{\nu}$ are set to zero along every null curve, that is, $\frac{d^2D_A}{d\lambda^2}=0$ along the entire light path. But it should be clear that we have $\frac{d^2D_A}{d\lambda^2} \ne 0$ for $z \ne 0$, and this changes everything. As we compare to their work, we should expect that the function $D_A(z)$ contains information about the local inhomogeneities such that the introduction of a single phenomenological constant adapted to the FLRW space-time could not describe.
Following the steps presented in Ref. [@villani], we consider Taylor expansions for $a$ and $\chi$ up to the second order in $\lambda$, as follows $$a(\lambda) = a_0\left(1 + a_1H_0\lambda + \frac{1}{2}\, a_2H_0^2\,\lambda^2\right) + O(\lambda^3)$$ and $$\chi(\lambda)=\chi_0\left(1+\chi_1H_0\lambda+\frac{1}{2}\, \chi_2H_0^2\,\lambda^2\right) + O(\lambda^3).$$ Using Eqs. (\[geo1\]) and (\[first\_int\_geo1\]), we see that the parameters $a_1$ and $a_2$ are related to the constants $\chi_i$ for $i=0,1,2$ through $$a_1=\frac{1}{\chi_0}\qquad\mbox{and}\qquad a_2=-\frac{1+q_0+2\chi_0\chi_1}{\chi_0^2},$$ where $q(t)=-\ddot a/aH^2$ is the FLRW deceleration parameter and $q_0$ its value in the moment of the observation.
At this order, we are able to compute the contribution of the inhomogeneity to the deceleration parameter and then try, in a first moment, to put bounds on such contributions based on observations. Thus, differentiating Eq. (\[z\_lambda\]) with respect to $\lambda$ and rewriting the outcome in terms of $z$ yields
$$\label{DerivadasDAZ0}
z=0 \quad \Rightarrow \quad
\frac{dz}{d\lambda} = -H_0\, (a_1+\chi_1)
\quad \text{and} \quad
\frac{d^2z}{d\lambda^2}=H_0^2\,[2(a_1^2+a_1\chi_1 + \chi_1^2)-\chi_2-a_2].$$
As we use the relation $D_L=(1+z)^2D_A$ between the luminosity and angular distances [@Ether], the Taylor expansion of order $z^2$ is expressed as [@visser] $$D=\frac{D^{(1)}}{H_0}\,z+\frac{D^{(2)}}{2H_0}\,z^2 ,$$ where the coefficients are linked through $$\begin{array}{lcl}
D_L^{(1)}&=&D_A^{(1)}=1,\\[1ex]
D_L^{(2)}&=&D_A^{(2)}+4\equiv 1-q_{{\rm eff}}
\end{array}$$ with $q_{{\rm eff}}$ the effective (observed) deceleration parameter. We get from the equations (\[DerivadaSegundaDA\]) and (\[DerivadasDAZ0\]) $$D_A^{(2)}=-\frac{2(a_1^2+a_1\chi_1 + \chi_1^2)-\chi_2-a_2}{(a_1+\chi_1)^2},$$ resulting for the effective deceleration parameter the expression $$\label{q_eff}
q_{{\rm eff}}=\frac{q_0 + 2\chi_1 - 4\chi_0\chi_1 - \chi_0^2\chi_1^2 -\chi_0^2\chi_2}{(1+\chi_0\chi_1)^2}.$$ Note that $q_{{\rm eff}}$ reduces to the usual FLRW value $q_0$ when we neglect the inhomogeneous terms $\chi_1$ and $\chi_2$, as expected.
In the next section we attempt to give a first approximation of the expressions found above relating the inhomogeneities to the cosmological parameters in order to verify whether they can fit observations.\
Accelerated expansion from inhomogeneities {#IV}
==========================================
We now provide a first analysis showing how our model can fit the data regarding the observations of the accelerated expansion of the universe. To that end, we consider the physical interpretation of the parameters composing $q_{{\rm eff}}$, where we let $(t_0,x_0,y_0,z_0)$ represent the event of observation at $z=0$ and set $f(t_0,x_0,y_0,z_0)=0$, $\rho_0:=\rho(t_0,x_0,y_0,z_0)$ and $\epsilon_0=\epsilon(a_0)=\rho_0$. This implies that $$\label{Xi0}
\chi_0 = 1 \quad \text{,} \quad E=a_0 \quad \text{and} \quad \epsilon_0=\rho_0=3H_0^2 \, .$$ We now use $\gamma_0=\gamma(a_0)$ and the relation $\epsilon = 3 H^2$ in the equation (\[eq\_rho\_a\]) to determine $q_0$ as $$q_0 = \frac{3}{2}\, \gamma_0 -1 + \frac{\kappa}{3\,a_0^2\, H_0^2} \, .$$ From Eqs. (\[first\_line\_el\]) and (\[first\_int\_geo1\]) we have for the norm of the wave vector $\|\vec k\| = \sqrt{\vec k\cdot\vec k}$ $$\|\vec k\| = \frac{1}{a_0} \left(\frac{a_0}{a}\right)^2\, \sqrt{\frac{\rho}{\epsilon}} \, .$$ As we use the definitions of $\chi$ and $\chi_1$, we obtain $$\label{Xi0}
\chi_1 = \left( \frac{1}{H \, \chi} \, \frac {d\chi}{d\lambda}\right)_{z=0} = -\left(\frac{1}{2 H \,\rho} \, \vec k \cdot \vec{\nabla}\rho \right)_{z=0} .$$ Therefore, denoting the angle between $\vec{\nabla}\rho$ and $\vec k$ at the event of observation by $\psi_0$, $$\label{Formula_X1}
\chi_1 = -\frac{1}{6\, a_0 \, H_0^3} \, \|\vec{\nabla}\rho\|_0 \, \cos \psi_0 ,$$ where $\|\vec{\nabla}\rho\|_0=\|\vec{\nabla}\rho(t_0,x_0,y_0,z_0)\|$. The definition of $\chi_2$, in its turn, is $$\label{definicao_X2}
\chi_2 = \left( \frac{1}{H^2 \, \chi} \, \frac {d^2\chi}{d\lambda^2}\right)_{z=0} =
\frac{1}{H_0^2} \left(
\frac{d\vec k}{d \lambda}\cdot \vec\nabla \chi + k^i k^j \,\partial_i\partial_j \chi
\right)_{z=0} .$$ If we use the equations (\[geo2\]), (\[first\_int\_geo1\]) and (\[Formula\_X1\]), we obtain $$\label{CalculoX2_1}
\frac{1}{H_0^2} \left(
\frac{d\vec k}{d \lambda}\cdot \vec\nabla \chi
\right)_{z=0} = \frac{\|\vec{\nabla}\rho\|_0}{3\, a_0 \, H_0^3} \, \cos \psi_0 - \left(\frac{\|\vec{\nabla}\rho\|_0}{6\, a_0 \, H_0^3} \, \right)^2 .$$ As $\Delta \chi$ is the trace of $\partial_i\partial_j \chi$, we define $B_{ij}$ to be its traceless part at the event of observation $(\delta^{ij}B_{ij} =0)$. Therefore, from equation (\[laplacian\]), we get $$\label{CalculoX2_2}
\partial_i\partial_j \chi (t_0,x_0,y_0,z_0) = - \frac{\kappa}{3} \delta_{ij} + B_{ij} .$$ Since the last term becomes diagonal by an Euclidean rotation, we can take it as the diagonal matrix $(B_{ij})= B \, \textbf{diag} \{\sin b, \sin (b + 2 \pi/3), \sin (b + 4 \pi/3) \}$, where $B$ measures the intensity of the anisotropy in the Hessian matrix of $\chi$ at the observation point and $b$ its angular distribution[^2]. Defining the polar and azimuthal angles in the usual way, with $k^1=\|k \| \cos \varphi$, $k^2=\|k \| \cos \theta \sin \varphi$ and $k^3=\|k \| \sin \theta \sin \varphi$, we substitute the equations (\[CalculoX2\_1\]) and (\[CalculoX2\_2\]) in (\[definicao\_X2\]) to get $$\chi_2 = - \, \frac{\kappa}{3\,a_0^2\, H_0^2} + \frac{\|\vec{\nabla}\rho\|_0}{3\, a_0 \, H_0^3}\, \cos \psi_0 - \left(\frac{\|\vec{\nabla}\rho\|_0}{6\, a_0 \, H_0^3} \, \right)^2 + \frac{B}{a_0^2\, H_0^2} \, A(\varphi, \theta,b) ,$$ where $$A(\varphi, \theta,b)= \left( 1 - \frac{3}{2}\sin^2 \varphi \right)\, \sin b - \frac{\sqrt{3}}{2}\, \cos(2\theta)\, \sin^2 \varphi\, \cos b .$$ Returning to the effective deceleration parameter in the equation (\[q\_eff\]), we get $$\label{q_eff_qs}
q_{{\rm eff}}=\frac{1}{\left(1-\frac{\|\vec{\nabla}\rho\|_0\,\cos\psi_0}{6\, a_0 \, H_0^3}\right)^2}\left[\frac{3}{2}\gamma_0 -1 + \frac{2\kappa}{3\,a_0^2\, H_0^2} + \left(\frac{\|\vec{\nabla}\rho\|_0\,\sin \psi_0}{6\, a_0 \, H_0^3} \, \right)^2 \, - \frac{B}{a_0^2\, H_0^2} \, A(\varphi,\theta,b)\right].$$
Let us assume that the observation point lies in what could be considered a local maximum of matter concentration, such that $\|\vec{\nabla}\rho\|_0$ vanishes. In addition, we should have $\nabla^2\rho(t_0, x_0, y_0, z_0)<0$, which implies the relation $\nabla^2\chi(t_0, x_0, y_0, z_0)>0$. Besides, we can assume that locally the anisotropic stresses vanish as the universe seems highly isotropic, which allows us to set the quantity $B$ to zero at the observation point. Therefore, the effective deceleration parameter reads $$q_{{\rm eff}}=\frac{3}{2}\gamma_0 -1 + \frac{2\kappa}{3\,a_0^2\, H_0^2}\, ,$$ where $\kappa$ should be negative, according to Eq. (\[laplacian\]).
In order to compare the role of our inhomogeneous model with the standard flat $\Lambda$CDM, we first set the dimensionless inhomogeneous constant $\Omega_I$ to be $$\Omega_I = -\, \frac{2 \, \kappa}{3\, H_0^2 \, a_0^2} \, .$$ It becomes clear from the equation (\[eq\_rho\_a\]) that the ratio $\kappa/a_0^2$ is invariant under scale changes in $a(t)$, and therefore it is invariant just like $H_0$. Under the suitable choice $\gamma=\gamma(a)$ in equation (\[eq\_rho\_a\]), we can write the homogeneous part of the energy density as a sum of contributions of matter ($\Omega_M$), dark energy $(\Omega_\Lambda)$ and an effective contribution from the inhomogeneity $(\Omega_I)$, with $\Omega_M + \Omega_I + \Omega_\Lambda = 1$. The effective deceleration parameter can then be rewritten as $$\label{Eq:qeff_Omegas}
q_{\rm eff} = \frac{1}{2} \left(\, \Omega_M - \Omega_I - 2 \Omega_\Lambda \,\right) \, .$$ Clearly, $\Omega_I=0$ suffices to recover the equivalent $\Lambda$CDM formula. If we assume $ q_{\rm eff} =-0.598$ [@lu11] for instance, we would have the constraint $$\label{Eq:Valor_Omega_Lambda}
\Omega_\Lambda = 0.732 - \frac{2}{3} \Omega_I \, .$$ Furthermore, this component might also diminish the contribution of the cold dark matter in $\Omega_M$, for we would have $$\label{Eq:Valor_Omega_Lambda}
\Omega_M = 0.268 - \frac{1}{3} \Omega_I \, .$$ If we add to this equation the condition $\Omega_M >0$, we get the inequality $$\label{Eq:Valor_Omega_I}
0 \le \Omega_I < 0.804 \, .$$
Concluding Remarks {#V}
==================
An exact solution of Einstein equations was derived which presents maximally symmetric submanifolds corresponding to flat tri-dimensional spaces of constant time. The matter content required for this is a viscous fluid where the dissipative terms are given through physically reasonable equations of state. The set of equations can be split into time and space, where the time evolution is driven by Friedmann equation with an effective curvature term ($\sim a^{-2}$) while the spatial dependence is given by a sort of a nonlinear Klein-Gordon equation. In comparison to the FLRW models, the expressions for the redshift and the luminosity distance are more complicate due to the presence of inhomogeneties. Thus, for small redshifts, we solve the Sachs equation perturbatively up to second order admitting a Taylor expansion of the solution. In this way, we were able to compute the angular diameter and luminosity distances and, consequently, we could find an equation for the cosmological deceleration parameter in which is clear how inhomogeneities could contribute to a late accelerated expansion.
Here we have shown, above all other possible conclusions, that models with nonlinear effects of inhomogeneity (and possibly anisotropy) beyond small perturbations of FLRW must be pushed forward and tested against the standard model. Just after such a tenacious scrutiny we could conclude that the Universe is of a $\Lambda$CDM type, or composed just by ordinary matter in an inhomogeneous and anisotropic way or somewhere in between.
Newman-Penrose (NP) invariants
==============================
In the $(a,x,y,z)$ coordinate system, we define a null tetrad basis $(l^\mu,n^\nu,m^\mu,\bar m^{\mu})$ in which the vectors are given by $$l^{\mu}=\left(a\sqrt{\frac{\rho}{3}},0,\frac{1}{a},0\right),\quad n^{\mu}=\left(\frac{a}{2}\sqrt{\frac{\rho}{3}},0,-\frac{1}{2a},0\right)\quad \mbox{and}\quad
m^{\mu}=\left(0,\frac{i}{\sqrt{2}a},0,\frac{1}{\sqrt{2}a}\right)$$ and $\bar m^{\mu}$ is the complex conjugate of $m^\mu$. This basis satisfies the relations $l^{\mu}m_{\mu}=0$, $n^{\mu}m_{\mu}=0$, $l^{\mu}n_{\mu}=-1$ and $m^{\mu}\bar m_{\mu}=1$. In possession of this, a direct calculation provides the NP scalars associated to the Weyl tensor as
$$\begin{array}{lcl}
\Psi_0&=&\frac{\left(\partial_z + i\partial_x\right)^2\chi}{2a^2\chi},\\[2ex]
\Psi_1&=&-\frac{\sqrt{2}\,\partial_y(\partial_z + i \partial_x)\chi}{4a^2\chi},\\[2ex]
\Psi_2&=&\frac{\nabla^2\chi}{6a^2\chi}-\frac{(\partial_z +i\partial_x)(\partial_z -i\partial_x)\chi}{4a^2\chi},\\[2ex]
\Psi_3&=&\frac{\sqrt{2}\,\partial_y(\partial_z - i \partial_x)\chi}{8a^2\chi},\\[2ex]
\Psi_4&=&\frac{\left(\partial_z - i\partial_x\right)^2\chi}{8a^2\chi}.
\end{array}$$
Following the scheme presented in [@acevedo] we are led to conclude that this geometry is algebraically general, that is, a Petrov type I spacetime. For completeness, the other invariants related to the Ricci tensor are found to be
$$\begin{array}{lcl}
\Phi_{00}&=&\frac{(\partial_z +i\partial_x)(\partial_z -i\partial_x)\chi}{2a^2\chi}-\frac{1}{\chi^2}\left(\frac{\ddot a}{a}-\frac{2\dot a\partial_y\chi}{a^2}-\frac{\dot a^2}{a^2}\right),\\[2ex]
\Phi_{01}&=&\frac{\sqrt{2}\,(2\dot a- \chi\partial_y)(\partial_z + i \partial_x)\chi}{4a^2\chi^2},\\[2ex]
\Phi_{02}&=&-\frac{\left(\partial_z + i\partial_x\right)^2\chi}{4a^2\chi}=-2\Psi_0,\\[2ex]
\Phi_{11}&=&\frac{\partial_{yy}\chi}{4a^2\chi}+\frac{1}{4\chi^2}\left(\frac{\dot a^2}{a^2} - \frac{\ddot a}{a}\right),\\[2ex]
\Phi_{12}&=&\frac{\sqrt{2}\,(2\dot a + \chi\partial_y)(\partial_z + i \partial_x)\chi}{8a^2\chi^2},\\[2ex]
\Phi_{22}&=&\frac{(\partial_z +i\partial_x)(\partial_z -i\partial_x)\chi}{8a^2\chi}-\frac{1}{4\chi^2}\left(\frac{\ddot a}{a}+\frac{2\dot a\partial_y\chi}{a^2}-\frac{\dot a^2}{a^2}\right),\\[2ex]
R&=&\frac{6}{\chi^2}\left(\frac{\ddot a}{a} +\frac{\dot a^2}{a^2}\right) -\frac{2\nabla^2\chi}{a^2\chi}.
\end{array}$$
G. B. Santos is supported by CAPES (Brazil) through the grant 88882.317979/2019-1.
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[^1]: It must be emphasized that the above condition is a consequence of the shear-free and flat spatial geometry, no other relation being allowed.
[^2]: This kind of treatment is common in Bianchi I spacetimes. See [@bitt17], for example.
|
---
abstract: 'We show how the existence of a PBW-basis and a large enough central subalgebra can be used to deduce that an algebra is Frobenius. This is done by considering the examples of rational Cherednik algebras, Hecke algebras, quantised universal enveloping algebras, quantum Borels and quantised function algebras. In particular, we give a positive answer to [@Rouquier Problem 6] stating that the restricted rational Cherednik algebra at the value $t=0$ is symmetric.'
address:
- 'Brown: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK'
- 'Gordon: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK'
- 'Stroppel: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK'
author:
- 'K.A. Brown, I.G. Gordon, C.H. Stroppel'
bibliography:
- 'ref7.bib'
title: 'Cherednik, Hecke and quantum algebras as free Frobenius and Calabi-Yau extensions'
---
Introduction
============
{#1.1}
In this note we will consider six types of algebras:
(I) the rational Cherednik algebra ${\operatorname}{H}_{0,{\bf c}}$ associated to the complex reflection group $W$;
(II) the graded (or degenerate) Hecke algebra $\mathbf{H}_{gr}$ associated to a complex reflection group $W$;
(III) the extended affine Hecke algebra ${{\mathcal H}}$ associated to a finite Weyl group $W$;
(IV) the quantised enveloping algebra ${{\mathcal U}}_\epsilon({\mathfrak{g}})$, at an $\ell$-th root of unity $\epsilon$, of a semisimple complex Lie algebra ${\mathfrak{g}}$;
(V) the corresponding quantum Borel ${{\mathcal U}}_\epsilon({\mathfrak{g}})^{\geq0}$;
(VI) the corresponding quantised function algebra ${{\mathcal O}}_\epsilon[G]$.
These algebras share two important properties: first, they have a regular central subalgebra ${{\mathcal{Z}}}$ over which they are free of finite rank, second, they - or a closely associated algebra in Case (VI) - have a basis of PBW type. The purpose of this paper is to show that these two properties are the key tools for defining an associative non-degenerate ${{\mathcal{Z}}}$-bilinear form for each of these algebras, and hence for deducing Frobenius and Calabi-Yau properties for the algebras in each class.
{#1.2}
We prove that each pair ${{\mathcal{Z}}}\subseteq R$ in the classes (I)-(VI) is a *free Frobenius extension*. The definition and basic properties are recalled in Section \[frob\] and Section \[Nak\] – in essence, one requires ${\operatorname{Hom}}_{{{\mathcal{Z}}}}(R,
{{\mathcal{Z}}}) \cong R$ as $({{\mathcal{Z}}}-R)$-bimodule.
{#1.3}
When an algebra $R$ is a free Frobenius extension of a central subalgebra ${{\mathcal{Z}}}$ then ${\operatorname{Hom}}_{{{\mathcal{Z}}}}(R,{{\mathcal{Z}}})$ is in fact isomorphic to $R$ *both* as a left *and* as a right $R$-module, but not necessarily as a bimodule. However, there is a ${{\mathcal{Z}}}$-algebra automorphism $\nu$ of $R$, the *Nakayama automorphism*, such that ${\operatorname{Hom}}_{{{\mathcal{Z}}}}(R,{{\mathcal{Z}}}) \cong \,{}^1
R^{\nu^{-1}}_{}$ as $R$-bimodules. This automorphism is unique up to an inner automorphism. We explicitly determine the Nakayama automorphisms for each case listed above: $\nu$ is trivial ([[*i.e.* ]{}]{}inner) in cases (I) and (IV); non-trivial in cases (II), (III) and (V) and (VI).
{#1.4}
The results summarised in Section \[1.2\] have immediate consequences regarding the *Calabi-Yau property* of the algebras in classes (I) - (VI). The definition and its relevance to Serre duality are recalled in Section \[2.3\]. In particular [@CGM], we get natural examples of so-called Frobenius functors - that is, functors which have a biadjoint. Frobenius algebras and Frobenius extensions play an important role in many different areas (see for example [@kadison]). They give rise to Frobenius functors which are the natural candidates for constructing interesting topological quantum field theories in dimension 2 and even 3 (for the latter see for example [@StTQFT]), and also provide connections between representation theory and knot theory (for example in the spirit of [@KadisonJones]).
{#1.5}
Let us assume for the moment that ${{\mathcal{Z}}}\subseteq R$ is a free Frobenius extension with Nakayama automorphism $\nu$. If $I$ is an ideal of ${{\mathcal{Z}}}$, then it’s clear from the definitions that ${{\mathcal{Z}}}/I \subseteq R/IR$ is a free Frobenius extension with Nakayama automorphism induced by $\nu$. This applies in particular when $I$ is a maximal ideal $\mathfrak{m}$ of ${{\mathcal{Z}}}$; since, for $R$ in classes (I) - (VI), every simple $R$-module is killed by such an ideal $\mathfrak{m}$, this is relevant to the finite dimensional representation theory of $R$. Thus $R/\mathfrak{m}R$ is a Frobenius algebra, which is symmetric provided the automorphism of $R/\mathfrak{m}R$ induced by $\nu$ is inner.
{#1.6}
To define the non-degenerate associative bilinear forms mentioned in Section \[1.1\], we follow in each case the approach of [@FP Proposition 1.2] to the study of the inclusion ${{\mathcal{Z}}}\subseteq R$ when $R$ is the enveloping algebra $U(\mathfrak{g})$ of a finite dimensional restricted Lie algebra $\mathfrak{g}$ over a field $k$ of characteristic $p > 0,$ and ${{\mathcal{Z}}}$ is the Hopf centre $k \langle x^p - x^{[p]} : x \in \mathfrak{g}
\rangle.$ In the language of the present paper, it is proved there that ${{\mathcal{Z}}}\subseteq U(\mathfrak{g})$ is a free Frobenius extension, with Nakayama automorphism $\nu$ the winding automorphism of the trace of the adjoint representation; in particular, $\nu$ is trivial when $U(\mathfrak{g})$ is semisimple. The parallel methods used here might suggest that an axiomatic approach covering all the cited cases simultaneously might be possible; but we have not found such a setting.
{#1.7}
The detailed results for classes (I) - (VI) are as follows.
1. (Theorem 3.5 and Corollary 3.6) The rational Cherednik algebra ${\bf H} = {\operatorname}{H}_{0,c}$ is a free Frobenius extension of its central subalgebra ${{\mathcal{Z}}}:=S(V)^W \otimes S(V^*)^W,$ with trivial Nakayama automorphism. Consequently ${\bf H}_{\chi}$ is a symmetric algebra for any central character $\chi$ (answering a question of Rouquier, [@Rouquier Problem 6]), and ${\bf H}$ is a Calabi–Yau $\mathcal{Z}-$algebra.
2. (Theorem \[gradedHecke\]) The graded Hecke algebra $\mathbf{H}_{gr}$ associated to a complex reflection group $W$ is a free Frobenius extension of its centre ${{\mathcal{Z}}}_{gr}:=S(V)^W$, but the Nakayama automorphism (which is determined explicitly) is non-trivial.
3. (Theorem \[affineHecke\]) The extended affine Hecke algebra ${{\mathcal H}}$ associated to a finite Weyl group $W$ is a free Frobenius extension of its centre ${{\mathcal{Z}}}_{{{\mathcal H}}}$, but the Nakayama automorphism is non-trivial.
4. (Theorem \[symmetricU\]) The quantised enveloping algebra ${{\mathcal U}}_\epsilon({\mathfrak{g}})$ is a free Frobenius extension of its $\ell$-centre ${{\mathcal{Z}}}$, with trivial Nakayama automorphism. Consequently, ${{\mathcal U}}_\epsilon({\mathfrak{g}})_\chi$ is symmetric for any central character $\chi$, and ${{\mathcal U}}_\epsilon({\mathfrak{g}})$ is a Calabi-Yau ${{\mathcal{Z}}}-$algebra.
5. (Theorem \[4.1\]) The quantum Borel ${{\mathcal U}}_\epsilon({\mathfrak{g}})^{\geq 0}$ is a free Frobenius extension of its $\ell$-centre $\mathcal{Z}_+$, but the Nakayama automorphism (which is determined explicitly) is non-trivial.
6. (Theorem \[Frobenius\]) There is an element $z$ of the central subalgebra ${{\mathcal O}}[G]$ of the quantised function algebra ${{\mathcal O}}_{\epsilon}[G]$ such that ${{\mathcal O}}_{\epsilon}[G][z^{-1}]$ is a free Frobenius extension of ${{\mathcal O}}[G][z^{-1}]$ with non-trivial Nakayama automorphism. The open set $\mathcal{O}_z = \{ g\in G: z\notin \mathfrak{m}_g\}$ meets every torus orbit of symplectic leaves in $G$. Thus, for any $g\in G$, the algebra ${{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g {{\mathcal O}}_\epsilon[G]$ is Frobenius but not, in general, symmetric.
{#1.8}
There is some overlap between this paper and [@Braun], a preliminary version of which we received while this paper was being written. The methods used in the two papers are completely different, and indeed complementary.
{#1.9}
In the following rings are always assumed to be unitary and, if not stated otherwise, modules are [*left*]{} modules. For any ring $S$ we denote by ${\operatorname{Hom}}_S(-,-)$, ${\operatorname{Hom}}_{-S}(-,-)$ and ${\operatorname{Hom}}_{S-S}(-,-)$ the morphism spaces in the category of (left) $S$-modules, right $S$-modules and $S$-bimodules respectively. Our algebras are all over ${\mathbb{C}}$; undoubtedly this hypothesis could be weakened. We abbreviate $\otimes=\otimes_{\mathbb{C}}$.
Frobenius and Calabi-Yau extensions {#Frobeniusext}
===================================
Definition {#frob}
----------
We first recall some basics on Frobenius extensions. For more details we refer for example to [@BF], [@K], [@NT], [@Pareigis]. A ring $R$ is a [*free Frobenius extension*]{} (of the first kind) over a subring $S$, if $R$ is a free $S$-module of finite rank, and there is an isomorphism of $R-S$-bimodules $F:R\longrightarrow {\operatorname{Hom}}_{S}(R,S)$. (The bimodule structure on the latter is defined as $r.f.s(x)=f(xr)s$ for $r,
x\in R$, $s\in S$, $f\in {\operatorname{Hom}}_{S}(R,S)$.) Equivalently, $R$ is a free right $S$-module of finite rank, and there is an isomorphism of $S-R$-bimodules $G:R\longrightarrow {\operatorname{Hom}}_{-S}(R,S)$ ([@NT Proposition 1]). The existence of $F$ provides a non-degenerate associative $S$-bilinear form ${{\mathbb B}}:R\times R\rightarrow S$, defined by ${{\mathbb B}}(r,t)=F(t)(r)$ for all $r,t \in R$. Given a basis $r_i$, $1\leq i\leq n$ of $R$ as an $S$-module, we find elements $r^i$, $1\leq i\leq n$ such that ${{\mathbb B}}(r_i, r^j)=\delta_{i,j}$ because $F$ is surjective. The two ordered sets $\{r_i: 1\leq i\leq n\}$ and $\{r^i: 1\leq i\leq n\}$ form a dual free pair (in the sense of [@BF Section 1]). Conversely, the existence of a non-degenerate associative bilinear form ${{\mathbb B}}:R\times R\rightarrow S$ together with a dual free pair implies that $R$ is a free Frobenius extension of $S$ with defining isomorphism $F$ given by $F(t)(r)={{\mathbb B}}(r,t)$ (see [@BF Section 1]).
The Nakayama automorphism {#Nak}
-------------------------
We recall some ideas from [@K]. Suppose for the rest of this section that $R$ is a free Frobenius extension of $
\mathcal{Z}$, with $ \mathcal{Z}$ now contained in the centre of $R$. The isomorphisms $F$ and $G$ defined in 2.1 induce isomorphisms of left respectively right $R$-modules $$\begin{aligned}
\label{eq:FG}
\begin{array}[thl]{ccccccc}
R&\cong&{\operatorname{Hom}}_{ \mathcal{Z}}(R, \mathcal{Z})&=&{\operatorname{Hom}}_{ \mathcal{Z}- \mathcal{Z}}(R, \mathcal{Z})&=&RF(1)\\
R&\cong&{\operatorname{Hom}}_{- \mathcal{Z}}(R, \mathcal{Z})&=&{\operatorname{Hom}}_{ \mathcal{Z}- \mathcal{Z}}(R, \mathcal{Z})&=&G(1)R.
\end{array}\end{aligned}$$ One can show [@K Section 2 (4)] that $h:=F(1)=G(1)$ as elements of ${\operatorname{Hom}}_{ \mathcal{Z}- \mathcal{Z}}(R,
\mathcal{Z})$. Thus we get a well-defined $\mathcal{Z}$-algebra automorphism $\nu:R \longrightarrow R$, defined by $rh = h\nu (r)$ for all $r\in R$. An easy calculation shows that $$\begin{aligned}
{{\mathbb B}}(x,y)&=&{{\mathbb B}}(\nu(y), x)\end{aligned}$$ for $x$, $y\in{{\mathcal B}}$. The automorphism $\nu$ is called the [*Nakayama automorphism*]{} (with respect to $F$, ${{\mathbb B}}$, or $G$). It’s clear that $\nu$ is uniquely determined up to an inner automorphism of $R$ by the pair $\mathcal{Z} \subseteq R$. It therefore makes sense to speak about the Nakayama automorphism attached to a free Frobenius extension. We call the extension [*symmetric*]{} if the Nakayama automorphism is inner.
Thanks to our assumption on $\mathcal{Z}$, there is now also a *right* $R$-action on ${\operatorname{Hom}}_{\mathcal{Z}}(R,\mathcal{Z})$, given by $fr(-) = f(r-)$ for $r \in R$ and $f \in {\operatorname{Hom}}_{\mathcal{Z}}(R,\mathcal{Z})$. Let $_{}^{1}R_{}^{\nu^{-1}}$ be the ring $R$ considered as an $R$-bimodule, but with its right $R$-module structure twisted by $\nu^{-1}$. Then the $R-\mathcal{Z}$-bimodule isomorphism $F$ is in fact an isomorphism of $R$-bimodules $$\begin{aligned}
\label{eq:CY}
_{}^{1}R_{}^{\nu^{-1}}\cong{\operatorname{Hom}}_{\mathcal{Z}}(R,\mathcal{Z}),
\end{aligned}$$ since $F(r\nu^{-1}(x))(y)=F(\nu^{-1}(x))(yr)={{\mathbb B}}(yr,\nu^{-1}(x))={{\mathbb B}}(x,yr)$ and $(F(r)x)(y)=F(r)(xy)=F(yr)(x)={{\mathbb B}}(x,yr)$ for all $x,y,r \in R$.
{#hyp}
We now highlight a condition which will allow us to prove that algebras are free Frobenius extensions. For this we let $R$ be free with a finite basis $\mathcal{B}$ over an affine central subalgebra $\mathcal{Z}$. The condition is:
[**Hypothesis:**]{} There exists a $\mathcal{Z}$–linear functional $\Phi : R\rightarrow \mathcal{Z}$ such that for any non-zero $a = \sum_{b \in \mathcal{B}} z_b b\in R$ there exists $x\in R$ with $\Phi(xa) = uz_b$ for some unit $u\in \mathcal{Z}$ and some non-zero $z_b\in \mathcal{Z}$.
\[cruc\] Let $R$ be a finitely generated free $\mathcal{Z}$–module with a basis $\mathcal{B}$ which satisfies the above hypothesis. Then $R$ is a free Frobenius extension of $\mathcal{Z}$ and for any maximal ideal $\mathfrak{m}$ of $\mathcal{Z}$, the finite dimensional quotient $R/\mathfrak{m}R$ is a finite dimensional Frobenius algebra.
Let $\theta : R \rightarrow {\operatorname{Hom}}_\mathcal{Z}(R,\mathcal{Z})$ be the $R-\mathcal{Z}$-bimodule homomorphism defined by $\theta(a)(a') = \Phi (a'a)$. Clearly $\theta$ is an injection since if $a\in R$ is non-zero then the displayed hypothesis implies that $\theta(a)(x) \neq 0$. Thus we have a short exact sequence $$\label{frobsequence} 0\rightarrow R \rightarrow {\operatorname{Hom}}_\mathcal{Z}(R,\mathcal{Z}) \rightarrow C
\rightarrow 0$$ of $R-\mathcal{Z}$-bimodules, where $C$ is the cokernel of $\theta$. We will prove that $C=0$ after showing that $\theta$ induces a Frobenius structure on each finite dimensional quotient $R/\mathfrak{m}R$.
Fix an arbitrary maximal ideal $\mathfrak{m}$ of $\mathcal{Z}$ and consider the mapping $$\overline{\theta} : \frac{R}{\mathfrak{m}R} \longrightarrow {\operatorname{Hom}}_\mathcal{Z}(R,\mathcal{Z}) \otimes_\mathcal{Z} \frac{\mathcal{Z}}{\mathfrak{m}}$$ which sends $a+\mathfrak{m}R$ to $\theta(a) \otimes 1$. Let $$\iota : {\operatorname{Hom}}_\mathcal{Z}(R,\mathcal{Z}) \otimes_\mathcal{Z} \frac{\mathcal{Z}}{\mathfrak{m}} \longrightarrow {\operatorname{Hom}}_{{\mathbb{C}}}( \frac{R}{\mathfrak{m}R}, {\mathbb{C}})$$ be the isomorphism sending $\psi\otimes 1$ to the mapping $(a + \mathfrak{m}R \mapsto \psi(a) + \mathfrak{m})$.
We claim that composition $\iota \overline{\theta}$ is an isomorphism. To prove this, we will show that $\iota \overline{\theta}$ is injective; then, since both the domain and codomain are vector spaces of the same dimension, the claim will follow. By construction, $$\iota\overline{\theta} ( a+ \mathfrak{m}R ) (a' + \mathfrak{m}R) = \Phi (a' a) + \mathfrak{m}.$$ Therefore, if $a+\mathfrak{m}R \in \ker \iota\overline{\theta}$ then $\Phi (a'a) \in \mathfrak{m}$ for all $a'\in R$. We assume that $a\neq 0$. Then, by hypothesis, if we write $a = \sum z_b b$, we can find $x\in R$ such that $\Phi (xa) = u z_b$ for some unit $u$ and some non-zero $z_b$. Thus $z_b \in \mathfrak{m}$. Now $a$ and $a-z_bb$ have the same image in $R/\mathfrak{m}R$ so we can replace $a$ by $a-z_bb$. Repeating this procedure shows that $a \in \mathfrak{m}R$ and hence that $\iota\overline{\theta}$ is injective.
As a first consequence we see that $\iota\overline{\theta}$ induces an $R/\mathfrak{m}R$-isomorphism $R/\mathfrak{m}R \cong (R/\mathfrak{m}R)^*$ so $R/\mathfrak{m}R$ is Frobenius. We also deduce that $\overline{\theta}$ is an isomorphism, and so from we see $C\otimes_\mathcal{Z} \mathcal{Z}/\mathfrak{m}\mathcal{Z} = 0$. Since this is true for an arbitrary maximal $\mathfrak{m}$ of $\mathcal{Z}$ and $C$ is finitely generated over $\mathcal{Z}$, it follows that $C = 0$. Hence $\theta : R \longrightarrow {\operatorname{Hom}}_\mathcal{Z}(R,\mathcal{Z})$ is an isomorphism and so $R$ is a free Frobenius extension of $\mathcal{Z}$.
Calabi-Yau algebras {#2.3}
-------------------
Let $d$ and $n$ be non-negative integers and let $R$ be a ring which has a commutative noetherian central subring $C$ of Krull dimension $d$, over which $R$ is a finitely generated module. Following for example [@IR], we say that $R$ is a *Calabi-Yau $C$-algebra of dimension $n$* if, for all $X,Y \in \mathcal{D}^b
(\mathrm{Mod} (\mathrm{fl-}R))$, the bounded derived category of $R-$modules of finite length, there is a natural isomorphism $$\begin{aligned}
{\operatorname{Hom}}_{\mathcal{D}(\mathrm{Mod} (R))}(X,Y[n]) \cong D{\operatorname{Hom}}_{\mathcal{D}(\mathrm{Mod} (R))}(Y,X). \end{aligned}$$ Here, $D$ denotes the *Matlis duality* functor $D = {\operatorname{Hom}}_C (-,E),$ where $E$ is the direct sum of the $C-$injective hulls of the simple $C-$modules. The following proposition is an immediate consequence of [@IR Theorems 3.1 and 3.2], once we note that if $C$ is regular then the Cohen-Macaulay $C-$modules coincide with the projective $C-$modules.
Let $C$, $R,$ $n$ and $d$ be as above, and suppose that $C$ is a regular domain. Then $R$ is a Calabi-Yau $C-$algebra of dimension $n$ if and only if $n=d,$ $R$ has finite global dimension, $R$ is a projective $C-$module, and ${\operatorname{Hom}}_C(R,C)$ is isomorphic to $R$ as $R-R-$bimodules. In this case, $R$ has global dimension $d.$
Hopf algebras
-------------
1\. When $H$ is a Hopf algebra which is a finite module over a central affine Hopf subalgebra $\mathcal{Z},$ Hopf-algebraic methods can be used to deduce that $H$ is a Frobenius extension of $\mathcal{Z}$. The result is due to Kreimer and Takeuchi [@KT Theorem 1.7]; the arguments are sketched in [@BGo Section III.4]. This provides an alternative approach to the algebras in classes (IV), (V) and (VI), but this does not provide an explicit description of the bilinear form, nor does it give immediate access to the Nakayama automorphism.
2\. The concept of the Nakayama automorphism was introduced also in a recent paper on noetherian Hopf algebras by Brown and Zhang [@BZ]. They showed that many noetherian Hopf algebras $H$ (including all those which are finite modules over their centres) have a rigid dualizing complex $R$ which is isomorphic (in the derived category of bounded complexes of $H-$bimodules) to ${}^{\hat{\nu}}H^1 [d]$; here, $d$ is the injective dimension of $H$, $[d]$ denotes the shift, and $\hat{\nu}$ is a certain algebra automorphism of $H$ which Brown and Zhang called the Nakayama automorphism. The automorphism $\hat{\nu}$ is trivial on the centre of $H$ and is uniquely determined by $H$, up to an inner automorphism.
When both usages of the term “Nakayama automorphism” are in play, they define the same map (bearing in mind that both definitions are only unique up to an inner automorphism of the algebra). To see this, suppose that $H$ is a free Frobenius extension of a smooth affine central subalgebra $\mathcal{Z}$, (as is the case for the algebras of (IV), (V) and (VI)). Then the injective dimension $d$ of $H$ equals the Krull dimension (of $H$ and of $\mathcal{Z}$). Thus the rigid dualizing complex of $\mathcal{Z}$ is $\mathcal{Z}[d]$, and, by [@Ye Proposition 5.9], [@YZ Example 3.11], $H$ has rigid dualizing complex $\mathrm{RHom}_{\mathcal{Z}}(H,\mathcal{Z}[d])$. From the free Frobenius property of $H$, and (2.2), we deduce that this latter complex is isomorphic to ${}^{\nu}H^1[d],$ where $\nu$ denotes the Nakayama automorphism of the present paper. By the uniqueness of the rigid dualizing complex of $H$ [@VdB Proposition 8.2], it follows that $\hat{\nu} = \nu$ up to an inner automorphism, as claimed.
The rational Cherednik algebra {#6}
==============================
In this section we show that the rational Cherednik algebra ${\bf H}$ is a Frobenius extension of its (what we call) bi-invariant centre, with trivial Nakayama automorphism, so that the reduced Cherednik algebras ${\bf H}_\chi$ are symmetric.
Rational Cherednik Algebras {#6.1}
---------------------------
Let $W$ denote an irreducible complex reflection group with identity element $e$ and set of complex reflections $S$. We fix $V$, a complex reflection representation of $W$, and set $n = \dim V$. Let $c$ be a conjugation invariant complex function on $S$. For $s\in
S$ let $\alpha_s$ (respectively $\check{\alpha}_s$) be a linear functional on $V$ (respectively $V^*$) which vanishes on the reflection hyperplane for $s$; we normalise these by the condition $\langle \alpha_s,\check{\alpha}_s\rangle=2$. [*The rational Cherednik algebra $\mathbf{H}=\mathbf{H}_{0,c}$*]{} is the ${\mathbb{C}}$-algebra generated by $\{w\in W, x\in V, y\in V^*\},$ with defining relations $$\begin{aligned}
&wxw^{-1}={}^{w}x,\quad wyw^{-1}={}^wy,&\label{def1}\\
&[x,x']=0,\quad [y,y']=0,&\label{def1.5}\\
&\left[x,y\right]=\sum_{s\in S}c(s)\langle
y, \alpha_s\rangle\langle\check{\alpha}_s,x\rangle s&,\label{def2}\end{aligned}$$ for $x, x'\in V$, $y, y'\in V^*$ and $w \in W.$ These are the algebras $\mathbf{H}_{0,c}$ from [@EG p.251].
The PBW-basis
-------------
The algebra $\mathbf{H}$ has a PBW-property in the following sense: multiplication induces an isomorphism $$S(V)\otimes_{\mathbb{C}}{\mathbb{C}}W\otimes_{\mathbb{C}}S(V^*)\tilde{\longrightarrow}\mathbf{H}$$ of vector spaces (see [@EG Theorem 1.3]). In particular, there is a PBW-basis given by the elements of the set ${{\mathcal B}}_{\mathbf{H}}=\{fwg\}$, where $w\in W,$ $f$ runs through a homogeneous basis of ${S(V)},$ and $g$ runs through a homogeneous basis of ${S(V^*)}$.
For $f$ in $S(V)$ or ${S(V^*)}$ we write $|f|$ for the degree of $f$. For $i \in {\mathbb{Z}}_{\geq 0}$ let ${{\mathcal B}}_{<i}$ be the span of all PBW-basis elements of the form $fxg$, where $f\in S(V)$, $x\in W$ and $g\in{S(V^*)},$ such that $f$ and $g$ are homogeneous with $|f| + |g| < i$: this induces a filtration of $\mathbf{H}$. Moreover, the commutation relation shows that $$\label{commforus} \text{$|[f,g]| \leq |f| + |g| - 2$ for all homogeneous $f\in S(V)$ and $g\in {S(V^*)}$.}$$
The central subalgebra {#6.2}
----------------------
The algebra $\mathbf{H}=\mathbf{H}_{0,c}$ has a large centre $Z({\bf H})$, isomorphic to the so-called spherical subalgebra ([@EG Theorem 3.1, Theorem 7.2]). In particular, $Z({\bf H})$ contains the [*bi-invariant centre*]{} $${{\mathcal{Z}}}\quad :\quad {S(V)}^W\otimes{S(V^*)}^W.$$ Now ${S(V)}$ (respectively ${S(V^*)}$) is a free ${S(V)}^W$-module (respectively ${S(V^*)}^W$-module) of rank $|W|$, see [@Kane V.18.3] for example. A basis can be obtained by taking arbitrary homogeneous preimages of any homogeneous basis of the coinvariant algebra $A:={S(V)}/({S(V)}^W_+)$.
Then $A$ is a local Frobenius algebra thanks to [@Kane Proposition VII.26.7] and its associated bilinear form is easy to describe. To do this, recall that the homogeneous component $A_N$ of $A$ of highest degree has dimension one and is skew invariant for the action of $W$ on $V$, [@Kane 20.3, Propositions A and B]. Let $\pi : A \rightarrow A_N$ be the projection map with $\pi (A_i) = 0$ for $i \neq N.$ Then the bilinear form is given by $$B(\overline{a}, \overline{a'}) = \pi(\overline{a'}\overline{a}).$$ Similar statements apply to ${S(V^*)}/({S(V^*)}^W_+)$: it is Frobenius and its highest degree component is skew invariant for the action of $W$ on $V^*$. Below, we shall use the notation $\epsilon_V, \epsilon_{V^*}$ for these two one-dimensional representations of $W$.
{#dualbases}
We fix a pair of homogeneous dual bases $\{\overline{{\bf a}}_i:1\leq i\leq
|W|\}$, $\{{\overline{\bf a}}^i:1\leq i\leq |W|\}$ for ${S(V)}/({S(V)}^W_+)$, and a pair of homogeneous dual bases $\{{\overline{\bf
b}}_i:1\leq i\leq |W|\}$, $\{{\overline{\bf b}}^i:1\leq i\leq |W|\}$ for ${S(V^*)}/({S(V^*)}^W_+)$. Then we lift them to homogeneous $S(V)^W$-bases, $\{{\bf
a}_i:1\leq i\leq |W|\}$, $\{{\bf
a}^i:1\leq i\leq |W|\}$ of ${S(V)}$, and homogeneous ${S(V^*)}^W$-bases $\{{\bf
b}_i:1\leq i\leq |W|\}$, $\{{\bf b}^i:1\leq i\leq |W|\}$ of ${S(V^*)}$. We set $|{\bf a}_i| d_i$ and $|{\bf b}_i|= e_i$; then $|{\bf a}^i| = N-d_i$ and $|{\bf
b}^i| = N - e_i$. Let ${\bf
a}_{max}$ and ${\bf b}_{max}$ be the elements of maximal degree $N$ amongst the ${\bf a}_i$ and ${\bf b}_i$ respectively.
The functional
--------------
For $f\in{S(V)}$ let ${\bf a}_{max}(f)$ be the coefficient of ${\bf
a}_{max}$ when $f$ is expressed in the chosen $S(V)^W$-basis of $S(V)$. Similarly, we define ${\bf b}_{max}(g)$ for $g\in{S(V^*)}$. Thanks to the PBW-property, $\mathbf{H}$ is a free ${{\mathcal{Z}}}$-module of finite rank with basis $${{\mathcal B}}_{\mathbf{H}} \quad := \quad \{{\bf
a}_{i}w{\bf b}_{j} : w\in W, 1\leq i,j\leq |W|\}.$$ We define a ${{\mathcal{Z}}}$-linear map $$\begin{aligned}
\Phi:\quad\quad\quad\mathbf{H}&\longrightarrow&{{\mathcal{Z}}}\\
{{\mathcal B}}_{\mathbf{H}}\ni {\bf a}_iw{\bf b}_j&\longmapsto&
\begin{cases}
1 \quad &\text{if $i=j=max$ and $w=e$},\\
0 \quad &\text{otherwise}.
\end{cases}\end{aligned}$$
\[6.3\]
\[main\] The functional $\Phi$ above satisfies Hypothesis \[hyp\].
Let $a = \sum_{b\in \mathcal{B}_{\mathbf{H}}} z_b b$ be a non-zero element of $\mathbf{H}$. Pick $b = {\bf a_i} w {\bf b_j} \in \mathcal{B}_{\mathbf{H}}$ of maximal degree $|{\bf a_i}|+|{\bf b_j}|$ such that $z_b \neq 0,$ and set $x = {\bf b}^jw^{-1} {\bf a}^i$. We claim that this choice of $x$ satisfies Hypothesis \[hyp\].
For indices $i', j'$ and for $u\in W$ we have, by and , $$\begin{aligned}
{\bf
a}_{i'} u{\bf b}_{j'}x={\bf a}_{i'} u{\bf b}_{j'} {\bf b}^jw^{-1}{\bf a}^i &=& {\bf a}_{i'}\cdot uw^{-1}\cdot
{}^w( {\bf b}_{j'}{\bf b}^j) {\bf a}^i \\ & = & {\bf a}_{i'} \, ({}^{uw^{-1}}{\bf a}^i) \cdot uw^{-1} \cdot
{}^{w}({\bf b}_{j'}{\bf b}^j) + \text{lower order terms}. \end{aligned}$$ Since $b$ was chosen to have maximal degree it follows that if ${\bf a}_{i'} u{\bf b}_{j'}$ appears in the expansion of $a$, then the lower order terms in the above expression have total degree less than $d_i + e_j + (N-d_i) + (N-e_i) = 2N$. Therefore we find that $$\begin{aligned}
\nonumber
\Phi ({\bf a}_{i'} u{\bf b}_{j'}x)\Phi ({\bf a}_{i'} u{\bf b}_{j'} {\bf
b}^jw^{-1}{\bf a}^i ) &=& \Phi ( {\bf a}_{i'} \, ({}^{uw^{-1}}{\bf a}^i) \cdot uw^{-1} \cdot {}^{w}({\bf
b}_{j'}{\bf b}^j))\\ \label{gut} & = & \delta_{u,w}\Phi ( {\bf a}_{i'} {\bf a}^i \cdot {}^{w}({\bf b}_{j'}{\bf
b}^j)).\end{aligned}$$ By definition of the dual basis we have, for $i,i',j,j' = 1, \ldots , N,$ $$\begin{aligned}
{\bf a}_{i'}{\bf a}^{i} = (\delta_{i,i'}+ r_{max}){\bf a}_{max} + \sum_{k\neq max} r_k {\bf a}_k \quad \mathrm{
and } \quad {\bf b}_{j'}{\bf b}^{j} = (\delta_{j,j'}+ r'_{max}){\bf b}_{max} + \sum_{k\neq max} r'_k {\bf
b}_k\end{aligned}$$ for some $r_{max}, r_k \in S(V)^W$ and $r'_{max}, r'_k \in S(V^*)^W$. Consideration of polynomial degrees in the above expressions shows that $r_k \in (S(V)^W_+)$ for $k = max,$ and for all $k$ when $i
= i',$ and that $r'_k \in (S(V^*)^W_+)$ for $k = max,$ and for all $k$ when $j = j'.$ Substituting in (\[gut\]) we find that there exists $0\not=c\in{\mathbb{C}}$ such that $$\begin{aligned}
\label{key} & \Phi ( {\bf a}_{i'} {\bf a}^i \cdot {}^{w}({\bf b}_{j'}{\bf b}^j)) = \\ \nonumber
&c\Phi((\delta_{i,i'}+ r_{max}){\bf a}_{max} + \sum_{k\neq max} r_k {\bf a}_k)((\delta_{j,j'}+ r'_{max}){\bf
b}_{max} + \sum_{k\neq max} r''_k {\bf b}_k)),\end{aligned}$$ where $r''_k \in S(V^*)^W$ and $r''_k \in (S(V^*)_+^W)$ when $j = j'.$ We claim that (\[key\]) is 0 except when $(i',j') = (i,j).$ To see this, suppose that $(i',j')$ is not equal to $(i,j)$, but (\[key\]) is non-zero. Our choice of $b$ to have maximal degree with $z_b \neq 0$ forces $$\begin{aligned}
\label{count} d_{i'} + e_{j'} = d_i + e_j, \end{aligned}$$ since otherwise the degree of ${\bf a}_{i'} {\bf a}^i
\cdot {}^{w}({\bf b}_{j'}{\bf b}^j)$ is strictly less than $2N$, and hence can’t involve ${\bf a}_{max}{\bf
b}_{max}.$
Suppose first that $i' \neq i$ and $j' \neq j.$ Then (\[key\]) becomes $$\begin{aligned}
\label{dubya} \Phi ( {\bf a}_{i'} {\bf a}^i \cdot {}^{w}({\bf b}_{j'}{\bf b}^j)) = r_{max}r'_{max} \Phi({\bf
a}_{max}{\bf b}_{max}).\end{aligned}$$ But $r_{max}, r'_{max}$ are in the ideals of positive degree invariants, and so have strictly positive degrees if they are not 0. Thus, comparing degrees in (\[dubya\]), using (\[count\]), shows that (\[dubya\]) is 0 in this case.
Suppose now that $i = i'$ but that $j \neq j'.$ Then, by (\[count\]), $e_{j'} = e_j.$ Therefore $$\begin{aligned}
\label{nixon} {\bf b}_{j'}{\bf b}^{j} = r'_{max}{\bf b}_{max} + \sum_{k\neq max} r'_k {\bf b}_k,\end{aligned}$$ and in this equation $r'_{max}=0$, since otherwise it has strictly positive degree, contradicting the homogeneity of degree $N$ of (\[nixon\]). Hence (\[key\]) becomes
$$\Phi ( {\bf a}_{i} {\bf a}^i \cdot {}^{w}({\bf b}_{j'}{\bf b}^j)) \Phi(({\bf a}_{max} + \sum_{k\neq max} r_k {\bf a}_k)(\sum_{k\neq max} r''_k {\bf b}_k)) = 0.$$ A similar argument applies if $i' \neq i$ but $j' = j.$ Thus the claim is proved. Therefore $$\Phi({\bf a}_{i'} u{\bf b}_{j'} {\bf
b}^jw^{-1}{\bf a}^i ) = \delta_{u,w}\delta_{i,i'}\delta_{j,j'}\epsilon_{V^*}(w).$$ It follows that, with $x {\bf b}^j w^{-1} {\bf a}^i,$ $$\Phi (ax) = z_b \epsilon_{V^*}(w)$$ where $b = {\bf a}_i w {\bf b}_j,$ confirming Hypothesis 2.3.
The theorem for Cherednik algebras
----------------------------------
Define the form ${{\mathbb B}}$ for $\mathbf{H}$ by ${{\mathbb B}}(a,b) = \Phi (ab),$ for $a,b \in \mathbf{H}.$ We can now deduce the
The rational Cherednik algebra $\mathbf{H}$ is a symmetric Frobenius extension of its central subalgebra ${{\mathcal{Z}}}= S(V)^W \otimes S(V^*)^W.$
It is immediate from Lemmas \[6.3\] and 2.3 that ${\bf H}$ is a free Frobenius extension of $\mathcal{Z}$ with form ${{\mathbb B}}$ as defined above. Therefore it remains only to prove that the Nakayama automorphism for ${\bf H}$ is inner.
We verify that ${{\mathbb B}}(Y,x)={{\mathbb B}}(x,Y)$, where $Y\in{{\mathcal B}}_{\mathbf{H}}$ and $x\in W$ or $V$ or $V^*$, since $W$, $V$ and $V^*$ generate $\mathbf{H}$ as a ${{\mathcal{Z}}}$-algebra. Let $fwg$ be a typical element from ${{\mathcal B}}_{\mathbf{H}}$. First, let $x\in W.$ Then $$\begin{aligned}
{{\mathbb B}}(fwg,x)=\Phi(fwgx)&=&\Phi(f \cdot wx\cdot {}^{x^{-1}}g)\label{1}\\
&=&\epsilon_{V^*}(x^{-1})\Phi(f\cdot wx \cdot g)\label{2}\\
&=&\epsilon_{V^*}(x^{-1})\Phi(f\cdot xw\cdot g)\label{3}\\
&=&\epsilon_{V}(x^{-1})\epsilon_{V^*}(x^{-1})\Phi({}^xf \cdot xw \cdot g)\label{4}\\
&=&\Phi(xfwg)\label{5} = {{\mathbb B}}(x,fwg)
\end{aligned}$$ The equalities follow from the definition of ${{\mathbb B}}$ and the defining relations of $\mathbf{H}$. To see the formulas and note that $x({\bf
a}_{max})=\epsilon_V(x){\bf a}_{max}+h$, where $h \in S(V)$ with ${\bf a}_{max}(h)=0$. Similarly for ${\bf b}_{max}$, and then invoke the definition of $\Phi$. The equality is true because both sides of the equation are trivial unless $x=w^{-1}$, in which case we have $xw=wx$. The relation holds because of the defining relations of ${\mathbf H}$ and thanks to the fact that $\epsilon_V(x)=\epsilon_{V^*}(x)^{-1}$. Finally, the last equation is clear by definition of ${{\mathbb B}}$, and hence ${{\mathbb B}}(fwg,x)= {{\mathbb B}}(x,fwg)$ holds.
If $a\in V$ we get $$\begin{aligned}
{{\mathbb B}}(fwg,a)&=&\Phi(fwga) \nonumber\\ & =& \Phi(fwag)\label{11}\\
&=&\Phi(f \, {}^wa \, wg)\nonumber\\
&=&\Phi(fawg)\label{22}\\
&=&\Phi(afwg)={{\mathbb B}}(a,fwg).\nonumber
\end{aligned}$$ The equality in arises since the degree of $fwga$ and $fwag$ is $|f| + |g| + 1$ and so both sides are zero unless $|f| +
|g| \geq 2N-1$. In the case $|g| = N$ or $N-1$ then $|[a,g]| < N$ by . This then means that $\Phi(fwga) = \Phi(
fwag - fw[a,g]) = \Phi ( fwag)$, as required. The equality is true, because we have zero on both sides if $w\not=e$. Hence ${{\mathbb B}}(fwg,a)={{\mathbb B}}(a,fwg)$ holds. If $b\in V^*$ the argument is similar, so we leave it to the reader.
Therefore we get ${{\mathbb B}}(x,y)={{\mathbb B}}(y,x)$ for any $x, y\in\mathbf{H}$, which means ${{\mathbb B}}$ is symmetric.
Consequences {#conseq}
------------
Given a maximal ideal $\mathfrak{m}_{\chi}$ of $\mathcal{Z}$ we define the [*reduced Cherednik algebra*]{} to be the $|W|^3$-dimensional algebra $${\bf H}_{\chi} \equiv \frac{\bf H}{\mathfrak{m}_{\chi} \bf H}.$$ Thanks to [@IGG] these algebras control a great deal of the geometry associated to the centre of [**H**]{}. The following corollary is immediate from Theorem 3.5 and the discussion in 2.4, after we have noted that ${\bf H}$ has finite global dimension by [@EG page 276]. The first part (for the case when $\mathfrak{m}_{\chi}$ is $(S(V)^W \otimes S(V^*)^W)_+$) answers [@Rouquier Problem 6].
1. The reduced Cherednik algebras ${\bf H}_{\chi}$ are symmetric, with dual bases the images of the bases $\mathcal{B} = \{{\bf a_i}w{\bf b_j}\}$ and $\mathcal{B}' \{{\bf a^i}w{\bf b^j}\}$ defined in Section \[6.2\] and \[dualbases\].
2. ${\bf H}$ is a Calabi–Yau ${{\mathcal{Z}}}-$algebra of dimension $2
\dim (V).$
The graded Hecke algebra
========================
In this section we show that the graded Hecke algebra ${\bf H}_{gr}$ is a Frobenius extension of its invariant centre, with non-trivial Nakayama automorphism, so that the reduced graded Hecke algebras ${{\bf H}_{gr}}_\chi$ are Frobenius but not, in general, symmetric.
Graded Hecke algebras {#GradedHecke}
---------------------
As in the previous section let $W$ be an irreducible complex reflection group with identity $e$, and $V$ the defining complex reflection representation of $W$. Let $\mathbf{H}_{gr}$ be the associative algebra generated by $V$ and ${\mathbb{C}}W$ with relations $$\begin{aligned}
wxw^{-1}={}^{w}x,\label{def1gr}\\
\left[x,y\right]=\sum_{w\in W}\Omega_w(x,y)w,\label{def2gr}\end{aligned}$$ for $x, y\in V$ and $w \in W.$ For each $w\in W$, $\Omega_w : V\times V \rightarrow {\mathbb{C}}$ is an alternating $2$-form on $V$; we insist these forms satisfy the coherence conditions of [@RS (1.6), (1.7)]. The algebra $\mathbf{H}_{gr}$ is a [*graded Hecke algebra*]{} for $W$ and $\mathbf{H}_{gr}\cong S(V)\otimes{\mathbb{C}}W$ as vector spaces ([@RS Lemma 1.5]). In particular, there is a PBW-basis given by the elements of the set $\{fw\}$, where $w\in W$, and $f$ runs through a homogeneous basis of ${S(V)}$. For $f$ in $S(V)$ we again write $|f|$ for the degree of $f$. For $i \in {\mathbb{Z}}_{\geq 0}$ let ${{\mathcal B}}_{<i}$ be the span of all PBW-basis elements of the form $fx$, where $f\in S(V)$, $x\in W$ such that $f$ is homogeneous with $|f|< i$: this induces a ${\mathbb{Z}}_{\geq0}$-filtration of $\mathbf{H}_{gr}$. Moreover, the commutation relation shows that $$\label{commforusgr} \text{$|[f,g]| \leq |f| + |g| - 2$ for all
homogeneous $f, g\in S(V)$.}$$ Recall that $s\in W$ is a [*bireflection*]{} if ${\operatorname}{codim} V^s:={\operatorname}{rank}({\operatorname}{id_V}-s)=2$. We denote by ${{\mathcal R}}$ the set of all bireflections $s$ such that for any $g\in Z_W(s)$, the $W$-centraliser of $s$, the action of $g$ restricted to $V/V^s$ has determinant equal to one. The set ${{\mathcal R}}$ plays an important role since $\Omega_g\not=0$ implies $g=e$ or $g\in{{\mathcal R}}$ ([@RS Theorem 1.9]). Moreover, since $V$ is the (faithful) defining reflection representation of $W$ and $\Omega_e\in((\wedge^2V)^*)^W$, we find $\Omega_e=0$. Hence relation becomes $$\begin{aligned}
\label{def2grb}
[x,y]=\sum_{w\in {{\mathcal R}}}\Omega_w(x,y)w.\end{aligned}$$ Let $N\triangleleft W$ be the normal subgroup generated by ${{\mathcal R}}$ and let ${{\bf H}_{gr}}(N)$ be the graded Hecke algebra associated with $N$ whose structure is inherited from ${{\bf H}_{gr}}$. The following fact illustrates once more that ${{\mathcal R}}$ controls ${{\bf H}_{gr}}$: there is ([@Passman Lemma 1.3]) an isomorphism of algebras $$\begin{aligned}
\label{crossedprod}
{{\bf H}_{gr}}\cong {{\bf H}_{gr}}(N)*' W/N,\end{aligned}$$ where ${{\bf H}_{gr}}(N)*' W/N$ is a crossed product algebra defined as follows. As a vector space it is just ${{\bf H}_{gr}}(N)\otimes {\mathbb{C}}[W/N]$. To define the commutator relations between these two subspaces we fix for each coset of $W/N$ one representative. Let $\{g_i\mid i\in J\}$ be the resulting complete system of coset representatives for $W/N$ with $g_i\in [g_i]\in W/N$.
Let $T(V)$ be the tensor algebra and $T(V)\ast W$ be the skew product algebra with the relations given by . Hence ${{\bf H}_{gr}}=(T(V)*W)/I$ where $I$ is given by the relations . These relations also define an ideal, $I(N)$, of $T(V)*N$ such that ${{\bf H}_{gr}}(N)=(T(V)*N)/I(N)$. If now $x=\sum_{n\in N}
v_nn\in T(V)\otimes{\mathbb{C}}N$ then define $$\begin{aligned}
\label{commcross}
[g_i]x=\sum_{n\in N} {}^{g_i}v_n\, g_i n g_i^{-1}\, [g_i].\end{aligned}$$ Passing to the quotient, this defines the commutator relations between ${{\bf
H}_{gr}}(N)$ and ${\mathbb{C}}[W/N]$ in ${{\bf H}_{gr}}(N)*W/N$. One can show that, up to isomorphism, this algebra does not depend on the choice of representatives. However, with these choices, the isomorphism is explicitly given as $fg\mapsto f\cdot g_ing_i^{-1}\cdot [g_i]$, where $f\in S(V)$, $g=g_in\in W$, $n\in N$. Since ${\bf H}_{gr}(N)$ is preserved by conjugation by the subgroup $W$ of ${\bf H}_{gr},$ we note:
\[Gaction\] Let $Z({{\bf H}_{gr}}(N))$ be the centre of ${{\bf H}_{gr}}(N)$ considered as a subalgebra of ${{\bf H}_{gr}}$ via the isomorphism . The $W$-action $g.h=ghg^{-1}$ for $g\in W$, $h\in{{\bf H}_{gr}}$ induces a $W$-action on $Z({{\bf H}_{gr}}(N))$.
The central subalgebra {#the-central-subalgebra}
----------------------
In the special case (see [@RS Section 3]) where $W$ is a Weyl group and ${\bf H}_{gr}$ is Lusztig’s graded Hecke algebra (as introduced in [@Lusztigaffine]) the following result is well-known ([@Lusztigaffine Proposition 4.5]). We retain the notation $\{{\bf a}_i : 1 \leq i \leq |W| \}$ from Section \[dualbases\].
\[centralsub\]
1. The algebra $\mathbf{H}_{gr}$ has finite global dimension.
2. The centre $Z({\bf H}_{gr})$ contains the subalgebra ${{\mathcal{Z}}}_{gr}:={S(V)}^W$.
3. With the notation from the previous section, $\mathbf{H}_{gr}$ becomes a free ${{\mathcal{Z}}}_{gr}$-module of finite rank with basis $${{\mathcal B}}_{\mathbf{H}_{gr}}
\quad := \quad \{{\bf a}_{i}w : w\in W, 1\leq i\leq |W|\}.$$
The proof of this proposition will occupy the rest of this subsection. We start with some preparations. Note that if $\Omega_w=0$ for all $w\in W$, then $\mathbf{H}_{gr}\cong S(V)* W$, the skew group algebra. Of course, the proposition holds in this case. For any filtered algebra $B$ we denote by ${\operatorname}{Gr}B$ its associated graded algebra. The following holds:
\[lemmaA\] Let $e_N=\frac{1}{|N|}\sum_{w\in N}w$ and consider $\mathbf{H}^{sph}:=e_N\mathbf{H}_{gr}(N)e_N$, the spherical subalgebra of $\mathbf{H}_{gr}(N)$. The ${\mathbb{Z}}_{\geq0}$-filtration on $\mathbf{H}_{gr}(N)$ induces a filtration on $\mathbf{H}^{sph}$ and also on its centre such that
1. \[ll1\] ${\operatorname}{Gr}\mathbf{H}^{sph}\cong S(V)^N$.
2. \[ll2\] There is an isomorphism of algebras $\Psi:Z(\mathbf{H}_{gr}(N))\cong Z(\mathbf{H}^{sph})$, $z\mapsto ze_N$.
3. \[ll3\] $\mathbf{H}^{sph}$ is commutative, in particular $Z(\mathbf{H}_{gr}(N))\cong \mathbf{H}^{sph}$.
4. \[ll4\] ${\operatorname}{Gr}Z(\mathbf{H}_{gr}(N))\cong S(V)^N$.
There is an isomorphism $S(V)^N\rightarrow e_N(S(V)*N)e_N$ via $f\mapsto fe$, and $e_N(S(V)*N)e_N\cong e_N({\operatorname}{Gr}\mathbf{H}_{gr}(N))e_N\cong {\it
Gr}(e_N\mathbf{H}_{gr}(N)e_N)={\operatorname}{Gr}\mathbf{H}^{sph}$. This proves . Statements and are analogous to [@EG Theorem 3.1] and [@EG Theorem 1.6] respectively; details can be found in [@Katrin]. Since $\Psi$ preserves the filtration and is surjective on each layer, the last statement follows from .
Let $R=S(V)*N$. Recall that an associative graded algebra $(A,\diamond)$, with multiplication $\diamond$, is called [*a graded deformation of $R$*]{} if $A\cong R\otimes_{\mathbb{C}}{\mathbb{C}}[h]$ as graded vector spaces where $h$ is an indeterminant concentrated in degree one, $\diamond$ is ${\mathbb{C}}[h]$-bilinear, and $r_1\diamond
r_2\equiv r_1r_2 \mod hA$ for any $r_1, r_2\in R$, considered as a subspace of $A$. Put $$\begin{aligned}
A=A(V,N):=(T(V)[h]*N)/I_N,\quad I_N:=\langle\left[x,y\right]-\sum_{w\in
\mathcal{R}}\Omega_w(x,y)wh^2:x,y\in V\rangle.
\end{aligned}$$ Note that $I_N$ becomes homogeneous, hence $A$ is graded. It follows directly that $A$ is a graded deformation of $R$ and $A/(h-1)A=\mathbf{H}_{gr}(N)$.
The first statement is clear from [@MR Corollary 7.6.18(i)], since $\mathbf{H}_{gr}$ is filtered such that ${\operatorname}{Gr}(\mathbf{H}_{gr})\cong S(V)* W$ and the latter has finite global dimension. The last statement will follow as soon as we established the second.
Recall (from Lemma \[Gaction\]) that $W$ and hence $W/N$ act on the centre of $\mathbf{H}_{gr}(N)$. We get ${\operatorname}{Gr}(Z(\mathbf{H}_{gr}(N))^{W/N})=({\operatorname}{Gr}Z(\mathbf{H}_{gr}(N)))^{W/N}=(S(V)^N)^{W/N}=S(V)^W$ by Lemma \[lemmaA\], and $e_NAe_N$ is a commutative graded deformation of $S(V)^N;$ the proof of this is analogous to the proof of [@EG Theorem 1.6], and is given in detail in [@Katrin]. The infinitesimal commutative graded deformations are controlled by the second Harrison cohomology ([@Harrison Theorem 8], [@Gerst Section 4]). In our situation $B:=(e_NAe_N)^{W/N}$ is a (global) commutative graded deformation of $S(V)^W$. On the other hand, $W$ is a complex reflection group, hence $S(V)^W$ is a polynomial ring, and so there are no non-trivial graded commutative deformations ([@Harrison Theorem 11]). Hence $B$ is a trivial deformation, and therefore $B/(h-1)B=S(V)^W$. On the other hand $B/(h-1)B=(e_N\mathbf{H}_{gr}(N)e_N)^{W/N}=(\mathbf{H}^{sph})^{W/N}$, hence $Z(\mathbf{H}_{gr}(N))^{W/N}=S(V)^W$ by Lemma \[lemmaA\]. The claim of the proposition follows then from as follows: Let $f\in S(V)^W$, in particular $fg=gf\in \mathbf{H}_{gr}$ for any $g\in W$. Since the centre of $\mathbf{H}_{gr}(N)$ is given by $S(V)^N$ and $f\in S(V)^W\subset S(V)^N$, we get $fh=hf$ for any $h\in\mathbf{H}_{gr}(N)$, considered as a subspace of $\mathbf{H}_{gr}$. Hence, $f$ is in the centre of $\mathbf{H}_{gr}$.
The centre
----------
Although it is not needed for the results of this paper, we record here the fact that the inclusion of $S(V)^W$ in the centre of $\mathbf{H}_{gr}$ is in fact an equality. In the special case where $W$ is a Weyl group, this result is [@Lusztigaffine Proposition 4.5].
Retain the notation of Sections \[GradedHecke\] and \[centralsub\]. Then $S(V)^W = Z(\mathbf{H}_{gr}).$
From Proposition \[centralsub\](2) we know that ${{\mathcal{Z}}}_{gr} := S(V)^W \subseteq Z:= Z(\mathbf{H}_{gr}).$ Let $F$ and $E$ be the quotient fields of ${{\mathcal{Z}}}_{gr}$ and $Z$ respectively, and let $Q$ be the (simple artinian) quotient ring of $\mathbf{H}_{gr},$ so $F \subseteq E \subseteq Q.$ Since $\mathbf{H}_{gr}$ is a finitely generated module over the commutative affine algebra ${{\mathcal{Z}}}_{gr},$ $Z \cap F$ is a finitely generated ${{\mathcal{Z}}}_{gr}-$module. Therefore, since ${{\mathcal{Z}}}_{gr}$ is integrally closed, $Z \cap F = {{\mathcal{Z}}}_{gr}.$ Suppose for a contradiction that ${{\mathcal{Z}}}_{gr} \subsetneq Z.$ Then $F \subsetneq E.$ It follows that $$\mathrm{dim}_E(Q) < \mathrm{dim}_F(Q) = |W|^2.$$ That is, the PI-degree of $\mathbf{H}_{gr}$ is strictly less than $|W|$, or - equivalently - the maximal dimension of an irreducible $\mathbf{H}_{gr}-$module is strictly less than $|W|,$ [@BGo Theorem I.13.5 and Lemma III.1.2].
We now claim that the maximal dimension of irreducible $\mathbf{H}_{gr}-$modules is $|W|.$ To see this, consider the algebra $\hat{\mathbf{H}}_{gr}$, which has the same generators as $\mathbf{H}_{gr},$ but is constructed as an algebra over a polynomial algebra $\mathbb{C} [h].$ Relations (4.1) are unchanged, but the right hand sides of the relations (4.2) are multiplied by $h^2$. Thus $\hat{\mathbf{H}}_{gr}$ is $\mathbb{N}-$graded, with $h$ and the elements of $V$ having degree 1, and elements of $W$ degree 0. As before, we can show that $\mathbb{C}[h]S(V)^W \subseteq Z(\hat{\mathbf{H}}_{gr}),$ so that $\hat{\mathbf{H}}_{gr}$ has PI-degree at most $|W|$ by the same argument as above. On the other hand, $\hat{\mathbf{H}}_{gr}/h\hat{\mathbf{H}}_{gr} \cong S(V)*W,$ the skew group algebra, and this has irreducible modules of dimension $|W|$ - for example, one has an irreducible $S(V)*W$-structure on $S(V)/\mathfrak{m}S(V)$ for any maximal ideal $\mathfrak{m}$ of $S(V)^W$ contained in a maximal orbit (of size $|W|$) of maximal ideals of $S(V).$ Therefore, $$\mathrm{PI-degree}(\hat{\mathbf{H}}_{gr}) = \mathrm{PI-degree}(S(V)*W) =|W|.$$ Now the Azumaya locus of $\hat{\mathbf{H}}_{gr}$ is dense in $\mathrm{maxspec}(Z(\hat{\mathbf{H}}_{gr}))$ ([@BGo Theorem III.1.7]); in particular, there must be an irreducible $\hat{\mathbf{H}}_{gr}-$module $U$ annihilated by $h - \lambda$ for some $0 \not= \lambda \in \mathbb{C}.$ This implies that $\mathrm{PI-degree}(\hat{\mathbf{H}}_{gr}/(h-\lambda)\hat{\mathbf{H}}_{gr}) = |W|,$ and so proves our claim, since all such factors, for $\lambda \not= 0,$ are isomorphic to $\mathbf{H}_{gr}.$ We have thus obtained the desired contradiction, so the proof is complete.
The bilinear form
-----------------
Consider the ${{\mathcal{Z}}}_{gr}$-linear map $$\begin{aligned}
\Phi_{gr}:\quad\quad\quad\mathbf{H}_{gr}&\longrightarrow&{{\mathcal{Z}}}_{gr}\\
{{\mathcal B}}_{\mathbf{H}_{gr}}\ni {\bf a}_{i}w &\longmapsto&
\begin{cases}
1&\text{if $w=e$, $i=max$},\\
0&\text{otherwise}.
\end{cases}\end{aligned}$$
Define the form ${{\mathbb B}}$ for $\mathbf{H}_{gr}$ by ${{\mathbb B}}(a,b) = \Phi_{gr}(ab),$ for $a,b \in \mathbf{H}_{gr}.$ We can now deduce the \[6.3gr\]
\[maingr\] The functional $\Phi_{gr}$ above satisfies Hypothesis \[hyp\].
The proof is completely analogous to Lemma \[6.3\].
\[gradedHecke\] The graded Hecke algebra $\mathbf{H}_{gr}$ is a free Frobenius extension of its central subalgebra ${{\mathcal{Z}}}_{gr}$ with Nakayama automorphism $\nu$ given by $\nu(w)=\epsilon_{V}(w)^{-1}w$, $\nu(v)=v$ for $w\in W$, $v\in V$.
It is immediate from Lemmas \[6.3\] and 2.3 that ${\bf H}_{gr}$ is a free Frobenius extension of ${{\mathcal{Z}}}_{gr}$ with form ${{\mathbb B}}$ as defined above. Therefore it remains only to determine the Nakayama automorphism. Let $\nu$ be as in the theorem, and let $fw$ be a typical element from ${{\mathcal B}}_{\mathbf{H}_{gr}}$. First, let $x\in W.$ Then ${{\mathbb B}}(fw,x)=\Phi_{gr}(fwx)=\delta_{w,x^{-1}}\Phi_{gr}(f)$ from the definition of $\Phi_{gr}$, and ${{\mathbb B}}(\nu(x),fw)=\Phi_{gr}(\nu(x)fw)=\Phi_{gr}(\epsilon_V(x)^{-1}xfw)=\Phi_{gr}(fxw)=\delta_{w,x^{-1}}\Phi_{gr}(f)$ using the defining relations of $\mathbf{H}_{gr}$ and again the definition of $\Phi_{gr}$. If $a\in V$ we get $$\begin{aligned}
{{\mathbb B}}(fw,a)&=&\Phi_{gr}(fwa)=\Phi(f \, {}^wa \, w)\stackrel{(*)}{=}\Phi(faw)\stackrel{(**)}{=}\Phi(afw)={{\mathbb B}}(a,fw).
\end{aligned}$$ The equality (\*\*) arises since the degree of $fa$ and $af$ is $|f| + 1$ and so both sides are zero unless $|f|\geq N-1$. In the case $|f| = N$ or $N-1$ then $|[a,f]| < N$ by . This then means that $\Phi(faw) = \Phi(
afw - [f,a]w) = \Phi (afw)$, as required. The equality (\*) is true, because we have zero on both sides if $w\not=e$. Hence ${{\mathbb B}}(fw,a)={{\mathbb B}}(a,fw)$ holds.
Since $\mathbf{H}_{gr}$ is generated by $V$ and $W$, ${{\mathbb B}}(x,y)={{\mathbb B}}(\nu(y),x)$ for any $x, y\in\mathbf{H}$, where $\nu$ is as claimed.
Just as in Section \[conseq\], we can immediately deduce the
The factor ${\mathbf{H}_{gr}}_{\chi}$ of the graded Hecke algebra $\mathbf{H}_{gr}$ by a maximal ideal $\mathfrak{m}_{\chi}$ of its central subalgebra $\mathcal{Z}_{gr}$ is a Frobenius algebra which in general is not symmetric.
The extended affine Hecke algebra
=================================
In this section we show that the extended affine Hecke algebra $\mathcal{H}$ is a Frobenius extension of its centre, with non-trivial Nakayama automorphism, so that the corresponding reduced algebras $\mathcal{H}_\chi$ are Frobenius but not, in general, symmetric.
{#section}
Let $W$ be a (finite) Weyl group with length function $l$ and integral weight lattice $X$, and let $v$ be an indeterminant. For a parameter set $L$ we denote by $\mathcal{H}$ the corresponding extended affine Hecke algebra over ${\mathbb{C}}[v,v^{-1}]$ as defined in [@Lusztigaffine 3.1]. With the notation from [@Lusztigaffine Lemma 3.4] ${{\mathcal H}}$ is a free ${\mathbb{C}}[v,v^{-1}]$-module with basis $T_w\theta_x$, for $w\in W$, $x\in X$, and the subalgebra $\mathbb{C}[v,v^{-1}]\langle \theta_x :x \in X \rangle$ is a Laurent polynomial algebra. Let ${{\mathcal{Z}}}_{{\mathcal H}}={\mathbb{C}}[v,v^{-1}][X]^W$ be the centre of ${{\mathcal H}}$ [@Lusztigaffine Proposition 3.11]. By the Pittie-Steinberg Theorem ([@Steinberg]), ${{\mathcal{Z}}}_{{\mathcal H}}$ is a polynomial ring over ${\mathbb{C}}[v,v^{-1}]$ and ${{\mathcal H}}$ is free over ${{\mathcal{Z}}}_{{\mathcal H}}$ of finite rank $|W|^2$. By abuse of language we denote by $({\mathbb{C}}[v,v^{-1}][X]^W_+)$ the augmentation ideal in ${{\mathcal{Z}}}_{{\mathcal H}}$. We consider the coinvariant ${\mathbb{C}}[v,v^{-1}]$-algebra ${\mathbb{C}}[v,v^{-1}][X]/({\mathbb{C}}[v,v^{-1}][X]^W_+)$ which we equip with a ${\mathbb{Z}}$-grading. This induces a ${\mathbb{Z}}_{\geq 0}$-filtration on ${\mathbb{C}}[v,v^{-1}][X]/({\mathbb{C}}[v,v^{-1}][X]^W_+)$. We fix again a pair of (homogeneous) dual bases $\{\overline{{\bf
a}}_i:1\leq i\leq |W|\}$, $\{{\overline{\bf a}}^i:1\leq i\leq |W|\}$ of the coinvariant algebra and lift these elements to bases $\{{\bf a}_i:1\leq i\leq |W|\}$, $\{{\bf
a}^i:1\leq i\leq |W|\}$ of the free ${{\mathcal{Z}}}$-module ${\mathbb{C}}[v,v^{-1}][X]$ such that the (filtered) degree of ${\bf a}_i$ agrees with the grading degree of $\overline{\bf a}_i$. Then ${{\mathcal H}}$ is free over ${{\mathcal{Z}}}_{{\mathcal H}}$ of rank $|W|^2$. Let ${{\mathcal B}}_{{\mathcal H}}$ be the basis given by the $T_w {\bf a}_i$.
Let ${{\mathcal H}}_i$ be the ${{\mathcal{Z}}}_{{\mathcal H}}$-span of all $T_w{\bf a}_j$, where $1\leq
j\leq |W|$ and $l(w)\leq i$. Then ${{\mathcal H}}=\bigcup_{i\geq 0}{{\mathcal H}}_i$ is a filtration of ${{\mathcal H}}$.
We have to show that ${{\mathcal H}}_i{{\mathcal H}}_j\subseteq{{\mathcal H}}_{i+j}$ for any $i,
j\in{\mathbb{Z}}_{\geq 0}$. With the notation from [@Lusztigaffine Proposition 3.9] we have $\theta_xT_s\equiv T_s\theta_{s(x)}\mod{{\mathcal H}}_0$, and then for any $w\in W$ $$\begin{aligned}
\label{eq:Lu}
\theta_xT_w\equiv T_w\theta_{w^{-1}(x)}\mod {{\mathcal H}}_{l(w)-1}
\end{aligned}$$ by induction. To establish the lemma we only have to show that $T_w\theta_x T_v\theta_y\in{{\mathcal H}}_{l(w)+l(v)}$ for any $v,w\in W$, $x,y\in X$. This is of course true if $l(v)=0$. From formula we get $T_w\theta_x T_v\theta_y\equiv
T_wT_v\theta_x\theta_y$ modulo $T_w{{\mathcal H}}_{l(v)-1}\subseteq{{\mathcal H}}_{l(w)+l(v)-1}\subset{{\mathcal H}}_{l(w)+l(v)}$. On the other hand $T_wT_v\theta_x\theta_y\in{{\mathcal H}}_{l(w)+l(v)}$ and we are done.
{#section-1}
Analogous to the cases above we define a ${{\mathcal{Z}}}_{{\mathcal H}}$-linear map $$\begin{aligned}
\Phi_{{\mathcal H}}:\quad\quad\quad{{\mathcal H}}&\longrightarrow&{{\mathcal{Z}}}_{{\mathcal H}}\\
{{\mathcal B}}_{{{\mathcal H}}}\ni T_w{\bf a}_i&\longmapsto&
\begin{cases}
1&\text{if $w=e$ and $i=max$},\\
0&\text{otherwise}.
\end{cases}\end{aligned}$$
\[6.3ext\]
\[mainext\] The functional $\Phi_{{\mathcal H}}$ defined above satisfies Hypothesis \[hyp\].
To prove this statement we need the following easily verified formulas:
Let $w, x\in W$, $f,g\in{\mathbb{C}}[v,v^{-1}][X]$.
1. \[TxTw\] Let $T_xT_w=\sum_{y\in W} h_y T_y$ in ${{\mathcal H}}$. If $h_e\not=0$ then $w=x^{-1}$.
2. \[stupid\] If $l(w)\geq l(x)$ then $\Phi_{{\mathcal H}}(T_wfT_xg)\not=0$ implies $x=w^{-1}$.
Statement is an easy induction argument using the defining relations of ${{\mathcal H}}$ and therefore omitted. (For a representation theoretic interpretation of this statement we refer to [@Stcompf Theorem 3.1]). To verify Statement note that if $x\in W$, $l(w)\geq l(x)$ then there exists some $h\in{\mathbb{C}}[v,v^{-1}][X]$ such that $T_wfT_x=T_wT_xh$ modulo $T_w{{\mathcal H}}_{l(x)-1}$ (by formula ). Therefore we get $T_wfT_xg=T_wT_xhg+r$, where $r\in T_w{{\mathcal H}}_{l(x)-1}$. Since $l(x)-1<l(w)$, using Statement we deduce that $\Phi_{{\mathcal H}}(r)=0$ and so $\Phi_{{\mathcal H}}(T_wfT_xg)=\Phi_{{\mathcal H}}(T_wT_xhg)$. The claim follows by applying Statement again.
Let $0 \not= u \in{{\mathcal H}}$, $u=\sum_{w,i}z_{w,i}T_w{\bf a}_i$, where $z_{w,i}\in{{\mathcal{Z}}}_{{\mathcal H}}$. Choose $x$ of minimal length such that $z_{x^{-1},i}\not=0$ for some $i$. From the lemma above and formula we get $$\begin{aligned}
\Phi_{{\mathcal H}}(uT_xf)=\Phi(\sum_{w,i}z_{w,i}T_w{\bf
a}_iT_xf)=\Phi(\sum_{i}z_{x^{-1},i}T_{x^{-1}}{\bf a}_iT_xf)
\end{aligned}$$ for any $f\in{\mathbb{C}}[v,v^{-1}][X]$. Using again the lemma above and formula we can rewrite the expression $\sum_{i}z_{x^{-1},i}T_{x^{-1}}{\bf a}_iT_x$ in the form $\sum_{i}c_{i} {\bf a}_i +r$, where $r\in {{\mathcal H}}$ is such that when expanded in the standard bases no $T_e$ occurs, and $c_i\in{{\mathcal{Z}}}_{{\mathcal H}}$ are not all zero. Since $\Phi_{{\mathcal H}}(rf)=0$ for any $f\in{\mathbb{C}}[v,v^{-1}][X]$, it is enough to verify the Hypothesis \[hyp\] for $u=\sum_{i}c_{i} {\bf
a}_i$. But now we are in a familiar situation, except that we have only filtered algebras instead of graded algebras. Nevertheless, the statement follows as in Lemma \[6.3\].
\[affineHecke\] The extended affine Hecke algebra ${{\mathcal H}}$ is a free Frobenius extension of its centre ${{\mathcal{Z}}}_{{\mathcal H}}$. In general, this extension is not symmetric.
We only have to verify that the Nakayama automorphism is non-trivial in general. This however follows directly from [@Lusztigaffine Theorem 9.3] and Theorem \[gradedHecke\].
{#section-2}
Just as in Section \[conseq\], we deduce the
The factor ${{\mathcal H}}_{\chi}$ of the extended affine Hecke algebra ${{\mathcal H}}$ by a maximal ideal $\mathfrak{m}_{\chi}$ of the centre $\mathcal{Z}_{{{\mathcal H}}}$ is a Frobenius algebra; in general it is not symmetric.
Nil-Hecke algebras
------------------
We would like to mention at least two related algebras, where our approach works, namely the [*affine Nil-Hecke algebra*]{} ${{\mathcal H}}^{nil}$ and the [*graded affine Nil-Hecke algebra*]{} ${{\mathcal H}}^{nil}_{gr}$ associated to a Weyl group $W$. (For the definitions see e.g. [@GR]). Analogous to the affine Hecke algebra case, the centre of ${{\mathcal H}}^{nil}$ is ${{\mathcal{Z}}}={\mathbb{C}}[X]^W$ and ${{\mathcal H}}^{nil}$ is a free ${{\mathcal{Z}}}$-module of rank $|W|^2$ ([@GR (1.9)]), similarly for the graded affine Nil-Hecke algebras. If we define the forms completely analogous to the affine and graded Hecke algebras we deduce that ${{\mathcal H}}^{nil}$ and ${{\mathcal H}}^{nil}_{gr}$ are free Frobenius extensions over their centres.
The quantised universal enveloping algebra
==========================================
In this section we show that the quantised enveloping algebra ${{\mathcal U}}_\epsilon({\mathfrak{g}})$ at a root of unity $\epsilon$ is a Frobenius extension of its Hopf centre, with trivial Nakayama automorphism, so that the reduced quantised enveloping algebras ${{\mathcal U}}_\epsilon({\mathfrak{g}})_\chi$ are symmetric.
The PBW-basis and the central subalgebra
----------------------------------------
Let ${\mathfrak{g}}$ be a complex semisimple Lie algebra. We fix a Borel and Cartan subalgebra of ${\mathfrak{g}}$, ${\mathfrak{b}}\supseteq{\mathfrak{h}},$ and denote the Weyl group by $W$ and the set of simple reflections by $S$. Let $\pi$ be the corresponding set of simple roots and $\rho$ the half-sum of positive roots. Let $\epsilon\in{\mathbb{C}}$ be an $l$-th root of unity, for some odd positive integer $l$, $l\not=3$ if ${\mathfrak{g}}$ has a summand of type $G_2$. Let $Q\subseteq P$ be, respectively, the root lattice and the weight lattice of ${\mathfrak{g}}$, with the $W$-equivariant bilinear form $(\,,\,):P\times Q\rightarrow {\mathbb{Z}}$.
The simply connected form of the quantised universal enveloping algebra ${{\mathcal U}}={{\mathcal U}}_\epsilon({\mathfrak{g}})$ is a ${\mathbb{C}}$-algebra with generators $E_\alpha$, $F_\alpha$, $K_{\lambda}$, for $\alpha\in\pi$ and ${\lambda}\in P$. For the defining relations and further details we refer for example to [@ChariPresley 9.1] or [@BGo I.6.3, III.6.1]. Let $w_0$ be the longest element of $W,$ and fix a reduced expression $$\begin{aligned}
\label{w0}
w_0=s_{i_1}s_{i_2}\ldots s_{i_N},\end{aligned}$$ where $s_{i_j}\in S$ for $1\leq j\leq N$. Let $\alpha_{i_j}$ be the simple root corresponding to $s_{i_j}\in S$. Recall that Lusztig defined an action on ${{\mathcal U}}$ of the braid group $B$ corresponding to $W,$ (see [@Luquantroot], [@ChariPresley Section 9], [@Jquant Section 8] or [@BGo I.6.7, I.6.8]). Let $T_i$ be the automorphism in $B$ corresponding to the simple reflection $s_i\in S$. We set $$\begin{aligned}
\label{eq:beta}
\beta_k:=s_{i_1}s_{i_2}\ldots s_{i_{k-1}}(\alpha_{i_k}),\end{aligned}$$ and put $E_{\beta_k}=T_{i_1}T_{i_2}\ldots
T_{i_{k-1}}(E_{\alpha_{i_k}})$ and $F_{\beta_k}=T_{i_1}T_{i_2}\ldots T_{i_{k-1}}(F_{\alpha_{i_k}})$. For any sequence $\mathbf{m}=(m_1,m_2,\ldots,
m_N)\in\mathbb{Z}^N_{\geq0}$ let $$\begin{aligned}
E^\mathbf{m}&=&E_{\beta_1}^{m_1}E_{\beta_2}^{m_2}\ldots E_{\beta_N}^{m_N},\\
F^\mathbf{m}&=&F_{\beta_N}^{m_N}F_{\beta_{N-1}}^{m_{N-1}}\ldots
F_{\beta_1}^{m_1}.
\end{aligned}$$ This yields a PBW-basis of ${{\mathcal U}}$ (associated with ), namely $$\begin{aligned}
{{\mathcal B}}\quad = \quad \{ F^{\bf k} K_{\lambda}E^{\bf m} : {\bf k}, {\bf m}\in{\mathbb{Z}}^{N}_{\geq0},{\lambda}\in P \},
\end{aligned}$$ see [@ChariPresley Theorem 9.3], [@BGo I.6.2, III.6.1]. The subspace ${{\mathcal{Z}}}$ of ${{\mathcal U}}$ spanned by the monomials $F^{l\bf k}
K_{l {\lambda}} E^{l \bf m}$ is a central Hopf subalgebra of ${{\mathcal U}}$, called the *$l$-centre*, and ${{\mathcal U}}$ is a free ${{\mathcal{Z}}}$-module of finite rank (see [@ChariPresley 19.1], [@BGo III.6.2]). As a ${{\mathcal{Z}}}$-basis of ${{\mathcal U}}$ one can choose the subset ${{\mathcal B}}'$ of ${{\mathcal B}}$ given by elements of the form $$\begin{aligned}
\label{B'} F^{\bf k} K_{\lambda}E^{\bf m},
\end{aligned}$$ where $0\leq k_i, l_i< l$ and the coefficients of ${\lambda}$ in terms of fundamental weights are non-negative integers less than $l$.
Filtrations, degrees and commutation formulas {#filt}
---------------------------------------------
To simplify formulas we set $E_{i}=E_{\beta_i}$ and $F_{i}=F_{\beta_i}$. (Note that $E_i$ is not $E_{\alpha_i}$ in general.) Let $i<j$. There are commutation formulas holding in ${{\mathcal U}}$ as follows [@BGo Proposition I.6.10, Theorem III.6.1(4)]: $$\begin{aligned}
\label{Es}
E_{i}E_{j}&=&\epsilon^{(\beta_i,\beta_j)}E_{j}E_{i}+r\\
\label{Fs}
F_{i}F_{j}&=&\epsilon^{-(\beta_i,\beta_j)}F_{j}F_{i}+r'\end{aligned}$$ where $r$ (resp. $r'$), written in the PBW-basis, involves no monomial containing any $E_{k}$ (resp. $F_{k}$) for $k\leq i$ or $k\geq j$.
The algebra ${{\mathcal U}}$ is $Q$-graded (see e.g. [@Jquant 4.7]), but also has several other filtrations, [@ChariPresley 10.1], [@BGo I.6.11, III.6.1]. First, there is the *degree filtration*, a ${\mathbb{Z}}_{\geq
0}$-filtration obtained by putting $F^{\bf k} K_{\lambda}E^{\bf m}\in{{\mathcal B}}$ in degree $$\begin{aligned}
{\operatorname}{deg}(F^{\bf k} K_{\lambda}E^{\bf m})=\sum_{i=1}^N(k_i+m_i){\operatorname}{ht}(\beta_i),\end{aligned}$$ where ${\operatorname}{ht}$ denotes the height function. One can refine this to a $({\mathbb{Z}}_{\geq 0})^{2N+1}$-filtration by putting $F^{\bf k}
K_{\lambda}E^{\bf m}\in{{\mathcal B}}$ in degree $$\begin{aligned}
{\operatorname}{d}\big(F^{\bf k} K_{\lambda}E^{\bf m}\big)=\big(k_N, k_{N-1},\ldots k_1,m_1,m_2\ldots
m_N, {\operatorname}{deg}(F^{\bf k} K_{\lambda}E^{\bf m})\big).\end{aligned}$$ Putting the reverse lexicographic ordering on $({\mathbb{Z}}_{\geq
0})^{2N+1}$ ([[*i.e.* ]{}]{}$e_1<e_2<\ldots$, where $(e_i)_j=\delta_{i,j}$) defines the filtration by *total degree*. The $E$’s and $F$’s commute up to terms of lower total degree, [@ChariPresley 10.1], [@BGo Proposition I.6.11]: $$\begin{aligned}
\label{EF}
E_{i}F_j=F_jE_i+\text{ terms of lower total degree}.
\end{aligned}$$ We denote by $${\operatorname}{max}\quad := \quad 2(l-1)\sum_{i=1}^N
{\operatorname}{ht}(\beta_i)$$ the maximal ${\operatorname}{deg}$-value on ${{\mathcal B}}'$.
The bilinear form {#form}
-----------------
In view of the ${{\mathcal{Z}}}$-freeness of ${{\mathcal U}}$ on the basis ${{\mathcal B}}',$ we can define a ${{\mathcal{Z}}}$-linear map $\Phi:{{\mathcal U}}\rightarrow
Z$ as follows. Set $\mathbf{l} := (l-1,l-1,\ldots l-1),$ and define $$\begin{aligned}
\Phi: {{\mathcal B}}' \longrightarrow {{\mathcal{Z}}}: F^{\bf k} K_{\lambda}E^{\bf m}&\longmapsto&
\begin{cases}
1&\text{ if $\mathbf{k}=\mathbf{m}=:\mathbf{l}$, ${\lambda}=0$},\\
0&\text{ otherwise,}
\end{cases}
\end{aligned}$$ and extend this ${{\mathcal{Z}}}$-linearly.
\[nondeg\] The functional $\Phi$ satisfies Hypothesis \[hyp\].
For $\mathbf{m}\in({\mathbb{Z}}_{\geq0})^N$, define $\mathbf{\tilde m}:=\mathbf{l}-\mathbf{m}\in({\mathbb{Z}})^N$. For $x= F^{\bf
k} K_{\lambda}E^{\bf m}\in {{\mathcal B}}$ and $\mu \in P$ set $\tilde{x}_\mu= F^{\bf \tilde k} K_\mu
E^{\bf\tilde m}$ and write $k_i(x)=k_i$, $m_i(x)=m_i$.\
[*Claim 1: Let $x$, $y\in{{\mathcal B}}$. If ${\operatorname}{deg}(x)+{\operatorname}{deg}(y)<{\operatorname}{max}$ then $\Phi(xy)=0$.*]{}\
This follows directly from the fact that the commutation relations (\[Es\]), (\[Fs\]) and (\[EF\]) do not increase the ${\operatorname}{deg}$-value and $\Phi$ annihilates every monomial in ${{\mathcal B}}'$ which is not of maximal ${\operatorname}{deg}$-value.\
[*Claim 2: Let $x, y\in{{\mathcal B}}'$, $\mu\in P$. If ${\operatorname}{d}(x)<{\operatorname}{d}(\tilde{y}_\mu)$ then $\Phi(yx)=0$.*]{}\
If ${\operatorname}{d}(x)<{\operatorname}{d}(\tilde{y}_\mu)$ then ${\operatorname}{deg}(x)\leq{\operatorname}{deg}(\tilde{y}_\mu)={\operatorname}{max}-{\operatorname}{deg}(y)$, hence ${\operatorname}{deg}(x)+{\operatorname}{deg}(y)\leq{\operatorname}{max}$. By Claim 1 we only have to deal with the case ${\operatorname}{deg}(x)+{\operatorname}{deg}(y)={\operatorname}{max}$. From our assumption and the definition of ${\operatorname}{d}$ it follows that *either*
- there is a $k_j(x)$ such that $k_j(x)\not=l-1-k_j(y)$ (so that $k_j(x)<l-1-k_j(y)$), *or*
- there is an $m_j(x)$ such that $m_j(x)<l-1-m_j(y)$ (so that $m_j(x)<l-1-m_j(y)$).
Let us assume the latter for a moment and choose $j$ maximal with this property. Recalling from and the commutation relations for the $K$s with the $E$s and the $K$s with the $F$s that the relevant generators of ${{\mathcal U}}$ commute up to nonzero scalars and terms of lower ${\operatorname}{deg}$-value, we see that it is enough to show that $$\begin{aligned}
\Phi(F^{{\bf k}(y)}F^{{\bf k}(x)}K_{{\lambda}(x)}K_{{\lambda}(y)}E^{{\bf m}(y)}E^{{\bf m}(x)})=0.
\end{aligned}$$From the relation we get $$\begin{aligned}
E_N^{m_n(y)}E_1^{m_1(x)}\cdots E_N^{m_N(x)}&=&cE_1^{m_1(x)}\cdots E_N^{m_N(x)+m_N(y)}+r,
\end{aligned}$$ for some $c\in{\mathbb{C}}^*$ and some $r \in {{\mathcal U}}$ such that $E_N$ occurs in every monomial in $r$ with power strictly smaller than $m_N(y)+m_N(x)\leq l-1$. In particular, $$\begin{aligned}
&& \Phi(F^{{\bf k}(y)}F^{{\bf k}(x)}K_{{\lambda}(x)}K_{{\lambda}(y)}E^{{\bf m}(y)}E^{{\bf
}{\bf{m}}(x)})\\
&=&c\Phi(F^{{\bf k}(y)}F^{{\bf k}(x)}K_{{\lambda}(x)}K_{{\lambda}(y)}E_1^{m_1(y)}\cdots E_{N-1}^{m_{N-1}(y)}E_1^{m_1(x)}\cdots
E_N^{m_N(x)+m_N(y)}).\end{aligned}$$ Repeating this argument we get $$\begin{aligned}
&& \Phi(F^{{\bf k}(y)}F^{{\bf k}(x)}K_{{\lambda}(x)}K_{{\lambda}(y)}E^{{\bf m}(y)}E^{{\bf m
}(x)})=c'\Phi(F^{{\bf k}(y)}F^{{\bf k}(x)}K_{{\lambda}(x)+{\lambda}(y)}X),\end{aligned}$$ where $X=E_1^{m_1(y)}\cdots
E_{j-1}^{m_{j-1}(y)}E_1^{m_1(x)}\cdots
E_{j-1}^{m_{j-1}(x)}E_j^{m_j(x)+m_j(y)}E_N^{m_N(x)+m_N(y)}$ and $c' \in {\mathbb{C}}^*$. The result is zero since any commutation of the $E_i$s for $i<j$ does not involve $E_j$ because of , and since $m_j(x)+m_j(y)<l-1$. The remaining case, where there is a $k_j(x)$ such that $k_j(x)\not=l-1-k_j(y),$ can be proved similarly and is therefore omitted.\
[*Claim 3: Let $x$, $y\in{{\mathcal B}}$, $x=F^{{\mathbf k}}K_{\lambda}E^{\mathbf{m}}$, $\mu \in P$ and ${\operatorname}{d}(x)={\operatorname}{d}(\tilde{y}_\mu)$ then we have $\Phi(yx)\not=0$ if and only if ${\lambda}+\mu\in lP$.*]{}\
With the arguments from the proof of Claim 2 we get $$\begin{aligned}
\Phi(yx)=c\Phi(F^{\mathbf{l}} K_{{\lambda}(x)+{\lambda}(y)} E^\mathbf{l}),\end{aligned}$$ for some nonzero number $c\in{\mathbb{C}}$. Claim 3 follows then from the definition of $\Phi$.\
To prove the proposition, let $x\in{{\mathcal U}}$ be arbitrary and write $x=\sum_{y\in{{\mathcal B}}'} z_y y$ with $z_y\in {{\mathcal{Z}}}$. We choose $b=F^{{\mathbf k}}K_{\lambda}E^{\mathbf{m}}\in{{\mathcal B}}'$ of maximal total degree such that $z_{b}\not=0$. If now $a=F^{{\mathbf r}}K_\nu
E^{\mathbf{s}}\in{{\mathcal B}}'$ with $z_a\not=0$ and $\nu\in P$ arbitrary, then we have have ${\operatorname}{deg}(a)\leq{\operatorname}{deg}(b)$, hence, for any $\mu\in P,$ $${\operatorname}{deg}(a)+{\operatorname}{deg}(\tilde{b}_\mu)={\operatorname}{deg}(a)+{\operatorname}{max}-{\operatorname}{deg}(b)\leq{\operatorname}{max}.$$ If this inequality is strict, Claim 1 implies $\Phi(\tilde{b}_\mu a)=0$. Let us assume equality. Then we either have ${\operatorname}{d}(a)<{\operatorname}{d}(b)$ which implies $\Phi(\tilde{b}_\mu a)=0$ by Claim 2, or ${\operatorname}{d}(a)={\operatorname}{d}(b)$. The latter means (because of Claim 3) that $\Phi(\tilde{b}_{\mu} a)=0$ except when $\nu+\mu\in
lP$. In particular, $\Phi(\tilde{b}_{-{\lambda}}a)=0$, except when $a=b$.
Summarising, we get $\Phi(\tilde{b}_{-{\lambda}}x)=\Phi(\tilde{b}_{-{\lambda}}b)=cz_b\not=0$ for some unit $c$, as required.
Symmetry of the form {#symm}
--------------------
\[NakU\] The Nakayama automorphism $\nu$ of ${{\mathcal U}}$ with respect to $\mathbb{B}$ is the identity.
We have to prove that $\nu$ fixes all generators. We will run through all possibilities $y$ for generators and prove ${{\mathbb B}}(x,y) = {{\mathbb B}}(y,x)$ for all $x\in \mathcal{B}'$.
First, let $y=K_{\lambda}$. From Claim 2 we have automatically ${{\mathbb B}}(x,y)=0={{\mathbb B}}(y,x)$ unless $x$ has maximal degree ${\operatorname}{max}$. But then $K_{\lambda}$ commutes with $x$ and hence $\nu(K_{\lambda})=K_{\lambda}$.
Now let $y=E_{\alpha}$ for some simple root $\alpha$, and let $x \in {{\mathcal B}}'.$ Claim 1 implies that ${{\mathbb B}}(x,y)=0={{\mathbb B}}(y,x)$ unless $\deg{x}\geq{\operatorname}{max}-1$, because ${\operatorname}{deg}(E_{\alpha})=1$. If $x = F^{\bf l}K_{\lambda} E^{\bf l}$ then both $yx$ and $xy$ have $Q$-grade equal to $\alpha$. Thus $\Phi(yx) = 0 = \Phi(xy)$ since, by definition, $\Phi$ is non-zero only on elements whose $Q$-grade belongs to $\ell Q$. We now have two possibilities for $x$: either ${\operatorname}{deg}(F^{{\bf k}(x)})\not=\frac{{\operatorname}{max}}{2}$ and ${\operatorname}{deg}(E^{{\bf m}(x)})=\frac{{\operatorname}{max}}{2},$ or vice versa. Let us consider the first case. Then ${{\mathbb B}}(x,y)=0={{\mathbb B}}(y,x)$, because $\Phi$ annihilates everything which does not have the same $Q/lQ$-grading as $F^{\mathbf{l}}E^{\mathbf{l}}$ by definition. In the second case, the $Q$-grading again implies ${{\mathbb B}}(x,y)=0={{\mathbb B}}(y,x)$, unless $m_j(x)\not=l-1$ implies $\beta_j=\alpha$. Let $j$ be such that this equation holds. That means we have to compare ${{\mathbb B}}(x,y)=\Phi(xE_{\alpha})$ and ${{\mathbb B}}(y,x)=\Phi(E_{\alpha}x)$, where $x=F^{\mathbf{l}}K_{\lambda}E_1^{l-1}\ldots
E_{j-1}^{l-1}E_{j}^{l-2}E_{j+1}^{l-1}\ldots E_N^{l-1}$. Both terms are trivial unless ${\lambda}=0$. From the commutator relation it follows that ${{\mathbb B}}(x,y)=\epsilon^{-(l-1)(\beta_{j+1}+\ldots+\beta_N, \beta_j)}$ and ${{\mathbb B}}(y,x)=\epsilon^{-(l-1)(\beta_{1}+\ldots+\beta_{j-1}, \beta_j)}$. It is now enough to show that the exponents are the same.
Put $w=s_{i_1}s_{i_2}\cdots s_{i_{j-1}}$. Then $M^-=\{\beta_r :
1\leq r\leq j-1\}$ (resp. $M^+=\{\beta_r : j\leq r\leq N\}$) is exactly the set of all positive roots such that $w^{-1}(\beta)$ is negative (resp. positive). Set $M_1=w^{-1}(M^+)$ and $M_2=-w^{-1}(M^-)$. The disjoint union of these two sets is exactly the set of all positive roots (see e.g. [@Kane I.4.3, Theorem B]). By definition (see ) we have $w^{-1}(\beta_{j})=\alpha_{i_j}\in M_1$. From the definition of $\rho$, the half-sum of positive roots, we get $$\begin{aligned}
(\alpha_{i_j},\alpha_{i_j})&=&(\alpha_{i_j},\alpha_{i_j}) +\sum_{\beta\in
M_1\backslash\{\alpha_{i_j}\}}
(\beta,\alpha_{i_j})+\sum_{\beta\in M_2}(\beta,\alpha_{i_j}).\end{aligned}$$ Since the bracket $(\,,\,)$ is non-degenerate and $W$-equivariant, we get $$\begin{aligned}
0&=&\sum_{\beta\in M_1\backslash\{\alpha_{i_j}\}}(w(\beta),w(\alpha_{i_j}))+\sum_{\beta\in
M_2}(w(\beta),w(\alpha_{i_j}))\\&=&\sum_{\beta\in
M^+\backslash\{\beta_{j}\}}(\beta,\beta_{j})-\sum_{\beta\in
M^-}(\beta,\beta_{j}).\end{aligned}$$ Hence we get the required equality for the exponents and therefore ${{\mathbb B}}(x,y)={{\mathbb B}}(y,x)$. We are left with the case $y=F_\alpha$ for some simple root $\alpha$. The arguments there are similar, and therefore omitted. This completes the proof of the lemma.
{#symmetricU}
Recall the terminology of \[frob\]. From Proposition \[nondeg\] together with Proposition \[cruc\] and Lemma \[NakU\], we have:
The quantised universal enveloping algebra ${{\mathcal U}}={{\mathcal U}}_\epsilon({\mathfrak{g}})$ at an $l$-th root of unity is a free Frobenius extension of its $l$-th centre ${{\mathcal{Z}}}$. The form ${{\mathbb B}}$ has a trivial Nakayama automorphism.
The following corollary is immediate from the theorem and the discussion in 2.4, noting that ${{\mathcal U}}_{\epsilon}({\mathfrak{g}})$ has finite global dimension by [@BGoo Theorem 2.3].
Let $\chi$ be a maximal ideal of ${{\mathcal{Z}}}$.
1. The reduced quantised enveloping algebra ${{\mathcal U}}_{\chi}:={{\mathcal U}}_\epsilon({\mathfrak{g}})/{{\mathcal U}}_\epsilon({\mathfrak{g}})\chi$ is a symmetric algebra.
2. ${{\mathcal U}}_\epsilon({\mathfrak{g}})$ is a Calabi-Yau ${{\mathcal{Z}}}-$algebra of dimension $\dim {\mathfrak{g}}.$
Quantum Borels
==============
In this section we show that the quantum Borel ${{\mathcal U}}^{\geq 0}$ at a root of unity $\epsilon$ is a Frobenius extension of its Hopf centre, with non-trivial Nakayama automorphism, so that the reduced quantum Borels ${{\mathcal U}}^{\geq 0}_\chi$ are Frobenius, but not in general symmetric.
{#Borels}
Let ${\mathfrak{g}}$ be as above. Let ${{\mathcal U}}_{\epsilon}^{\geq 0}$ be the subalgebra of ${{\mathcal U}}_{\epsilon}(\mathfrak{g})$ generated by all the $E$s and $K$s. The PBW-basis of ${{\mathcal U}}_{\epsilon}(\mathfrak{g})$ gives rise to a PBW-basis of ${{\mathcal U}}_{\epsilon}^{\geq 0}$ given by the elements of the form $K_{\lambda}E^{\bf
m}$, where ${\bf m}\in{\mathbb{Z}}^{N}_{\geq0}$ and ${\lambda}\in P,$ [@ChariPresley 9.3]. Moreover, ${{\mathcal U}}_{\epsilon}^{\geq 0}$ is free over $Z_+:={{\mathcal{Z}}}\cap {{\mathcal U}}_{\epsilon}^{\geq 0}$ with basis ${{\mathcal B}}'_+$ given by all elements of the form $K_{\lambda}E^{\bf m}$, where $0\leq
m_i< l$ and the coefficients of ${\lambda}$ in terms of fundamental weights are non-negative integers less than $l$, [@ChariPresley 19.1].
The bilinear form and its Nakayama automorphism {#4.1}
-----------------------------------------------
Analogously to Section 4, we define a $Z_+$-linear map $\Phi_+:{{\mathcal U}}_{\epsilon}^{\geq 0}\rightarrow Z_+$ by $$\begin{aligned}
{{\mathcal B}}_+'\ni K_{\lambda}E^{\bf m}&\longmapsto&
\begin{cases}
1&\text{ if $\mathbf{m}=\mathbf{l}$, ${\lambda}=0$}\\
0&\text{ otherwise.}
\end{cases}
\end{aligned}$$ Define a $Z_+$-bilinear associative form ${{\mathbb B}}_+$ on ${{\mathcal U}}_{\epsilon}^{\geq 0}$ by putting ${{\mathbb B}}_+(x,y)=\Phi(xy)$ for any $x$, $y\in{{\mathcal U}}^{\geq 0}$.
\[NakBorels\] Let ${{\mathcal U}}_{\epsilon}^{\geq 0}$ be the quantum Borel defined in (\[Borels\]), with central subalgebra $Z_+$ as defined there. Let $\Phi_+$ and ${{\mathbb B}}_+$ be as above.
1. $\Phi_+$ satisfies Hypothesis 2.3.
2. The form ${{\mathbb B}}_+$ is non-degenerate and has a dual free pair of bases, so that ${{\mathcal U}}_{\epsilon}^{\geq 0}$ is a free Frobenius extension of $Z_+ .$
3. The corresponding Nakayama automorphism $\nu_+$ of ${{\mathcal U}}_{\epsilon}^{\geq 0}$ is given by $\nu_+(E_\alpha)=E_\alpha$ for simple roots $\alpha$ and $\nu_+(K_{\lambda})=\epsilon^{(2\rho,{\lambda})}K_{\lambda}$ for ${\lambda}\in P$.
The proofs of 1.) and 2.) are similar to, but easier than the corresponding arguments for ${{\mathcal U}}_\epsilon({\mathfrak{g}})$, so we leave the details to the reader.
Consider now part 3.). As in the proof of Lemma \[NakU\], $\nu_+(E_\alpha)=E_\alpha$ for any simple root $\alpha$. By the degree argument from the same proof, the value of $\nu_+(K_{\lambda})$ is determined by $E^\mathbf{l}K_{\lambda}=\nu_+(K_{\lambda})E^{\mathbf{l}}$. Hence $$\nu_+(K_{\lambda})=\epsilon^{-(l-1)(\beta_1+\cdots\beta_N,{\lambda})}K_{\lambda}=\epsilon^{(2\rho,{\lambda})}K_{\lambda}.$$ The result follows.
\[winding\]
1\. With the standard comultiplication of [@BGo I.6], [@Jquant Chapter 4], $E_{\alpha}\mapsto E_\alpha\otimes
1+K_\alpha\otimes E_\alpha$, $F_{\alpha}\mapsto F_\alpha\otimes
K_\alpha^{-1}+1\otimes F_\alpha$, $K_{\lambda}\mapsto K_{\lambda}\otimes K_{\lambda}$ for $\alpha\in\pi$, ${\lambda}\in P$, then $\nu_+$ is nothing else than the right winding automorphism [@BGo I.9.25] $\tau^r_{2\rho}$ of ${{\mathcal U}}_{\epsilon}(\mathfrak{g})$ associated with the representation $2\rho,$ restricted to ${{\mathcal U}}_{\epsilon}^{\geq 0}$.
2\. Calculations parallel to the above will of course handle ${{\mathcal U}}_{\epsilon}^{\leq 0},$ the Hopf subalgebra of ${{\mathcal U}}_\epsilon({\mathfrak{g}})$ generated by the $F_{\alpha}$s and the $K_{\lambda}$s. A more elegant approach is to make use of the Chevalley involution $\omega$ [@Jquant Lemma 4.6(a)]
: $\omega(E_{\alpha}) = F_{\alpha}$ and $\omega(K_i) = K_{i}^{-1},$ so $\omega$ is an algebra automorphism and a coalgebra anti-automorphism. Thus one calculates that the Nakayama automorphism of ${{\mathcal U}}_{\epsilon}^{\leq 0},$ namely $\omega \circ \tau^r_{2\rho} \circ \omega^{-1},$ is the restriction of the automorphism $\tau^{\ell}_{-2\rho}$ of ${{\mathcal U}}_{\epsilon}(\mathfrak{g}).$
Quantised function algebras {#qfun}
===========================
In this section we show that the quantised function algebra ${{\mathcal O}}_{\epsilon}[G]$ at a root of unity $\epsilon$ is a Frobenius extension of its Hopf centre, with non-trivial Nakayama automorphism, so that the reduced quantised function algebras ${{\mathcal O}}_{\epsilon}[G](g)$ are Frobenius but not, in general, symmetric.
Preliminaries {#prefun}
-------------
Let $G$ be the simply connected, semisimple algebraic group over ${\mathbb{C}}$ associated with the semisimple Lie algebra ${\mathfrak{g}}$. Let $B$ be the Borel subalgebra of $G$ associated with $\pi$ and let $B^-$ be the opposite Borel. Let $T$ be the corresponding maximal torus. let $\epsilon$ be as in (6.1), and let ${{\mathcal O}}_\epsilon[G]$ be the quantised function algebra of $G$ at the root of unity $\epsilon.$ For the definition and basic properties of ${{\mathcal O}}_\epsilon [G]$, see [@CL] or [@BGo III.7.1].[Note, however, that the algebra in [@CL] is the *opposite algebra* to that in [@BGo]; put in another way, there is a switch between $\epsilon$ and $\epsilon^{-1}$ in going from [@BGo III.7.1] to [@CL].]{} Recall that de Concini and Lyubashenko show [@CL] that ${{\mathcal O}}_\epsilon [G]$ is a noetherian Hopf ${\mathbb{C}}$-algebra which is a finitely generated module over its centre. (An outline proof is also provided in [@BGo Theorem III.7.2, III.7.3].) Indeed, more specifically, ${{\mathcal O}}_\epsilon [G]$ contains a copy of the coordinate ring of G, ${{\mathcal O}}[G]$, as a central Hopf subalgebra, and, by [@IK Proposition 2.2], ${{\mathcal O}}_\epsilon [G]$ is a free ${{\mathcal O}}[G]$-module of rank $l^{\mathrm{dim}G}$.
Calculations with ${{\mathcal O}}_\epsilon[G]$ are most easily carried out by embedding it as a subalgebra of ${{\mathcal U}}_{\epsilon}^{\leq 0} \otimes {{\mathcal U}}_{\epsilon}^{\geq 0}$, as in [@CL Section 4.3]. But in fact [@CL] works with $({{\mathcal O}}_\epsilon [G])^{op}$, in terms of the definition of the function algebra of [@BGo] or [@Jquant]; the simplest way to accommodate this here is to include a map from $\epsilon$ to $\epsilon^{-1}$ into the embedding. Once this is done, the inclusion $\mu''$ of [@CL 4.3] is given by the composite $$\begin{aligned}
i': {{\mathcal O}}_\epsilon[G]\stackrel{\text{comult}}\longrightarrow
{{\mathcal O}}_\epsilon[G]\otimes {{\mathcal O}}_\epsilon[G]\rightarrow
{{\mathcal O}}_\epsilon[B]\otimes {{\mathcal O}}_\epsilon[B^-]\longrightarrow {{\mathcal U}}_{\epsilon^{-1}}^{\leq 0} \otimes_{\mathbb{C}}{{\mathcal U}}_{\epsilon^{-1}}^{\geq 0},
\end{aligned}$$ where the second map is the canonical one (given by “restriction”) and the last map combines the isomorphism from [@CL Lemma 3.4] with the parameter switch explained above. Note in passing that this embedding shows that ${{\mathcal O}}_\epsilon [G]$ is a domain. Moreover, by [@CL Theorem 4.6, Lemma 4.3 and Proposition 6.5], there is a nonzero element $z$ of ${{\mathcal O}}[G]$, such that $i'$ extends to an inclusion $$\begin{aligned}
i: {{\mathcal O}}_\epsilon[G][z^{-1}]\longrightarrow {{\mathcal U}}_{\epsilon^{-1}}^{\leq 0}\otimes_{\mathbb{C}}{{\mathcal U}}_{\epsilon^{-1}}^{\geq 0},\end{aligned}$$ with image generated by the elements $1\otimes E_\alpha$, $F_\alpha\otimes 1$ and $K_{-{\lambda}}\otimes K_{{\lambda}}$, for simple roots $\alpha$ and integral weights ${\lambda}$. In the following we will often identify ${{\mathcal O}}_\epsilon[G][z^{-1}]$ with its image under $i$.
In particular, making this identification, a basis ${{\mathcal B}}_{{\mathcal O}}$ of ${{\mathcal O}}_\epsilon [G][z^{-1}]$ as a free ${{\mathcal O}}[G][z^{-1}]$-module is given by the set of elements $$\begin{aligned}
F^{\bf k} K_{-{\lambda}}\otimes K_{\lambda}E^{\bf m},\end{aligned}$$ where $0\leq k_i, m_i< l$ and the coefficients of ${\lambda}$ in terms of fundamental weights are non-negative integers less than $l;$ for this, see the proof of [@CL Proposition 7.2].
The bilinear form {#5.2}
-----------------
We can define a ${{\mathcal O}}[G][z^{-1}]$-linear map $$\Phi:\quad{{\mathcal O}}_\epsilon[G][z^{-1}]\times {{\mathcal O}}_\epsilon[G][z^{-1}]\rightarrow {{\mathcal O}}[G][z^{-1}]$$ by mapping $$\begin{aligned}
{{\mathcal B}}_{{\mathcal O}}\ni F^{\bf k} K_{-{\lambda}} \otimes K_{\lambda}E^{\bf m}&\longmapsto&
\begin{cases}
1&\text{ if $\mathbf{k}=\mathbf{m}=\mathbf{l}$, ${\lambda}=0$},\\
0&\text{ otherwise,}
\end{cases}
\end{aligned}$$ and extending ${{\mathcal O}}[G][z^{-1}]$-linearly. Since ${{\mathcal O}}[G][z^{-1}]$ is central, we get an associative ${{\mathcal O}}[G][z^{-1}]$-bilinear form ${{\mathbb B}}:{{\mathcal O}}_\epsilon[G][z^{-1}]\times{{\mathcal O}}_\epsilon[G][z^{-1}]\rightarrow
{{\mathcal O}}_\epsilon[G][z^{-1}]$ by putting ${{\mathbb B}}(x,y)=\Phi(xy)$ for $x$, $y\in{{\mathcal O}}_\epsilon[G][z^{-1}]$.
Frobenius extension {#Frobenius}
-------------------
We can now record the key
The functional $\Phi$ satisfies Hypothesis 2.3.
The argument is similar to the ones used to prove Lemma 6.3 and Theorem 7.2, and is therefore left to the reader.
As usual, the above lemma yields at once the first part of the following
1. ${{\mathcal O}}_\epsilon[G][z^{-1}]$ is a free Frobenius extension of ${{\mathcal O}}[G][z^{-1}]$ with the form ${{\mathbb B}}$ defined in Section \[5.2\].
2. In the notation of Remarks 7.2, the Nakayama automorphism of ${{\mathcal O}}_\epsilon[G][z^{-1}]$ is the restriction of the automorphism $\tau^{\ell}_{-2\rho} \otimes \tau^r_{2\rho}$ of ${{\mathcal U}}_{\epsilon^{-1}}^{\leq 0} \otimes_{\mathbb{C}}{{\mathcal U}}_{\epsilon^{-1}}^{\geq 0}.$ In particular, it fixes $F_{\alpha} \otimes 1$ and $1 \otimes E_{\alpha}$ for all simple roots $\alpha$, and maps $K_{\lambda} \otimes K_{-\lambda}$ to $\epsilon^{2(2\rho,\lambda)}K_{\lambda} \otimes K_{-\lambda}$
3. There is a non-degenerate ${{\mathcal O}}[G]$-bilinear form ${{\mathbb B}}'$ on ${{\mathcal O}}_\epsilon[G]$ with values in ${{\mathcal O}}[G]$ and Nakayama automorphism $\nu_{\mathcal{O}} = \tau_{-2\rho}^l \otimes \tau_{2\rho}^r$.
\(2) This is clear from Theorem 7.2 and Remarks 7.2(2).
\(3) Choose a finite generating set $\mathcal{F}$ of ${{\mathcal O}}_\epsilon[G]$ as a ${{\mathcal O}}[G]$-module. There is a non-negative integer $k$ such that ${{\mathbb B}}(u,v) \in z^{-k}{{\mathcal O}}[G]$ for all $u,v \in \mathcal{F}.$ Let $k_0$ be the minimal such integer, and define ${{\mathbb B}}' := z^{k_0} {{\mathbb B}}.$ Then ${{\mathbb B}}'$ has the stated properties.
Suppose that $G = SL(n,\mathbb{C}),$ so that $\mathcal{O}_{\epsilon}[G]$ is generated by $\{X_{ij} : 1 \leq i,j
\leq n \}$, with the relations given at [@BGo I.2.2,I.2.4]. Then it is easy to calculate that the automorphism $\nu_{\mathcal{O}}$ of the theorem is given by $\nu_{\mathcal{O}}(X_{ij}) \epsilon^{2(n+1-i-j)}X_{ij},$ for $i,j = 1, \ldots , n.$
Finite dimensional factors {#factors}
--------------------------
Corollary \[Frobenius\] is sufficient to yield the desired applications to the finite dimensional representation theory of ${{\mathcal O}}_\epsilon[G],$ as follows:
Let $g \in G$ and let $\mathfrak{m}_g$ be the corresponding maximal ideal of ${{\mathcal O}}[G]$. Then the algebra ${{\mathcal O}}_\epsilon[G](g):={{\mathcal O}}_\epsilon[G]/{{\mathcal O}}_\epsilon[G]\mathfrak{m}_g$ is a Frobenius algebra with Nakayama automorphism induced from $\nu_{\mathcal{O}}.$
First let $\mathfrak{m}$ be a maximal ideal of the algebra ${{\mathcal O}}[G][z^{-1}]$ of Proposition \[Frobenius\]. Then Proposition \[Frobenius\] implies that there is a non-degenerate ${\mathbb{C}}$-bilinear form $\overline{{{\mathbb B}}}$ on ${{\mathcal O}}_\epsilon[G][z^{-1}]/\mathfrak{m}{{\mathcal O}}_\epsilon[G][z^{-1}],$ with Nakayama automorphism induced also from $\nu_{\mathcal{O}}.$
Now suppose that $z$ is not in $\mathfrak{m}_g$. Then $$\begin{aligned}
{{\mathcal O}}_\epsilon[G][z^{-1}]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G][z^{-1}] \cong
({{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G])[z^{-1}]
= {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G],\end{aligned}$$ using [@GW Exercise 9L] for the isomorphism, and the fact that $z$ is a unit modulo $\mathfrak{m}_g{{\mathcal O}}_\epsilon[G]$ for the equality. In particular, by the first paragraph of the proof, $$\begin{aligned}
\label{conc}
\textit{the desired conclusions apply to } {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G].\end{aligned}$$
To extend this conclusion to arbitrary $g$ in $G$ we apply the results of [@CL]. Recall that there is a Poisson bracket on $\mathcal{O}[G]$, under which $G$ decomposes as a disjoint union of symplectic leaves. Moreover, if $g,h \in G$ belong to the same symplectic leaf, then $$\begin{aligned}
\label{symp} {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G] \cong
{{\mathcal O}}_\epsilon[G]/\mathfrak{m}_h{{\mathcal O}}_\epsilon[G]\end{aligned}$$ by [@CL Corollary 9.4]. In fact, ${{\mathcal O}}_\epsilon[G]$ is a Poisson ${{\mathcal O}}[G]$-order in the sense of [@IKPoisson 2.1], and we can, if preferred, quote [@IKPoisson Theorem 4.2] to obtain (\[symp\]). By [@CL Proposition 9.3 and Proposition 8.7 (b)] there is an action of the torus $T$ as automorphisms of ${{\mathcal O}}_\epsilon[G],$ restricting to Poisson automorphisms of the subalgebra ${{\mathcal O}}[G]$ induced by right and left multiplication by $T$ on $G$, preserving the Poisson order structure in the sense of [@IKPoisson 3.8]. Therefore, if $g \in G$ and $t \in T$, then $$\begin{aligned}
\label{wind} {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G] \cong {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_{tg}{{\mathcal O}}_\epsilon[G]
\cong {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_{gt}{{\mathcal O}}_\epsilon[G].\end{aligned}$$ Since the action of $T$ preserves the leaves we can conclude from (\[symp\]) and (\[wind\]) that $$\begin{aligned}
\label{comb} {{\mathcal O}}_\epsilon[G]/\mathfrak{m}_g{{\mathcal O}}_\epsilon[G] \cong
{{\mathcal O}}_\epsilon[G]/\mathfrak{m}_h{{\mathcal O}}_\epsilon[G]\end{aligned}$$ if $g$ and $h$ are in the same $T$-orbit of symplectic leaves [@CL Corollary 9.4], [@IKPoisson 4.2 and 4.3].
Recall from [@HL Theorem A.3.2 and Theorem A.2.1] (see also [@CL Section 9.3]) that the $T$-orbits of symplectic leaves are indexed by the elements of $W\times W$, where $W$ is the Weyl group of $G.$ To be precise, they are the double Bruhat cells $$\begin{aligned}
\label{HL}
X_{w_1,w_2}:=B\dot{w_1}B\cap B^-\dot{w_2}B^-,
\end{aligned}$$ where $\dot{w_1}$, $\dot{w_2}$ are chosen from the normaliser $N_G(T)$ to represent $w_1, w_2\in W$.
Note that the localisation with respect to $z$ corresponds exactly to the localisation over the big cell $BB^-$, as explained in [@CL proof of Theorem 7.2]. In view of (\[conc\]) and (\[comb\]) it is therefore enough to show that every $T$-orbit of leaves in $G$ has non-empty intersection with the big cell. That is, by (\[HL\]), we must check that every double Bruhat cell $X_{w_1,w_2}$ has non-empty intersection with the big cell. This is easy to verify as follows: Consider the double Bruhat cells $X_{w_1,e}=Bw_1B\cap B^-$ and $X_{e,
w_2}=B\cap B^-w_2 B^-$. Let $a\in X_{w_1,e}$ and $b\in X_{e,
w_2}.$ Then $ab\in B^-B\cap Bw_1B \cap B^-w_2 B^-\subseteq B^-B\cap X_{w_1,w_2}$.
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank Ami Braun for telling us about the results in [@Braun] and for supplying us with a preliminary version of his paper. The third author acknowledges the support of the EPSRC grant number GR/S14900/01. All of us benefited from the support of Leverhulme research Interchange F/00158/X (UK).
|
---
abstract: |
Convex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an important question is thus whether such regularization techniques preserve the properties of the underlying probability measures. We focus here on a case which has produced a very lively debate in the cosmological literature, namely Gaussian and isotropic spherical random fields, and we prove that Gaussianity and isotropy are not conserved in general under convex regularization over a Fourier dictionary, such as the orthonormal system of spherical harmonics.
**Keywords and Phrases**: Random Fields, Spherical Harmonics, Gaussianity, Isotropy, Convex Regularization, Inpainting.
**AMS Classification**: 60G60; 33C53, 43A90
author:
- |
Valentina Cammarota and Domenico Marinucci\
Department of Mathematics, University of Rome Tor Vergata
title: 'The Stochastic Properties of $\ell ^{1}$-Regularized Spherical Gaussian Fields'
---
Introduction
============
Let $T:M\rightarrow \mathbb{R}$ be a square integrable function on a manifold $M$, and assume that the following is observed:$$T^{obs}:=\mathcal{A}T+n\text{ ,} \label{invprob}$$where $\mathcal{A}:L^{2}(M)\rightarrow $ $L^{2}(M)$ is a linear operator that can represent, for instance, a blurring convolution or a mask setting some values of the function $T$ to zero, while $n:M\rightarrow \mathbb{R}$ denotes observational noise. Recovering $T$ from observations on $T^{obs}$ is a standard example of a linear inverse problem, and it is now classical to pursue a solution for (\[invprob\]) by means of convex/$\ell ^{1}$-regularization procedures. More precisely, we can proceed by postulating that the signal $T$ can be sparsely represented in a given dictionary $\Psi
, $ e.g., $T=\Psi \alpha _{0}$ where the vector $\alpha _{0}$ is assumed to be sparse in a suitable sense, and then solving the $\ell ^{1}$-regularized problem$$\alpha ^{reg}:=\arg \min_{\alpha }\left\{ \lambda \left\Vert \alpha
\right\Vert _{\ell ^{1}}+\frac{1}{2}\left\Vert T^{obs}-\mathcal{A}\Psi
\alpha \right\Vert _{L^{2}(S^{2})}^{2}\right\} \text{ ,} \label{min1}$$which can be viewed for instance as a form of Basis Pursuit Denoising [chen]{} or a variation of the Lasso algorithm introduced in the statistical literature by [@tibshirani]. Often the following alternative formulation is considered:$$\alpha ^{reg}:=\arg \min_{\alpha }\left\{ \left\Vert \alpha \right\Vert
_{\ell ^{1}}\right\} \text{ subject to }\left\Vert T^{obs}-\mathcal{A}\Psi
\alpha \right\Vert _{L^{2}(S^{2})}\leq \varepsilon \text{ }, \label{min2}$$for some $\varepsilon >0;$ it is known that there exist a bijection $\lambda
\leftrightarrow \varepsilon $ such that (\[min1\]) and (\[min2\]) have the same solution [@StarckBook]. Many authors have worked on related regularization problems over the last two decades - a very incomplete list includes [@osborne], [@daubechies], [@efron], [@fornasier], [@mcewen], [@wright], see for instance [@StarckBook], Chapter 7 for more references and a global overview. These results are also connected to the rapidly growing literature on compressive sensing, see, e.g., [donoho,baraniuk,Candes,rauhut,rauhut2]{}.
In many applied fields, it is customary to view $T$ as the realization of a random field, and the reconstruction problems (\[min1\]) and (\[min2\]) are usually just the first steps before statistical data analysis (e.g., estimation and testing) is implemented. In other words, $T$ is viewed as a random object on a probability space ($\Omega ,\Im ,P)$, $T(\omega
,x):=T:\Omega \times M\rightarrow \mathbb{R}$; hence it becomes important to verify that $T^{reg}:=\Psi \alpha ^{reg}$, $T^{reg}:\Omega \times
M\rightarrow \mathbb{R}$, is close to $T$ in a meaningful probabilistic sense. For instance, let $M$ be a homogeneous space of a compact group $\mathcal{G};$ a natural question is the following:
\[problem1\] Assume that the field $T$ is Gaussian and isotropic, e.g., the probability laws of $T(.)$ and $T^{g}(.)=T(g.)$ are the same for all $g\in \mathcal{G}.$ Is the random field $T^{reg}$ Gaussian and isotropic?
The scenario we have described fits very well, for instance, the current situation in the Cosmological literature, in particular in the field of Cosmic Microwave Background (CMB) data analysis. The latter can be viewed as a snapshot picture of the Universe at the so-called age of recombination, e.g. $3.7\times 10^{5}$ years after the Big Bang (some 13 billion years ago); its observation has been made possible by satellite experiments such as WMAP [@WMAP] and Planck [@Planck], which have raised an enormous amount of theoretical and applied interest. CMB is usually viewed as a single realization of a Gaussian isotropic random field on the sphere, e.g., $M=S^{2}$ and $\mathcal{G=}SO(3),$ the group of rotations in $\mathbb{R}^{3}$; observations are corrupted by observational noise and various forms of convolutions (e.g., instrumental beams, masked regions) and a number of efforts have been devoted to solving (\[invprob\]) under these circumstances. In this setting, algorithms such as (\[min1\]) and ([min2]{}) have been widely proposed, in some cases (see e.g., [@abrial1], [@dupe; @StarckBook; @Starck1] and the references therein) taking as a dictionary the orthonormal system of spherical harmonics $\left\{ Y_{\ell
m}\right\} .$ As well-known, the latter are eigenfunctions of the spherical Laplacians $\Delta _{S^{2}}Y_{\ell m}=-\ell (\ell +1)Y_{\ell m}$ and lead to the spectral representation$$T(x)=\sum_{\ell =0}^{\infty }T_{\ell }(x)=\sum_{\ell =0}^{\infty
}\sum_{m=-\ell }^{\ell }a_{\ell m}Y_{\ell m}(x)\text{ .}$$Under Gaussianity and isotropy, this representation holds in the mean square sense and the random coefficients are Gaussian and independent with variance $Ea_{\ell m}\overline{a}_{\ell ^{\prime }m^{\prime }}=C_{\ell }\delta _{\ell
}^{\ell ^{\prime }}\delta _{m}^{m^{\prime }},$ the sequence $\left\{ C_{\ell
}\right\} $ representing the angular power spectrum (see for instance [MarPecBook]{}). A very lively debate has then developed, to ascertain whether in this setting the solution to the issue raised in Problem (\[problem1\]) should allow for a positive or negative answer, see for instance [Starck1]{}, [@StarckBayes] and the references therein. In particular, the recent paper [@feeney] provides from an astrophysical perspective some arguments and a large amount of numerical evidence to suggest that isotropy will not hold in general.
The purpose of this paper is to address this question from a mathematical point of view. To this aim, we will focus on idealistic circumstances where $\mathcal{A}$ is just the identity operator and noise $n$ is set identically to zero, so that $T$ and $T^{obs}$ coincide. Of course, under these circumstances the inverse problem would not really arise: however for our aims these assumptions suffice, as we will show that even in this idealistic setting stochastic properties such as Gaussianity and isotropy are not preserved by regularization according to (\[min1\]) or (\[min2\]).
Statement of the main results
-----------------------------
To establish our results, we shall first reformulate (\[min1\]) and ([min2]{}) in a form which is more directly amenable to stochastic analysis; in particular, we shall show that:
\[equivalence\] Let $T$ be a Gaussian isotropic spherical random field, and denote by $\Psi $ the spherical harmonic dictionary. Then for any given $\delta ,\varepsilon >0,$ there exist a positive $\lambda =\lambda (\delta
,\varepsilon )$ such that the solution$$\alpha ^{reg}:=\arg \min_{\alpha }\left\{ \lambda (\delta ,\varepsilon
)\left\Vert \alpha \right\Vert _{\ell ^{1}}+\frac{1}{2}\left\Vert T-\Psi
\alpha \right\Vert _{L^{2}(S^{2})}^{2}\right\} \text{ } \label{min4}$$satisfies $$\Pr \left\{ \left\Vert T-\Psi \alpha ^{reg}\right\Vert _{L^{2}(S^{2})}\leq
\varepsilon \right\} \geq 1-\delta \text{ }. \label{min3}$$
The previous result is stating that for a suitable choice of $\lambda $ the solution to (\[min1\]) satisfies the constraint in (\[min2\]) with probability arbitrarily close to one, so that the two problems can be seen as substantially equivalent in a stochastic setting. Let us now write$$T_{\delta ,\varepsilon }^{reg}(x):=\sum_{\ell m}a_{\ell m}^{reg}(\delta
,\varepsilon )Y_{\ell m}(x)=\sum_{\ell }T_{\ell ;\delta ,\varepsilon
}^{reg}(x)\text{ .}$$The main claim of this paper is the following
\[theo-aniso\] The random fields $T_{\delta ,\varepsilon }^{reg}(.)$ are necessarily anisotropic and nonGaussian, for any (arbitrarily small but positive) values of $\delta ,\varepsilon .$
To make this claim more concrete, we shall also focus on the normalized fourth-moment$$\kappa _{\ell }(\theta ,\phi ):=\frac{E\{T_{\ell }^{reg}(\theta ,\phi )^{4}\}}{(E\{T_{\ell }^{reg}(0,0)^{2}\})^{2}}\text{ ,}$$which of course should be constant for all $(\theta ,\phi )\in S^{2}$ under isotropy, and identically equal to 3 under Gaussianity. On the contrary, we will provide an analytic expression for the value of $\kappa _{\ell }(\theta
,\phi )$ at the North Pole $N:(\theta ,\phi )=(0,0)$, as a function of the angular power spectrum $C_{\ell }$ and the penalization parameter $\lambda
(\delta ,\varepsilon ).$ In particular, for the so-called complex-valued regularization procedure (to be defined below), we shall show that
\[theo-trisp1\] As $\lambda/\sqrt{C_\ell} \to \infty$, we have $$\lim_{\lambda/\sqrt{C_\ell} \to \infty} \frac{\log k_\ell(0,0)}{\lambda^2/C_{\ell}}=1.$$
Because the sequence $C_{\ell }$ is summable, the previous result entails that the kurtosis of the field diverges exponentially at the North Pole as $\ell \rightarrow \infty $, showing an extremely nonGaussian behavior at high frequencies. Under the same setting, we shall show that
\[theo-trisp2\] As $\lambda /\sqrt{C_{\ell }}\rightarrow \infty ,$ we have that$$\lim_{\ell \rightarrow \infty }\frac{\kappa _{\ell }(\theta ,\phi )}{\kappa
_{\ell }(0,0)P_{\ell }^{4}(\cos \theta )}=1\text{ , for all }(\theta ,\phi
)\in S^{2}\text{ ,}$$$P_{\ell }(.)$ denoting the usual Legendre polynomial.
The latter result entails that the so-called trispectrum of the random field is not constant over the sphere, as required by isotropy, but it rather exhibits anisotropic oscillations. Under the so-called real-norm regularization procedure (to be defined later), the asymptotic behavior is slightly different, but anisotropy remains and the oscillations of the trispectrum can again be predicted analytically, see below.
One heuristic intuition behind these results can be summarized as follows. To understand the relationship between convex regularization and isotropy, it can be convenient to view a problem like (\[min1\]) as resulting from the maximization of a Bayesian posterior distribution on the spherical harmonic coefficients $a_{\ell m}$, assuming a Laplacian/Exponential prior on these coefficients. We can now recall some earlier results from [BaMa]{} (see also [@BMV; @MarPecBook; @BT]), showing that a random field generated by sampling such independent non-Gaussian coefficients is necessarily anisotropic; it can then be natural to conjecture that this implicit anisotropy in the prior fields will persist in the regularized maps. However, while this interpretation led us to conjecture the results of this paper, it should be noted that it plays no role in the arguments that follow. We refer again to [@feeney] for further discussion on these issues and for a large set of numerical results.
The plan of the paper is as follows: in Section \[regularized\], we discuss regularized estimates in a stochastic setting, and we establish Proposition \[equivalence\]; in Section \[anisotropy\], we prove that regularized fields with the spherical harmonics dictionary are necessarily anisotropic and nonGaussian, while in Section \[trispectra\] the trispectra and their asymptotic behavior are studied. Some final remarks are collected in Section \[conclusions\].
Acknowledgements
----------------
We thank Stephen Feeney, Jason McEwen, Hiranya Peiris, Jean-Luc Starck and Benjamin Wandelt for useful and lively discussions. This research is supported by the European Research Council under the European Community Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 277742 *Pascal*.
$\ell ^{1}-$Regularized Random Fields \[regularized\]
=====================================================
As motivated in the Introduction, we wish to consider the $\ell _{1}$ minimization problem $$\left\{ a_{\ell m}^{reg}\right\} =\arg \min_{{\{a_{\ell m}\}}}\left\{
\lambda \sum_{\ell m}|a_{\ell m}|+\frac{1}{2}\left\Vert T^{obs}-\sum_{\ell
m}a_{\ell m}Y_{\ell m}\right\Vert _{L^{2}{(S^{2})}}^{{2}}\right\} \text{ ,}
\label{minprob0}$$where as usual$$T^{obs}=\sum_{\ell m}a_{\ell m}^{obs}Y_{\ell m}\text{ .} \label{specrap}$$and [ $\left\Vert .\right\Vert _{L^{2}(S^{2})}$ denotes the $L^{2}$ -norm for functions on the sphere , e.g., $$\left\Vert T^{obs}-\sum_{\ell m}a_{\ell m}Y_{\ell m}\right\Vert
_{L^{2}(S^{2})}^{2}=\int_{S^{2}}\left\vert T^{obs}-\sum_{\ell m}a_{\ell
m}Y_{\ell m}\right\vert ^{2}dx=\sum_{\ell m}\left\vert a_{\ell
m}^{obs}-a_{\ell m}\right\vert ^{2}\text{ .}$$]{} In equations (\[minprob0\]) and (\[specrap\]), we take as [usual]{} $\left\{ Y_{\ell m}\right\} $ to denote complex-valued spherical harmonics, so that $Y_{\ell m}=(-1)^{m}\overline{Y}_{\ell {,-m}},$ the bar denoting complex conjugations, and $\left\vert .\right\vert $ the complex modulus $|a_{\ell m}|:=\sqrt{[{\func{Re}}(a_{\ell m})]^{2}+[{\func{Im}}(a_{\ell
m})]^{2}};$ we label this case as the *complex-valued regularization scheme*. As an alternative, an orthonormal expansion into a real-valued basis can be obtained by simply taking $$T^{obs}=\sum_{\ell m}a_{\ell m}^{obs;\mathcal{R}}Y_{\ell m}^{\mathcal{R}}\text{ ,} \label{specrap_real}$$where [ $a_{\ell 0}^{obs;\mathcal{R}}=a_{\ell 0}^{obs}$, $Y_{\ell 0}^{\mathcal{R}}=Y_{\ell 0}$, ]{} $$a_{\ell m}^{obs;\mathcal{R}}=\sqrt{2}{\func{Re}}({a_{\ell m}^{obs}})\text{
for }{m>0}\text{ , }a_{\ell m}^{\mathcal{R}}={-}\sqrt{2}{\func{Im}}({a_{\ell
,-m}^{obs}})\text{ for }m<0\text{ ,}$$and$$Y_{\ell m}^{\mathcal{R}}=\sqrt{2}{\func{Re}}(Y_{\ell m})\text{ for }{m>0}\text{ , }Y_{\ell m}^{\mathcal{R}}=\sqrt{2}{\func{Im}}(Y_{\ell {,-m}})\text{
for }m<0\text{ .}$$We are then led to the *real-valued regularization scheme*$$\left\{ a_{\ell m}^{reg\ast }\right\} =\arg \min_{{\left\{ a_{\ell
m}\right\} }}\left\{ \lambda \sum_{\ell m}\left\vert a_{\ell m}^{\mathcal{R}}\right\vert +\frac{1}{2}\sum_{\ell m}\left\vert a_{\ell m}^{obs;\mathcal{R}}-a_{\ell m}^{\mathcal{R}}\right\vert ^{2}\right\} \text{ ,}
\label{minprob0_real}$$where $\left\vert .\right\vert $ is standard absolute value for real numbers. We shall consider both schemes in what follows.
The following two lemmas are standard, but nevertheless we report their straightforward proofs for completeness. We shall use below the standard polar coordinates for complex-valued random variables$$a_{\ell m}=\rho _{\ell m}\exp (i\psi _{\ell m})\text{ ,}$$$$\rho _{\ell m}:=\sqrt{[({\func{Re}}(a_{\ell m})]^{2}+[\func{Im}(a_{\ell
m})]^{2}}\text{ , }\psi _{\ell m}:=\arctan \frac{{\func{Re}}(a_{\ell m})}{{\func{Im}}(a_{\ell m})}\text{ .}$$Also, we denote by $\left\vert x\right\vert _{+}$ the positive part of the real number $x.$
If $T^{obs}$ is Gaussian and isotropic, we have that for $m\neq 0,$ $a_{\ell
m}^{obs}=\rho _{\ell m}^{obs}\exp (i\psi _{\ell m}^{obs}),$ where $\psi
_{\ell m}^{obs}\sim U[0,2\pi ]$ and the density of ${\rho _{\ell m}^{obs}}$ is given by$$\Pr \left\{ {\rho _{\ell m}^{obs}} \leq R\right\} =\int_{0}^{R}f_{\rho ;\ell
}(r)dr\text{ ; }f_{\rho ;\ell }(r)=2\frac{r}{C_{\ell }}\exp\{-\frac{r^{2}}{C_{\ell }}\}\text{ .}$$
It suffices to notice that$$\Pr \left\{ {\rho _{\ell m}^{obs}}\leq R\right\} =\Pr \left\{ {(\rho _{\ell
m}^{obs})}^{2}\leq R^{2}\right\}$$$$=\Pr \left\{ \frac{1}{2}\frac{[{\func{Re}}({a_{\ell m}^{obs}})]^{2}+[{\func{Im}}({a_{\ell m}^{obs}})]^{2}}{C_{\ell }/2}\leq \frac{R^{2}}{C_{\ell }}\right\}$$$$=\Pr \left\{ \frac{\chi _{2}^{2}}{2}\leq \frac{R^{2}}{C_{\ell }}\right\}
=1-\exp (\frac{R^{2}}{C_{\ell }})\text{ ,}$$[ where $\chi _{2}^{2}$ is a Chi-squared with 2 degrees of freedom.]{} Whence the result follows from differentiation.
The solution to (\[minprob0\]) is provided by$$a_{\ell m}^{reg}:={\func{Re}}(a_{\ell m}^{reg})+i{\func{Im}}(a_{\ell
m}^{reg})\text{ ,}$$where, for $\ell =1,...,\ell _{\max }$ and $m=-\ell ,...,\ell $ we have$${\func{Re}}(a_{\ell m}^{reg})=\left\vert \rho _{\ell m}^{obs}-\lambda
\right\vert _{+}\cos \psi _{\ell m}^{obs}\text{ , }{\func{Im}}(a_{\ell
m}^{reg})=\left\vert \rho _{\ell m}^{obs}-\lambda \right\vert _{+}\sin \psi
_{\ell m}^{obs}\text{ .}$$
We can rewrite (\[minprob0\]) as $$\begin{aligned}
\left\{ a_{\ell m}^{reg}\right\} &=&\arg \min_{{\left\{ a_{\ell m}\right\} }}\left\{ \lambda \sum_{\ell m}\left\vert a_{\ell m}\right\vert +\frac{1}{2}\left\Vert T^{obs}-\sum_{\ell m}a_{\ell m}Y_{\ell m}\right\Vert
_{L^{2}(S^{2})}^{2}\right\} \notag \\
&=&\arg \min_{{\left\{ a_{\ell m}\right\} }}\left\{ \lambda \sum_{\ell
m}\left\vert a_{\ell m}\right\vert +\frac{1}{2}\sum_{\ell m}\left\vert
a_{\ell m}^{obs}-a_{\ell m}\right\vert ^{2}\right\} \text{.}
\label{jointinp}\end{aligned}$$We can hence rewrite$$\begin{aligned}
&\frac{1}{2}\sum_{\ell m}\left\vert a_{\ell m}^{obs}-a_{\ell m}\right\vert
^{2}+\lambda \sum_{\ell m}\left\vert a_{\ell m}\right\vert \\
&=\frac{1}{2}\sum_{\ell m}\left\{ [{\func{Re}}(a_{\ell m}^{obs})-{\func{Re}}(a_{\ell m})]^{2}+[{\func{Im}}(a_{\ell m}^{obs})-\func{Im}(a_{\ell
m})]^{2}\right\} +\lambda \sum_{\ell m}\left\vert a_{\ell m}\right\vert \\
&=\sum_{\ell m}v_{\ell m}\text{ ,}\end{aligned}$$where$$\begin{aligned}
v_{\ell m}&=\frac{1}{2}(\rho _{\ell m}^{obs})^{2}\cos ^{2}\psi _{\ell
m}^{obs}+\frac{1}{2}\rho _{\ell m}^{2}\cos ^{2}\psi _{\ell m}-\rho _{\ell
m}^{obs}\rho _{\ell m}\cos \psi _{\ell m}^{obs}\cos \psi _{\ell m} \\
&+\frac{1}{2}(\rho _{\ell m}^{obs})^{2}\sin ^{2}\psi _{\ell m}^{obs}+\frac{1}{2}\rho _{\ell m}^{2}\sin ^{2}\psi _{\ell m}-\rho _{\ell m}^{obs}\rho
_{\ell m}\sin \psi _{\ell m}^{obs}\sin \psi _{\ell m}+\lambda \rho _{\ell m}
\\
&=\frac{1}{2}(\rho _{\ell m}^{obs})^{2}+\frac{1}{2}\rho _{\ell m}^{2}-\rho
_{\ell m}^{obs}\rho _{\ell m}\cos (\psi _{\ell m}^{obs}-\psi _{\ell
m})+\lambda \rho _{\ell m}\text{ .}\end{aligned}$$It is obvious that for any value of $\rho _{\ell m}^{obs},$ $v_{\ell m}$ is minimized at $\psi _{\ell m}^{obs}=\psi _{\ell m};$ we are then led to the following optimization problem:$$\min_{{\{\rho _{\ell m}\}}}\sum_{\ell m}\phi (\rho _{\ell m}^{obs},\rho
_{\ell m};\lambda )\text{ ,}$$where$$\phi (\rho _{\ell m}^{obs},\rho _{\ell m};\lambda )=\frac{1}{2}(\rho _{\ell
m}^{obs})^{2}+\frac{1}{2}\rho _{\ell m}^{2}-\rho _{\ell m}^{obs}\rho _{\ell
m}+\lambda \rho _{\ell m}\text{ .}$$Now it is standard calculus to show that, for $\lambda >\rho _{\ell m}^{obs}$$$\frac{d\phi }{d\rho _{\ell m}}=\rho _{\ell m}+\lambda -\rho _{\ell m}^{obs}>0\text{ ,}$$while for $\lambda \leq \rho _{\ell m}^{obs}$$$\frac{d\phi }{d\rho _{\ell m}}=\rho _{\ell m}+\lambda -\rho _{\ell
m}^{obs}=0\Longleftrightarrow \rho _{\ell m}=\rho _{\ell m}^{obs}-\lambda
\text{ .}$$The solution now follows immediately, given the global convexity of the function $\phi (.).$
The previous Lemma provides a simple generalization of the very well-known fact that soft-thresholding provides the solution to (\[min1\]) when the dictionary is represented by an orthonormal basis of real valued functions. In particular, for (\[minprob0\_real\]) the solution is immediately seen to be given by $${\ a_{\ell m}^{reg\ast }=sign(a_{\ell m}^{obs;\mathcal{R}})\left\vert
|a_{\ell m}^{obs;\mathcal{R}}|-\lambda \right\vert _{+}\text{ .} }$$It is important to note that the solution for the coefficient corresponding to $m=0$ is exactly the same for both regularization schemes.
The next result shows that, in the simplified circumstances we are considering and for a [suitable]{} choice of the penalization parameter $\lambda $, the reconstruction error can be made arbitrarily small, with probability arbitrarily close to one. For finite variance fields we have $E\{T^{2}\}=\sum_{\ell }\frac{2\ell +1}{4\pi }C_{\ell }<\infty $. To enforce this condition and for notational convenience, in what follows we shall assume that for all $\ell $, $0<C_{\ell }\leq K\ell ^{-\alpha }$, for some $K>0$ and $\alpha >2$. This condition is minimal and fulfilled for instance by all physically relevant models for CMB radiation.
Under the above conditions, for all $\delta ,\varepsilon >0$ there exists [a positive]{} $\lambda =\lambda (\delta ,\varepsilon )$ such that $$T_{{\delta ,\varepsilon }}^{reg}(x)=\sum_{\ell m}a_{\ell m}^{reg}(\delta
,\varepsilon )Y_{\ell m}(x)$$$$\left\{ a_{\ell m}^{reg}(\delta ,\varepsilon )\right\} =\arg \min_{\left\{
a_{\ell m}\right\} }\left\{ {\lambda (\delta ,\varepsilon )}\sum_{\ell
m}\left\vert a_{\ell m}\right\vert +\frac{1}{2}\sum_{\ell m}\left\vert
a_{\ell m}^{obs}-a_{\ell m}\right\vert ^{2}\right\} \text{ }$$and the solution satisfies$$\Pr \left\{ \left\Vert T^{obs}-T_{{\delta ,\varepsilon }}^{reg}\right\Vert
_{L^{2}(S^{2})}<\varepsilon \right\} \geq 1-\delta \text{ .}$$The same result holds when the real-valued regularization scheme is adopted.
Note that $$\begin{aligned}
E\left\Vert T^{obs}-T^{reg}\right\Vert _{L^{2}(S^{2})}^{2}&=\sum_{\ell
m}E|a_{\ell m}^{obs}-a_{\ell m}^{reg}(\lambda )|^{2} \\
&=\sum_{\ell m}E{\{|a_{\ell m}^{obs}|^{2}\mathbb{I(}\left\vert a_{\ell
m}^{obs}\right\vert \leq \lambda )\}}+\lambda ^{2}\sum_{\ell m}E\mathbb{I(}\left\vert a_{\ell m}^{obs}\right\vert >\lambda )\text{ .}\end{aligned}$$Now fix $\ell ^{\ast }$ such that$$\sum_{\ell {>}\ell ^{\ast }}(2\ell +1)C_{\ell }\leq \frac{\varepsilon }{4}\text{ ,}$$and note that$$\sum_{\ell =1}^{\infty }\sum_{m}E{\{|a_{\ell m}^{obs}|^{2}\mathbb{I(}\left\vert a_{\ell m}^{obs}\right\vert \leq \lambda )\}}\leq \sum_{\ell
=1}^{\ell ^{\ast }}\sum_{m}E{\{|a_{\ell m}^{obs}|^{2}\mathbb{I(}\left\vert
a_{\ell m}^{obs}\right\vert \leq \lambda )\}}+\frac{\varepsilon }{4}$$where $$\begin{aligned}
\sum_{\ell =1}^{\ell ^{\ast }}\sum_{m}E{\{|a_{\ell m}^{obs}|^{2}\mathbb{I(}\left\vert a_{\ell m}^{obs}\right\vert \leq \lambda )\}} &\leq &\lambda
^{2}\sum_{\ell =1}^{\ell ^{\ast }}\sum_{m}E\{\mathbb{I(}\left\vert a_{\ell
m}^{obs}\right\vert {\leq }\lambda )\} \\
&{=}&{\lambda ^{2}\sum_{\ell =1}^{\ell ^{\ast }}\big[\Pr \{|a_{\ell
0}^{obs}|\leq \lambda \}+\sum_{m\neq 0}\Pr \{|a_{lm}^{obs}|\leq \lambda \}\big]} \\
&{=}&{\lambda ^{2}\sum_{\ell =1}^{\ell ^{\ast }}\big[\int_{-\lambda
}^{\lambda }\frac{1}{\sqrt{2\pi C_{\ell }}}\exp \{-\frac{u^{2}}{2C_{\ell }}\}du+\sum_{m\neq 0}\int_{0}^{\lambda }\frac{2u}{C_{\ell }}\exp \{-\frac{u^{2}}{C_{\ell }}\}du\big]} \\
&{=}&{\lambda ^{2}\sum_{\ell =1}^{\ell ^{\ast }}\big[\int_{-\lambda
}^{\lambda }\frac{1}{\sqrt{2\pi C_{\ell }}}\exp \{-\frac{u^{2}}{2C_{\ell }}\}du+\sum_{m\neq 0}\big(1-\exp \{-\frac{\lambda ^{2}}{C_{\ell }}\}\big)\big]}
\\
&{\leq }&{\lambda ^{2}\sum_{\ell =1}^{\ell ^{\ast }}\big[\int_{-\lambda
}^{\lambda }\frac{1}{\sqrt{2\pi C_{\ell }}}du+\sum_{m\neq 0}\frac{\sqrt{2}\lambda }{\sqrt{\pi C_{\ell }}}\big]} \\
&\leq &\lambda ^{2}\sum_{\ell =1}^{\ell ^{\ast }}\frac{(2\ell +1)\sqrt{2}\lambda }{\sqrt{\pi C_{\ell }}}\leq {\frac{\lambda ^{3}\sqrt{2}}{\sqrt{\pi {C_{\ell }}^{\ast }}}(2\ell ^{\ast }+{\ell ^{\ast }}^{2})}\leq \frac{\varepsilon }{4}\text{ , }\end{aligned}$$provided that $C_{\ell }^{\ast }:=\min_{\ell =1,..,\ell ^{\ast }}C_{\ell }$ and $${\lambda ^{3}\leq {\frac{\varepsilon \sqrt{\pi C_{\ell }^{\ast }}}{4\sqrt{2}(2{\ell ^{\ast }}+{\ell ^{\ast }}^{2})}}\text{ .}}$$Let $\text{Erfc}$ be the complementary error function defined by $\text{Erfc}(x)=\frac{2}{\sqrt{\pi }}\int_{x}^{\infty }\exp \{-x^{2}\}dx$. Since for $x>0 $, $\text{Erfc}(x)$ is bounded by $\text{Erfc}(x)\leq \frac{2}{\sqrt{\pi
}}\frac{1}{x+\sqrt{x^{2}+\frac{4}{\pi }}}\exp \{-x^{2}\}\leq \exp \{-x^{2}\}$, to bound the second term, we note that $$\begin{aligned}
\lambda ^{2}\sum_{\ell m}E\{\mathbb{I(}\left\vert a_{\ell
m}^{obs}\right\vert >\lambda )\} &=&{\lambda ^{2}\sum_{\ell }\big[\Pr
\{|a_{\ell ,0}^{obs}|>\lambda \}+\sum_{m\neq 0}\Pr \{|a_{\ell
m}^{obs}|>\lambda \}\big]} \\
&=&{\lambda ^{2}\sum_{\ell }\big[2\int_{\lambda }^{\infty }\frac{1}{\sqrt{2\pi C_{\ell }}}\exp \{-\frac{u^{2}}{2C_{\ell }}\}du+\sum_{m\neq
0}\int_{\lambda }^{\infty }\frac{2u}{C_{\ell }}\exp \{-\frac{u^{2}}{C_{\ell }}\}du\big]} \\
&=&\lambda ^{2}\sum_{\ell }\big[\text{Erfc}(\frac{\lambda }{\sqrt{2C_{\ell }}})+\sum_{m\neq 0}\exp \{-\frac{\lambda ^{2}}{C_{\ell }}\}\big] \\
&=&\lambda ^{2}\sum_{\ell }(2\ell +1)\exp \{-\frac{\lambda ^{2}}{2C_{\ell }}\},\end{aligned}$$now, for a fixed $\ell ^{+}>1$, we write $$\lambda ^{2}\sum_{\ell m}E\{\mathbb{I(}\left\vert a_{\ell
m}^{obs}\right\vert >\lambda )\}\leq \lambda ^{2}(2\ell ^{+}+{\ell ^{+}}^{2})+\lambda ^{2}\sum_{\ell >\ell ^{+}}(2\ell +1)\exp \{-\frac{\lambda ^{2}}{2C_{\ell }}\}.$$Here we apply the integral test to the remainder of the series; since $f(\ell )=(2\ell +1)\exp \{-\frac{\lambda ^{2}}{2C_{\ell }}\}$ for $C_{\ell
}\le K \ell ^{-\alpha }$, $\alpha >2$, is a positive and monotonically decreasing function for all $\ell \geq 1$, we have $$\begin{aligned}
\sum_{\ell >\ell ^{+}}(2\ell +1)\exp \{-\frac{\lambda ^{2}\ell ^{\alpha }}{2}\}& \leq \int_{\ell ^{+}}^{\infty }(2x+1)\exp \{-\frac{\lambda ^{2}x^{\alpha
}}{2}\}dx \\
& \leq \int_{0}^{\infty }(2x+1)\exp \{-\frac{\lambda ^{2}x^{2}}{2}\}dx=\frac{4+\lambda \sqrt{2\pi }}{2\lambda ^{2}}\text{ for all }\lambda \geq 0\text{ ,
}\alpha >2\text{ .}\end{aligned}$$Therefore $$\lambda ^{2}\sum_{\ell m}E\{\mathbb{I(}\left\vert a_{\ell
m}^{obs}\right\vert >\lambda )\}\leq \lambda ^{2}(2\ell ^{+}+{\ell ^{+}}^{2})+\lambda ^{2}\frac{4+\lambda \sqrt{2\pi }}{2\lambda ^{2}}\leq \frac{\varepsilon }{2}\text{ ,}$$provided that we take $\lambda $ such that $$\lambda \leq \min \left\{ \sqrt{\frac{\varepsilon }{4(2\ell ^{+}+{\ell ^{+}}^{2})}},\left( \frac{\varepsilon }{2}-4\right) \frac{1}{\sqrt{2\pi }},\sqrt[3]{{\frac{\varepsilon \sqrt{\pi C_{\ell }^{\ast }}}{4\sqrt{2}(2{\ell ^{\ast }}+{\ell ^{\ast }}^{2})}}}\right\} .$$The proof for the real-valued regularization scheme is entirely analogous.
The previous result is straightforward, but it has some important consequenc[e]{}s for the interpretation of the results to follow in the next Sections. In particular, it entails that the presence of nonGaussianity and anisotropy after convex regularization is not due to poor approximation properties of the reconstructed maps. The regularized fields we shall deal with can indeed be viewed as solutions to the optimization problem: for $\delta ,\varepsilon
>0,$$$\{a_{\ell m}^{reg}(\delta ,\varepsilon )\}:=\arg \min_{\{a_{\ell
m}\}}\left\{ \lambda (\delta ,\varepsilon )\sum_{\ell m}\left\vert a_{\ell
m}\right\vert +\sum_{\ell m}\left\vert a_{\ell m}^{obs}-a_{\ell
m}\right\vert^{2} \right\}$$where $\lambda (\delta ,\varepsilon )$ is such that$$\Pr \left\{ \left\Vert T^{obs}-T^{reg}_{ \delta, \varepsilon} \right\Vert
_{L^{2}(S^{2})}>\varepsilon \right\} \leq \delta \text{ .}$$We shall show that even for $T$ Gaussian and $\delta ,\varepsilon $ arbitrary small (but positive), $T^{reg}_{ \delta, \varepsilon}$ exhibits nonGaussian statistics which diverge to infinity at the highest frequencies.
Anisotropy and NonGaussianity \[anisotropy\]
============================================
Let us write as before$$T^{reg}=\sum_{\ell }T_{\ell }^{reg}=\sum_{\ell m}a_{\ell m}^{reg}Y_{\ell m}\text{ , }T^{reg\ast }=\sum_{\ell }T_{\ell }^{reg\ast }=\sum_{\ell m}a_{\ell
m}^{reg\ast }Y_{\ell m}^{\mathcal{R}}\text{ ,}$$e.g., $T^{reg},T^{reg\ast }$ represent, respectively the $\ell ^{1}-$regularized maps under the complex and real-valued optimization schemes. For the discussion to follow, we need to recall briefly the following result:
\[BaldiM\] (See Ref. [@BaMa]) Assume the spherical harmonic coefficients $\left\{ a_{\ell m}\right\} $ of an isotropic random field are independent for $\ell =1,2,...$ and $m=0,1,...,\ell .$ Then they are necessarily Gaussian.
This result was established in [@BaMa], see also [@BMV], [@BT] for extensions to homogeneous spaces of more general compact groups and [MarPecBook]{}, Theorem 6.12 for a proof. An obvious consequence is that a sequence of independent, but nonGaussian, random coefficients $\left\{
a_{\ell m}\right\} $ will necessarily lead to anisotropic random fields.
We are now in the position to state and prove the first result of this paper; here and in what follows, we use $\Phi (x)=(2\pi
)^{-1/2}\int_{-\infty }^{x}\exp \{-\frac{1}{2}u^{2}\}du$ to denote the cumulative distribution function of a standard Gaussian variable.
Let $T^{obs}$ be a Gaussian and isotropic spherical random field. Then the fields $T^{reg},T^{reg\ast }$ are necessarily nonGaussian and anisotropic. In particular, in the complex-valued regularization scheme we have $$\begin{aligned}
E\left\{ a_{\ell 0}^{reg}(\lambda )^{2}\right\}&=\gamma _{0}(\frac{\lambda }{\sqrt{C_{\ell }}}) :=C_{\ell }\left\{ (1+\frac{\lambda ^{2}}{C_{\ell }})2(1-\Phi (\frac{\lambda }{\sqrt{C_{\ell }}}))-\exp (-\frac{\lambda ^{2}}{2C_{\ell }})\frac{\lambda }{\sqrt{C_{\ell }}}\sqrt{\frac{2}{\pi }}\right\}
\text{ ,} \label{gamma0}\end{aligned}$$while for $m\neq 0$$$E\{\left\vert a_{\ell m}^{reg}(\lambda )\right\vert ^{2}\}=\gamma _{1}(\frac{\lambda }{\sqrt{C_{\ell }}}) :=C_{\ell }\left\{ \exp (-\frac{\lambda ^{2}}{C_{\ell }})-\frac{\lambda }{\sqrt{C_{\ell }}}\sqrt{\pi }2(1-\Phi (\frac{\sqrt{2}\lambda }{\sqrt{C_{\ell }}}))\right\} \text{ .} \label{gamma1}$$Moreover, for all $m\neq 0,$ we have that$$-\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{2C_{\ell }}{\lambda ^{2}}\log \frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}}{E\{\left\vert a_{\ell 0}^{reg}\right\vert ^{2}\}}=1\text{ .} \label{ghione3}$$ and$$\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{E\left\{ T_{\ell
}^{reg}(\theta ,\phi )^{2}\right\} }{E\left\{ T_{\ell }^{{reg}}(0,0)^{2}\right\} P_{\ell }^{2}(\cos \theta )}=1\text{ .}$$Finally, in the real-valued regularization scheme$$E\left\{ a_{\ell 0}^{reg\ast }(\lambda )^{2}\right\} =E\{\left\vert a_{\ell
m}^{reg\ast }(\lambda )\right\vert ^{2}\}=\gamma _{0}(\frac{\lambda }{\sqrt{C_{\ell }}})\text{ ,}$$for all $m=-\ell ,...,\ell .$
By assumption, the input coefficients $\left\{ a_{\ell m}\right\} ,$ are Gaussian and independent. The inpainted coefficients can be written $a_{\ell
m}^{reg}=j(a_{\ell m};\lambda ),$ where the function $j(.;\lambda )$ is nonlinear; it follows immediately that they are independent and nonGaussian. Hence the fields $T^{reg},T^{reg\ast }$ are necessarily anisotropic, in view of Theorem \[BaldiM\]. Focussing on $m=0,$ we have in particular$$\Pr \left\{ a_{\ell 0}^{reg}=0\right\} =p_{\ell }(\lambda ):=\int_{-\lambda
}^{\lambda }\frac{1}{\sqrt{2\pi C_{\ell }}}\exp (-\frac{x^{2}}{2C_{\ell }})dx>0$$so that the distribution of $a_{\ell 0}^{reg}=a_{\ell 0}^{reg}(\lambda )$ is given by the mixture$$p_{\ell }(\lambda )\delta _{0}+(1-\frac{p_{\ell }(\lambda )}{2})\Phi
^{+}(.;\lambda ,C_{\ell })+(1-\frac{p_{\ell }(\lambda )}{2})\Phi
^{-}(.;\lambda ,C_{\ell })\text{ ,}$$where $\Phi ^{+}(.;\lambda ,C_{\ell })$ is the distribution [ of a]{} Gaussian random variable with mean $-\lambda $ and conditioned to be positive$,$ and likewise $\Phi ^{-}(.;\lambda ,C_{\ell })$ is the distribution [ of a]{} Gaussian random variable with mean $\lambda $ and conditioned to be negative. It is simple to see that we have$$E\{a_{\ell 0}^{reg}(\lambda )\}=0\text{ ,}$$and$$\begin{aligned}
E\left\{ a_{\ell 0}^{reg}(\lambda )^{2} \right\} &=&\frac{2}{\sqrt{2\pi
C_{\ell }}}\int_{0}^{\infty }y^{2}\exp \left\{ -\frac{(y+\lambda )^{2}}{2C_{\ell }}\right\} dy \\
&=&\frac{2}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }(x-\lambda
)^{2}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx \\
&=&\frac{2}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }(x^{2}-2\lambda
x+\lambda ^{2})\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx \\
&=&\frac{2}{\sqrt{\pi }}C_{\ell }\int_{\lambda }^{\infty }\frac{x}{\sqrt{2C_{\ell }}}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x^{2}}{2C_{\ell }}-\frac{4}{\sqrt{2\pi }}\sqrt{C_{\ell }}\int_{\lambda }^{\infty
}\lambda \exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x^{2}}{2C_{\ell }} \\
&&+\frac{2}{\sqrt{\pi }}\lambda ^{2}\int_{\lambda }^{\infty }\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x}{\sqrt{{\ 2}C_{\ell }}} \\
&=&\frac{2C_{\ell }}{\sqrt{\pi }}\Gamma (\frac{3}{2};\frac{\lambda ^{2}}{2C_{\ell }})-\frac{4\sqrt{C_{\ell }}}{\sqrt{2\pi }}\lambda \exp \left\{ -\frac{\lambda ^{2}}{2C_{\ell }}\right\} +2\lambda ^{2}\left\{ 1-\Phi (\frac{\lambda }{\sqrt{C_{\ell }}})\right\} \\
&=&2C_{\ell }\left\{ \frac{1}{\sqrt{\pi }}\Gamma (\frac{3}{2};\frac{\lambda
^{2}}{2C_{\ell }})-\frac{2}{\sqrt{2\pi }}\frac{\lambda }{\sqrt{C_{\ell }}}\exp \left\{ -\frac{\lambda ^{2}}{2C_{\ell }}\right\} +\frac{\lambda ^{2}}{C_{\ell }}\left\{ 1-\Phi (\frac{\lambda }{\sqrt{C_{\ell }}})\right\}
\right\} \text{ ,}\end{aligned}$$where$$\Gamma (p;c)=\int_{c}^{\infty }x^{p-1}\exp (-x)dx$$denotes the incomplete Gamma function. Now using $$\Gamma (\frac{3}{2};c)=\sqrt{c}e^{-c}+\frac{1}{2}\sqrt{\pi }\; \text{Erfc}
\left( \sqrt{c}\right) \text{ , }\text{Erfc} (u):=2(1-\Phi (\sqrt{2}u))\text{
, } \label{iniez1}$$the previous expression can be further developed to obtain $$\begin{aligned}
E\left\{ a_{\ell 0}^{reg}(\lambda )^{2} \right\} &=2C_{\ell }\left\{ \frac{1}{\sqrt{\pi }} {\ \frac{\lambda}{\sqrt{2 C_\ell}}} \exp\{-\frac{\lambda ^{2}}{2C_{\ell }}\}+\frac{1}{2}\text{Erfc} \left( {\ \frac{\lambda}{\sqrt{2
C_\ell}}} \right) -\frac{2}{\sqrt{2\pi }}\frac{\lambda }{\sqrt{C_{\ell }}}\exp \left\{ -\frac{\lambda ^{2}}{2C_{\ell }}\right\} +\frac{\lambda ^{2}}{C_{\ell }}\left\{ 1-\Phi (\frac{\lambda }{\sqrt{C_{\ell }}})\right\} \right\}
\\
&=2C_{\ell }\left\{ 1-\Phi (\frac{\lambda }{\sqrt{C_{\ell }}})-\sqrt{\frac{1}{2\pi }}\frac{\lambda }{\sqrt{C_{\ell }}}\exp \left\{ -\frac{\lambda ^{2}}{2C_{\ell }}\right\} +\frac{{\lambda ^{2}}}{C_{\ell }}\left\{ 1-\Phi (\frac{\lambda }{\sqrt{C_{\ell }}})\right\} \right\} \\
&=C_{\ell }\left\{ (1+\frac{\lambda ^{2}}{C_{\ell }})2(1-\Phi (\frac{\lambda
}{\sqrt{C_{\ell }}}))-\exp (-\frac{\lambda ^{2}}{2C_{\ell }})\frac{\lambda }{\sqrt{C_{\ell }}}\sqrt{\frac{2}{\pi }}\right\} \text{ .}\end{aligned}$$ It can be checked easily that $\lim_{\lambda \rightarrow 0}E\left\{ a_{\ell
0}^{reg}(\lambda )^{2} \right\}=C_{\ell }$ , as expected. Similarly, by moving to polar coordinates it is easy to see that we have$$\begin{aligned}
E\{\left\vert a_{\ell m}^{reg}(\lambda )\right\vert ^{2} \}&=&\frac{1}{2\pi }\int_{0}^{2\pi }\int_{\lambda }^{\infty }(r-\lambda )^{2}\frac{r}{C_{\ell }/2}\exp (-\frac{r^{2}}{C_{\ell }})drd\varphi \notag \\
&=&C_{\ell }\int_{\lambda }^{\infty }(\frac{r-\lambda }{\sqrt{C_{\ell }}})^{2}\frac{2r}{C_{\ell }}\exp (-\frac{r^{2}}{C_{\ell }})dr \notag \\
&=&C_{\ell }\int_{\lambda }^{\infty }(\frac{r-\lambda }{\sqrt{C_{\ell }}})^{2}\exp (-\frac{r^{2}}{C_{\ell }})d\frac{r^{2}}{C_{\ell }} \notag \\
&=&C_{\ell }\left\{ \int_{\lambda ^{2}/C_{\ell }}^{\infty }u\exp (-u)du+\frac{\lambda ^{2}}{C_{\ell }}\int_{\lambda ^{2}/C_{\ell }}^{\infty }\exp
(-u)du\right\} \notag \\
&&+C_{\ell }\left\{ -2\frac{\lambda }{\sqrt{C_{\ell }}}\int_{\lambda
^{2}/C_{\ell }}^{\infty }\sqrt{u}\exp (-u)du\right\} \text{ .}
\label{iniez3}\end{aligned}$$Now using (\[iniez1\]) and$$\Gamma (2;c)=\int_{c}^{\infty }u\exp (-u)du=e^{-c}+ce^{-c}, \label{iniez2}$$we have$$\begin{aligned}
E\{\left\vert a_{\ell m}^{reg}(\lambda )\right\vert ^{2} \}&=&C_{\ell
}\left\{ \exp (-\frac{\lambda ^{2}}{C_{\ell }})+\frac{\lambda ^{2}}{C_{\ell }}\exp (-\frac{\lambda ^{2}}{C_{\ell }}) +\frac{\lambda ^{2}}{C_{\ell }}\exp
(-\frac{\lambda ^{2}}{C_{\ell }}) \right\} \\
&&-2C_{\ell }\left\{ \frac{\lambda ^{2}}{C_{\ell }} \exp (-\frac{\lambda ^{2}}{C_{\ell }}) +\frac{1}{2}\sqrt{\pi }\text{Erfc} \left(\frac{\lambda}{\sqrt{C_\ell}} \right) \frac{\lambda }{\sqrt{C_{\ell }}}\right\} \\
&=&C_{\ell }\left\{ \exp (-\frac{\lambda ^{2}}{C_{\ell }})-\sqrt{\pi }\text{Erfc} \left( \frac{\lambda}{\sqrt{C_\ell}} \right) \frac{\lambda }{\sqrt{C_{\ell }}}\right\} \\
&=&C_{\ell }\left\{ \exp (-\frac{\lambda ^{2}}{C_{\ell }})-\frac{\lambda }{\sqrt{C_{\ell }}}\sqrt{\pi }2(1-\Phi (\frac{\sqrt{2}\lambda }{\sqrt{C_{\ell }}}))\right\} \text{ .}\end{aligned}$$Hence we have$$\begin{aligned}
\frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}}{E\{\left\vert
a_{\ell 0}^{reg}\right\vert ^{2}\}}&=&\frac{\int_{\lambda }^{\infty
}(r-\lambda )^{2}2\frac{r}{C_{\ell }}\exp (-\frac{r^{2}}{C_{\ell }})dr}{\frac{2}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }(x-\lambda )^{2}\exp
\left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx} \\
&=&\frac{C_{\ell }\int_{\lambda }^{\infty }(\frac{r}{\sqrt{C_{\ell }}}-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}2\frac{r}{\sqrt{C_{\ell }}}\exp (-\frac{r^{2}}{C_{\ell }})d\frac{r}{\sqrt{C_{\ell }}}}{\frac{2}{\sqrt{2\pi }}C_{\ell
}\int_{\lambda }^{\infty }(\frac{x}{\sqrt{C_{\ell }}}-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x}{\sqrt{C_{\ell }}}} \\
&=&\frac{\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}u\exp (-u^{2})du}{\frac{1}{\sqrt{2\pi }}\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}\exp \left\{ -\frac{u^{2}}{2}\right\} du} \\
&\leq &K_\varepsilon \exp (-\frac{\lambda ^{2}}{2C_{\ell }}(1-\varepsilon ))\frac{\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}\exp (-\frac{u^{2}}{2})du}{\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}\exp \left\{ -\frac{u^{2}}{2}\right\} du} \\
&=&K_{\varepsilon }\exp (-\frac{\lambda ^{2}}{2C_{\ell }}(1-\varepsilon ))\text{ },\text{ }\end{aligned}$$for some constant $K_{\varepsilon }>0,$ any $\varepsilon >0,$ because $u\exp
(-\frac{\varepsilon u^{2}}{2})\leq K_{\varepsilon }$ for all $u\geq \frac{\lambda }{\sqrt{C_{\ell }}}.$ Now$$\begin{aligned}
&-\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{2C_{\ell }}{\lambda ^{2}}\log \frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}}{E\{\left\vert a_{\ell 0}^{reg}\right\vert ^{2}\}} \\
&=-\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{2C_{\ell }}{\lambda ^{2}}\log K_{\varepsilon }-\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{2C_{\ell }}{\lambda ^{2}}\log \exp (-\frac{\lambda
^{2}}{2C_{\ell }}(1-\varepsilon )) \\
&=(1-\varepsilon )\text{ ,}\end{aligned}$$and because $\varepsilon $ is arbitrary, the first result follows. To conclude, it is then sufficient to note that$$\frac{E\{T_{\ell }^{{reg}}(\theta ,\phi )^2 \}}{E\{T_{\ell }^{{reg}}(0,0)^2
\}P_{\ell }^{2}(\cos \theta )}=\frac{\sum_{m}E\{|a_{\ell
m}^{reg}|^{2}\}\left\vert Y_{\ell m}(\theta ,\phi )\right\vert ^{2}}{\frac{2\ell +1}{4\pi }E\{(a_{\ell 0}^{{reg}})^{2}\}P_{\ell }^{2}(\cos \theta )}$$$$=1+\frac{\sum_{m\neq 0}E\{|a_{\ell m}^{reg}|^{2}\}\left\vert Y_{\ell
m}(\theta ,\phi )\right\vert ^{2}}{\frac{2\ell +1}{4\pi }E\{(a_{\ell 0}^{{reg}})^{2}\}P_{\ell }^{2}(\cos \theta )}\rightarrow 1\text{ ,}$$because$$\frac{\sum_{m\neq 0}E\{|a_{\ell m}^{reg}|^{2}\} \left\vert Y_{\ell m}(\theta
,\phi )\right\vert ^{2}}{\frac{2\ell +1}{4\pi }E\{(a_{\ell 0}^{{reg}})^2
\}P_{\ell }^{2}(\cos \theta )}\leq {2\ell} \max_{m\neq 0}\frac{E\{|a_{\ell
m}^{reg}|^{2}\} \left\vert Y_{\ell m}(\theta ,\phi )\right\vert ^{2}}{\frac{2\ell +1}{4\pi } E\{(a_{\ell 0}^{{reg}})^{2}\} {P_{\ell}^2(\cos \theta)} }\rightarrow 0\text{ ,}$$as $\lambda /\sqrt{C_{\ell }}\rightarrow \infty$. The proof in the real-valued scheme is analogous.
As a consequence of the previous Theorem, in the complex-valued regularization scheme the ratio $E\left\vert a_{\ell 0}^{reg}\right\vert
^{2}/E\left\vert a_{\ell m}^{reg}\right\vert ^{2}$ diverges to infinity super-exponentially as $C_{\ell }\rightarrow 0,$ and the covariance function is dominated by a single random coefficient, thus oscillating over the sphere as the square of a Legendre polynomial. For the real-valued algorithm, this is not the case, and the variance is constant; nevertheless, this field is still anisotropic, as confirmed by the analysis of higher order power spectra which we entertain in the next Sections.
High-Frequency Behavior of Trispectra \[trispectra\]
====================================================
The Trispectrum at the North Pole
---------------------------------
A natural tool to explore non-Gaussian/anisotropic features of a spherical random fields is provided by the expected values of higher-order moments of its multipole components. For instance, fourth order moments lead to so-called trispectra, see [@hu; @MarPecBook; @planckNG] for properties and applications; for our aims, it suffices to recall that, under Gaussianity and isotropy, we should have$$\frac{ET_{\ell }^{4}(x)}{(ET_{\ell }^{2}(x))^{2}}\equiv 3\text{ , for all }x\in S^{2},\text{ }\ell =0,1,2,...$$In the following result we show instead that the normalized trispectrum of convexly regularized fields diverges to infinity at the North Pole.
The normalized trispectrum of a convexly regularized random field at the North Pole is given by$$\frac{E\{T_{\ell }^{reg} {(0,0)}^{4}\}}{[E\{T_{\ell }^{reg} {(0,0)}^{2}\}]^{2}}=\frac{E\left\{ a_{\ell 0}^{reg}(\lambda )^{4} \right\} }{[E\{
a_{\ell 0}^{reg}(\lambda )^{2} \} ]^{2}}={\ \sqrt{\frac{\pi}{2}}} \psi (\frac{\lambda }{\sqrt{C_{\ell }}})\text{ ,}$$where$$\psi (\frac{\lambda }{\sqrt{C_{\ell }}}):=\frac{\int_{0}^{\infty }v^{4}\exp
\left\{ -\frac{1}{2}\left[ v+\frac{\lambda }{\sqrt{C_{\ell }}}\right]
^{2}\right\} dv}{\left[ \int_{0}^{\infty }v^{2}\exp \left\{ -\frac{1}{2}\left[ v+\frac{\lambda }{\sqrt{C_{\ell }}}\right] ^{2}\right\} dv\right] ^{2}}\text{ .}$$The function $\psi (.)$ is strictly positive and increasing, with$$\lim_{x\rightarrow 0}\psi (x)=3\text{ , }\lim_{x\rightarrow \infty }\left[
\frac{15}{4} x^3 \exp\left\{\frac{x^2}{2}\right\} \right]^{-1}\psi(x)=1
\text{ .}$$
Similarly to the proof of the previous Theorem and using the same notation, we have$$\begin{aligned}
E\left\{ a_{\ell 0}^{reg}(\lambda )^{4} \right\} &=\frac{2}{\sqrt{2\pi
C_{\ell }}}\int_{\lambda }^{\infty }(x-\lambda )^{4}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx \\
&=\frac{2}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }x^{4}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx-8\frac{1}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }x^{3}\lambda \exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx \\
&\;\;+12\frac{1}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }x^{2}\lambda
^{2}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx-8\frac{1}{\sqrt{2\pi
C_{\ell }}}\int_{\lambda }^{\infty }x\lambda ^{3}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx \\
&\;\;+\frac{2}{\sqrt{2\pi C_{\ell }}}\int_{\lambda }^{\infty }\lambda
^{4}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} dx \\
&=\frac{2}{\sqrt{2\pi }}\sqrt{8}C_{\ell }^{2}\int_{\lambda }^{\infty }\frac{x^{3}}{\sqrt{8C_{\ell }^{3}}}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x^{2}}{2C_{\ell }} \\
&\;\;-8\frac{1}{\sqrt{2\pi }}2\sqrt{C_{\ell }^{3}}\lambda \int_{\lambda
}^{\infty }(\frac{x}{\sqrt{2C_{\ell }}})^{2}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x^{2}}{2C_{\ell }} \\
&\;\;+12\frac{1}{\sqrt{2\pi }}\sqrt{2}C_{\ell }\lambda ^{2}\int_{\lambda
}^{\infty }\frac{x}{\sqrt{2C_{\ell }}}\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x^{2}}{2C_{\ell }} \\
&\;\;-8\frac{{\ \sqrt{C_\ell}}}{\sqrt{2\pi }}\lambda ^{3}\int_{\lambda
}^{\infty }\exp \left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x^{2}}{2C_{\ell }} +\frac{2}{\sqrt{\pi }}\lambda ^{4}\int_{\lambda }^{\infty }\exp
\left\{ -\frac{x^{2}}{2C_{\ell }}\right\} d\frac{x}{\sqrt{2C_{\ell }}} \\
&=2C_{\ell }^{2}\frac{\sqrt{8}}{\sqrt{2\pi }}\left\{ \Gamma (\frac{5}{2};\frac{\lambda ^{2}}{2C_{\ell }})-\frac{4}{\sqrt{2}}\frac{{\ \sqrt{2}}\lambda
}{\sqrt{2C_{\ell }}}\Gamma (2;\frac{\lambda ^{2}}{2C_{\ell }})+\frac{6}{\sqrt{2^{2}}}(\frac{\lambda }{\sqrt{{C_{\ell }}}})^{2}\Gamma (\frac{3}{2};\frac{\lambda ^{2}}{2C_{\ell }})\right. \\
&\;\;\left. - {\ \frac{\sqrt{2} \lambda^3}{C_\ell \sqrt{C_\ell}} \exp\left\{-\frac{\lambda^2}{2 C_\ell}\right\} }+\frac{4 {\ \sqrt{2 \pi}} }{\sqrt{2^{3}}}(\frac{\lambda }{\sqrt{2C_{\ell }}})^{4}(1-\Phi (\frac{\lambda }{\sqrt{{\ 2
C_{\ell }}}}))\right\}.\end{aligned}$$ Observing that $\Gamma(1;c)=e^{-c}$, we obtain $$\begin{aligned}
&E\left\{ a_{\ell 0}^{reg}(\lambda )^{4} \right\} \\
&=C_{\ell }^{2}\frac{4}{\sqrt{\pi }} \left\{ \sum_{k=2}^{5} (-1)^{k+1} \left(\frac{\nu _{\ell }}{\sqrt{2}}\right)^{5-k} \binom{4}{5-k} \Gamma \left(\frac{k}{2};\left(\frac{\nu _{\ell }}{\sqrt{2}}\right)^2 \right)+ 2 \sqrt \pi
\left(\frac{\nu _{\ell }}{\sqrt{2}}\right)^{4} (1-\Phi (\frac{\nu _{\ell }}{\sqrt{2}}) \right\} \text{ ,}\end{aligned}$$where $\nu _{\ell }:=\frac{\lambda }{\sqrt{C_{\ell }}}.$ Again, it is simple to check that $$\begin{aligned}
\lim_{\lambda/ \sqrt{C_\ell} \rightarrow 0}E\left\{ a_{\ell 0}^{reg}(\lambda
)^{4} \right\} &=&2C_{\ell }^{2}\frac{\sqrt{8}}{\sqrt{2\pi }}\lim_{\lambda/
\sqrt{C_\ell} \rightarrow 0}\Gamma (\frac{5}{2};\frac{\lambda ^{2}}{2C_{\ell
}}) \\
&=&2C_{\ell }^{2}\frac{\sqrt{8}}{\sqrt{2\pi }}\frac{3}{4}\sqrt{\pi }=3C_{\ell }^{2}\text{ ,}\end{aligned}$$as expected, because $\lim_{\lambda/ \sqrt{C_\ell} \rightarrow 0}E\left\{
a_{\ell 0}^{reg}(\lambda )^{4} \right\} /[E\left\{ a_{\ell 0}^{reg}(\lambda
)^{2} \right\} ]^{2}=3$ provides the fourth moment of a standard Gaussian variable. Note also that$$\begin{aligned}
\psi(\frac{\lambda}{\sqrt{C_\ell}})&=&\frac{\int_{0}^{\infty }v^{4}\exp
\left\{ -\frac{1}{2}\left[ v+\frac{\lambda }{\sqrt{C_{\ell }}}\right]
^{2}\right\} dv}{ \left[ \int_{0}^{\infty }v^{2}\exp \left\{ -\frac{1}{2}\left[ v+\frac{\lambda }{\sqrt{C_{\ell }}}\right] ^{2}\right\} dv\right] ^{2}} \\
&=&\exp \left\{ \frac{1}{2}\frac{\lambda^2 }{C_{\ell } } \right\} \frac{\int_{0}^{\infty }v^{4}\exp \left\{ -\frac{1}{2}\left[ v^{2}+\frac{2\lambda {v} }{\sqrt{C_{\ell }}}\right] \right\} dv}{\left[ \int_{0}^{\infty
}v^{2}\exp \left\{ -\frac{1}{2}\left[ v^{2}+\frac{2\lambda {v} }{\sqrt{C_{\ell }}}\right] \right\} dv\right] ^{2}} \\
&=&\exp \left\{ \frac{1}{2} \frac{\lambda^2 }{ C_{\ell } } \right\} \frac{-5
\frac{\lambda }{\sqrt{C_{\ell }}} - (\frac{\lambda }{\sqrt{C_{\ell }}})^3+\exp\{\frac{\lambda^2 }{2 C_{\ell }}\} \sqrt{\frac{\pi}{2}}(3+ 6 \frac{\lambda^2 }{C_{\ell }} +\frac{\lambda^4 }{C_{\ell }^2} )\text{Erfc}(\frac{\lambda }{\sqrt{ 2 C_{\ell }}}) }{\left[- \frac{\lambda }{\sqrt{C_{\ell }}}
+\exp\{\frac{\lambda^2 }{2 C_{\ell }}\} \sqrt{\frac{\pi}{2}}(1+ \frac{\lambda^2 }{C_{\ell }} )\text{Erfc}(\frac{\lambda }{\sqrt{ 2 C_{\ell }}}) \right]^2} \text{ ,}\end{aligned}$$ here we use the classical asymptotic expansion of the complementary error function, i.e., for large $x$ we have $\text{Erfc}(x)=\frac{e^{-x^2}}{x
\sqrt{\pi}} (1- \frac{1}{2 x^2}+ \frac{3}{4 x^4}+O(x^{-5}))$, then $$\begin{aligned}
&&\lim_{\lambda / \sqrt{C_\ell} \to \infty} \psi(\frac{\lambda}{\sqrt{C_\ell}}) \left[ \exp \left\{ \frac{1}{2}\frac{\lambda^2 }{{C_{\ell }}}\right\}
\frac{15}{4} \frac{\lambda^3}{C_\ell^{\frac 3 2}} \right]^{-1} \\
&=&\lim_{\lambda / \sqrt{C_\ell} \to \infty}\frac{-5 \frac{\lambda }{\sqrt{C_{\ell }}} - (\frac{\lambda }{\sqrt{C_{\ell }}})^3+\exp\{\frac{\lambda^2 }{2 C_{\ell }}\} \sqrt{\frac{\pi}{2}}(3+ 6 \frac{\lambda^2 }{C_{\ell }} +\frac{\lambda^4 }{C_{\ell }^2} )\text{Erfc}(\frac{\lambda }{\sqrt{ 2 C_{\ell }}})
}{ \frac{15}{4} \frac{\lambda^3}{C_\ell^{\frac 3 2}} \left[- \frac{\lambda }{\sqrt{C_{\ell }}} +\exp\{\frac{\lambda^2 }{2 C_{\ell }}\} \sqrt{\frac{\pi}{2}}(1+ \frac{\lambda^2 }{C_{\ell }} )\text{Erfc}(\frac{\lambda }{\sqrt{ 2
C_{\ell }}}) \right]^2} \\
&=&\lim_{\lambda / \sqrt{C_\ell} \to \infty} \frac{-5 \frac{\lambda }{\sqrt{C_{\ell }}} - (\frac{\lambda }{\sqrt{C_{\ell }}})^3+\exp\{\frac{\lambda^2 }{2 C_{\ell }}\} \sqrt{\frac{\pi}{2}}(3+ 6 \frac{\lambda^2 }{C_{\ell }} +\frac{\lambda^4 }{C_{\ell }^2} ) \frac{e^{-\frac{\lambda^2}{2 C_\ell }}}{\frac{\lambda}{\sqrt{C_\ell}} \sqrt{\frac \pi 2}} (1- \frac{1}{\frac{\lambda^2}{C_\ell}}+ \frac{3}{2 \frac{\lambda^4}{C_\ell^2}})}{ \frac{15}{4} \frac{\lambda^3}{C_\ell^{\frac 3 2}} \left[- \frac{\lambda }{\sqrt{C_{\ell }}}
+\exp\{\frac{\lambda^2 }{2 C_{\ell }}\} \sqrt{\frac{\pi}{2}}(1+ \frac{\lambda^2 }{C_{\ell }} )\frac{e^{-\frac{\lambda^2}{2 C_\ell }}}{\frac{\lambda}{\sqrt{C_\ell}} \sqrt{\frac \pi 2}} (1- \frac{1}{\frac{\lambda^2}{C_\ell}}+
\frac{3}{2 \frac{\lambda^4}{C_\ell^2}}) \right]^2} \\
&=&\lim_{\lambda / \sqrt{C_\ell} \to \infty} \frac{3 \frac{\lambda^5}{{C_\ell}^\frac 5 2} (3+5 \frac{\lambda^2}{{C_\ell}})}{\frac{15}{4} \frac{\lambda^3}{C_\ell^{\frac 3 2}} (3+2 \frac{\lambda^2}{{C_\ell}})^2}=1 \text{.}\end{aligned}$$
(Some Numerical Examples) It is instructive to provide some numerical evidence on the kurtosis of the multipole components at the North Pole, as a function of the penalization parameter $\lambda $ and the angular power spectrum $C_{\ell }.$ It should be recalled that $\sum_{\ell }(2\ell
+1)C_{\ell }<\infty $ by finite variance, whence $C_{\ell }$ must decay at least as fast as $\ell ^{-2-\tau },$ some $\tau >0,$ as $\ell \rightarrow
\infty .$ For instance, considering some physically realistic values for the power spectrum of CMB and a fixed penalization parameter $\lambda =1,$ we have
$$\begin{array}{cccccccccc}
\ell & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 200 \\
C_{\ell } & 48.20 & 13.7 & 7.17 & 4.8 & 3.7 & 2.9 & 2.4 & 2.1 & 0.76 \\
\kappa_{\ell} & 3.50 & 4.08 & 4.65 & 5.19 & 5.65 & 6.21 & 6.76 & 7.22 & 15.39\end{array}$$
Asymptotic Behavior of the Angular Trispectrum
----------------------------------------------
Exploiting the computations developed so far, it is also possible to provide analytic expressions for the trispectra at the various multipoles, as follows. We start recalling the results we have earlier established on the moments of the spherical harmonic coefficients $\left\{ a_{\ell m}\right\} ,$ under the complex and real-valued regularization schemes. More precisely, we have
- For all $\ell ,m$ we have$$E\{a_{\ell m}^{reg}\}=E\{a_{\ell m}^{reg\ast }\}=0\text{ ;} \label{zero}$$
- For all $\ell ,m=0$ $$E\left\{ (a_{\ell 0}^{reg})^{2} \right\} =E\left\{ (a_{\ell 0}^{reg\ast
})^{2}\right\}=\gamma _{0}(\frac{\lambda }{\sqrt{C_{\ell }}})\text{ ;}$$
- Under the complex-valued regularization scheme, for $m\neq 0$$$E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}=\gamma _{1}(\frac{\lambda }{\sqrt{C_{\ell }}})\text{ ,}$$while in the real-valued framework$$E\{( a_{\ell m}^{reg {*}})^{2}\}=\gamma _{0}(\frac{\lambda }{\sqrt{C_{\ell }}})\text{ ;}$$
- Finally for the fourth-order moments, for all ${\ell}$$$E\left\{ (a_{\ell 0}^{reg}) ^{4}\right\}=E\{(a_{\ell {0}}^{reg\ast
})^4\}=\gamma _{2}(\frac{\lambda }{\sqrt{C_{\ell }}})$$and for $m\neq 0$$${E\{\left\vert a_{\ell m}^{reg }\right\vert ^{4}\}}=E\{(a_{\ell m}^{reg\ast
})^{4}\}=\gamma _{3}(\frac{\lambda }{\sqrt{C_{\ell }}})$$where $$\begin{aligned}
\gamma _{2}(\nu_\ell)&:=C_{\ell }^{2}\frac{4}{\sqrt{\pi }} \left\{
\sum_{k=2}^{5} (-1)^{k+1} (\frac{\nu_\ell}{\sqrt 2})^{5-k} \binom{4}{5-k}
\Gamma (\frac{k}{2};(\frac{\nu_\ell}{\sqrt 2})^2)+ 2 \sqrt \pi (\frac{\nu_\ell}{\sqrt 2})^{4} (1-\Phi (\frac{\nu_\ell}{\sqrt 2})) \right\}\end{aligned}$$ and$$\begin{aligned}
\gamma _{3}(\nu_\ell)&= C_\ell^2 \sum_{k=0}^4 (-1)^k \binom{4}{k}
\nu_\ell^{4-k} \Gamma(\frac {k+2} 2; \nu_\ell^2) \text{,}\end{aligned}$$
where $\nu_\ell=\frac{\lambda}{\sqrt{C_\ell}}$. For [(\[zero\])]{}, we note first that it would be trivially true for an isotropic random field, but it requires to be checked under anisotropy. In any case, the proof is straightforward, indeed we have $$\begin{aligned}
E\{a_{\ell m}^{reg} \}&=&E\left\{ \rho_{\ell m}^{{\ reg}} \exp (i\psi _{\ell
m}^{{obs}})\right\} \\
&=&{\ \frac{1}{2\pi}}\int_{0}^{\infty }\int_{0}^{2\pi }r\exp (i\theta
)f_{\rho; {\ell }}(r)drd\theta \\
&=&{\ \frac{1}{2\pi}}\int_{0}^{\infty }r f_{\rho; {\ell}}(r)\left\{
\int_{0}^{2\pi }\exp (i\theta )d\theta \right\} dr=0\text{ .}\end{aligned}$$A similar argument will actually cover any product of an odd number of spherical harmonic coefficients, because $\int_{0}^{2\pi }\exp (ik\theta
)d\theta =0$ for all non-zero integers $k$. We only need to study $E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\},$ for which we have$$\begin{aligned}
E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4} \} &=&\int_{\lambda }^{\infty
}(r-\lambda )^{4}2\frac{r}{C_{\ell }}\exp (-\frac{r^{2}}{C_{\ell }})dr \\
&=&\int_{\lambda }^{\infty }(r-\lambda )^{4}\exp (-\frac{r^{2}}{C_{\ell }})d\frac{r^{2}}{C_{\ell }} \\
&=&C_{\ell }^{2}\int_{\lambda ^{2}/C_{\ell }}^{\infty }(u^{2}-4u^{3/2}\frac{\lambda }{\sqrt{C_{\ell }}}+6u\frac{\lambda ^{2}}{C_{\ell }}-4\sqrt{u}\frac{\lambda ^{3}}{\sqrt{C_{\ell }^{3}}}+\frac{\lambda ^{4}}{C_{\ell }^{2}})\exp
(-u)du \\
&=&C_{\ell }^{2}\left\{ \Gamma (3;\frac{\lambda ^{2}}{C_{\ell }})-4\frac{\lambda }{\sqrt{C_{\ell }}}\Gamma (\frac{5}{2};\frac{\lambda ^{2}}{C_{\ell }})+6\frac{\lambda ^{2}}{C_{\ell }}\Gamma ({2};\frac{\lambda ^{2}}{C_{\ell }})-4\frac{\lambda ^{3}}{\sqrt{C_{\ell }^{3}}}\Gamma ({\ \frac 3 2};\frac{\lambda ^{2}}{C_{\ell }})+\frac{\lambda ^{4}}{C_{\ell }^{2}}\exp (-\frac{\lambda ^{2}}{C_{\ell }}))\right\} \text{ ,}\end{aligned}$$using repeatedly integration by parts on the incomplete Gamma function.
It should be noted that, as expected,$$\lim_{\lambda \rightarrow 0}E\{\left\vert a_{\ell m}^{reg}\right\vert
^{2}\}=C_{\ell }\text{ ,}$$and more generally$$\lim_{C_{\ell }/\lambda ^{2}\rightarrow \infty }\frac{1}{C_{\ell }}E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}=1\text{ .}$$Moreover $$\lim_{\lambda \rightarrow 0}{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}}/{C_{\ell }^{2}}=2\text{ ,}$$again as expected, because in the limiting Gaussian case$$\begin{aligned}
E\{\left\vert a_{\ell m}\right\vert ^{4}\}&=E\{[{\func{Re}}(a_{lm})^{2}+{\func{Im}}(a_{lm})^{2}]^{2}\} \\
&=E\{{\func{Re}}(a_{lm})^{4}\}+E\{{\func{Im}}(a_{lm})^{4}\}+2E\{{\func{Re}}(a_{lm})^{2}\}E\{{\func{Im}}(a_{lm})^{2}\} \\
&=\frac{3}{4}C_{\ell }^{2}+\frac{3}{4}C_{\ell }^{2}+\frac{2}{4}C_{\ell
}^{2}=2C_{\ell }^{2}\text{ .}\end{aligned}$$
The previous result can be generalized as follows.
For all $p=1,2,3,...$ and for $m \ne 0$, we have$$E\left\{ \left\vert a_{\ell m}^{reg}\right\vert ^{2p}\right\} =C_{\ell
}^{p}\sum_{k=0}^{2p}(-1)^{k} \binom{2p}{k} \nu_\ell ^{{2 p-k }}\Gamma (\frac{{k+2}}{2}; \nu_\ell^2)\text{ ,}$$where $\binom{2p}{k}=\frac{(2p)!}{(2p-k)!k!}$ is the standard binomial coefficient.
The proof is identical to the previous arguments, and hence it is not repeated for brevity’s sake.
An important consequence of these results is the following
We have that$$\lim_{C_{\ell }\rightarrow 0} \exp\left\{-\frac{\lambda^2}{C_\ell}\right\}
\frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}}{\left[ E\{\left\vert
a_{\ell m}^{reg}\right\vert ^{2}\}\right] ^{2}} = 6\text{ ,}$$whence $\frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}}{\left[
E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}\right] ^{2}}$ diverges superexponentially.
It suffices to notice that $$\begin{aligned}
\frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}}{\left[ E\{\left\vert
a_{\ell m}^{reg}\right\vert ^{2}\}\right] ^{2}}&=&\frac{C_{\ell }^{2}
\int_{\lambda }^{\infty } (\frac{r-\lambda }{\sqrt{C_{\ell }}})^{4} {\ \frac{2r}{\sqrt{C_\ell}}} \exp(-\frac{r^{2}}{C_{\ell }})d {\ \frac{r}{\sqrt{
C_{\ell }}}}}{\left\{ C_{\ell }\int_{\lambda }^{\infty }(\frac{r-\lambda }{\sqrt{C_{\ell }}})^{2}2\frac{r}{\sqrt{C_{\ell }}}\exp (-\frac{r^{2}}{C_{\ell
}})d\frac{r}{\sqrt{C_{\ell }}}\right\} ^{2}} \\
&=&\frac{\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{4} {2u} \exp (-u^{2})du}{\left\{ 2\int_{\lambda /\sqrt{C_{\ell
}}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{2}u\exp
(-u^{2})du\right\} ^{2}} \\
&=&\frac{\int_{0}^{\infty }v^{4} {\ 2 (v+\frac{\lambda}{\sqrt{C_\ell}})}\exp
(-(v+\frac{\lambda }{\sqrt{C_{\ell }}})^{2})dv}{\left\{ 2\int_{0}^{\infty
}v^{2}(v+\frac{\lambda }{\sqrt{C_{\ell }}})\exp (-(v+\frac{\lambda }{\sqrt{C_{\ell }}})^{2})d{v}\right\} ^{2}} \\
&=&\exp (\frac{\lambda ^{2}}{C_{\ell }}) \frac{\int_{0}^{\infty }v^{4} {(v+\frac{\lambda }{\sqrt{C_{\ell }}}) } \exp (-v^{2}-2\frac{\lambda v}{\sqrt{C_{\ell }}})dv}{{2}\left\{ \int_{0}^{\infty }v^{2}(v+\frac{\lambda }{\sqrt{C_{\ell }}})\exp (-v^{2}-2\frac{\lambda v}{\sqrt{C_{\ell }}})d {v} \right\}
^{2}}\text{ ,}\end{aligned}$$ where, by applying the expansion of the $\text{Erfc}$ function $\frac{e^{-x^2}}{x \sqrt{\pi}} (1- \frac{1}{2 x^2}+ \frac{3}{4 x^4}-\frac{15}{8 x^6}+O(x^{-7}))$, for large $x$, we have $$\begin{aligned}
&&\lim_{\lambda / \sqrt{C_\ell} \to \infty}\frac{\int_{0}^{\infty }v^{4} {(v+\frac{\lambda }{\sqrt{C_{\ell }}}) } \exp (-v^{2}-2\frac{\lambda v}{\sqrt{C_{\ell }}})dv}{{2}\left[ \int_{0}^{\infty }v^{2}(v+\frac{\lambda }{\sqrt{C_{\ell }}})\exp (-v^{2}-2\frac{\lambda v}{\sqrt{C_{\ell }}})d {v} \right]
^{2}} \\
&=&\lim_{\lambda / \sqrt{C_\ell} \to \infty} \frac{1+\frac{\lambda^2 }{{C_{\ell }}} -\frac {\sqrt{\pi}}{2} \frac{\lambda }{\sqrt{C_{\ell }}} \exp\{\frac{\lambda^2}{{C_{\ell }}}\} (3+2 \frac{\lambda^2}{{C_{\ell }}}) \text{Erfc}(\frac{\lambda }{\sqrt{C_{\ell }}})}{2 \left[ \frac{1}{2} (1-\exp\{\frac{\lambda^2 }{{C_{\ell }}}\} \sqrt{\pi} \frac{\lambda }{\sqrt{C_{\ell }}}
erfc(\frac{\lambda }{\sqrt{C_{\ell }}}))\right]^2} \\
&=&\lim_{\lambda / \sqrt{C_\ell} \to \infty} \frac{1+\frac{\lambda^2 }{{C_{\ell }}} -\frac {\sqrt{\pi}}{2} \frac{\lambda }{\sqrt{C_{\ell }}} \exp\{\frac{\lambda^2}{{C_{\ell }}}\} (3+2 \frac{\lambda^2}{{C_{\ell }}}) \frac{e^{-\frac{\lambda^2}{ C_\ell }}}{\frac{\lambda}{\sqrt{C_\ell}} \sqrt{\pi}}
(1- \frac{1}{2 \frac{\lambda^2}{C_\ell}}+ \frac{3}{4 \frac{\lambda^4}{C_\ell^2}}-\frac{15}{8 x^6})}{2 \left[ \frac{1}{2} (1-\exp\{\frac{\lambda^2
}{{C_{\ell }}}\} \sqrt{\pi} \frac{\lambda }{\sqrt{C_{\ell }}} \frac{e^{-\frac{\lambda^2}{ C_\ell }}}{\frac{\lambda}{\sqrt{C_\ell}} \sqrt{\pi}} (1-
\frac{1}{2 \frac{\lambda^2}{C_\ell}}+ \frac{3}{4 \frac{\lambda^4}{C_\ell^2}}-\frac{15}{8 x^6}))\right]^2} \\
&=&\lim_{\lambda / \sqrt{C_\ell} \to \infty} \frac{24 \frac{\lambda^6}{{C_\ell}^3}(15+4 \frac{\lambda^2}{{C_\ell}})}{(15-6 \frac{\lambda^2}{{C_\ell}}+4 \frac{\lambda^4}{{C_\ell}^2})^2}=6 \text{ . }\end{aligned}$$
In view of the previous results, it is simple to provide exact analytic expressions for the expected trispectra $E\{T_{\ell }\}$ under both regularization schemes, and to study their asymptotic behavior as the frequencies increase. We obtain
We have$$\begin{aligned}
E\left\{ T_{\ell }^{reg}(\theta ,\phi ) ^{4}\right\}&=\gamma _{2}(\frac{\lambda }{\sqrt{C_{\ell }}})\left\vert Y_{\ell 0}(\theta ,\phi )\right\vert
^{4}+\gamma _{3}(\frac{\lambda }{\sqrt{C_{\ell }}})\sum_{m\neq 0}\left\vert
Y_{\ell m}(\theta ,\phi )\right\vert ^{4} \\
&\:\:+{2}\gamma _{0}(\frac{\lambda }{\sqrt{C_{\ell }}})\gamma _{1}(\frac{\lambda }{\sqrt{C_{\ell }}})\left\vert Y_{\ell 0}(\theta ,\phi )\right\vert
^{2}\left\{ \frac{2\ell +1}{4\pi }-\left\vert Y_{\ell 0}(\theta ,\phi
)\right\vert ^{2}\right\} \\
&\:\:+\gamma _{1}^{2}(\frac{\lambda }{\sqrt{C_{\ell }}}){\sum_{m^{\prime
}\neq m}, {\ m,m^{\prime }\neq 0}}\left\vert Y_{\ell m}(\theta ,\phi
)\right\vert ^{2}\left\vert Y_{\ell m^{\prime }}(\theta ,\phi )\right\vert
^{2}.\end{aligned}$$Likewise$$\begin{aligned}
E\left\{ T_{\ell }^{reg\ast }(\theta ,\phi )\right\} ^{4}&=\gamma _{2}(\frac{\lambda }{\sqrt{C_{\ell }}})\sum_{m}\left\vert Y_{\ell m}^{\mathcal{R}}(\theta ,\phi )\right\vert ^{4}+\gamma _{0}^{2}(\frac{\lambda }{\sqrt{C_{\ell }}})\sum_{m^{\prime }\neq m}\left\vert Y_{\ell m}(\theta ,\phi
)\right\vert ^{2}\left\vert Y_{\ell m^{\prime }}(\theta ,\phi )\right\vert
^{2}.\end{aligned}$$As $\lambda /\sqrt{C_{\ell }}\rightarrow \infty $, the trispectrum is then asymptotic to $$\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{E\left\{ T_{\ell
}^{reg}(\theta ,\phi )^{4}\right\} }{E\left\{ T_{\ell
}^{reg}(0,0)^{4}\right\} P_{\ell }^{4}(\cos \theta )}=1\text{ .}$$For the real-valued regularization scheme we get$$\lim_{\lambda /\sqrt{C_{\ell }}\rightarrow \infty }\frac{E\left\{ T_{\ell
}^{reg\ast }(\theta ,\phi )^{4}\right\} }{E\{T_{\ell }^{reg\ast
}(0,0)^{4}\}V_{\ell }(\theta ,\phi )}=1\text{,}$$where as $\ell \to \infty$ $$V_{\ell }(\theta ,\phi )=\left( \frac{4\pi }{2\ell +1}\right)
^{2}\sum_{m}(Y_{\ell ,m}^{\mathcal{R}})^{4}\rightarrow
\begin{cases}
1, & \text{for }(\theta ,\phi )=(0,0), \\
0\text{ a.e.}, & \text{otherwise }.\end{cases}$$
Recall that $E\left\{ T_{\ell }^{reg}(0,0)^{4}\right\} =E\{\left\vert
a_{\ell 0}^{reg}\right\vert ^{4}\}\left\{ \frac{2\ell +1}{4\pi }\right\} ^{2}
$, and note that$$\begin{aligned}
&E\left\{ T_{\ell }^{reg}(\theta ,\phi )^{4}\right\} \notag \\
&=\sum_{m}E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}\left\vert Y_{\ell
m}(\theta ,\phi )\right\vert ^{4}+\sum_{m\neq m^{\prime }}E\{\left\vert
a_{\ell m}^{reg}\right\vert ^{2}\}E\{\left\vert a_{\ell m^{\prime
}}^{reg}\right\vert ^{2}\}\left\vert Y_{\ell m}(\theta ,\phi )\right\vert
^{2}\left\vert Y_{\ell {m^{\prime }}}(\theta ,\phi )\right\vert ^{2} \notag
\\
&=\sum_{m\neq 0}E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}\left\vert
Y_{\ell m}(\theta ,\phi )\right\vert ^{4} +\sum_{m\neq m^{\prime
}}E\{\left\vert a_{\ell m}^{reg}\right\vert ^{2}\}E\{\left\vert a_{\ell
m^{\prime }}^{reg}\right\vert ^{2}\}\left\vert Y_{\ell m}(\theta ,\phi
)\right\vert ^{2}\left\vert Y_{\ell {m^{\prime }}}(\theta ,\phi )\right\vert
^{2} \label{ghione2} \\
&\;\;+ E\{( a_{\ell 0}^{reg})^{4}\}\left\{ \frac{2\ell +1}{4\pi }\right\}
^{2}P_{\ell }^{4}(\cos \theta ) \notag\end{aligned}$$whence it suffices to notice that the expected values in (\[ghione2\]) are all of smaller order with respect to $E\{( a_{\ell 0}^{reg})^{4}\}$ . Indeed from (\[ghione3\]) it follows easily that$$\lim_{{\lambda }/\sqrt{C_{\ell }}\rightarrow \infty }\frac{\sum_{m\neq
m^{\prime }}E\{\left\vert a_{\ell m}^{reg}\right\vert^{2} \} E\{\left\vert
a_{\ell m^{\prime }}^{reg}\right\vert^{2} \}}{E\left\{ (a_{\ell
0}^{reg})^4\right\} }\leq \lim_{{\lambda }/\sqrt{C_{\ell }}\rightarrow
\infty }{{\ (4 \ell^2+2 \ell) \max_{m\neq m^{\prime }}}}\frac{E\{\left\vert
a_{\ell m}^{reg}\right\vert ^{2}\}E\{\left\vert a_{\ell m^{\prime
}}^{reg}\right\vert ^{2}\}}{E\left\{ (a_{\ell 0}^{reg})^{4}\right\} }=0\text{
.}$$Note also that$$\begin{aligned}
E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\} &=&\int_{0}^{\infty
}r^{4}f_{\rho _{\ell m}}(r)dr \\
&=&\int_{\lambda }^{\infty }(r-\lambda )^{4}2\frac{r}{C_{\ell }}\exp (-\frac{r^{2}}{C_{\ell }})dr \\
&=&2C_{\ell }^{2}\int_{{\lambda }}^{\infty }(\frac{r-\lambda }{\sqrt{C_{\ell
}}})^{4}\frac{r}{\sqrt{C_{\ell }}}\exp (-\frac{r^{2}}{C_{\ell }})d\frac{r}{\sqrt{C_{\ell }}} \\
&=&2C_{\ell }^{2}\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda
}{\sqrt{C_{\ell }}})^{4}u\exp (-u^{2})du\text{ .}\end{aligned}$$It follows that$$\begin{aligned}
\frac{E\{\left\vert a_{\ell m}^{reg}\right\vert ^{4}\}}{E\{( a_{\ell
0}^{reg})^{4}\}} &=&\frac{2C_{\ell }^{2}\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{4}u\exp (-u^{2})du}{\frac{{2}}{\sqrt{2\pi }}C_{\ell }^{2}\int_{\lambda /\sqrt{C_{\ell }}}^{\infty }(u-\frac{\lambda }{\sqrt{C_{\ell }}})^{4}\exp \left\{ -\frac{u^{2}}{2}\right\}
du} \\
&\leq &K\exp \left\{ -\frac{\lambda ^{2}(1-\varepsilon )}{2C_{\ell }}\right\} ,\text{ any }\varepsilon >0\text{ ,}\end{aligned}$$so that $$\lim_{{\lambda }/\sqrt{C_{\ell }}\rightarrow \infty }\frac{E\{\left\vert
a_{\ell m}^{reg}\right\vert ^{4}\}}{E\left\{ (a_{\ell 0}^{reg})^{4}\right\} }=0\text{ }.$$It is then immediate to see that $\left[ (\ref{ghione2})/E\left\{ T_{\ell
}^{reg}(0,0)^{4}\right\} \right] \rightarrow 1$ as $\lambda /\sqrt{C_{\ell }}\rightarrow \infty ,$ whence our first result is established. By an analogous argument, it is easy to see that$$\begin{aligned}
\frac{E\left\{ T_{\ell }^{reg *}(\theta ,\phi )^{4}\right\} }{E\{T_{\ell
}^{reg\ast }(0,0)^{4}\}V_{\ell }(\theta ,\phi )}&=\frac{\sum_{m}E\{\left\vert a_{\ell m}^{reg*}\right\vert ^{4}\}\left\vert Y_{\ell m}^{\mathcal{R}}(\theta ,\phi )\right\vert ^{4}}{E\{T_{\ell }^{reg\ast }(0,0)^{4}\}V_{\ell
}(\theta ,\phi )} \\
&\;\;+\frac{\sum_{m\neq m^{\prime }}E\{\left\vert a_{\ell
m}^{reg*}\right\vert ^{2}\}E\{\left\vert a_{\ell m^{\prime
}}^{reg*}\right\vert ^{2}\}\left\vert Y_{\ell m}^{\mathcal{R}}(\theta ,\phi
)\right\vert ^{2}\left\vert Y_{\ell m^{\prime }}^{\mathcal{R}}(\theta ,\phi
)\right\vert ^{2}}{E\{T_{\ell }^{reg\ast }(0,0)^{4}\}V_{\ell }(\theta ,\phi )}\rightarrow 1\text{ ,}\end{aligned}$$so that we need only investigate the asymptotic behavior of $$V_{\ell }(\theta ,\phi ):=\left\{ \frac{4\pi }{2\ell +1}\right\}
^{2}\sum_{m=-\ell }^{\ell }(Y_{\ell ,m}^{\mathcal{R}})^{4}\text{ .}$$To this aim, we recall the following recent result by Sogge and Zelditch [@sogge]; as $\ell \rightarrow \infty $$$\frac{1}{2\ell +1}\int_{S^{2}}\sum_{m=-\ell }^{\ell }\left\vert Y_{\ell
,m}(x)\right\vert ^{4}dx=o(\left\{ \log \ell \right\} ^{1/4})\text{ .}$$Now of course$$\begin{aligned}
\sum_{m=-\ell }^{\ell }\left\vert Y_{\ell ,m}(x)\right\vert ^{4}
&=&\sum_{m=-\ell }^{\ell }\left\{ \left\vert \func{Re}(Y_{\ell
,m}(x))\right\vert ^{2}+\left\vert \func{Im}(Y_{\ell ,m}(x))\right\vert
^{2}\right\} ^{2} \\
&=&\sum_{m=-\ell }^{\ell }\left\{ \left\vert \func{Re}(Y_{\ell
,m}(x))\right\vert ^{4}+\left\vert \func{Im}(Y_{\ell ,m}(x))\right\vert
^{4}+2\left\vert \func{Re}(Y_{\ell ,m}(x))\right\vert ^{2}\left\vert \func{Im}(Y_{\ell ,m}(x))\right\vert ^{2}\right\} \\
&\geq &\frac{1}{2}\sum_{m=-\ell }^{\ell }(Y_{\ell ,m}^{\mathcal{R}})^{4}\text{ ,}\end{aligned}$$from which we obtain immediately$$\left\{ \frac{4\pi }{2\ell +1}\right\} ^{2}\sum_{m=-\ell }^{\ell }(Y_{\ell
,m}^{\mathcal{R}})^{4}=o(\frac{\left\{ \log \ell \right\} ^{1/4}}{\ell })\text{ for almost all }x\in S^{2},$$as claimed.
The previous Theorem can be expressed in plain words as follows: under the complex-valued regularization scheme, after normalization, the trispectrum behaves asymptotically as the fourth power of the Legendre polynomial. In the real-valued case, the normalized trispectrum behaves as the averaged sum of the fourth-powers of (real-valued) spherical harmonics. The first result is heuristically explained considering that sparsity will enforce the choice of the single coefficient $a_{\ell 0}^{reg}$ more and more often,as $\lambda
/\sqrt{C_{\ell }}\rightarrow 0;$ in the latter case, each coefficient $a_{\ell m}^{\mathcal{R}}$ has the same probability to be selected, as they are all identically distributed: however in the limit at most one of them will be nonzero, so the trispectrum will reproduce the oscillations of a single (randomly chosen) functions $Y_{\ell m}^{\mathcal{R}}$. Note that as $\ell \rightarrow \infty ,$ $P_{\ell }(\cos \theta )\rightarrow 0$ for all $\theta \neq 0,\pi ,$ whence in both cases the trispectrum at the Poles has a dominating behavior with respect to almost all other directions.
Some Concluding Remarks \[conclusions\]
=======================================
In this paper, we have shown that convex regularization of spherical isotropic Gaussian fields with a Fourier dictionary does not preserve in general the Gaussianity and isotropy properties of the input random fields. We refer to [@feeney] for more discussions form a physical point of view and ample numerical evidence to illustrate these claims, in a setting related to Cosmological data analysis.
In a nutshell, our arguments can be summarized as follows. The result of convex regularization is basically a form of soft-thresholding on the spherical harmonic coefficients $\left\{
a_{\ell m}\right\}.$ These coefficients are hence independent and nonGaussian, whence anisotropy follows. Indeed, this finding complements earlier results from [@BaMa; @BMV; @MarPecBook; @BT], entailing that independent coefficients in a spherical harmonic basis are necessarily Gaussian under isotropy, and therefore cannot be sparse in the usual meaning with which this concept is understood.
It seems hence quite natural to extend our results and to suggest that for Gaussian isotropic random fields defined on homogeneous spaces of noncommutative groups, sparsity cannot be imposed on the random coefficients of a Fourier basis. The crucial difficulty here is the choice of a Fourier basis as a sparsity dictionary in a noncommutative setting; in particular it should be noted that our arguments do not entail that anisotropy will arise when choosing, for instance, a wavelet frame as a dictionary. Likewise, no anisotropy would arise for homogeneous spaces of commutative groups: for instance, soft- or hard- thresholding the random Fourier coefficients of isotropic random fields on the circle does not make these random fields anisotropic. It is indeed the noncommutative manifold structure of the sphere, and in particular the multiplicity of eigenfunctions corresponding to the same eigenvalue, which brings in a conflict between independence and nonGaussianity, under isotropy assumptions; because the random spherical harmonics coefficients arising from convex regularization are independent and nonGaussian, anisotropy follows.
The paper has presented a number of characterizations of such anisotropy as a function of the angular power spectra of the input fields; the relevance of these findings for the areas where convex regularization is exploited as a preliminary step for spherical data analysis is going to be investigated elsewehere.
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---
abstract: 'We give a class of examples of reducible threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such stable maps occur in infinitely many degrees.'
address: 'Department of Mathematics, UC Davis, One Shields Ave, Davis, CA 95616'
author:
- 'A. Zahariuc'
title: 'Reducible Calabi-Yau threefolds with countably many rational curves'
---
Introduction
============
A rational curve on a smooth complex projective variety $X$ is an integral curve $C \subset X$ of geometric genus zero. We say that the rational curve $C \subset X$ is **isolated** if the map $\smash{\tilde{C} \to X}$ from the normalization doesn’t fit in any nonconstant family of maps $\varphi:T \times {\mathbb P}^1 \to X$ over an integral base $T$, where such a family of maps ought to be considered constant if the image of $\varphi$ is a curve.
\[hard question\] Does there exist a Calabi-Yau threefold which contains infinitely many rational curves and all its rational curves are isolated?
A negative answer would be shocking because the property is expected to hold for “most” Calabi-Yau threefolds. However, to the best of the author’s knowledge, the answer is not actually known and the proof is likely currently out of reach. In other words, it doesn’t seem to be known whether any nontrivial analogues of Clemens’ conjecture [@[Cl86]] hold, although it is known that some fail [@[Vo03]].
Further motivation for studying this question comes from work of Voisin proving that Clemens’ conjecture is incompatible with a longstanding conjecture of Lang [@[Vo04]] now widely believed to be false. It is possible that examples which confirm Question \[hard question\] could disprove Lang’s conjecture.
The main purpose of this note is to formulate a weaker version of the question above by allowing reducible threefolds and to give an affirmative answer to it.
\[reducible CY\] A reducible Calabi-Yau threefold $X$ is a reducible, reduced, connected, separated, proper scheme over ${\mathbb C}$ of dimension $3$ with simple normal crossing singularities such that
(T1) $X$ is d-semistable [@[Fr83]];
(T2) the dualizing sheaf $\omega_X$ is trivial;
(T3) $\mathrm{H}^1(X,{{\mathcal}O}_X)=0$.
We say that $X$ is a *two-piece reducible Calabi-Yau threefold* if it consists of two smooth irreducible components $Y_1$ and $Y_2$ intersecting transversally along a smooth surface $S$.
D-semistability is a necessary, but not sufficient [@[PP83]], condition for the existence of smoothings. Of course, if $X$ is a two-piece reducible (Calabi-Yau) threefold, d-semistability amounts to ${{\mathcal}N}_{S/Y_1} \otimes {{\mathcal}N}_{S/Y_2} \cong {{\mathcal}O}_S$.
Even phrasing the correct analogue of Question \[hard question\] for singular or reducible threefolds is a hard technical matter. Luckily, since our examples will only have two components, we may use the language of relative stable maps [@[Li01]; @[Li02]] to achieve that in satisfactory generality.
Given a two-piece reducible Calabi-Yau threefold $X=Y_1 \cup_S Y_2$, let $\overline{{\mathscr}M}({\mathfrak Y}_i^\mathrm{rel},\Gamma_i)$ be the spaces of relative stable maps to $Y_i$ of topological type $\Gamma_i$. The data in $\Gamma_i$ includes an edgeless graph $G(\Gamma_i)$ with vertices $V(\Gamma_i)$ corresponding to the connected components of the relative stable maps and roots $R(\Gamma_i)$ corresponding to the distinguished marked points, as well as the data of the arithmetic genera and degrees relative to some ample line bundles of the connected components and weights $m_{\rho}$ of the roots corresponding to the multiplicities at the distinguished marked points. Denote the evaluation morphisms at the distinguished marked points by $${\mathbf q}_i : \overline{{\mathscr}M}({\mathfrak Y}_i^\mathrm{rel},\Gamma_i) \longrightarrow S^{r(\Gamma_i)},$$ where $r(\Gamma_i) = |R(\Gamma_i)|$. Two topological types $\Gamma_1$ and $\Gamma_2$ are compatible if $r(\Gamma_1)=r(\Gamma_2)$ and any two identically indexed roots in $R(\Gamma_1)$ and $R(\Gamma_2)$ (thus corresponding to distinguished marked points which are to be glued) have equal weights.
Since we are morally concerned with limits of genus zero maps, it is understood that all compatible pairs $(\Gamma_1,\Gamma_2)$ we will consider have zero arithmetic genus on all vertices and moreover, the glued graph $G(\Gamma_1) \cup_{R(\Gamma_1) \cong R(\Gamma_2)} G(\Gamma_2)$ is a tree.
\[main theorem\] There exists a two-piece reducible Calabi-Yau threefold $X=Y_1 \cup_S Y_2$ which contains a countable infinite number of “admissible rational curves.” More precisely, it satisfies the following sufficient conditions:
(T4) $S$ contains no rational curves;
(C1) no family of relative stable maps in any geometric fiber of the distinguished evaluation morphisms ${\mathbf q}_i$ sweeps out a surface on $Y_i$ for $i \in \{1,2\}$;
(C2) for any pair $(\Gamma_1,\Gamma_2)$ of *compatible* topological types, the intersection of the images of ${\mathbf q}_1$ and ${\mathbf q}_2$ is a finite set of points of $S^r$, where $r=r(\Gamma_1)=r(\Gamma_2)$;
(C3) there exist infinitely many pairs $(\Gamma_1,\Gamma_2)$ of compatible topological types for which there exist $[f_i] \in \overline{{\mathscr}M}({\mathfrak Y}_i^\mathrm{rel},\Gamma_i)({\mathbb C})$ with smooth sources, $i=1,2$, such that each $f_i$ maps birationally onto its image and without contracted components into the trivial expansion $Y_i[0] = Y_i$ and ${\mathbf q}_1([f_1]) = {\mathbf q}_2([f_2])$.
In our example, $S$ is an abelian surface, so (T4) is trivially satisfied. Granted, this condition is not strictly necessary, but it allows to phrase the others in a cleaner way. Conditions (C1) and (C2) imply that all “admissible rational curves” are isolated and (C3) says that such “curves” occur infinitely often.
Regarding (C1), note that by (T4), families of genus zero relative stable maps inside a given geometric fiber of the distinguished evaluation morphism ${\mathbf q}_i$ trivially also cannot sweep out surfaces on any additional component of an expansion of $Y_i$. Condition (C3) is not particularly elegant or interesting and perhaps could be skipped for the (somewhat open-ended) purposes of paper. However, it is required in the proof of the remark below, which justifies why the reducible threefolds in Theorem \[main theorem\] count as having an “infinite countable number of rational curves.”
If $W \to B$ is a one-dimensional family of smooth Calabi-Yau threefolds degenerating at $0 \in B({{{\mathbb C}}})$ to a two-piece reducible Calabi-Yau threefold $W_0$ which satisfies the conditions in Theorem \[main theorem\], then the (very) general member $W_b$ of the family contains a countable infinite number of rational curves.
The fact that all rational curves on $W_b$ are isolated follows from (C1) and (C2). The fact that there are infinitely many such rational curves follows from (C3). Indeed, the main point is that, by rigidity of the degenerate stable maps, the fact that $\overline{{\mathscr}M}({\mathfrak W},\Gamma)$ admits an obstruction theory of the expected dimension $1$ [@[Li02] §§1.2] implies that rigid degenerate stable maps do deform to the smooth members of the family. Imitating the argument in the second half of the proof of [@[Za15] Theorem 1.1], we deduce that all the nodes of the degenerate stable maps are smoothed out in this process, so the sources are all irreducible. It is also clear that no multiple covers are obtained in this way.
Of course, the reason why this doesn’t answer Question \[hard question\] is that d-semistability is not sufficient to ensure the existence of smoothings. The author does not know if any of the reducible threefolds constructed in the proof of Theorem \[main theorem\] are actually smoothable. On one hand, since it is conjectured that there are only finitely many deformation classes of Calabi-Yau threefolds, the fact that the discrete parameter in our construction can a priori take infinitely many values could be interpreted as a red flag, though not necessarily. On the other hand, the fact that the general idea of the construction is simply to imitate that in [@[Za15]], which is automatically smoothable, might be a positive sign. Also, it is likely possible to tweak the current construction in many ways.
Construction of the threefolds
==============================
Let $(E,q)$ be an elliptic curve with a marked point $q \in E({{\mathbb C}})$. Let $S = E^2$ be the square with the projections to the two factors denoted by $\pi_1$ and $\pi_2$, diagonal $\Delta \subset S$ and second diagonal $$\Delta'=\{(p_1,p_2) \in S({{\mathbb C}}): p_1+p_2 \sim 2q\}^\mathrm{cl},$$ where the “cl” superscript means closure. Note that $$\label{diags}
\Delta +\Delta' \sim 2\pi_1^{-1}(q) + 2\pi_2^{-1}(q).$$ Indeed, the difference between the two divisors is linearly equivalent to $0$ on any geometric fiber of either projection, which implies that it is both a pullback of a divisor from $E$ via $\pi_1$ and a pullback of a class divisor from $E$ via $\pi_2$. However, this is clearly only possible if the two divisors classes are $0$.
Fix an integer $m$ and consider the line bundle $${{\mathcal}L} = {{\mathcal}O}_S((m+3) \Delta - m \pi_2^{-1}(q)).$$ It is clear that $\mathrm{R}^1(\pi_1)_* {{\mathcal}L} = 0$ and that $(\pi_1)_* {{\mathcal}L}$ is a locally free sheaf of rank $3$. First, we check that $$\label{eq1}
\det (\pi_1)_* {{\mathcal}L} = {{\mathcal}O}_E(-m(m+3)q).$$ Since the relative tangent bundle of $\pi_1$ is trivial, the Grothendieck-Riemann-Roch theorem implies $\mathrm{ch}((\pi_1)_*{{\mathcal}L}) = (\pi_1)_* \mathrm{ch}({{\mathcal}L})$, hence $$c_1((\pi_1)_*{{\mathcal}L}) = (\pi_1)_* \frac{c_1({{\mathcal}L})^2}{2} = -m(m+3)[q] \in A_0(E),$$ confirming (\[eq1\]) since the first Chern class of a vector bundle coincides with the first Chern class of the determinant bundle.
Consider the projective bundle $$\pi:P_m := \mathbf{Proj}_E \mathrm{Sym} (\pi_1)_* {{\mathcal}L}\longrightarrow E.$$ The bundle comes equipped with a tautological invertible sheaf ${{\mathcal}O}_{P_m}(-1)$.
\[normall\] If $L_{i,q} = \pi_i^{-1}(q)$, then $${{\mathcal}N}_{S/P_m} = {{\mathcal}O}_S(3(m+3)\Delta - 3mL_{2,q} + m(m+3)L_{1,q}).$$
The short exact sequence for the normal bundle of $S$ in $P_m$ implies that $${{\mathcal}N}_{S/P_m} = \det ({{\mathcal}T}_{P_m}|_S).$$ The short exact sequence for the relative tangent bundle of $\pi$ gives $\det {{\mathcal}T}_{P_m} = \det {{\mathcal}T}_{\pi}$, hence ${{\mathcal}N}_{S/P_m} = \det ({{\mathcal}T}_\pi|_S)$. Moreover, $$\begin{aligned}
\det {{\mathcal}T}_\pi &= \det {{\mathcal}Hom}({{\mathcal}O}_{P_m}(-1), \pi^*[(\pi_1)_*{{\mathcal}L}]^\vee/{{\mathcal}O}_{P_m}(-1)) \\
&= {{\mathcal}O}_{P_m}(3) \otimes \pi^* \det [(\pi_1)_*{{\mathcal}L}]^\vee \\
&= {{\mathcal}O}_{P_m}(3) \otimes \pi^* {{\mathcal}O}_E(m(m+3)q)
\end{aligned}$$ by (\[eq1\]), hence $$\begin{aligned}
{{\mathcal}N}_{S/P_m} &= {{\mathcal}L}^{\otimes 3} \otimes \pi_1^* {{\mathcal}O}_E(m(m+3)q) \\
&= {{\mathcal}O}_S(3(m+3)\Delta - 3mL_{2,q} + m(m+3)L_{1,q}),
\end{aligned}$$ as desired.
\[anticanonical\] $K_P+S = 0$, that is, $S$ is an anticanonical divisor on $P$.
By construction, $\{t\} \times E$ is a cubic hence an anticanonical divisor in $\pi^{-1}(t)$, so $\omega_{P/E}(S) = \pi^*{{\mathcal}M}$ for some line bundle ${{\mathcal}M} \in \mathrm{Pic}(E)$. However, $\det {{\mathcal}T}_{P_m} = \det {{\mathcal}T}_{\pi}$ as in the proof of Lemma \[normall\], hence $\omega_P(S) = \pi^*{{\mathcal}M}$ as well. By adjunction $$\pi_1^*{{\mathcal}M} = \omega_P(S)|_S = \omega_S = {{\mathcal}O}_S,$$ hence ${{\mathcal}M} = {{\mathcal}O}_E$ and $\omega_P(S) = {{\mathcal}O}_P$, as desired.
Assume that $m=2k$, where $k$ is a positive integer. Let $D_1$ be a divisor on $S \cong S_1$ consisting of very general[^1] translates of $\Delta$ such that $D_1 \sim (2k^2+12k+18) \Delta$ and $D_2 \subset S_2$ consisting of very general translates of $\Delta'$ such that $D_2 \sim 2k^2\Delta'$. Let $$Y_i:= \text{Blowup}_{D_i} P_{2k}$$ for $i=1,2$. Let $S_i \cong S$ be the proper transform of $S$ and $\varphi_i:Y_i \to E$ the natural morphism whose fibers are rational surfaces.
We construct the reducible threefold $X$ to be the transversal gluing of the disjoint threefolds $Y_1$ and $Y_2$ along $S_1 \cong S_2$ in such a way that the induced automorphism of $S \cong E^2$ is the involution which exchanges the two factors. One possible reference for gluing along closed subschemes is [@[Sch05] Corollary 3.9]. By a slight abuse of notation, we will continue to denote the intersection $Y_1 \cap Y_2$ by $S$. The notation for the projections to the two factors respects the notation in $Y_1$ and is opposite to that in $Y_2$. We will verify that $X$ satisfies the conditions of Theorem \[main theorem\].
The reducible threefold $X$ is a two-piece reducible Calabi-Yau threefold in the sense of Definition \[reducible CY\].
We simply need to verify conditions (T1)–(T3).
(T1) A straightforward calculation using (\[diags\]) and Lemma \[normall\] shows that $${{\mathcal}N}_{S/P_{2k}}^{\otimes 2} \cong {{\mathcal}O}_{S}(D_1 + D_2),$$ hence the d-semistability condition is verified since ${{\mathcal}N}_{S/Y_i} = {{\mathcal}N}_{S/P_{2k}}(-D_i)$ and $D_2$ is invariant under the involution of $S$ exchanging the factors.
(T2) This follows from Lemma \[anticanonical\] and the fact that the proper transform of an anticanonical divisor remains anticanonical if the center of the blowup was contained inside the former divisor.
(T3) By the usual short exact sequence, it suffices to verify that the map $${\mathrm H}^*({{\mathcal}O}_{Y_1}) \oplus {\mathrm H}^*({{\mathcal}O}_{Y_2}) \longrightarrow {\mathrm H}^*({{\mathcal}O}_S)$$ is surjective in degree zero, which is trivial, and injective in degree one. The latter is a consequence of the following easy facts: the diagram
\(a) at (0,0) [${\mathrm H}^1({{\mathcal}O}_E)$]{}; (b) at (3,0) [${\mathrm H}^1({{\mathcal}O}_S)$]{}; (c) at (3,1.5) [${\mathrm H}^1({{\mathcal}O}_{Y_i})$]{};
\(a) – (b); (a) – (c); (c) – (b);
at (1.5, 0.3) [$\pi_i^*$]{}; at (1.5, 1.1) [$\varphi_i^*$]{};
is commutative, $\varphi_i^*$ is an isomorphism, essentially because the cohomology of the structure sheaf is a birational invariant, and $\pi_1^* \oplus \pi_2^*$ is an isomorphism.
The genus zero stable maps
==========================
The rigidity requirement
------------------------
Of course, the key basic observation is that, because there are no nonconstant maps from a rational curve to a genus one curve, all connected components of the relative stable maps are sent to (expansions of the) fibers of the maps $\varphi_i$.
Condition (C1) is proved completely analogously to [@[Za15] Proposition 3.2]. However, we point out that the argument ultimately is entirely equivalent to applying the following elementary lemma.
If $E \subset W$ is a smooth connected divisor on a smooth rational projective surface $W$ such that $E + K_W \sim 0$, then the image-cycles of maps from rational trees into $W$ which intersect $E$ in any predetermined divisor do not cover the surface $W$ birationally.
If such a family exists, consider a semistable completion over a proper base. Properness mandates that some image-cycle of a map in the complete family contains $E$, but this is clearly impossible.
Assume by way of contradiction that (C2) fails. Let ${\mathbf q}$ be the fiber product of ${\mathbf q}_1$ and ${\mathbf q}_2$ over their common target. Then we may find a nonsingular connected affine curve $ \iota:Z \to \overline{{\mathscr}M}({\mathfrak Y}_1^\mathrm{rel},\Gamma_1) \times_{S^r} \overline{{\mathscr}M}({\mathfrak Y}_2^\mathrm{rel},\Gamma_2)$ such that ${\mathbf q} \iota$ is not constant. For each $v \in V:=V(\Gamma_1) \sqcup V(\Gamma_2)$, let $$\lambda_v:Z \longrightarrow E$$ be the function which specifies the fiber of $\varphi_i$ containing the connected component of the source of $(f_i)_z$ corresponding to the vertex $v \in V(\Gamma_i)$. Moreover, by further shrinking $Z$ if necessary, we may assume that all $(f_i)_z$ map to the same expansion $Y_i[n_i]_0$ of $Y_i$ for $i=1,2$.
For two vertices $v$ and $w$ which become adjacent after gluing the graphs in $\Gamma_1$ and $\Gamma_2$, denote by $[vw$ the root at $v$ “pointing towards” $w$. Let $R(v)$ be the set of all roots incident to $v$. Let $$\begin{aligned}
R_0(v) & := \left\{ \rho \in R(v): (\pi_{\rho} {\mathbf q} \iota)(z) \notin D_1 \cup D_2 \right\} \\
R_+(v) & := \left\{ \rho \in R(v): (\pi_{\rho} {\mathbf q} \iota)(z) \in D_j \backslash D_i \right\} \\
\end{aligned}$$ where $\pi_\rho:S^r \to S$ is the projection corresponding to the root $\rho$, and let $R_{\geq 0}$ be their union. Shrinking $Z$ even more, we may assume that the properties whether $(\pi_{\rho} {\mathbf q} \iota)(z) \in D_1$ or $D_2$ or neither hold either for all $z \in Z$ or for none.
Assume that $v \in V(\Gamma_i)$. Let $d_v \geq 0$ be the degree of the pullback of ${{\mathcal}O}_{P_{2k}}(1)$ to the connected component corresponding to $v $. Let $\smash{ L_v = \pi_i^{-1}(\lambda_v(z)) \subset S }$ and $\smash{ P_v = \pi^{-1}(\lambda_v(z))}$ living inside $P_{2k}$. Let $C_v \subset P_v$ be the image-cycle of the restriction of the map $f_i$ to the connected component corresponding to the vertex $v$, so that $C_v$ has degree $d_v$. For each $\alpha \in L_v$, let $$m_{v,\alpha} = (C_v \cdot L_v)_{\alpha,P_v},$$ the intersection multiplicity at $\alpha$. If $\alpha \in L_v \cap D_i$, let $\hat{\lambda}_{v,\alpha}(z) = \pi_j(\alpha) \in E$, where $\{j\} = \{1,2\} \backslash \{i\}$.
\[linear equivalence\] The following linear equivalence holds $$\label{main}
\sum_{[vw \in R_{\geq 0}(v)} m_{[vw} \lambda_w(z) + \sum_{ \alpha \in L_v \cap D_i} m_{v,\alpha}\hat{\lambda}_{v,\alpha}(z) \sim_{\mathrm{rat}} d_v((2k+3)\lambda_v(z) - 2kq).$$ In particular, the sum of the coefficients on the left is $3d_v$.
The basic idea is that the hyperplane divisor class of $P_v$ restricts on $L_v \cong E$ to ${{\mathcal}O}_E((2k+3)\lambda_v(z) - 2kq)$. Let $p_{[vw} \in L_v \subset S$ be the image of the distinguished marked point corresponding to the root $[vw$. Because all bubbles of the expansions are ${\mathbb P}^1$-bundles over $S$, condition (T4) and the admissibility condition *along the chain of bubbles* imply that for each closed point $\alpha \in L_v \backslash D_i$, $$\sum_{p_{[vw} = \alpha} m_{[vw} = m_{v,\alpha}$$ and (\[main\]) follows after summing over all $\alpha \in L_v$.
Fix $z \in Z({{\mathbb C}})$ and a nonzero tangent vector $\mathbf{u}$ to $Z$ at $z$. For each $v \in V$, let $$\lambda_v'=\frac{\mathrm{d}\lambda_v}{\mathrm{d}z} := \mathrm{d}\lambda_v({\mathbf u}) \in {{\mathbb C}},$$ where a trivialization of ${{\mathcal}T}_E$ has been fixed beforehand. The derivative of $\smash{ \hat{\lambda}_{v,\alpha}(z) }$ can be defined similarly. However, note that $\smash{ \hat{\lambda}'_{v,\alpha} \in \{\pm \lambda'_v\} }$ and also $\lambda'_w \in \{\pm \lambda'_v\}$ if $[vw \in R_+(v)$. Taking derivatives in (\[main\]), we obtain $$d_v(2k+3)\frac{\mathrm{d}\lambda_v}{\mathrm{d}z} = \sum_{[vw \in R_{\geq 0}(v)} m_{[vw} \frac{\mathrm{d}\lambda_w}{\mathrm{d}z} + \sum_{ \alpha \in L_v \cap D_i} m_{v,\alpha}\frac{\mathrm{d}\hat{\lambda}_{v,\alpha}}{\mathrm{d}z}.$$ The key idea is to choose $v \in V$ for which $|\lambda'_v|$ is *maximal*. It is possible to choose such a $v$ such that $d_v \neq 0$ thanks to the kissing condition, the connectivity of the glued graph and the fact that $D_1 \cap D_2$ is finite. By the triangle inequality, $$\begin{aligned}
d_v(2k+3) \left| \frac{\mathrm{d}\lambda_v}{\mathrm{d}z} \right| & \leq \sum_{[vw \in R_{\geq 0}(v)} m_{[vw} \left| \frac{\mathrm{d}\lambda_w}{\mathrm{d}z} \right| + \sum_{ \alpha \in L_v \cap D_i} m_{v,\alpha} \left| \frac{\mathrm{d}\hat{\lambda}_{v,\alpha}}{\mathrm{d}z} \right| \\
& \leq \left[ \sum_{[vw \in R_{\geq 0}(v)} m_{[vw} + \sum_{ \alpha \in L_v \cap D_i} m_{v,\alpha} \right] \left| \frac{\mathrm{d}\lambda_v}{\mathrm{d}z} \right| = 3d_v \left| \frac{\mathrm{d}\lambda_v}{\mathrm{d}z} \right|,
\end{aligned}$$ hence $\lambda'_v = 0$ for the chosen $v$ and thus $\lambda'_v = 0$ for all $v$. Then all $\lambda_v$ are constant which contradicts the assumption that ${\mathbf q} \iota$ is not constant. This completes the proof that (C2) is satisfied.
The existence requirement
-------------------------
Finally, we need to check that condition (C3) holds. There is a lot of leeway in constructing the desired pairs of relative stable maps. Below is one possibility.
We start by introducing some overdue notation. For simplicity, write $N_1 = 2k^2+12k+18$ and $N_2 = 2k^2$. Let $D_1 = \Delta_1+\Delta_2+...+\Delta_{N_1}$ and $D_2 = \Delta'_1+\Delta'_2+...+\Delta'_{N_2}$, $$\begin{aligned}
\Delta_j &=\{(p_1,p_2) \in S({{\mathbb C}}): p_1-p_2 \sim q_j-q\}^\mathrm{cl}, \\
\Delta'_j &=\{(p_1,p_2) \in S({{\mathbb C}}): p_1+p_2 \sim q'_j+q\}^\mathrm{cl}.
\end{aligned}$$ All that we need from the assumption that the translates are very general is that $$\label{sole}
q_1+q_2+...+q_{N_1} \sim_E N_1q \text{ and } q'_1+q'_2+...+q'_{N_2} \sim_E N_2q$$ and their combinations are the only nontrivial relations involving the $q$s.
We will construct a chain of lines in $X$. More accurately, we desire to find curves $\ell_1,\ell_3,...,\ell_{n-1} \subset Y_1$ and $\ell_2,\ell_4,...,\ell_n \subset Y_2$ and points $p_1,p_2,...,p_{n-1} \in S$ satisfying the following properties:
$\bullet$ $p_j \in \ell_j \cap \ell_{j+1}$ for all $j$;
$\bullet$ $\ell_1$ is the proper transform under the blowup $Y_1 \to P_{2k}$ of a line inside some fiber of $\pi$ and $\ell_1$ intersects the exceptional divisors corresponding to $\Delta_1$ and $\Delta_2$;
$\bullet$ $\ell_j$ is the proper transform under the blowup $Y_i \to P_{2k}$ of a line inside some fiber of $\pi$ and $\ell_j$ intersects the exceptional divisor corresponding to $\Delta_3$ or $\Delta'_3$, for $2 \leq j \leq n-1$;
$\bullet$ $\ell_n$ is the proper transform under the blowup $Y_2 \to P_{2k}$ of a line inside some fiber of $\pi$ and $\ell_n$ intersects the exceptional divisors corresponding to $\Delta'_1$ and $\Delta'_2$;
$\bullet$ all $\ell_j$ live inside different fibers of $\varphi_1$ and $\varphi_2$. The only closed conditions come from Lemma \[linear equivalence\], which in this case reads $$\label{mess}
(2k+3) \lambda_j - 2kq \sim_E
\begin{cases}
2\lambda_1 + \lambda_2 + 2q -q_1-q_2 & \text{if } j=1, \\
\lambda_{j-1} -\lambda_j + \lambda_{j+1} +q + q'_3 & \text{if } j \leq n-2 \text{ even}, \\
\lambda_{j-1} +\lambda_j + \lambda_{j+1} +q - q_3 & \text{if } j \geq 3 \text{ odd}, \\
\lambda_{n-1} - 2\lambda_n + q'_1+ q'_2 +2q & \text{if } j = n. \\
\end{cases}$$ Let $f_1$ and $f_2$ be the inclusions $$\begin{aligned}
f_1 &: \ell_1 \sqcup \ell_3 \sqcup \ell_5 \sqcup ... \hookrightarrow Y_1 \\
f_2 &: \ell_2 \sqcup \ell_4 \sqcup \ell_6 \sqcup ... \hookrightarrow Y_2.
\end{aligned}$$ The fact that the construction above is sound, which amounts to saying that the points $p_j$ don’t belong to the exceptional divisors of the blowups $Y_i \to P_{2k}$, and that the conditions in (C3) are satisfied turns out to boil down to the following straightforward computation due to the uniqueness of the relations (\[sole\]): the $n \times n$ symmetric tridiagonal matrix $$\begin{bmatrix*}[c]
2k+1 & -1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\
-1 & 2k+4 & -1 & 0 & 0 & \cdots & 0 & 0 & 0 \\
0 & -1 & 2k+2 & -1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 0 & -1 & 2k+4 & -1 & \cdots & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 2k+2 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & 0 & \cdots & 2k+4 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & \cdots & -1 & 2k+2 & -1 \\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & -1 & 2k+5 \\
\end{bmatrix*}$$ is invertible and all the entries in the first row/column of the inverse matrix are pairwise distinct. The calculation is uninteresting and as such will be skipped.
*Acknowledgements.* This paper is morally related to the earlier work [@[Za15]] and I would like to reiterate my thanks to Joe Harris, Xi Chen and the others acknowledged in that paper. I would also like to thank Brian Osserman for many valuable suggestions and Andreas Knutsen for useful discussions.
[pppppp]{}
H. Clemens, *Curves on higher-dimensional complex projective manifolds*, Proc. Int. Cong. Math., Berkeley 634–640 (1986) R. Friedman, *Global smoothings of varieties with normal crossings*, Ann. of Math. (2) **118**, 75–114 (1983) J. Li, *Stable morphisms to singular schemes and relative stable morphisms*, J. Diff. Geom. **57**, 509–578 (2001) J. Li, *A degeneration formula of GW-invariants*, J. Diff. Geom. **60**, 199–293 (2002) U. Persson and H. Pinkham, *Some examples of nonsmoothable varieties with normal crossings*, Duke Math. J. **50**, 477 – 486 (1983) K. Schwede, *Gluing schemes and a scheme without closed points*, Recent progress in arithmetic and algebraic geometry, Contemp. Math. **386**, 157–172 (2005) C. Voisin, *On some problems of Kobayashi and Lang; algebraic approaches*, in “Current developments in mathematics” 53–125 (2003) C. Voisin, *A geometric application of Nori’s connectivity theorem*, Ann. Sc. Norm. Super. Pisa Cl. Sci. **3**, 637–656 (2004) A. Zahariuc, *Deformation of quintic threefolds to the chordal variety*, Trans. Amer. Math. Soc. (to appear)
[^1]: The “very general” assumption is actually *not* essential. It is only used in §§3.2 and even there it is likely avoidable at the expense of further complications.
|
---
abstract: 'We find a new relation among codimension $2$ algebraic cycles in the moduli space ${\overline{\mathcal{M}}}_{1,4}$, and use this to calculate the elliptic Gromov-Witten invariants of projective spaces ${\mathbb{CP}}^2$ and ${\mathbb{CP}}^3$.'
address: 'Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, D-53225 Bonn, Germany'
author:
- 'E. Getzler'
title: 'Intersection theory on ${\overline{\mathcal{M}}}_{1,4}$ and elliptic Gromov-Witten invariants'
---
In this paper, we find a new relation among codimension $2$ algebraic cycles in ${\overline{\mathcal{M}}}_{1,4}$. The main application of the new relation is to the calculation of elliptic Gromov-Witten invariants. For example, we show that if $V$ has no primitive cohomology in degrees above $2$, the elliptic Gromov-Witten invariants are determined by the elliptic Gromov-Witten invariants $${\langle}I_{1,1,\beta}^V{\rangle}: H^{2(i+1)}(V,{\mathbb{Q}})\to{\mathbb{Q}}, \quad 0\le
i=c_1(V)\cap\beta<\dim(V) ,$$ together with the rational Gromov-Witten invariants.
In [@elliptic3], we will prove, using mixed Hodge theory, that the cycles $[{\overline{\mathcal{M}}}(G)]$, as $G$ ranges over all stable graphs of genus $1$ and valence $n$, span the even dimensional homology of ${\overline{\mathcal{M}}}_{1,n}$, and that the new relation, together with those already known in genus $0$, generate all relations among these cycles. This result is the analogue, in genus $1$, of a theorem of Keel [@Keel] in genus $0$.
Our new relation is closely related to a relation in $A_2({\overline{\mathcal{M}}}_3){\otimes}{\mathbb{Q}}$ discovered by Faber (Lemma 4.4 of [@Faber]); the image of his relation in $H_4({\overline{\mathcal{M}}}_3,{\mathbb{Q}})$ under the cycle map is the same as the push-forward of our relation under the map ${\overline{\mathcal{M}}}_{1,4}\to{\overline{\mathcal{M}}}_3$ obtained by contracting the $4$ tails pairwise. This suggests that our new relation should actually be a rational equivalence[^1].
Let us illustrate our results with the case of the projective plane. The genus $0$ and genus $1$ potentials of ${\mathbb{CP}}^2$ equal $$\begin{aligned}
F_0({\mathbb{CP}}^2) &= \frac12 (t_0t_1^2 + t_0^2t_2) + \sum_{n=1}^\infty N^{(0)}_n q^n
e^{nt_1} \frac{t_2^{3n-1}}{(3n-1)!} , \\
F_1({\mathbb{CP}}^2) &= - \frac{t_1}{8} + \sum_{n=1}^\infty N^{(1)}_n q^n e^{nt_1}
\frac{t_2^{3n}}{(3n)!} , \\\end{aligned}$$ where $t_0$, $t_1$ and $t_2$ are formal variables, of degree $-2$, $0$ and $2$ respectively, dual to the classes $1\in H^0({\mathbb{CP}}^2,{\mathbb{Q}})$, ${\omega}\in
H^2({\mathbb{CP}}^2,{\mathbb{Q}})$ and ${\omega}^2\in H^4({\mathbb{CP}}^2,{\mathbb{Q}})$ respectively, and $N^{(0)}_n$ and $N^{(1)}_n$ are the number of rational, respectively elliptic, plane curves of degree $n$ which meet $3n-1$, respectively $3n$, generic points. Kontsevich and Manin [@KM] establish the recursion relation $$N^{(0)}_n = \sum_{n=i+j} \textstyle
\bigl( \binom{3n-4}{3i-2} i^2j^2 - i^3j \binom{3n-4}{3i-1} \bigr) N^{(0)}_i
N^{(0)}_j ,$$ which, together with the initial condition $N^{(0)}_1=1$, determines the coefficients $N^{(0)}_n$. In Section 2, we prove that the coefficients $N^{(1)}_n$ satisfy the recursion $$\begin{aligned}
\label{recursion}
6N^{(1)}_n & = \sum_{n=i+j+k} {\textstyle
\binom{3n-2}{3j-1,3k-1} ij^3k^3 (2i-j-k) N^{(1)}_{i}
N^{(0)}_{j} N^{(0)}_{k} } \\
& {} + 2 \sum_{n=i+j} \Bigl( {\textstyle \binom{3n-2}{3i} ij^2(8i-j) -
\binom{3n-2}{3i-1} 2(i+j)j^3 } \Bigr) N^{(1)}_{i} N^{(0)}_{j} \notag \\
& {} - \frac{1}{24} \biggl( \sum_{n=i+j} {\textstyle
\binom{3n-2}{3i-1} (n^2-3n-6ij)i^3j^3 N^{(0)}_{i} N^{(0)}_{j} } + 6n^3(n-1)
N^{(0)}_n \biggr) . \notag\end{aligned}$$ In Table 1, we list the first few coefficients $N^{(1)}_n$; for comparison, we also include the corresponding rational Gromov-Witten invariants. We have checked that our results for $N^{(1)}_n$ agree in degrees up to $6$ with those obtained by Caporaso and Harris [@CH].
Recently, Eguchi, Hori and Ziong [@EHX] have proposed a bold conjecture, generalizing the conjectured of Witten [@Witten] and proved by Kontsevich [@KdV] that the Gromov-Witten invariants of a point (“in the large phase space”) are the highest weight vector for a certain Virasoro algebra. Their conjecture implies in particular the recursion $$N^{(1)}_n = {\textstyle\frac{1}{12} \binom{n}{3} N^{(0)}_n + \frac{1}{9}}
\sum_{n=i+j} \textstyle \binom{3n-1}{3i-1} (3i^2-2i)j N^{(0)}_i N^{(1)}_j ,$$ which is far simpler than ours. Pandharipande [@Pandharipande] has proved that this recursion is a formal consequence of .
\[CP2\]
$$\begin{tabular}{|C|R|R|} \hline
n & N^{(0)}_n \quad\quad\quad\quad & N^{(1)}_n \quad\quad\quad\quad \\ \hline
1 & 1 & 0 \\
2 & 1 & 0 \\
3 & 12 & 1 \\
4 & 620 & 225 \\
5 & 87\,304 & 87\,192 \\
6 & 26\,312\,976 & 57\,435\,240 \\
7 & 14\,616\,808\,192 & 60\,478\,511\,040 \\
8 & 13\,525\,751\,027\,392 & 96\,212\,546\,526\,096 \\ \hline
\end{tabular}$$
The situation for the elliptic Gromov-Witten invariants of ${\mathbb{CP}}^3$ is a little more complicated. The genus $0$ and $1$ potentials of ${\mathbb{CP}}^3$ have the form $$\begin{aligned}
F_0({\mathbb{CP}}^3) &= \frac{t_0^2t_3}{2} + t_0t_1t_2 + \frac{t_1^3}6 +
\sum_{n=1}^\infty \sum_{4n=a+2b} N^{(0)}_{ab} q^n e^{nt_1}
\frac{t_2^at_3^b}{a!b!} , \\
F_1({\mathbb{CP}}^3) &= - \frac{t_1}{4} + \sum_{n=1}^\infty \sum_{4n=a+2b}
N^{(1)}_{ab} q^n e^{nt_1} \frac{t_2^at_3^b}{a!b!} ,\end{aligned}$$ where $t_i$ is the formal variable, of degree $2i-2$, dual to ${\omega}^i\in
H^{2i}({\mathbb{CP}}^3,{\mathbb{Q}})$, and $N^{(g)}_{ab}$ is the Gromov-Witten invariant which “counts” the stable maps of genus $g$ and degree $n$ to ${\mathbb{CP}}^3$ which meet $a$ generic lines and $b$ generic points. As we show in Section 6, the elliptic Gromov-Witten invariants are no longer positive integers: for example, $N^{(1)}_{02}=-1/12$. In [@cp3], we use the methods of this paper to prove that the linear combination $N^{(1)}_{ab}+(2n-1)N^{(0)}_{ab}/12$ counts the number of elliptic space curves which meet $a$ generic planes and $b$ generic points.
**Acknowledgments.** Conversations with K. Behrend, E. Looijenga, Yu. Manin and especially with C. Faber, enabled me to write this paper at all. T. Graber and R. Pandharipande informed the author of some erroneous statements in the preprint of the paper.
I am very grateful to Yu. Manin, D. Zagier and the Max-Planck-Institut für Mathematik in Bonn, where this paper was conceived, and to A. Kupiainen and the Finnish Mathematical Society for an invitation to Helsinki University, where much of it was finished.
The author is partially supported by the NSF.
Intersection theory on ${\overline{\mathcal{M}}}_{1,4}$
=======================================================
In this section, we calculate the relations among certain codimension two cycles in ${\overline{\mathcal{M}}}_{1,4}$; one such relation was known, and we find that there is one new one.
First, we assign names to the codimension $1$ strata of ${\overline{\mathcal{M}}}_{1,4}$. Denote by $\Delta_0$ the boundary stratum of irreducible curves in ${\overline{\mathcal{M}}}_{1,4}$, associated to the stable graph $$\begin{picture}(80,45)(40,745)
\put( 20,765){$\Delta_0 =$}
\put( 80,775){\circle{30}}
\put( 80,760){\line(-3,-4){ 15}}
\put( 80,760){\line(-1,-4){ 5}}
\put( 80,760){\line( 1,-4){ 5}}
\put( 80,760){\line( 3,-4){ 15}}
\end{picture}$$ For each subset $S$ of $\{1,2,3,4\}$ of cardinality at least $2$, let $\Delta_S$ be the boundary stratum associated to the stable graph with two vertices, of genus $0$ and $1$, one edge connecting them, and with those tails labelled by elements of $S$ attached to the vertex of genus $0$; there are $11$ such graphs. In our pictures, we denote genus $1$ vertices by a hollow dot, leaving genus $0$ vertices unmarked. For example, $$\begin{picture}(35,80)(60,722)
\put( 20,760){$\Delta_{\{1,2\}} =$}
\put( 80,780){\circle{5}}
\put( 80,777){\line( 0,-1){ 38}}
\put( 80,740){\line(-2,-3){ 10}}
\put( 80,740){\line( 2,-3){ 10}}
\put( 83,782){\line( 2, 3){ 10}}
\put( 77,782){\line(-2, 3){ 10}}
\put( 65,715){$1$}
\put( 89,715){$2$}
\put( 65,800){$3$}
\put( 89,800){$4$}
\end{picture}$$ We only need the three ${S}_4$-invariant combinations of these $11$ strata, which are as follows: $$\begin{aligned}
\Delta_2 &= \Delta_{\{1,2\}} + \Delta_{\{1,3\}} + \Delta_{\{1,4\}} +
\Delta_{\{2,3\}} + \Delta_{\{2,4\}} + \Delta_{\{3,4\}} , \\
\Delta_3 &= \Delta_{\{1,2,3\}} + \Delta_{\{1,2,4\}} + \Delta_{\{1,3,4\}} +
\Delta_{\{2,3,4\}} , \\
\Delta_4 &= \Delta_{\{1,2,3,4\}} .\end{aligned}$$ In summary, there are four invariant combinations of boundary strata: $\Delta_0$, $\Delta_2$, $\Delta_3$ and $\Delta_4$.
We now turn to enumeration of the codimension two strata. These fall into two classes, distinguished by whether they are contained in the irreducible stratum $\Delta_0$ or not. We start by listing those which are not; each of them is the intersection of a pair of boundary strata $\Delta_S{\cdot}\Delta_T$. We give four examples: from these, the other strata may be obtained by the action of ${S}_4$: $$\begin{picture}(100,95)(30,715)
\put(-20,755){$\Delta_{\{1,2\}} {\cdot}\Delta_{\{3,4\}} =$}
\put( 80,760){\circle{5}}
\put( 80,757){\line( 0,-1){ 18}}
\put( 80,762){\line( 0, 1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 10}}
\put( 80,780){\line( 1, 2){ 10}}
\put( 80,780){\line(-1, 2){ 10}}
\put( 67,803){$1$}
\put( 87,803){$2$}
\put( 67,710){$3$}
\put( 87,710){$4$}
\end{picture}
\begin{picture}(100,85)(-20,700)
\put(-30,740){$\Delta_{\{1,2\}} {\cdot}\Delta_{\{1,2,3\}} =$}
\put( 80,760){\circle{5}}
\put( 80,757){\line( 0,-1){ 18}}
\put( 80,762){\line( 0, 1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 20}}
\put( 70,720){\line( 1,-2){ 10}}
\put( 56,690){$1$}
\put( 77,690){$2$}
\put( 87,710){$3$}
\put( 77,783){$4$}
\end{picture}$$
$$\begin{picture}(100,75)(30,690)
\put(-40,740){$\Delta_{\{1,2\}} {\cdot}\Delta_{\{1,2,3,4\}} =$}
\put( 80,760){\circle{5}}
\put( 80,757){\line( 0,-1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 20}}
\put( 70,720){\line( 1,-2){ 10}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 56,690){$1$}
\put( 77,690){$2$}
\put( 77,710){$3$}
\put( 87,710){$4$}
\end{picture}
\begin{picture}(100,75)(-40,690)
\put(-50,740){$\Delta_{\{1,2,3\}} {\cdot}\Delta_{\{1,2,3,4\}} =$}
\put( 80,760){\circle{5}}
\put( 80,757){\line( 0,-1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 20}}
\put( 70,720){\line( 1,-2){ 10}}
\put( 70,720){\line( 0,-1){ 20}}
\put( 56,690){$1$}
\put( 67,690){$2$}
\put( 77,690){$3$}
\put( 87,710){$4$}
\end{picture}$$
The ${S}_4$-invariant combinations of these strata are as follows: $$\begin{aligned}
\Delta_{2,2} &= \Delta_{\{1,2\}} {\cdot}\Delta_{\{3,4\}} + \Delta_{\{1,3\}} {\cdot}\Delta_{\{2,4\}} + \Delta_{\{1,4\}} {\cdot}\Delta_{\{2,3\}} , \\
\Delta_{2,3} &= \Delta_{\{1,2\}} {\cdot}\Delta_{\{1,2,3\}} + \Delta_{\{1,2\}}
{\cdot}\Delta_{\{1,2,4\}} + \Delta_{\{1,3\}} {\cdot}\Delta_{\{1,2,3\}} +
\Delta_{\{1,3\}} {\cdot}\Delta_{\{1,3,4\}} \\
& + \Delta_{\{1,4\}} {\cdot}\Delta_{\{1,2,4\}} + \Delta_{\{1,4\}} {\cdot}\Delta_{\{1,3,4\}} + \Delta_{\{2,3\}} {\cdot}\Delta_{\{1,2,3\}} +
\Delta_{\{2,3\}} {\cdot}\Delta_{\{2,3,4\}} \\
& + \Delta_{\{2,4\}} {\cdot}\Delta_{\{1,2,4\}} + \Delta_{\{2,4\}} {\cdot}\Delta_{\{2,3,4\}} + \Delta_{\{3,4\}} {\cdot}\Delta_{\{1,3,4\}} +
\Delta_{\{3,4\}} {\cdot}\Delta_{\{2,3,4\}} , \\
\Delta_{2,4} &= \Delta_{\{1,2\}} {\cdot}\Delta_{\{1,2,3,4\}} + \Delta_{\{1,3\}}
{\cdot}\Delta_{\{1,2,3,4\}} + \Delta_{\{1,4\}} {\cdot}\Delta_{\{1,2,3,4\}} \\
& + \Delta_{\{2,3\}} {\cdot}\Delta_{\{1,2,3,4\}} + \Delta_{\{2,4\}} {\cdot}\Delta_{\{1,2,3,4\}} + \Delta_{\{3,4\}} {\cdot}\Delta_{\{1,2,3,4\}} , \\
\Delta_{3,4} &= \Delta_{\{1,2,3\}} {\cdot}\Delta_{\{1,2,3,4\}} +
\Delta_{\{1,2,4\}} {\cdot}\Delta_{\{1,2,3,4\}} \\
& + \Delta_{\{1,3,4\}} {\cdot}\Delta_{\{1,2,3,4\}} + \Delta_{\{2,3,4\}} {\cdot}\Delta_{\{1,2,3,4\}} .\end{aligned}$$
Each of the intersections $\Delta_0{\cdot}\Delta_S$ is a codimension two stratum in $\Delta_0$; for example $$\begin{picture}(80,80)(60,690)
\put(0,740){$\Delta_0{\cdot}\Delta_{\{1,2\}} =$}
\put( 80,755){\circle{30}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 20}}
\put( 70,720){\line( 1,-2){ 10}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 56,690){$1$}
\put( 77,690){$2$}
\put( 77,710){$3$}
\put( 87,710){$4$}
\end{picture}
\begin{picture}(80,80)(0,690)
\put(-10,740){$\Delta_0{\cdot}\Delta_{\{1,2,3\}} =$}
\put( 80,755){\circle{30}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 20}}
\put( 70,720){\line( 1,-2){ 10}}
\put( 70,720){\line( 0,-1){ 20}}
\put( 56,690){$1$}
\put( 67,690){$2$}
\put( 77,690){$3$}
\put( 87,710){$4$}
\end{picture}
\begin{picture}(90,80)(-60,690)
\put(-15,740){$\Delta_0{\cdot}\Delta_{\{1,2,3,4\}} =$}
\put( 80,755){\circle{30}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 80,720){\line(-3,-4){ 15}}
\put( 80,720){\line(-1,-4){ 5}}
\put( 80,720){\line( 1,-4){ 5}}
\put( 80,720){\line( 3,-4){ 15}}
\put( 61,690){$1$}
\put( 72,690){$2$}
\put( 83,690){$3$}
\put( 93,690){$4$}
\end{picture}$$ From these, we may form the ${S}_4$-invariant combinations $$\begin{aligned}
\Delta_{0,2} &= \Delta_0{\cdot}\Delta_{\{1,2\}} + \Delta_0{\cdot}\Delta_{\{1,3\}} +
\Delta_0{\cdot}\Delta_{\{1,4\}} \\
& + \Delta_0{\cdot}\Delta_{\{2,3\}} + \Delta_0{\cdot}\Delta_{\{2,4\}} +
\Delta_0{\cdot}\Delta_{\{3,4\}} , \\
\Delta_{0,3} &= \Delta_0 {\cdot}\Delta_{\{1,2,3\}} + \Delta_0 {\cdot}\Delta_{\{1,2,4\}} + \Delta_0 {\cdot}\Delta_{\{1,3,4\}} +
\Delta_0 {\cdot}\Delta_{\{2,3,4\}} , \\
\Delta_{0,4} &= \Delta_0 {\cdot}\Delta_{\{1,2,3,4\}} .\end{aligned}$$ There remain seven strata which are not expressible as intersections, which we denote by $\Delta_{\alpha,i}$, $1\le i\le 4$, and $\Delta_{\beta,12|34}$, $\Delta_{\beta,13|24}$ and $\Delta_{\beta,14|24}$. We illustrate the stable graphs for two of these strata: $$\begin{picture}(80,100)(40,720)
\put( 20,775){$\Delta_{\alpha,1} =$}
\put( 80,775){\circle{30}}
\put( 80,760){\line(-1,-2){ 10}}
\put( 80,760){\line( 0,-1){ 20}}
\put( 80,760){\line( 1,-2){ 10}}
\put( 80,790){\line( 0, 1){ 20}}
\put( 77,815){$1$}
\put( 66,727){$2$}
\put( 77,727){$3$}
\put( 87,727){$4$}
\end{picture}
\begin{picture}(100,100)(-20,720)
\put( 05,775){$\Delta_{\beta,12|34}=$}
\put( 80,775){\circle{30}}
\put( 80,760){\line(-1,-2){ 10}}
\put( 80,760){\line( 1,-2){ 10}}
\put( 80,790){\line(-1, 2){ 10}}
\put( 80,790){\line( 1, 2){ 10}}
\put( 66,815){$1$}
\put( 87,815){$2$}
\put( 66,727){$3$}
\put( 87,727){$4$}
\end{picture}$$ Denote by $\Delta_\alpha$ and $\Delta_\beta$ the ${S}_4$-invariant combinations of strata: $$\Delta_\alpha = \Delta_{\alpha,1} + \Delta_{\alpha,2} + \Delta_{\alpha,3} +
\Delta_{\alpha,4} , \quad \Delta_\beta =
\Delta_{\beta,12|34} + \Delta_{\beta,13|24} + \Delta_{\beta,14|24} .$$
For each of these strata, let $\delta_x=[\Delta_x]$ be the corresponding cycle in $H_{\bullet}({\overline{\mathcal{M}}}_{1,4},{\mathbb{Q}})$, in the sense of orbifolds. (This is sometimes denoted $[\Delta_x]_Q$ instead, but we omit the letter $Q$ from the notation.) If the generic point of $\Delta_x$ has an automorphism group of order $e$, then $\delta_x$ is $e^{-1}$ times the scheme-theoretic fundamental class of $\Delta_x$; this occurs, with $e=2$, for the cycles $\delta_{2,3}$, $\delta_{2,4}$ and $\delta_{0,4}$.
\[trivial\] The following relation among cycles holds in $H_4({\overline{\mathcal{M}}}_{1,4},{\mathbb{Q}})$: $$\delta_{0,2} + 3 \delta_{0,3} + 6 \delta_{0,4} = 3 \delta_\alpha + 4
\delta_\beta .$$
The two strata $${\setlength{\unitlength}{0.01in}
\begin{picture}(90,85)(60,685)
\put( 80,755){\circle{30}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 80,720){\line(-1,-2){ 10}}
\put( 80,720){\line( 1,-2){ 10}}
\put( 63,683){$1$}
\put( 87,683){$2$}
\end{picture}
\begin{picture}(65,85)(60,727)
\put( 80,775){\circle{30}}
\put( 80,760){\line( 0,-1){ 20}}
\put( 80,790){\line( 0, 1){ 20}}
\put( 77,814){$1$}
\put( 77,725){$2$}
\end{picture}}$$ define the same cycle, and are even rationally equivalent. (This is an instance of the WDVV equation.) We obtain the lemma by lifting this relation by the $6$ distinct projections ${\overline{\mathcal{M}}}_{1,4}\to{\overline{\mathcal{M}}}_{1,2}$ and summing the answers.
We can now state the main result of this section.
\[main\] The first seven rows of the intersection matrix of the nine ${S}_4$-invariant codimension two cycles in ${\overline{\mathcal{M}}}_{1,4}$ introduced above equals $$\begin{tabular}{C|CCCC|CCC|CC}
& \delta_{2,2} & \delta_{2,3} & \delta_{2,4} & \delta_{3,4} &
\delta_{0,2} & \delta_{0,3} & \delta_{0,4} & \delta_\alpha & \delta_\beta \\
\hline
\delta_{2,2} & 1/8 & 0 & 0 & 0 & -3 & 0 & 3/2 & 0 & 3/2 \\
\delta_{2,3} & 0 & 0 & 0 & 0 & 0 & -6 & 6 & 6 & 0 \\
\delta_{2,4} & 0 & 0 & 0 & -1/2 & 0 & 6 & -3 & 0 & 0 \\
\delta_{3,4} & 0 & 0 & -1/2 & 1/6 & 6 & -2 & 0 & 0 & 0
\\ \hline
\delta_{0,2} & -3 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 \\
\delta_{0,3} & 0 & -6 & 6 & -2 & 0 & 0 & 0 & 0 & 0 \\
\delta_{0,4} & 3/2 & 6 & -3 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{tabular}$$
The following lemma shows that many of the intersection numbers vanish. (The use of this lemma simplifies our original proof of Theorem \[main\], and was suggested to us by C. Faber.)
Let $\delta$ be a cycle in $\Delta_0$. Then $\delta_0{\cdot}\delta=0$.
Consider the projection $\pi:{\overline{\mathcal{M}}}_{1,n}\to{\overline{\mathcal{M}}}_{1,1}$ which forgets all but the first marked point, and stabilizes the marked curve which results. The divisor $\Delta_0$ is the inverse image under $\pi$ of the compactification divisor of ${\overline{\mathcal{M}}}_{1,1}$; thus, we may replace it in calculating intersections by any cycle of the form $\pi^{-1}(x)$, where $x\in{\mathcal{M}}_{1,1}$. The resulting cycle has empty intersection with $\delta$, proving the lemma.
This lemma shows that all intersections among the cycles $\delta_{0,2}$, $\delta_{0,3}$ and $\delta_{0,4}$, and between these and $\delta_\alpha$ and $\delta_\beta$ vanish.
A number of other entries in the intersection matrix vanish because the associated strata do not meet: thus, $$\begin{aligned}
& \delta_{2,2}{\cdot}\delta_{2,3} = \delta_{2,2}{\cdot}\delta_{3,4} =
\delta_{2,2}{\cdot}\delta_{0,3} = \delta_{2,2}{\cdot}\delta_\alpha = 0 , \\
& \delta_{2,3}{\cdot}\delta_{2,4} = \delta_{2,3}{\cdot}\delta_\beta = 0 , \\
& \delta_{2,4}{\cdot}\delta_\alpha = \delta_{2,4}{\cdot}\delta_\beta =
\delta_{3,4}{\cdot}\delta_\alpha = \delta_{3,4}{\cdot}\delta_\beta = 0 .\end{aligned}$$
To calculate the remaining entries of the intersection matrix, we need the excess intersection formula (Fulton [@Fulton], Section 6.3).
\[excess\] Let $Y$ be a smooth variety, let $X\hookrightarrow Y$ be a regular [embedding]{}of codimension $d$, and let $V$ be a closed subvariety of $Y$ of dimension $n$. Suppose that the inclusion $W=X\cap V\hookrightarrow V$ is a regular [embedding]{}of codimension $d-e$. Then $$[X]{\cdot}[V] = c_e(E) \cap [W] \in A_{n-d}(W) ,$$ where $E=(N_XY)|_W/(N_WV)$ is the *excess bundle* of the intersection.
Observe that in calculating the top four rows of our intersection matrix, at least one of the cycles which we intersect with has a regular [embedding]{}in ${\overline{\mathcal{M}}}_{1,4}$, since its dual graph is a tree. This makes the application of the excess intersection formula straightforward.
It remains to give a formula for the normal bundles to the strata of ${\overline{\mathcal{M}}}_{1,4}$.
The *tautological line bundles* are defined by $${\omega}_i = \sigma_i^*{\omega}_{{\overline{\mathcal{M}}}_{g,n+1}/{\overline{\mathcal{M}}}_{g,n}} ,$$ where $\sigma_i:{\overline{\mathcal{M}}}_{g,n}\to{\overline{\mathcal{M}}}_{g,n+1}$, $1\le i\le n$, are the $n$ canonical sections of the universal stable curve ${\overline{\mathcal{M}}}_{g,n+1}\to{\overline{\mathcal{M}}}_{g,n}$. Denote the Chern class $c_1({\omega}_i)$ by $\psi_i$.
To apply the excess intersection formula, we need to know the normal bundles of strata ${\overline{\mathcal{M}}}(G)\subset{\overline{\mathcal{M}}}_{g,n}$. The following result gives a partial answer to this question, and is all that we need for the calculations in this paper: a proof may be found in Section 4 of Hain-Looijenga [@HL].
\[normal\] Let $G$ be a stable graph of genus $g$ and valence $n$, and let $$p: \prod_{v\in\operatorname{\mathit{V}}(G)} {\overline{\mathcal{M}}}_{g(v),n(v)} \to {\overline{\mathcal{M}}}_{g,n}$$ be the ramified cover (of degree $|\operatorname{Aut}(G)|$) of the closed stratum ${\overline{\mathcal{M}}}(G)$ of ${\mathcal{M}}_{g,n}$. Each edge $e$ of the graph determines two flags $s(e)$ and $t(e)$, and hence two tautological line bundles ${\omega}_{s(e)}$ and ${\omega}_{t(e)}$ on $\prod_{v\in\operatorname{\mathit{V}}(G)} {\overline{\mathcal{M}}}_{g(v),n(v)}$, and the normal bundle of $p$ is given by the formula $$N_p = \bigoplus_{e\in\operatorname{\mathit{E}}(G)} {\omega}_{s(e)}^{\vee}{\otimes}{\omega}_{t(e)}^{\vee}.
\qed$$
In particular, if the graph $G$ has no automorphisms, so that $p$ is an [embedding]{}, the bundle $N_p$ may be identified with the normal bundle of the stratum ${\overline{\mathcal{M}}}(G)$.
It is now straightforward to calculate the remaining entries of the intersection matrix. We will use the integrals $$\label{tau}
\int_{{\overline{\mathcal{M}}}_{0,4}} \psi_i = 1 , \quad \int_{{\overline{\mathcal{M}}}_{1,1}} \psi_1 =
\int_{{\overline{\mathcal{M}}}_{1,2}} \psi_1 \cup \psi_2 = \frac{1}{24} ,$$ which are proved in Witten [@Witten].
In performing the calculations, it is helpful to introduce a graphical notation for the cycle obtained from a stratum by capping with a monomial in the Chern classes $-\psi_i$: we point a small arrow along each flag $i$ where we intersect by the class $-\psi_i$. (This notation generalizes that of Kaufmann [@Kaufmann], who considers the case of trees where the genus of each vertex is $0$. The minus signs come from the inversion accompanying the tautological line bundles in the formula of Proposition \[normal\].) One then calculates the contribution of such a graph by multiplying together factors for each vertex equal to the integral over ${\overline{\mathcal{M}}}_{g(v),n(v)}$ of the appropriate monomial in the classes $-\psi_i$, and dividing by the order of the automorphism group $\operatorname{Aut}(G)$: in particular, this vanishes unless there are $3(g(v)-1)+n(v)$ arrows at each vertex $v$.
We illustrate the sort of enumeration which arises with one of the most complicated of these calculations, that of $\delta_{2,4}{\cdot}\delta_{2,4}$. Two sorts of terms contribute: $6$ terms of the form $$\bigl( \delta_{\{1,2\}}{\cdot}\delta_{\{1,2,3,4\}} \bigr)^2 = \frac{1}{24} ,$$ and $6$ terms of the form $$\delta_{\{1,2\}}{\cdot}\delta_{\{1,2,3,4\}}{\cdot}\delta_{\{3,4\}}{\cdot}\delta_{\{1,2,3,4\}}
= - \frac{1}{24} .$$ Applying the excess intersection formula, we see that $$\bigl(\delta_{\{1,2\}}{\cdot}\delta_{\{1,2,3,4\}}\bigr)^2 =
c_2\bigl(N_{\Delta_{\{1,2\}}\cap\Delta_{\{1,2,3,4\}}}{\overline{\mathcal{M}}}_{1,4}\bigr) \cap
\bigl( \delta_{\{1,2\}}{\cdot}\delta_{\{1,2,3,4\}} \bigr) .$$ Expanding the second Chern class of the normal bundle, we see that each term contributes the sum of four graphs: $$\begin{picture}(80,90)(60,690)
\put( 80,760){\circle{5}}
\put( 80,740){\vector( 0, 1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 70,720){\vector( 1, 2){ 10}}
\put( 70,720){\line(-1,-2){ 10}}
\put( 70,720){\line( 1,-2){ 10}}
\end{picture}
\begin{picture}(80,90)(60,690)
\put( 80,760){\circle{5}}
\put( 80,757){\vector( 0,-1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 70,720){\vector( 1, 2){ 10}}
\put( 70,720){\line(-1,-2){ 10}}
\put( 70,720){\line( 1,-2){ 10}}
\end{picture}
\begin{picture}(80,90)(60,690)
\put( 80,760){\circle{5}}
\put( 80,740){\vector( 0, 1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 80,740){\vector(-1,-2){ 10}}
\put( 70,720){\line(-1,-2){ 10}}
\put( 70,720){\line( 1,-2){ 10}}
\end{picture}
\begin{picture}(50,90)(60,690)
\put( 80,760){\circle{5}}
\put( 80,757){\vector( 0,-1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line( 0,-1){ 20}}
\put( 80,740){\vector(-1,-2){ 10}}
\put( 70,720){\line(-1,-2){ 10}}
\put( 70,720){\line( 1,-2){ 10}}
\end{picture}$$ Only the first graph is nonzero, since in the other cases, the wrong number of arrows point towards the vertices. And the first graph contributes $$\int_{{\overline{\mathcal{M}}}_{0,4}} (-\psi_1) {\cdot}\int_{{\overline{\mathcal{M}}}_{1,1}} (-\psi_1) = \frac{1}{24} .$$
In the case of terms of the form $\delta_{\{1,2\}}{\cdot}\delta_{\{1,2,3,4\}}
{\cdot}\delta_{\{3,4\}}{\cdot}\delta_{\{1,2,3,4\}}$, the excess dimension $e$ equals $1$, and we must calculate the degree of the excess bundle on the stratum $\Delta_{\{1,2\}}\cap\Delta_{\{3,4\}}\cap\Delta_{\{1,2,3,4\}}$. Two graphs contribute: $$\begin{picture}(120,90)(60,690)
\put( 80,760){\circle{5}}
\put( 80,740){\vector( 0, 1){ 18}}
\put( 80,740){\line( 1,-1){ 20}}
\put( 80,740){\line(-1,-1){ 20}}
\put( 60,720){\line(-1,-2){ 10}}
\put( 60,720){\line( 1,-2){ 10}}
\put(100,720){\line(-1,-2){ 10}}
\put(100,720){\line( 1,-2){ 10}}
\end{picture}
\begin{picture}(60,90)(60,690)
\put( 80,760){\circle{5}}
\put( 80,757){\vector( 0,-1){ 18}}
\put( 80,740){\line( 1,-1){ 20}}
\put( 80,740){\line(-1,-1){ 20}}
\put( 60,720){\line(-1,-2){ 10}}
\put( 60,720){\line( 1,-2){ 10}}
\put(100,720){\line(-1,-2){ 10}}
\put(100,720){\line( 1,-2){ 10}}
\end{picture}$$ Only the first of these graphs gives a nonzero value, namely $$\int_{{\overline{\mathcal{M}}}_{1,1}} (-\psi_1) = - \frac{1}{24} .$$ This completes our outline of the proof of Theorem \[main\].
The intersection matrix of Theorem \[main\] has rank $7$. We now apply the results of [@genus1], where we calculated the character of the ${S}_n$-modules $H^i({\overline{\mathcal{M}}}_{1,n},{\mathbb{Q}})$: these calculations show that $\dim
H^4({\overline{\mathcal{M}}}_{1,4},{\mathbb{Q}})^{{S}_4}=7$. This shows that our $9$ cycles span $H^4({\overline{\mathcal{M}}}_{1,4},{\mathbb{Q}})^{{S}_4}$, and that the nullspace of the intersection matrix gives relations among them. We already know one such relation, by Lemma \[trivial\]. Calculating the remaining null-vector of the intersection matrix, we obtain the main theorem of this paper.
\[relation\] The following new relation among cycles holds: $$12\delta_{2,2} - 4\delta_{2,3} - 2\delta_{2,4} + 6\delta_{3,4} +
\delta_{0,3} + \delta_{0,4} - 2\delta_\beta = 0 .
\qed$$
Using this theorem, it is easy to calculate the remaining intersections among our $9$ strata: $$\delta_\alpha{\cdot}\delta_\alpha=16 , \quad \delta_\alpha{\cdot}\delta_\beta=-12 ,
\quad \delta_\beta{\cdot}\delta_\beta=9 .$$ C. Faber informs us that the direct calculation of these intersection numbers is not difficult. This would allow a different approach to the proof of Theorem \[relation\], using the theorem of [@elliptic3] that the strata of ${\overline{\mathcal{M}}}_{1,n}$ span the even-dimensional rational cohomology.
Gromov-Witten invariants
========================
In the remainder of this paper, we apply the new relation to the calculation of elliptic Gromov-Witten invariants: we will do this explicitly for curves and for the projective plane ${\mathbb{CP}}^2$, and prove some general results in other cases.
The Novikov ring
----------------
Let $V$ be a smooth projective variety of dimension $d$. In studying the Gromov-Witten invariants, it is convenient to work with cohomology with coefficients in the Novikov ring ${\Lambda}$ of $V$, which we now define.
Let $\operatorname{N}_1(V)$ be the abelian group $$\operatorname{N}_1(V) = \operatorname{Z}_1(V) / \text{numerical equivalence} ,$$ and let $\operatorname{NE}_1(V)$ be its sub-semigroup $$\operatorname{NE}_1(V) = \operatorname{ZE}_1(V) / \text{numerical equivalence} ,$$ where $\operatorname{Z}_1(V)$ is the abelian group of $1$-cycles on $V$, and $\operatorname{ZE}_1(V)$ is the semigroup of effective $1$-cycles. (Recall that two $1$-cycles $x$ and $y$ are numerically equivalent $x\equiv y$ when $x{\cdot}Z=y{\cdot}Z$ for any Cartier divisor $Z$ on $V$.)
The Novikov ring is $$\begin{aligned}
{\Lambda}&= {\mathbb{Q}}[\operatorname{N}_1(V)] {\otimes}_{{\mathbb{Q}}[\operatorname{NE}_1(V)]} {\mathbb{Q}}{{[\![}}\operatorname{NE}_1(V){{]\!]}}\\
&= \textstyle \bigl\{ a = \sum_{\beta\in\operatorname{N}_1(V)} a_\beta q^\beta \mid
\text{ $\operatorname{supp}(a) \subset \beta_0+\operatorname{NE}_1(V)$ for some $\beta_0\in\operatorname{N}_1(V)$}
\bigr\} ,\end{aligned}$$ with product $q^{\beta_1}q^{\beta_2}=q^{\beta_1+\beta_2}$ and grading $|q^\beta|=-2c_1(V)\cap\beta$. That the product is well-defined is shown by the following proposition (Kollár [@Kollar], Proposition II.4.8).
\[Mori\] If $V$ is a projective variety with Kähler form ${\omega}$, the set $$\{\beta\in\operatorname{NE}_1(V)\mid {\omega}\cap\beta\le c\}$$ is finite for each $c>0$.
For example, if $V={\mathbb{CP}}^n$, then $\operatorname{N}_1({\mathbb{CP}}^n)={\mathbb{Z}}{\cdot}[L]$, where $[L]$ is the cycle defined by a line $L\subset{\mathbb{CP}}^n$, and ${\Lambda}\cong{\mathbb{Q}}{{(\!(}}q{{)\!)}}$, with grading $|q|=-2(n+1)$, since $c_1({\mathbb{CP}}^n)\cap[L]=n+1$.
If $V=E$ is an elliptic curve, then $\operatorname{N}_1(E)={\mathbb{Z}}{\cdot}[E]$, and ${\Lambda}\cong{\mathbb{Q}}{{(\!(}}q{{)\!)}}$, concentrated in degree $0$.
Stable maps
-----------
The definition of Gromov-Witten invariants is based on the study of the moduli stacks ${\overline{\mathcal{M}}}_{g,n}(V,\beta)$ of stable maps of Kontsevich, which have been shown by Behrend and Manin [@BM] to be complete Deligne-Mumford stacks (though not in general smooth).
For each $N\ge0$, let $\pi_{n,N}:{\overline{\mathcal{M}}}_{g,n+N}(V,\beta) \to
{\overline{\mathcal{M}}}_{g,n}(V,\beta)$ be the projection which forgets the last $N$ marked points of the stable curve, and stabilizes the resulting map.
In the special case $N=1$, we obtain a fibration $$\pi: {\overline{\mathcal{M}}}_{g,n+1}(V,\beta) \to {\overline{\mathcal{M}}}_{g,n}(V,\beta)$$ which is shown by Behrend and Manin to be the universal curve; that is, its fibre over a stable map $(f:C\to V,x_i)$ is the curve $C$. Denote by $f:{\overline{\mathcal{M}}}_{g,n+1}(V,\beta)\to V$ the universal stable map, obtained by evaluation at $x_{n+1}$.
The virtual fundamental class
-----------------------------
There are projections ${\overline{\mathcal{M}}}_{g,n}(V,\beta)\to{\overline{\mathcal{M}}}_{g,n}$, when $2(g-1)+n>0$, which send the stable map $(f:C\to V,x_i)$ to the stabilization of $(C,x_i)$. If the sheaf $R^1\pi_*f^*TV$ vanishes on ${\overline{\mathcal{M}}}_{g,n}(V,\beta)$, the Riemann-Roch theorem predicts that the fibres of the projection ${\overline{\mathcal{M}}}_{g,n}(V,\beta)\to{\overline{\mathcal{M}}}_{g,n}$ have dimension $$d(1-g)+c_1(V)\cap\beta ,$$ and hence that ${\overline{\mathcal{M}}}_{g,n}(V,\beta)$ has dimension $$d(1-g)+c_1(V)\cap\beta + \dim{\overline{\mathcal{M}}}_{g,n} = (3-d)(1-g)+c_1(V)\cap\beta + n .$$ This hypothesis is only rarely true, and in any case only in genus $0$. However, Behrend-Fantecchi [@B; @BF] and Li-Tian [@LT] show that there is a bivariant class $$[{\overline{\mathcal{M}}}_{g,n}(V,\beta)/{\overline{\mathcal{M}}}_{g,n},R^{\bullet}\pi_*f^*TV] \in
A^{d(1-g)+c_1(V)\cap\beta}({\overline{\mathcal{M}}}_{g,n}(V,\beta)\to{\overline{\mathcal{M}}}_{g,n}) ,$$ the virtual relative fundamental class, which stands in for $[{\overline{\mathcal{M}}}_{g,n}(V,\beta)/{\overline{\mathcal{M}}}_{g,n}]$ in the obstructed case.
The following result is proved in [@B], and sometimes permits the explicit calculation of Gromov-Witten invariants, as we will see later.
\[excess-virtual\] If the coherent sheaf $R^1\pi_*f^*TV$ on ${\overline{\mathcal{M}}}_{g,n}(V,\beta)$ is locally trivial of dimension $e$ (the *excess dimension*), then ${\overline{\mathcal{M}}}_{g,n}(V,\beta)$ is smooth of dimension $$(3-d)(1-g)+c_1(V)\cap\beta+n+e ,$$ and $[{\overline{\mathcal{M}}}_{g,n}(V,\beta)/{\overline{\mathcal{M}}}_{g,n},R^{\bullet}\pi_*f^*TV] =
c_e(R^1\pi_*f^*TV) \cap [{\overline{\mathcal{M}}}_{g,n}(V,\beta)/{\overline{\mathcal{M}}}_{g,n}]$.
Gromov-Witten invariants
------------------------
The Gromov-Witten invariant of genus $g\ge0$, valence $n\ge0$ and degree $\beta\in\operatorname{NE}_1(V)$ is a cohomology operation $$I_{g,n,\beta}^V : H^{2d(1-g)+2c_1(V)\cap\beta+{\bullet}}(V^n,{\mathbb{Q}}) \to
H^{\bullet}({\overline{\mathcal{M}}}_{g,n},{\mathbb{Q}}) ,$$ defined by the formula $$I_{g,n,\beta}^V(\alpha_1,\dots,\alpha_n) =
[{\overline{\mathcal{M}}}_{g,n}(V,\beta)/{\overline{\mathcal{M}}}_{g,n},R^{\bullet}\pi_*f^*TV] \cap
\operatorname{ev}^*(\alpha_1\boxtimes\dots\boxtimes\alpha_n) ,$$ where $\operatorname{ev}:{\overline{\mathcal{M}}}_{g,n}(V,\beta)\to V^n$ is evaluation at the marked points: $$\operatorname{ev}: (f:C\to V,x_i) \mapsto (f(x_1),\dots,f(x_n)) \in V^n .$$
Capping $I_{g,n,\beta}^V$ with the fundamental class $[{\overline{\mathcal{M}}}_{g,n}]$, we obtain a numerical invariant $${\langle}I_{g,n,\beta}^V{\rangle}: H^{2(d-3)(1-g)+2c_1(V)\cap\beta+2n}(V^n,{\mathbb{Q}}) \to {\mathbb{Q}}.$$ This is the $n$-point correlation function of two-dimensional topological gravity with the topological $\sigma$-model associated to $V$ as a background [@Witten]. Note that if $\beta\ne0$, ${\langle}I_{g,n,\beta}^V{\rangle}$ may be defined even when $2(g-1)+n\le0$, even though $I_{g,n,\beta}^V$ does not exist.
Introducing the Novikov ring, we may define the generating function $$I_{g,n}^V = \sum_{\beta\in\operatorname{NE}_1(V)} q^\beta I_{g,n,\beta}^V :
H^*(V,{\Lambda})^{{\otimes}n} \to H^{\bullet}({\overline{\mathcal{M}}}_{g,n},{\Lambda}) ,$$ along with its integral over the fundamental class $[{\overline{\mathcal{M}}}_{g,n}]$ $${\langle}I_{g,n}^V{\rangle}= \sum_{\beta\in\operatorname{NE}_1(V)} q^\beta {\langle}I_{g,n,\beta}^V{\rangle}:
H^*(V,{\Lambda})^{{\otimes}n} \to {\Lambda},$$ Note that $I_{g,n}^V$ and ${\langle}I_{g,n}^V{\rangle}$ are invariant under the action of the symmetric group ${S}_n$ on $H^*(V,{\Lambda})^{{\otimes}n}$.
In the special case of zero degree, the moduli space ${\overline{\mathcal{M}}}_{g,n}(V,\beta)$ is isomorphic to ${\overline{\mathcal{M}}}_{g,n}\times V$. This allows us to calculate the Gromov-Witten invariants ${\langle}I_{0,3,0}^V{\rangle}$ and ${\langle}I_{1,1,0}^V{\rangle}$. The former is given by the explicit formula $${\langle}I_{0,3,0}^V(\alpha_1,\alpha_2,\alpha_3){\rangle}= \int_V
\alpha_1\cup\alpha_2\cup\alpha_2 .$$ This formula is very simple to prove, since the moduli space ${\overline{\mathcal{M}}}_{0,3}(V,0)\cong V$ is smooth, with dimension equal to its virtual dimension $d$, and thus the virtual fundamental class $[{\overline{\mathcal{M}}}_{0,3}(V,0),R^{\bullet}\pi_*f^*TV]$ may be identified with the fundamental class of $V$. A similar proof shows that ${\langle}I_{0,n,0}^V{\rangle}$ vanishes if $n>3$.
The calculation of the Gromov-Witten invariant ${\langle}I_{1,1,0}^V{\rangle}$ (see Bershadsky et al. [@BCOV]) is a good illustration of the application of Proposition \[excess-virtual\].
\[BCOV\] $${\langle}I_{1,1,0}^V(\alpha){\rangle}= -\frac{1}{24} \int_V c_{d-1}(V)\cup\alpha ,$$ while ${\langle}I_{1,n,0}^V{\rangle}=0$ if $n>1$.
The moduli stack ${\overline{\mathcal{M}}}_{1,n}(V,0)$ is isomorphic to ${\overline{\mathcal{M}}}_{1,n}\times V$, and the obstruction bundle $R^1\pi_*f^*TV$ is isomorphic to the vector bundle ${\mathbb{E}}^{\vee}\boxtimes TV$, of rank $d$, where ${\mathbb{E}}=\pi_*{\omega}_{{\overline{\mathcal{M}}}_{1,n+1}/{\overline{\mathcal{M}}}_{1,n}}$. Hence $R^1\pi_*f^*TV$ has top Chern class $$c_d({\mathbb{E}}^{\vee}{\otimes}f^*TV) = 1\boxtimes f^*c_d(V) - \lambda_1 \boxtimes
f^*c_{d-1}(V) ,$$ where $\lambda_1=c_1({\mathbb{E}})$. By Proposition \[excess-virtual\], $$\begin{aligned}
{\langle}I_{1,n,0}^V(\alpha_1,\dots,\alpha_n){\rangle}&= \int_{{\overline{\mathcal{M}}}_{1,n}\times V}
c_d({\mathbb{E}}^{\vee}{\otimes}f^*TV) \boxtimes (\alpha_1\cup\dots\cup\alpha_n) \\
&= - \int_{{\overline{\mathcal{M}}}_{1,n}} \lambda_1 {\cdot}\int_V c_{d-1}(V) \cup
\alpha_1\cup\dots\cup\alpha_n .\end{aligned}$$ On dimensional grounds, ${\langle}I_{1,n,0}^V{\rangle}$ vanishes if $n>1$, while the formula follows when $n=1$ from $\lambda_1\cap[{\overline{\mathcal{M}}}_{1,1}]=\frac{1}{24}$.
The puncture axiom
------------------
One of the basic axioms satisfied by Gromov-Witten invariants is expressed in the relationship between virtual fundamental classes $$[{\overline{\mathcal{M}}}_{g,n+1}(V,\beta)/{\overline{\mathcal{M}}}_{g,n+1},R^{\bullet}\pi_*f^*TV] = \pi^*
[{\overline{\mathcal{M}}}_{g,n}(V,\beta)/{\overline{\mathcal{M}}}_{g,n},R^{\bullet}\pi_*f^*TV] .$$ Here, $\pi^*:A^k({\overline{\mathcal{M}}}_{g,n}(V,\beta)\to{\overline{\mathcal{M}}}_{g,n}) \to
A^k({\overline{\mathcal{M}}}_{g,n+1}(V,\beta)\to{\overline{\mathcal{M}}}_{g,n+1})$ is the operation of flat pullback associated to the diagram $$\begin{CD}
{\overline{\mathcal{M}}}_{g,n+1}(V,\beta) @>>> {\overline{\mathcal{M}}}_{g,n+1} \\
@V{\pi}VV @V{\pi}VV \\
{\overline{\mathcal{M}}}_{g,n}(V,\beta) @>>> {\overline{\mathcal{M}}}_{g,n}
\end{CD}$$ This axiom implies that if $\alpha$ is a cohomology class on $V$ of degree at most $2$ and $2(g-1)+n>0$, $$\label{low}
I_{g,n+1,\beta}^V(\alpha,\alpha_1,\dots,\alpha_n) =
\begin{cases}
0 , & |\alpha|=0,1 , \\
(\alpha\cap\beta) I_{g,n,\beta}^V(\alpha_1,\dots,\alpha_n) , &
|\alpha|=2 .
\end{cases}$$
Generating functions
--------------------
Let $\Lambda{{[\![}}H{{]\!]}}$ be the power series ring $\Lambda{{[\![}}H_{{\bullet}+2}(V,{\mathbb{Q}}){{]\!]}}$. Let $\{\gamma^a\}_{a=0}^k$ be a homogeneous basis of the graded vector space $H^{\bullet}(V,{\mathbb{Q}})$, with $\gamma^0=1$, and let $\{t_a\}_{a=0}^k$ be the dual basis; the (homological) degree of $t_a$ equals the (cohomological) degree of $\gamma^a$ minus $2$. We may identify the ring $\Lambda{{[\![}}H{{]\!]}}$ with $\Lambda{{[\![}}t_0,\dots,t_k{{]\!]}}$.
Let $F_g(V)$ be the generating function $$F_g(V) = \sum_{n=0}^\infty {\langle}I_{g,n}^V{\rangle}\in \Lambda{{[\![}}H{{]\!]}}.$$ This is a power series of degree $2(d-3)(1-g)$. This suggests assigning to Planck’s constant $\hbar$ the degree $2(d-3)(g-1)$, and forming the total generating function, homogeneous of degree $0$, $$F(V) = \sum_{g=0}^\infty \hbar^{g-1} F_g(V) .$$
The composition axiom
---------------------
The composition axiom for Gromov-Witten invariants gives a formula for the integral of the Gromov-Witten invariant $I_{g,n}^V$ over the cycle $[{\overline{\mathcal{M}}}(G)]$ associated to a stable graph $G$ which bears a strong resemblance to the Feynman rules of quantum field theory:
Let $\eta_{ab}$ be the Poincaré form of $V$ with respect to the basis $\{\gamma^a\}_{a=0}^k$ of $H^{\bullet}(V,{\mathbb{Q}})$. Then $$\int_{{\overline{\mathcal{M}}}(G)} I_{g,n}^V(\alpha_1,\dots,\alpha_n) = \frac{1}{\operatorname{Aut}(G)}
\sum_{\substack{a(e),b(e)=0 \\ e\in\operatorname{\mathit{E}}(G)}}^k \prod_{e\in\operatorname{\mathit{E}}(G)}
\eta_{a(e),b(e)} \prod_{v\in\operatorname{\mathit{V}}(G)} {\langle}I_{g(v),n(v)}^V(\dots){\rangle}.$$ Here, the Gromov-Witten invariant ${\langle}I_{g(v),n(v)}^V(\dots){\rangle}$ is evaluated on the cohomology classes $\alpha_i$ corresponding to the tails of $G$ which meet the vertex $v$, on the $\gamma^{a(e)}$ corresponding to edges $e$ which start at the vertex $v$, and on the $\gamma^{b(e)}$ corresponding to edges $e$ which end at $v$. (The right-hand side is independent of the chosen orientation of the edges, by the symmetry of the Poincaré form.)
Relations among Gromov-Witten invariants
----------------------------------------
Let $G$ be a stable graph of genus $g$ and valence $n$. The subvariety $\pi_{n,N}^{-1}\bigl({\overline{\mathcal{M}}}(G)\bigr) \subset {\overline{\mathcal{M}}}_{g,n+N}$ is the union of strata associated to the set of stable graphs obtained from $G$ by adjoining $N$ tails $\{n+1,\dots,n+N\}$ in all possible ways to the vertices of $G$.
For example, consider the stratum $\Delta_{12|34}\subset{\overline{\mathcal{M}}}_{0,4}$, associated to the stable graph $$\begin{picture}(35,85)(60,715)
\put( 20,760){$\Delta_{12|34} =$}
\put( 80,775){\line( 0,-1){ 30}}
\put( 80,745){\line(-2,-3){ 10}}
\put( 80,745){\line( 2,-3){ 10}}
\put( 80,775){\line( 2, 3){ 10}}
\put( 80,775){\line(-2, 3){ 10}}
\put( 65,715){$1$}
\put( 89,715){$2$}
\put( 65,795){$3$}
\put( 89,795){$4$}
\end{picture}$$ The inverse image $\pi_{4,N}^{-1}(\Delta_{12|34})$ consists of the union of all strata in ${\overline{\mathcal{M}}}_{0,4+N}$ associated to stable graphs $$\begin{picture}(35,85)(60,715)
\put( 5,760){$\Delta_{12I|34J} =$}
\put( 80,780){\line( 0,-1){ 40}}
\put( 80,740){\line(-2,-3){ 10}}
\put( 80,740){\line( 2,-3){ 10}}
\put( 80,780){\line( 2, 3){ 10}}
\put( 80,780){\line(-2, 3){ 10}}
\put( 65,715){$1$}
\put( 89,715){$2$}
\put( 65,800){$3$}
\put( 89,800){$4$}
\put( 80,780){\line( 3,-1){ 30}}
\put( 80,780){\line( 3, 1){ 30}}
\put( 95,780){\dots}
\put(115,777){$J$}
\put( 80,740){\line( 3,-1){ 30}}
\put( 80,740){\line( 3, 1){ 30}}
\put( 95,740){\dots}
\put(115,737){$I$}
\end{picture}$$ where $I$ and $J$ form a partition of the set $\{5,\dots,N+4\}$.
If $\delta$ is a cycle in ${\overline{\mathcal{M}}}_{g,n}$, define the generating function $$F(\delta,V) = \sum_{N=0}^\infty \int_{\pi^{-1}(\delta)} I_{g,n+N}^V :
H^{{\bullet}+2}(V,{\Lambda})^{{\otimes}n} \to {\Lambda}{{[\![}}H{{]\!]}}.$$ More explicitly, $$\begin{gathered}
F(\delta,V)(\alpha_1,\dots,\alpha_n) \\ = \sum_{N=0}^\infty \frac{1}{N!}
\sum_{a_1,\dots,a_N} t_{a_N}\dots t_{a_1} \int_\delta \bigl( \pi_{n,N}
\bigr)_*
I_{g,n+N}^V(\gamma^{a_1},\dots,\gamma^{a_N},\alpha_1,\dots,\alpha_n) .\end{gathered}$$ In particular, if $\delta=[{\overline{\mathcal{M}}}(G)]$ where $G$ is a stable graph, we set $$F(G,V)=F([{\overline{\mathcal{M}}}(G)],V) .$$ If $g>1$, $F_g(V)$ is a special case of this construction, with $\delta=[{\overline{\mathcal{M}}}_{g,0}]$.
A little exercise involving Leibniz’s rule shows that the composition axiom implies the following formula for these generatings functions: $$\label{composition}
F(G,V) = \frac{1}{\operatorname{Aut}(G)} \sum_{\substack{a(e),b(e)=0 \\ e\in\operatorname{\mathit{E}}(G)}}^k
\prod_{e\in\operatorname{\mathit{E}}(G)} \eta_{a(e),b(e)} \prod_{v\in\operatorname{\mathit{V}}(G)}
{\partial}^{n(v)}F_{g(v)}(V) (\dots) ,$$ where as before, the multilinear form ${\partial}^{n(v)}F_{g(v)}(V)$ is evaluated on the cohomology classes $\alpha_i$ corresponding to the tails of $G$ meeting the vertex $v$, on the $\gamma^{a(e)}$ corresponding to edges $e$ which start at the vertex $v$, and on the $\gamma^{b(e)}$ corresponding to edges $e$ which end at $v$.
The composition axiom implies that any relation among the cycles $[{\overline{\mathcal{M}}}(G)]$ is reflected in a relation among Gromov-Witten invariants, which, by may be translated into a differential equation among generating functions $F_g(V)$. An example is the rational equivalence of the cycles associated to the three strata of ${\overline{\mathcal{M}}}_{0,4}$ of codimension $1$: $$\begin{picture}(35,85)(60,715)
\put( 80,775){\line( 0,-1){ 30}}
\put( 80,745){\line(-1,-2){ 10}}
\put( 80,745){\line( 1,-2){ 10}}
\put( 80,775){\line( 2, 3){ 10}}
\put( 80,775){\line(-2, 3){ 10}}
\put( 65,710){$1$}
\put( 89,710){$2$}
\put( 65,795){$3$}
\put( 89,795){$4$}
\end{picture}
\hskip0.5in
\begin{picture}(35,85)(60,715)
\put( 40,755){$\sim$}
\put( 80,775){\line( 0,-1){ 30}}
\put( 80,745){\line(-1,-2){ 10}}
\put( 80,745){\line( 1,-2){ 10}}
\put( 80,775){\line( 2, 3){ 10}}
\put( 80,775){\line(-2, 3){ 10}}
\put( 65,710){$1$}
\put( 89,710){$3$}
\put( 65,795){$2$}
\put( 89,795){$4$}
\end{picture}
\hskip0.5in
\begin{picture}(35,85)(60,715)
\put( 40,755){$\sim$}
\put( 80,775){\line( 0,-1){ 30}}
\put( 80,745){\line(-1,-2){ 10}}
\put( 80,745){\line( 1,-2){ 10}}
\put( 80,775){\line( 2, 3){ 10}}
\put( 80,775){\line(-2, 3){ 10}}
\put( 65,710){$1$}
\put( 89,710){$4$}
\put( 65,795){$2$}
\put( 89,795){$3$}
\end{picture}$$ The equality of the Gromov-Witten invariant $F(\delta,V)$ when evaluated on these cycles is the WDVV equation.
In order to express the relation among the Gromov-Witten invariants implied by Theorem \[relation\], it is useful to introduce certain operators which act on elements of ${\Lambda}[H]{\otimes}{\Lambda}{{[\![}}H{{]\!]}}$ through differentiation in the first factor: the Laplacian $$\Delta = \frac12 \sum_{a,b=0}^k \eta_{ab} \frac{{\partial}^2}{{\partial}t_a{\partial}t_b} ,$$ and the sequence of bilinear differential operators $\Gamma_n$ by $\Gamma_0(f,g)=fg$ and $$\Gamma_n(f,g) = \frac{1}{n} \bigl( \Delta\Gamma_{n-1}(f,g) -
\Gamma_{n-1}(\Delta f,g) - \Gamma_{n-1}(f,\Delta g) \bigr) .$$ (We will abbreviate $\Gamma_1(f,g)$ to $\Gamma(f,g)$.)
\[Relation\] Denote the derivative ${\partial}^{n(v)}F_{g(v)}(V)/n(v)!\in{\Lambda}[H]{\otimes}{\Lambda}{{[\![}}H{{]\!]}}$ by $f_{g,n}$. (Note that $f_{g,n}=F([{\overline{\mathcal{M}}}_{g,n}],V)$.) Then $$\begin{aligned}
6 \, \Gamma(\Gamma_1(f_{1,2},f_{0,3}),f_{0,3})
&- 5 \, \Gamma(f_{1,2},\Gamma(f_{0,3},f_{0,3})) \\
& {} - 2 \, \Gamma(f_{0,3},\Gamma(f_{1,1},f_{0,4}))
+ 6 \, \Gamma(f_{0,4},\Gamma(f_{1,1},f_{0,3})) \\
& {} + \Gamma(f_{0,4},\Delta f_{0,4})
+ \Gamma(f_{0,5},\Delta f_{0,3})
- \Gamma_2(f_{0,4},f_{0,4}) = 0 .\end{aligned}$$
This follows from the following table, which is obtained by application of . $$\begin{tabular}{|L|L||L|L|}
\hline
\delta & F(\delta,V) & & \\ \hline
\delta_{2,2} & \frac12 \Gamma(\Gamma(f_{1,2},f_{0,3}),f_{0,3}) &
\delta_{0,2} & \Gamma(f_{0,3},\Delta f_{0,5}) \\
& {} - \frac14 \Gamma(f_{1,2},\Gamma(f_{0,3},f_{0,3})) &
\delta_{0,3} & \Gamma(f_{0,4},\Delta f_{0,4}) \\
\delta_{2,3} & \frac12 \Gamma(f_{1,2},\Gamma(f_{0,3},f_{0,3})) &
\delta_{0,4} & \Gamma(f_{0,5},\Delta f_{0,3}) \\
\delta_{2,4} & \Gamma(f_{0,3},\Gamma(f_{1,1},f_{0,4})) &
\delta_\alpha & \Gamma_2(f_{0,3},f_{0,5}) \\
\delta_{3,4} & \Gamma(f_{0,4},\Gamma(f_{1,1},f_{0,3})) &
\delta_\beta & \frac12 \Gamma_2(f_{0,4},f_{0,4}) \\[1pt] \hline
\end{tabular}$$
When we apply Proposition \[Relation\] with $V={\mathbb{CP}}^2$ and evaluate the resulting multilinear form to ${\omega}^{\boxtimes4}$, we obtain the recursion relation for the elliptic Gromov-Witten invariants $N^{(1)}_n$ of ${\mathbb{CP}}^2$.
The symbol of the new relation
==============================
We may introduce a filtration on Gromov-Witten invariants with respect to which the leading order of our new relation takes a relatively simple form; by analogy with the case of differential operators, we call this leading order relation the symbol of the full relation. In some cases, this symbol may be used to prove that elliptic Gromov-Witten invariants are determined by rational ones.
The *symbol* of a relation $\delta=0$ among cycles of strata in ${\overline{\mathcal{M}}}_{g,n}$ is the set of relations among Gromov-Witten invariants obtained by taking, for each $\beta\in\operatorname{NE}_1(V)$, the coefficient of $q^\beta$ in $I_{g,n}^V\cap[\delta]$, expanding in Feynman diagrams using the composition axiom, and setting all Gromov-Witten invariants ${\langle}I_{g',n',\beta'}^V{\rangle}$ other than ${\langle}I_{g,n,\beta}^V{\rangle}$ and ${\langle}I_{0,3,0}^V{\rangle}$ to zero.
We define a total order on the symbols ${\langle}I_{g,n,\beta}^V{\rangle}$ by setting ${\langle}I_{g',n',\beta'}^V{\rangle}\prec{\langle}I_{g,n,\beta}^V{\rangle}$ if $g'<g$, or $g'=g$ and $n'<n$, or $g'=g$, $n'=n$ and $\beta=\beta'+\beta''$ where $\beta''\in\operatorname{NE}_1(V)$ is non-zero. Thus, knowledge of the symbol determines relations among Gromov-Witten invariants such that the error in the relation on ${\langle}I_{g,n,\beta}^V{\rangle}$ involves invariants ${\langle}I_{g',n',\beta'}^V{\rangle}$ with ${\langle}I_{g',n',\beta'}^V{\rangle}\prec{\langle}I_{g,n,\beta}^V{\rangle}$. (Here, we must of course exclude ${\langle}I_{0,3,0}^V{\rangle}$.) We use the symbol $\sim$ to denote this equivalence relation.
For example, the symbol of the WDVV equation is $$(a,b,c\cup d) + (a\cup b,c,d) \sim (-1)^{|a|(|b|+|c|)} \bigl( (b,c,a\cup d)
+ (b\cup c,a,d) \bigr) ,$$ where we have abbreviated ${\langle}I_{0,n,\beta}^V
(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\dots,\alpha_n){\rangle}$ to $(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$.
Next, consider the symbol of the relation $$\pi_{4,n-4}^{-1}\bigl(12\delta_{2,2} - 4\delta_{2,3} - 2\delta_{2,4} +
6\delta_{3,4} + \delta_{0,3} + \delta_{0,4} - 2\beta\bigr) = 0$$ in $H_{2n-4}({\overline{\mathcal{M}}}_{1,n},{\mathbb{Q}})$. Only the cycles $\delta_{2,2}$ and $\delta_{2,3}$ contribute terms to the symbol. Abbreviate the Gromov-Witten class ${\langle}I_{1,n,\beta}^V(\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n){\rangle}$ to $\{\alpha_1,\alpha_2\}$. Up to a numerical factor to be determined, the cycle $\delta_{2,2}$ contributes the expression $$\{a\cup b,c\cup d\} + (-1)^{|b|\,|c|} \{a\cup c,b\cup d\} +
(-1)^{(|b|+|c|)|d|} \{a\cup d,b\cup c\} .$$ This numerical factor equals $$\frac{1}{24} {\cdot}3 {\cdot}12 {\cdot}8 = 12 .$$ The factor $1/24$ comes from symmetrization over the four inputs, the factor of $3$ from the three strata making up $\delta_{2,2}$, the factor of $12$ is the coefficient of the cycle in the relation, and the factor $8$ is illustrated by listing all of the graphs which contribute a term $\{a\cup b
,c\cup d\}$: $$\def\nnn{\begin{picture}(48,95)(56,715)
\put( 80,760){\circle{5}}
\put( 80,757){\line( 0,-1){ 18}}
\put( 80,762){\line( 0, 1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 10}}
\put( 80,780){\line( 1, 2){ 10}}
\put( 80,780){\line(-1, 2){ 10}}}
\nnn
\put( 67,805){$a$}
\put( 87,805){$b$}
\put( 67,709){$c$}
\put( 87,709){$d$}
\end{picture}
\nnn
\put( 67,805){$a$}
\put( 87,805){$b$}
\put( 67,709){$d$}
\put( 87,709){$c$}
\end{picture}
\nnn
\put( 67,805){$b$}
\put( 87,805){$a$}
\put( 67,709){$c$}
\put( 87,709){$d$}
\end{picture}
\nnn
\put( 67,805){$b$}
\put( 87,805){$a$}
\put( 67,709){$d$}
\put( 87,709){$c$}
\end{picture}
\nnn
\put( 67,805){$c$}
\put( 87,805){$d$}
\put( 67,709){$a$}
\put( 87,709){$b$}
\end{picture}
\nnn
\put( 67,805){$d$}
\put( 87,805){$c$}
\put( 67,709){$a$}
\put( 87,709){$b$}
\end{picture}
\nnn
\put( 67,805){$c$}
\put( 87,805){$d$}
\put( 67,709){$b$}
\put( 87,709){$a$}
\end{picture}
\nnn
\put( 67,805){$d$}
\put( 87,805){$c$}
\put( 67,709){$b$}
\put( 87,709){$a$}
\end{picture}$$ Similarly, the cycle $\delta_{2,3}$ contributes the expression $$\begin{gathered}
\{a,b\cup c\cup d\} + (-1)^{|a|\,|b|} \{b,a\cup c\cup d\} \\
+ (-1)^{(|a|+|b|)|c|} \{c,a\cup b\cup d\} + (-1)^{(|a|+|b|+|c|)|d|}
\{d,a\cup b\cup c\} ,\end{gathered}$$ with numerical factor $$\frac{1}{24} {\cdot}12 {\cdot}(-4) {\cdot}6 = - 12 ;$$ the factor $12$ counts the strata making up $\delta_{2,3}$, $-4$ is the coefficient of the cycle in the relation, and we illustrate the factor $6$ by listing all of the graphs which contribute a term $\{a,b\cup c\cup d\}$: $$\def\nnn{\begin{picture}(60,95)(50,695)
\put( 80,760){\circle{5}}
\put( 80,757){\line( 0,-1){ 18}}
\put( 80,762){\line( 0, 1){ 18}}
\put( 80,740){\line( 1,-2){ 10}}
\put( 80,740){\line(-1,-2){ 20}}
\put( 70,720){\line( 1,-2){ 10}}
}
\nnn
\put( 56,688){$b$}
\put( 77,688){$c$}
\put( 87,708){$d$}
\put( 77,783){$a$}
\end{picture}
\nnn
\put( 56,688){$c$}
\put( 77,688){$b$}
\put( 87,708){$d$}
\put( 77,783){$a$}
\end{picture}
\nnn
\put( 56,688){$b$}
\put( 77,688){$d$}
\put( 87,708){$c$}
\put( 77,783){$a$}
\end{picture}
\nnn
\put( 56,688){$d$}
\put( 77,688){$b$}
\put( 87,708){$c$}
\put( 77,783){$a$}
\end{picture}
\nnn
\put( 56,688){$c$}
\put( 77,688){$d$}
\put( 87,708){$b$}
\put( 77,783){$a$}
\end{picture}
\nnn
\put( 56,688){$d$}
\put( 77,688){$c$}
\put( 87,708){$b$}
\put( 77,783){$a$}
\end{picture}$$ In conclusion, we obtain the following result.
\[symbol\] Abbreviating ${\langle}I_{1,n,\beta}^V(\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n){\rangle}$ to $\{\alpha_1,\alpha_2\}$, we have $$\begin{aligned}
\Psi(a,b,c,d) & = \{a\cup b,c\cup d\} + (-1)^{|b|\,|c|} \{a\cup c,b\cup d\} +
(-1)^{(|b|+|c|)|d|} \{a\cup d,b\cup c\} \\
& {} - \{a,b\cup c\cup d\} - (-1)^{|a|\,|b|} \{b,a\cup c\cup d\} \\
& {} - (-1)^{(|a|+|b|)|c|} \{c,a\cup b\cup d\} - (-1)^{(|a|+|b|+|c|)|d|}
\{d,a\cup b\cup c\} \sim 0 .\end{aligned}$$
Note that the linear form $\Psi(a,b,c,d)$ is (graded) symmetric in its four arguments, and vanishes if any of them equals $1$.
\[reduce\] If ${\omega}\in H^2(V,{\mathbb{Q}})$ and $a,b\in H^{\bullet}(V,{\mathbb{Q}})$, then for $j>2$, $$\textstyle
\{{\omega}^i\cup a,{\omega}^{j-i}\cup b\} = \binom{i+2}{2} \{a,{\omega}^j\cup b\} .$$
By Theorem \[symbol\], we have for $i\ge0$ and $j>2$, $$\begin{gathered}
\Psi({\omega},{\omega}^{i+1}\cup a,{\omega},{\omega}^{j-i-3}\cup b) -
\Psi({\omega},{\omega}^i\cup a,{\omega},{\omega}^{j-i-2}\cup b) \\
\begin{aligned} \sim & \quad \bigl(
2\{{\omega}^{i+2}\cup a,{\omega}^{j-i-2}\cup b\} +
\{{\omega}^2,{\omega}^{j-2}\cup a\cup b\} \\
{} & {} - \{{\omega}^{i+1}\cup a,{\omega}^{j-i-1}\cup b\} -
\{{\omega}^{i+3}\cup a,{\omega}^{j-i-3}\cup b\} \bigr) \\
{} & {} - \bigl( 2\{{\omega}^{i+1}\cup a,{\omega}^{j-i-1}\cup b\} +
\{{\omega}^2,{\omega}^{j-2}\cup a\cup b\} \\
{} & {} - \{{\omega}^i\cup a,{\omega}^{j-i}\cup b\}
- \{{\omega}^{i+2}\cup a,{\omega}^{j-i-2}\cup b\} \bigr) \\
\sim & \quad \{{\omega}^i\cup a,{\omega}^{j-i}\cup b\} -
3\{{\omega}^{i+1}\cup a,{\omega}^{j-i-1}\cup b\} \\
{} & \quad {} + 3\{{\omega}^{i+2}\cup a,{\omega}^{j-i-2}\cup b\}
- \{{\omega}^{i+3}\cup a,{\omega}^{j-i-3}\cup b\} \sim 0 .
\end{aligned}\end{gathered}$$ This implies that the function $a(i,j)=\{{\omega}^i\cup a,{\omega}^{j-i}\cup b\}$ satisfies the difference equation $$a(i,j) - 3 a(i+1,j) + 3a(i+2,j) - a(i+3,j) \sim 0$$ with solution $a(i,j)\sim\binom{i+2}{2} a(0,j)$.
We can now prove a weak analogue for elliptic Gromov-Witten invariants of the (first) Reconstruction Theorem of Kontsevich-Manin (Theorem 3.1 of [@KM]). For $0\le j\le d$, let $P_j(V)=\operatorname{coker}( H^{j-2}(V,{\mathbb{Q}})
\xrightarrow{{\omega}\cup{\cdot}} H^j(V,{\mathbb{Q}}) )$ be the $j^{\text{th}}$ primitive cohomology group of $V$.
\[reconstruction\] If $P^i(V)=0$ for $i>2$, the elliptic Gromov-Witten invariants of $V$ are determined by its rational Gromov-Witten invariants together with the Gromov-Witten invariants ${\langle}I_{1,1,\beta}(-){\rangle}:H^{2i+2}(V,{\mathbb{Q}})\to{\mathbb{Q}}$ for $0\le c_1(V)\cap\beta=i<d$. (These are all of the non-vanishing Gromov-Witten invariants ${\langle}I_{1,1,\beta}(\alpha){\rangle}$.)
We proceed by induction: by hypothesis, ${\langle}I_{g,n,\beta}^V{\rangle}$ is known for $g=0$ or $g=1$ and $n=1$. Now consider the Gromov-Witten invariant ${\langle}I_{1,n,\beta}^V(\alpha_1,\dots,\alpha_n){\rangle}$, where $n>1$. By , we may assume that $|\alpha_i|>2$, and under the hypotheses of the proposition, we may write it as ${\omega}^{p_i}\cup\gamma_i$ where $|\gamma_i|\le2$ is a primitive cohomology class.
Step 1: If any two indices $p_i$ and $p_j$ satisfy $p_i+p_j>2$, we may apply Corollary \[reduce\] to replace the pair $({\omega}^{p_i}\cup\gamma_i,{\omega}^{p_j}\cup\gamma_j)$ by $(\gamma_i,{\omega}^{p_i+p_j}\cup\gamma_j)$. If $|\gamma_1|=1$, the result vanishes by , while if $|\gamma_1|=2$, we may apply to reduce $n$ by $1$.
Step 2: We are reduced to considering ${\langle}I_{1,n,\beta}^V({\omega}\cup\gamma_1,\dots,{\omega}\cup\gamma_n){\rangle}$, where the classes $\gamma_i$ have degree $1$ or degree $2$. Applying Theorem \[symbol\], we see that $$\Psi({\omega},\gamma_1,{\omega},\gamma_2) = 2\{{\omega}\cup\gamma_1,{\omega}\cup\gamma_2\} +
\{{\omega}^2,\gamma_1\cup\gamma_2\} \sim 0 .$$ In particular, we may assume that $n=2$, since otherwise, we would be able to return to Step 1. There are two cases.
Step 2a: If the classes $\gamma_i$ are both of degree $1$, we see that $\{{\omega}\cup\gamma_1,{\omega}\cup\gamma_2\}\sim0$, since in that case $\gamma_1\cup\gamma_2$ has degree $2$ and we may apply .
Step 2b: If the classes $\gamma_i$ are both of degree $2$, there is a class $\gamma\in H^2(V,{\mathbb{Q}})$ such that $\gamma_1\cup\gamma_2 = {\omega}\cup\gamma$, since $P^4(V)=0$. We must calculate $${\langle}I_{1,2,\beta}^V({\omega}^2,\gamma_1\cup\gamma_2){\rangle}= {\langle}I_{1,2,\beta}^V({\omega}^2,{\omega}\cup\gamma){\rangle}\sim 6
{\langle}I_{1,2,\beta}^V(1,{\omega}^3\cup\gamma){\rangle}= 0 ,$$ where we have applied Corollary \[reduce\] and .
Two special cases of this result are worth singling out:
1. If $V$ is a surface, the elliptic Gromov-Witten invariants are determined by the rational invariants together with ${\langle}I_{1,1,\beta}(-){\rangle}:H^2(V,{\mathbb{Q}})\to{\mathbb{Q}}$ for $c_1(V)\cap\beta=0$ and ${\langle}I_{1,1,\beta}(-){\rangle}:H^4(V,{\mathbb{Q}})\to{\mathbb{Q}}$ for $c_1(V)\cap\beta=1$. If $V$ is the blow-up of ${\mathbb{CP}}^2$ at a finite number of points, only $\beta=0$ satisfies $c_1(V)\cap\beta<2$, and by Proposition \[BCOV\], ${\langle}I_{1,1,0}{\rangle}$ is determined by $c_1(V)$, while the rational Gromov-Witten invariants are determined by the WDVV equation (Göttsche-Pandharipande [@GP]).
2. If $V={\mathbb{CP}}^d$, the elliptic Gromov-Witten invariants are determined by the rational Gromov-Witten invariants.
Gromov-Witten invariants of curves
==================================
To illustrate our new relation, we start with the case where $V$ is a curve. We will only discuss curves of genus $0$ and $1$, since for curves of higher genus, $I_{g,n,\beta}^V=0$ if $\beta\ne0$, and the new relation is identically satisfied.
The projective line
-------------------
When $V={\mathbb{CP}}^1$, the potential $F_g$ is a power series of degree $4g-4$ in variables $t_0$ and $t_1$ (of degree $-2$ and $0$) and the generator $q$ of $\Lambda$, of degree $-4=-2c_1({\mathbb{CP}}^1)\cap[{\mathbb{CP}}^1]$. By degree counting, together with , we see that $$F_g({\mathbb{CP}}^1) =
\begin{cases} \displaystyle
t_0^2t_1/2 + q e^{t_1} , & g=0 , \\
\displaystyle -t_1/24 , & g=1 , \\ 0 , & g>1 ;
\end{cases}$$ the only thing which is not immediate is the coefficient of $q$ in $F_0({\mathbb{CP}}^1)$, which is the number of maps of degree $1$ from ${\mathbb{CP}}^1$ to itself, up to isomorphism, and clearly equals $1$.
It is easy to calculate $F(\delta,{\mathbb{CP}}^1)$ for $\delta$ equal to one of our nine $2$-cycles: all of them vanish except $$F(\delta_{3,4},{\mathbb{CP}}^1) = \frac{t_1^4}{24} {\otimes}(-qe^{t_1}/6) ;
F(\delta_{0,4},{\mathbb{CP}}^1) = \frac{t_1^4}{24} {\otimes}qe^{t_1} ;
F(\delta_\alpha,{\mathbb{CP}}^1) = \frac{t_1^4}{24} {\otimes}2qe^{t_1} .$$ We see that the new relation holds among these potentials.
Elliptic curves
---------------
Let $E$ be an elliptic curve. Denote by $\xi,\eta$ variables of degree $-1$ corresponding to a basis of $H_1(E,{\mathbb{Z}})$ such that ${\langle}\xi,\eta{\rangle}=1$. The ring $\Lambda$ has one generator $q$, of degree $0$ (since $c_1(V)=0$). Since there are no rational curves in $E$ of positive degree, we have $$F_0(E) = t_0^2t_1/2 + t_0\xi\eta .$$ It is shown in [@BCOV] that $$\label{Eisenstein}
F_1(E) = - \frac{t_1}{24} + \sum_{\beta=1}^\infty \frac{\sigma(\beta)}{\beta}
q^\beta \bigl(e^{\beta t_1} - 1\bigr) ,$$ since ${\langle}I_{1,1,\beta}^E({\omega}){\rangle}=\sigma(\beta)$ counts the number of unramified covers of degree $\beta$ of the curve $E$ up to automorphisms, which are easily enumerated. An equivalent form of is $$\frac{{\partial}F_1(E)}{{\partial}t_1} = G_2(qe^{t_1}) ,$$ where $$G_2(q) = - \frac{1}{24} + \sum_{\beta=1}^\infty \sigma(\beta) q^\beta$$ is the Eisenstein series of weight $2$. By degree counting, we also see that $F_g(E)=0$ for $g>1$.
Note that the Gromov-Witten invariants of an elliptic curve are invariant under deformation; this is true for any smooth projective variety $V$ (Li-Tian [@LT]). In fact, the definition of Gromov-Witten invariants extends to any almost-Kähler manifold (a symplectic manifold with compatible almost-complex structure), and the resulting invariants are independent of the almost-complex structure (Li-Tian [@LT:symp]).
It is simple to calculate the Gromov-Witten potentials $F(\delta,E)$ for our nine $2$-cycles in ${\overline{\mathcal{M}}}_{1,4}$.
\[elliptic\] We have $$F(\delta_{2,2},E) = \Bigl( \frac{5}{12} G_4(qe^{t_1}) - G_2(qe^{t_1})^2
\Bigr) (t_0t_1+\xi\eta)^2 = \frac{q}{2} (t_0t_1+\xi\eta)^2 + O(q^2) ,$$ $F(\delta_{2,3},E)=3F(\delta_{2,2},E)$, while the remaining $7$ potentials vanish.
Again, we see that the new relation holds.
The Gromov-Witten invariants of ${\mathbb{CP}}^2$
=================================================
The Gromov-Witten potential $F_g({\mathbb{CP}}^2)$ is a power series of degree $2g-2$ in variables $t_0$, $t_1$ and $t_2$, of degrees $-2$, $0$ and $2$, where $t_i$ is dual to ${\omega}^i$, and the generator $q$ of $\Lambda$, of degree $-6=-2c_1({\mathbb{CP}}^2)\cap[L]$.
By degree counting, together with , we see that $$F_g({\mathbb{CP}}^2) =
\begin{cases}
\displaystyle \frac{1}{2} (t_0t_1^2 + t_0^2t_2) + \sum_{\beta=1}^\infty
N^{(0)}_\beta q^\beta e^{\beta t_1} \frac{t_2^{3\beta-1}}{(3\beta-1)!} , &
g=0 , \\
\displaystyle - \frac{t_1}{8} + \sum_{\beta=1}^\infty N^{(1)}_\beta q^\beta
e^{\beta t_1} \frac{t_2^{3\beta}}{(3\beta)!} , & g=1 , \\
\displaystyle \sum_{\beta=1}^\infty N^{(g)}_\beta q^\beta e^{\beta t_1}
\frac{t_2^{3\beta+g-1}}{(3\beta+g-1)!} , & g>1 , \end{cases}$$ where $N^{(g)}_\beta$ are certain rational coefficients.
Using the Severi theory of plane curves, we will show that $N^{(g)}_\beta$ is the answer to an enumerative problem for plane curves; in particular, it is a non-negative integer. This phenomenon is special to del Pezzo surfaces: we have already seen that the elliptic Gromov-Witten invariants of an elliptic curve are non-integral, while for ${\mathbb{CP}}^3$, they are not even positive.
We apply the following result, which is Proposition 2.2 of Harris [@Harris].
\[Harris\] Let $S$ be a smooth rational surface. Let $\pi:{\mathcal{C}}\to{\mathcal{M}}$ be a family of curves of geometric genus $g$ with ${\mathcal{M}}$ irreducible, and let $f:{\mathcal{C}}\to{\mathcal{M}}$ be a map such that on each component of a general fibre ${\mathcal{C}}_z$ of $\pi$, the restriction $f_z$ of $f$ to ${\mathcal{C}}_z$ is not constant and $f_z^*{\omega}_S$ has negative degree.
Let $W$ be the image of the map from ${\mathcal{M}}$ to the Chow variety of curves on $S$ defined by sending $z\in{\mathcal{M}}$ to the curve ${\mathcal{C}}_z$. Then $\dim(W)\le-\deg(f_z^*{\omega}_S)+g-1$, and if equality holds, then $f_z$ is birational for all $z\in{\mathcal{M}}$.
\[Severi\] The coefficient $N^{(g)}_\beta$ equals the number of irreducible plane curves of arithmetic genus $g$ and degree $\beta$ passing through $3\beta+g-1$ general points in ${\mathbb{CP}}^2$.
Let ${\mathcal{M}}$ be a component of the boundary ${\overline{\mathcal{M}}}_{g,n}({\mathbb{CP}}^2,\beta)\setminus{\mathcal{M}}_{g,n}({\mathbb{CP}}^2,\beta)$, and consider the family of curves ${\mathcal{C}}\to{\mathcal{M}}$ obtained by restricting the universal curve ${\overline{\mathcal{M}}}_{g+1,n}({\mathbb{CP}}^2,\beta)\to{\overline{\mathcal{M}}}_{g,n}({\mathbb{CP}}^2,\beta)$ to ${\mathcal{M}}$ and contracting to a point all components of the fibres on which $f$ has degree $0$.
The geometric genus of the fibres of this family is bounded above by $g-1$. Applying Proposition \[Harris\], we see that the image of ${\mathcal{M}}$ in the Chow variety of plane curves has dimension at most $3\beta+g-2$.
On the other hand, if ${\mathcal{M}}$ is a component of ${\mathcal{M}}_{g,n}({\mathbb{CP}}^2,\beta)$, and ${\mathcal{C}}\to{\mathcal{M}}$ is the universal family of curves ${\mathcal{C}}\to{\mathcal{M}}$, we see that the image of ${\mathcal{M}}$ in the Chow variety of plane curves has dimension less than $3\beta+g-1$ unless the stable maps parametrized by ${\mathcal{M}}$ are birational to their image.
The Gromov-Witten invariant $N^{(g)}_\beta$ equals the degree of the intersection of the image of ${\overline{\mathcal{M}}}_{g,3\beta+g-1}({\mathbb{CP}}^2,\beta)$ in the Chow variety of curves in ${\mathbb{CP}}^2$ with the cycle of curves passing through $3\beta+g-1$ general points. By Bertini’s theorem for homogenous spaces [@Kleiman], we see that the points of intersection are reduced and lie in the components of ${\mathcal{M}}_{g,n}({\mathbb{CP}}^2,\beta)$ on which the map $f$ is birational to its image, and hence an [embedding]{}. (This argument is borrowed from Section 6 of Fulton-Pandharipande [@FP].) The result follows.
Comparison with the formulas of Caporaso and Harris
---------------------------------------------------
Caporaso and Harris [@CH] have calculated the numbers $N^{(g)}_\beta$ for all $g\ge0$, and we now turn the comparison of our results for $N^{(1)}_\beta$ . We have not been able to find a proof that our answers agree, but we have verified that this is so for $\beta\le6$.
The recursion of Caporaso and Harris for the Gromov-Witten invariants of ${\mathbb{CP}}^2$ is more easily applied if it is recast in terms of generating functions.
If $\alpha$ is a partition, denote by $\ell(\alpha)$ the number of parts of $\alpha$ and by $|\alpha|$ the sum $\alpha_1+\dots+\alpha_{\ell(\alpha)}$ of the parts of $\alpha$. Let $\alpha!$ be the product $\alpha! = \alpha_1!
\dots \alpha_{\ell(\alpha)}!$.
Fix a line $L$ in ${\mathbb{CP}}^2$. If $\alpha$ and $\beta$ are partitions with $|\alpha|+|\beta|=d$, and ${\Omega}$ is a collection of $\ell(\alpha)$ general points of $L$, let $V^{d,\delta}(\alpha,\beta)({\Omega})=V^{d,\delta}(\alpha,\beta)$ be the generalized Severi variety: the closure of the locus of reduced plane curves of degree $d$ not containing $L$, smooth except for $\delta$ double points, having order of contact $\alpha_i$ with $L$ at ${\Omega}_i$, and to order $\beta_1,\dots,\beta_{\ell(\beta)}$ at $\ell(\beta)$ further unassigned points of $L$. For example, $V^{d,\delta}(0,1^d)$ is the classical Severi variety of plane curves of degree $d$ with $\delta$ double points, while $V^{d,\delta}(0,21^{d-1})$ is the closure of the locus of plane curves tangent to $L$ at a smooth point.
Denote by $V_0^{d,\delta}(\alpha,\beta)$ the union of the components of $V^{d,\delta}(\alpha,\beta)$ whose general point is an irreducible curve. Let $N^{d,\delta}(\alpha,\beta)$ be the degree of $V^{d,\delta}(\alpha,\beta)$ and let $N_0^{d,\delta}(\alpha,\beta)$ be the degree of $V_0^{d,\delta}(\alpha,\beta)$. Form the generating functions $$\begin{aligned}
Z &= \sum \frac{z^{\binom{d+1}{2}-\delta+\ell(\beta)}}
{\bigl(\binom{d+1}{2}-\delta+\ell(\beta)\bigr)!} \frac{p^\alpha}{\alpha!}
q^\beta N^{d,\delta}(\alpha,\beta) , \\
F &= \sum \frac{z^{\binom{d+1}{2}-\delta+\ell(\beta)}}
{\bigl(\binom{d+1}{2}-\delta+\ell(\beta)\bigr)!} \frac{p^\alpha}{\alpha!}
q^\beta N_0^{d,\delta}(\alpha,\beta) .\end{aligned}$$ The integer $\binom{d+1}{2}-\delta+\ell(\beta)$ is the dimension of the variety $V^{d,\delta}(\alpha,\beta)$. The union of curves of degree $d_i$, $1\le i\le n$, with $\delta_i$ double points and partitions $\alpha_i$ and $\beta_i$ is a (reducible) curve has degree $d=d_1+\dots+d_n$ with $$\delta = \delta_1 + \dots + \delta_n + \sum_{i<j} \delta_i\delta_j$$ double points and partitions $\alpha=(\alpha_1,\dots,\alpha_n)$ and $\beta=(\beta_1,\dots,\beta_n)$. This formula for $\delta$ amounts to the condition that the sum of the dimensions of the generalized Severi varieties $V_0^{d_i,\delta_i}(\alpha_i,\beta_i)$ equals the dimension of $V^{d,\delta}(\alpha,\beta)$. The proof of the relationship $Z = \exp(F)$ between these two generating functions is an exercise in the definition of degree (see Ran [@Ran]).
Caporaso and Harris prove a recursion which in terms of the generating function $Z$ may be written $$\frac{{\partial}Z}{{\partial}z} = \sum_{k=1}^\infty kq_k\frac{{\partial}Z}{{\partial}p_k} + \operatorname{res}_{t=0}
\biggl[ \exp\Bigl( \sum_{k=1}^\infty t^{-k} p_k + \sum_{k=1}^\infty k t^k
\frac{{\partial}}{{\partial}q_k} \Bigr) \biggr] Z ,$$ where $\operatorname{res}_{t=0}$ is the residue with respect to the formal variable $t$, in other words, the coefficient of $t^{-1}$ when the exponential is expanded.[^2] Dividing by $Z$, we obtain $$\frac{{\partial}F}{{\partial}z} = \sum_{k=1}^\infty kq_k\frac{{\partial}F}{{\partial}p_k} + \operatorname{res}_{t=0}
\biggl[ \exp\Bigl( \sum_{k=1}^\infty t^{-k} p_k + F|_{q_k\mapsto q_k+kt^k}
- F \Bigr) \biggr] ,$$ which clearly allows the recursive calculation of the coefficients $N_0^{d,\delta}(\alpha,\beta)$.
As a special case of $Z=\exp(F)$, we have $$1 + \sum \frac{z^{\binom{d+2}{2}-\delta-1} q^d N^{d,\delta}}
{\bigl(\binom{d+2}{2}-\delta-1\bigr)!} = \exp \biggl( \sum
\frac{z^{\binom{d+2}{2}-\delta-1} q^d N_0^{d,\delta}}
{\bigl(\binom{d+2}{2}-\delta-1\bigr)!} \biggr) ,$$ since $\binom{d+1}{2}-\delta+d=\binom{d+2}{2}-\delta-1$. Expanding the exponential, we obtain $$N^{d,\delta} = \sum_{n=1}^\infty \frac{1}{n!} \sum_{d=d_1+\dots+d_n} \\
\sum_{\substack{\delta=\sum_{i<j}\delta_i\delta_j\\+\delta_1+\dots+\delta_n}}
\frac{\bigl(\binom{d+2}{2}-\delta-1\bigr)! N_0^{d_1,\delta_1} \dots
N_0^{d_n,\delta_n}} {\bigl(\binom{d_1+2}{2}-\delta_1-1\bigr)! \dots
\bigl(\binom{d_i+2}{2}-\delta_i-1\bigr)!} .$$ For example, with $d=5$, we obtain $$\begin{aligned}
N_0^{5,4} &= N^{5,4} - \frac{16!}{14!2!} N_0^{4,0}N_0^{1,0} = 36975 - 120 {\cdot}1 = 36855 , \\
N_0^{5,5} &= N^{5,5} - \frac{15!}{13!2!} N_0^{4,1}N_0^{1,0} = 90027 - 105
{\cdot}27 = 87192 ,\end{aligned}$$ while with $d=6$ and $\delta=9$, we obtain $$\begin{aligned}
N_0^{6,9} &= N^{6,9} - 18! \biggl( \frac{N_0^{5,4}}{16!}
\frac{N_0^{1,0}}{2!} - \frac{N_0^{4,1}}{13!} \frac{N_0^{2,0}}{3!}
- \frac{1}{2} \Bigl( \frac{N_0^{3,0}}{9!} \Bigr)^2 - \frac{1}{2}
\frac{N_0^{4,0}}{14!} \Bigl( \frac{N_0^{1,0}}{2!} \Bigr)^2 \biggr) \\
&= 63338881 - 153 {\cdot}36855 {\cdot}1 + 8568 {\cdot}27 {\cdot}1 + {\tfrac12}{\cdot}48620 {\cdot}1^2 +
{\tfrac12}{\cdot}18360 {\cdot}1 {\cdot}1^2 \\ &= 57435240\end{aligned}$$ in agreement with the recursion .
By Proposition \[Severi\], the relation between the numbers $N_0^{d,\delta}$ and the Gromov-Witten invariants is very simple: $N^{(g)}_d=N_0^{d,\delta}$ where $g=\binom{d-1}{2}-\delta$. In terms of $F$, the Gromov-Witten potentials $F_g({\mathbb{CP}}^2)$ are given by the formula $$\sum_{g=0}^\infty \hbar^{g-1} F_g({\mathbb{CP}}^2) = \frac{1}{2\hbar}
(t_0^2t_2+t_0t_1^2) - \frac{t_1}{8} +
F\big|_{\substack{(q_1,q_2,\dots)=(\hbar^{-3}qe^{t_1},0,\dots) \\
(p_1,p_2,\dots)=(0,0,\dots) , z=\hbar t_2}} .$$
The elliptic Gromov-Witten invariants of ${\mathbb{CP}}^3$
==========================================================
For $g=0$ and $g=1$, the Gromov-Witten potentials of the projective space ${\mathbb{CP}}^3$ have the form $$F_g({\mathbb{CP}}^3) = \begin{cases}
\bigl( \frac{1}{2} t_0^2t_3 + t_0t_1t_2 + \frac{1}{6} t_1^3 \bigr) +
\displaystyle \sum_{4\beta=a+2b} N^{(0)}_{ab} q^\beta e^{\beta t_1}
\frac{t_2^at_3^b}{a!b!} , & g=0 , \\
\displaystyle - \frac{t_1}{4} + \sum_{4\beta=a+2b}
N^{(1)}_{ab} q^\beta e^{\beta t_1} \frac{t_2^at_3^b}{a!b!} , & g=1 .
\end{cases}$$ Here, $t_i$ is the formal variable of degree $2i-2$ dual to ${\omega}^i\in
H^{2i}({\mathbb{CP}}^3,{\mathbb{Q}})$ and $q$ is the generator of the Novikov ring $\Lambda\cong{\mathbb{Q}}{{(\!(}}q{{)\!)}}$ of ${\mathbb{CP}}^3$. By Proposition \[BCOV\], the coefficient of $t_1$ in $F_1({\mathbb{CP}}^3)$ equals $-c_2({\mathbb{CP}}^3)/24$.
Thus, the coefficient $N^{(g)}_{ab}$ is a rational number which “counts” the number of stable maps of degree $\beta$ from a curve of genus $g$ to ${\mathbb{CP}}^3$ meeting $a$ generic lines and $b$ generic points.
It is shown by Fulton and Pandharipande [@FP] that $N^{(0)}_{ab}$ equals the number of rational space curves of degree $\beta$ which meet $a$ generic lines and $b$ generic points. In particular, they are non-negative integers. By contrast, the coefficients $N^{(1)}_{ab}$ are neither positive nor integral: for example, $N^{(1)}_{02}=-1/12$. In [@cp3], we prove the following result.
The number of elliptic space curves of degree $\beta$ passing through $a$ generic lines and $b$ generic points, where $4\beta=a+2b$, equals $N^{(1)}_{ab} + (2\beta-1)N^{(0)}_{ab}/12$.
By evaluating the equation of Proposition \[Relation\] on ${\omega}\boxtimes{\omega}\boxtimes{\omega}\boxtimes{\omega}$, we obtain the following relation among the elliptic Gromov-Witten for ${\mathbb{CP}}^3$: if $a\ge2$, then $$\begin{gathered}
3 N^{(1)}_{ab} = 4 nN^{(1)}_{a-2,b+1} - \tfrac{1}{4} n^2 N^{(0)}_{ab} +
\tfrac{1}{6} n^3 (n-3) N^{(0)}_{a-2,b+1} \\
\shoveleft{ {} - 2 \sum_{\substack{a-2=a_1+a_2\\b+1=b_1+b_2}}
\textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} n_2^2 (n-3n_1)
\binom{a-2}{a_1} \Bigr\{ n_1 \binom{b}{b_1} + n_2 \binom{b}{b_1-1} \Bigr\} } \\
\shoveleft{ {} + \sum_{\substack{a=a_1+a_2\\b=b_1+b_2}} N^{(1)}_{a_1b_1}
N^{(0)}_{a_2b_2} \textstyle \Bigl\{ n_1n_2 (n+3n_1) \binom{a-2}{a_1} +
n_2^2 (3n_1-n) \binom{a-2}{a_1-1} - 6 n_2^3 \binom{a-2}{a_1-2} \Bigr\}
\binom{b}{b_1} } \\
\shoveleft{{} + \tfrac{1}{12} {\displaystyle
\sum_{\substack{a=a_1+a_2\\b=b_1+b_2}}} N^{(0)}_{a_1b_1} N^{(0)}_{a_2b_2}
n_1 n_2^2 } \\
\textstyle \Bigl\{ n_1^2 (3-n_1) \binom{a-2}{a_1} + n_1n_2(n-3n_1-3)
\binom{a-2}{a_1-1} + n_2^2 (-n_1+n_2-6) \binom{a-2}{a_1-2} \Bigr\}
\binom{b}{b_1} \\
\shoveleft{{} + \tfrac{1}{2} \sum_{\substack{a=a_1+a_2+a_3\\b=b_1+b_2+b_3}}
\textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} N^{(0)}_{a_3b_3} \Bigl\{
2n_1n_2^3n_3(n+3n_1-3n_2) \binom{a-2}{a_2,a_3-2} - 6 n_2^3n_3^3
\binom{a-2}{a_2,a_3} } \\
\textstyle {} + n_2^2n_3^2 (3n_1-n) \Bigl( n_1
\binom{a-2}{a_2-1,a_3-1} + n_2 \binom{a-2}{a_2,a_3-1} + n_3
\binom{a-2}{a_2-1,a_3} \Bigr) \Bigr\} \binom{b}{b_2,b_3} .\end{gathered}$$ This relation determines the elliptic coefficient $N^{(1)}_{ab}$ for $a>0$ in terms of $N^{(1)}_{0,\frac{1}{2}a+b}$, the elliptic coefficients of lower degree, and the rational coefficients.
To determine the coefficients $N^{(1)}_{0,b}$, we need the relation obtained by evaluating Proposition \[Relation\] on ${\omega}^2\boxtimes{\omega}^2\boxtimes{\omega}\boxtimes{\omega}$: if $b\ge2$, then $$\begin{gathered}
0 = N^{(1)}_{ab} + \tfrac{1}{24} n(2n-1) N^{(0)}_{a+2,b-1}
+ \tfrac{1}{48} N^{(0)}_{a+4,b-2} \\
\shoveleft{ {} + \sum_{\substack{a+2=a_1+a_2\\b-1=b_1+b_2}} \textstyle
N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} \textstyle \Bigl\{ n_2 \Bigl( n
\binom{a}{a_1} + n_2 \binom{a}{a_1-1} \Bigr) \binom{b-2}{b_1-1} } \\[-10pt]
\shoveright{ \textstyle {} - \frac{1}{6} \Bigl( n_1(6n_1-n_2)
\binom{a}{a_1} + n_2 (16n_1-n_2) \binom{a}{a_1-1} + 6n_2^2 \binom{a}{a_1-2}
\Bigr) \binom{b-2}{b_1} \Bigr\} } \\
\shoveleft{{} - \tfrac{1}{12} \sum_{\substack{a+4=a_1+a_2\\b-2=b_1+b_2}}
\textstyle
N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} \Bigl( n_1 \binom{a}{a_1} + (2n_1-5n_2)
\binom{a}{a_1-1} + 6n_2 \binom{a}{a_1-2} \Bigr) \binom{b-2}{b_1} } \\
\shoveleft{ {} - \tfrac{1}{48} \sum_{\substack{a+4=a_1+a_2\\b-2=b_1+b_2}}
N^{(0)}_{a_1b_1} N^{(0)}_{a_2b_2} \textstyle \Bigl( n_1^3(n_1-1) \binom{a}{a_1}
+ n_1^2n_2(2n_1-2n_2+1) \binom{a}{a_1-1} } \\
\textstyle {} + n_1n_2^2(2n_1-2n_2+7) \binom{a}{a_1-2}
+ n_2^3(2n_1+5) \binom{a}{a_1-3} + n_2^4 \binom{a}{a_1-4} \Bigr)
\binom{b-2}{b_1} \\
\shoveleft{{} - \tfrac{1}{12}
\sum_{\substack{a+4=a_1+a_2+a_3\\b-2=b_1+b_2+b_3}}
\textstyle N^{(1)}_{a_1b_1} N^{(0)}_{a_2b_2} N^{(0)}_{a_3b_3}
\textstyle \Bigl\{ 3n_2n_3 \Bigl( n_2^2 \binom{a}{a_2,a_3-2} + n_3^2
\binom{a}{a_2-2,a_3} \Bigr) } \\
\shoveleft{ {} + \textstyle n_1 \Bigl( n_2^3 \binom{a}{a_2,a_3-4}
+ n_2^2(6n_1-n_3) \binom{a}{a_2-1,a_3-3} - 7n_2n_3^2 \binom{a}{a_2-2,a_3-2}
- 5n_3^3 \binom{a}{a_2-3,a_3-1} \Bigr) } \\
\shoveleft{\textstyle {} + \Bigl( n_2^3(n_1-5n_3) \binom{a}{a_2,a_3-3}
+ n_2^2n_3(5n_1-7n_3) \binom{a}{a_2-1,a_3-2} } \\
\textstyle {}+ n_2n_3^2 (5n_1-n_3) \binom{a}{a_2-2,a_3-1}
+ n_3^3(n_1+n_3) \binom{a}{a_2-3,a_3}\Bigr) \Bigr\} \binom{b-2}{b_2,b_3} .\end{gathered}$$ This relation determine the coefficient $N^{(1)}_{0b}$ in terms of elliptic coefficients of lower order and the rational coefficients, and thus ultimately in terms of $N^{(0)}_{02}=1$, the number of lines between two points.
Using these relation, we obtain the results of Table 2. Up to degree $3$, Theorem A is easily seen to hold, since there are no elliptic space curves of degrees $1$ and $2$, while all elliptic space curves of degree $3$ lie in a plane.
It is well-known that there is one quartic elliptic space curve through $8$ general points, while the number of elliptic quartic space curves through $16$ general lines was calculated by Vainsencher and Avritzer ([@Vainsencher]; see also [@Avritzer], which contains a correction to [@Vainsencher], bringing it into agreement with our calculation!).
\[CP3\]
$$\begin{tabular}{|R|C|R|D{.}{}{2}|R|} \hline
n & (a,b) & N^{(0)}_{ab} & N^{(1)}_{ab} &
{\scriptstyle N^{(1)}_{ab}+(2n-1)N^{(0)}_{ab}/12} \\ \hline
1 & (0,2) & 1 & -.\frac{1}{12} & 0 \\
& (2,1) & 1 & -.\frac{1}{12} & 0 \\
& (4,0) & 2 & -.\frac{1}{6} & 0 \\[5pt]
2 & (0,4) & 0 & .0 & 0 \\
& (2,3) & 1 & -.\frac{1}{4} & 0 \\
& (4,2) & 4 & -1. & 0 \\
& (6,1) & 18 & -4.\frac{1}{2} & 0 \\
& (8,0) & 92 & -23. & 0 \\[5pt]
3 & (0,6) & 1 & -.\frac{5}{12} & 0 \\
& (2,5) & 5 & -2.\frac{1}{12} & 0 \\
& (4,4) & 30 & -12.\frac{1}{2} & 0 \\
& (6,3) & 190 & -78.\frac{1}{6} & 1 \\
& (8,2) & 1\,312 & -532.\frac{2}{3} & 14 \\
& (10,1) & 9\,864 & -3\,960. & 150 \\
& (12,0) & 80\,160 & -31\,900. & 1\,500 \\[5pt]
4 & (0,8) & 4 & -1.\frac{1}{3} & 1 \\
& (2,7) & 58 & -29.\frac{5}{6} & 4 \\
& (4,6) & 480 & -248. & 32 \\
& (6,5) & 4\,000 & -2\,023.\frac{1}{3} & 310 \\
& (8,4) & 35\,104 & -17\,257.\frac{1}{3} & 3\,220 \\
& (10,3) & 327\,888 & -156\,594. & 34\,674 \\
& (12,2) & 3259\,680 & -1\,515\,824. & 385\,656 \\
& (14,1) & 34\,382\,544 & -15\,620\,216. & 4\,436\,268 \\
& (16,0) & 383\,306\,880 & -170\,763\,640. & 52\,832\,040 \\[5pt]
5 & (0,10) & 105 & -36.\frac{3}{4} & 42 \\
& (2,9) & 1\,265 & -594.\frac{3}{4} & 354 \\
& (4,8) & 13\,354 & -6\,523.\frac{1}{2} & 3\,492 \\
& (6,7) & 139\,098 & -66\,274.\frac{1}{2} & 38\,049 \\
& (8,6) & 1\,492\,616 & -677\,808. & 441\,654 \\
& (10,5) & 16\,744\,080 & -7\,179\,606. & 5\,378\,454 \\
& (12,4) & 197\,240\,400 & -79\,637\,976. & 68\,292\,324 \\
& (14,3) & 2\,440\,235\,712 & -928\,521\,900. & 901\,654\,884 \\
& (16,2) & 31\,658\,432\,256 & -11\,385\,660\,384. & 12\,358\,163\,808 \\
& (18,1) & 429\,750\,191\,232 & -146\,713\,008\,096. & 175\,599\,635\,328 \\
& (20,0) & 6\,089\,786\,376\,960 & -1\,984\,020\,394\,752. &
2\,583\,319\,387\,968 \\ \hline
\end{tabular}$$
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[^1]: Since this paper was written, Pandharipande [@Pandharipande] has found a direct geometric proof of the relation of Theorem \[relation\], showing that it is a linear equivalence, by means of an auxilliary moduli space of admissible covers of ${\mathbb{CP}}^1$.
[^2]: The resemblance of the right-hand side to the Hamiltonian of the Liouville model is striking — we have no idea why operators so closely resembling vertex operators make their appearance here.
|
---
abstract: 'We prove a necessary and sufficient condition for a symmetric association scheme to be a $Q$-polynomial scheme.'
author:
- 'Hirotake Kurihara, Hiroshi Nozaki'
title: 'A characterization of $Q$-polynomial association schemes'
---
**Key words**: $Q$-polynomial association scheme, $s$-distance set.
Introduction
============
A [*symmetric association scheme*]{} of class $d$ is a pair $\mathfrak{X} = (X, \{R_i \}_{i=0}^d)$, where $X$ is a finite set and each $R_i$ is a nonempty subset of $X \times X$ satisfying the following:
1. $R_0 = \{(x, x) \mid x \in X \}$,
2. $X \times X = \bigcup_{i=0}^d R_i$ and $R_i \cap R_j$ is empty if $i\ne j$,
3. $^tR_i = R_i$ for any $i \in\{ 0, 1,\ldots , d\}$, where $^t R_i = \{(y, x) \mid (x, y) \in R_i\}$,
4. for all $i,j,k\in \{0,1,\ldots ,d\}$, there exist integers $p_{i j}^k$ such that for all $x,y \in X$ with $(x,y) \in R_k$, $$p_{i j}^k= |\{ z \in X \mid (x,z) \in R_i, (z,y) \in R_j \}|.$$
The integers $p_{i j}^k$ are called the [*intersection numbers*]{}.
Let $\mathfrak{X}$ be a symmetric association scheme. The $i$-th [*adjacency matrix*]{} $A_i$ of $\mathfrak{X}$ is the matrix with rows and columns indexed by $X$ such that the $(x,y)$-entry is $1$ if $(x,y)\in R_i$ or $0$ otherwise. The [*Bose–Mesner algebra*]{} of $\mathfrak{X}$ is the algebra generated by the adjacency matrices $\{A_i \}_{i=0}^d$ over the complex field $\mathbb{C}$. Then $\{A_i\}_{i=0}^d$ is a natural basis of the Bose–Mesner algebra. By [@Bannai-Ito page 59], the Bose–Mesner algebra has a second basis $\{E_i\}_{i=0}^d$ such that
1. $E_0=|X|^{-1}J$, where $J$ is the all-ones matrix,
2. $I=\sum_{i=0}^d E_i$, where $I$ is the identity matrix,
3. $E_i E_j= \delta_{ij}E_i$, where $\delta_{ij}=1$ if $i =j$ and $\delta_{ij}=0$ if $i \ne j$.
The basis $\{E_i\}_{i=0}^d$ is called the [*primitive idempotents*]{} of $\mathfrak{X}$. We have the following equations: $$\begin{aligned}
A_i&= \sum_{j=0}^d p_i(j)E_j, \label{eq:basic} \\
E_i&=\frac{1}{|X|}\sum_{j=0}^d q_i(j) A_j, \\
A_i A_j&= \sum_{k=0}^d p_{i j}^k A_k, \\
E_i \circ E_j&= \frac{1}{|X|} \sum_{k=0}^d q_{i j}^k E_k,
\end{aligned}$$ where $\circ$ denotes the [*Hadamard product*]{}, that is, the entry-wise matrix product. The matrices $P=(p_j(i))^d_{i,j=0}$ and $Q=(q_j(i))^d_{i,j=0}$ are called the first and second [*eigenmatrices*]{}, respectively. The numbers $q_{i j}^k$ are called the [*Krein parameters*]{}. The Krein parameters are nonnegative real numbers (the Krein condition) [@Scott] [@Bannai-Ito page 69].
A symmetric association scheme is called a [*$P$-polynomial scheme*]{} (or a [*metric scheme*]{}) with respect to the ordering $\{A_i \}_{i=0}^d$ if for each $i \in \{0,1,\ldots, d \}$, there exists a polynomial $v_i$ of degree $i$ such that $p_i(j)=v_i(p_1(j))$ for any $j \in \{0,1,\ldots, d \}$. We say a symmetric association scheme is a $P$-polynomial scheme with respect to $A_1$ if it has the $P$-polynomial property with respect to some ordering $A_0,A_1, A_{i_2},A_{i_3},\ldots ,A_{i_d}$. Dually a symmetric association scheme is called a [*$Q$-polynomial scheme*]{} (or a [*cometric scheme*]{}) with respect to the ordering $\{E_i \}_{i=0}^d$ if for each $i \in \{0,1,\ldots, d \}$, there exists a polynomial $v_i^{\ast}$ of degree $i$ such that $q_i(j)=v_i^{\ast}(q_1(j))$ for any $j \in \{0,1,\ldots, d \}$. Moreover a symmetric association scheme is called a $Q$-polynomial scheme with respect to $E_1$ if it has the $Q$-polynomial property with respect to some ordering $E_0,E_1, E_{i_2},E_{i_3},\ldots ,E_{i_d}$. Note that both $\{v_i\}^d_{i=0}$ and $\{v_i^{\ast}\}^d_{i=0}$ form systems of orthogonal polynomials. Throughout this paper, we use the notation $m_i=q_i(0)$ and $\theta_i^{\ast}=q_1(i)$ for $0 \leq i \leq d$. If an association scheme is $Q$-polynomial, then $\{ \theta_i^{\ast} \}_{i=0}^d$ are mutually distinct because the second eigenmatrix $Q=(v_i^{\ast}(\theta_j^{\ast}))^d_{j,i=0}$ is non-singular. For a univariate polynomial $f$ and a matrix $M$, we denote by $f(M^\circ)$ the matrix obtained by substituting $M$ into $f$ with multiplication the Hadamard product. We introduce known equivalent conditions of the $Q$-polynomial property of symmetric association schemes [@Bannai-Ito page 193]. The following are equivalent:
1. $\mathfrak{X}$ is a $Q$-polynomial scheme with respect to the ordering $\{E_i \}_{i=0}^d$.
2. $(q_{1,i}^j)^d_{i,j=0}$ is an irreducible tridiagonal matrix.
3. For each $i \in \{0,1,\ldots,d \}$, there exists a polynomial $f_i$ of degree $i$ such that $E_i=f_i(E_1^{\circ})$.
In the present paper, we prove a new necessary and sufficient condition for a symmetric association scheme to be $Q$-polynomial. Since the $Q$-polynomial property of a symmetric association scheme of class $1$ is trivial, we assume that $d$ is greater than $1$.
\[main\] Let $\mathfrak{X}$ be a symmetric association scheme of class $d\ge 2$. Suppose that $\{\theta_j^{\ast}\}^d_{j=0}$ are mutually distinct. Then the following are equivalent:
1. $\mathfrak{X}$ is a $Q$-polynomial scheme with respect to $E_1$.
2. There exists $l \in \{2,3, \ldots ,d\}$ such that for any $i \in \{1,2,\ldots, d\}$, $$\prod_{\substack{j=1 \\ j \ne i}}^d \frac{\theta_0^\ast-\theta_j^\ast}{\theta_i^\ast-\theta_j^\ast}= -p_i(l).$$
Moreover if $(2)$ holds, then $l=i_d$.
We call a finite set $X$ in $\mathbb{R}^m$ a [*$d$-distance set*]{} if the number of the Euclidean distances between distinct two points in $X$ is equal to $d$. Larman–Rogers–Seidel [@Larman-Rogers-Seidel] proved that if the size of a two-distance set with the distances $a,b$ ($a<b$) is greater than $2m+3$, then there exists a positive integer $k$ such that $a^2/b^2=(k-1)/k$, [*i.e.*]{} $k=b^2/(b^2-a^2)$. Bannai–Bannai [@Bannai-Bannai] proved that the ratio $k$ of the spherical embedding of a primitive association scheme of class $2$ coincides with $-p_i(2)$. The research of the present paper is motivated by [@Bannai-Bannai]. For a symmetric association scheme satisfying that $\{\theta_j^{\ast}\}^d_{j=0}$ are mutually distinct, the values $K_i:=\prod_{j=1, j \ne i}^d (\theta_0^\ast-\theta_j^\ast)(\theta_i^\ast-\theta_j^\ast)^{-1}$ ($1\leq i \leq d$) are the generalized Larman–Rogers–Seidel’s ratios [@Nozaki] of the spherical embedding of this association scheme with respect to $E_1$. Theorem \[main\] is an extension of Bannai–Bannai’s result to $Q$-polynomial schemes of any class. Furthermore Theorem \[main\] is a new characterization of the $Q$-polynomial property on the spherical embedding of a symmetric association scheme.
At the end of this paper, we give some sufficient conditions for the integrality of $K_i$.
Proof of Theorem \[main\]
=========================
First we give several lemmas that will be needed to prove Theorem \[main\].
\[sum\_Ki\_2\] For any mutually distinct real numbers $\beta_1,\beta_2, \ldots, \beta_s$, the following identity holds. $$\sum_{i=1}^s \beta_i^j \prod_{\substack{k=1\\ k \ne i}}^s \frac{x-\beta_k}{\beta_i-\beta_k}=x^j$$ for any $j \in \{0,1, \ldots ,s-1\}$, where $x$ is a variable.
For each $j\in \{0,1,\ldots,s-1\}$, the polynomial $$L_j(x):=\sum^s_{i=1}\beta_i^j\prod_{\substack{k=1\\ k \ne i}}^s\frac{x-\beta_k}{\beta_i-\beta_k}$$ of degree at most $s-1$ is known as the interpolation polynomial in the Lagrange form (see [@lag]). Namely, the property $L_j(\beta_i)=\beta_i^j$ holds for any $i\in \{1,2,\ldots ,s\}$. Therefore $L_j(x)=x^j$, and the lemma follows.
We say $E_j$ is a [*component*]{} of an element $M$ of the Bose–Mesner algebra if $E_jM\neq0$. Let $N_h$ denote the set of indices $j$ such that $E_j$ is a component of $E_1^{\circ h}$ but not of $E_1^{\circ l}$ $(0 \leq l \leq h-1)$. Note that $N_0=\{0\}$ and $N_1=\{1\}$.
\[key\] Suppose $\mathfrak{X}$ is a symmetric association scheme of class $d \ge 2$. Then the following are equivalent.
1. $\mathfrak{X}$ is a $Q$-polynomial scheme with respect to $E_1$.
2. The cardinality of $N_d$ is equal to $1$.
3. $N_d$ is nonempty.
$(2)\Rightarrow (3)$: Clear.\
$(1) \Rightarrow (2)$: Without loss of generality, we assume that $\mathfrak{X}$ is a $Q$-polynomial scheme with respect to $\{E_i\}^d_{i=0}$. By noting that $\{ \theta_i^{\ast} \}_{i=0}^{d}$ are mutually distinct, $\{ E_1^{\circ i} \}_{i=0}^d$ are linearly independent, and a basis of the Bose–Mesner algebra. We have $$E_i=f_i(E_1^{\circ})= \sum_{j=0}^i \alpha_{i,j} E_1^{\circ j},$$ where $\alpha_{i,j}\in \mathbb{R}$ are the coefficients of a polynomial $f_i$ of degree $i$. The upper triangular matrix $(\alpha_{i,j})^d_{i,j=0}$ is non-singular because $\alpha_{i,i} \ne 0$ for each $i$. Since the inverse matrix $(\alpha'_{i,j})^d_{i,j=0}$ of $(\alpha_{i,j})^d_{i,j=0}$ is also an upper triangular matrix with $\alpha'_{i,i} \ne 0$ for each $i$, we can express $$E_1^{\circ i}= \sum_{j=0}^i \alpha'_{i,j} E_j.$$ Therefore $(2)$ follows.\
$(3) \Rightarrow (1)$: First we prove that if $N_{i}$ is empty for some $i\in\{1,2,\ldots,d-1\}$, then $N_{i+1}$ is also empty. Let $\mathcal{I}=\cup_{j=0}^{i-1} N_j$. We consider the expression $
\sum_{j =0}^{i-1} E_1^{\circ j} = \sum_{j \in \mathcal{I}} \beta_j E_j
$. Note that $\beta_j>0$ for any $j\in \mathcal{I}$ by the Krein condition. Then we have $$E_1 \circ (\sum_{h =0}^{i-1} E_1^{\circ h}) = \sum_{j \in \mathcal{I}} \beta_j \sum_{k=0}^d q_{1,j}^k E_k
= \sum_{k=0}^d \sum_{j \in \mathcal{I}} \beta_j q_{1,j}^k E_k.$$ If $N_{i}$ is empty, then $$\label{q}
q_{1,j}^k=0 \text{ for any $j\in \mathcal{I}$ and any $k \not\in \mathcal{I}$}$$ because $\beta_j>0$ holds for any $j\in \mathcal{I}$. We can express $E_1^{\circ i}= \sum_{j \in \mathcal{I}} \beta_j' E_j $, where $\beta_j'$ are non-negative integers for any $j\in \mathcal{I}$. By and the equalities $$E_1^{\circ (i+1)}=E_1 \circ E_1^{\circ i}= E_1 \circ \sum_{j \in \mathcal{I}} \beta_j' E_j= \sum_{k=0}^d \sum_{j \in \mathcal{I}} \beta_j' q_{1,j}^k E_k,$$ we obtain $\sum_{j \in \mathcal{I}} \beta_j' q_{1,j}^k=0$ for $k \not\in \mathcal{I}$. Hence $N_{i+1}$ is also empty. This means that if $N_d$ is not empty, then the cardinalities of $N_h$ is equal to $1$ for any $h \in \{0,1,\ldots,d \}$. Put $N_h=\{i_h\}$ and order $E_0,E_1,E_{i_2},E_{i_3},\ldots, E_{i_d}$. Then we can construct polynomials $f_h$ of degree $h$ such that $f_h(E_1^{\circ})=E_{i_h}$ for any $h \in \{0,1,\ldots, d\}$. Hence $(1)$ follows.
Now we prove Theorem \[main\].
\(1) $\Rightarrow$ (2): Without loss of generality, we assume that $\mathfrak{X}$ is a $Q$-polynomial scheme with respect to $\{E_i\}^d_{i=0}$. For each $i \in \{1,2,\ldots ,d \}$, we define the polynomial $$F_i(t):=\prod_{\substack{j=1\\ j \ne i}}^d \frac{|X|t- \theta^{\ast}_j }{\theta^{\ast}_i-\theta^{\ast}_j}$$ of degree $d-1$. Set $M_i=F_i(E_1^{\circ})$. Then $|X|E_1=\sum_{j=0}^d \theta_j^\ast A_j$ yields that the $(x,y)$-entries of $M_i$ are $$M_i(x,y)=\begin{cases}
K_i \text{\qquad if $(x,y) \in R_0$},\\
1 \text{\qquad if $(x,y) \in R_i$},\\
0 \text{\qquad otherwise},
\end{cases}$$ where $K_i:=\prod_{j=1, j \ne i}^d (\theta_0^\ast-\theta_j^\ast)(\theta_i^\ast-\theta_j^\ast)^{-1}$. Since $F_i$ is a polynomial of degree $d-1$, the matrix $M_i$ is a linear combination of $\{ E_i \}_{i=0}^{d-1}$. This means that $M_i E_d=0$. By , $$0=M_i E_d=(K_iI+A_i)E_d=(K_i+p_i(d))E_d$$ for any $i\in \{1,2,\ldots,d\}$. Therefore the desired result follows.
\(2) $\Rightarrow$ (1): From the equation $A_i= \sum_{j=0}^d p_i(j)E_j$ and our assumptions, we have $$A_iE_l= p_i(l)E_l=-K_iE_l.$$ By Lemma \[sum\_Ki\_2\], $$(|X|E_1)^{\circ j} E_l=\bigr( (\theta_0^\ast)^j I + \sum_{i=1}^d (\theta^{\ast}_i)^j A_i \bigr) E_l=
\bigr( (\theta_0^\ast)^j - \sum_{i=1}^d (\theta^{\ast}_i)^j K_i \bigr) E_l=0$$ for any $j \leq d-1$. This means that $l$ is not an element of $N_j$ for any $j \leq d-1$. Note that the following equality holds: $$\prod^{d}_{j=1}\frac{|X|E_1-\theta^\ast_j J}{\theta_0^\ast-\theta^\ast_j}=I,$$ where the multiplication is the Hadamard product. Obviously, $I$ has $E_l$ as a component. Since $l \notin N_i$ for any $i\in \{0,1,\ldots ,d-1\}$, we have $l \in N_d$. By Lemma \[key\], the desired result follows.
Integrality of $K_i$
====================
In this section, we consider when $K_i=-p_i(d)$ is an integer for each $i \in \{1,2,\ldots,d \}$ for a $Q$-polynomial scheme. The following theorem is important in this section.
\[Suzuki\] Let $\mathfrak{X}$ with $m_1>2$ be a $Q$-polynomial scheme with respect to the ordering $\{E_i\}_{i=0}^d$. Suppose $\mathfrak{X}$ is $Q$-polynomial with respect to another ordering. Then the new ordering is one of the following:
1. $E_0,E_2,E_4,E_6, \ldots , E_5,E_3,E_1$,
2. $E_0,E_d,E_1,E_{d-1},E_2,E_{d-2},E_3,E_{d-3}, \ldots $,
3. $E_0,E_d,E_2,E_{d-2},E_4,E_{d-4}, \ldots , E_{d-5},E_5,E_{d-3},E_3,E_{d-1},E_1$,
4. $E_0,E_{d-1},E_2,E_{d-3},E_4,E_{d-5}, \ldots , E_5,E_{d-4},E_3,E_{d-2},E_1,E_d$, or
5. $d = 5$ and $E_0,E_5,E_3,E_2,E_4,E_1$.
Note that $Q$-polynomial schemes with $m_1=2$ are the ordinary $n$-gons as distance-regular graphs.
Let $\mathfrak{X}$ with $m_1>2$ be a $Q$-polynomial association scheme with respect to the ordering $\{ E_i \}_{i=0}^d$. If there exists $t$ such that $t\leq d/2$, $t \equiv 1 \pmod 2$ and $m_t\ne m_{d-t+1}$, then $K_j$ is an integer for any $j$.
Let $\mathbb{F}$ be the splitting field of the scheme, generated by the entries of the first eigenmatrix $P$. Then $\mathbb{F}$ is a Galois extension of the rational field. Let $G$ be the Galois group $\operatorname{Gal}(\mathbb{F}/\mathbb{Q})$. We consider the action of $G$ on the primitive idempotents $E_i$, where elements of $G$ are applied entry-wise. Then the action of $G$ on $\{E_i\}_{i=0}^d$ is faithful and $|G| \leq 2$ [@Martin-Williford].
Suppose $K_j$ is not an integer for some $j$. Since $-K_j=p_j(d)$ is an eigenvalue of $A_j$, $K_j$ is an algebraic integer. By the basic number theory, $K_j$ is irrational. Therefore $|G|\ne 1$ and hence $|G|=2$. Let $\sigma$ be the non-identity element of $G$. From the definition of $K_j$, $E_1$ must have an irrational entry, and $E_1^{\sigma} \ne E_1$. Therefore $\{E_i^{\sigma} \}_{i=0}^d$ is another $Q$-polynomial ordering with the same polynomials $f_i$. Hence $\{E_i^{\sigma} \}_{i=0}^d$ coincides with one of (1)–(5) in Theorem \[Suzuki\].
For $d=2$, it is known that $K_i$ is an integer for each $i=1,2$ if $m_1 \ne m_2$ [@Bannai-Bannai]. For (1) and (2) with $d>2$, $(E_1^{\sigma})^{\sigma} \ne E_1$, this contradicts that $\sigma^2$ is the identity. Since $p_j(d)$ is irrational and $A_j E_d=p_j(d) E_d$, $E_d$ has an irrational entry. Therefore $E_d^{\sigma} \ne E_d$. For (4), $\sigma$ fixes $E_d$, a contradiction. Therefore the ordering $\{E_i^{\sigma}\}_{i=0}^{d}$ coincides with (3) or (5).
Suppose that there exists $t$ such that $t\leq d/2$, $t \equiv 1 \pmod 2$ and $m_t\ne m_{d-t+1}$. Since $E_t\circ I= (m_t/|X|) I$, we have $E_t^{\sigma} \circ I^{\sigma} = (m_t/|X|) I^{\sigma}$ and hence $E_t^{\sigma} \circ I = (m_t/|X|) I \ne (m_{d-t+1}/|X|) I$. Therefore $E_t^{\sigma} \ne E_{d-t+1}$. Thus, the ordering $\{E_i^{\sigma} \}_{i=0}^d$ does not coincide with (3) for $d \geq 2$. If $d=5$, then $m_1 \ne m_5$ and hence $E_1^{\sigma} \ne E_{5}$. Therefore $\{E_i^{\sigma} \}_{i=0}^5$ does not coincide with (5). Thus the proposition follows.
Remark that the known $Q$-polynomial schemes with some irrational $K_i$ and $d>2$ are the ordinary $n$-gons and the association scheme obtained from the icosahedron [@Kiyota-Suzuki; @Martin-Muzychuk-Williford]. We can give a similar equivalent condition of the $P$-polynomial property of symmetric association schemes [@Kurihara-Nozaki]. Let $\theta_i=p_1(i)$ for $0 \leq i \leq d$.
Let $\mathfrak{X}$ be a symmetric association scheme of class $d\ge 2$. Suppose $\{\theta_j\}_{j=0}^d$ are mutually distinct. Then the following are equivalent:
1. $\mathfrak{X}$ is a $P$-polynomial association scheme with respect to $A_1$.
2. There exists $l \in \{2,3,\ldots,d\}$ such that for any $i \in \{1,2,\ldots d\}$, $$\prod_{\substack{j=1\\ j \ne i}}^d \frac{\theta_0-\theta_j}{\theta_i-\theta_j}= -q_i(l).$$
Moreover if $(2)$ holds, then $l=i_d$.
**Acknowledgments.** Both of authors are supported by the fellowship of the Japan Society for the Promotion of Science. The authors would like to thank Eiichi Bannai, Edwin van Dam, Tatsuro Ito, William J. Martin, Akihiro Munemasa, Hiroshi Suzuki, Makoto Tagami, Hajime Tanaka, Paul Terwilliger and Paul-Hermann Zieschang for useful discussions and comments.
[9]{} E. Bannai and E. Bannai, A note on the spherical embeddings of strongly regular graphs, [*European J. Combin.*]{} 26 (2005), no. 8, 1177–1179.
E. Bannai and T. Ito, [*Algebraic Combinatorics I: Association Schemes*]{}, Benjamin/Cummings, Menro Park, CA, 1984.
J.-P. Berrut and L.N. Trefethen, Barycentric Lagrange Interpolation, [*SIAM Review*]{} 46 (3) (2004), 501–517.
P. Delsarte, J.M. Goethals, and J.J. Seidel, Spherical codes and designs, [*Geom. Dedicata*]{} 6 (1977), no. 3, 363–388.
M. Kiyota and H. Suzuki, Character products and $Q$-polynomial group association schemes, [*J. Algebra*]{} 226 (2000), no. 1, 533–546.
H. Kurihara and H. Nozaki, A spectral equivalent condition of the $P$-polynomial property for association schemes, in preparation.
D.G. Larman, C.A. Rogers, and J.J. Seidel, On two-distance sets in Euclidean space, *Bull. London Math. Soc.* 9 (1977), 261–267.
W.J. Martin, M. Muzychuk and J. Williford, Imprimitive cometric association schemes: constructions and analysis, [*J. Algebraic Combin.*]{} 25 (2007), no. 4, 399–415.
W.J. Martin and J.S. Williford, There are finitely many $Q$-polynomial association schemes with given first multiplicity at least three, [*European J. Combin.*]{} 30 (3) (2009) 698–704,
H. Nozaki, A generalization of Larman–Rogers–Seidel’s theorem, [*Discrete Math.*]{} 311 (2011), 792–799.
L.L. Scott, A conditions on Higman’s parameters, [*Amer. Math. Soc. Notices*]{} 701 (1973), 20–45.
H. Suzuki, Association schemes with multiple $Q$-polynomial structures, [*J. Algebraic Combin.*]{} 7 (2) (1998), 181–196
[*Hirotake Kurihara*]{}\
Mathematical Institute,\
Tohoku University\
Aramaki-Aza-Aoba 6-3,\
Aoba-ku,\
Sendai 980-8578,\
Japan\
[email protected]\
\
[*Hiroshi Nozaki*]{}\
Graduate School of Information Sciences,\
Tohoku University\
Aramaki-Aza-Aoba 6-3-09,\
Aoba-ku,\
Sendai 980-8579,\
Japan\
[email protected]\
|
---
abstract: |
This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.
We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We handle several cases of phase spaces, including those associated to specular reflection in a bounded domain, or to a compact Riemannian manifold.
address:
- 'CNRS and École Polytechnique, Centre de Mathématiques Laurent Schwartz UMR7640, 91128 Palaiseau cedex France'
- 'UniversitŽ Paris Diderot, Institut de MathŽmatiques de Jussieu UMR7586, Paris Rive Gauche B‰timent Sophie Germain, 75205 Paris Cedex 13 France'
author:
- 'Daniel Han-Kwan'
- Matthieu Léautaud
bibliography:
- 'HKL4.bib'
title: |
Geometric analysis of the linear Boltzmann equation I.\
Trend to equilibrium
---
Introduction
============
This paper is concerned with the study of the linear Boltzmann equation $$\label{B}
\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv',$$ for $x \in \Omega, \, v \in {{\mathbb R}}^d$, $d \in {{\mathbb N}}^*$, where $\Omega$ is either the flat torus ${{\mathbb T}}^d := {{\mathbb R}}^d/{{\mathbb Z}}^d$ or an open bounded subset of ${{\mathbb R}}^d$, in which case we add some appropriate boundary conditions to the equation. The linear Boltzmann equation is a classical model of statistical physics, allowing to describe the interaction between particles and a fixed background [@Cer-book; @DL1english; @DL]. Among many possible applications, we mention the modeling of semi-conductors, cometary flows, or neutron transport. We refer the reader interested by further physical considerations or by a discussion of the validity of in these contexts to [@Cer-book Chapter IV, 3] or [@DL1english Chapter I, 5]. We also point out that this equation can be derived in various settings: see for instance [@EY] in the context of quantum scattering, or [@BGSR] in the context of a gas of interacting particles.
In , the unknown function $f = f(t,x,v)$ is the so-called distribution function; the quantity $f(t,x,v) \, dv dx$ can be understood as the (non-negative) density at time $t$ of particles whose position is close to $x$ and velocity close to $v$. The function $V$ is a potential which drives the dynamics of particles; we shall assume throughout this work that $V$ is smooth, more precisely that $ V \in W^{2,\infty}(\Omega)$.
The linear Boltzmann equation is a typical example of a so-called hypocoercive equation, in the sense of Villani [@V-Hypo]. It is made of a conservative part, namely the kinetic transport operator $v \cdot \nabla_x - \nabla_x V \cdot \nabla_v$ associated to the hamiltonian $H(x,v)= \frac{1}{2} |v|^2 + V(x)$, and a degenerate dissipative part which is the collision operator (i.e. the right hand-side of ). According to the hypocoercivity mechanism of [@V-Hypo], only the interaction between the two parts can lead to convergence to some global equilibrium.
The function $k$ is the so-called collision kernel, which describes the interaction between the particles and the background. In the following, we shall denote by $C(x,v,f)$ the collision operator, which can be split as $$C(x,v,f) = C^+(x,v,f) + C^-(x,v,f),$$ where $$C^+(x,v,f)= \int_{{{\mathbb R}}^d}k(x,v' , v) f(v') \, dv' ,\quad C^-(x,v,f)=- \left( \int_{{{\mathbb R}}^d} k(x,v , v') \, dv' \right) f(v)$$ are respectively the *gain* and the *loss* term. A first property of this operator is that, due to symmetry reasons, the formal identity holds: $$\label{ConvMass}
\text{for all } x \in \Omega, \quad \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv' \, dv = 0.$$ This, together with the fact that the vector field $v \cdot \nabla_x - \nabla_x V \cdot \nabla_v$ is divergence free, implies that the mass is conserved: any solution $f$ of satisfies $$\text{ for all } t \geq 0, \quad \frac{d}{dt} \int_{\Omega \times {{\mathbb R}}^d} f(t,x,v) \, dv dx =0.$$
We shall now list the assumptions we make on the collision kernel $k$.
[**A1.**]{} The collision kernel $k$ belong to the class $C^0(\overline{\Omega} \times {{\mathbb R}}^d \times {{\mathbb R}}^d)$ and is nonnegative.
[**A2.**]{} Introducing the Maxwellian distribution: $${\mathcal{M}}(v) := \frac{1}{(2\pi)^{d/2} }e^{-\frac{|v|^2}{2}},$$ we assume that ${\mathcal{M}}$ cancels the collision operator, that is $$\label{Mannule}
\text{for all } (x,v) \in \Omega \times {{\mathbb R}}^d, \quad \int_{{{\mathbb R}}^d} \left[k(x,v' , v) {\mathcal{M}}(v') - k(x,v , v') {\mathcal{M}}(v)\right] \, dv' = 0.$$
[**A3.**]{} We assume that $$x \mapsto \int_{{{\mathbb R}}^d \times {{\mathbb R}}^d} k^2(x,v',v) \frac{{\mathcal{M}}(v')}{{\mathcal{M}}(v)} \, dv' dv \in L^\infty(\Omega).$$ It will sometimes be convenient to work with the function $$\label{bornek}
\tilde{k}(x,v',v) := \frac{k(x,v' , v) }{{\mathcal{M}}(v)} .$$ With this notation, Assumptions [**A2**]{} and [**A3**]{} may be rephrased in a more symmetric way as $$\text{for all }(x,v) \in \Omega \times {{\mathbb R}}^d ,\quad
\int_{{{\mathbb R}}^d}\tilde{k}(x,v',v) {\mathcal{M}}(v')\, dv' = \int_{{{\mathbb R}}^d}\tilde{k}(x,v,v') {\mathcal{M}}(v')\, dv' ;$$ $$x \mapsto \int_{{{\mathbb R}}^d \times {{\mathbb R}}^d} \tilde{k}^2(x,v',v) {\mathcal{M}}(v') {\mathcal{M}}(v)\, dv' dv \in L^\infty(\Omega).$$ Assumption [**A3**]{} is in particular satisfied if $\tilde{k}$ is bounded or has a polynomial growth in the variables $v$ and $v'$.
Note that with assumption [**A2**]{}, the function $(x,v)\mapsto {\mathcal{M}}(v)e^{-V}=\frac{1}{(2\pi)^{d/2} }e^{-H}$, which we shall call the *Maxwellian equilibrium*, cancels both the transport operator and the collision operator and thus is a stationary solution of .
Before going further, let us present usual classes of examples of collision kernels covered by Assumptions [**A1**]{}–[**A3**]{} and addressed in the present article.
[**E1. “Symmetric” collision kernels.**]{} Let $k$ be a collision kernel verifying [**A1**]{} and [**A3**]{}. We moreover require $\tilde{k}$ to be symmetric with respect to $v$ and $v'$, i.e. $\tilde{k}(x,v,v') = \tilde{k}(x,v',v)$ for all $(x,v,v') \in \Omega \times {{\mathbb R}}^d \times {{\mathbb R}}^d$. Notice that for these kernels, [**A2**]{} is automatically satisfied.
A classical example of such a kernel is the following.
[**E1’. Linear relaxation kernel.**]{} Taking ${k}(x,v,v')=\sigma(x){\mathcal{M}}(v')$, with $\sigma \geq 0, \, \sigma \neq 0$ and $\sigma \in C^0(\overline{\Omega})$ provides the simplest example of kernel in the class [**E1**]{}. This corresponds to the following equation (often called linearized BGK): $$\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= \sigma(x) \left( \left(\int_{{{\mathbb R}}^d} f \, dv\right) {\mathcal{M}}(v) - f \right) .$$ This example also belongs to the following class.
[**E2. “Factorized” collision kernels**]{} Let $k$ be a collision kernel verifying [**A1**]{}–[**A3**]{}. We require $k$ to be of the form $$k(x,v,v')= \sigma(x) {k}^*(x,v,v'),$$ with $\sigma \in C^0(\overline{\Omega})$, $\sigma \geq 0, \, \sigma \neq 0$ and $k^* \in C^0(\overline{\Omega}\times {{\mathbb R}}^d \times {{\mathbb R}}^d)$, satisfying for some $\lambda>0$, for all $x \in \Omega$, $v,v' \in {{\mathbb R}}^d$, $$\frac{{k}^*(x,v',v) }{{\mathcal{M}}(v)}+ \frac{{k}^*(x,v,v') }{{\mathcal{M}}(v')} \geq \lambda.$$ The sub-class of [**E2**]{} which is the most studied in the literature (see e.g. [@DMS]) consists in the following non-degenerate case.
[**E2’. Non-degenerate collision kernels.**]{} Let $k$ be a collision kernel verifying [**A1**]{}–[**A3**]{}. The classical *non-degeneracy* condition consists in assuming that there exists $\lambda>0$ such that for all $x \in \Omega$, $v,v' \in {{\mathbb R}}^d$ $$\frac{{k}(x,v',v) }{{\mathcal{M}}(v)}+ \frac{{k}(x,v,v') }{{\mathcal{M}}(v')}= \tilde{k}(x,v',v) + \tilde{k}(x,v,v') \geq \lambda .$$
Later in the paper (see Section \[secexamples\]), we will introduce other classes of collision kernels, that are interesting for our purposes.
Under assumptions [**A1**]{}-[**A3**]{}, the linear Boltzmann equation is well-posed in appropriate Lebesgue spaces, and the weighted $L^2$ norm of its solutions, that is $\int_{\Omega \times {{\mathbb R}}^d}|f(t,x,v)|^2 \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx $, is dissipated (i.e. decreasing with respect to time, see Lemma \[lemdissip\]).
This work aims at describing the large time behavior of solutions of , under assumptions [**A1**]{}–[**A3**]{}. The main feature is that we thus allow the collision operator $k$ to be degenerate in the following two senses:
$\bullet$ the collision kernel $k$ may vanish in a large subset of the phase space $\Omega \times {{\mathbb R}}^d$;
$\bullet$ we do not assume that $\tilde{k}$ is bounded below by a fixed positive constant at infinity in velocity.
However, still in the spirit of Villani’s hypocercivity, one may hope that the transport term in compensates for this strong degeneracy. Our goal is to find geometric criteria (on the hamiltonian $H$ and the collision kernel $k$) to characterize:
$\bullet$ [**P1**]{} convergence to a global equilibrium,
$\bullet$ [**P1’**]{} exponential convergence to this global equilibrium.
The study of these questions naturally leads to another problem:
$\bullet$ [**P2**]{} describe the structure and the localization properties of the spectrum of the underlying linear Boltzmann operator.
In recent works [@DS; @BS1; @BS2; @BS3], Bernard, Desvillettes and Salvarani investigated [**P1**]{} and [**P1’**]{} in a framework close to that of [**E2**]{}. In particular, in [@BS1], the authors have shown that in the case where $V=0$, $(x, v) \in {{\mathbb T}}^d\times {{\mathbb S}}^{d-1}$, and $k^*(x,v,v') = k^*(v,v')$ (where $k^*$ is defined in [**E2**]{}), the exponential convergence to equilibrium (in the Lebesgue space $L^1$) was equivalent to a *geometric control condition* (similar to that of Bardos-Lebeau-Rauch-Taylor in control theory [@RT:74; @BLR:92]).
Previous works on this topic, for the non-degenerate class of collision kernels [**E2’**]{} include [@Vid; @Vid70; @U74; @UPG; @MK] (spectral approach), [@DVcpam; @V-Hypo; @CCG; @MN; @DMS] (hypocoercivity methods), [@Her] (Lie techniques), and references therein. There are also several related works which concern the *non-linear* Boltzmann equation, but we do not mention them since that equation is not studied in this paper.
In this article, we introduce another point of view on these questions (in particular different from [@DS; @BS1; @BS2; @BS3]), by implementing in this context different methods coming from control theory. We borrow several ideas from the seminal paper of Lebeau [@Leb], which concerns the decay rates for the damped wave equation.
The goal of this paper is to give necessary and sufficient *geometric* conditions ensuring [**P1**]{} and [**P1’**]{}, in several settings: we mostly focus on the torus case $\Omega = {{\mathbb T}}^d$ and on the case of specular reflection in bounded domains. We also show that the methods we develop here are sufficiently robust to handle a general Riemannian setting. The related question [**P2**]{} is studied in the companion paper [@HKL2].
We now give a more detailed overview of the main results of this work.
Overview of the paper {#main}
=====================
In this Section, we give an overview of the results contained in this paper. For readability, we focus on the torus case, i.e. when the phase space is ${{\mathbb T}}^d \times {{\mathbb R}}^d$. Several generalizations (bounded domains with specular reflection, Riemannian manifolds) are actually provided in the following.
Some definitions
----------------
In this section, we introduce the notions needed to characterize convergence and exponential convergence to equilibrium. Given a collision kernel $k$ satisfying Assumptions [**A1**]{}–[**A3**]{}, we first introduce the set $\omega$ where the collisions are effective.
\[def-om\] Define the open set of ${{\mathbb T}}^d \times {{\mathbb R}}^d$ $$\label{omega}
\omega : = \left\{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d, \, \int_{{{\mathbb R}}^d} k(x,v,v') \, dv' >0 \right\}.$$
Note that because of [**A1**]{}–[**A2**]{}, we also have $$\label{omegabis}
\omega = \left\{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d, \, \exists v' \in {{\mathbb R}}^d, k(x,v,v') >0 \right\}= \left\{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d, \, \exists v' \in {{\mathbb R}}^d, k(x,v',v) >0 \right\}.$$
Let us recall the definition of the hamiltonian flow associated to $H$, and associated characteristic curves (or characteristics) in the present setting.
\[def-carac\] The hamiltonian flow $(\phi_t)_{t\in{{\mathbb R}}}$ associated to $H(x,v) = \frac{|v|^2}{2} + V(x)$ is the one parameter family of diffeomorphisms on ${{\mathbb T}}^d \times {{\mathbb R}}^d$ defined by $\phi_t(x,v) := (X_t (x,v), \, \Xi_t(x,v))$ with $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$ and $$\label{hamilflow}
\left\{
\begin{aligned}
&\frac{dX_t(x,v)}{dt} = \Xi_t(x,v), \\
&\frac{d\Xi_t(x,v)}{dt} = - \nabla_x V(X_t(x,v)), \\
&X_{t=0}=x, \quad \Xi_{t=0}=v.
\end{aligned}
\right.$$ The characteristic curve stemming from $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$ is the curve $\{\phi_t(x,v), t \in {{\mathbb R}}^+\}$.
Recall that throughout the paper, we assume that $V \in W^{2,\infty} ({{\mathbb T}}^d)$, so that the Cauchy-Lipshitz theorem ensures the local existence and uniqueness of the solutions of . Global existence follows from the fact that $H$ is preserved along any characteristic curve. [Note in particular that each energy level $\{H = R\}$ is compact ($V$ being continuous on ${{\mathbb T}}^d$, it is bounded from below). Hence, each characteristic curve is contained in a compact set of ${{\mathbb T}}^d \times {{\mathbb R}}^d$.]{}
The notions needed to understand the interaction between collisions and transport are of two different nature. We start by expressing purely geometric definitions. Then, we formulate structural-geometric definitions. We finally introduce the weighted Lebesgue spaces used in this paper, as well as a definition of a “unique continuation type” property.
### Geometric definitions {#defgeo}
We start by introducing the following definitions:
- The Geometric Control Condition of [@BLR:92; @RT:74], in Definition \[def: GCC\],
- The Lebeau constants of [@Leb], $C^-(\infty)$ and $C^+(\infty)$, in Definition \[definitionCinfini\],
- The almost everywhere in infinite time Geometric Control Condition, in Definition \[defaeitgcc\].
Let us first recall the Geometric Control Condition, which is a classical notion in the context of control theory. It is due to Rauch-Taylor [@RT:74] and Bardos-Lebeau-Rauch [@BLR:92].
\[def: GCC\] Let $U$ be an open subset of ${{\mathbb T}}^d \times {{\mathbb R}}^d$ and $T>0$. We say that $(U,T)$ satisfies the Geometric Control Condition (GCC) with respect to the hamiltonian $H(x,v) = \frac{|v|^2}{2} + V(x)$ if for any $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$, there exists $t \in [0,T]$ such that $\phi_t(x,v) = (X_t(x,v),\Xi_t(x,v)) \in U$.
We shall say that $U$ satisfies the Geometric Control Condition with respect to the hamiltonian $H(x,v) = \frac{|v|^2}{2} + V(x)$ if there exists $T>0$ such that the couple $(U,T)$ does.
We now define two important constants in view of the study of the large time behavior of the linear Boltzmann equation, which involve averages of the *damping function* (usually called *collision frequency* in kinetic theory) $b(x,v):= \int_{{{\mathbb R}}^d} k(x,v,v') \, dv'$ along the flow $\phi_t$.
\[definitionCinfini\] Define the Lebeau constants ([@Leb]) in ${{\mathbb R}}^+ \cup \{+\infty\}$ by $$\begin{aligned}
C^-(\infty) &:= \sup_{T\in {{\mathbb R}}^+} C^-(T), \qquad C^-(T) = \inf_{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d} \frac{1}{T} \int_0^T \left( \int_{{{\mathbb R}}^d} k(\phi_t (x,v), v')\, dv'\right)\, dt, \\
C^+(\infty) &:= \inf_{T\in {{\mathbb R}}^+} C^+(T), \qquad C^+(T) = \sup_{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d} \frac{1}{T} \int_0^T \left( \int_{{{\mathbb R}}^d} k(\phi_t (x,v), v')\, dv'\right)\, dt,
\end{aligned}$$ where $\phi_t$ denotes the hamiltonian flow of Definition \[def-carac\].
It is not clear at first sight that $C^-(\infty)$ and $C^+(\infty)$ are well defined: see [@Leb] and the beginning of Section \[expocon\] for a short explanation. It turns out that only $C^-(\infty)$ will be useful in this paper (but $C^+(\infty)$ will be interesting in the companion paper [@HKL2]).
Finally, we introduce a weaker version of the Geometric Control Condition, which will also play an important role in this work.
\[defaeitgcc\] Let $U$ be an open subset of ${{\mathbb T}}^d \times {{\mathbb R}}^d$. We say that $U$ satisfies the almost everywhere infinite time (a.e.i.t.) Geometric Control Condition with respect to the hamiltonian $H(x,v) = \frac{|v|^2}{2} + V(x)$ if for almost any $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$, there exists $s\geq 0$ such that the characteristics $(X_t(x,v), \, \Xi_t(x,v))_{t \geq 0}$ associated to $H$ satisfy $(X_{t=s},\Xi_{t=s}) \in U$.
Using this terminology, the usual Geometric Control Condition of Definition \[def: GCC\] could be called “everywhere finite time” GCC.
\[rem-reecriture\] We have the following characterization of the different geometric properties introduced here.
- The couple $(U, T)$ satisfies GCC if and only if $\bigcup_{s \in (0,T)}\phi_{-s}(U)= {{\mathbb T}}^d \times {{\mathbb R}}^d$.
- The set $U$ satisfies the a.e.i.t. GCC if and only if there exists ${\mathcal{N}}\subset {{\mathbb T}}^d \times {{\mathbb R}}^d$ with zero Lebesgue measure such that $\bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(U)\cup {\mathcal{N}}= {{\mathbb T}}^d \times {{\mathbb R}}^d$. Note that this implies in particular that ${\mathcal{N}}$ is a closed subset of ${{\mathbb T}}^d \times {{\mathbb R}}^d$ satisfying $\phi_s({\mathcal{N}}) \subset {\mathcal{N}}$ for all $s\geq0$.
### Structural-geometric definitions {#defstrugeo}
In the sequel, the above geometric definitions are used with $U = \omega$. They hence involve joint properties of the flow $\phi_t$ together with the open set $\omega$, i.e. of the damping function $b =\int_{{{\mathbb R}}^d} k( \cdot , \cdot, v')\, dv'$. As such, they do not take into account the fine structure of the Boltzmann operator, and in particular the non-local property with respect to the velocity variable of the gain operator $f \mapsto \Big( (x,v) \mapsto \int_{{{\mathbb R}}^d} k( x , v', v) f(x,v')\, dv' \Big)$.
The next definitions aim at describing how the information may travel between the different connected component of $\omega$. Let us first define two basic binary relations on the open sets of ${{\mathbb T}}^d \times {{\mathbb R}}^d$.
Let $U_1$ and $U_2$ be two open subsets of ${{\mathbb T}}^d \times {{\mathbb R}}^d$. We say that $U_1 {\, \mathcal{R} }_{\phi} \, U_2$ if there exist $s \in {{\mathbb R}}$ such that $\phi_s(U_1) \cap U_2 \neq \emptyset$.
Let $U_1$ and $U_2$ be two open subsets of ${{\mathbb T}}^d \times {{\mathbb R}}^d$. We say that $U_1 {\, \mathcal{R} }_k \, U_2$ if there exist $(x,v_1,v_2) \in {{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d$ with $(x,v_1) \in U_1$, $(x,v_2) \in U_2$ such that $k(x,v_1,v_2)>0$ or $k(x,v_2,v_1)>0$.
Both relations are symmetric and ${\, \mathcal{R} }_{\phi}$ is moreover reflexive. When restricted to open sets intersecting $\omega$, the relation ${\, \mathcal{R} }_k$ also becomes reflexive. The relation ${\, \mathcal{R} }_{\phi}$ expresses the fact that the open sets are “connected through” the flow $\phi_s$, whereas the relation ${\, \mathcal{R} }_{k}$ means that the open sets are “connected through” a collision.
We also define another convenient $x$-dependent binary relation.
Let $x \in {{\mathbb T}}^d$ and $O_1$ and $O_2$ be two open subsets of ${{\mathbb R}}^d$. We say that $O_1 {\, \mathcal{R} }_k^x \, O_2$ if there exists $(v_1,v_2) \in {{\mathbb R}}^d \times {{\mathbb R}}^d$ with $v_1 \in O_1$, $v_2\in O_2$ such that $k(x,v_1,v_2)>0$ or $k(x,v_2,v_1)>0$.
Given now an open subset $U$ of ${{\mathbb T}}^d \times {{\mathbb R}}^d$, we define $U(x) = \{v \in {{\mathbb R}}^d, (x,v) \in U\}$. With this notation, notice that $U_1 {\, \mathcal{R} }_k \, U_2$ if and only if there exists $x \in {{\mathbb T}}^d$ such that $U_1(x) {\, \mathcal{R} }_k^x \, U_2(x)$.
Given $U$ an open set of ${{\mathbb T}}^d \times {{\mathbb R}}^d$, we denote by ${\mathcal{C}\mathcal{C}}(U)$ the set of connected components of $U$. Note that from the separability of ${{\mathbb T}}^d \times {{\mathbb R}}^d$, it follows that for any open set $U \subset {{\mathbb T}}^d \times {{\mathbb R}}^d$, the cardinality of the set ${\mathcal{C}\mathcal{C}}(U)$ is at most countable.
In the sequel, the main open sets $U$ we are interested in are $\omega$ and $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$. We now define the key equivalence relation on ${\mathcal{C}\mathcal{C}}(\omega)$.
\[def-sim-o\] Given $\omega_1$ and $\omega_2$ two connected components of $\omega$, we say that $\omega_1 \Bumpeq \omega_2$ if there is $N \in {{\mathbb N}}$ and $N$ connected components $(\omega^{(i)})_{1\leq i \leq N}$ of $\omega$ such that
- we have $\omega_1 {\, \mathcal{R} }_{\phi} \, \omega^{(1)}$ or $\omega_1 {\, \mathcal{R} }_k \, \omega^{(1)}$,
- for all $1\leq i \leq N-1$, we have $\omega^{(i)} {\, \mathcal{R} }_{\phi} \, \omega^{(i+1)}$ or $\omega^{(i)} {\, \mathcal{R} }_k \, \omega^{(i+1)}$,
- we have $\omega^{(N)} {\, \mathcal{R} }_{\phi} \, \omega_2$ or $\omega^{(N)} {\, \mathcal{R} }_k \, \omega_2$.
The relation $\Bumpeq$ is an equivalence relation on the set ${\mathcal{C}\mathcal{C}}(\omega)$ of connected components of $\omega$. For $\omega_1\in {\mathcal{C}\mathcal{C}}(\omega)$, we denote its equivalence class for $\Bumpeq$ by $[\omega_1]$.
This definition means that the two connected components $\omega_1$ and $\omega_2$ are linked by ${\, \mathcal{R} }_{\phi}$ or ${\, \mathcal{R} }_k$ through a chain of connected components of $\omega$.
We will introduce later in Section \[autre-rel-equiv\] a related equivalence relation on ${\mathcal{C}\mathcal{C}}\left( \bigcup_{t \geq 0} \phi_{-t} \omega\right)$.
### Weighted Lebesgue spaces and a unique continuation type property
Let us now introduce the weighted Lebesgue spaces that will be used throughout this paper.
\[weightedLp\] We define the Banach spaces ${\mathcal{L}}^p({{\mathbb T}}^d \times {{\mathbb R}}^d)$ (for $p \in [1,+\infty)$) and ${\mathcal{L}}^\infty({{\mathbb T}}^d \times {{\mathbb R}}^d)$ by $$\begin{aligned}
&{\mathcal{L}}^p({{\mathbb T}}^d \times {{\mathbb R}}^d) := \Big\{f \in L^1_{loc}({{\mathbb T}}^d \times {{\mathbb R}}^d), \, \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f|^p \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx < + \infty \Big\}, \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \|f\|_{{\mathcal{L}}^p} = \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f|^p \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx \right)^{1/p}.
\\
& {\mathcal{L}}^\infty({{\mathbb T}}^d \times {{\mathbb R}}^d) : = \Big\{f \in L^1_{loc}({{\mathbb T}}^d \times {{\mathbb R}}^d), \, \sup_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f| \frac{e^V}{{\mathcal{M}}(v)} < + \infty \Big\},
\quad \|f\|_{{\mathcal{L}}^\infty} = \sup_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f| \frac{e^V}{{\mathcal{M}}(v)} .
\end{aligned}$$ The space ${\mathcal{L}}^2$ is a (real) Hilbert space endowed with the inner product $$\langle f, g \rangle_{{\mathcal{L}}^2} := \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} e^{V} \frac{f \, g}{{\mathcal{M}}(v)} \, dv \, dx.$$
We finally define a Unique Continuation Property for .
\[def:UCP\] We say that the set $\omega$ satisfies the Unique Continuation Property if the only solution $f \in C^0_t({\mathcal{L}}^2)$ to $$\label{eq:UCP}
\left\{
\begin{array}{l}
\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
C(f)=0,
\end{array}
\right.$$ is $f= \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx\right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}$.
It is actually possible to reformulate in a more explicit form the second equation in , involving the value of $f$ on connected components of $\omega$ (see Remark \[def:UCP-expli\]).
Convergence to equilibrium
--------------------------
Recall that the main goal of this paper is to provide necessary and sufficient geometric conditions to ensure [**P1**]{} and [**P1’**]{}. In the case of the torus, these results can be formulated as follows.
We first give a general characterization of convergence to some equilibrium.
\[thmconvgene-intro\] The following statements are equivalent.
1. The set $\omega$ satisfies the a.e.i.t. GCC with respect to $H$.
2. For all $f_0 \in {\mathcal{L}}^2$, there exists a stationary solution $Pf_0$ of such that $$\| f(t)- Pf_0 \|_{{\mathcal{L}}^2} \to_{t \to + \infty} 0,$$ where $f(t)$ is the solution of with initial datum $f_0$.
Theorem \[thmconvgene-intro\] is actually a weak version of Theorem \[thmconv-general\], which is our main result in this direction. If $(1)$ or $(2)$ holds, we can actually describe precisely the stationary solution $Pf_0$. This involves the equivalence classes of another equivalence relation, which is related to $\Bumpeq$ (see Definition \[def-sim\] and Lemma \[equiv-equiv\]): we refer to the statement of Theorem \[thmconv-general\]. In particular, in several cases, the stationary solution $Pf_0$ is not the Maxwellian equilibrium $$\label{glomax}
\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx\right) \frac{e^{-V(x)}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v) .$$ As a matter of fact, we will see that the dimension of the vector space of stationary solutions is equal to the number of equivalence classes for $\Bumpeq$. An explicit example of collision kernel for which this is relevant is given in Section \[secE3’\].
Among all possible stationary solutions of the linear Boltzmann equation, the Maxwellian equilibrium of course particularly stands out. In the next theorem (which is actually a particular case of Theorem \[thmconv-general\]), we characterize the situation for which the stationary solution ultimately reached is precisely the projection to the Maxwellian.
\[thmconv-intro\] The following statements are equivalent.
1. The set $\omega$ satisfies the Unique Continuation Property.
2. The set $\omega$ satisfies the a.e.i.t. GCC and there exists one and only one equivalence class for $\Bumpeq$.
3. For all $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\label{convergeto0}
\left\|f(t)- \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \right)\frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}\right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0.$$
As already mentioned in the introduction, our proofs are inspired by ideas which originate from control theory. For the sake of brevity, we shall not give a detailed explanation of the proof of these results in this introduction. Nevertheless, we would like to comment on an important aspect of the proof of (i.) implies (iii.) in Theorem \[thmconv-intro\]. Our approach is based on the fact that the square of the ${\mathcal{L}}^2$ norm of a solution $f(t)$ of , which we shall sometimes refer to as the *energy*, is damped via an explicit dissipation identity, see Lemma \[lemdissip\]: $$\label{dissip-intro}
\text{ for all } t \geq 0, \quad \frac{d}{dt} \|f(t)\|_{{\mathcal{L}}^2}^2 = -D(f),$$ with $D(f) \geq0$, which we shall call the *dissipation* term.
The idea of the proof is to assume by contradiction that there exists an initial condition $g_0$ in ${\mathcal{L}}^2$, with zero mean, such that the associated solution $g(t)$ to does not decay to $0$. This yields the existence of ${{\varepsilon}}>0$ and of a sequence of times $(t_n)_{n \geq 0}$ going to $+\infty$ such that $\| g(t_n)\|_{{\mathcal{L}}^2} \geq {{\varepsilon}}$.
We then study the sequence of shifted functions $h_n(t):= g(t+t_n)$. This is the core of our analysis, which basically consists in a *uniqueness-compactness* argument. We study the weak limit of $h_n$ and show, using the identity and the *unique continuation property*, that it is necessarily trivial. Then, we consider the associated sequence of *defect measures* and prove that it is also necessarily trivial, yielding a contradiction.
A difficulty in the analysis comes from the fact that in general, the dissipation term does not control neither the ${\mathcal{L}}^2$ distance to the projection on the set of stationary solutions, nor the ${\mathcal{L}}^2$ norm of the collision operator. However, what holds true is the *weak coercivity* property $$\label{wc-intro}
\text{for } f \in {\mathcal{L}}^2, \quad D(f)=0 \implies C(f)=0,$$ see Lemma \[collannule\]. This turns out to be sufficient for our needs. Denoting by $$\label{boltzop}
A := T + C ,$$ the linear Boltzmann operator, where $T f = (v\cdot {{\nabla}}_x - {{\nabla}}_x V\cdot {{\nabla}}_v) f$ and $C$ is the collision operator, the property together with the skew-adjointness of $T$ then implies that $\operatorname{Ker}(A) = \operatorname{Ker}(T) \cap \operatorname{Ker}(C)$. This precise structure, together with the equivalence relation $\Bumpeq$, allows to identify $\operatorname{Ker}(A)$, i.e. the space of stationary solutions of .
Besides, when studying defect measures, the analysis relies on another peculiar structure of , which is, loosely speaking, made of a propagative and dissipative part (transport and the loss term) and a *relatively compact* part (the gain term). That the gain term is relatively compact is proved via *averaging lemmas* (see Appendix \[section-AL\] and the references therein).
Exponential convergence to equilibrium
--------------------------------------
For what concerns exponential convergence, we need to introduce a technical assumption, which is slightly stronger than [**A3**]{}:
[**A3’.**]{} Assume that there exists a continuous function $\varphi(x,v):= \Theta \circ H(x,v)$ with $\Theta : {{\mathbb R}}\to [ 1, +\infty)$, such that for all $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$, we have $$\int_{{{\mathbb R}}^d} k(x,v,vÕ) \, dvÕ \leq \varphi(x,v) ,$$ and $$\sup_{x \in {{\mathbb T}}^d} \int_{{{\mathbb R}}^d \times {{\mathbb R}}^d} k^2(x,vÕ,v) \frac{{\mathcal{M}}(vÕ)}{{\mathcal{M}}(v)} \left(\frac{\varphi(x,v)}{\varphi(x,v')}-1\right)^2 \, dv dvÕ <+\infty .$$
This assumption is for instance satisfied in the standard case where $\tilde{k}$ has a polynomial growth in the variables $v$ and $v'$ (taking for example $\Theta(t)= \lambda \exp(\frac{1}{4} t)$ and $\lambda>0$ large enough).
We have the following criterion, assuming that [**A3’**]{} is satisfied in addition to [**A1–A3**]{}.
\[thmexpo-intro\] Assume that the collision kernel satisfies [**A3’**]{}. The following statements are equivalent:
1. $C^-(\infty) > 0$.
2. There exist $C>0, \gamma>0$ such that for any $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, the unique solution to with initial datum $f_0$ satisfies for all $t \geq 0$, $$\begin{gathered}
\label{decexpo-intro}
\left\|f(t)-\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}\right\|_{{\mathcal{L}}^2} \\
\leq C e^{-\gamma t} \left\|f_0-\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx\right)\frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}\right\|_{{\mathcal{L}}^2}.\end{gathered}$$
3. There exists $C>0, \gamma>0$ such that for any $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, there exists a stationary solution $Pf_0$ of such that the unique solution to with initial datum $f_0$ satisfies for all $t \geq 0$, $$\label{decexpogene-intro}
\left\|f(t)-Pf_0\right\|_{{\mathcal{L}}^2}
\leq C e^{-\gamma t} \left\|f_0-Pf_0\right\|_{{\mathcal{L}}^2}.$$
If we do not assume that [**A3’**]{} is satisfied, then we still have that (a.) implies (b.) and (c.).
If $C^-(\infty)>0$, note in particular that the Geometric Control Condition of Definition \[def: GCC\] is satisfied.
One interesting consequence of Theorem \[thmexpo-intro\] is a rigidity property of the Maxwellian equilibrium with respect to exponential convergence: loosely speaking, given a linear Boltzmann equation with a collision kernel satisfying [**A1-A2-A3-A3’**]{}, if (c.) holds, then the stationary solution ultimately reached is necessarily the projection of the initial datum on the Maxwellian equilibrium.
As an immediate consequence of Theorem \[thmexpo-intro\], we deduce the following result.
\[coroexpo-gen\] Assume that there is $x \in {{\mathbb T}}^d$ such that ${{\nabla}}V(x)=0$ and $\int k(x,0,v') \, dv' =0$. Then $C^-(\infty)=0$ and there is no uniform exponential rate of convergence to equilibrium.
In particular we get in the free transport case:
\[coroexpo-libre\] Assume that $V=0$ and that $p_x(\omega) \neq {{\mathbb T}}^d$, where $$\label{defpx}
p_x(\omega) = \{x \in {{\mathbb T}}^d , \text{ there exists }v \in {{\mathbb R}}^d \text{ such that }(x,v) \in \omega\}$$ denotes the projection on the space of positions. Then there is no uniform exponential rate of convergence to equilibrium.
The proof of Theorem \[thmexpo-intro\] is as well inspired by ideas coming from control theory. The proof of $(a.)\implies (b.)$ relies on the following facts.
- By , the exponential decay (i.e. $(ii.)$ in Theorem \[thmexpo-intro\]) can be rephrased as a certain *observability* inequality relating the dissipation and the energy at time $0$, see Lemma \[lemfondamental\]: there exist $K,T>0$ such that for all $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d\times {{\mathbb R}}^d)$ with zero mean, $$K\int_0^T D(f) \, dt \geq \| f_0 \|_{{\mathcal{L}}^2}^2,$$ where $f$ is the solution of with initial datum $f_0$.
- This inequality is proved using a contradiction argument, following Lebeau [@Leb], which also consists in a uniqueness-compactness argument. The analysis follows the same lines as those of $(i.)$ implies $(iii.)$ in the proof of Theorem \[thmconv-intro\]. In particular, it also relies on the weak coercivity property . The main difference is that we need to use here the fact that the Lebeau constant is positive in order to show that the sequence of defect measures becomes trivial at the limit, yielding a contradiction.
For what concerns $(b.)\implies (a.)$, the idea is to contradict the observability condition: we construct a sequence of initial conditions in ${\mathcal{L}}^2({{\mathbb T}}^d\times {{\mathbb R}}^d)$ for , which concentrate to a *trapped* ray (whose existence is guaranteed by the cancellation of the Lebeau constant). Loosely speaking, this corresponds to a geometric optics type construction. In order to justify this procedure, we need that the collision kernel satisfies [**A3’**]{}. Finally, we mention that the proof of $(c.)\implies (a.)$ is similar but relies on an additional argument based on the precise version of Theorem \[thmconvgene-intro\].
1. As for the damped wave equation [@RT:74; @BLR:92; @Leb; @KoTa], the study of asymptotic decay rates relies on “phase space” analysis. However, as opposed to the wave equation, the Boltzmann equation is directly set on the phase space. As a consequence, the study of associated propagation and damping phenomena only uses “local” analysis, whereas that of the wave equation (see [@RT:74; @BLR:92; @Leb; @KoTa]) requires the use of microlocal analysis.
2. One technical difficulty here is to handle the lack of compactness of the phase space ${{\mathbb T}}^d \times {{\mathbb R}}^d$ in the variable $v$. It is also possible to consider the equations on a compact phase space. In this case, all our proofs apply, sometimes with significant simplifications. We refer to Section \[compactPS\].
Organization of the paper
-------------------------
[Part \[LTB\]]{} is mainly dedicated to the proof of Theorems \[thmconvgene-intro\], \[thmconv-intro\] and \[thmexpo-intro\]. In Section \[preliminaries\], we give some preliminaries in the analysis; in Section \[sec:wp\], we start by proving the dissipation identity and recalling the classical well-posedness result for , while Section \[sec:wc\] is dedicated to a detailed study of the kernel of the collision operator, which leads to the weak coercivity property . Section \[subsectiondecroissance\] is mainly devoted to the proofs of Theorems \[thmconv-intro\] and \[thmconvgene-intro\]; in Section \[sec:conv\], we start by proving Theorem \[thmconv-intro\]. Then, in Section \[sec:conv2\], we state and prove Theorem \[thmconv-general\], which is the precise version of Theorem \[thmconvgene-intro\]. Section \[secexamples\] is dedicated to the application of these results to some particular classes of collision kernels. In Section \[expocon\], we prove Theorem \[thmexpo-intro\]. Finally in Section \[lowerbounds\], we briefly revisit the recent work of Bernard and Salvarani [@BS2] in our framework, in order to give some abstract lower bounds on the convergence rate when $C^-(\infty)=0$.
Part \[Boundary\] is dedicated to the case of specular reflection in a bounded and piecewise $C^1$ domain of ${{\mathbb R}}^d$. In Section \[prelimbound\], we state preliminary definitions and results in this setting (including the geometric definitions and well-posedness). Then in Section \[convboun\], we state and sketch the proof of the exact analogues of Theorems \[thmconv-intro\] and \[thmconv-general\], see Theorems \[thm-convboun\] and \[thmconv-general-bord\]. In Section \[expoboun\], we study exponential convergence to equilibrium. For a more restrictive class of collision kernels (namely [**E2**]{} with an additional $L^\infty$ bound) and under a technical regularity assumption on $p_x(\omega)$ near $ \partial \Omega$ (here $p_x$ denotes the projection on $\Omega$ defined in ; the technical assumption is automatically satisfied when $\overline{p_x({\omega})} \subset \Omega$), we prove in Theorem \[thmexpo-specular\] the analogue of Theorem \[thmexpo-intro\] $(a.)\implies (b.)$. This technical assumption, as well as the fact that we do not prove the analogue of Theorem \[thmexpo-intro\], $(b.)\implies (a.)$, is due to the lack of compactness up to the boundary of $\Omega$ in averaging lemmas. We are nevertheless able to overcome this difficulty for proving $(a.)\implies (b.)$, by adapting an argument of Guo [@Guo]. The fact that the collision kernel belongs to the class [**E2**]{} allows us to obtain a control of the distance of a solution to its projection on Maxwellians, see Lemma \[cerci2\]. This is a quantitative coercivity estimate which is much stronger than the weak coercivity property we use in torus case. Finally, in Section \[otherboun\], we give some remarks on some other possible boundary conditions.
In Part \[part3\], we adapt our analysis in order to handle other geometric situations. In Section \[sec:manifold\], we deal with the case of a general compact Riemannian manifold (without boundary): we first explain how to express the linear Boltzmann equation in this setting and generalize Theorems \[thmconv-general\] and \[thmexpo-intro\] to this context. We provide to this end a version of averaging lemmas for kinetic transport equations on a manifold, see Lemma \[lem-moyenne-variete\]. Finally, in Section \[compactPS\], we explain very shortly how to adapt all these results to the case of compact phase spaces.
This paper ends with five appendices. In Appendix \[stabob\], we give the equivalence between exponential decay and the observability inequality, used crucially in the proof of Theorem \[thmexpo-intro\]. In Appendix \[section-AL\], we give a reminder about classical averaging Lemmas and adapt them to our purposes. In Appendix \[GCCother\], we provide reformulations of some geometric properties. In Appendix \[proofCEXucp\], we provide the proof of Proposition \[CEXucp\], which relates the two equivalence relations, which are key notions in our analysis. In Appendix \[Other\], to stress the robustness of our methods, we explain how the results of Part \[LTB\] of this paper concerning large time behavior can be adapted to other Boltzmann-like equations (e.g. relativistic Boltzmann equation or general linearized BGK equation).
Remarks and Examples
====================
In this section, we provide several comments on the different geometric definitions introduced in Sections \[defgeo\] and \[defstrugeo\].
About a.e.i.t. GCC in the torus
-------------------------------
A first natural question is to understand the a.e.i.t. GCC in the usual situation of free transport (i.e. $V=0$), when $\omega$, i.e. the set where collisions are effective, is of the simple form $\omega_x \times {{\mathbb R}}^d$. We prove that this condition is satisfied for [*any*]{} nonempty $\omega_x \subset {{\mathbb T}}^d$. We also prove that this situation is very particular, and unstable with respect to small perturbations of the potential.
\[coroUCP\] Suppose that $V = 0$ and that $ \omega = \omega_x \times {{\mathbb R}}^d$, where $\omega_x$ is a non-empty open subset of ${{\mathbb T}}^d$. Then $(i.)-(ii.)-(iii.)$ in Theorem \[thmconv-intro\] hold.
Such a result is in particular relevant for the study of the linearized BGK equation (class [**E1’**]{}). Proposition \[coroUCP\] shows that there is convergence to the Maxwellian equilibrium as soon as $\sigma \neq 0$.
On the other hand, $\omega$ being fixed, we give an example of dynamics (i.e. exhibit a potential $V$) for which the a.e.i.t. GCC fails. More precisely, we prove that this property is very unstable with respect to small perturbations of the potential: for $\omega = \omega_x \times {{\mathbb R}}^d \neq {{\mathbb T}}^d \times {{\mathbb R}}^d$ there exist arbitrary small potentials (in any $C^k$-norm) such that $\omega$ does not satisfy a.e.i.t. GCC for the associated Hamiltonian. This illustrates the fact that the free transport on the torus is a very uncommon situation.
\[CEXucp\] Assume that $\overline{p_x(\omega)} \neq {{\mathbb T}}^d$, where $p_x(\omega)$ denotes the projection of $\omega$ on ${{\mathbb T}}^d$ defined in . Then there exists a potential $V \in C^\infty({{\mathbb T}}^d)$ such that for any ${{\varepsilon}}>0$, $\omega$ does not satisfy a.e.i.t. GCC for the Hamiltonian $H_{{\varepsilon}}(x,v) = \frac{|v|^2}{2}+ {{\varepsilon}}V(x)$.
The proof of Proposition \[coroUCP\] and Proposition \[CEXucp\] are given respectively in Section \[secE3pp\] and Appendix \[proofCEXucp\].
The equivalence relation on ${\mathcal{C}\mathcal{C}}\left(\bigcup_{t \geq 0} \phi_{-t} (\omega)\right)$ {#autre-rel-equiv}
--------------------------------------------------------------------------------------------------------
We define here another key equivalence relation $\sim$ on the set of connected components of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$. We then explain the link between the two equivalence relations $\sim$ on ${\mathcal{C}\mathcal{C}}\left(\bigcup_{t \geq 0} \phi_{-t} (\omega)\right)$ and $\Bumpeq$ on ${\mathcal{C}\mathcal{C}}(\omega)$.
\[def-sim\] Given $\Omega_1, \Omega_2$ two connected components of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$, we say that $\Omega_1 \sim \Omega_2$ if there is $N \in {{\mathbb N}}$ and $N$ connected components $(\Omega^{i})_{1\leq i \leq N}$ of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ such that
- we have $\Omega_1 {\, \mathcal{R} }_{\phi} \, \Omega^{(1)}$ or $\Omega_1 {\, \mathcal{R} }_k \, \Omega^{(1)}$,
- for all $1\leq i \leq N-1$, we have $\Omega^{(i)} {\, \mathcal{R} }_{\phi} \, \Omega^{(i+1)}$ or $\Omega^{(i)} {\, \mathcal{R} }_k \, \Omega^{(i+1)}$,
- we have $\Omega^{(N)} {\, \mathcal{R} }_{\phi} \, \Omega_2$ or $\Omega^{(N)} {\, \mathcal{R} }_k \, \Omega_2$.
The relation $\sim$ is an equivalence relation on the set of ${\mathcal{C}\mathcal{C}}\left(\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega) \right)$ of connected components of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$. For $\Omega_1 \in {\mathcal{C}\mathcal{C}}\left(\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega) \right)$, we denote its equivalence class for $\sim$ by $[\Omega_1]$.
Observe that the two equivalence relations $\Bumpeq$ and $\sim$ are defined the same way, except that $\Bumpeq$ is considered as a relation on ${\mathcal{C}\mathcal{C}}(\omega)$ and $\sim$ on ${\mathcal{C}\mathcal{C}}(\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega))$. Note also that if the phase space is without boundary, in the definition of $\sim$ above, one can omit that $\Omega^{(i)} {\, \mathcal{R} }_{\phi} \, \Omega^{(i+1)}$ and require only $$\Omega_1 {\, \mathcal{R} }_k \, \Omega^{(1)}, \quad \text{for all } 1\leq i \leq N-1, \, \Omega^{(i)} {\, \mathcal{R} }_k \, \Omega^{(i+1)},\quad \text{and } \Omega^{(N)} {\, \mathcal{R} }_k \, \Omega_2 .$$ However, this precise Definition \[def-sim\] will be useful when considering the case of a boundary value problem for which the associated flow $\phi_t$ is no longer continuous in time.
The following lemma gives the link between the two equivalence relations. We define the function $$\begin{array}{rrcl}
\Psi : & {\mathcal{C}\mathcal{C}}(\omega) & \to &{\mathcal{C}\mathcal{C}}\Big(\bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)\Big) \\
& \omega_0 & \mapsto & \Omega_0 , \text{ such that } \omega_0 \subset \Omega_0.
\end{array}$$ The application $\Psi$ maps $\omega_0 \in {\mathcal{C}\mathcal{C}}(\omega)$ to the connected component $\Omega_0$ of $\bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)$ containing $\omega_0$.
\[equiv-equiv\] Given $\omega_1, \omega_2 \in {\mathcal{C}\mathcal{C}}(\omega)$, we have $\omega_1 \Bumpeq \omega_2$ if and only if $\Psi(\omega_1) \sim \Psi(\omega_2)$. As a consequence, $\Psi$ goes to the quotient defining a [*bijection*]{} $\tilde\Psi$ between the equivalence classes of $\Bumpeq$ and $\sim$: $$\begin{array}{rrcl}
\tilde\Psi : & {\mathcal{C}\mathcal{C}}(\omega) / \Bumpeq & \to &{\mathcal{C}\mathcal{C}}\Big(\bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)\Big)/\sim.
\end{array}$$ In particular, the number of equivalence classes for $\Bumpeq$ in ${\mathcal{C}\mathcal{C}}(\omega)$ and for $\sim$ in ${\mathcal{C}\mathcal{C}}\Big(\bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)\Big)$ are equal.
The proof of Lemma \[equiv-equiv\] is given in Appendix \[prooflemequiv\]. As a consequence of this lemma, all the results of this paper (together with their proofs) can be formulated with $\Bumpeq$ or with $\sim$ equivalently.
Comparing $C^-(\infty)>0$ and GCC
---------------------------------
The statement that $C^-(\infty)>0$ is in general stronger to the fact that $\omega$ satisfies the Geometric Control Condition with respect to the hamiltonian $H(x,v) = \frac{|v|^2}{2} + V(x)$. This is due to the the non-compactness of the phase space ${{\mathbb T}}^d \times {{\mathbb R}}^d$.
Assume for instance that $V = 0$ so that $\phi_t(x,v) = ( x+tv ,v )$: the characteristic curves are straight lines and the velocity component $v$ is preserved by the flow. Take $k(x,v',v)>0$ on the whole ${{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d$. In this situation, $\omega ={{\mathbb T}}^d \times {{\mathbb R}}^d$ and so, it satisfies automatically GCC in any positive time. Assume further that $k$ does not depend on the space variable, i.e. $k(x,v',v) = k(v',v)$ and that there exists a sequence $(v_n)$ such that $\int_{{{\mathbb R}}^d} k(v_n,v') dv' \to 0$. Then, we have $\left(\int_{{{\mathbb R}}^d} k( \cdot ,v') dv' \right) \circ \phi_t (x,v) = \int_{{{\mathbb R}}^d} k( v ,v') dv'$ (as the flow preserves $v$) and hence, for any $n \in {{\mathbb N}}$, we have $C^-(\infty) \leq \int_{{{\mathbb R}}^d} k(v_n,v') dv'$. This yields $C^-(\infty)= 0$ although $\omega$ satisfies GCC. As an explicit example, one can take $k(x, v', v ) = {\mathcal{M}}(v') {\mathcal{M}}(v)^2$.
Note finally that if $\int_{{{\mathbb R}}^d} k( x,v ,v') dv'$ is uniformly bounded from below at infinity (i.e. there exists $C,R>0$ such that $\int_{{{\mathbb R}}^d} k( x,v ,v') dv' \geq C$ for all $(x,v) \in {{\mathbb T}}^d \times B(0,R)^c$), then $C^-(\infty)>0$ and GCC become equivalent.
The next paragraph shows that our result is indeed more general.
Example of exponential convergence without a bound from below at infinity
-------------------------------------------------------------------------
Here, we produce a simple example of dynamics and collision kernel such that $C^-(\infty)>0$, but neither $\tilde{k}$ nor $\int_{{{\mathbb R}}^d} k( x,v ,v') dv'$ are uniformly bounded from below at infinity.
For this, assume $(x,v) \in {{\mathbb T}}\times {{\mathbb R}}$ and take $V=0$, so that $\phi_t(x,v) = ( x+tv ,v )$. We identify ${{\mathbb T}}$ to $[-1/2,1/2)$ with periodic boundary conditions. Define $\alpha \in C^0({{\mathbb T}}; {{\mathbb R}}^+)$ with support contained in $(-1/3,1/3)$ and satisfying $\alpha = 1$ on $[-1/4,1/4]$ and $\psi \in C^0({{\mathbb R}}; {{\mathbb R}}^+)$ such that $\psi(v) \to_{|v| \to +\infty} 0$ and $\psi = 1$ on $[-2,2]$. Consider the collision kernel in the class [**E1**]{} $$k(x,v,v') = \tilde{k}(x,v,v'){\mathcal{M}}(v') , \qquad \tilde{k}(x,v,v') = \left[ \alpha(x) + \psi(v) \psi(v')\right].$$ We first readily check that $\tilde{k}>0$ on ${{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d$ and hence $\omega={{\mathbb T}}\times {{\mathbb R}}$. We also remark that for any $R>0$, $$\inf_{(x,v,v') \in {{\mathbb T}}^d \times B(0,R)^c \times B(0,R)^c} \tilde{k}(x,v,v') =0, \quad \text{and} \quad \inf_{(x,v) \in {{\mathbb T}}^d \times B(0,R)^c} \int_{{{\mathbb R}}^d} k( x,v ,v') dv' = 0 .$$ Nevertheless, we can prove that $C^-(\infty) \geq C^-(1)>0$, and thus, by Theorem \[thmexpo-intro\], there is exponential convergence to the Maxwellian equilibrium. We set $\beta := \int_{{{\mathbb R}}^d} \psi(v') {\mathcal{M}}(v') \, dv'>0$ and take $(x,v) \in {{\mathbb T}}\times {{\mathbb R}}$.
- If $v\in [-2,2]$, then we have $$\int_0^1 \int_{{{\mathbb R}}^d} k(\phi_t(x,v),v') \, dv' \, dt \geq \beta \int_0^1 \psi(v) \,dt = \beta>0.$$
- If $v\notin [-2,2]$, then, denoting by $\lfloor v \rfloor$ the integer part of $v$, we have $$\begin{aligned}
\int_0^1 \int_{{{\mathbb R}}^d} k(\phi_t(x,v),v') \, dv' \, dt &\geq \int_0^1 \alpha(x+tv) \, dt
\geq \int_0^{\frac{\lfloor v \rfloor}{|v|}} \alpha(x+tv) \, dt
\geq \frac{1}{2} \frac{\lfloor v \rfloor}{|v|} \geq \frac{1}{4}.\end{aligned}$$
This proves that $C^-(1)>0$ and thus $C^-(\infty)>0$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We wish to thank Diogo Arsénio for several interesting and stimulating discussions related to this work.
\[LTB\]
Throughout this section, $\Omega = {{\mathbb T}}^d$.
Preliminary results {#preliminaries}
===================
Well-posedness and dissipation {#sec:wp}
------------------------------
For readability, we shall sometimes denote $$C(f):=C(x,v,f) = \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv'.$$
The following dissipation identity holds for solutions to .
Let $k$ be collision kernel satisfying [**A1**]{}–[**A3**]{}. \[lemdissip\]Let $f \in C^0({{\mathbb R}};{\mathcal{L}}^2)$ be a solution to . The following identity holds, for all $t\in {{\mathbb R}}$: $$\label{eqdissipation}
\frac{d}{dt} \|f(t)\|_{{\mathcal{L}}^2}^2 = - D(f),$$ where $$\begin{aligned}
D(f) &= - 2\langle C(f), f \rangle_{{\mathcal{L}}^2} \nonumber \\
&= \frac{1}{2} \int_{{{\mathbb T}}^d} e^{V} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \left( \frac{k(x,v' , v)}{{\mathcal{M}}(v)} + \frac{k(x,v , v')}{{\mathcal{M}}(v')} \right) {\mathcal{M}}(v) {\mathcal{M}}(v') \left(\frac{f(t,x,v)}{{\mathcal{M}}(v)}- \frac{f(t,x,v')}{{\mathcal{M}}(v')}\right)^2 \, dv' \, dv \, dx . \label{defD}
\end{aligned}$$
The term $D(f)$ will often be referred to as the dissipation term in the following. The proof is rather classical and follows [@DGP].
Multiply by $f \, \frac{e^V}{{\mathcal{M}}(v)}$ and integrate with respect to $x$ and $v$. This yields $$\begin{aligned}
\frac{1}{2} \frac{d}{dt} \|f(t)\|_{{\mathcal{L}}^2}^2 + \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \left(v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f\right) f \, \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx
= \int_{{{\mathbb T}}^d} e^V \int_{{{\mathbb R}}^d} C(f) \frac{f}{{\mathcal{M}}} \, dv \, dx.\end{aligned}$$ On the one hand, the contribution of the transport term vanishes $$\begin{aligned}
\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \left(v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f\right) f \, \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx
& = \frac{1}{2}\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \left(v \cdot \nabla_x - \nabla_x V \cdot \nabla_v \right) |f|^2 \, \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx \\
&= - \frac{1}{2}\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} |f|^2 \left(v \cdot \nabla_x - \nabla_x V \cdot \nabla_v \right) \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx \\
& =0,
\end{aligned}$$ since $\left(v \cdot \nabla_x - \nabla_x V \cdot \nabla_v \right) \frac{e^V}{{\mathcal{M}}(v)} =0$. On the other hand, following [@DGP], we have for any $x \in {{\mathbb T}}^d$ the identity $$\begin{aligned}
\label{eq: DGP1}
\int_{{{\mathbb R}}^d} C(x,v,f) \frac{f}{{\mathcal{M}}} \, dv &= \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv' \frac{f(v)}{{\mathcal{M}}(v)} \, dv \nonumber \\
&= \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v' , v) f(v')\frac{f(v)}{{\mathcal{M}}(v)} \, dv' \, dv - \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v') \frac{|f(v)|^2}{{\mathcal{M}}(v)} \, dv' \, dv .\end{aligned}$$ Symmetrizing the first term in the right hand-side of yields $$\begin{gathered}
\int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v' , v) f(v')\frac{f(v)}{{\mathcal{M}}(v)} \, dv' \, dv =\\
\frac{1}{2} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v' , v) f(v')\frac{f(v)}{{\mathcal{M}}(v)} \, dv' \, dv
+\frac{1}{2} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v') f(v)\frac{f(v')}{{\mathcal{M}}(v')} \, dv' \, dv .\end{gathered}$$ Concerning the second term in the right hand-side of , we use to obtain $$\begin{aligned}
& - \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v') \frac{|f(v)|^2}{{\mathcal{M}}(v)} \, dv' \, dv \\
&=-\frac{1}{2} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v') \frac{|f(v)|^2}{{\mathcal{M}}(v)} \, dv' \, dv -\frac{1}{2} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v'){\mathcal{M}}(v) \frac{|f(v)|^2}{{\mathcal{M}}(v)^2} \, dv' \, dv \\
&=-\frac{1}{2} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v') \frac{|f(v)|^2}{{\mathcal{M}}(v)} \, dv' \, dv -\frac{1}{2} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v', v){\mathcal{M}}(v') \frac{|f(v)|^2}{{\mathcal{M}}(v)^2} \, dv' \, dv \\
&=- \frac{1}{4} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v , v') \frac{|f(v)|^2}{{\mathcal{M}}(v)} \, dv' \, dv - \frac{1}{4} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v', v) \frac{|f(v')|^2}{{\mathcal{M}}(v')} \, dv' \, dv\\
& \quad - \frac{1}{4} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v' , v){\mathcal{M}}(v') \frac{|f(v)|^2}{{\mathcal{M}}(v)^2} \, dv' \, dv - \frac{1}{4} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} k(x,v, v'){\mathcal{M}}(v) \frac{|f(v')|^2}{{\mathcal{M}}(v')^2} \, dv' \, dv .\end{aligned}$$ Combining the last two identities, we can now collect together the terms with $k(x,v',v)$ (resp. $k(x,v, v')$) and rewrite the right hand-side of as a sum of two squares. Namely, this provides $$\begin{aligned}
\int_{{{\mathbb R}}^d} C(x,v,f) \frac{f}{{\mathcal{M}}} \, dv = - \frac{1}{4} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \left( \frac{k(x,v' , v)}{{\mathcal{M}}(v)} + \frac{k(x,v , v')}{{\mathcal{M}}(v')} \right) {\mathcal{M}}(v) {\mathcal{M}}(v') \left(\frac{f(v)}{{\mathcal{M}}(v)}- \frac{f(v')}{{\mathcal{M}}(v')}\right)^2 \, dv' \, dv .\end{aligned}$$ This yields and concludes the proof of the Lemma.
We have the following standard well-posedness result for the linear Boltzmann equation .
\[prop:WP\] Assume that $f_0 \in {\mathcal{L}}^2$. Then there exists a unique $f\in C^0({{\mathbb R}};{\mathcal{L}}^2)$ solution of satisfying $f|_{t = 0} =f_0$, and we have $$\text{for all } t \geq 0, \quad \frac{d}{dt} \| f(t)\|_{{\mathcal{L}}^2}^2 = - D(f(t)),$$ where $D(f)$ is defined in . If moreover $f_0 \geq 0$ a.e., then for all $t \in {{\mathbb R}}$ we have $f(t, \cdot,\cdot)\geq 0$ a.e. (Maximum principle).
We denote $$\begin{aligned}
(A_0 f)(x, v) &= (v\cdot {{\nabla}}_x - {{\nabla}}_x V\cdot {{\nabla}}_v) f(x,v) +
\left(\int_{{{\mathbb R}}^d} k(x,v,v') \, dv'\right) f(x,v), \\
(K f)(x, v) &= - \int_{ {{\mathbb R}}^d} k(x,v',v) f(x,v') \, dv', \\
Af &= A_0 f + Kf\end{aligned}$$ with domain $$D(A) = D(A_0) = \left\{f\in {\mathcal{L}}^2 , (v\cdot {{\nabla}}_x - {{\nabla}}_x V\cdot {{\nabla}}_v) f \in {\mathcal{L}}^2, \, \left(\int_{{{\mathbb R}}^d} k(\cdot,v') \, dv'\right) f \in {\mathcal{L}}^2 \right\}.$$ The operator $A_0$ generates a strongly continuous group on ${\mathcal{L}}^2$ given by $$\label{semigpA0}
e^{-tA_0} u = \exp\left(- \int_0^t \int_{{{\mathbb R}}^d} k(\phi_{-(t-s)}(x,v),v') \, dv' \, ds \right) u \circ \phi_{-t},$$ where $\phi_s(x,v) = (X_s(x,v),\Xi_s(x,v))$ denotes the hamiltonian flow of Definition \[def-carac\].
On the other hand, the Cauchy-Schwarz inequality yields $$\begin{aligned}
\|Kf\|_{{\mathcal{L}}^2}^2 & \leq \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \frac{e^{V(x)}}{{\mathcal{M}}(v)} \left(\int_{{{\mathbb R}}^d}k(x,v',v)^2{\mathcal{M}}(v') dv' \right) \left(\int_{{{\mathbb R}}^d}\frac{f(x,v')^2}{{\mathcal{M}}(v')} dv' \right) dx \, dv \\
& \leq\left(\sup_{x \in {{\mathbb T}}^d}\int_{{{\mathbb R}}^d}\int_{{{\mathbb R}}^d} k^2(x, v',v)\frac{{\mathcal{M}}(v')}{{\mathcal{M}}(v)}\, dv'\, dv \right) \|f\|_{{\mathcal{L}}^2}^2\end{aligned}$$ The operator $K$ is hence bounded in ${\mathcal{L}}^2$, with $$\|K\|_{{\mathcal{L}}^2 \to {\mathcal{L}}^2}\leq\left(\sup_{x \in {{\mathbb T}}^d}\int_{{{\mathbb R}}^d}\int_{{{\mathbb R}}^d} k^2(x, v',v)\frac{{\mathcal{M}}(v')}{{\mathcal{M}}(v)}\, dv'\, dv \right)^\frac12,$$ which is finite by [**A3**]{}.
According to [@Paz Chapter 3, Theorem 1.1], this implies the well-posedness of the Cauchy problem associated to .
For the maximum principle, we recall that we have the classical representation formula: $$f = \sum_{n=0}^{+\infty} \mathcal{K}^n ( e^{-t A_0} f_0),$$ where $$\begin{gathered}
\mathcal{K} g = \int_0^t \left(\int_{{{\mathbb R}}^d} [k(X_{-(t-s)} (x,v) ,v' , \Xi_{-(t-s)} (x,v)) g(s,X_{-(t-s)} (x,v),v') \, dv'\right)\\
\times \exp \left( - \int_s^t \left( \int_{{{\mathbb R}}^d} k(\phi_{-(t-u)}(x,v),v') \, dv' \right) \, du\right) \, ds.\end{gathered}$$ Recall that $k\geq 0$. If $f_0 \geq 0$ a.e., one can observe from that for all $t\geq 0$, $ e^{-t A_0} f_0 \geq 0$ a.e., and for all $n \in {{\mathbb N}}$, $\mathcal{K}^n ( e^{-t A_0}f_0) \geq 0$ a.e.. This concludes the proof.
A useful consequence of the maximum principle, of the linearity of the equation, and of Assumption [**A2**]{} is the following statement. If $f_0 \in {\mathcal{L}}^2 \cap {\mathcal{L}}^\infty$, then the unique solution of starting from $f|_{t = 0} =f_0$, satisfies $$\sup_{t\geq 0} \| f(t)\|_{{\mathcal{L}}^\infty} \leq \| f_0\|_{{\mathcal{L}}^\infty} .$$
Weak coercivity {#sec:wc}
---------------
In this section, we describe some properties of the collision kernel $C$ and associated dissipation $D$. In several proofs of the paper, we shall need to exploit some local coercivity properties of the dissipation. In particular, we would like to have the *weak coercivity* property $$\label{wc}
\forall f \in {\mathcal{L}}^2, \quad D(f)=0 \implies C(f)=0$$ (and thus, $D(f)=0$ is equivalent to $C(f)=0$). A difficulty comes from the fact that in general the dissipation term does not control neither the ${\mathcal{L}}^2$ distance to the projection on the set of stationary solutions, nor the ${\mathcal{L}}^2$ norm of the collision operator.
The main result is the following lemma.
\[collannule\] Let $k$ be a collision kernel satisfying [**A1**]{}–[**A3**]{}. Let $T \in (0, +\infty]$ and denote $\omega = \cup_{i \in I} \omega_i$ the partition of $\omega$ in connected components. Then, the following three properties are equivalent
1. \[toto1\] $f \in C^0(0,T; {\mathcal{L}}^2)$ satisfies ${C}(f(t))=0$ for all $t \in [0,T]$.
2. $f \in C^0(0,T; {\mathcal{L}}^2)$ satisfies ${D}(f(t))=0$ for all $t \in [0,T]$.
3. - for all $i \in I$, we have $f(t,x,v)=\rho_i (t,x){\mathcal{M}}(v)$ on $[0,T] \times \omega_i$;
- for $i,j \in I$ and $x \in {{\mathbb T}}^d$, we have: $\omega_i(x) {\, \mathcal{R} }_k^x \, \omega_j(x) \Longrightarrow \rho_i(t,x) = \rho_j(t,x)$ for all $t \in [0,T]$.
This lemma only states properties of the collision kernel. As such, it is not concerned with the time dependence, that we shall drop in the proof.
By definition, we have $D(f)=\langle C(f) , f\rangle_{{\mathcal{L}}^2}$ so that $(1) \Longrightarrow (2)$.
Then, from ${D}(f)=0$, Equation implies that $$\label{rho-i-j}
\frac{f(x,v)}{{\mathcal{M}}(v)} = \frac{f(x,v')}{{\mathcal{M}}(v')} \quad \text{almost everywhere in } S:=\{(x,v, v'), \, {k}(x,v',v) + k(x,v',v)>0\} ,$$ Let $(x,v) \in \omega$. Thus, there exists $v' \in {{\mathbb R}}^d$ such that $(x,v,v') \in S$. By continuity of ${k}$, there exists a neighborhood $U$ of $(x,v)$ such that for all $(y,w) \in U$, we have $(y,w,v') \in S$. Thus, for all $(y,w) \in U$, we have $$\frac{f(y,w)}{{\mathcal{M}}(w)} = \frac{f(y,v')}{{\mathcal{M}}(v')}$$ that is to say that locally, $(y,w) \mapsto \frac{f(y,w)}{{\mathcal{M}}(w)}$ is function of $y$ only. Therefore, for all $i \in I$, there is a function $\rho_i$ such that $\frac{f(x,v)}{{\mathcal{M}}(v)} = \rho_i (x)$ on $ \omega_i$.
Furthermore, take $i,j \in I$ and $x \in {{\mathbb T}}^d$ such that $\omega_i(x){\, \mathcal{R} }_k^x \, \omega_j(x)$. There exists $v_i,v_j \in {{\mathbb R}}^d \times {{\mathbb R}}^d$ such that $(x,v_i) \in \omega_i$, $(x,v_j) \in \omega_j$, and $(x,v_i,v_j) \in S$. It then follows from and $\frac{f(x,v_i)}{{\mathcal{M}}(v_i)} = \rho_i (x)$, $\frac{f(x,v_j)}{{\mathcal{M}}(v_j)} = \rho_j (x)$ that $\rho_i(x) = \rho_j(x)$. This concludes the proof of $(2)\Longrightarrow (3)$.
Finally, let us check that a function satisfying the two assumptions of $(3)$ cancels the collision operator, i.e. that for all $x,v$, we have $C(x,v,f)=0$.
Let $(x,v) \in \omega$ (if $(x,v) \notin \omega$, then $C(x,v,f)=0$). Let $i \in I$ such that $(x,v) \in \omega_i$. We have $$\begin{aligned}
C(x,v,f) &= \int \tilde{k}(x,v',v) f(x,v') \, dv' {\mathcal{M}}(v) - \int \tilde{k}(x,v,v') {\mathcal{M}}(v') \, dv' f(x,v) \\
&= \int \tilde{k}(x,v',v) f(x,v') \, dv' {\mathcal{M}}(v) - \int \tilde{k}(x,v,v') {\mathcal{M}}(v') \, dv' \rho_i(x) {\mathcal{M}}(v) \\
&= \sum_{j \in J_i} \int \tilde{k}(x,v',v) \mathds{1}_{\omega_j}(x,v') f(x,v') \, dv' {\mathcal{M}}(v) - \int \tilde{k}(x,v,v') {\mathcal{M}}(v') \, dv' \rho_i(x) {\mathcal{M}}(v) \\
&= \sum_{j \in J_i} \int \tilde{k}(x,v',v) \mathds{1}_{\omega_j}(x,v') {\mathcal{M}}(v') \, dv' \rho_j(x) {\mathcal{M}}(v) - \int \tilde{k}(x,v,v') {\mathcal{M}}(v') \, dv' \rho_i(x) {\mathcal{M}}(v),\end{aligned}$$ where $J_i$ is the largest subset of $I$ such that for all $j \in J_i$, there exists $v' \in {{\mathbb R}}^d$ such that $(x,v') \in\omega_j$ and $\tilde{k}(x,v',v)>0$. According to the second property satisfied by $f$, for all $j \in J_i$, $$\rho_j(x) =\rho_i(x),$$ and thus we deduce $$\begin{aligned}
C(x,v,f) &= \sum_{j \in J_i} \int \tilde{k}(x,v',v) \mathds{1}_{\omega_j}(x,v') {\mathcal{M}}(v') \, dv' \rho_i(x) {\mathcal{M}}(v) - \int \tilde{k}(x,v,v') {\mathcal{M}}(v') \, dv' \rho_i(x) {\mathcal{M}}(v) \\
&= \left(\int \tilde{k}(x,v',v) {\mathcal{M}}(v') \, dv' - \int \tilde{k}(x,v,v') {\mathcal{M}}(v') \, dv' \right) \rho_i(x) {\mathcal{M}}(v)\\
&= 0.\end{aligned}$$ The last line comes from the fact that $k$ satisfies [**A2**]{}. This concludes the proof of $(3) \Longrightarrow (1)$.
\[def:UCP-expli\] Another benefit of Lemma \[collannule\] is that it allows to rephrase the Unique Continuation Property, in a slightly more explicit way.
Denote $\omega = \cup_{i \in I} \omega_i$ the partition of $\omega$ in connected components. The set $\omega$ satisfies the Unique Continuation Property if and only if the following holds. The only solution $f \in C^0({{\mathbb R}}; {\mathcal{L}}^2)$ to $$\label{eq:UCP-expli}
\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0,$$ satisfying the following properties
- for all $i \in I$, $f(t,x,v)=\rho_i (t,x){\mathcal{M}}(v)$ on $[0,T] \times \omega_i$;
- for $i,j \in I$ and $x \in {{\mathbb T}}^d$, $\omega_i(x) {\, \mathcal{R} }_k^x \, \omega_j(x) \Longrightarrow \rho_i(t,x) = \rho_j(t,x)$ for all $t \in [0,T]$,
is $f= \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx\right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)$.
If $\tilde{k}$ is bounded in $L^\infty$ (but only in this case), we can actually prove that the dissipation controls the norm of a “symmetrized” collision operator. To state and prove such a result, we introduce the symmetrized collision kernel $$\label{barrekstar}
\overline{k}(x,v',v) := \frac{\tilde{k}(x,v',v)+ \tilde{k}(x,v,v')}{2}, \quad k^*(x,v',v) := \overline{k}(x,v',v){\mathcal{M}}(v) .$$ Note in particular that $ \overline{k}(x,v',v) \in L^\infty ({{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d)$ if $\tilde{k} \in L^\infty ({{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d)$.
We also introduce the associated collision operator $$\label{mathcolop}
\mathcal{C} (f) := \int_{{{\mathbb R}}^d} \left[{k^*}(x,v' , v) f(v') - {k^*}(x,v , v')) f(v)\right] \, dv'.$$
Note that we have $ \mathcal{C} (f) = C(f)$ if and only if $\overline{k}(x,v',v)=\tilde{k}(x,v',v)$, i.e. if $\tilde{k}(x,v',v)$ is symmetric with respect to $v$ and $v'$ (this corresponds to the class [**E1**]{}).
\[cerci1\] Let $k$ be a collision kernel satisfying [**A1**]{}–[**A2**]{}, and such that $\tilde{k} \in L^\infty$. For any $f \in {\mathcal{L}}^2$, we have $$\|\overline{k}\|_{L^\infty} D(f) \geq \left\|\mathcal{C} (f) \right\|_{{\mathcal{L}}^2}^2,$$
According to the definition of the dissipation and the symmetry of $\overline{k}$, we have $$\begin{aligned}
D(f) &\geq \int_{{{\mathbb T}}^d} e^{V} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \overline{k}(x,v',v) {\mathcal{M}}{\mathcal{M}}' \left(\frac{f}{{\mathcal{M}}}- \frac{f'}{{\mathcal{M}}'}\right)^2 \, dv' \, dv \, dx ,
\end{aligned}$$ and hence $$\begin{aligned}
\|\overline{k}\|_{L^\infty} D(f) \geq \int_{{{\mathbb T}}^d} e^{V} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \overline{k}^2(x,v',v) {\mathcal{M}}{\mathcal{M}}' \left(\frac{f}{{\mathcal{M}}}- \frac{f'}{{\mathcal{M}}'}\right)^2 \, dv' \, dv \, dx.
\end{aligned}$$ By Jensen’s inequality it follows that $$\begin{aligned}
\|\overline{k}\|_{L^\infty} D(f) &\geq \int_{{{\mathbb T}}^d} e^{V} \int_{{{\mathbb R}}^d} {\mathcal{M}}(v)\left(\int_{{{\mathbb R}}^d} \overline{k}(x,v',v) {\mathcal{M}}(v') \left(\frac{f(v)}{{\mathcal{M}}(v)}- \frac{f(v')}{{\mathcal{M}}(v')}\right) \, dv'\right)^2 dv \, dx \\
&= \int_{{{\mathbb T}}^d} e^{V} \int_{{{\mathbb R}}^d} \frac{1}{{\mathcal{M}}(v)} \mathcal{C} (f)^2 \, dv \, dx\\
&= \left\| \mathcal{C} (f) \right\|_{{\mathcal{L}}^2}^2,
\end{aligned}$$ where we again used the symmetry of $\overline{k}$. This concludes the proof of the lemma.
Characterization of convergence to equilibrium {#subsectiondecroissance}
==============================================
In this Section, we shall first give a proof of Theorem \[thmconv-intro\]; then we will provide a proof of Theorem \[thmconvgene-intro\], which will be a consequence of our main result in this direction, namely Theorem \[thmconv-general\].
We start with a technical lemma concerning the evolution under the flow of the connected components of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$.
\[leminvarflow\] Set $\tilde\Omega = \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ and denote by $(\Omega_i)_{i\in I}$ the partition of $\tilde\Omega$ in connected components, and ${\mathcal{A}}= {{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \tilde\Omega$. Then we have for all $t \geq 0$, for all $i \in I$, $$\label{inclusionsphi}
\phi_{-t}(\tilde\Omega) \subset \tilde\Omega , \quad \phi_{t}({\mathcal{A}}) \subset {\mathcal{A}}, \quad \phi_{-t}(\Omega_i) \subset \Omega_i .$$ If moreover $\omega$ satisfies a.e.i.t. GCC (i.e. ${\mathcal{A}}$ has zero Lebesgue measure), then for all $i \in I$, for all $t \in {{\mathbb R}}$, $$\label{eq-propconn}
\phi_{t}(\Omega_i) = \Omega_i \quad \text{almost everywhere}.$$
First, we just remark that for all $t \geq 0$, we have $\phi_{-t}(\tilde\Omega) = \bigcup_{s\in {{\mathbb R}}^+}\phi_{-t-s}(\omega) \subset \tilde\Omega$. Taking the complement of this inclusion yields for $t \geq 0$, $\phi_{t}({\mathcal{A}}) \subset {\mathcal{A}}$.
Let us now fix $i \in I$ and prove that $$\label{omegaiphit}
\text{for all } t \geq 0 , \quad
\phi_{-t}(\Omega_i)\subset \Omega_i.$$ Take $(x,v) \in \Omega_i$ and $t> 0$. If $\phi_{-t}(x, v) \notin \Omega_i$, then there exists $t_0 \in (0,t]$ such that $\phi_{-t_0}(x, v) \notin \tilde\Omega$ since $\Omega_i$ is a connected component of this set. This is in contradiction with $\phi_{-t_0}(\tilde\Omega) \subset \tilde\Omega$. This implies .
Finally, if ${\mathcal{A}}$ has zero Lebesgue measure, then $|\phi_{-t}({\mathcal{A}})| = 0$ as well. As a consequence, the identity $${{\mathbb T}}^d \times {{\mathbb R}}^d = \phi_{-t} ({{\mathbb T}}^d \times {{\mathbb R}}^d) = \phi_{-t} \left(\bigcup_{i \in I} \Omega_i \cup {\mathcal{A}}\right)
= \bigcup_{i \in I} \phi_{-t} (\Omega_i ) \cup \phi_{-t}({\mathcal{A}})$$ yields ${{\mathbb T}}^d \times {{\mathbb R}}^d = \bigcup_{i \in I} \phi_{-t} (\Omega_i )$ almost everywhere. Since for all $i \in I$ and $t\geq 0$, $\phi_{-t} (\Omega_i ) \subset \Omega_i$, we obtain (still for $t \geq 0$) $\phi_{-t} (\Omega_i ) = \Omega_i$ almost everywhere, from which the conclusion of the lemma follows.
Note that if $V=0$ and $\omega$ satisfies the following property: $(x,v) \in \omega \Leftrightarrow (x,-v) \in \omega$, then the inclusions in become equalities. Hence, all sets considered in are then invariant by $\phi_t$ for all $t\in {{\mathbb R}}$.
Proof of Theorem \[thmconv-intro\] {#sec:conv}
----------------------------------
We shall prove that $(i.) \implies (iii.)$, that $(iii.) \implies (ii.)$ and finally that $(ii.) \implies (i.)$.
$(i.) \implies (iii.)$ We prove that the Unique Continuation Property (of Definition \[def:UCP\]) implies the decay.
We first prove the expected convergence for data enjoying more regularity, i.e. we prove $$\label{convergeto0'}
\text{for all } f_0 \in {\mathcal{L}}^2\cap {\mathcal{L}}^\infty, \quad \left\|f(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0 .$$
Since is linear, $g(t):= f(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)$ is a solution to $\eqref{B}$ with initial datum $g(0) = f(0)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v) \in {\mathcal{L}}^2\cap {\mathcal{L}}^\infty$, satisfying $\int g(0) \, dv \, dx =0$. Therefore proving is equivalent to proving that $\|g(t)\|_{{\mathcal{L}}^2} \to 0$.
We argue by contradiction. Assume that there exists an initial datum $g_{0}$ in ${\mathcal{L}}^2 \cap {\mathcal{L}}^\infty$ (with $\int g_0 \, dv \,dx = 0 $), ${{\varepsilon}}>0$ and an increasing sequence $(t_n)_{n \in {{\mathbb N}}}$ such that: $$\label{borne}
t_n \geq e^n , \quad \text{and} \quad \left\|g_0\right\|_{{\mathcal{L}}^2} \geq \left\|g(t_n)\right\|_{{\mathcal{L}}^2} > {{\varepsilon}}.$$ From this sequence, we may extract a subsequence (still denoted $(t_n)$) satisfying $$\label{tn+1 tn}
t_{n+1}-t_n \to + \infty.$$ According to the Maximum Principle of Proposition \[prop:WP\], we have $$\label{eqequi}
\text{for all } t \geq 0, \quad \|g(t) \|_{{\mathcal{L}}^\infty} \leq \|g_0 \|_{{\mathcal{L}}^\infty}.$$
We introduce the shifted function $h_n(t,x,v) := g(t_n + t, x,v)$. By the time translation invariance of , $h_n$ is still a solution to , with initial datum $h_n(0)= g(t_n)$. Using , up to some extraction, we can assume that there is $\alpha \in [{{{\varepsilon}}}, \left\|g_0\right\|_{{\mathcal{L}}^2}]$ such that $$\label{limithn}
\| h_n(0) \|_{{\mathcal{L}}^2} \to_{n \rightarrow + \infty} \alpha.$$ Note also that by conservation of the mass, for all $n \in {{\mathbb N}}$ and all $t \geq 0$, $$\label{eqconsmas}
\int h_n(t) \, dv dx = \int g_0 \, dv dx =0.$$ Using the dissipation identity of Lemma \[lemdissip\] for $g$, we have: $$\| g(t_{n+1}) \|_{{\mathcal{L}}^2}^2 - \| g(t_{n}) \|_{{\mathcal{L}}^2}^2 = - \int_{t_n}^{t_{n+1}} D(g) \, dt,$$ that is (using the time translation invariance): $$\| h_{n+1}(0) \|_{{\mathcal{L}}^2}^2 - \| h_{n}(0) \|_{{\mathcal{L}}^2}^2 = - \int_{0}^{t_{n+1}-t_n} D(h_n) \, dt .$$ This, together with and , implies that for any $T>0$, $$\label{Dzero}
\int_{0}^{T} D(h_n) \, dt \to 0.$$ Now, up to another extraction, since for any $n \in {{\mathbb N}}$, $$\label{hng0}
\text{for all } t \geq 0, \quad \| h_n (t) \|_{{\mathcal{L}}^2} \leq \| g_0 \|_{{\mathcal{L}}^2},$$ we can assume that $h_n \rightharpoonup h$ weakly in $L^2_{t, loc}{\mathcal{L}}^2$. Let us now prove that $h=0$. First, since $h_n$ is a solution to , by linearity, $h$ also satisfies .
We denote now $d\lambda := \mathds{1}_{[0,T]}(t)\left( \frac{k(x,v' , v)}{{\mathcal{M}}(v)} + \frac{k(x,v , v')}{{\mathcal{M}}(v')} \right) {\mathcal{M}}(v) {\mathcal{M}}(v') \, dv dv' dx dt$. Then, introducing $$\tilde{h}_n(t,x,v,v') := \frac{h_n(t,x,v)}{{\mathcal{M}}(v)}- \frac{h_n(t,x,v')}{{\mathcal{M}}(v')},$$ we observe that $\| \tilde{h}_n \|_{L^2(d\lambda)}^2= \int_0^TD(h_n) \, dt$ and thus, by , we deduce that $\tilde{h}_n(t,x,v,v')$ is uniformly bounded in $L^2(d\lambda)$. Consequently, up to a another extraction, $\tilde{h}_n$ weakly converges in $L^2(d\lambda)$ to some $\tilde{h}$. By uniqueness of the limit in the sense of distributions, we have $\tilde{h}= \frac{h(t,x,v)}{{\mathcal{M}}(v)}- \frac{h(t,x,v')}{{\mathcal{M}}(v')}$.
Then, by and weak lower semi-continuity, we deduce that for any $T>0$, we have $$\left\|{D}(h) \right\|_{L^1(0,T)} = \| \tilde{h} \|_{L^2(d\lambda)}^2\leq \liminf_{n\to +\infty} \| \tilde{h}_n \|_{L^2(d\lambda)}^2 = \liminf_{n\to +\infty} \left\|{D}(h_n) \right\|_{L^1(0,T)}= 0.$$ Thus, by weak coercivity (see Lemma \[collannule\]), we infer that ${C}(h)=0$ on $[0,T]$, for any $T>0$, and therefore $h$ satisfies the kinetic transport equation . Using the Unique Continuation Property (see Definition \[def:UCP\]), we deduce that $$h=\left(\int h \, dv dx \right) \frac{e^V}{{\mathcal{M}}}.$$ Since $h_{n} \rightharpoonup h$ weakly in $L^2_t {\mathcal{L}}^2$, using , we obtain in particular that for any $T>0$ $$\int_0^{T} \left(\int h \, dv dx \right) dt =0.$$ Since $\int_0^{T} \left(\int h \, dv dx \right) dt = T \left(\int h(0) \, dv dx \right)$, we deduce that $\int h \, dv dx=0$ so that $h=0$.
Let us now consider the sequence of defect measures $\nu_n := |h_n|^2$, which, according to and satisfies, for all $n \in {{\mathbb N}}$, $$\text{for all } t \geq 0, \quad \| \nu_n(t) \|_{{\mathcal{L}}^{1}} \leq \| g(0)\|_{{\mathcal{L}}^2}^2, \quad \| \nu_n(t) \|_{{\mathcal{L}}^{\infty}} \leq C_0\| g(0)\|_{{\mathcal{L}}^\infty}^2,$$ for $C_0 = \max_{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d} e^{-V(x)}{\mathcal{M}}(v)$. We have that, up to another subsequence $\nu_n \rightharpoonup \nu$ weakly-$\star$ in $L^{\infty}_{t,loc} {\mathcal{L}}^\infty$. Let us compute the equation satisfied by $\nu$: to this purpose, we consider satisfied by $h_n$ and multiply it by $h_n$. We obtain: $$\label{eq:nunnun}
\begin{aligned}
\partial_t \nu_n &+ v \cdot \nabla_x \nu_n - \nabla_x V \cdot \nabla_v \nu_n \\
&= 2 \left[ \int_{{{\mathbb R}}^d} \left[k(x,v' , v) h_n(v') - k(x,v , v') h_n(v)\right] \, dv' \right]
h_n\\
&= 2 \left(\int_{{{\mathbb R}}^d} k(x,v' , v) h_n(v') \, dv'\right) h_n - 2 \left( \int_{{{\mathbb R}}^d} k(x,v , v')\, dv' \right) \nu_n .
\end{aligned}$$ Using the averaging lemma of Corollary \[lemmoyenne\] and the fact that $h_n$ weakly converges to $0$, we deduce that $$\label{termecompacite}
\int_{{{\mathbb R}}^d} k(x,v' , v) h_n(t,x,v') \, dv' \rightarrow 0$$ strongly in $L^2_{t,loc}{\mathcal{L}}^2$. On the other hand, according to , the sequence $(h_n)$ is uniformly bounded in $L^2_{t,loc}{\mathcal{L}}^2$. We hence obtain $$\label{compacitemoyenne}
\left(\int_{{{\mathbb R}}^d} k(x,v' , v) h_n(t,x,v') dv'\right) h_n \to 0 \quad \text{strongly in } L^1_{t,loc}{\mathcal{L}}^1.$$ The second term, by definition of $\nu$, weakly converges to $- 2 \left( \int_{{{\mathbb R}}^d} k(x,v , v')\, dv' \right) \nu$ in the sense of distributions, so that $\nu$ satisfies the equation $$\label{transnu}
\partial_t \nu + v \cdot \nabla_x \nu - \nabla_x V \cdot \nabla_v \nu = - 2 \left( \int_{{{\mathbb R}}^d} k(x,v , v')\, dv' \right) \nu.$$ Moreover, writing $$\label{buternu-pre}
\left( \int_{{{\mathbb R}}^d} k(x,v , v')\, dv' \right) |h_n|^2 = -C(h_n) h_n + \left(\int_{{{\mathbb R}}^d} k(x,v' , v) h_n(t,x,v') dv'\right) h_n,$$ and using together with , we deduce that $$\label{buternu}
\left( \int_{{{\mathbb R}}^d} k(x,v , v')\, dv' \right) \nu =0.$$ Thus, combined with entails that $\nu$ satisfies the kinetic transport equation $$\partial_t \nu + v \cdot \nabla_x \nu - \nabla_x V \cdot \nabla_v \nu =0,$$ which also shows that $\nu \in C^0_t({\mathcal{L}}^{2})$. This, combined with the fact that $\nu=0$ on ${{\mathbb R}}^+ \times \omega$ (again coming from ) and the Unique Continuation Property, implies $$\nu = \left(\int \nu(0) \, dv \,dx\right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v).$$ According to , this means that $\nu=0$.
We now prove that there is no loss of mass at infinity. Let $\delta >0$ and $R>0$. We have: $$\begin{aligned}
\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu_n(0) \frac{e^{V}}{{\mathcal{M}}} \mathds{1}_{v \in {{\mathbb R}}^d \setminus B(0,R)} \, dv \,dx &\leq \| g(0)\|_{{\mathcal{L}}^\infty}^2 \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \frac{{\mathcal{M}}}{e^{V}}\mathds{1}_{v \in {{\mathbb R}}^d \setminus B(0,R)} \, dv \,dx \\
&\leq \| g(0)\|_{{\mathcal{L}}^\infty}^2 \int_{{{\mathbb T}}^d } {e^{-V}} \,dx \int_{|v| \geq R} {\mathcal{M}}(v) \, dv,\end{aligned}$$ which is exponentially converging to zero as $R\to \infty$. This yields $$\label{nolossofmass}
\lim_{R \to \infty} \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu_n(0) \frac{e^{V}}{{\mathcal{M}}} \mathds{1}_{v \in {{\mathbb R}}^d \setminus B(0,R)} \, dv \,dx = 0 .$$ Therefore, on the one hand, up to a subsequence, we can assume that $\nu_n(0)\frac{e^{V}}{{\mathcal{M}}} \rightharpoonup \nu(0)\frac{e^{V}}{{\mathcal{M}}}$ tightly in $ \mathcal{M}^+_{x,v}$ and thus $$\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu_n(0) \frac{e^{V}}{{\mathcal{M}}} dv \, dx \rightarrow \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu(0) \frac{e^{V}}{{\mathcal{M}}} dv \, dx =0.$$ On the other hand, using , we have $$\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu_n(0) \frac{e^{V}}{{\mathcal{M}}} dv \, dx \rightarrow \alpha \, >0.$$ This yields a contradiction, and concludes the proof of .
We finally deduce by an approximation argument. Let $f_0 \in {\mathcal{L}}^2$ and $f(t)$ be the solution associated to $f_0$. Let ${{\varepsilon}}>0$. There exists $f_{0,{{\varepsilon}}} \in {\mathcal{L}}^2 \cap {\mathcal{L}}^\infty$ such that $\|f_0 - f_{0,{{\varepsilon}}}\|_{{\mathcal{L}}^2} \leq {{\varepsilon}}$. Let $f_{{\varepsilon}}(t)$ be the solution associated to $f_{0,{{\varepsilon}}}$. Since $f- f_{{\varepsilon}}$ is a solution of with initial datum $f_0 -f_{0,{{\varepsilon}}}$ we also have for any $t\geq 0$, $\|f(t) - f_{{\varepsilon}}(t)\|_{{\mathcal{L}}^2} \leq {{\varepsilon}}$.
By , there exists $t_0 \geq 0$ such that for all $t\geq t_0$, $$\left\|f_{{\varepsilon}}(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_{0,{{\varepsilon}}} \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \leq {{\varepsilon}}.$$ Thus, it follows that for all $t\geq t_0$, $$\begin{aligned}
&\left\|f(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_{0} \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \\
&\leq \| f(t)- f_{{\varepsilon}}(t) \|_{{\mathcal{L}}^2} + \left\|f_{{\varepsilon}}(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_{0,{{\varepsilon}}} \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \\
& \quad+
\left\| \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} ( f_{0, {{\varepsilon}}}- f_{0}) dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \\
& \leq 2 {{\varepsilon}}+ \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} |f_{0}-f_{0,{{\varepsilon}}}| \, dv \,dx .\end{aligned}$$ Besides, we have $$\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} |f_{0}-f_{0,{{\varepsilon}}}| \, dv \,dx \right)^2 \leq \|f_{0}-f_{0,{{\varepsilon}}}\|_{{\mathcal{L}}^2}^2 \left( \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d}e^{-V}{\mathcal{M}}(v) \, dx \, dv \right).$$ The last two inequalities together yield, for all $t \geq t_0$, $$\begin{aligned}
\left\|f(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_{0} \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2}
\leq \left(2+ \left(\int_{{{\mathbb T}}^d}e^{-V} \, dx\right)^\frac12 \right){{\varepsilon}},\end{aligned}$$ which concludes the proof of .
$(iii) \Rightarrow (ii.)$ Assume that $(ii.)$ does not hold. Either ${{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)$ has positive Lebesgue measure, or the equivalence relation $\Bumpeq$ has two or more equivalence classes (or equivalently, by Lemma \[equiv-equiv\], the binary relation $\sim$ has two or more equivalence classes).
Suppose first that ${\mathcal{A}}:= {{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)$ has positive Lebesgue measure. We set $$f_0 (x,v) = \mathds{1}_{{\mathcal{A}}}(x,v)\, e^{-V(x)} {\mathcal{M}}(v).$$ Note that $f_0$ satisfies $\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d}f_0(x,v) dx \, dv > 0$ as ${\mathcal{A}}$ is of positive Lebesgue measure. We consider $f(t, \cdot)$ the solution to with initial datum $f_0$, given by $$f(t,x,v) = f_0 \circ \phi_{-t}(x,v) = \mathds{1}_{\phi_{t}({\mathcal{A}})}(x,v)\, e^{-V(x)} {\mathcal{M}}(v)$$ Moreover, for all $t\geq0$, we have $\phi_{t}({\mathcal{A}}) \cap \omega = \emptyset$ since ${\mathcal{A}}\cap \phi_{-t}(\omega) = \emptyset$. Therefore, $f =0$ on ${{\mathbb R}}^+\times \omega$ so that, according to the characterization of $\omega$ in , $C(f) = 0$ and $f$ is also a solution of . Moreover, this implies that $$\left\| f(t) - \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv dx\right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v) \right\|_{{\mathcal{L}}^2} \not\to_{t \to +\infty} 0$$ Thus, $(iii.)$ does not hold.
Suppose now that ${{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)$ has zero Lebesgue measure and that the equivalence relation $\sim$ has (at least) two distinct equivalence classes, say $[\Omega_1]$ and $[\Omega_2]$.
We define now a function $f(x,v)$ as follows $$f(x,v) = \sum_{\Omega' \in [\Omega_1]} \mathds{1}_{\Omega'} (x,v) e^{-V(x)}{\mathcal{M}}(v).$$ Using in Lemma \[leminvarflow\], we deduce that for all $t \geq 0$, $$f \circ \phi_t (x,v) =\sum_{\Omega' \in [\Omega_1]} \mathds{1}_{\Omega'} \big(\phi_t (x,v)\big) e^{-V(x)}{\mathcal{M}}(v) =
\sum_{\Omega' \in [\Omega_1]} \mathds{1}_{\Omega'} (x,v) e^{-V(x)}{\mathcal{M}}(v) = f (x,v) ,$$ so that $f$ is a stationary solution of $\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0$.
There remains to prove that $f$ cancels the collision operator. Denote $\omega = \cup_{i \in I} \omega_i$ the partition of $\omega$ in connected components.
By Lemma \[collannule\], $f$ cancels the collision operator if and only if $f$ satisfies the following two properties.
1. For all $i \in I$, $$f=\rho_i (x){\mathcal{M}}(v) \quad \text{on } \omega_i,$$
2. For $i,j \in I$, and $x \in {{\mathbb T}}^d$ $\omega_i(x), {\, \mathcal{R} }_k^x \, \omega_j(x) \Longrightarrow \rho_i(x) = \rho_j(x)$.
We check now that our function $f$ satisfies these two properties.
Let $i \in I$. If for all $\Omega' \in [\Omega_1]$, $\omega_i \cap \Omega' =\emptyset$, then $f=0$ on $\omega_i$ (and hence satisfies $(1)$ with $\rho_i=0$). If there is $\Omega' \in [\Omega_1]$ such that $\omega_i \cap \Omega' \neq \emptyset$, then since $\Omega'$ is a connected component of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$, we have $\omega_i \subset \Omega'$. Thus we have $f=e^{-V}{\mathcal{M}}$ on $\omega_i$ (and hence $f$ satisfies $(1)$ with $\rho_i=e^{-V}$). We deduce that for all $i \in I$, $f$ is of the form $\rho_i (x){\mathcal{M}}(v)$ on $\omega_i$.
Take now $i,j \in I$ and $x\in {{\mathbb T}}^d$ such that $\omega_i(x) {\, \mathcal{R} }_k^x \, \omega_j(x)$. Let $\Omega^{(i)}$ (resp. $\Omega^{(j)}$) be the connected component of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ which contains $\omega_i$ (resp. $\omega_j$). By definition of the relations ${\, \mathcal{R} }_k$ and ${\, \mathcal{R} }_k^x$, this directly yields $\Omega^{(i)} {\, \mathcal{R} }_k \, \Omega^{(j)}$, and [*a fortiori*]{} we deduce that $\Omega^{(i)}\sim \Omega^{(j)}$: in other words, these are in the same equivalence class for $\sim$. According to the definition of $f$, this implies $\rho_i(x) = \rho_j(x)$ (which is equal to $ e^{-V(x)}$ if $\Omega^{(i,j)} \in [\Omega_1]$ and to $0$ if not).
Therefore, the function $f$ satisfies the two properties and, by Lemma \[collannule\], cancels the collision operator.
However, we have $$\cup_{\Omega' \in [\Omega_2]} \Omega' \subset {{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \cup_{\Omega' \in [\Omega_1]} \Omega'$$ and $\cup_{\Omega' \in [\Omega_2]} \Omega' $ has a positive Lebesgue measure. Consequently, the measure of ${{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \cup_{\Omega' \in [\Omega_1]} \Omega'$ is positive, so that $f$ is a stationary solution of which is not a uniform Maxwellian. As a consequence, $(iii.)$ does not hold.
$(ii.) \Rightarrow (i.)$ Assume that $(ii.)$ holds. Let $f \in C^0_t({\mathcal{L}}^2)$ be a solution to $$\begin{aligned}
\label{eqhypo1} \partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
\label{eqhypo2} C(f)=0.\end{aligned}$$ As usual, without loss of generality, we can assume that $\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx=0$. The goal is to show that $f=0$.
Since $f$ cancels the collision operator, by Lemma \[collannule\], the restriction of $f$ to $\omega$ is necessarily of the form $f_{|\omega}(t,x,v) = \sum_{i \in I}\mathds{1}_{\omega_i}(x,v) \rho_i(t,x) {\mathcal{M}}(v)$, where $\omega = \bigcup_{i \in I} \omega_i$ is the partition of $\omega$ in connected components. Furthermore, for $i,j \in I$, if there is $x \in {{\mathbb T}}^d$ such that $\omega_i(x) {\, \mathcal{R} }_k^x \, \omega_j(x)$, then $\rho_i(t,x) = \rho_j(t,x)$.
Consider now $\tilde{\omega}$ a connected component of $\omega$. Consider for $(t,x, v) \in {{\mathbb R}}^+ \times \tilde\omega$, $g(t,x) := \frac{e^V}{{\mathcal{M}}(v)} \, f$. We have, in the sense of distributions in ${{\mathbb R}}^+ \times \tilde\omega$, $$\partial_t g + v \cdot \nabla_x g = \frac{e^V}{{\mathcal{M}}(v)} \left[\partial_t f + v \cdot \nabla_x f + (v\cdot \nabla_x V ) f\right].$$ Since $f$ satisfies , this implies that $g$ satisfies the free transport equation $$\label{transg}
\partial_t g + v \cdot \nabla_x g =0 ,$$ in the sense of distributions in ${{\mathbb R}}^+ \times \tilde\omega$.
Let $(x, v) \in \tilde\omega$. Since $\tilde\omega$ is open, there exists $\delta>0$ such that $B(x,\delta)\times B(v,\delta) \subset \tilde\omega$ and $\eta>0$ such that for all $t \in (-\eta,\eta)$, for all $(x', v')\in B(x,\delta)\times B(v,\delta)$, $(x'+tv' ,v') \in \tilde\omega$. Integrating along characteristics we obtain $$\label{gpropcons}
g(t,x') = g(0,x'+tv') , \quad \text{for } (t,x',v') \in (-\eta,\eta) \times B(x,\delta) \times B(v,\delta).$$ Setting $U_x := \{ x - \frac{\eta}{2} v + \frac{\eta}{2} v', v' \in B(v, \delta)\}$, we remark that $U_x$ is an open set containing $x$. Moreover, for all $y \in U_x$, we have $y = x - \frac{\eta}{2} v + \frac{\eta}{2} v'$ for some $v' \in B(v, \delta)$ so that, using , we have $$g(0, y) = g\left(0,x - \frac{\eta}{2} v + \frac{\eta}{2} v'\right) = g\left(\eta/2, x- \frac{\eta}{2} v\right).$$ Hence, $g(0, \cdot)$ is constant on $U_x$, and therefore constant on $\tilde\omega$ (since $\tilde\omega$ is connected).
Using the time translation invariance of , we also have that for all $t\geq 0$, $g(t, \cdot)$ is locally constant on $\omega$. As a consequence, for all $t \geq 0$, $\frac{e^V}{{\mathcal{M}}} \, f(t,\cdot)$ is locally constant on $\omega$, which means that $\rho_i(t,x) = \kappa_i (t)$, i.e. $f_{|\omega}(t,x,v) = e^{-V(x)} {\mathcal{M}}(v) \sum_{i \in I} \kappa_i(t) \mathds{1}_{\omega_i}(x,v)$. Since $f$ satisfies the transport equation on ${{\mathbb R}}^+ \times \omega_i$, we infer that $\kappa_i$ is constant, so that $f_{|\omega}(t,x,v) =f_{|\omega}(0,x,v)= e^{-V(x)} {\mathcal{M}}(v) \sum_{i \in I} \kappa_i \mathds{1}_{\omega_i}(x,v)$. Using again the transport equation , we deduce that $\frac{e^V}{{\mathcal{M}}} \, f(t,x,v) = \frac{e^V}{{\mathcal{M}}} \, f(0,\cdot) \circ \phi_{-t}(x,v)$. Since there is only one equivalence class for $\Bumpeq$, we first deduce that $f= \kappa e^{-V(x)} {\mathcal{M}}(v) $ on $\omega$ and then that $f= \kappa e^{-V(x)} {\mathcal{M}}(v) $ on $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$. Since $\omega$ satisfies a.e.i.t. GCC, this is a full measure set and we deduce that $f =\kappa e^{-V} {\mathcal{M}}(v)$. Since $\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx=0$, necessarily $f=0$.
This concludes the proof of Theorem \[thmconv-intro\].
Note that the proof of $(i.) \implies (iii.)$ relies on the maximum principle for the linear Boltzmann equation (i.e. the ${\mathcal{L}}^\infty$ bound). This was in particular useful to prevent loss of mass at infinity for the sequence of solutions under study. This will turn out to be also very useful to overcome another issue in the proof of the analogous theorem in the case of a bounded domain of ${{\mathbb R}}^d$ with specular reflection, see Section \[convboun\].
If $\omega$ satisfies the Geometric Control Condition of Defintion \[def: GCC\], we can actually show a slightly stronger property than the Unique Continuation Property (with the same proof as $(ii.)$ implies $(i.)$ above, up to some slight modifications), that we state for convenience as a Proposition.
\[remUCP\] Assume that $(\omega,T)$ satisfies the Geometric Control Condition.
If $f \in C^0_t({\mathcal{L}}^2)$ is a solution to $$\left\{
\begin{aligned}
&\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
&C(f) =0 \text{ on } I \times \omega,
\end{aligned}
\right.$$ where $I$ is an interval of time of length larger than $T$, then $f= \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx\right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)$.
This will be useful for the proof of Theorem \[thmexpo-intro\].
Proof of Theorem \[thmconvgene-intro\] {#sec:conv2}
--------------------------------------
We start by describing the vector space of stationary solutions of the linear Boltzmann equation , when the associated set $\omega$ satisfies the a.e.i.t. GCC.
For the sake of readability, we set here $$\tilde\Omega:= \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega).$$ We denote $\tilde\Omega= \cup_{i \in I} \Omega_i$ the partition of $\tilde\Omega$ in connected components. We write $([\Omega_j])_{j\in J}$ the equivalence classes for the equivalence relation $\sim$. We denote for all $j \in J$ $$\label{def-Uj}
U_j := \bigcup_{\Omega' \in [\Omega_j]} \Omega'.$$
We have the following description of the vector space of stationary solutions of .
\[cardinal\] Assume that $\omega$ satisfies the a.e.i.t. GCC. Then a Hilbert basis of the subspace of stationary solutions to the linear Boltzmann equation (or, equivalently of $\operatorname{Ker}(A)$, where $A$ is the linear Boltzmann operator defined in ) is given by the family $(f_j)_{j \in J}$, with $$\label{def-fj}
f_j = \frac{\mathds{1}_{U_j} e^{-V}{\mathcal{M}}}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}}.$$ In particular, the cardinality of the set of equivalence classes for $\sim$ is equal to the dimension of the vector space of stationary solutions to the linear Boltzmann equation , i.e. $$\dim (\operatorname{Ker}(A)) = \sharp({\mathcal{C}\mathcal{C}}(\tilde\Omega) / \sim) = \sharp({\mathcal{C}\mathcal{C}}(\omega) / \Bumpeq).$$
We can introduce a [**generalized**]{} Unique Continuation Property, as follows.
\[def:UCPgene\] We say that the set $\omega$ satisfies the [**generalized**]{} Unique Continuation Property if the only solutions $f \in C^0_t({\mathcal{L}}^2)$ to $$\label{eq:UCPgene}
\left\{
\begin{array}{l}
\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
C(f) = 0,
\end{array}
\right.$$ are of the form $f=\sum_{j\in J} \langle f, f_j \rangle_{{\mathcal{L}}^2} \, f_j =\sum_{j \in J} \frac{1}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} \left( \int_{U_j} f \, dv dx\right) f_j$.
We can now state the precise version of Theorem \[thmconvgene-intro\]:
\[thmconv-general\] We keep the notations of Proposition \[cardinal\]. The following statements are equivalent.
1. The set $\omega$ satisfies the [**generalized**]{} Unique Continuation Property.
2. The set $\omega$ satisfies the a.e.i.t. GCC.
3. For all $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\label{convergeto0-general}
\left\|f(t)-Pf_0 \right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$ where $$\label{def-equimulti}
P f_0 (x,v) = \sum_{i\in J} \frac{1}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} \left( \int_{U_i} f_0 \, dv dx\right)f_j ,$$ with $(U_j)_{j \in J}$ defined in and $(f_j)_{j \in J}$ defined in .
4. For all $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, there exists a stationary solution $Pf_0$ of such that we have $$\label{convergeto0-general'}
\left\|f(t)-Pf_0 \right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$ where $f(t)$ the unique solution to with initial datum $f_0$. .
Note that Theorem \[thmconv-intro\] is a particular case of Theorem \[thmconv-general\], when there is only one equivalence class for $\sim$ (or equivalently for $\Bumpeq$).
This section is organized as follows: in Paragraph \[para1\], we prove Proposition \[cardinal\]. Then, in Paragraph \[para1\], we prove that (iv.) implies (ii.). Finally, in Paragraph \[para3\], we show that (i.)–(ii.)–(iii.) are equivalent. Since (iii) implies (iv.) is straightforward, this will conclude the proof of Theorem \[thmconv-general\].
### Proof of Proposition \[cardinal\] {#para1}
We start by checking that for all $i \in J$, $f_j$ is a stationary solution of . From Lemma \[leminvarflow\], we know that for any connected component $\Omega'$ of $\tilde\Omega$ and any $t\geq0$, $\phi_{-t} (\Omega') = \Omega'$ almost everywhere. Thus for all $t\geq0$, $\phi_{-t} (U_j) = U_j$ almost everywhere. The function $f_j$ hence cancels the kinetic transport part.
We now check that $f_j$ cancels the collision operator, i.e. $C(\mathds{1}_{U_i} e^{-V}{\mathcal{M}})=0$. We use for this Lemma \[collannule\].
Denote $\omega = \cup_{i \in I} \omega_i$ the partition of $\omega$ in connected components. Let $i \in I$. If $\omega_i \cap U_j =\emptyset$, then $f_j=0$ on $\omega_i$. If $\omega_i \cap U_j \neq \emptyset$, then there exists $\Omega' \in [\Omega_j]$ such that $\omega_i \cap \Omega' \neq \emptyset$. Since $\Omega'$ is a connected component of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$, we have $\omega_i \subset \Omega'$ and thus $f_j=\frac{ e^{-V(x)}{\mathcal{M}}(v)}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} = \rho_j(x) {\mathcal{M}}(v)$ on $\omega_i$, with $\rho_j(x)=\frac{ e^{-V(x)}}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}}$.
Assume now that there exist $k,l \in I$ and $x\in {{\mathbb T}}^d$ such that $\omega_k(x) {\, \mathcal{R} }_k^x \, \omega_l(x)$. Denote by $\Omega' \in [\Omega_j]$, the connected component of $\tilde\Omega$ such that $\omega_l \subset \Omega'$, and $\Omega''$ the connected component of $\tilde\Omega$ such that $\omega_k \subset \Omega''$. Note then that $\omega_k(x) {\, \mathcal{R} }_k^x \, \omega_l(x)$ implies $\Omega' \sim \Omega''$. By definition of $f_j$, this implies that $\rho_k(x)= \rho_l(x)$. Therefore, by Lemma \[collannule\], we infer that the function $f_j$ cancels the collision operator. We deduce that $f_j$ is a stationary solution of .
Furthermore, since the supports of the $(f_j)_{j \in J}$ are disjoint, $(f_j)_{j \in J}$ is an orthonormal family of ${\mathcal{L}}^2$.
Finally, let $\varphi$ be a stationary solution of . Then $\varphi$ satisfies $$v\cdot {{\nabla}}_x \varphi - {{\nabla}}_x V \cdot {{\nabla}}_v \varphi = C(\varphi).$$ Taking the ${\mathcal{L}}^2$ scalar product with $\varphi$, we deduce that $D(\varphi)= \langle C(\varphi), \varphi \rangle_{{\mathcal{L}}^2}=0$, so that by Lemma \[collannule\], ${C}(\varphi)=0$. Then, with the same analysis as the proof of $(ii.) \implies (i.)$ in Theorem \[thmconv-intro\], we deduce that $\frac{e^{V}}{{\mathcal{M}}}\varphi$ is constant on each $U_j$. Using the fact that $\omega$ satisfies a.e.i.t. GCC, we deduce that we can write $$\varphi= \sum_{j\in J} \lambda_j \mathds{1}_{U_j} e^{-V} {\mathcal{M}}(v), \quad \lambda_j \in {{\mathbb R}},$$ that is $$\varphi= \sum_{j\in J} \langle \varphi, f_j \rangle_{{\mathcal{L}}^2} \, f_j,$$ and this concludes the proof.
### Necessity of the a.e.i.t. Geometric Control Condition {#para2}
We prove here that (iv.) implies (ii.).
We argue by contradiction. Assume that the a.e.i.t. Geometric Control Condition does not hold. Then ${\mathcal{A}}:= {{\mathbb T}}^d \times {{\mathbb R}}^d \setminus \bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)$ has positive Lebesgue measure.
We set $$f_0 (x,v) = \Psi(x) \mathds{1}_{{\mathcal{A}}} (x,v) \, e^{-V(x)} {\mathcal{M}}(v),$$ with $\Psi$ to be determined later. We define $$\label{f0psi}
f(t,x,v) := f_0 \circ \phi_{-t}(x,v) = \Psi \circ \phi_{-t}(x,v) \mathds{1}_{\phi_t({\mathcal{A}})}(x,v) \, e^{-V(x)} {\mathcal{M}}(v)$$ which satisfies, by construction, $$\label{f0psieq}
{{\partial}}_t f + v \cdot {{\nabla}}_x f - {{\nabla}}_x V \cdot {{\nabla}}_v f =0.$$ Note that for all $t\geq0$, we have $\phi_{t}({\mathcal{A}}) \cap \omega = \emptyset$ since ${\mathcal{A}}\cap \phi_{-t}(\omega) = \emptyset$, which yields $C(f(t)) = 0$ for all $t \geq 0$ and thus $f$ is also a solution of . We now fix $\Psi$ in order to ensure that $f(t)$ is not stationary.
- If $(v \cdot {{\nabla}}_x - {{\nabla}}_x V\cdot {{\nabla}}_v)(\mathds{1}_{{\mathcal{A}}}) \neq 0$, then we take $\Psi =1$.
- If $(v \cdot {{\nabla}}_x - {{\nabla}}_x V\cdot {{\nabla}}_v )(\mathds{1}_{{\mathcal{A}}}) = 0$, we take for $\Psi$ a Morse function on ${{\mathbb T}}^d$, so that, in particular, $\Psi$ is smooth and $\nabla \Psi(x) \neq 0$ for almost every $x \in {{\mathbb T}}^d$. Note that with such a function $\Psi$, we have $f_0 \in {\mathcal{L}}^2$. We compute $$\left[v \cdot {{\nabla}}_x \Psi - {{\nabla}}_x V \cdot {{\nabla}}_v \Psi\right] \mathds{1}_{{\mathcal{A}}} = (v \cdot {{\nabla}}_x \Psi(x) )\mathds{1}_{{\mathcal{A}}}(x,v).$$ Therefore for almost all $(x,v) \in {\mathcal{A}}$, this is not null, which shows that $f(t)$ is not stationary.
Finally if there existed a stationary solution $Pf_0$ of such that $$\| f(t)- Pf_0 \|_{{\mathcal{L}}^2} \to_{t \to + \infty} 0,$$ then since for all $t\geq 0$, $f(t)$ is supported in ${\mathcal{A}}$, we also have $P f_0$ supported in ${\mathcal{A}}$. Thus $P f_0$ cancels the collision operator, i.e. $C(Pf_0)=0$. We deduce that $f(t)- Pf_0$ satisfies $${{\partial}}_t (f -Pf_0)+ v \cdot {{\nabla}}_x (f -Pf_0) - {{\nabla}}_x V \cdot {{\nabla}}_v (f -Pf_0) =0 .$$ This yields for all $t\geq 0$, $$\| f(t) -Pf_0\|_{{\mathcal{L}}^2} = \| f_0 -Pf_0\|_{{\mathcal{L}}^2} .$$ Moreover, the solution $f$ defined in is a non-stationary solution of according to the definition of $\Psi$. In conclusion, we have $ f_0 \neq Pf_0$. This yields $\| f_0 -Pf_0\|_{{\mathcal{L}}^2} >0$ so that we cannot have $\| f(t)- Pf_0 \|_{{\mathcal{L}}^2} \to_{t \to + \infty} 0$. This concludes the proof.
### End of the proof of Theorem \[thmconv-general\] {#para3}
We have the following key lemma.
\[lem-proj\] Let $f$ be a solution in $C^0_t({\mathcal{L}}^2)$ of . Then for all $j \in J$, $$\frac{d}{dt} \langle f, f_j\rangle_{{\mathcal{L}}^2} =\frac{1}{\| \mathds{1}_{U_j}e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} \frac{d}{dt} \int_{U_j} f \, dv dx =0.$$
Let $f$ be a solution in $C^0_t({\mathcal{L}}^2)$ of ; denote by $f_0$ its initial datum. Let $j \in J$. We take the ${\mathcal{L}}^2$ scalar product with $f_j$ in to obtain $$\begin{aligned}
\frac{d}{dt} \langle f, f_j\rangle_{{\mathcal{L}}^2} &= - \langle (v\cdot {{\nabla}}_x -{{\nabla}}_x V\cdot {{\nabla}}_v) f, f_j\rangle_{{\mathcal{L}}^2}
+ \langle C(f), f_j\rangle_{{\mathcal{L}}^2} \\
&= \langle f, (v\cdot {{\nabla}}_x -{{\nabla}}_x V\cdot {{\nabla}}_v) f_j\rangle_{{\mathcal{L}}^2}
+ \langle f, C^*(f_j) \rangle_{{\mathcal{L}}^2}\end{aligned}$$ where $C^*$ is the collision operator of collision kernel defined by $$C^*(g) = \int_{{{\mathbb R}}^d} \left[\tilde{k}(x,v,v') {\mathcal{M}}(v) g(v') - \tilde{k}(x,v',v) {\mathcal{M}}(v') g(v)\right] \, dv'$$ with $\tilde{k}(x,v,v') = \frac{k(x,v,v')}{{\mathcal{M}}(v')}$. This follows from Property [**A2**]{} satisfied by $k$.
We then use the following two facts.
[**1.**]{} We have $(v\cdot {{\nabla}}_x -{{\nabla}}_x V\cdot {{\nabla}}_v) f_j =0$ (see the proof of Proposition \[cardinal\]).
[**2.**]{} We have $C^*(f_j)=0$. This follows from Property [**A2**]{}, the fact that $C(f_j)= 0$ (see again the proof of Proposition \[cardinal\]) and Lemma \[collannule\].
We conclude that $\frac{d}{dt} \langle f, f_j\rangle_{{\mathcal{L}}^2}=0$.
We therefore infer that if $f(t)$ satisfies with an initial datum $f_0$, then for all $t\geq 0$, $$\int_{U_j} f(t) \, dv \, dx = \int_{U_j} f_0 \, dv \, dx$$ Equipped with this result, we prove the equivalence between (i.)–(ii.)–(iii.) exactly as for Theorem \[thmconv-intro\], with only minor adaptations.
Application to particular classes of collision kernels {#secexamples}
======================================================
In this section, we introduce the following classes of collision kernels to illustrate the main results of the previous sections. We then draw consequences of the additional assumptions made in these examples.
[**E3.**]{} Let $k$ be a collision kernel verifying [**A1**]{}–[**A3**]{}. Let $\omega$ be the set where collisions are effective, defined in . We moreover require that for all $(x,v), (x,v') \in \omega$, there exist $N \in {{\mathbb N}}^*$ and a “chain” $v_1,\cdots, v_N \in {{\mathbb R}}^d$ such that the following hold.
- For all $i$, $1\leq i \leq N$, $(x,v_i) \in \omega$.
- The points $(x,v)$ and $(x,v_1)$ belong to the same connected component of $\omega$.
- The points $(x,v')$ and $(x,v_N)$ belong to the same connected component of $\omega$.
- For all $i$, $1\leq i \leq N-1$, we have $$k(x,v_i,v_{i+1}) > 0 \text{ or } k(x,v_{i+1},v_i) > 0.$$
As a subclass of [**E3**]{}, we have
[**E3’.**]{} Let $k$ be a collision kernel verifying [**A1**]{}–[**A3**]{}. We require that for all $y \in p_x(\omega)$ (where $p_x(\omega)$ is the projection of $\omega$ on ${{\mathbb T}}^d$), the set $p_x^{-1}( \{y\})$ is included in one single connected component of $\omega$.
A trivial subclass of [**E3’**]{} is the case where $\omega$ is connected. Another subclass of [**E3’**]{} is given in the following example.
[**E3”.**]{} Let $k$ be a collision kernel verifying [**A1**]{}–[**A3**]{}. We require that $$\omega = \omega_x \times {{\mathbb R}}^d,$$ where $\omega_x$ is an open subset of ${{\mathbb T}}^d$.
Remark that [**E2**]{} is a subclass of [**E3”**]{}.
In what follows, we explain the interest of these classes of collision kernels regarding the geometric definitions introduced before.
The case of collision kernels in the class [**E3**]{}
-----------------------------------------------------
The interest of [**E3**]{} lies in the simple description of the kernel of the associated collision operator $C$.
Using Lemma \[collannule\] and the “chain” in the definition of a collision kernel in [**E3**]{}, we deduce the following result.
\[lemE3\] Let $k$ be a collision kernel in the class [**E3**]{}. Let $T \in (0, +\infty]$ and assume that $f \in L^2(0,T; {\mathcal{L}}^2)$ satisfies ${C}(f(t))=0$ for almost every $t \in [0,T]$. Then there is a function $\rho \in L^2(0,T ;L^2({{\mathbb T}}^d))$ such that $$f=\rho (t,x){\mathcal{M}}(v) \quad \text{on }[0,T] \times \omega.$$ Reciprocally, any function $f$ satisfying this property satisfies $C(f) =0$.
In other words, the kernel of the associated collision operator is equal to the set of functions which are Maxwellians on $\omega$: $$\label{eqkerC}
\operatorname{Ker}(C) = \{ f \in {\mathcal{L}}^2 , f_{|\omega} = \rho(x){\mathcal{M}}(v) \} .$$ We recall that this property, which is usual in the non degenerate case $\omega= {{\mathbb T}}^d \times {{\mathbb R}}^d$, is not true in general for collision kernels satisfying merely [**A1**]{}, [**A2**]{} and [**A3**]{}.
This allows us to reformulate in a very simple way the Unique Continuation Property for collision kernels in [**E3**]{}.
\[def:UCP-E3\] Let $k$ be a collision kernel in the class [**E3**]{}. Then the set $\omega$ satisfies the Unique Continuation Property if and only if the only solution $f \in C^0_t({\mathcal{L}}^2)$ to $$\label{eq:UCP-E3}
\left\{
\begin{array}{l}
\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
f = \rho(t,x) {\mathcal{M}}(v) \text{ on } {{\mathbb R}}^+ \times \omega ,
\end{array}
\right.$$ is $f= \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx\right) \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v)$.
Conversely, using again Lemma \[collannule\], we have the following result.
Let $k$ be a collision kernel satisfying [**A1**]{}–[**A3**]{}. If any function $f \in {\mathcal{L}}^2$ canceling the collision operator has its restriction to $\omega$ satisfying $$f_{\vert \omega} = \mathds{1}_\omega \rho(x) {\mathcal{M}}(v)$$ for some $\rho \in L^2({{\mathbb T}}^d)$, then $k$ necessarily belongs to the class [**E3**]{}.
Therefore, [**E3**]{} is the largest class of collision kernels such that the kernel of the associated collision operator is equal to the set of functions which are Maxwellians on $\omega$, i.e. for which holds.
The case of collision kernels in the class [**E3’**]{} {#secE3'}
------------------------------------------------------
To explain the interest of [**E3’**]{}, let us introduce now another geometric condition:
1. The set $\omega$ satisfies the a.e.i.t. GCC and $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ is connected.
This condition is compared to other geometric conditions in Appendix \[appaeitgccconnected\]. It has to be confronted to the geometric condition of Theorem \[thmconv-intro\] (rephrased using Lemma \[equiv-equiv\]):
1. The set $\omega$ satisfies the a.e.i.t. GCC and there is only one equivalence class for $\sim$.
In what follows, we shall adopt the notations of Theorem \[thmconv-intro\]. It is clear that $(iv.)$ implies $(ii.)$, as $(iv.)$ means that there is a single equivalence class for an equivalence relation defined as in Definition \[def-sim-o\] with ${\, \mathcal{R} }_\phi$ only. However, it is false in general that $(i.)$–$(iii.)$ and $(iv.)$ are equivalent; see Proposition \[prop-cex-connected\] below for an example of collision kernel such that $(i.)$ is satisfied, but not $(iv.)$.
\[prop-cex-connected\] For $V=0$, there exists a collision kernel $k$ in the class [**E1**]{} and [**E3**]{}, for which $(ii)$ holds, but not $(iv)$.
Consider $(x,v) \in {{\mathbb T}}\times {{\mathbb R}}$ (a similar construction in higher dimension could also be performed). We take a function $\varphi \in C^0({{\mathbb R}})$, such that $\varphi(0)=0$ and $\varphi(v) >0$ for all $v \in {{\mathbb R}}\setminus \{0\}$. Define $$k(x,v',v)= {\mathcal{M}}(v) \varphi(v) \varphi(v').$$ By construction, [**A1**]{} and [**A3**]{} are satisfied and we notice that $\tilde{k}$ is symmetric (so that ${k}$ is in [**E1**]{}). We can also readily check that $k$ is in [**E3**]{} (with $N=2$).
We have $$\omega := \{{{\mathbb T}}\times {{\mathbb R}}^-_*\} \cup \{{{\mathbb T}}\times {{\mathbb R}}^+_*\}$$ and so $$\bigcup_{t\geq 0} \phi_{-t}(\omega) =\{ {{\mathbb T}}\times {{\mathbb R}}^-_*\} \cup \{{{\mathbb T}}\times {{\mathbb R}}^+_*\},$$ from which we deduce that a.e.i.t. GCC is satisfied, but $\cup_{t\geq 0} \phi_{-t}(\omega)$ is not connected.
On the other hand, the set $\omega$ satisfies the unique continuation property. Indeed, let $f$ satisfying $$\label{f-trans-eq}
{{\partial}}_t f + v \cdot {{\nabla}}_x f =0, \quad \forall (x,v) \in {{\mathbb T}}\times {{\mathbb R}}$$ and $C(f) =0$. Using Lemma \[lemE3\] and the definition of $k$, we deduce that $f= \rho(t,x) {\mathcal{M}}(v)$ on ${{\mathbb R}}^+ \times {{\mathbb T}}\times {{\mathbb R}}\setminus \{0\}$, and thus almost everywhere in ${{\mathbb R}}^+ \times {{\mathbb T}}\times {{\mathbb R}}$.
As $f$ satisfies the transport equation , this implies that $f = C {\mathcal{M}}(v)$ for some $C>0$ and we conclude that the unique continuation property holds.
Therefore, by Theorem \[thmconv-intro\], we deduce that $(ii.)$ holds.
However, when restricting to collision kernels in the class [**E3’**]{}, (ii.) and (iv.) become equivalent.
\[prop-E3’\] Let $k$ be a collision kernel in the class [**E3’**]{}. Then $(i.)$–$(iv.)$ are equivalent.
We consider $k$ a collision kernel in the class [**E3’**]{}. Assume that $(ii.)$ holds. The aim is to prove that $(iv.)$ holds. By contradiction, assume that there are at least two connected components $\Omega_1$, $\Omega_2$ of $\bigcup_{t\geq 0} \phi_{-t}(\omega)$.
By $(ii.)$, $\Omega_1$ and $\Omega_2$ belong to the same equivalence class for $\sim$. Thus, there exist $x,v_1,v_2$ with $(x,v_1) \in \Omega_1$, $(x,v_2) \in \Omega_2$ and $$k(x,v_1,v_2)>0 \text{ or } k(x,v_2,v_1)>0.$$ But since $k$ is in the class [**E3’**]{}, the set $p_x^{-1}( \{x\})$ is included in one connected component of $\omega$. Thus we can not have $(x,v_1) \in \Omega_1$ and $(x,v_2) \in \Omega_2$. This is a contradiction and this concludes the proof.
More generally, we observe that for collision kernels in [**E3’**]{}, the equivalence classes for $\sim$ are exactly the connected components of $\bigcup_{t\geq 0} \phi_{-t}(\omega)$. Thus Theorem \[thmconv-general\] can be reformulated as follows.
\[thmconv-general-E3’\] Let $k$ be a collision kernel in the class [**E3’**]{}. The following statements are equivalent.
1. The set $\omega$ satisfies the [generalized]{} Unique Continuation Property.
2. The set $\omega$ satisfies the a.e.i.t. GCC.
3. Let $(\Omega_i)_{i \in I}$ be the connected components of $\cup_{t\geq 0} \phi_{-t}(\omega)$. For all $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\label{convergeto0-general-E3'}
\left\|f(t)-P f_0\right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$ where $$P f_0 = \sum_{i\in I} \frac{1}{\| \mathds{1}_{\Omega_i} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} \left( \int_{\Omega_i} f_0 \, dv dx\right)g_j,$$ with for all $i \in I$, $$g_i = \frac{\mathds{1}_{\Omega_i} e^{-V}{\mathcal{M}}}{\| \mathds{1}_{\Omega_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}}.$$
We close this section by exhibiting an example of collision kernel in [**E3’**]{}, for which Corollary \[thmconv-general-E3’\] (and thus Theorem \[thmconv-general\]) are relevant.
We restrict ourselves to the case ${{\mathbb T}}\times {{\mathbb R}}$ (this can be easily adapted to higher dimensions). We consider the free transport case, i.e. $V=0$. We identify ${{\mathbb T}}$ to $[-1/2,1/2)$. Consider $\alpha \in C^0({{\mathbb T}})$ supported in $[-1/2,0)$ and $\beta \in C^0({{\mathbb T}})$ supported in $[0,1/2)$.
Let $\varphi \in L^\infty\cap C^0({{\mathbb R}})$ such that $\varphi>0$ on ${{\mathbb R}}^-_*$ and $\varphi =0$ on ${{\mathbb R}}^+$. Likewise, let $\Psi \in L^\infty\cap C^0({{\mathbb R}})$ such that $\Psi>0$ on ${{\mathbb R}}^+_*$ and $\Psi =0$ on ${{\mathbb R}}^-$. We define the collision kernel $$k(x,v,v') := \left[ \alpha(x) \varphi(v) \varphi(v') + \beta(x) \Psi(v) \Psi(v') \right] {\mathcal{M}}(v').$$ Note that $\tilde{k}(x,v,v') = \alpha(x) \varphi(v) \varphi(v') + \beta (x)\Psi(v) \Psi(v')$ is symmetric in $v$ and $v'$, and belongs to $L^\infty$. Thus $k$ is in the class [**E1**]{}. Furthermore, we readily check $k$ is in the class [**E3’**]{}.
Moreover, we have $\omega = \{\{\alpha>0\} \times {{\mathbb R}}^-_*\} \cup \{\{\beta>0\} \times {{\mathbb R}}^+_*\}$, and $$\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega) = \{{{\mathbb T}}\times {{\mathbb R}}^-_*\} \cup \{{{\mathbb T}}\times {{\mathbb R}}^+_*\}.$$ Thus $\omega$ satisfies the a.e.i.t. GCC but $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ is not connected.
The basis of the subspace of stationary solutions of is given by $(f_j)_{j=1,2}$ with $$f_1 = \frac{\mathds{1}_{{{\mathbb T}}\times {{\mathbb R}}^-_*} e^{-V}{\mathcal{M}}}{\| \mathds{1}_{{{\mathbb T}}\times {{\mathbb R}}^-_*} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}}, \quad
f_2 = \frac{\mathds{1}_{{{\mathbb T}}\times {{\mathbb R}}^+_*} e^{-V}{\mathcal{M}}}{\| \mathds{1}_{{{\mathbb T}}\times {{\mathbb R}}^+_*} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}}.$$
The case of collision kernels in the class [**E3”**]{} {#secE3pp}
------------------------------------------------------
We finally study collision kernels in the class [**E3”**]{}. Because of the remarkable properties of the geodesic flow on the torus ${{\mathbb T}}^d$, the following holds.
\[lemUCP1\] Suppose that $V = 0$ and that the collision kernel belongs to the class [**E3”**]{}. Then $\omega$ satisfies a.e.i.t. GCC.
Define $T_v^t : \, x \mapsto x + t \, v$; then $(T_v^t \, x)_{t \geq 0}$ is dense in ${{\mathbb T}}^d$ for almost every $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$ (with respect to the Lebesgue measure). This proves the lemma.
We deduce a proof of Proposition \[coroUCP\].
Take $\omega_x^0$ a connected component of $\omega_x$. According to Lemma \[lemUCP1\], $\omega_x^0 \times {{\mathbb R}}^d$ satisfies a.e.i.t. GCC. The result then follows from Proposition \[aeitgccconnected\], $(i.) \Rightarrow (iii.)$, and Theorem \[thmconv-intro\].
Characterization of exponential convergence {#expocon}
===========================================
Let us first briefly recall why $C^-(\infty)$ is well-defined (see [@Leb]). We can first define for all $T>0$, $$C^-(T) := \text{inf}_{(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d} \frac{1}{T} \int_0^T \left( \int_{{{\mathbb R}}^d} k(\phi_t (x,v), v')\, dv'\right)\, dt,$$ which is well-defined (and nonnegative) since $k$ is non-negative. We then remark that the function $T \mapsto - T C^-(T)$ is subadditive. This entails that $C^-(\infty) = \lim_{T \to +\infty} C^-(T)$ exists (one proves similarly that $C^+(\infty)$ is well defined).
In this section, we assume that $k$ satisfies [**A3’**]{} and provide the proof of Theorem \[thmexpo-intro\]. To this end, we first prove that (a.) and (b.) are equivalent, and finally that (c.) implies (a.). This will conclude the proof, noticing that (b.) implies (c.) is straightforward. One can also readily check that in the proof of (a.) implies (b.), the assumption [**A3’**]{} is not used.
Proof of the equivalence between (a.) and (b.)
----------------------------------------------
Since the equation is linear, if $f(t)$ satisfies then $$g(t) := f(t)-\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f(0) \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v),$$ is still a solution to . Thus we can deal only with initial data which have zero mean.
By conservation of the mass, the Boltzmann equation is well-posed in the space $${\mathcal{L}}^2_0:= \left\{f \in {\mathcal{L}}^2, \, \int f \, dv dx =0\right\}$$ and we can use Lemma \[lemfondamental\] for solutions in ${\mathcal{L}}^2_0$, which yields that $(b.)$ is equivalent to
1. There exists $T>0$ and $K>0$ such that for all $f_0 \in {\mathcal{L}}^2_0$, the associated solution $f$ to satisfies $$K \int_0^T D(f(t)) \, dt \geq \| f_0\|_{{\mathcal{L}}^2}^2.$$
We first prove that $(a.)$ implies $(b'.)$, then that $(b'.)$ implies $(a.)$
${(a.) \implies (b'.)}$ Assume that $(a.)$ holds.
We argue by contradiction. Denying $(b'.)$ is equivalent to assume for all $T>0$ and all $C>0$, the existence of $g_0^{C,T} \in {\mathcal{L}}^2_0$, such that $$C \int_0^T D(g^{C,T}(t)) \, dt < \| g_0^{C,T}\|_{{\mathcal{L}}^2}^2,$$ where $g^{C,T}(t)$ is the unique solution to with initial datum $g_0^{C,T}$. For all $n \in {{\mathbb N}}^*$, there exists $g_{0,n} \in {\mathcal{L}}^2_0$, such that $$\label{dissipgn}
\int_0^n D(g_n(t)) \, dt < \frac{1}{n} \| g_{0,n}\|_{{\mathcal{L}}^2}^2.$$ where $g_n(t)$ is the unique solution to with initial datum $g_{0,n}$. Furthermore, by linearity of , we can normalize the initial data so that for all $n \in {{\mathbb N}}^*$, $$\label{normalization}
\| g_{0,n}\|_{{\mathcal{L}}^2} =1.$$ Recall that by Lemma \[lemdissip\], we have, for all $t\geq0$, $$\label{eqdiss}
\| g_n(t) \|^2_{{\mathcal{L}}^2} - \| g_{0,n} \|^2_{{\mathcal{L}}^2} = - \int_0^t D(g_n(s)) \, ds.$$ In particular, the sequence $(g_n)_{n \in {{\mathbb N}}^*}$ is uniformly bounded in $L^\infty_t {\mathcal{L}}^2$; thus, up to some extraction, we can assume that $g_n \rightharpoonup g$ weakly in $L^2_t {\mathcal{L}}^2$. Let us prove that $g=0$. By linearity of , $g$ still satisfies since $g_n$ does.
Note also that by conservation of the mass, for all $n \in {{\mathbb N}}$ and all $t \geq 0$, $$\label{eqconsmas2}
\int g_n(t) \, dv dx = \int g_{0,n} \, dv dx =0.$$
Take $T'>0$ such that $(\omega,T')$ satisfies GCC. By , we have $\int_0^{T'} D(g_n(t)) \, dt \to 0$ and therefore by weak lower semi-continuity, we deduce $$\left\|{D}(g) \right\|_{L^1(0,T')} \leq \liminf_{n\to +\infty} \left\|{D}(g_n) \right\|_{L^1(0,T')}= 0.$$ As a consequence, by weak coercivity (see Lemma \[collannule\]), we infer that ${C}(h)=0$ on $[0,T']$, and therefore $h$ satisfies the kinetic transport equation . The Unique Continuation Property of Proposition \[remUCP\] then implies that $$g= \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} g \, dv \,dx \frac{e^{-V}}{\int_{{{\mathbb T}}^d} e^{-V} \, dx} {\mathcal{M}}(v).$$ Since $g_{n} \rightharpoonup g$ weakly in $L^2_t {\mathcal{L}}^2$, using , we obtain in particular that $$\int_0^{T'} \left(\int g \, dv dx \right) dt =0.$$ Since $\int_0^{T'} \left(\int g \, dv dx \right) dt = T' \left(\int g(0) \, dv dx \right)$, we deduce that $\int g \, dv dx=0$, so that $g=0$. Therefore, this leads to $g=0$.
Now, let us study the sequence of defect measures $\nu_n := |g_n|^2$ and $\nu_{0,n} := |g_{0,n}|^2$. Consider the equation satisfied by $g_n$ and multiply it by $g_n$ to get: $$\begin{aligned}
\label{eq:nun}
\partial_t \nu_n &+ v \cdot \nabla_x \nu_n - \nabla_x V \cdot \nabla_v \nu_n \\ \nonumber
&= 2 \left(\int_{{{\mathbb R}}^d} k(x,v' , v) g_n(v') \, dv'\right) g_n - 2 \left( \int_{{{\mathbb R}}^d} k(x,v , v')\, dv' \right) \nu_n
\end{aligned}$$ By Duhamel’s formula, we infer $$\begin{gathered}
\nu_n(t,x,v)= e^{-2 \int_0^t \int_{{{\mathbb R}}^d} k(\phi_{s-t}(x,v) , v')\, dv' \, ds } \nu_{0,n} (\phi_{-t} (x,v)) \\
+ \int_0^t 2 \left(\int_{{{\mathbb R}}^d} k(X_{s-t}(x,v),v' , \Xi_{s-t}(x,v)) g_n(s,X_{s-t}(x,v),v') \, dv'\right) g_n(s, \phi_{s-t}(x,v)) \\
\times e^{-2 \int_s^t \int_{{{\mathbb R}}^d} k(\phi_{\tau-t}(x,v) , v')\, dv' \, d\tau } \, ds ,
\end{gathered}$$ and thus for all $t\geq 0$, $$\begin{gathered}
\label{duhamelnun}
\| \nu_n(t)\|_{{\mathcal{L}}^1} \leq \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} e^{-2 \int_0^t \int_{{{\mathbb R}}^d} k(\phi_{s-t}(x,v) , v')\, dv' \, ds } \nu_{0,n} (\phi_{-t}(x,v)) \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx \\
+\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \Bigg( \int_0^t 2 \left|\int_{{{\mathbb R}}^d} k(X_{s-t}(x,v),v' , \Xi_{s-t}(x,v)) g_n(s,X_{s-t}(x,v),v') \, dv'\right| |g_n(s, \phi_{s-t}(x,v))| \\
\times e^{-2 \int_s^t \int_{{{\mathbb R}}^d} k(\phi_{\tau-t}(x,v) , v')\, dv' \, d\tau } \, ds \Bigg)\frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx
\end{gathered}$$
By definition of $C^-(\infty)$, there exists $T_0>0$ large enough such that for all $t\geq T_0, C^-(t)\geq C^-(\infty)/2>0$. We have, after the change of variables $\phi_{-t}(x,v) \mapsto (x,v)$, which has unit Jacobian (recall also that the hamiltonian is left invariant by this transform), for all $t\geq T_0$, $$\begin{aligned}
&\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} e^{-2 \int_0^t \int_{{{\mathbb R}}^d} k(\phi_{s-t}(x,v) , v')\, dv' \, ds } \nu_{0,n} (\phi_{-t} (x,v)) \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx \\
&=\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} e^{-2 \int_0^t \int_{{{\mathbb R}}^d} k(\phi_{s}(x,v) , v')\, dv' \, ds } \nu_{0,n} (x,v) \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx\\
& \leq e^{-t C^-(t)} \| \nu_{0,n} \|_{{\mathcal{L}}^1} \leq e^{-t C^-(\infty)/2} \| \nu_{0,n} \|_{{\mathcal{L}}^1}
\end{aligned}$$ and thus we can choose $T_1 \geq T_0$ large enough such that the left-handside is less than $\frac{1}{4} \| \nu_{0,n} \|_{{\mathcal{L}}^1}$ for $t=T_1$.
On the other hand, since $g_n \rightharpoonup 0$, by the averaging lemma of Corollary \[lemmoyenne\], we deduce that in $L^2(0,T_1;{\mathcal{L}}^2)$, $$\label{eq-compact}
\left(\int_{{{\mathbb R}}^d} k(x,v' , v) g_n(v') \, dv'\right) \to 0 .$$ Hence, by the weak/strong convergence principle, $$\left(\int_{{{\mathbb R}}^d} k(x,v' , v) g_n(v') \, dv'\right) g_n \to 0,$$ in $L^1(0,T_1;{\mathcal{L}}^1)$. Therefore, using the change of variables $\phi_{s-t}(x,v) \mapsto (x,v)$, which has unit Jacobian, we infer that $$\begin{gathered}
\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \Bigg( \int_0^t 2 \left|\int_{{{\mathbb R}}^d} k(X_{s-t}(x,v),v' , \Xi_{s-t}(x,v)) g_n(s,X_{s-t}(x,v),v') \, dv'\right| |g_n(s, \phi_{s-t}(x,v))| \\
\times e^{-2 \int_s^t \int_{{{\mathbb R}}^d} k(\phi_{\tau-t}(x,v) , v')\, dv' \, d\tau } \, ds \Bigg)\frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx \to 0\end{gathered}$$ and thus, coming back to , for $n$ large enough, we finally obtain (we recall that $ \| \nu_{0,n} \|_{{\mathcal{L}}^1}^2 =1$) $$\label{eq:beforecontract}
\| \nu_n(T_1)\|_{{\mathcal{L}}^1}^2 \leq \frac{1}{2} .$$
But integrating with respect to time and using -, we also have $$\begin{aligned}
\| \nu_n({T_1} )\|_{{\mathcal{L}}^1}^2 &= 1 - \int_0^{T_1} D(g_n(s)) \, ds \\
&\geq \frac{3}{4} \text{ for } n \text{ large enough},
\end{aligned}$$ which is a contradiction with .
${(b'.) \implies (a.)}$ We show that if $(a.)$ does not hold (i.e. $C^-(\infty)=0$), then $(b'.)$ does not either. Assume that $(a.)$ does not hold. The goal is to show that for all $T>0$, for all ${{\varepsilon}}>0$, there exists $g_{0,{{\varepsilon}}} \in {\mathcal{L}}^2_0$ such that $$\label{nonii}
\| g_{0,{{\varepsilon}}} \|_{{\mathcal{L}}^2} =1, \quad \int_0^T {D}(g_{{\varepsilon}})(t) \, dt < {{\varepsilon}},$$ where $g_{{\varepsilon}}$ is the solution to with initial datum $g_{0,{{\varepsilon}}}$.
Fix $T>0$ and ${{\varepsilon}}>0$. Since $C^-(\infty)=0$, there exists $(x_0,v_0) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$, such that $$\label{hyp}
\int_0^T \int_{{{\mathbb R}}^d} k(\phi_t(x_0,v_0), v') \, dv' \,dt < {{\varepsilon}}/2.$$
Let $\chi$ be a smooth compact cutoff function defined from ${{\mathbb R}}^+$ to ${{\mathbb R}}$ such that $\chi \equiv 1$ on $[0,1]$ and $\chi \equiv 0$ on $[2,\infty)$ and such that $\int_{{{\mathbb R}}^+} \chi(r) r^{d-1} \, dr =0$. Consider $$\tilde{g}_{0,n} = \chi(n|x-x_0|) \chi(n|v-v_0|).$$ Then notice that there is $\alpha>0$ such that $$\|\tilde{g}_{0,n} \|_{{\mathcal{L}}^2}^2 = n^{-2d} \alpha.$$ There, in order to normalize, we take $g_{0,n} := \frac{n^{d}}{\alpha} \tilde{g}_{0,n}$. Note that by construction, $$\int g_{0,n} \, dv dx = \frac{n^{d}}{\alpha} \left(\int \chi(n|x-x_0|) \, dx\right) \left(\int \chi(n|v-v_0|) \, dv\right) =0,$$ and thus $g_{0,n} \in {\mathcal{L}}^2_0$.
We call $g_n$ the solution to with initial datum $g_{0,n}$. By construction, we observe that $g_{0,n} \rightharpoonup 0$ weakly in ${\mathcal{L}}^2$ and we deduce that $g_n\rightharpoonup 0$ weakly in $L^2_{t,loc} {\mathcal{L}}^2$. As in the previous proofs, by the averaging lemma of Corollary \[lemmoyenne\], this implies that $$\label{convforte}
\int k(x,v',v) g_n(t,x,v') \, dv' \to 0, \text{ strongly in } L^2_{t,loc} {\mathcal{L}}^2.$$
Now, consider $\nu_n := |g_n|^2$. By construction, we have: $$\label{eq:nu0n}
\nu_n(0) \rightharpoonup \delta_{x=x_0,v=v_0},$$ where $\delta$ denotes as usual the Dirac measure. As in , we get the Duhamel’s formula $$\begin{gathered}
\nu_n(t,x,v)= e^{-2 \int_0^t \int_{{{\mathbb R}}^d} k(\phi_{s-t}(x,v) , v')\, dv' \, ds } \nu_{0,n} (\phi_{-t} (x,v)) \\
+ \int_0^t 2 \left(\int_{{{\mathbb R}}^d} k(X_{s-t}(x,v),v' , \Xi_{s-t}(x,v)) g_n(s,X_{s-t}(x,v),v') \, dv'\right) g_n(s, \phi_{s-t}(x,v)) \\
\times e^{-2 \int_s^t \int_{{{\mathbb R}}^d} k(\phi_{\tau-t}(x,v) , v')\, dv' \, d\tau } \, ds.
\end{gathered}$$
Define now the weighted $L^2$ norm as follows $$\| f \|_{{\mathbb{L}}^2}^2: = \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} |f|^2 \varphi^2(x,v) \frac{e^{V}}{{\mathcal{M}}} \, dvdx .$$ We have the following ${\mathcal{L}}^2$ estimate for the Boltzmann equation .
\[lemtech\] For any function $f_0 \in {\mathcal{L}}^2$ with $\| f_0 \|_{{\mathbb{L}}^2}<+\infty$, the solution $f(t)$ to the Boltzmann equation with initial datum $f_0$ satisfies, for all $t\geq 0$, $$\label{tech}
\| f(t) \|_{{\mathbb{L}}^2}^2 \leq \| f_0 \|_{{\mathbb{L}}^2}^2 + (1+ \Gamma) \int_0^t \| f(s) \|_{{\mathbb{L}}^2}^2 \, ds,$$ where $$\Gamma:= \sup_{x \in {{\mathbb T}}^d} \int_{{{\mathbb R}}^d \times {{\mathbb R}}^d} k^2(x,vÕ,v) \frac{{\mathcal{M}}(vÕ)}{{\mathcal{M}}(v)} \left(\frac{\varphi(x,v)}{\varphi(x,v')}-1\right)^2\, dv dvÕ ,$$ which is finite by [**A3’**]{}.
The proof follows from an energy estimate for . We first multiply by $f \, \varphi^2(x,v) \frac{e^{V}}{{\mathcal{M}}}$. Recalling that $\varphi$ is a function of the hamiltonian, we can integrate and argue as in Lemma \[lemdissip\] to treat the terms coming from the collision operator to obtain: $$\begin{aligned}
\frac{1}{2} \frac{d}{dt} \| f(t) \|_{{\mathbb{L}}^2}^2 + 0 &= \int_{{{\mathbb T}}^d\times {{\mathbb R}}^d} \left( \int_{ {{\mathbb R}}^d} k(x,v',v) \left(1-\frac{\varphi(x,v')}{\varphi(x,v)}\right) f(t,x,v') \, dv' \right) f \, \varphi^2(x,v) \frac{e^{V}}{{\mathcal{M}}} \, dv dx \\
& \qquad + \langle C(f\varphi), f\varphi\rangle_{{\mathcal{L}}^2} \\
&= \int_{{{\mathbb T}}^d\times {{\mathbb R}}^d} \left( \int_{ {{\mathbb R}}^d} k(x,v',v) \left(1-\frac{\varphi(x,v')}{\varphi(x,v)}\right) f(t,x,v') \, dv' \right) f \, \varphi^2(x,v) \frac{e^{V}}{{\mathcal{M}}} \, dv dx - \frac{1}{2} D(f \varphi) \\
&\leq \frac{1}{2} \int_{{{\mathbb T}}^d\times {{\mathbb R}}^d} \left( \int_{ {{\mathbb R}}^d} k(x,v',v) \left(1-\frac{\varphi(x,v')}{\varphi(x,v)}\right) f(t,x,v') \, dv' \right)^2 \, \varphi^2(x,v) \frac{e^{V}}{{\mathcal{M}}} \, dv dx \\
& \qquad + \frac{1}{2} \| f(t) \|_{{\mathbb{L}}^2}^2 .
\end{aligned}$$ Above we have used that the dissipation term $D(f \varphi)$ is non-negative. Using the Cauchy-Schwarz inequality, we thus deduce $$\begin{aligned}
\frac{1}{2} \frac{d}{dt} \| f(t) \|_{{\mathbb{L}}^2}^2 \leq \frac{1}{2}\Gamma \| f(t) \|_{{\mathbb{L}}^2}^2 + \frac{1}{2} \| f(t) \|_{{\mathbb{L}}^2}^2,
\end{aligned}$$ which yields .
By construction of the sequence $(g_{0,n})$, it is uniformly compactly supported and we observe that there exists $C_0>0$ such that for all $n \in {{\mathbb N}}$, $$\| g_{0,n}\|_{{\mathbb{L}}^2} \leq C_0.$$ We thus use Lemma \[lemtech\] to infer that there exists $C_T>0$, such that for all $n \in {{\mathbb N}}$, for all $t \in [0,T]$, $$\label{amelior}
\| g_n(t)\|_{{\mathbb{L}}^2} \leq C_T.$$ We now study the term $$\begin{gathered}
A_n:= \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \int_0^t 2 \left(\int_{{{\mathbb R}}^d} k(X_{s-t}(x,v),v' , \Xi_{s-t}(x,v)) g_n(s,X_{s-t}(x,v),v') \, dv'\right) \\
\times e^{-2 \int_s^t \int_{{{\mathbb R}}^d} k(\phi_{\tau-t}(x,v) , v')\, dv' \, d\tau} g_n(s, \phi_{s-t}(x,v))
\int_{{{\mathbb R}}^d} k(x,v, v') \, dv' \frac{e^{V}}{{\mathcal{M}}} \, ds \, dvdx\end{gathered}$$ We notice that since $\varphi$ is a function of the hamiltonian, we have $$\int_{{{\mathbb R}}^d} k(x,v, v') \, dv' \leq \varphi(x,v)= \varphi(\phi_{s-t}(x,v)).$$ Therefore, using the change of variables $\phi_{s-t}(x,v) \mapsto (x,v)$ and the Cauchy-Schwarz inequality, we obtain $$A_n \leq C \left\|\int_{{{\mathbb R}}^d} k(x,v' ,v) g_n(s,x,v') \, dv'\right\|_{L^1([0,t]; {\mathcal{L}}^2)} \sup_{[0,t]} \|g_n \|_{{\mathbb{L}}^2}.$$ By and , we deduce that $\|A_n \|_{L^1[0,T]} \to 0$ as $n \to + \infty$.
Consequently, by , we get $$\begin{aligned}
& \int_0^T \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu_n(t,x,v) \int_{{{\mathbb R}}^d} k(x,v, v') \, dv' \frac{e^{V(x)}}{{\mathcal{M}}(v)}dx \, dv \, dt\\
&= \int_0^T \exp\left( -2 \int_0^t \int_{{{\mathbb R}}^d} k(\phi_s(x_0,v_0),v') \, dv' \, ds\right) \int_{{{\mathbb R}}^d} k(\phi_t(x_0,v_0), v') \, dv' \, dt + \|A_n \|_{L^1[0,T]} \\
&<{{\varepsilon}}/2,\end{aligned}$$ for $n$ large enough. The last inequality follows by definition of $(x_0,v_0)$ and by the assumption . But by definition of $D(g_n)$, and using again , for $n$ large enough, we thus have $$\int_0^T {D}(g_n)(t) \, dt = 2 \int_0^T \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \nu_n(t,x,v) \int_{{{\mathbb R}}^d} k(x,v, v') \, dv' \frac{e^{V(x)}}{{\mathcal{M}}(v)}dx \, dv \, dt + o_{n\to +\infty}(1)<{{\varepsilon}},$$ We finally take $g_{0,{{\varepsilon}}} := g_{0,n}$ (with $n$ large enough), which satisfies . This concludes the proof.
About the rigidity with respect to exponential convergence of the Maxwellian
----------------------------------------------------------------------------
We prove here that (c.) implies (a.) in Theorem \[thmexpo-intro\].
Assume that (c.) holds. By (1) implies (2) in Theorem \[thmconvgene-intro\], $\omega$ satisfies the a.e.i.t. GCC. Therefore, by Theorem \[thmconv-general\], this means that $Pf_0$ is of the form defined in . We use these notations again.
Recall by Lemma \[lem-proj\] that given an equivalence class $[\Omega_j]$ for $\sim$, denoting as usual $U_j= \bigcup_{\Omega' \in [\Omega_j]} \Omega'$, we have for all $t\geq 0$, $$\int_{U_j} f(t) \, dv \, dx = \int_{U_j} f_0 \, dv \, dx$$ where $f(t)$ is the solution of with initial condition $f_0$.
Thus, the linear Boltzmann equation is well-posed in the space $${\mathcal{L}}^2_{00}:= \left\{f \in {\mathcal{L}}^2, \, \forall j \in J, \, \int_{U_j} f \, dv dx =0\right\},$$ and we can use Lemma \[lemfondamental\] for solutions in ${\mathcal{L}}^2_{00}$, which yields that the exponential convergence property is equivalent to
1. There exists $T>0$ and $K>0$ such that for all $f_0 \in {\mathcal{L}}^2_{00}$, the associated solution $f$ to satisfies $$K \int_0^T D(f(t)) \, dt \geq \| f_0\|_{{\mathcal{L}}^2}^2.$$
We can then make the same proof as $(b'.) \implies (a.)$ in Theorem \[thmexpo-intro\] in order to conclude that $C^-(\infty)=0$. We keep the notations of that proof. The only thing to check is that $g_{0,n}$ defined there belongs to ${\mathcal{L}}^2_{00}$ for $n$ large enough. Let $j \in J$ such that $(x_0,v_0) \in U_j$. Then for $n$ large enough, ${{\rm supp}\,}g_{0,n} \subset U_j$. Thus for all $i \neq j$, we have $ \int_{U_i} g_{0,n}\, dv \, dx =0$ and $$\int_{U_j} g_{0,n} \, dv \, dx = \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} g_{0,n} \, dv \, dx =0,$$ by definition of $g_{0,n}$. Thus $g_{0,n} \in {\mathcal{L}}^2_{00}$ for $n$ large enough.
Remarks on lower bounds for convergence when $C^-(\infty)=0$ {#lowerbounds}
============================================================
In the situation where $\omega$ satisfies a.e.i.t. GCC but $C^-(\infty)=0$, we know by Theorem \[thmconv-general\] that for all data in ${\mathcal{L}}^2$ there is convergence to some $Pf_0$ (defined in ). It is natural to wonder if there is a uniform decay rate for smoother data (e.g. in the domain of the generator of the semigroup). If so, then the question of the convergence rate one can obtain becomes particularly interesting.
Let us provide here some [*a priori*]{} results in this direction. The following is nothing but a rephrasing in a general framework of a result of Bernard and Salvarani [@BS2]. Note that they consider in their work free transport ($V=0$) and velocities on the sphere ${{\mathbb S}}^{d-1}$, but one can readily check that their methods are relevant for . In their computations, one should add the weight $e^{V}/{\mathcal{M}}$ in the integrals.
\[thmabstract\] Denote $\tau(x,v) := \inf\{ t \geq 0, \phi_{-t}(x,v) \in \omega)\}$. Assume that there is a function of time $\varphi(t)$ such that $$\operatorname{Leb}\{ (x,v) \in {{\mathbb T}}^d\times {{\mathbb R}}^d, \, \tau(x,v) >t\} {\,\raisebox{-0.6ex}{$\buildrel > \over \sim$}\,}\varphi(t).$$ Then, there exists a non-negative initial datum $f_0$ of $C^\infty$ class and $C>0$ such that for any $t \geq 0$, denoting by $f(t)$ the solution of the linear Boltzmann equation with initial datum $f_0$, $$\left\|f(t) - P f_0\right\|_{{\mathcal{L}}^2} \geq C \varphi(t),$$ where $Pf_0$ is defined in .
In particular, we obtain
\[coro-lower-free\] Assume that $V=0$ and that $\overline{p_x(\omega)} \neq {{\mathbb T}}^d$, where $p_x$ denotes the projection on the space of positions. Then there exists a non-negative initial datum $f_0$ of $C^\infty$ class and $C>0$ such that for any $t \geq 0$, denoting by $f(t)$ the solution of the linear Boltzmann equation with initial datum $f_0$, $$\left\|f(t) - Pf_0\right\|_{{\mathcal{L}}^2} \geq C/(1+t)^{d/2},$$ where $Pf_0$ is defined in .
Take $x_0 \in {{\mathbb T}}^d \setminus \overline{p_x(\omega)}$. Let $\delta := \text{dist}(x_0, \overline{p_x(\omega)})$. Consider $U:=B(x_0,\delta/2)\times B(0,1)$; here $\tau(x,v) := \inf\{ t \geq 0, x-tv \in p_x(\omega)\}$. Then the crucial point is the straightforward lower bound $$\operatorname{Leb}\{ (x,v) \in B(x_0,\delta/2)\times B(0,1), \, \tau(x,v) >t\} {\,\raisebox{-0.6ex}{$\buildrel > \over \sim$}\,}1/(1+t)^{d/2}$$ and we can thus apply Theorem \[thmabstract\].
Combining with Bernard-Salvarani’s theorem which concerns the case with trapped trajectories [@BS2], that we recall below, one may deduce that the “worst” lower bound in the free transport case is due to trapped trajectories, and not to low velocities.
Let $k$ a collision kernel belonging to the class [**E3”**]{} and $V=0$. Assume that there is $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$ such that for all $t\in {{\mathbb R}}^+$, $x+tv \notin \omega_x$. Then there exists a non-negative initial datum $f_0$ of $C^\infty$ class and $C>0$ such that for any $t \geq 0$, denoting by $f(t)$ the solution of the linear Boltzmann equation with initial datum $f_0$, $$\left\|f(t) - \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx\right) {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \geq C/(1+t).$$
It is natural to conjecture that in this case, the bound in $1/t$ is optimal (this is supported by numerical evidence, as shown by De Vuyst and Salvarani [@DVS]).
\[Boundary\]
In this section, $\Omega$ is a bounded and piecewise $C^1$ domain of ${{\mathbb R}}^d$. For all $x \in \partial \Omega$ (except for a set of zero Lebesgue measure, referred to as $\mathcal{B}$ below), we can consider the outward unit normal to $\partial\Omega$ at the point $x$, denoted by $n(x)$. In what follows, we also denote by $d\Sigma(x)$ the standard surface measure on ${{\partial}}\Omega$. Consider the following partition of $\partial \Omega \times {{\mathbb R}}^d$: $$\left\{
\begin{aligned}
&\mathcal{B} = \left\{(x,v) \in \partial \Omega \times {{\mathbb R}}^d, \, n(x) \text{ is not well defined} \right\}, \\
&\Sigma_- = \left\{(x,v) \in \partial \Omega \times {{\mathbb R}}^d, \, v\cdot n(x) <0 \right\}, \\
& \Sigma_+= \left\{(x,v) \in \partial \Omega \times {{\mathbb R}}^d, \, v\cdot n(x) >0 \right\}, \\
&\Sigma_0 = \left\{(x,v) \in \partial \Omega \times {{\mathbb R}}^d, \, v\cdot n(x) =0 \right\}.
\end{aligned}
\right.$$
There are several relevant boundary conditions that can be considered for kinetic equations. In this paper, we focus only on the specular boundary condition case (maybe the most natural one). For this, let us first define the symmetry with respect to the tangent hyperplane to $\partial \Omega$ as $$\label{speculBC}
R_x v = v - 2 (v \cdot n(x)) n(x), \quad (x,v) \in (\partial \Omega\times {{\mathbb R}}^d) \setminus \mathcal{B} .$$ Remark that for any point $(x,v) \in \Sigma_\pm$ the reflection $R_x$ associate the point $(x,R_x v) \in \Sigma_\mp$, and that $R_x R_x v = v$. The linear Boltzmann equation with specular boundary condition then reads as follows: $$\label{B-specular}
\left\{
\begin{aligned}
&\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv', \\
&f(t,x,R_x v) = f(t,x,v), \quad (x ,v) \in \partial \Omega \times {{\mathbb R}}^d.
\end{aligned}
\right.$$ Note that the boundary conditions used here translate the fact that the particles reflect against the boundary of $\Omega$ according to the laws of geometric optics.
In this equation, we still assume that $V \in W^{2,\infty}(\Omega)$.
We now revisit the results which were obtained previously in the torus case.
Characteristics, well-posedness and dissipation {#prelimbound}
===============================================
We start by defining the “broken characteristics” $(\phi_t)_{t \geq 0}$ which will allow us to express the different relevant geometric control conditions. Note first that the force $-\nabla_x V$ can be extended to $\overline \Omega$ by uniform continuity.
$\bullet$ Let $(x,v) \in \Omega \times {{\mathbb R}}^d$. For small enough values of $t \geq0$, we can consider the characteristics $\psi_t(x,v) := (X_t (x,v), \, \Xi_t(x,v))$, where $$\label{hamilflow-boun}
\left\{
\begin{aligned}
&\frac{dX_t(x,v)}{dt} = \Xi_t(x,v), \quad \frac{d\Xi_t(x,v)}{dt} = - \nabla_x V(X_t(x,v)), \\
&X_{t=0}=x, \quad \Xi_{t=0}=v.
\end{aligned}
\right.$$ We define $\tau(x,v) := \inf \{t \geq 0, X_t(x,v) \in \partial \Omega\}$. Note then that for all $t \in (0,\tau(x,v))$, $\psi_t(x,v) \in \Omega \times {{\mathbb R}}^d$ and $\psi_{t=\tau(x,v)} (x,v) \in \Sigma_+ \cup \Sigma_0 \cup \mathcal{B} $.
$\bullet$ Let $(x,v) \in \Sigma_-$. Then the same construction of forward characteristics $\psi_t(x,v)$ can be performed for $t \geq 0$.
$\bullet$ We now define the broken characteristics $(\phi_t)_{t \in {{\mathbb R}}^+}$ associated to the hamiltonian $H = \frac{1}{2} |v|^2 + V(x)$ and specular boundary conditions for almost every point of $\Omega \times {{\mathbb R}}^d$ as follows.
Let $(x,v) \in \Omega \times {{\mathbb R}}^d$. If for any $t\geq 0$, $\psi_t(x,v) \in \Omega \times {{\mathbb R}}^d$ (i.e. $\tau(x,v)=+\infty$), then we set $\phi_t(x,v)= \psi_t(x,v)$ for all $t\geq 0$. If not, for all $t\in [0,\tau(x,v)]$, set $\phi_t(x,v)= \psi_t(x,v)$. If $(x',v') := (X_{t= \tau(x,v)}(x,v), \Xi_{t= \tau(x,v)}(x,v)) \in \mathcal{B} \cup \Sigma_0$, then stop the construction here. Otherwise, consider $(x_1,v_1) := (x',R_{x'} v')$. Note that $(x',v') \in \Sigma_+$ and thus $(x_1,v_1) \in \Sigma_-$.
If for any $t> 0$, $\psi_t(x_1,v_1) \in \Omega \times {{\mathbb R}}^d$, then set for $t> \tau(x,v)$, $\phi_t(x,v) = \psi_{t- \tau(x,v)} (x_1,v_1)$. If $\tau(x_1,v_1)<\infty$, for all $t\in (\tau(x,v),\tau(x,v)+\tau(x_1,v_1)]$, set $\phi_t(x,v)= \psi_{t-\tau(x,v)} (x_1,v_1)$.
Then if $(x'_1,v'_1):= (X_{t= \tau(x_1,v_1)} (x_1,v_1), \Xi_{t= \tau(x_1,v_1)}(x_1,v_1)) \in \mathcal{B} \cup \Sigma_0$, stop the construction here. Otherwise, consider $(x_2, v_2) := (x'_1, R_{x'_1} v'_1)$, and so on.
There are two possibilities:
- either on any interval of time, there is only a finite number of such intersections with the boundary, in which case the construction can be carried on by recursion,
- or there is an interval of time in which there is an infinite number of such intersections with the boundary.
Nevertheless, as shown in [@Tab Section 1.7] (this is an application of Poincaré’s recurrence lemma), the measure of the points of the phase space for which the second possibility occurs is equal to $0$.
We now recall that by a classical result by Bardos [@Bar Proposition 2.3], which is basically an elegant application of Sard’s lemma, the Lebesgue measure of the set $$\mathcal{S} := \left\{ (x,v) \in \Omega \times {{\mathbb R}}^d, \, \exists t >0, \, \psi_t (x,v) \in \mathcal{B} \cup \Sigma_0 \right\}$$ is equal to zero. This shows that that “pathological” trajectories can actually be neglected. Indeed, remark that $\phi_t(x,v)$ is well defined for all $t\geq 0$ for all $(x,v) \in \Omega \times {{\mathbb R}}^d$, except the set of zero measure evoked in the construction and the set $$\left\{ (x,v) \in \Omega \times {{\mathbb R}}^d, \, \exists t >0, \, \phi_t (x,v) \in \mathcal{B} \cup \Sigma_0 \right\},$$ but by the above property, this set has zero Lebesgue measure (as the countable union of sets with zero Lebesgue measure).
We define likewise the characteristics for almost every point of the phase space on negative times.
We finally recall the following very useful lemma (see [@Wec] and also [@SS]).
For all $s\in {{\mathbb R}}$, $\phi_s$ is measure preserving.
Note that in the case where $\Omega$ is $C^1$, the Hamiltonian flow of $H(x, v) = \frac{|v|^2}{2} + V(x)$ with specular boundary conditions can be made continuous on the appropriate phase space. The latter can for instance be seen as the quotient ${W}= \overline{\Omega}\times {{\mathbb R}}^d / \approx$, where $(x,v ) \approx (x,R_x v) $ for $x \in \partial \Omega$. A continuous function $f$ on $\overline{\Omega}\times {{\mathbb R}}^d$ satisfying $f (x,v) = f(x,R_x v)$ for $(x,v) \in \partial \Omega\times {{\mathbb R}}^d$ can be identified with a continuous function $f$ on ${W}$, so that Equation can be viewed as an equation on ${W}$.
We are in position to state the relevant definitions for the problem of convergence to equilibrium. We can define
- the set $\omega$ where collisions are effective, as in Definition \[def-om\],
- a.e.i.t. GCC, as in Definition \[defaeitgcc\],
- the equivalence relation $\sim$, as in Definition \[def-sim\],
- the Lebesgue spaces ${\mathcal{L}}^p = {\mathcal{L}}^p(\Omega \times {{\mathbb R}}^d)$, as in Definition \[weightedLp\],
- the Unique Continuation Property, as in Definition \[def:UCP\].
We can introduce as in Definition \[definitionCinfini\] the following Lebeau constant (note that we need to consider an essential infimum here, because characteristics are defined only almost everywhere).
We define the Lebeau constant: $$C^-_{b}(\infty) := \sup_{T\in {{\mathbb R}}^+} \text{ess inf}_{(x,v) \in \Omega \times {{\mathbb R}}^d} \frac{1}{T} \int_0^T \left( \int_{{{\mathbb R}}^d} k(\phi_t (x,v), v')\, dv'\right)\, dt.$$
Our next task is to study the well-posedness of in ${\mathcal{L}}^2$ spaces, which are defined as in Definition \[weightedLp\]. One important feature is that with specular reflection, the dissipation identity still holds. The key point is to observe that for symmetry reasons, it is exactly the same as in the torus case (that is without boundary).
Besides, as checked by Weckler [@Wec Theorem 3 and Lemma 3.3], we have the following Duhamel formula.
\[repformu\] Let $\nu \in C(\overline\Omega\times {{\mathbb R}}^d)$ and $g \in L^\infty {\mathcal{L}}^2$. Let $f_0 \in {\mathcal{L}}^2$. The unique weak solution to the kinetic transport equation $$\label{Liouville-specular}
\left\{
\begin{aligned}
&\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= -b(x,v) f(t,x,v) + g , \\
&f(t,x,R_x v) = f(t,x,v), \quad (x,v) \in \partial \Omega \times {{\mathbb R}}^d, \\
&f(0,x,v)= f_0(x,v)
\end{aligned}
\right.$$ is given by $$\begin{aligned}
f(t,x,v)=& \exp\left(- \int_0^t b(\phi_{-(t-s)}(x,v)) \, ds \right) f_0 \circ \phi_{-t}(x,v) \\
& + \int_0^t g(s, \phi_{s-t}(x,v)) \exp\left(- \int_s^t b(\phi_{\tau-t}(x,v)) \, d\tau \right) \, ds.\end{aligned}$$
In turn, this Duhamel formula allows in particular to prove well-posedness for .
Note that well-posedness for can be proved by other means (see for instance Mischler [@Mis], where much weaker assumptions on the force field are considered); however, having the representation formula of Lemma \[repformu\] is a key point for the subsequent analysis (as seen in the torus case).
\[prop:WP-specular\] Assume that $f_0 \in {\mathcal{L}}^2$. Then there exists a unique $f\in C^0({{\mathbb R}};{\mathcal{L}}^2)$ solution of the initial boundary value problem satisfying $f|_{t = 0} =f_0$, and we have $$\text{ for all } t \geq 0, \quad \frac{d}{dt} \| f(t)\|_{{\mathcal{L}}^2}^2 = - D(f(t)),$$ where $D(f)$ is defined as follows: $$\label{defD-boun}
D(f) = \frac{1}{2} \int_{\Omega} e^{V} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \left( \frac{k(x,v' , v)}{{\mathcal{M}}(v)} + \frac{k(x,v , v')}{{\mathcal{M}}(v')} \right) {\mathcal{M}}(v) {\mathcal{M}}(v') \left(\frac{f(v)}{{\mathcal{M}}(v)}- \frac{f(v')}{{\mathcal{M}}(v')}\right)^2 \, dv' \, dv \, dx.$$ If moreover $f_0 \geq 0$ a.e., then for all $t \in {{\mathbb R}}$ we have $f(t, \cdot,\cdot)\geq 0$ a.e. (Maximum principle).
We shall not dwell on the proof of Proposition \[prop:WP-specular\], as it follows the lines of that of Proposition \[prop:WP\]. More generally, all results of Section \[preliminaries\] are still relevant.
Convergence to equilibrium {#convboun}
==========================
As in the torus case, the following holds:
\[thm-convboun\]
The following statements are equivalent.
1. The set $\omega$ satisfies the Unique Continuation Property.
2. The set $\omega$ satisfies the a.e.i.t. GCC and there exists one and only one equivalence class for the equivalence relation $\sim$.
3. For all $f_0 \in {\mathcal{L}}^2$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\label{convergeto0-boundary}
\left\|f(t)-\left(\int_{\Omega \times {{\mathbb R}}^d} f_0 \, dv \,dx\right)\frac{e^{-V}}{\int_{\Omega} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$
One can check that the methods given in Section \[subsectiondecroissance\] for proving Theorem \[thmconv-intro\] are still relevant. We only highlight here the main differences.
$(i.) \implies (iii.)$ One can check that the beginning of the proof given for Theorem \[thmconv-intro\] also applies here, *mutatis mutandis*.
The only problem comes from the fact that at some point, we want to use an averaging lemma to obtain compactness of some averages in $v$ of the solutions, see . The issue is that we a priori need “global” compactness, that is to say on the whole open set $\Omega$; unfortunately, classical averaging lemmas (see Corollary \[thmmoyenne-domain\]) only provide “local” compactness, i.e. only for restrictions of the average on compact subsets of $\Omega$.
The key point is that the proof of $(i.) \implies (iii.)$ in Theorem \[thmconv-intro\] is performed with a sequence of solutions enjoying a uniform ${\mathcal{L}}^\infty$ bound, which means that it also enjoys uniform equi-intregrability (in the sense of Definition \[thmmoyenne-domain\]). Therefore, we can apply the “improved” averaging lemma of Corollary \[thmmoyenne-domain2\] which does yield compactness in the term . This argument allows us to complete the proof of $(i.) \implies (iii.)$ in Theorem \[thm-convboun\].
$(iii.) \implies (ii.)$ Once again, the beginning of the previous proof is still relevant.
Suppose that $\sim$ has at least two equivalence classes. The property of Lemma \[leminvarflow\] is not true in this context, but can replaced by the following one. For any equivalence class $[\Omega_0]$ (for $\sim$), we have, by definition of $\sim$, $$\text{for all } t \geq 0 , \quad
\phi_{-t}\left(\bigcup_{\Omega' \in [\Omega_0]} \Omega' \right) = \bigcup_{\Omega' \in [\Omega_0]} \Omega' \quad \text{almost everywhere},$$ and we can argue as before to conclude.
$(ii.) \implies (i.)$ The previous proof is still relevant, *mutatis mutandis*.
Similarly, we also have
\[thmconv-general-bord\] The following statements are equivalent.
1. The set $\omega$ satisfies the a.e.i.t. GCC.
2. For all $f_0 \in {\mathcal{L}}^2$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\label{convergeto0-general-bord}
\left\|f(t)-Pf_0 \right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$ where $$\label{def-equimulti-bord}
P f_0 (x,v) = \sum_{j\in J} \frac{1}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} \left( \int_{U_j} f_0 \, dv dx\right)f_j.$$ with $([\Omega_j])_{j \in J}$ the equivalence classes of the equivalence relation $\sim$, $$U_j = \bigcup_{\Omega' \in [\Omega_j]} \Omega', \qquad f_j := \frac{\mathds{1}_{U_j} e^{-V} {\mathcal{M}}}{\|\mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2} }.$$
Note that Theorem \[thm-convboun\] has an interesting consequence as soon as we know that the geodesic flow enjoys ergodic properties, and thus in particular a.e.i.t. GCC is satisfied by any $\omega$ of the form $\omega_x \times {{\mathbb R}}^d$, where $\omega_x$ is a non-empty subset of $\Omega$.
\[coroconvergeto0-boundaryre\] Assume that $V=0$ and and $\omega=\omega_x \times {{\mathbb R}}^d$, where $\omega_x $ is a non-empty subset of $\Omega$. Consider $$S \Omega = \left\{ (x,v) \in \Omega \times {{\mathbb R}}^d, \, \frac12 |v|^2 = 1 \right\},$$ and assume that the dynamics $(\phi_t)_{t \geq 0}$ defined on $S\Omega$ is ergodic.
Then for all $f_0 \in {\mathcal{L}}^2$, denoting by $f(t)$ the unique solution to with initial datum $f_0$, we have $$\label{convergeto0-boundaryre}
\left\|f(t)-\left(\int_{\Omega \times {{\mathbb R}}^d} f_0 \, dv \,dx\right)\frac{1}{|\Omega|} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$
Examples of domains $\Omega$ where this corollary applies are the Bunimovich stadium, the Sinai billiard or the interior of a cardioid. Note that the homogeneity of the flow implies that ergodicity on the unit sphere is equivalent to ergodicity on any sphere bundle over $\Omega$. Note that the conclusion of Corollary \[coroconvergeto0-boundaryre\] remains valid in the more general case where $\omega = \omega_x \times \omega_v$ where $\omega_v$ is an open set in ${{\mathbb R}}^d$ such that $\omega_v \cap S(0,R) \neq \emptyset$ for all $R>0$ (where $S(0,R)\subset {{\mathbb R}}^d$ is sphere centered in $0$ of radius $R$).
Exponential convergence to equilibrium {#expoboun}
======================================
In this paragraph (and here only), we consider a more restricted class of collision kernels. Precisely, we assume that the collision kernel $k$ belongs to the subclass [**E2\***]{} of [**E2**]{}, which we define below (note that we have slightly changed the original notations of [**E2**]{}).
[**E2\*. “Factorized” collision kernels**]{} Let $k$ be a collision kernel verifying [**A1**]{}–[**A3**]{}. We suppose that there exist $k^* \in C^0(\overline{\Omega} \times {{\mathbb R}}^d \times {{\mathbb R}}^d)$, $\sigma \in C^0(\overline{\Omega})$, and $\lambda_0>0$ such that
- for all $(x, v,v') \in \Omega \times {{\mathbb R}}^d \times {{\mathbb R}}^d$, $$\label{bornek-improv}
k(x,v,v')= \sigma(x) {k}^*(x,v,v'){\mathcal{M}}(v'),
\quad \text{with} \quad
{{k}^*(x,v, v') } + {{k}^*(x,v', v) } \geq \lambda_0.$$
- we have $$(x,v) \mapsto \int k^*(x,v,v'){\mathcal{M}}(v') \,dv' \in L^\infty(\Omega \times {{\mathbb R}}^d)$$
In this situation, the set $\omega$ (where the collisions are effective) is of the form $\omega_x \times {{\mathbb R}}^d$, where $$\omega_x= \{x \in \Omega, \, \sigma(x) >0\}.$$ For the sake of readability, in this section, we shall write in the following ${\omega}$ instead of ${\omega}_x$. To (slightly) simplify the statements, we shall assume here that $\omega$ is connected.
For collision kernels in [**E2\***]{}, we have the improvement of Lemma \[cerci1\]:
\[cerci2\] For any $f \in {\mathcal{L}}^2$, we have $$D(f) \geq \lambda_0 \left\| \sqrt{\sigma(x)} ( f- \rho_f {\mathcal{M}}(v)) \right\|_{{\mathcal{L}}^2}^2$$
It is much stronger than what we have used in the case without boundary.
By and by symmetry in $v$ and $v'$, using , we have: $$D(f) \geq {\lambda_0} \int_{\Omega} \sigma(x) e^{V} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} {\mathcal{M}}{\mathcal{M}}' \left(\frac{f}{{\mathcal{M}}}- \frac{f'}{{\mathcal{M}}'}\right)^2 \, dv' \, dv \, dx.$$ By Jensen’s inequality it follows that $$\begin{aligned}
D(f) &\geq {\lambda_0}\int_{\Omega}\sigma(x) e^{V} \int_{{{\mathbb R}}^d} {\mathcal{M}}(v)\left(\int_{{{\mathbb R}}^d} {\mathcal{M}}(v') \left(\frac{f(v)}{{\mathcal{M}}(v)}- \frac{f(v')}{{\mathcal{M}}(v')}\right) \, dv'\right)^2 dv \, dx \\
&= {\lambda_0}\int_{\Omega} \sigma(x) e^{V} \int_{{{\mathbb R}}^d} \frac{1}{{\mathcal{M}}(v)} \left(f- \rho_f \, {\mathcal{M}}(v) \right)^2 \, dv \, dx\\
&= {\lambda_0} \left\| \sqrt{\sigma(x)} ( f- \rho_f {\mathcal{M}}(v)) \right\|_{{\mathcal{L}}^2}^2,
\end{aligned}$$ which proves our claim.
The main result of this section is the following theorem.
\[thmexpo-specular\] Let $k$ be a collision kernel in the class [**E2\***]{}. Let $\Omega$ be a bounded and piecewise $C^1$ domain. Assume that $\omega$ is connected. Assume also that there exists a neighborhood $\mathcal{V}$ of $\partial \omega \cap \partial \Omega$ in ${{\mathbb R}}^d$ such that $\partial \omega $ is of class $C^2$ in $\mathcal{V}$.
Consider the following two statements:
1. $C^-_{b}(\infty) > 0$.
2. There exists $C>0, \gamma>0$ such that for any $f_0 \in {\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d)$, the unique solution to with initial datum $f_0$ satisfies for all $t\geq 0$ $$\begin{gathered}
\label{decexpo-specular}
\left\|f(t)- \left(\int_{\Omega \times {{\mathbb R}}^d} f_0 \, dv \,dx\right) \frac{e^{-V}}{\int_{\Omega} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \\
\leq C e^{-\gamma t} \left\|f_0-\left(\int_{\Omega \times {{\mathbb R}}^d} f_0 \, dv \,dx\right) \frac{e^{-V}}{\int_{\Omega} e^{-V} \, dx} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2}.\end{gathered}$$
Then $(a.)$ implies $(b.)$.
\[domain-ghost\]
If $C^-_{b}(\infty) > 0$, we first remark that the analogue of Proposition \[remUCP\] holds in this setting:
Assume that $C^-_{b}(\infty) > 0$. Let $T>0$ such that $$\text{ess inf}_{(x,v) \in \Omega \times {{\mathbb R}}^d} \frac{1}{T} \int_0^T \left( \int_{{{\mathbb R}}^d} k(\phi_t (x,v), v')\, dv'\right)\, dt >0.$$ If $f \in C^0_t({\mathcal{L}}^2)$ is a solution to $$\left\{
\begin{aligned}
&\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
&f(t,x,R_x v) = f(t,x,v), \quad R_x v = v - 2 (v \cdot n(x)) n(x), \quad x \in \partial \Omega, v \in {{\mathbb R}}^d, \\
&f = \rho(t,x) {\mathcal{M}}(v) \text{ on } I \times \omega,
\end{aligned}
\right.$$ where $I$ is an interval of length larger than $T$, then $f= \left(\int_{\Omega \times {{\mathbb R}}^d} f \, dv \,dx\right) \frac{e^{-V}}{\int_{\Omega} e^{-V} \, dx} {\mathcal{M}}(v)$.
As already pointed out in the proof of Theorem \[thm-convboun\], the difficulty in proving Theorem \[thmexpo-specular\] comes from the fact that the regularity provided by the classical averaging lemmas is a priori not valid up to the boundary. This is the main reason why we have not been able to prove neither that $(a.)$ and $(b.)$ are equivalent, nor that the equivalence holds under the far less restrictive assumptions of Theorem \[thmexpo-intro\]. Nevertheless, it seems natural to conjecture that this is indeed the case.
In order to prove $(a.)$ implies $(b.)$, we shall overcome this difficulty by adapting some arguments due to Guo in [@Guo], which forces us to make a regularity assumption on $\omega$ near $\partial \Omega$.
Note that this regularity assumption is for instance satisfied as long as $\partial \Omega$ is $C^2$ and $\sigma$ is positive on the whole boundary $\partial \Omega$.
The regularity assumption is also automatically satisfied in the case where $\overline{\omega} \subset \Omega$. As a matter of fact, if it is the case and if $\partial \Omega$ is $C^1$ then we recover the exact analogue of Theorem \[thmexpo-intro\].
The paper [@Guo] concerns the decay of classical Boltzmann equations (that is with collisions “everywhere”), set in bounded domains and a similar issue has to be faced at some point. Here, we provide a slight generalization of the analysis by handling non zero potentials $V$ (in [@Guo], the dynamics is dictated by free transport); furthermore, we have to modify Guo’s strategy since collisions are only effective in $\omega$ in our framework. Loosely speaking, we will show that there can not be concentration of mass near the boundary ${{\partial}}\omega \cap {{\partial}}\Omega$. Note also that this point of the proof actually does not depend on the boundary condition chosen for the kinetic equation.
Assume that $C^-_{b}(\infty) > 0$. In order to show that $(b.)$ holds, the beginning of the proof is the same as that given for Theorem \[thmexpo-intro\], $(a.) \implies (b.)$. We keep the same notations as those of that proof. The only difference appears in the justification of the compactness property , which we shall perform now. To summarize, the property which remains to be shown is the following one. Given a sequence $(g_n)$ of solutions of in $C^0_t([0,\infty[;{\mathcal{L}}^2)$, satisfying $$\label{assumptions-gn}
\begin{aligned}
&\sup_{t\geq 0} \| g_n(t) \|_{{\mathcal{L}}^2}\leq 1,
&g_n \rightharpoonup 0 \text{ weakly}-\star \text{ in } L^\infty_t {\mathcal{L}}^2, \\
& \int_0^{T_1} D(g_n) \, dt \to 0 ,
\end{aligned}$$ prove that $$\int k(x,v',v) g_n(t,x,v') \, dv' \to 0, \text{ strongly in } L^2(0,T_1;{\mathcal{L}}^2).$$
First remark that as in the torus case, by , Lemma \[cerci2\] and the averaging lemma of Corollary \[thmmoyenne-domain\], for all compact sets $K \subset \Omega$, we have $$\left\|\mathds{1}_{K}(x) \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{L^2(0,T_1;{\mathcal{L}}^2)} \to 0,$$ We shall refer to this property as *interior compactness*.
According to the assumption on $\Omega$ and $\omega$, there exists an open subset $\tilde\omega$, of class $C^2$, included in $\omega$ such that ${{\partial}}\tilde\omega \cap {{\partial}}\Omega = {{\partial}}\omega \cap {{\partial}}\Omega$.
We can write $\tilde\omega = \{x \in {{\mathbb R}}^d, \, \eta(x)<0\}$, where $\eta$ is a $C^2$ function such that $$\label{def-tilde-n}\tilde n(x) := \frac{{{\nabla}}_x \eta(x)}{|{{\nabla}}_x \eta(x)|}$$ is well defined on a neighborhood of ${{\partial}}\tilde\omega = \{x \in {{\mathbb R}}^d, \, \eta(x)=0\}$.
We denote $$\tilde\omega_{{{\varepsilon}}} =\{x \in \tilde\omega, \, \eta(x) < - {{\varepsilon}}^4\}.$$
\[domain-ghost-zoom\]
Then, according to the decomposition $$\begin{gathered}
\left\|\int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2\\
= \left\| \mathds{1}_{\tilde\omega} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2 + \left\|\mathds{1}_{\omega \setminus \tilde\omega} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2,\end{gathered}$$ and since by interior compactness (as $\overline{\omega\setminus\tilde\omega} \subset \Omega$), $$\left\|\mathds{1}_{\omega \setminus \tilde\omega} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{L^2(0,T_1;{\mathcal{L}}^2)} \to 0,$$ it only remains to prove $$\left\| \mathds{1}_{\tilde\omega} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2} \to 0 \text{ strongly in } L^2(0,T_1).$$ We also have the decomposition $$\begin{gathered}
\left\| \mathds{1}_{\tilde\omega} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2\\
= \left\| \mathds{1}_{\tilde\omega_{{\varepsilon}}} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2 + \left\|\mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2 .\end{gathered}$$ Again according to interior compactness (as $\overline{\tilde\omega\setminus\tilde\omega_{{\varepsilon}}} \subset \Omega$), we only have to prove that $$\label{pregoal}
\left\| \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int k(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2} \to 0 \text{ strongly in } L^2(0,T_1).$$ Using and the Cauchy-Schwarz inequality, we have
$$\begin{aligned}
\Big\| \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} &\int k(x,v',v) g_n(t,x,v') \, dv' \Big\|_{{\mathcal{L}}^2} \\
& = \left\| \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int \sigma(x) {\mathcal{M}}(v) {k}^*(x,v',v) g_n(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}\\
& \leq \| \sqrt{\sigma}\|_\infty \left[\int \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}}\sigma(x) {\mathcal{M}}(v)\left(\int |k^*|^2(x,v',v) {\mathcal{M}}(v') \, dv'\right) \right.\\
& \qquad \times \left. \left(\int \frac{|g_n|^2(t,x,v')}{{\mathcal{M}}(v')} \, dv'\right) {e^{V}} \, \, dv dx \right]^{1/2} \\
& \leq \| \sqrt{\sigma}\|_\infty \left\| \int |k^*|^2(x,v',v) {\mathcal{M}}(v) {\mathcal{M}}(v') \, dv' \, dv \right\|_{L^\infty(\Omega)}^{1/2} \\
& \qquad \times \left[\int \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \sigma(x) {|g_n|^2(t,x,v')} \frac{e^{V}}{{\mathcal{M}}(v')} \,dv'\, dx \right]^{1/2}\end{aligned}$$
Therefore, in order to prove , this is sufficient to prove that $$\label{goal}
\| \mathds{1}_{\tilde\omega\setminus \tilde\omega_{{\varepsilon}}} \sqrt{\sigma(x)} g_n \|_{L^2(0,T_1; {\mathcal{L}}^2)} \to 0.$$ The following is dedicated to the proof of this convergence.
Let $m>0$ satisfying $2m<1$. We have the following lemma, adapted from Guo [@Guo Lemma 9], which shows that the contribution of high velocities and “grazing” trajectories is negligible.
\[lem:grandesvitessesourasantes\] There exist ${{\varepsilon}}_0>0$, $C>0$ and a nonnegative function $\varphi(n)$ going to $0$ as $n$ goes to $+\infty$, such that for any $n \in {{\mathbb N}}$ and any ${{\varepsilon}}\in (0,{{\varepsilon}}_0)$, $$\int_0^{T_1} \int_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d } \mathds{1}_{\{ |v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}}(x,v) \sigma(x) |g_n (s,x,v) |^2 \frac{e^V}{{\mathcal{M}}(v)}\, dv \, dx \,ds \leq C {{\varepsilon}}+ \varphi(n).$$
We define now the cut-off functions $$\begin{aligned}
\chi_+(x,v) \: &= \: \mathds{1}_{\tilde\omega\setminus \tilde\omega_{{\varepsilon}}}(x) \, \mathds{1}_{\{|v| \leq {\varepsilon}^{-m}, \, \tilde{n}(x)\cdot v >{{\varepsilon}}\}} (x,v) \, \sqrt{\sigma(x)}, \\
\chi_-(x,v) \: &= \: \mathds{1}_{\tilde\omega\setminus \tilde\omega_{{\varepsilon}}}(x) \, \mathds{1}_{\{|v| \leq {\varepsilon}^{-m}, \, \tilde{n}(x)\cdot v <-{{\varepsilon}}\}} (x,v) \, \sqrt{\sigma(x)}.\end{aligned}$$ Let $u \in [{{\varepsilon}},T_1-{{\varepsilon}}]$. Let $X(t,u,x,v)$ and $\Xi(t,u,x,v)$ be the solution to $$\label{caraccarac}
\left\{
\begin{aligned}
&\frac{dX}{dt}(t,u,x,v) = \Xi(t,u,x,v), \\
&\frac{d\Xi}{dt}(t,u,x,v) = -\nabla_x V(X(t,u,x,v)),
\end{aligned}
\right.$$ with $X(u,u,x,v)= x, \, \Xi(u,u,x,v)=v$. We define for $t \in [{{\varepsilon}},s]$, $$\label{chiplus}
\tilde{\chi}_+(t,x,v) = \chi_+\Big(X(s,t,x,v), \, \Xi(s,t,x,v)\Big),$$ and for $t\in [s,1-{{\varepsilon}}]$, $$\label{chimoins}
\tilde{\chi}_- (t,x,v) = \chi_-\Big(X(s,t,x,v), \, \Xi(s,t,x,v)\Big),$$ which are built in order to satisfy the transport equation $${{\partial}}_t \tilde\chi_\pm + v \cdot {{\nabla}}_x \tilde\chi_\pm - {{\nabla}}_x V \cdot {{\nabla}}_v \tilde\chi_\pm = 0, \quad \tilde\chi_\pm(s,x,v) = \chi_\pm(x,v).$$
The following lemma is an adaptation of [@Guo Lemma 10], with an additional potential $V$.
\[lem:carac\] There exists ${{\varepsilon}}_0>0$ such that if ${{\varepsilon}}\in (0,{{\varepsilon}}_0)$, the following statements hold.
1. For $t \in [s-{{\varepsilon}}^2,s]$, if $\tilde\chi_+(t,x,v) \neq 0$, then $\tilde{n}(x)\cdot v>{{\varepsilon}}/2$ and $|v|<2 {{\varepsilon}}^{-m}$.
Moreover, $\tilde{\chi}_+(s-{{\varepsilon}}^2,x,v) =0$ for $x \in \tilde\omega\setminus \tilde\omega_{{\varepsilon}}$.
2. For $t \in [s,s+ {{\varepsilon}}^2]$, if $\tilde\chi_-(t,x,v) \neq 0$, then $\tilde{n}(x)\cdot v<-{{\varepsilon}}/2$ and $|v|<2 {{\varepsilon}}^{-m}$.
Moreover, $\tilde{\chi}_-(s+{{\varepsilon}}^2,x,v) =0$ for $x \in \tilde\omega\setminus \tilde\omega_{{\varepsilon}}$.
For the sake of readability, we postpone the proofs of Lemmas \[lem:grandesvitessesourasantes\] and \[lem:carac\] to the end of the section.
We finally have the last adaptation from [@Guo Lemma 11].
\[lem:reste\] There exists ${{\varepsilon}}_1>0$, $C>0$ such that for all ${{\varepsilon}}\in (0,{{\varepsilon}}_1)$, there exists a function $\Psi_{{\varepsilon}}: {{\mathbb N}}\to {{\mathbb R}}^+$ satisfying $\lim_{n \to +\infty} \Psi_{{\varepsilon}}(n) = 0$ such that $$\int_0^{T_1} [\| \chi_+ g_n(s) \|_{{\mathcal{L}}^2} ^2 + \| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2] \, ds \leq C {{\varepsilon}}+ \Psi_{{\varepsilon}}(n).$$
Let $s \in [{{\varepsilon}},T_1-{{\varepsilon}}]$. By construction of $\tilde\chi_\pm(t,x,v)$ (see and ), we have $$\label{eqchign}
{{\partial}}_t (\tilde\chi_\pm g_n)+ v \cdot {{\nabla}}_x (\tilde \chi_\pm g_n) - {{\nabla}}_x V \cdot {{\nabla}}_v (\tilde\chi_\pm g_n) = \tilde\chi_\pm C(g_n)$$ We deal with $\tilde\chi_+$. We multiply by $\tilde\chi_+ g_n \frac{e^V}{{\mathcal{M}}(v)}$ and integrate on $[s-{{\varepsilon}}^2, s] \times \tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d$ to get $$\begin{aligned}
\| \chi_+ g_n(s) \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \|_{{\mathcal{L}}^2} ^2 -
\| \tilde{\chi}_+ g_n (s-{{\varepsilon}}^2)\mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \|_{{\mathcal{L}}^2}^2,\\
+ A \:
= \: 2 \int_{s-{{\varepsilon}}^2}^s \langle \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \tilde\chi_+ C(g_n), g_n
\rangle_{{\mathcal{L}}^2} \, du\end{aligned}$$ where $A$ denotes the contributions from the boundary of $\tilde\omega \setminus \tilde\omega_{{\varepsilon}}$, $$A:=\int_{s-{{\varepsilon}}^2}^s \int_{{{\partial}}(\tilde\omega \setminus \tilde\omega_{{\varepsilon}})\times {{\mathbb R}}^d} v\cdot N(x) (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt.$$ Here $N(x)$ is the outer normal on ${{\partial}}(\tilde\omega \setminus \tilde\omega_{{\varepsilon}})$ and $d\Sigma_{{\varepsilon}}(x)$ is the surface measure on ${{\partial}}(\tilde\omega \setminus \tilde\omega_{{\varepsilon}})$.
By Lemma \[lem:carac\], if $x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}$, then $\tilde\chi_+(s-{{\varepsilon}}^2, x,v) = 0$. We deduce that $$\| \tilde{\chi}_+ g_n (s-{{\varepsilon}}^2)\mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \|_{{\mathcal{L}}^2}^2=0.$$
Now, we partition ${{\partial}}(\tilde\omega \setminus \tilde\omega_{{\varepsilon}}) \times {{\mathbb R}}^d$ as follows $${{\partial}}(\tilde\omega \setminus \tilde\omega_{{\varepsilon}}) \times {{\mathbb R}}^d= \gamma^- \cup \gamma^+ \cup \gamma^-_{\varepsilon}\cup \gamma^+_{\varepsilon},$$ where $$\left\{
\begin{aligned}
&\gamma = {{\partial}}\tilde\omega, \\
&\gamma^-= \{ (x,v) \in \gamma\times {{\mathbb R}}^d, \, \eta(x) = 0, \,N(x)\cdot v < 0\},\\
&\gamma^+= \{ (x,v) \in \gamma\times {{\mathbb R}}^d, \, \eta(x) = 0, \,N(x)\cdot v\geq 0\},\\
&\gamma_{{\varepsilon}}= {{\partial}}\tilde\omega_{{\varepsilon}}, \\
&\gamma_{{\varepsilon}}^-= \{ (x,v) \in \gamma_{{\varepsilon}}\times {{\mathbb R}}^d, \, \eta(x) = -{{\varepsilon}}^4, \,N(x)\cdot v <0\},\\
&\gamma_{{\varepsilon}}^+= \{ (x,v) \in \gamma_{{\varepsilon}}\times {{\mathbb R}}^d, \, \eta(x) = -{{\varepsilon}}^4,\, N(x)\cdot v \geq 0\}.
\end{aligned}
\right.$$ Therefore we obtain $$\begin{aligned}
A \: = \int_{s-{{\varepsilon}}^2}^s \int_{\gamma^+} |v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x)
\\
+ \int_{s-{{\varepsilon}}^2}^s \int_{\gamma^+_{{\varepsilon}}} |v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x)
\\
-\int_{s-{{\varepsilon}}^2}^s \int_{ \gamma^-} |v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x)
\\
- \int_{s-{{\varepsilon}}^2}^s \int_{ \gamma^-_{{\varepsilon}}} |v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x).\end{aligned}$$ Recall the definition of $\tilde{n}$ in . Observe that on $\gamma$, we have $N(x)=\tilde{n}(x)$, while on $\gamma_{{\varepsilon}}$, we have $N(x)=-\tilde{n}(x)$.
Using Lemma \[lem:carac\], for all $t \in [s-{{\varepsilon}}^2,s]$, if $\tilde{\chi}(t,x,v) \neq 0$, then $\tilde{n}(x)\cdot v >0$. Therefore, $$\int_{s-{{\varepsilon}}^2}^s \int_{ \gamma^-} |v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x)=0.$$ We thus obtain the bound $$\begin{aligned}
\| \chi_+ g_n(s) \|_{{\mathcal{L}}^2} ^2 &\leq \int_{s-{{\varepsilon}}^2}^s \int_{ \gamma^-_{{\varepsilon}}} |v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt \nonumber\\
&+ 2 \int_{s-{{\varepsilon}}^2}^s \langle \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \tilde\chi_+ C(g_n), g_n
\rangle_{{\mathcal{L}}^2} \, du.\end{aligned}$$ Similarly, for $\tilde\chi_-$, we multiply by $\tilde\chi_- g_n \frac{e^V}{{\mathcal{M}}(v)}$ and integrate on $[s,s+{{\varepsilon}}^2] \times \tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d$ to get $$\begin{aligned}
\| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2 &\leq \int_{s}^{s+{{\varepsilon}}^2}\int_{ \gamma^+_{{\varepsilon}}} |v\cdot N(x)| (\tilde\chi_-(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt \nonumber \\&- 2 \int_s^{s+{{\varepsilon}}^2} \langle \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \tilde\chi_- C(g_n), g_n
\rangle_{{\mathcal{L}}^2} \, du.\end{aligned}$$ According to Lemma \[lem:carac\] and the definitions of $\chi_\pm$ and $\tilde\chi_\pm$, we have $$\begin{aligned}
\int_{s-{{\varepsilon}}^2}^s \int_{ \gamma^-_{{\varepsilon}}} &|v\cdot N(x)| (\tilde\chi_+(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt \\
&\leq \|\sigma\|_\infty \int_{s-{{\varepsilon}}^2}^s \int_{ \gamma_{{\varepsilon}}} |v\cdot N(x)| (\mathds{1}_{\{|v| \leq 2 {{\varepsilon}}^{-m}, \, \tilde{n}(x)\cdot v >{{\varepsilon}}/2\}} (x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt \\
&\leq \|\sigma\|_\infty \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \int_{ \gamma_{{\varepsilon}}} |v\cdot \tilde{n}(x)| (\mathds{1}_{\{|v| \leq 2 {{\varepsilon}}^{-m}, \, |\tilde{n}(x)\cdot v |>{{\varepsilon}}/2\}} (x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt,\end{aligned}$$ and likewise $$\begin{aligned}
\int_s^{s+{{\varepsilon}}^2} \int_{ \gamma^+_{{\varepsilon}}} &|v\cdot N(x)| (\tilde\chi_-(t,x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt \\
&\leq \|\sigma\|_\infty \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \int_{ \gamma_{{\varepsilon}}} |v\cdot \tilde{n}(x)| (\mathds{1}_{\{|v| \leq 2 {{\varepsilon}}^{-m}, \, |\tilde{n}(x)\cdot v| >{{\varepsilon}}/2\}} (x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt.\end{aligned}$$ We thus have to study $$\begin{aligned}
&\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \int_{ \gamma_{{\varepsilon}}} |v\cdot \tilde{n}(x)| (\mathds{1}_{\{|v| \leq 2 {{\varepsilon}}^{-m}, \, |\tilde{n}(x)\cdot v| >{{\varepsilon}}/2\}} (x,v) g_n(t,x,v))^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) dt \\
&\leq \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \int_{ \gamma_{{\varepsilon}}} (\mathds{1}_{\{|v| \leq 2 {{\varepsilon}}^{-m}, \, |\tilde{n}(x)\cdot v| >{{\varepsilon}}/2\}} (x,v) g_n(t,x,v))^2 \frac{(\tilde{n}(x)\cdot v)^2}{1+|v|} \frac{4(1+2{{\varepsilon}}^{-m})}{{{\varepsilon}}^2} \frac{e^V}{{\mathcal{M}}(v)} \, dv d\Sigma_{{\varepsilon}}(x) du\\
&\leq C_{{\varepsilon}}\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \int_{ \gamma_{{\varepsilon}}} g_n^2(u) \frac{e^V}{{\mathcal{M}}(v)} \frac{(N(x)\cdot v)^2}{1+|v|} \, dv d\Sigma_{{\varepsilon}}(x) du,\end{aligned}$$ with $C_{{\varepsilon}}= \frac{4(1+2{{\varepsilon}}^{-m})}{{{\varepsilon}}^2}$.
We recall below Cessenat’s trace theorem (see [@Ces] or [@SR Proposition B.2]):
Let $U$ be a $C^1$ domain. Let $T_1, T_2>0$. There exists a constant $C>0$ such that, for any function $f \in L^2(T_1,T_2;L^2(U \times {{\mathbb R}}^d))$ satisfying $({{\partial}}_t + v \cdot {{\nabla}}_x - {{\nabla}}_x V \cdot {{\nabla}}_v) f \in L^2(T_1,T_2;L^2(U \times {{\mathbb R}}^d))$, we have $$\begin{gathered}
\| f\vert_{{{\partial}}U} \|_{L^2(T_1,T_2; L^2( |v \cdot n(x)|^2 (1+ |v|)^{-1} d\Sigma(x) dv))} \\
\leq C \left( \|f \|_{L^2(T_1,T_2;L^2(U \times {{\mathbb R}}^d))} +
\|({{\partial}}_t + v \cdot {{\nabla}}_x - {{\nabla}}_x V \cdot {{\nabla}}_v) f \|_{L^2(T_1,T_2;L^2(U \times {{\mathbb R}}^d))} \right),\end{gathered}$$ where $d\Sigma$ denotes the surface measure and $n$ is the outward unit normal on ${{\partial}}U$.
This allows us to obtain the control: $$\begin{aligned}
\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \int_{ \gamma_{{\varepsilon}}} g_n^2(u) \frac{e^V}{{\mathcal{M}}(v)} \frac{(N(x)\cdot v)^2}{1+|v|} \, dv d\Sigma_{{\varepsilon}}(x) du \leq K_{{\varepsilon}}\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} [\|g_n(u) \|^2_{{\mathcal{L}}^2( \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d)} + \|C(g_n)(u) \|^2_{{\mathcal{L}}^2( \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d)}] \, du.\end{aligned}$$
By interior compactness, since $\overline{\tilde\omega_{{\varepsilon}}} \subset \Omega$, we know that $$\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \|g_n(u) \|^2_{{\mathcal{L}}^2( \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d)} \, du \to 0,$$ as $n \to + \infty$.
On the other hand, using the assumption [**A2**]{} on the collision kernel, we have $$\begin{aligned}
\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \|C(g_n)(u) \|^2_{{\mathcal{L}}^2} \, du&= \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \| C(g_n- \rho_{g_n} {\mathcal{M}}(v))(u) \|^2_{{\mathcal{L}}^2} \, du \nonumber\\
& \leq E_1 +E_2, \end{aligned}$$ where $$\begin{aligned}
E_1 &:= \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \left\| \left( \int k(x,v,v') \, dv'\right) [g_n - \rho_{g_n} {\mathcal{M}}(v)] \right\|_{{\mathcal{L}}^2}^2 \, du ,\\
E_2 &:= \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \left\| \int k(x,v',v) [g_n(v') - \rho_{g_n} {\mathcal{M}}(v')] \, dv' \right\|_{{\mathcal{L}}^2}^2 \, du .\end{aligned}$$ We first study $E_1$. Using , we have $$\begin{aligned}
E_1 &= \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \left\| \left( \int \sigma (x) k^*(x,v,v') {\mathcal{M}}(v') \, dv'\right) [g_n - \rho_{g_n} {\mathcal{M}}(v)] \right\|_{{\mathcal{L}}^2}^2 \, du ,\\
&\leq \left(\sup_{(x,v) \in \Omega \times {{\mathbb R}}^d} \int k^*(x,v,v'){\mathcal{M}}(v') \,dv'\right)^2 \| \sqrt\sigma\|_\infty \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \left\| \sqrt{\sigma} [g_n - \rho_{g_n} {\mathcal{M}}(v)] \right\|_{{\mathcal{L}}^2}^2 \, du.\end{aligned}$$ We argue likewise for $E_2$ and obtain a similar bound.
Therefore, using Lemma \[cerci2\] and the dissipation bound in , we deduce: $$\begin{aligned}
\int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \|C(g_n)(u) \|^2_{{\mathcal{L}}^2} \, du
& \leq C \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \|\sqrt{\sigma(x)} (g_n- \rho_{g_n} {\mathcal{M}}(v)) \|^2_{{\mathcal{L}}^2} \, du
\leq C \int_0^{T_1} D(g_n(t)) dt\nonumber \\
& \leq \Psi_1(n), \label{bouboun}\end{aligned}$$ with $\Psi_1(n)$ a function tending to $0$ as $n$ goes to $+\infty$.
Moreover, we have the rough bound, obtained by Cauchy-Schwarz inequality: $$\begin{aligned}
\int_{s-{{\varepsilon}}^2}^s \langle \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \tilde\chi_+ C(g_n), g_n \rangle_{{\mathcal{L}}^2} \, du &- \int_s^{s+{{\varepsilon}}^2} \langle \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \tilde\chi_- C(g_n), g_n
\rangle_{{\mathcal{L}}^2} \, du \\
&\leq \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \| C(g_n)\|_{{\mathcal{L}}^2} \| g_n\|_{{\mathcal{L}}^2} \,du \\
&\leq \sup_{t\geq 0} \| g_n\|_{{\mathcal{L}}^2} \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \| C(g_n)\|_{{\mathcal{L}}^2} \,du \\
& \leq \int_{s-{{\varepsilon}}^2}^{s+{{\varepsilon}}^2} \| C(g_n)\|_{{\mathcal{L}}^2} \,du,\end{aligned}$$ where we have used on the last line. We can use again to bound this by $\Psi_1(n)$.
Summarizing (${{\varepsilon}}$ being fixed), we have proved the existence of $\Psi_{{\varepsilon}}(n)$ tending to $0$ as $n$ tends to infinity such that $$\int_{{\varepsilon}}^{T_1-{{\varepsilon}}} [\| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2 + \| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2] \, ds \leq \Psi_{{\varepsilon}}(n).$$ We also have, by , $$\begin{aligned}
\int_0^{{\varepsilon}}[\| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2 + \| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2] \, ds &\leq 2 \|\sigma\|_{\infty} {{\varepsilon}}, \\
\int_{T_1 - {{\varepsilon}}}^{T_1} [\| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2 + \| \chi_- g_n(s) \|_{{\mathcal{L}}^2} ^2] \, ds &\leq 2 \|\sigma\|_{\infty} {{\varepsilon}}.\end{aligned}$$ which yields the claimed result of Lemma \[lem:reste\].
[**End of the proof of Theorem \[thmexpo-specular\].**]{} We are now ready to prove , which is a consequence of Lemmas \[lem:grandesvitessesourasantes\] and \[lem:reste\]. Indeed, we can write $$\begin{aligned}
&\| \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \sqrt{\sigma} g_n \|_{L^2(0,T_1; {\mathcal{L}}^2)} ^2 \\
&=\| \mathds{1}_{\{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}}\mathds{1}_{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} \sqrt{\sigma} g_n \|_{L^2(0,T_1;{\mathcal{L}}^2)} ^2 + \| \mathds{1}_{\{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}}\mathds{1}_{|v| \leq {\varepsilon}^{-m} \text{ and } |\tilde{n}(x)\cdot v| >{{\varepsilon}}\}} \sqrt{\sigma} g_n \|_{L^2(0,T_1; {\mathcal{L}}^2)} ^2 \\
&=\| \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}}\mathds{1}_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} \sqrt{\sigma} g_n \|_{L^2(0,T_1;{\mathcal{L}}^2)} ^2 + \|\chi_+ g_n \|_{L^2(0,T_1; {\mathcal{L}}^2)} ^2 + \|\chi_- g_n \|_{L^2(0,T_1; {\mathcal{L}}^2)} ^2 \\
&\leq C {{\varepsilon}}+ \varphi(n) + \Psi_{{\varepsilon}}(n).\end{aligned}$$ Let $\delta>0$. Fix ${{\varepsilon}}>0$ small enough such that ${{\varepsilon}}<{{\varepsilon}}_0$ and $C{{\varepsilon}}< \delta/2$. Once this parameter ${{\varepsilon}}$ is fixed, choose $n_0$ large enough such that for all $n \geq n_0$, $ \varphi(n) + \Psi_{{\varepsilon}}(n) \leq \delta/2$. Then for all $n \geq n_0$, we have $\| \mathds{1}_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \sqrt{\sigma} g_n \|_{L^2(0,T_1; {\mathcal{L}}^2)} ^2 \leq \delta$, which proves the convergence to $0$.
Therefore, this concludes the proof of Theorem \[thmexpo-specular\].
Now, it only remains to prove Lemmas \[lem:grandesvitessesourasantes\] and \[lem:carac\] for the proof of Theorem \[thmexpo-specular\] to be complete. This is the aim of the next two sections.
Proof of Lemma \[lem:grandesvitessesourasantes\]
------------------------------------------------
First take ${{\varepsilon}}_0$ small enough so that $\tilde{n}(x)$ is well defined for $x \in \tilde\omega \setminus \tilde\omega_{{{\varepsilon}}_0}$. We write the decomposition $$\begin{aligned}
\int_0^{T_1} \int_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \mathds{1}_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} \sigma(x) |g_n (s,x,v) |^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx \,ds
\leq A_1 + A_2,\end{aligned}$$ with $$\begin{aligned}
A_1&:= 2\|\sigma\|_{L^\infty(\Omega)} \int_0^{T_1} \int_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \mathds{1}_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} |\rho_{g_n} (s,x) |^2 e^V {\mathcal{M}}(v) \, dv \, dx \,ds,\\
A_2&:= 2 \int_0^{T_1} \int_{\tilde\omega \setminus \tilde\omega_{{\varepsilon}}\times {{\mathbb R}}^d} \mathds{1}_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} \sigma(x) |g_n (s,x,v)- \rho_{g_n}(s,x) {\mathcal{M}}(v) |^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx \,ds,\end{aligned}$$ where $$\rho_{g_n} := \int g_n \, dv.$$ For $A_1$, we use the uniform bound $\sup_{t \geq 0}\| g_n \|_{{\mathcal{L}}^2} \leq 1$ and Cauchy-Schwarz inequality to obtain $$\begin{aligned}
A_1 &\leq 2 \|\sigma\|_{L^\infty(\Omega)} \int_0^{T_1} \left(\int_\Omega |\rho_{g_n}|^2 e^{V} \, dx \right) dt \sup_{x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} {\mathcal{M}}(v) \, dv \\
&\leq 2 \|\sigma\|_{L^\infty(\Omega)} \int_0^{T_1} \left(\int_\Omega \left(\int \frac{|{g_n(x,v')}|^2}{{\mathcal{M}}(v')} \, dv' \right)\left(\int {\mathcal{M}}(v') \, dv' \right) e^{V(x)} \, dx \right) dt \\
& \qquad\qquad\qquad \times \sup_{x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} {\mathcal{M}}(v) \, dv \\
&\leq 2T_1 \|\sigma\|_{L^\infty(\Omega)} \left(\sup_{t \geq 0} \| g_n \|_{{\mathcal{L}}^2} \right)^2 \sup_{x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} {\mathcal{M}}(v) \, dv \\
&\leq 2T_1 \|\sigma\|_{L^\infty(\Omega)} \sup_{x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}} \int_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} {\mathcal{M}}(v) \, dv \\end{aligned}$$ One can check that there exists $C>0$ independent of ${{\varepsilon}}$ such that for all $x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}$, we have $$\int_{\{|v| > {\varepsilon}^{-m} \text{ or } |\tilde{n}(x)\cdot v| \leq{{\varepsilon}}\}} {\mathcal{M}}(v) \, dv \leq C {{\varepsilon}},$$ so that we have $$A_1 \leq 2 C T_1\|\sigma\|_{L^\infty(\Omega)} {{\varepsilon}}.$$ On the other hand, we have the rough bound $$A_2 \leq 2\int_0^{T_1} \int_{\Omega} \int_{{{\mathbb R}}^d} \sigma(x) |g_n (s,x,v)- \rho_n(s,x) {\mathcal{M}}(v) |^2 \frac{e^V}{{\mathcal{M}}(v)} \, dv \, dx \,ds ,$$ which also goes to $0$ as $n$ goes to infinity, thanks to Lemma \[cerci2\] and the dissipation bound in . This concludes the proof of Lemma \[lem:grandesvitessesourasantes\].
Proof of Lemma \[lem:carac\]
----------------------------
First take ${{\varepsilon}}_0$ small enough such that $\tilde{n}(x)$ is well defined on $\{x \in \tilde\omega, \, -\sqrt{{{\varepsilon}}_0}<\eta(x) < \sqrt{{{\varepsilon}}_0} \}$.
We only prove Item $(1)$ (the treatment of Item $(2)$ being identical).
For the first statement of Item $(1)$, assume that $s,t,x,v$ are such that $\tilde\chi_+(t,x,v) \neq 0$ and $t \in [s-{{\varepsilon}}^2 ,s]$. We have $$\begin{aligned}
\label{B1B2B3}
& \tilde{n}(x) \cdot v = B_1 +B_2 +B_3, \quad \text{with} \quad
B_1 = \tilde{n} (X(s,t,x,v)) \cdot \Xi(s,t,x,v) ,\\
& B_2 = - \tilde{n} (X(s,t,x,v)) \cdot (\Xi(t,s,x,v)- v) ,
\quad B_3 = - (\tilde{n} (X(s,t,x,v))-\tilde{n}(x))\cdot v . \nonumber\end{aligned}$$ Since $\tilde\chi_+(t,x,v) \neq 0$, we have $B_1 \geq {{\varepsilon}}$. For $B_2$ and $B_3$, using , we obtain the estimates $$\begin{aligned}
|B_2 |&\leq |\Xi(s,t,x,v)- v| \leq |t-s| \|{{\nabla}}_x V\|_\infty \leq C {{\varepsilon}}^2, \\
|B_3 |&\leq |\tilde{n} (X(s,t,x,v))-\tilde{n}(x)| |v| \\
&\leq |{{\nabla}}\tilde{n}(\overline{x})||X(s,t,x,v))-x| |v|,\end{aligned}$$ where $\overline{x}$ belongs to the segment $[x,X(s,t,x,v)]$. Since $\tilde\chi_+(t,x,v) \neq 0$, we have $|\Xi(s,t,x,v)|\leq {{\varepsilon}}^{-m}$. We deduce, using again , that $$\begin{aligned}
\nonumber |X(s,t,x,v))-x| &\leq \int_t^s |\Xi (s,u,x,v)| \, du \\
\nonumber&\leq |t-s| |\Xi(s,t,x,v)| + \frac{|t-s|^2}{2} \| {{\nabla}}_x V\|_\infty\\
\nonumber&\leq {{\varepsilon}}^2 {{\varepsilon}}^{-m} + \frac{ \| {{\nabla}}_x V\|_\infty}{2} {{\varepsilon}}^4 \\
\label{utileX}&\leq 2{{\varepsilon}}^2 {{\varepsilon}}^{-m},\end{aligned}$$ for ${{\varepsilon}}< {{\varepsilon}}_0$ with ${{\varepsilon}}_0$ small enough, as $m>0$.
Thus, we infer that $|\overline{x}-X(s,t,x,v)| \leq |{x}-X(s,t,x,v)|\leq 2 {{\varepsilon}}^2 {{\varepsilon}}^{-m} < {{\varepsilon}}$, for ${{\varepsilon}}< {{\varepsilon}}_0$ with ${{\varepsilon}}_0$ small enough, as $2m<1$. Moreover, since $\tilde\chi_+(t,x,v) \neq 0$, we have $X(s,t,x,v) \in \tilde\omega\setminus \tilde\omega_{{\varepsilon}}$ and thus $\eta(\overline{x}) \in (-\sqrt{{{\varepsilon}}_0}, \sqrt{{{\varepsilon}}_0})$. Therefore, there is a constant $C>0$ (depending only on ${{\varepsilon}}_0$ and $\eta$) such that $|{{\nabla}}\tilde{n}(\overline{x})|\leq C$. We deduce that $$\begin{aligned}
|B_3 |&\leq C {{\varepsilon}}^2 {{\varepsilon}}^{-2m}.\end{aligned}$$ Coming back to , this yields $$\tilde{n}(x) \cdot v \geq {{\varepsilon}}- C {{\varepsilon}}^{2(1-m)} - C {{\varepsilon}}^2 \geq {{\varepsilon}}/2.$$ for ${{\varepsilon}}< {{\varepsilon}}_0$ with ${{\varepsilon}}_0$ small enough, since $2m<1$.
Likewise using the decomposition $$v = \Xi(s,t,x,v) + [v- \Xi(s,t,x,v)],$$ we prove that $|v|<2 {{\varepsilon}}^{-m}$ for ${{\varepsilon}}< {{\varepsilon}}_0$ with ${{\varepsilon}}_0$ small enough.
Now let us prove the second statement of Item $(1)$. Let $x \in \tilde\omega \setminus \tilde\omega_{{\varepsilon}}$. We argue by contradiction. Assuming that $\tilde{\chi}_+(s-{{\varepsilon}}^2,x,v)>0$, we have $$\label{contraeq}
\begin{array}{c}
-{{\varepsilon}}^4 < \eta(X(s,s-{{\varepsilon}}^2,x,v)) < 0, \quad |\Xi(s,s-{{\varepsilon}}^2,x,v)|<{{\varepsilon}}^{-m}, \\
\text{and} \quad
\tilde{n}(X(s,s-{{\varepsilon}}^2,x,v)) \cdot \Xi(s,s-{{\varepsilon}}^2,x,v) >{{\varepsilon}}.
\end{array}$$ As before, we deduce that $|v|<2 {{\varepsilon}}^{-m}$.
We write the Taylor-Lagrange formula $$\begin{aligned}
\eta(X(s,s-{{\varepsilon}}^2,x,v))
& = \eta(x) - {{\nabla}}\eta(X(s,s-{{\varepsilon}}^2,x,v)) \cdot (x-X(s,s-{{\varepsilon}}^2,x,v)) \\
&\quad - ({{\nabla}}^2 \eta(\overline{x}) [x-X(s,s-{{\varepsilon}}^2,x,v)],x-X(s,s-{{\varepsilon}}^2,x,v)),\end{aligned}$$ where $\overline{x}$ belongs to the segment $[x,X(s,s-{{\varepsilon}}^2,x,v)]$.
The Taylor formula, together with yields $$x-X(s,s-{{\varepsilon}}^2,x,v) = {{\varepsilon}}^2 \Xi(s,s-{{\varepsilon}}^2,x,v) -\int_{s-{{\varepsilon}}^2}^s (s-w) {{\nabla}}_x V(X(s,w,x,v)) \, dw.$$ As a consequence, we have $$- {{\nabla}}\eta(X(s,s-{{\varepsilon}}^2,x,v)) \cdot (x-X(s,s-{{\varepsilon}}^2,x,v)) = D_1 +D_2,$$ where $$\begin{aligned}
D_1 &:= -{{\varepsilon}}^2 {{\nabla}}\eta(X(s,s-{{\varepsilon}}^2,x,v)) \cdot \Xi(s,s-{{\varepsilon}}^2,x,v) ,\\
D_2 &:= \int_{s-{{\varepsilon}}^2}^s (s-w) {{\nabla}}_x V(X(s,w,x,v)) \cdot {{\nabla}}\eta(X(s,s-{{\varepsilon}}^2,x,v)) \, dw.\end{aligned}$$ Recalling the definition of $\tilde{n}$ in , there is a constant $C>1$ (depending only on ${{\varepsilon}}_0$ and $\eta$) such that $$1/C\leq|{{\nabla}}\eta|(X(s,s-{{\varepsilon}}^2,x,v)) \leq C, \quad |{{\nabla}}^2 \eta(\overline{x})|\leq C.$$ We thus have $$\begin{aligned}
D_1 &= -{{\varepsilon}}^2 |{{\nabla}}\eta(X(s,s-{{\varepsilon}}^2,x,v))| \tilde{n}(X(s,s-{{\varepsilon}}^2,x,v)) \cdot \Xi(s,s-{{\varepsilon}}^2,x,v) \\
&\leq -{{{\varepsilon}}^3} |{{\nabla}}\eta(X(s,s-{{\varepsilon}}^2,x,v))| \\
&\leq -1/C {{\varepsilon}}^3,\end{aligned}$$ together with $$|D_2| \leq C {{\varepsilon}}^4.$$ Furthermore, using the second equation of and arguing as for , we obtain the bound $$({{\nabla}}^2 \eta(\overline{x}) [X(s,s-{{\varepsilon}}^2,x,v)-x],X(s,s-{{\varepsilon}}^2,x,v)-x) \leq C [ {{\varepsilon}}^2 {{\varepsilon}}^{-m} + {{\varepsilon}}^4]^2 \leq C {{\varepsilon}}^{2(2-m)}.$$ We deduce that $$\eta(X(s,s-{{\varepsilon}}^2,x,v)) < 0 -1/C {{\varepsilon}}^3 + {C {{\varepsilon}}^4 + {{\varepsilon}}^{2(2-m)}} < - {{\varepsilon}}^4,$$ using again $2m<1$ and taking ${{\varepsilon}}$ small enough. This is a contradiction with the first equation of . This concludes the proof of Lemma \[lem:carac\].
About other boundary conditions {#otherboun}
===============================
[**1.**]{} We could have considered slightly more general reflection laws of the form $$\label{eq-refl}
f(t,x,T_x v) = f(t,x,v), \quad \text{for } (x,v) \in \Sigma_+,$$ for any family of transformations $(T_x)_{x \in \partial\Omega}$ such that $T_x: p_v(\Sigma_+) \to p_v(\Sigma_-)$ (here $p_v$ denotes the projection on the $v$ space) satisfies $\|T_x(v)\|=\|v\|$, where $\| \cdot \|$ denotes the standard euclidian norm. The only thing to do is to modify accordingly the definition of characteristics. This includes for instance the (sometimes used) *bounce-back* boundary condition: $$\label{eq-bounce}
f(t,x,v) = f(t,x,-v), \quad \text{for } (x,v) \in \partial \Omega \times {{\mathbb R}}^d.$$
[**2.**]{} Another important class of boundary conditions for kinetic equations is given by the:
$\bullet$ *Diffusive boundary condition* (or Maxwellian diffusion), which reads: $$f(t,x, v) = \frac{\int_{v'\cdot n(x) >0} f(t,x,v') \, v' \cdot n(x) \, dv'}{\int_{v'\cdot n(x) <0}{\mathcal{M}}_w(v') |v' \cdot n(x)| \, dv'} {\mathcal{M}}_w(v), \quad (x,v) \in \Sigma_-.$$ where ${\mathcal{M}}_w(v) $ is some Maxwellian distribution characterizing the state of the wall (depending on its temperature).
We restrict ourselves to the case ${\mathcal{M}}_w(v) = {\mathcal{M}}(v)$ (which means that we consider that the wall has reached a global equilibrium compatible with the linear Boltzmann equation). Then, contrary to the specular reflection case, there is a non-trivial contribution of the boundary in the dissipation identity.
\[lemdissip-diffusive\]Let $f \in C^0_t({\mathcal{L}}^2)$ be a solution to with diffusive boundary conditions. The following identity holds, for all $t\geq 0$: $$\label{eqdissipation-diffusive}
\frac{d}{dt} \|f(t)\|_{{\mathcal{L}}^2}^2 = - \tilde{D}(f),$$ with: $$\tilde{D}(f) = D(f) + \int_{v\cdot n(x)>0} \left( f- \frac{\int_{v'\cdot n(x) >0} f(t,x,v') \, v' \cdot n(x) \, dv'}{\int_{v'\cdot n(x) <0}{\mathcal{M}}(v') |v' \cdot n(x)| \, dv'} {\mathcal{M}}(v)\right)^2 v \cdot n(x) \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, d\Sigma.$$ where $D(f)$ is defined in and $d\Sigma$ denotes the surface measure on $\partial \Omega$.
It is also possible to study
$\bullet$ *A combination of specular and diffusive boundary conditions*. Let $\alpha \in (0,1)$. $$f(t,x, v) = \alpha \frac{\int_{v'\cdot n(x) >0} f(t,x,v') \, v' \cdot n(x) \, dv'}{\int_{v'\cdot n(x) <0}{\mathcal{M}}_w(v') |v' \cdot n(x)| \, dv'} {\mathcal{M}}_w(v) + (1- \alpha) f(t,x,R_x v)
\quad (x,v) \in \Sigma_-.$$
The last two boundary conditions are very relevant from the physical point of view. As such, they are worth being studied. We leave this problem for future studies.
\[part3\]
The case of a general compact Riemannian manifold {#sec:manifold}
=================================================
Let $(M,g)$ be a smooth compact connected $d$-dimensional Riemannian manifold (without boundary). In local coordinates, the metric $g$ is a symmetric positive definite matrix such that for all $x \in M$ and $u, w \in T_x M$, we have $$(u,w)_{g(x)} = g_{i,j}(x) u^i w^j ,$$ where the Einstein summation notations are used. This provides a canonical identification between the tangent bundle $TM$ and the cotangent bundle $T^*M$ [*via*]{} the following formula. For any vector $u \in T_x M$ there exists a unique covector $\eta \in T^*_x M$ satisfying $$\langle \eta , w \rangle_{T^*_x M , T_x M} = (u,w)_{g(x)}, \quad \text{ for all } w \in T_x M .$$ In local coordinates, we have $$\eta_i = g_{i,j}(x) u^j .$$ We can define an inner product on $T^*_x M$ using the above identification, denoted by $(\cdot,\cdot)_{g^{-1}(x)}$. In local coordinates, we have $$(\eta,\xi)_{g^{-1}(x)} = g^{i,j}(x) \eta_i \xi_j, \quad \text{ where } g^{i,j}(x) = (g(x)^{-1})^{i,j} .$$ For all $x \in M$ and all $\eta \in T^*_x M$, we denote by $|\eta|_x = (\eta,\eta)_{g^{-1}(x)}^{\frac12}$ the associated norm. Let $d\operatorname{Vol}(x)$ be the canonical Riemannian measure on $M$. In local charts this reads $$d\operatorname{Vol}(x) = \sqrt{|\det(g(x))|} dx_1 \cdots dx_d.$$ The cotangent bundle $T^*M$ is canonically endowed with a symplectic 2-form $\omega$ (in local charts, $\omega = \sum_{j=1}^d dx_j \wedge d\xi_j$). Let $\omega^d$ be the canonical symplectic volume form on $T^*M$ and by a slight abuse of notation $d \omega^d$ the associated [*normalized*]{} measure on $T^*M$ (see for instance [@Hor p.274]). In local coordinates, we have $$d \omega^d = d \xi_1 \cdots d \xi_d \, d x_1 \cdots d x_d.$$ The canonical projection $\pi : T^*M \to M$ is measurable from $(T^*M , d\omega^d)$ to $(M, d\operatorname{Vol})$. For $f \in L^1 (T^*M , d\omega^d)$, we define $\pi_* f \in L^1 (M , d\operatorname{Vol})$ by $$\int_M \varphi(x) (\pi_* f)(x) d\operatorname{Vol}(x) = \int_{T^* M} \varphi \circ \pi (x, \xi) f(x, \xi) d\omega^d(x, \xi), \quad \text{ for all } \varphi \in C^0(M) .$$ In local charts, we have $$(\pi_* f)(x) = \frac{1}{\sqrt{\det(g(x))}}\int_{{{\mathbb R}}^d} f(x, \xi) d\xi_1 \cdots d\xi_d .$$
Note also that we have the following desintegration formula $$\int_{T^*M} f(x, \xi) d\omega^d(x, \xi) = \int_M d\operatorname{Vol}(x) \int_{T^*_x M} f(x, \xi)dm_x(\xi) ,$$ where the measure $dm_x$ on $T^*_xM$ is given in local charts by $$dm_x = \frac{1}{\sqrt{\det(g(x))}} d\xi_1 \cdots d\xi_d .$$
Let $V \in W^{2, \infty}(M)$ and define on $T^*M$ the hamiltonian $$H(x, \xi) = \frac12|\xi|_x^2 + V(x) , \quad x \in M , \quad \xi \in T^*_x M .$$ We define the associated Hamilton vector field $X_H$, given in local coordinates by $$X_H = \nabla_{\xi}H \cdot \nabla_x - \nabla_{x} H \cdot {{\nabla}}_\xi .$$ Using the $2$-form $\omega$, we can also define the Poisson bracket $\{\cdot , \cdot\}$, see again [@Hor p.271]. We have $X_H f = \{H, f\}$.
We denote by $\Lambda = \{(x, \xi, \xi'), x \in M, (\xi, \xi') \in T^*_x M \times T^*_x M\}$ the vector bundle over $M$ whose fiber above $x$ is $T^*_x M \times T^*_x M$.
With these notations, the Boltzmann equation on $T^*M$ can be written as follows, for $(t,x, \xi) \in {{\mathbb R}}\times T^*M$, $$\begin{aligned}
\label{Bmanifold}
{{\partial}}_t f (t,x, \xi)+ X_H f (t,x, \xi)=
\int_{T^*_x M} \left[k(x,\xi',\xi) f(t,x,\xi') - k(x,\xi,\xi') f(t,x,\xi)\right] \, dm_x(\xi')
.\end{aligned}$$
We recover the key properties of the usual linear Boltzmann collision operator on flat spaces. We have, for all $x \in M$, $$\int_{T^*_x M} \int_{T^*_x M} \left[k(x,\xi',\xi) f(x,\xi') - k(x,\xi,\xi') f(x,\xi)\right] \, dm_x(\xi') dm_x(\xi) = 0 .$$ Besides, $$\int_{T^* M} (X_H f) (x, \xi) d\omega^d (x, \xi) = 0 ,$$ since $(X_H f) (x, \xi) d\omega^d$ is an exact form (since $X_H$ is hamiltonian). As a consequence, the mass is conserved: any solution $f$ of satisfies $$\text{ for all } t \geq 0, \quad \frac{d}{dt} \int_{T^* M} f(t,x,\xi) \, d\omega^d (x, \xi) =0.$$ Consider now the (generalized) Maxwellian distribution: $${\mathcal{M}}(x, \xi) := \frac{1}{(2\pi)^{d/2}}e^{-\frac{|\xi|_x^2}{2}}.$$ Note that for all $x \in M$, $\int_{T^*_x M} \frac{1}{(2\pi)^{d/2}}e^{-\frac{|\xi|_x^2}{2}} \, dm_x (\xi) = 1$. As usual, we make the following assumptions on the collision kernel $k$.
[**A1.**]{} The collision kernel $k \in C^0(\Lambda)$, is nonnegative.
[**A2.**]{} We assume that the Maxwellian cancels the collision operator, that is: $$\int_{T^*_x M} \left[k(x,\xi' , \xi) {\mathcal{M}}(x, \xi') - k(x,\xi , \xi') {\mathcal{M}}(x, \xi)\right] \, dm_x(\xi') = 0, \quad \text{for all } (x,\xi) \in T^*M.$$
[**A3.**]{} We assume that $$x \mapsto \int_{T_x^*M \times T_x^*M} k^2(x,\xi',\xi) \frac{{\mathcal{M}}(x,\xi')}{{\mathcal{M}}(x,\xi)} \, dm_x(\xi') dm_x(\xi) \in L^\infty(M).$$
We can define the characteristics in this Riemannian setting as follows.
\[def-caracmanifold\] Given $(x, \xi) \in T^*M$, we define the hamiltonian flow associated to $H$ by $s \mapsto \phi_s(x , \xi) \in T^*M $: $$\label{eq: geodesic flow}
\frac{d}{ds}\phi_s(x, \xi) = X_{H} \big(\phi_s(x, \xi)\big), \quad \phi_0(x, \xi) = (x, \xi) \in T^*M,$$ The characteristics associated to $H$ are the integral curves of this flow.
Note also for any function $g$ defined on ${{\mathbb R}}$, $g \circ H$ is preserved along these integral curves, as $$\left(\frac{d}{ds}g \circ H \circ \phi_s\right)|_{s = s_0} = X_H(g \circ H)(\phi_{s_0}) = \{H,g \circ H\}(\phi_{s_0}) =0 .$$ In particular, this holds for the function $\frac{e^V}{{\mathcal{M}}} = \frac{1}{(2\pi)^{d/2}} e^{H}$.
With this definition of characteristics, we can then properly define
- the set $\omega$ where collisions are effective, as in Definition \[def-om\],
- $C^-(\infty)$, as in Definition \[definitionCinfini\],
- a.e.i.t. GCC, as in Definition \[defaeitgcc\],
- the Unique Continuation Property, as in Definition \[def:UCP\],
- the [*generalized*]{} Unique Continuation Property, as in Definition \[def:UCPgene\],
- the equivalence relations $\sim$ and $\Bumpeq$, as in Definitions \[def-sim\] and \[def-sim-o\],
- the sets $U_j$ as in .
Note that in this Riemannian setting, the classes of collision operators [**E1**]{}, [**E2**]{}, [**E3**]{} still make sense, up to some obvious adaptations.
Let us now introduce the relevant weighted Lebesgue spaces.
We define the Banach spaces ${\mathcal{L}}^2$ and ${\mathcal{L}}^\infty$ by $$\begin{aligned}
&{\mathcal{L}}^2 (T^*M):= \Big\{f \in L^2_{loc}(T^*M), \, \int_{T^*M} | f|^2 \frac{e^V}{{\mathcal{M}}} \, d\omega^d < + \infty \Big\} ,
\quad \|f\|_{{\mathcal{L}}^2} = \left(\int_{T^*M} | f|^2 \frac{e^V}{{\mathcal{M}}} \, d\omega^d \right)^{1/2}.
\\
& {\mathcal{L}}^\infty(T^*M) : = \Big\{f \in L^1_{loc}(T^*M), \, \sup_{T^*M} | f| \frac{e^V}{{\mathcal{M}}} < + \infty \Big\} ,
\quad \|f\|_{{\mathcal{L}}^\infty} = \sup_{T^*M} | f| \frac{e^V}{{\mathcal{M}}}
\end{aligned}$$ The space ${\mathcal{L}}^2$ is a Hilbert space endowed with the inner product $$\langle f, g \rangle_{{\mathcal{L}}^2} := \int_{T^*M} f \, g \frac{e^{V}}{{\mathcal{M}}} \, d\omega^d.$$
As usual, we have the following well-posedness result for the Boltzmann equation .
\[prop:WP-manifold\] Assume that $f_0 \in {\mathcal{L}}^2$. Then there exists a unique $f\in C^0({{\mathbb R}};{\mathcal{L}}^2)$ solution of satisfying $f|_{t = 0} =f_0$, and we have $$\text{ for all } t \geq 0, \quad \frac{d}{dt} \| f(t)\|_{{\mathcal{L}}^2}^2 = - D(f(t)),$$ where $$\begin{gathered}
D(f) = \frac{1}{2} \int_{M} e^{V(x)} \int_{T^*_x M} \int_{T^*_x M} \left( \frac{k(x,\xi' , \xi)}{{\mathcal{M}}(x,\xi)} + \frac{k(x,\xi , \xi')}{{\mathcal{M}}(x,\xi')} \right) \\ \times {\mathcal{M}}(x,\xi) {\mathcal{M}}(x,\xi') \left(\frac{f(x,\xi)}{{\mathcal{M}}(x,\xi)}- \frac{f(x,\xi')}{{\mathcal{M}}(x,\xi'))}\right)^2 \, dm_x (\xi) \,dm_x(\xi') \, d\operatorname{Vol}(x).
\end{gathered}$$ If moreover $f_0 \geq 0$ a.e., then for all $t \in {{\mathbb R}}$ we have $f(t, \cdot,\cdot)\geq 0$ a.e. (Maximum principle).
More generally, all results of Section \[preliminaries\] (up to obvious adaptations) are still relevant.
The crucial point we have to check now concerns velocity averaging lemmas for kinetic transport equations on a Riemannian manifold.
\[lem-moyenne-variete\] Let $H$ be defined as above, and $X_H$ the associated vector field. Let $T>0$ and $\Psi \in C^\infty_c(T^*M)$. There exists $C>0$ a constant such that the following holds. For any $f,g \in L^2((0,T)\times T^*M)$ satisfying $$\partial_t f + X_H f = g,$$ we have $$\| \pi_*(f \Psi) \|_{H^{1/4}((0,T) \times M)} \leq C (\| f|_{t=0}\|_{L^2((0,T) \times T^*M)} + \|g\|_{L^2((0,T) \times T^*M)} ).$$ i.e. $$\left\| \int_{T_x^*M}f \Psi dm_x \right\|_{H^{1/4}((0,T) \times M)} \leq C (\| f|_{t=0}\|_{L^2((0,T) \times T^*M)} + \|g\|_{L^2((0,T) \times T^*M)}).$$
Assuming that $V$ is smooth enough, we may obtain the optimal Sobolev regularity $H^{1/2}$ (instead of $H^{1/4}$), see Remark \[rk-smooth\].
In local charts, we have $$\pi_*(f \Psi)(x,\xi) = \int_{{{\mathbb R}}^d} {f}(x,\xi) \Psi(x,\xi) \frac{1}{\sqrt{\det(g(x))}} \, d\xi.$$ and $f$ satisfies the kinetic equation $${{\partial}}_t f + g^{i,j}(x) \xi_j {{\partial}}_{x_i} f - \left(\frac12 {{\partial}}_{x_i} g^{j,k}(x) \xi_j \xi_k + {{\partial}}_{x_i} V(x) \right) {{\partial}}_{\xi_i} f = g.$$ We use the change of variables $\overline{f}(t,x,v^i) = {f}(t,x,g_{i,j}v^j)$ (we define as well $\overline{g}$ and $\overline{\Psi}$), which satisfies the equation $${{\partial}}_t \overline{f} + v^j {{\partial}}_{x_i} \overline{f} - \left(\Gamma^i_{j,k}(x)v^j v^k+ {{\partial}}_{x_i} V(x) \right) {{\partial}}_{v_i} \overline{f} = \overline{g},$$ where $\Gamma^i_{j,k}(x) = \frac12 g^{i \ell}(x)\left({{\partial}}_{x_j}g_{k \ell}(x) + {{\partial}}_{x_k}g_{j \ell}(x) - {{\partial}}_{x_\ell}g_{j k}(x) \right)$ are the Christoffel symbols. Using a classical averaging lemma (see in Theorem \[thmmoyenne\] in Appendix \[section-AL\] with $m=1$ and $s=0$), we deduce that $$\left\|\int_{{{\mathbb R}}^d} \overline{f} \, \overline{\Psi} \sqrt{\det(g(x))} \, dv \right\|_{H^{1/4}((0,T) \times {{\mathbb R}}^d \times {{\mathbb R}}^d)} \leq C (\| \overline{f}|_{t=0}\|_{L^2((0,T) \times {{\mathbb R}}^d\times {{\mathbb R}}^d)} + \|\overline{g}\|_{L^2((0,T) \times{{\mathbb R}}^d\times {{\mathbb R}}^d)} ) .$$ Going back to the original variables, we deduce that $$\left\|\int_{{{\mathbb R}}^d} {f} \, {\Psi} \frac{1}{\sqrt{\det(g(x))}} \, dv \right\|_{H^{1/4}((0,T) \times {{\mathbb R}}^d \times {{\mathbb R}}^d)} \leq C (\| {f}|_{t=0}\|_{L^2((0,T) \times{{\mathbb R}}^d\times {{\mathbb R}}^d)} + \|{g}\|_{L^2((0,T) \times{{\mathbb R}}^d\times {{\mathbb R}}^d)} ),$$ which proves our claim.
Equipped with this tool (more generally the analogues of all averaging lemmas of Appendix \[section-AL\] can be obtained as well), we have the following analogue of the general convergence result of Theorem \[thmconv-general\] (which includes Theorems \[thmconv-intro\] and \[thmconvgene-intro\]). The same proof applies with only minor adaptations. Recall that the sets $(U_j)_{j \in J}$ are defined in .
\[thmconv-manifold\] The following statements are equivalent.
1. The set $\omega$ satisfies the [**generalized**]{} Unique Continuation Property.
2. The set $\omega$ satisfies the a.e.i.t. GCC.
3. For all $f_0 \in {\mathcal{L}}^2$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\left\|f(t)-Pf_0 \right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$ where $$P f_0 (x,v) = \sum_{j\in J} \frac{1}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}} \left( \int_{U_j} f_0 \, d\omega^d \right)f_j.$$ and the $(U_j)_{j \in J}$ are defined in and the $f_j = \frac{ \mathds{1}_{U_j} e^{-V} {\mathcal{M}}}{\| \mathds{1}_{U_j} e^{-V} {\mathcal{M}}\|_{{\mathcal{L}}^2}}$.
4. For all $f_0 \in {\mathcal{L}}^2$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\left\|f(t)-Pf_0 \right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$ where $Pf_0$ is a stationary solution of .
We obtain as well the analogue of Theorem \[thmexpo-intro\]. As in the torus case, we make the additional technical assumption: [**A3’.**]{} Assume that there exists a continuous function $\varphi(x,\xi):= \Theta \circ H(x,\xi)$, with $\varphi \geq 1$, such that for all $(x,\xi) \in T^*M$, we have $$\int_{T^*_xM} k(x,\xi,\xiÕ) \, dm_x(\xiÕ) \leq \varphi(x,\xi)$$ and $$\sup_{x \in M } \int_{T^*_xM \times T^*_xM} k^2(x,\xiÕ,\xi) \frac{{\mathcal{M}}(\xiÕ)}{{\mathcal{M}}(\xi)} \left(\frac{\varphi(x,\xi)}{\varphi(x,\xi')}-1\right)^2 \, dm_x(\xi) dm_x(\xiÕ) <+\infty$$
\[thmexpo-manifold\] Assume that the collision kernel satisfies [**A3’**]{}. The following statements are equivalent:
1. $C^-(\infty) > 0$.
2. There exists $C>0, \gamma>0$ such that for any $f_0 \in {\mathcal{L}}^2$, the unique solution to with initial datum $f_0$ satisfies for all $t\geq 0$ $$\begin{gathered}
\label{decexpo-manifold}
\left\|f(t)-\left( \int_{T^*M} f_0 \, d\omega^d \right) \frac{e^{-V(x)}}{\int_{M} e^{-V(x)} \, d\operatorname{Vol}(x)} {\mathcal{M}}\right\|_{{\mathcal{L}}^2} \\
\leq C e^{-\gamma t}\left\|f_0-\left( \int_{T^*M} f_0 \, d\omega^d \right) \frac{e^{-V(x)}}{\int_{M} e^{-V(x)} \, d\operatorname{Vol}(x)} {\mathcal{M}}\right\|_{{\mathcal{L}}^2} .\end{gathered}$$
3. There exists $C>0, \gamma>0$ such that for any $f_0 \in {\mathcal{L}}^2$, there exists $Pf_0$ a stationary solution of such that the unique solution to with initial datum $f_0$ satisfies for all $t \geq 0$, $$\label{decexpogene-manifold}
\left\|f(t)-Pf_0\right\|_{{\mathcal{L}}^2}
\leq C e^{-\gamma t} \left\|f_0-Pf_0\right\|_{{\mathcal{L}}^2}.$$
As a particular case of Theorem \[thmconv-manifold\], we have the following corollary.
Assume that $V=0$ and $\omega=T^*\omega_x$, where $\omega_x$ is a non-empty open subset of $M$. Suppose that the dynamics associated to $(\phi_t)_{t \geq 0}$ on $$S^* M = \left\{ (x,\xi) \in T^*M, \, \frac12 |\xi|_x^2 = 1 \right\},$$ is ergodic. Then for all $f_0 \in {\mathcal{L}}^2$, denoting by $f(t)$ the unique solution to with initial datum $f_0$, we have $$\label{convergeto0-manifold-re}
\left\|f(t)-\left(\int_{T^*M} f_0 \, d\omega^d\right)\frac{1}{|\operatorname{Vol}(M)|} {\mathcal{M}}(v)\right\|_{{\mathcal{L}}^2} \to_{t \to +\infty} 0,$$
Note that if the dynamics of $(\phi_t)_{t \geq 0}$ is ergodic on $S^* M$, then it is also ergodic on cosphere bundles of any positive radius (since for $V=0$, the flow is homogeneous of degree one).
Classical examples of Riemannian manifolds satisfying this dynamical assumption are given by compact Riemannian manifolds with negative curvature.
The case of compact phase spaces {#compactPS}
================================
Instead of studying the linear Boltzmann equation on the “whole” phase space $T^*M$ or $\Omega \times {{\mathbb R}}^d$, it is possible to consider this equation set on the “reduced” phase spaces $$\begin{aligned}
B_H^*M &= \{(x, \xi) \in T^*M, H(x, \xi) \leq R\},\quad
\quad
S_H^*M = \{(x, \xi) \in T^*M , H(x, \xi) = R\}, \\
&\text{or} \quad \mathcal{R}_H^*M = \{(x, \xi) \in T^*M , R\leq H(x, \xi) \leq R'\},\end{aligned}$$ for $R'>R>0$, or (with similar definitions) on $B_H^*\Omega$, $S_H^*\Omega$ or $ \mathcal{R}_H^*\Omega$. Note that by continuity, the potential $V$ is always bounded from below (and above), so that $B_H^*M$, $ S_H^*M$ and $\mathcal{R}_H^*M$ (as well as $B_H^*\Omega$, $S_H^*\Omega$ and $\mathcal{R}_H^*\Omega$) are compact. The linear Boltzmann equation is well-posed in $L^2(B_H^*M)$ (resp. $L^2(B_H^*\Omega)$), $L^2(S_H^*M)$ (resp. $L^2(B_H^*\Omega)$) or $L^2(\mathcal{R}_H^*M)$ (resp. $L^2(\mathcal{R}_H^*\Omega)$), in particular because the hamiltonian is preserved by the dynamics. The case of $S_H^*M$ is for instance relevant for the equations of radiative transfer or neutronics.
The analogues of Theorems \[thmconv-intro\], \[thmexpo-intro\], \[thmconv-general\] still hold in this framework, replacing the former phase space by $B_H$, $S_H$ or $\mathcal{R}_H$ in the various geometric conditions. For the sake of conciseness, we do not write these results again. All proofs remain valid, with some simplifications, since the phase space is now compact. Note that the fact that $C^-(\infty)>0$ is equivalent to GCC in this compact case.
A stabilization criterion {#stabob}
=========================
We provide here a characterization of exponential decay for dissipative evolution equations. The following lemma is very classical and we reproduce it here for the convenience of the reader.
\[lemfondamental\] Consider the evolution equation $$\label{pde}
\left\{
\begin{aligned}
\partial_t f + L f = 0, \\
f_{|t=0} = f_0,
\end{aligned}
\right.$$ assumed to be:
- globally wellposed in some functional space $X$ in the sense that for any $f_0 \in X$, there is a unique $f \in C^0_t(X)$ solution to ,
- invariant by translation in time, in the sense that if $f \in C^0_t(X)$ is the solution of , then for all $t_0\geq 0$, $g(t) := f(t+t_0)$ is the unique solution of $$\left\{
\begin{aligned}
\partial_t g + L g = 0, \\
g_{|t=0} = f_{|t=t_0}.
\end{aligned}
\right.$$
Let $E(f)$ and $D(f)$ be two non-negative functionals defined for all $f \in X$, and such that if $f$ is a solution to , $$\label{eqidentity}
\text{ for all } t \geq 0, \quad \frac{d}{dt} E(f(t)) = - D(f(t)).$$
Then, the following two properties are equivalent:
1. There exist $C,\gamma>0$ such that for all $f(0) \in X$, the associated solution $f$ to satisfies $$\label{eqfonda1}
\text{ for all } t \geq 0, \quad E(f(t))\leq C e^{-\gamma t} E(f(0)).$$
2. There exists $T>0$ and $K>0$ such that for all $f(0) \in X$, the associated solution $f$ to satisfies $$\label{eqfonda2}
K \int_0^T D(f(t)) \, dt \geq E(f(0)).$$
For the sake of completeness, we provide a short proof of this lemma.
$(1) \Rightarrow (2)$ Assume that $(1)$ holds. Let $T_0>0$ such that $C e^{-\gamma T_0} = \frac{1}{2}$. Then, after integrating betwen $0$ and $T_0$, we have: $$E(f(T_0))- E(f(0)) = - \int_0^{T_0} D(f(t)) \, dt,$$ so that, by , $$\int_0^{T_0} D(f(t)) \, dt \geq E(f(0)) - C e^{-\gamma T_0} E(f(0)) = \frac{1}{2} E(f(0)),$$ and we can therefore take $T =T_0$ and $C=2$ in .
$(2) \Rightarrow (1)$ Assume that $(2)$ holds. Here (and only here), we need the property of invariance by time translations for . By and , we have $$E(f(T)) \leq \left(1-\frac{1}{K} \right) E(f(0)).$$ Note that the assumption $E(f) \geq 0$ implies in particular that $K\geq 1$. We may assume that $K>1$. Indeed, for $K=1$, we have $E(f(t)) = 0$ for all $t \geq T$ so that for any $\gamma>0$, there exists $C>0$ such that (1) holds. By invariance by translation in time of , one likewise obtains $$E(f(2T)) \leq \left(1-\frac{1}{K} \right) E(f(T)).$$ Thus, by a straightforward induction, for any $k \in {{\mathbb N}}$, we have the bound: $$E(f(kT)) \leq \left(1-\frac{1}{K} \right)^k E(f(0)).$$ Defining $\gamma_0 := \frac{- \log\left(1-\frac{1}{K} \right)}{T}>0$ and $C_0 := \left(1-\frac{1}{K} \right)^{-1} = e^{\gamma_0 T}>0$, we can now check that $$\text{ for all } t \geq 0, \quad E(f(t))\leq C_0 e^{-\gamma_0 t} E(f(0)).$$ Indeed, let $t\geq 0$ and $k \in {{\mathbb N}}$ such that $t \in [kT, (k+1) T[$; since $E(f(\cdot))$ is decreasing (see ), we have $$E(f(t)) \leq E(f(kT)) \leq \left(1-\frac{1}{K} \right)^k E(f(0)) = e^{-\gamma_0 k T} E(f(0)) \leq C_0 e^{-\gamma_0 t} E(f(0)),$$ which concludes the proof.
Velocity averaging lemmas {#section-AL}
=========================
Velocity averaging lemmas play an important role in many proofs of this paper. In this appendix, we recall some classical results and also state the versions precisely adapted to our needs.
Kinetic transport equations are hyperbolic partial differential equations and as it can be seen from Duhamel’s formula, there is propagation of potential singularities at initial time and/or from a source in the equations. Thus there is no hope that the solution of a kinetic equation becomes more regular than the initial condition.
It was nevertheless observed by Golse, Perthame and Sentis [@GPS] that the averages in velocity of the solution of a kinetic transport equation enjoy extra regularity/compactness properties (see also the independent paper of Agoshkov [@Ago]). We refer to the by now classical paper of Golse, Lions, Perthame, Sentis [@GLPS], DiPerna, Lions [@DPL1], DiPerna, Lions, Meyer [@DPLM], Bézard [@B94] for quantitative estimates of this compactness property in various settings of increasing complexity.
We also refer to the review paper of Jabin [@Jab] and to the recent work of Arsénio and Saint-Raymond [@ASR].
Velocity averaging lemmas in ${{\mathbb R}}^d$
----------------------------------------------
We start by recalling classical averaging lemmas in the whole space ${{\mathbb R}}^d$. There are also versions of these lemmas for $p\in (1,\infty)$, but we stick to the case $p=2$, which is sufficient for our needs.
\[thmmoyenne\] Let $s\in [0,1)$ and $m \in {{\mathbb R}}^+$.
1. For any $T>0$ and any [*bounded*]{} open sets $\Omega_x ,\Omega_v \subset {{\mathbb R}}^d$, there exists a constant $C>0$ such that for all $\Psi \in C^{\infty}_c(\mathbb{R}^d)$ supported in $\Omega_v$ and all $f,g \in L^2_{loc}({{\mathbb R}}\times {{\mathbb R}}^d \times {{\mathbb R}}^d)$ satisfying $$\partial_t f + v \cdot \nabla_x f = (1- \Delta_{t,x})^{s/2}(1- \Delta_{v})^{m/2} g,$$ we have $$\label{avlemmaRd}
\| \rho_{\Psi} \|_{H^{\alpha}([0,T] \times \Omega_x)} \leq C \left(\| f \|_{L^2([0,T] \times \Omega_x \times \Omega_v )} + \| g \|_{L^2([0,T] \times \Omega_x \times \Omega_v )}\right),$$ where $\rho_{\Psi}(t,x):=\int_{{{\mathbb R}}^d} f(t,x,v)\Psi(v)dv$ and $\alpha= {\frac{(1-s)}{2(1+m)}}$.
2. Let $T>0$ and $(f_n)_{n\in {{\mathbb N}}}$ and $(g_n)_{n\in {{\mathbb N}}}$ be two sequences of $L^2(0,T ; L^2_{loc}( {{\mathbb R}}^d \times {{\mathbb R}}^d))$ such that the following holds $$\partial_t f_n + v \cdot \nabla_x f_n = (1- \Delta_{t,x})^{s/2}(1- \Delta_{v})^{m/2} g_n,$$ with $s \in [0,1), m \geq 0$. Assume that for any [*bounded*]{} open sets $\Omega_x ,\Omega_v \subset {{\mathbb R}}^d$, there exists $C_1>0$, such that for all $n \in {{\mathbb N}}$, $$\label{ass2.0}
\| f_n \|_{L^2((0,T) \times \Omega_x \times \Omega_v)} + \| g_n \|_{L^2((0,T) \times \Omega_x \times \Omega_v)} \leq C_1.$$ Then, for any $\Psi \in C^{\infty}_c(\mathbb{R}^d)$, the sequence $(\rho_{\Psi,n})_{n\in {{\mathbb N}}}$ defined for $n \in {{\mathbb N}}$ by $$\rho_{\Psi,n}(t,x):=\int_{{{\mathbb R}}^d} f_n(t,x,v)\Psi(v)dv$$ is relatively compact in $L^2(0,T ; L^2_{loc}( {{\mathbb R}}^d))$.
\[rk-smooth\] In the main part of the paper, we apply this averaging lemma to the Boltzmann equation by writing it under the form $$\partial_t f + v \cdot \nabla_x f = \nabla_x V \cdot \nabla_v f + \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv' .$$ To this end, we consider the case $s=0, m=1$ in Theorem \[thmmoyenne\]. This implies that the averages in $v$ belong to the Sobolev space $H^{1/4}$.
Nevertheless, assuming that the potential $V$ is smooth enough, we can also use the approach of Berthelin-Junca [@BJ] to obtain the optimal Sobolev space $H^{1/2}$ for these averages (which is not needed in this paper).
We also have a version of these lemmas for kinetic transport equations set in general Riemannian manifolds, see Lemma \[lem-moyenne-variete\].
We now state the result as needed in the main part of this work. Assuming an extra uniform integrability, we can deduce some compactness on moments of $f$ without having to consider compactly supported test functions in $v$. This is the purpose of the next result, which is actually the version of averaging lemmas used most of the time in this work.
\[lemmoyenne\] Let $\Omega_x$ be a bounded open set of ${{\mathbb R}}^d$, $T>0$, and $(f_n)_{n\in {{\mathbb N}}}$, $(g_n)_{n\in {{\mathbb N}}}$ be two sequences of $L^2(0,T ; L^2_{loc}( \Omega_x \times {{\mathbb R}}^d))$ satisfying $
\partial_t f_n + v \cdot \nabla_x f_n = (1- \Delta_{v})^{m/2} g_n,
$ for some $ m \geq 0$. Suppose that there exists $V \in L^\infty$ such that for any [bounded]{} open set $\Omega_v \subset {{\mathbb R}}^d$, there exists $C_0>0$ such that, for any $n \in {{\mathbb N}}$, $$\label{eqmoybound1}
\text{ for all } t\geq 0,\quad \| f_n\|_{{\mathcal{L}}^2(\Omega_x \times {{\mathbb R}}^d)}^2 := \int_{\Omega_x} \int_{{{\mathbb R}}^d} |f_n|^2 \, \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx \leq C_0, \qquad \| g_n \|_{L^2((0,T) \times \Omega_x \times \Omega_v)} \leq C_0 .$$ Assume moreover that there is $f \in L^\infty(0,T ;L^2(\Omega_x \times {{\mathbb R}}^d))$ such that $f_n \rightharpoonup f$ weakly$-\star$ in $L^\infty(0,T ; {\mathcal{L}}^2(\Omega_x \times {{\mathbb R}}^d))$. Consider $\rho_{n}(t,x):=\int_{{{\mathbb R}}^d} f_n(t,x,v) \,dv$. Then up to a subsequence, we have $$\label{eqconclu1}
\rho_n {\mathcal{M}}(v) \rightarrow \left(\int_{{{\mathbb R}}^d} f \, dv\right) {\mathcal{M}}(v) , \quad \text{strongly in } L^2(0,T ;{\mathcal{L}}^2(\Omega_x \times {{\mathbb R}}^d)) ,$$ and for any continuous kernel $k(\cdot,\cdot,\cdot): {{\mathbb R}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d \to {{\mathbb R}}$ satisfying [**A3**]{}, we have $$\label{eqconclu2}
\int_{{{\mathbb R}}^d} k(x,v',v) f_n(t,x,v') \, dv' \rightarrow \int_{{{\mathbb R}}^d} k(x,v',v) f(t,x,v') \, dv' , \quad \text{strongly in } L^2(0,T ; {\mathcal{L}}^2(\Omega_x \times {{\mathbb R}}^d)) .$$
First note that by Fatou’s lemma, $f$ satisfies: $$\label{eqmoybound2}
\text{ for all } t\geq 0, \quad \|f\|_{{\mathcal{L}}^2}^2 = \int_{\Omega_x} \int_{{{\mathbb R}}^d} |f|^2 \, \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx \leq C_0.$$
Since $\rho_n$ does not depend on $v$, proving is equivalent to show that: $$\rho_n e^{V} \rightarrow \int_{{{\mathbb R}}^d} f \, dv \, e^{V}, \quad \text{strongly in } L^2((0,T) \times \Omega_x).$$ Let $\Psi \in C^{\infty}_c({{\mathbb R}})$ such that $\Psi = 1$ in a neighborhood of $0$, and define $\Psi_R(v) = \Psi(\frac{|v|}{R})$, $v\in {{\mathbb R}}^d$.
By Theorem \[thmmoyenne\], we can assume, up to a subsequence, that $$\label{convcutoff}
\rho_{\Psi_R,n} := \int_{{{\mathbb R}}^d} f_n \Psi_R \, dv \rightarrow \int_{{{\mathbb R}}^d} f \Psi_R \, dv, \quad \text{strongly in } L^2((0,T) \times \Omega_x).$$ Let ${{\varepsilon}}>0$. We can write the decomposition: $$\begin{aligned}
& \rho_n - \int_{{{\mathbb R}}^d} f \, dv = A_1 + A_2 +A_3 , \quad \text{with }\\
& A_1 = \left( \rho_{\Psi_R,n} - \int_{{{\mathbb R}}^d} f \Psi_R \, dv \right) , \quad
A_2 = \int_{{{\mathbb R}}^d} f_n (1-\Psi_R) \, dv , \quad
A_3 = - \int_{{{\mathbb R}}^d} f (1-\Psi_R) \, dv .
\end{aligned}$$ By Cauchy-Schwarz inequality, using , we have for all $n \in {{\mathbb N}}$ and all $t \in (0,T)$, $$\begin{aligned}
\left\| \int_{{{\mathbb R}}^d} f_n (1-\Psi_R) \, dv \,e^{V} \right \|_{L^2(\Omega_x)}^2 &\leq \int \left(\int_{{{\mathbb R}}^d} |f_n|^2 \frac{1}{{\mathcal{M}}(v)} \, dv\right) \left(\int_{{{\mathbb R}}^d} (1-\Psi_R)^{2}{{\mathcal{M}}(v)} \, dv\right) e^{2V(x)} \, dx\\
&\leq C_0 \|e^V\|_{L^\infty(\Omega_x)} \left( \int_{{{\mathbb R}}^d} (1-\Psi_R)^{2}{{\mathcal{M}}(v)} \, dv \right).
\end{aligned}$$ As a consequence, there exists $R_0>0$ large enough such that for all $R\geq R_0$ and all $n \in {{\mathbb N}}$, we have $$\|e^V A_2\|^2_{L^2((0,T) \times \Omega_x)} \leq \frac{{{\varepsilon}}}{3} .$$ Likewise, we use to get for all $n \in {{\mathbb N}}$, $$\| e^{V} A_3 \|_{L^2((0,T) \times \Omega_x)}^2 \leq \frac{{{\varepsilon}}}{3}.$$ Using ($R$ is now fixed), there is $N\geq 0$ such that for any $n\geq N$, $$\left\| \left( \rho_{\Psi_R,n} - \int_{{{\mathbb R}}^d} f \Psi_R \, dv\right) e^{V}\right\|_{L^2((0,T)\times \Omega_x)} \leq {{\varepsilon}}/3,$$ from which we infer that $$\left\| \left( \rho_n - \int_{{{\mathbb R}}^d} f \, dv\right) e^{V}\right\|_{L^2((0,T)\times \Omega_x)} \leq {{\varepsilon}}$$ and this concludes the proof of .
For the proof of , let us first assume for a while that $k$ is smooth (namely for all $x$, $k(x, \cdot, \cdot)$ belongs to the $C^\infty$ class). We first have to be careful about the integration in the velocity variable. The convergence in results from the following two facts:
- For all $v \in {{\mathbb R}}^d$, we have the following convergence $$\int_{{{\mathbb R}}^d} k(x,v',v) f_n(t,x,v') \, dv' \rightarrow \int_{{{\mathbb R}}^d} k(x,v',v) f(t,x,v') \, dv' \quad \text{strongly in } L^2(0,T;L^2_{x}(\Omega_x)).$$ This follows from a truncation argument and Theorem \[thmmoyenne\], exactly as for $\rho_n$. Keeping the same notations, the only difference is that we have to study $$\begin{gathered}
\int_{{{\mathbb R}}^d} (1-\Psi_R)^{2}k(x,v',v){{\mathcal{M}}(v')} \, dv' \\
\leq \left( \int_{{{\mathbb R}}^d} (1-\Psi_R)^{2}k^2(x,v',v) {{\mathcal{M}}(v')} \, dv' \right)^{1/2} \left( \int_{{{\mathbb R}}^d} (1-\Psi_R)^{2}{{\mathcal{M}}(v')} \, dv' \right)^{1/2},
\end{gathered}$$ which is, using [**A3**]{}, small for $R$ large enough.
- By the Cauchy-Schwarz inequality and the bound , we have $$\begin{aligned}
&\int_0^T \int_{\Omega_x} \left(\int_{{{\mathbb R}}^d} k(x,v',v) (f_n-f)(t,x,v') \, dv' \right)^2 \frac{e^V}{{\mathcal{M}}(v)}\, dx \, dt \\
& \leq \int_0^T \left(\sup_{x \in \Omega_x} \int_{{{\mathbb R}}^d} k^2(x,v',v) \frac{{\mathcal{M}}(v')}{{\mathcal{M}}(v)} \, dv' \right) \int_{\Omega_x}\left(\int_{{{\mathbb R}}^d} \frac{|f_n-f|^2(t,x,v')}{{\mathcal{M}}(v')}\, dv' \right) e^V \, dx \, dt \\
& \leq C_0 T \sup_{x \in \Omega_x} \int_{{{\mathbb R}}^d} k^2(x,v',v) \frac{{\mathcal{M}}(v')}{{\mathcal{M}}(v)} \, dv' ,
\end{aligned}$$ which is independent of $n$ and in $L^1(dv)$, since by [**A3**]{}, we have $$\sup_{x \in \Omega_x} \int_{{{\mathbb R}}^d \times {{\mathbb R}}^d} k^2(x,v',v) \frac{{\mathcal{M}}(v')}{{\mathcal{M}}(v)} \, dv' \, dv < +\infty.$$
Hence, by Lebesgue dominated convergence theorem, we deduce .
We now use an approximation argument to handle the general case, i.e. when $k$ is only assumed to be continuous. Consider $(\phi_\delta)_{\delta>0}$ a family of mollifiers in $C^\infty_c({{\mathbb R}}^d \times {{\mathbb R}}^{d})$ for the measure $ {\mathcal{M}}(v){\mathcal{M}}(v') \, dv' dv$. We set for all $x,v,v'$ $$\tilde{k}_\delta(x,v,v') =\left(\tilde{k}(x,\cdot,\cdot) \star \phi_\delta(\cdot,\cdot)\right)(v,v'), \quad k_\delta(x,v,v')= \tilde{k}_\delta(x,v,v') {\mathcal{M}}(v').$$ We use the following classical properties of mollifiers:
- for all $x$, $k_\delta(x, \cdot,\cdot)$ is in the $C^\infty$ class;
- we have for all $\delta>0$ $$\label{prop1mol}
\sup_{x \in \Omega_x} \| \tilde k- k_\delta \|_{L^2({\mathcal{M}}(v){\mathcal{M}}(v') \, dv' dv)} \to_{\delta\to0} 0.$$
Let ${{\varepsilon}}>0$. We write the decomposition $$\left\|\int_{{{\mathbb R}}^d} k(x,v',v) f_n(t,x,v') \, dv' - \int_{{{\mathbb R}}^d} k(x,v',v) f(t,x,v') \, dv' \right\|_{{\mathcal{L}}^2}^2
\leq 2A_1 + 2 A_2 ,$$ with $$\begin{aligned}
A_1 &= \left\|\int_{{{\mathbb R}}^d} k_\delta(x,v',v) f_n(t,x,v') \, dv' - \int_{{{\mathbb R}}^d} k_\delta(x,v',v) f(t,x,v') \, dv' \right\|^2_{{\mathcal{L}}^2}, \\
A_2 &= \left\|\int_{{{\mathbb R}}^d} (k-k_\delta)(x,v',v) (f_n-f)(t,x,v') \, dv' \right\|^2_{{\mathcal{L}}^2}.\end{aligned}$$ We estimate $A_2$ as follows, using $$\begin{aligned}
A_2 &= \int_{{{\mathbb R}}^d} {\mathcal{M}}(v) \int_{\Omega_x} \left(\int_{{{\mathbb R}}^d} (\tilde{k}-\tilde{k}_\delta) (x,v',v) (f_n-f)(t,x,v') \, dv' \right)^2 e^V \, dx \, dv \\
& \leq \sup_{x\in \Omega_x} \int_{{{\mathbb R}}^d\times {{\mathbb R}}^d} |\tilde{k}-\tilde{k}_\delta|^2 (x,v',v) {\mathcal{M}}(v'){\mathcal{M}}(v) \, dv' \, dv \left( \int_{\Omega_x} \int_{{{\mathbb R}}^d} \frac{|f_n-f|^2(t,x,v')}{{\mathcal{M}}(v')} e^V \, dv' \, dx \right) \\
&\leq 4 C_0^2 \sup_{x\in \Omega_x} \int_{{{\mathbb R}}^d\times {{\mathbb R}}^d} |\tilde{k}-\tilde{k}_\delta|^2 (x,v',v) {\mathcal{M}}(v'){\mathcal{M}}(v) \, dv' \, dv\end{aligned}$$ Using , we fix $\delta>0$ small enough so that for all $n\in {{\mathbb N}}$, $$A_2 \leq \big({{\varepsilon}}/(4T)\big)^{1/2}.$$ and thus for all $n\in {{\mathbb N}}$, we have $$\| A_2 \|^2_{L^2(0,T)} \leq {{\varepsilon}}/4.$$ For $A_1$, we use the above analysis in the smooth case to deduce that we can take $N$ large enough to get for all $n\geq N$, $$\| A_1 \|^2_{L^2(0,T)} \leq {{\varepsilon}}/4.$$ Finally, we have proven that for any ${{\varepsilon}}>0$, there is $N$ such that for all $n\geq N$, $$\left\|\int_{{{\mathbb R}}^d} k(x,v',v) f_n(t,x,v') \, dv' - \int_{{{\mathbb R}}^d} k(x,v',v) f(t,x,v') \, dv' \right\|_{L^2(0,T;{\mathcal{L}}^2)}^2 \leq {{\varepsilon}},$$ which concludes the proof of the convergence.
Velocity averaging lemmas in open sets with boundary
----------------------------------------------------
We now consider the case of equations set in open sets of ${{\mathbb R}}^d$ with boundary.
A first result is the following localized averaging lemma, which shows the interior regularity of velocity averages. It is obtained from the whole space case after a standard localization procedure and does not depend on the prescribed boundary conditions.
\[thmmoyenne-domain\] Let $\Omega$ be an open set of ${{\mathbb R}}^d$ and $m\geq 0$. Let $f,g \in L^2_{loc}({{\mathbb R}}^+, L^2(\Omega \times {{\mathbb R}}^d))$ satisfying $$\begin{aligned}
&\partial_t f + v \cdot \nabla_x f = (1- \Delta_{v})^{m/2}g, \quad (x,v) \in \Omega \times {{\mathbb R}}^d. \end{aligned}$$
Then for all $\psi \in C^{1}_c(\Omega \times \mathbb{R}^d)$, and all $T>0$, there exists $C>0$, such that $$\rho_{\psi}(t,x):=\int_{{{\mathbb R}}^d} f(t,x,v)\psi(x,v)dv$$ satisfies $$\| \rho_{\psi} \|_{H^{\frac{1}{2(1+m)}}((0,T) \times \Omega)} \leq C \left(\| f \|_{L^2(0,T; L^2(\Omega \times {{\mathbb R}}^d))} + \| g \|_{L^2(0,T ;L^2(\Omega \times {{\mathbb R}}^d))}\right).$$
We now formulate another compactness result with an additional uniform equi-integrability assumption on the sequence. Note here that only compactness is obtained, not uniform regularity up to the boundary, and that we again do not use the boundary conditions.
The problem of finding uniform regularity up to the boundary (without the equi-integrability assumption) seems difficult and is clearly beyond the scope of this paper. For this question, the precise boundary condition satisfied by the solution of the kinetic transport equation should play a key role (whereas it does not play any role in the results of this section).
\[def-equi\] Let $d\mu$ be a positive measure on the phase space $\Omega \times {{\mathbb R}}^d$. We say that a sequence $(g_n)_{n \in {{\mathbb N}}}$ of $L^1(d\mu)$ is equi-integrable (with respect to $d\mu$) if for any ${{\varepsilon}}>0$, there exists $\delta >0$ such that for any measurable subset $A \subset \Omega \times {{\mathbb R}}^d$ satisfying $\mu(A) \leq \delta$, we have $$\sup_{n \in {{\mathbb N}}} \int_{A} |f_n| d \mu \leq {{\varepsilon}}.$$
\[thmmoyenne-domain2\] Let $\Omega$ be an open subset of ${{\mathbb R}}^d$ and $V \in L^\infty(\Omega)$. Fix $T>0$ and $m \geq 0$. Let $(f_n)_{n\in {{\mathbb N}}}$ and $(g_n)_{n\in {{\mathbb N}}}$ be two sequences of $L^2(0,T ; L^2(\Omega \times {{\mathbb R}}^d))$ satisfying in $D'({{\mathbb R}}^+ \times \Omega \times {{\mathbb R}}^d)$ the equation $$\partial_t f_n + v \cdot \nabla_x f_n = (1- \Delta_{v})^{m/2} g_n .$$ Assume that for all open sets $\Omega_x \subset \Omega, \Omega_v \subset {{\mathbb R}}^d$ such that $\overline{\Omega_x} \subset \Omega$, there exists $C_1>0$, such that for all $n \in {{\mathbb N}}$, $$\label{ass1-boun}
\| f_n \|_{L^2((0,T) \times \Omega_x \times \Omega_v)} + \| g_n \|_{L^2((0,T) \times \Omega_x \times \Omega_v)} \leq C_1$$ and that for any $n \in {{\mathbb N}}$, $$\label{eqmoybound1-boun}
\sup_{t\in (0,T)} \|f_n\|_{{\mathcal{L}}^2}^2 = \sup_{t\in (0,T)} \int_{\Omega} \int_{{{\mathbb R}}^d} |f_n|^2 \, \frac{e^{V(x)}}{{\mathcal{M}}(v)} \, dv \, dx \leq C_0.$$ Assume in addition that the sequence $(|f_n|^2)_{n\in {{\mathbb N}}}$ is equi-integrable with respect to the measure $d \mu:=e^{V}/{\mathcal{M}}\, dv dx$. Consider $\rho_{n}(t,x):=\int_{{{\mathbb R}}^d} f_n(t,x,v) \,dv$. Suppose that $f_n \rightharpoonup f$ weakly$-\star$ in $L^\infty(0,T; {\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d))$. Then up to a subsequence, we have $$\label{eqconclu1-boun}
\rho_n {\mathcal{M}}(v) \rightarrow \left(\int_{{{\mathbb R}}^d} f \, dv\right) {\mathcal{M}}(v) , \quad \text{strongly in } L^2(0,T; {\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d))$$ and for any continuous kernel $k(\cdot,\cdot,\cdot): \overline{\Omega}\times {{\mathbb R}}^d \times {{\mathbb R}}^d \to {{\mathbb R}}$ satisfying [**A3**]{}, we have $$\label{eqconclu2-boun}
\int_{{{\mathbb R}}^d} k(x,v',v) f_n(t,x,v') \, dv' \rightarrow \int_{{{\mathbb R}}^d} k(x,v',v) f(t,x,v') \, dv' , \quad \text{strongly in } L^2(0,T;{\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d)).$$
The result follows from a localization and approximation argument. Consider $(\Phi_k(x))_{k \in {{\mathbb N}}}$ a sequence of smooth approximations of unity in $\Omega$, such that for any $k \in {{\mathbb N}}$, there exists $C_k>0$ with $$\| \Phi_k \|_{L^{\infty}} \leq 1 , \quad \| \Phi_k \|_{W^{1,\infty}} \leq C_k.$$ Fix $T>0$. Take ${{\varepsilon}}>0$ and write the decomposition $$\rho_n - \rho = (\rho_n -\rho) \Phi_k + (\rho_n- \rho) (1-\Phi_k) =: B_1 + B_2.$$ For $B_2$, we write $$\begin{aligned}
\| \rho_n (1- \Phi_k ) {\mathcal{M}}(v)\|_{{\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d))}^2 &\leq \int |f_n|^2 (1-\Phi_k)^2 \frac{e^V}{{\mathcal{M}}} \, dv dx, \\
&\leq \int_{{{\rm supp}\,}(1-\Phi_k)\times {{\mathbb R}}^d} |f_n|^2 \frac{e^V}{{\mathcal{M}}} \, dv dx.\end{aligned}$$ Using the fact that $\operatorname{Leb}({{\rm supp}\,}(1-\Phi_k)) \to 0$ and the equiintegrability of $|f_n|^2$, we can consider $k$ large enough so that for all $n\in {{\mathbb N}}$, $$\| B_2\|_{ L^2(0,T;{\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d))} \leq {{\varepsilon}}/2.$$ Once $k$ is fixed, consider $\tilde{f}_n^k = \Phi_k f_n$, which satisfies the transport equation $$\partial_t \tilde{f}_n^k + v \cdot \nabla_x \tilde{f}_n^k = (1- \Delta_{v})^{m/2} (\Phi_k g_n) + v \cdot \nabla_x \Phi_k f_n,$$ For $B_1$, we can apply Corollary \[lemmoyenne\] and take $n$ large enough to ensure $$\| B_1\|_{ L^2(0,T;{\mathcal{L}}^2(\Omega \times {{\mathbb R}}^d))} \leq {{\varepsilon}}/2,$$ which concludes the first part of the proof.
The rest of the proposition is proved exactly as for Corollary \[lemmoyenne\].
Reformulation of some geometric properties {#GCCother}
==========================================
Proof of Lemma \[equiv-equiv\] {#prooflemequiv}
------------------------------
We first define another convenient equivalence relation.
\[def-sim-phi\] Given $\omega_1$ and $\omega_2$ two connected components of $\omega$, we say that $\omega_1 \bumpeq \omega_2$ if there is $N \in {{\mathbb N}}$ and $N$ connected components $(\omega^{i})_{1\leq i \leq N}$ of $\omega$ such that
- we have $\omega_1 {\, \mathcal{R} }_{\phi} \, \omega^{(1)}$,
- for all $1\leq i \leq N-1$, we have $\omega^{(i)} {\, \mathcal{R} }_{\phi} \, \omega^{(i+1)}$,
- we have $\omega^{(N)} {\, \mathcal{R} }_{\phi} \, \omega_2$.
The relation $\bumpeq$ is an equivalence relation on the set of connected components of $\omega$. For $\omega_1$ a connected component of $\omega$, we denote its equivalence class for $\bumpeq$ by $\{\omega_1\}$.
Then, the proof of Lemma \[equiv-equiv\] relies on the following lemma.
\[lem-classeq\] Let $\Omega_0$ be a connected component of $\bigcup_{s \in {{\mathbb R}}^+}\phi_{-s}(\omega)$ and let $(\omega_\ell)_{\ell \in L}$ be the connected components of $\omega$ such that for all $\ell \in L$, there exists $t\geq 0$ with $\phi_{-t} (\omega_\ell) \cap \Omega_0 \neq \emptyset$. Then, for all $\ell,\ell' \in L$, we have $\omega_\ell \bumpeq \omega_{\ell'}$.
Assume that there exist at least two equivalence classes for $\bumpeq$ among the $\omega_\ell$, $\ell \in L$. Let $\ell_0 \in L$ and consider $\{\omega_{\ell_0}\}$ the equivalence class of $\omega_{\ell_0}$ for $\bumpeq$. Defining $$U_1 := \bigcup_{U \in {\mathcal{C}\mathcal{C}}(\omega), \, U \in \{\omega_{\ell_0}\}} \bigcup_{t\geq 0} \phi_{-t} (U) \cap \Omega_0 \quad \text{ and } \quad U_2 := \bigcup_{U \in {\mathcal{C}\mathcal{C}}(\omega), \, U \notin \{\omega_{\ell_0}\}}\bigcup_{t\geq 0} \phi_{-t} (U) \cap \Omega_0,$$ we have by construction that $U_1, U_2 $ are two open non-empty subsets of $\Omega_0$ and that $U_1 \cup U_2 = \Omega_0$.
Let us check $U_1 \cap U_2 \neq \emptyset$: otherwise there would exist two connected components of $\omega$, $U_1 \in \{\omega_{\ell_0}\}$, $U_2 \notin \{\omega_{\ell_0}\}$ such that $U_1 {\, \mathcal{R} }_\phi U_2$, which is excluded by definition of the equivalence class.
This is a contradiction with the fact that $\Omega_0$ is connected.
We are now in position to prove Lemma \[equiv-equiv\].
Let us first prove that $\omega_1 \Bumpeq \omega_2 \Longrightarrow \Psi(\omega_1) \sim \Psi(\omega_2)$. It suffices to prove that $$\label{equivequiv}
\Big(\omega_1 {\, \mathcal{R} }_k \, \omega_2 \text{ or } \omega_1 {\, \mathcal{R} }_\phi \, \omega_2 \Big) \Longrightarrow \left(\Psi(\omega_1) {\, \mathcal{R} }_k \Psi(\omega_2) \text{ or } \Psi(\omega_1) {\, \mathcal{R} }_\phi \Psi(\omega_2) \right) .$$ The conclusion then follows from the iterative use of this argument.
If $\omega_1 {\, \mathcal{R} }_k \, \omega_2$, then $\Psi(\omega_1) {\, \mathcal{R} }_k \Psi(\omega_2)$. This follows from the fact that $\omega_j \subset \Psi(\omega_j)$ and the definition of ${\, \mathcal{R} }_k$. Similarly, if $\omega_1 {\, \mathcal{R} }_\phi \, \omega_2$, then $\Psi(\omega_1) {\, \mathcal{R} }_\phi \, \Psi(\omega_2)$.
Let us now prove that $\Psi(\omega_1) \sim \Psi(\omega_2) \Longrightarrow \omega_1 \Bumpeq \omega_2 $. According to Lemma \[lem-classeq\], it is sufficient to prove that $\Omega^{(1)}, \Omega^{(2)} $ being two given connected components of $\bigcup_{t\geq 0} \phi_{-t} (\omega)$, $$\label{equivequivbis}
\Omega^{(1)} {\, \mathcal{R} }_k \, \Omega^{(2)}
\Longrightarrow
\begin{array}{l}
\text{there exits two connected components } \omega_1^* \text{ and } \omega_2^* \text{ of } \omega \\
\text{such that }
\omega_1^* {\, \mathcal{R} }_k \, \omega_2^* \text{ and } \omega_1^* \subset \Omega^{(1)} , \, \omega_2^* \subset \Omega^{(2)} .
\end{array}$$ The conclusion then follows from an iterative use of this argument.
By definition of ${\, \mathcal{R} }_k$, there exist $(x,v_1,v_2) \in {{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d$ with $(x,v_1) \in\Omega^{(1)}$ and $(x,v_2) \in \Omega^{(2)}$ such that $k(x,v_1,v_2)>0$ or $k(x,v_2,v_1)>0$.
Note in particular that this implies $(x,v_1) ,(x,v_2) \in \omega$. Denoting by $\omega_1^*$ (resp. $\omega_2^*$) the connected component of $\omega$ such that $(x,v_1) \in \omega_1^*$ (resp. $(x,v_2) \in \omega_2^*$), we hence have $\omega_1^* {\, \mathcal{R} }_k \, \omega_2^*$. The conclusion of follows from the fact that $\omega_j^* \subset\Omega^{(j)}$, for $j=1,2$ and the definition of ${\, \mathcal{R} }_k$.
Almost everywhere geometric control conditions and connectedness {#appaeitgccconnected}
----------------------------------------------------------------
\[aeitgccconnected\] Here, $\Omega$ is either ${{\mathbb T}}^d$ or an open subset of ${{\mathbb R}}^d$. Let $\omega \subset \Omega \times {{\mathbb R}}^d$ be an open subset. Consider the following geometric properties:
(i) There exists $\tilde{\omega} \subset \omega$, $\tilde{\omega}$ [*connected*]{} satisfying the a.e.i.t. Geometric Control Condition;
(ii) The set $\omega$ satisfies the a.e.i.t. GCC and for any connected components $(\omega_1,\omega_2)$ of $\omega$, there exists $(x_0,v_0) \in \omega_1$ and $s \in {{\mathbb R}}$ such that $\phi_s(x_0,v_0) \in \omega_2$;
(iii) The set $\omega$ satisfies the a.e.i.t. GCC and $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ is connected.
Then $(i) \implies (iii)$ and $(ii) \implies (iii)$.
Before starting the proof, let us remark that is $\omega$ is a connected open subset of $\Omega \times {{\mathbb R}}^d$, then $ \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ is also a connected open subset. Indeed it is first an open subset of $\Omega \times {{\mathbb R}}^d$, and it is equivalent to show that it is path-connected. Let $y_1, y_2 \in \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$; there exists $s_1,s_2 \geq 0$ and $z_1, z_2 \in \omega$ such that $y_1 = \phi_{-s_1} (z_1)$ and $y_2 = \phi_{-s_2} (z_2)$. Since $\omega$ is a connected open subset of $\Omega \times {{\mathbb R}}^d$, it is also path-connected and one can find a continuous path in $\omega$ between $z_1$ and $z_2$ in $\omega$. Using the application $\phi_{-s}$, we also get a continuous path between $y_1$ and $z_1$ in $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ (resp. between $y_2$ and $z_2$).
Gluing these paths together, this yields a continuous path in $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ between $y_1$ and $y_2$.
$\bullet$ $(i) \implies (iii)$. Since $\tilde{\omega} \subset \omega$, we have $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\tilde\omega) \subset \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$. Denote by $(\Omega_i)_{i \in I}$ the connected components of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ . The sets $\Omega_i$ are connected open sets so that the inclusion $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\tilde\omega) \subset \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ together with the connectedness of $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\tilde\omega)$ yields the existence of $i_0 \in I$ such that $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\tilde\omega) \subset \Omega_{i_0}$. Since $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\tilde\omega)$ is of full measure, this is also the case for $\Omega_{i_0}$. As $\Omega_i$ is open, we obtain that $\Omega_i = \emptyset$ for $i \neq i_0$, so that $\Omega_{i_0} = \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ is connected (and of full measure).
$\bullet$ $(ii) \implies (iii)$. Let $y_1, y_2 \in \bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$; there exists $s_1,s_2 \geq 0$ and $z_1, z_2 \in \omega$ such that $y_1 = \phi_{-s_1} (z_1)$ and $y_2 = \phi_{-s_2} (z_2)$. If $z_1, z_2$ belong to the same connected component $\tilde \omega$ of $\omega$, then since $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\tilde\omega)$ is connected, one can find a continuous path between $z_1$ and $z_2$.
If $z_1, z_2$ belong to two different connected components $\omega_1$ and $\omega_2$, apply $(i)$ to find, up to a permutation between the indices $1$ and $2$, $u \in \omega_1$ and $s \in {{\mathbb R}}^+$ such that $\phi_{-s}(u) \in \omega_2$. Then one can find a continuous path between $u$ and $z_1$ in $\omega_1$, and another between $\phi_{-s}(u)$ and $z_2$ in $\omega_2$. We conclude as in the previous sub case by gluing the paths together.
Note in particular that $(i)-(ii)-(iii)$ hold as soon as $\omega$ is connected and satisfies a.e.i.t. GCC.
Proof of Proposition \[CEXucp\] {#proofCEXucp}
===============================
In this section, we prove Proposition \[CEXucp\].
Let $x_0 \in {{\mathbb T}}^d \setminus \overline{p_x(\omega)} \neq \emptyset$ and take $\eta>0$ such that $B(x_0 , 2\eta) \cap \overline{p_x(\omega)} = \emptyset$. Define the potential $V (x) := \frac{|x-x_0|^2}{2} \Psi(x)$, where $\Psi$ is a “corrector” to ensure $V\in C^\infty ({{\mathbb T}}^d)$, and such that $\Psi \equiv 1$ on $B(x_0 , 2\eta)$ (reduce $\eta$ if necessary). Denote $V_{{\varepsilon}}= {{\varepsilon}}V$ and notice that $\nabla V_{{\varepsilon}}(x) = {{\varepsilon}}(x-x_0)$ on $B(x_0 , 2\eta)$. As a consequence, the hamiltonian flow $(\phi_t)_{t \in {{\mathbb R}}}$ associated to the vector field $v \cdot \nabla_x - \nabla_x V_{{\varepsilon}}\cdot \nabla_v$ may be explicited in the set $B(x_0,2\eta) \times {{\mathbb R}}^d$: we have $$\phi_t(x,v) = \left(x_0 + (x-x_0) \cos(\sqrt{{{\varepsilon}}} t) + \frac{v}{\sqrt{{{\varepsilon}}}}\sin (\sqrt{{{\varepsilon}}} t) , - (x-x_0) \sqrt{{{\varepsilon}}} \sin(\sqrt{{{\varepsilon}}} t) + v \cos (\sqrt{{{\varepsilon}}} t) \right) ,$$ as long as $\phi_t(x,v) \in B(x_0,2\eta) \times{{\mathbb R}}^d$. In particular, note that if $(x,v)\in B(x_0,\eta) \times B(0, \sqrt{{{\varepsilon}}}\eta)$, then $\phi_t(x,v)$ remains in $B(x_0,2\eta) \times B(0, 2 \sqrt{{{\varepsilon}}}\eta)$ for all $t \in {{\mathbb R}}^+$. This reads $\phi_t(B(x_0,\eta) \times B(0,\sqrt{{{\varepsilon}}}\eta)) \subset B(x_0,2\eta) \times B(0,2\sqrt{{{\varepsilon}}}\eta)$, i.e. in particular $\phi_t(B(x_0,\eta) \times B(0,\sqrt{{{\varepsilon}}}\eta)) \cap \omega = \emptyset$ for all $t \in {{\mathbb R}}^+$. This proves that a.e.i.t. GCC is not satisfied.
Notice that in the previous proof, to handle small potential, we consider small speeds, i.e. $v \in B(0,\sqrt{{{\varepsilon}}}\eta)$. In the opposite direction, if one fixes the speeds in a large Hamiltonian sphere $v \in S_H(0,R)$ (note that with the particular potential used in the proof, on the set $\{\Psi=1\}$ we have $S_H(0,R) = S(0,R)$) for some $R>0$, then one can find a (large) potential (namely $R^2/\eta^2 V$ where $V$ is that of the previous proof) such that a.e.i.t. GCC fails.
Other linear Boltzmann type equations {#Other}
=====================================
The goal of this appendix is to show that the methods developed in Part \[LTB\] can be adapted to handle other types of Boltzmann-like equations.
Generalization to a wider class of kinetic transport equations
--------------------------------------------------------------
Consider now the equation $$\label{B-general}
\partial_t f + a(v) \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= \int_{{{\mathbb R}}^d} \left[k(x,v' , v) f(v') - k(x,v , v') f(v)\right] \, dv',$$ where $a(v) = \nabla_v A(v)$ with $A: {{\mathbb R}}^d \to {{\mathbb R}}$ is such that $\int_{{{\mathbb R}}^d} e^{-A(v)} \, dv<+\infty$.
For simplicity, we assume that is set on ${{\mathbb T}}^d\times {{\mathbb R}}^d$ (but it is also possible to consider the case of bounded domains with specular reflection, as in Section \[Boundary\]).
Assume that $a(v)$ satisfies a *non degeneracy* property: there exists $\gamma \in (0,2)$ and $C>0$ such that, for all $\xi \in {{\mathbb S}}^{d-1}$, $$\label{nondege}
\operatorname{Leb}\left\{v \in {{\mathbb R}}^d, \, |a(v) \cdot \xi| \leq {{\varepsilon}}\right\} \leq C {{\varepsilon}}^\gamma.$$ This prevents concentrations of $a(v)$ in any direction of ${{\mathbb S}}^{d-1}$.
The hamiltonian associated to the transport equation is then the following: $$H(x,v)= A(v) + V(x).$$
Define the global Maxwellian associated to $a(v)$: $${\mathcal{M}}_A(v) = C_A e^{-A(v)},$$ with $C_A = 1/ \left( \int_{{{\mathbb R}}^d} e^{-A(v)} \, dv \right)$.
In addition to the usual assumption [**A1**]{} on the collision kernel $k$, we shall assume the following (which replace [**A2**]{} –[**A3**]{}):
[**A2’.**]{} We assume that ${\mathcal{M}}$ cancels the collision operator, that is $$\label{M_Aannule}
\text{for all } (x,v) \in \Omega \times {{\mathbb R}}^d, \quad \int_{{{\mathbb R}}^d} \left[k(x,v' , v) {\mathcal{M}}_A(v') - k(x,v , v') {\mathcal{M}}_A(v)\right] \, dv' = 0.$$
[**A3’.**]{} We assume that $$\label{bornek-A}
\tilde{k}(x,v',v) := \frac{k(x,v' , v) }{{\mathcal{M}}_A(v)} \in L^\infty({{\mathbb T}}^d \times {{\mathbb R}}^d \times {{\mathbb R}}^d).$$
For $a(v)=v$ (for which $\gamma=1$ in ), we recover the framework which has been already treated before. One physically relevant case is $$a_{rel}(v):= \frac{v}{\sqrt{1+ |v|^2}},$$ (for which we also have $\gamma=1$ in ), which allows to model relativistic transport. Note that in this case, we have $A_{rel}(v) :=\sqrt{1+|v|^2}$, and the related Maxwellian is then the so-called *relativistic Maxwellian*: $${\mathcal{M}}_{rel}(v) := C_{rel} e^{-\sqrt{1+|v|^2}},$$ where $C_{rel}$ is a normalizing constant, so that $\int_{{{\mathbb R}}^d} {\mathcal{M}}_{rel}(v) \, dv =1$.
Our aim in this paragraph is to show that the methods developed in Part \[LTB\] are still relevant here. The characteristics of the equation are defined in the following way:
\[def-carac-mod\] Let $V \in W^{2,\infty}_{loc} ({{\mathbb T}}^d)$. Let $(x_0,v_0) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$. The characteristics $\phi_t(x_0,v_0) := (X_t (x_0,v_0), \, \Xi_t(x_0,v_0))$ associated to the hamiltonian $H(x,v) = A(v) + V(x)$ are defined as the solutions to the system: $$\left\{
\begin{aligned}
&\frac{dX_t}{dt} = a(\Xi_t), \quad \frac{d\Xi_t}{dt} = - \nabla_x V(X_t), \\
&X_{t=0}=x_0, \quad \Xi_{t=0}=v_0.
\end{aligned}
\right.$$
With this definition of characteristics, we can then properly define
- the set $\omega$ where collisions are effective, as in Definition \[def-om\],
- the Unique Continuation Property, as in Definition \[def:UCP\]
- $C^-(\infty)$, as in Definition \[definitionCinfini\],
- a.e.i.t. GCC, as in Definition \[defaeitgcc\],
- the equivalence relation $\sim$, as in Definition \[def-sim\].
The next thing to do concerns the local well-posedness of in some relevant weighted spaces, which we introduce below.
We define the Banach spaces ${\mathcal{L}}^2_A$ and ${\mathcal{L}}^\infty_A$ by $$\begin{aligned}
&{\mathcal{L}}^2_A := \Big\{f \in L^1_{loc}({{\mathbb T}}^d \times {{\mathbb R}}^d), \, \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f|^2 \frac{e^V}{ {\mathcal{M}}_{A}(v)} \, dv \, dx < + \infty \Big\} , \\
&\qquad \qquad \qquad \|f\|_{{\mathcal{L}}^2_A} = \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f|^2 \frac{e^V}{ {\mathcal{M}}_{A}(v)} \, dv \, dx \right)^{1/2}, \\
& {\mathcal{L}}^\infty_A : = \Big\{f \in L^1_{loc}({{\mathbb T}}^d \times {{\mathbb R}}^d), \, \sup_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f| \frac{e^V}{ {\mathcal{M}}_A(v)} < + \infty \Big\} ,
\quad \|f\|_{{\mathcal{L}}^\infty_A} = \sup_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f| \frac{e^V}{ {\mathcal{M}}_A(v)}
\end{aligned}$$ The space ${\mathcal{L}}^2_A$ is a Hilbert space endowed with the inner product $$\langle f, g \rangle_{{\mathcal{L}}^2_A} := \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} e^{V} \frac{f \, g}{ {\mathcal{M}}_A(v)} \, dv \, dx.$$
As usual, we have
\[prop:WP-mod\] Assume that $f_0 \in {\mathcal{L}}^2_A$. Then there exists a unique $f\in C^0({{\mathbb R}};{\mathcal{L}}^2_A)$ solution of satisfying $f|_{t = 0} =f_0$, and we have $$\text{ for all } t \geq 0, \quad \frac{d}{dt} \| f(t)\|_{{\mathcal{L}}^2_A}^2 = - D_A(f(t)),$$ where $$\label{defD-A}
D_A(f) = \frac{1}{2} \int_{\Omega} e^{V} \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \left( \frac{k(x,v' , v)}{ {\mathcal{M}}_A(v)} + \frac{k(x,v , v')}{ {\mathcal{M}}_A(v')} \right) {\mathcal{M}}_A(v) {\mathcal{M}}_A(v') \left(\frac{f(v)}{ {\mathcal{M}}_A(v)}- \frac{f(v')}{ {\mathcal{M}}_A(v')}\right)^2 \, dv' \, dv \, dx.$$ If moreover $f_0 \geq 0$ a.e., then for all $t \in {{\mathbb R}}$ we have $f(t, \cdot,\cdot)\geq 0$ a.e. (Maximum principle).
Then the analogues of Theorems \[thmconvgene-intro\], \[thmconv-intro\] and \[thmexpo-intro\] hold in this setting (with some obvious modifications); for the sake of conciseness, we omit these statements.
Such results can be proved exactly as Theorems \[thmconvgene-intro\], \[thmconv-intro\] and \[thmexpo-intro\]. The crucial additional ingredient is the fact that averaging lemmas for the operator $a(v) \cdot {{\nabla}}_x$ still hold, precisely when $a$ satisfies the non degeneracy condition , see [@GLPS]. Note that in this case, the gain of regularity on averages depends on the index $\gamma$ in , but in any case, this is always sufficient to optain compactness.
Generalization to linearized BGK operators {#sec-BGK}
------------------------------------------
Once again, for simplicity, we assume that $(x,v) \in {{\mathbb T}}^d \times {{\mathbb R}}^d$. Let $V \in W^{2,\infty}_{loc}({{\mathbb T}}^d)$. Let $\varphi: {{\mathbb R}}\to {{\mathbb R}}^+_*$ be a function in $L^\infty({{\mathbb R}})$ such that $$\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \varphi\left( \frac{|v|^2}{2} + V(x) \right)\, dv \, dx < +\infty,$$ Denote $F(x,v) = \varphi\left( \frac{|v|^2}{2} + V(x) \right)$ and $\rho_F(x) = \int F(x,v) \, dv$.
Let $\sigma \in C^0({{\mathbb T}}^d)$ be a non-negative function. We study in this paragraph the following degenerate linearized BGK equation: $$\label{BGK}
\partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f = \sigma(x) \left( \frac{\int_{{{\mathbb R}}^d} f\, dv}{\rho_F(x)} F(x,v) - f \right).$$ with an initial condition $f_0$ at time $0$. The natural equilibrium is given by $$(x,v) \mapsto \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \, dx \frac{F(x,v)}{\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} F(x,v) \, dv \, dx}.$$ The main feature of this equilibrium is that there is no separation of variables contrary to the Maxwellian case.
Our aim in this paragraph is again to show that the methods developed in Part \[LTB\] are still relevant here. For what concerns well-posedness, we introduce the relevant weighted $L^p$ spaces and have the usual result.
We define the Banach spaces ${\mathcal{L}}^2_{bgk}$ and ${\mathcal{L}}^\infty_{bgk}$ by $$\begin{aligned}
&{\mathcal{L}}^2_{bgk} := \Big\{f \in L^1_{loc}({{\mathbb T}}^d \times {{\mathbb R}}^d), \, \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f|^2\frac{1}{F(x,v)} \, dv \, dx < + \infty \Big\} , \\
&\qquad \qquad \qquad \|f\|_{{\mathcal{L}}^2_{bgk}} = \left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f|^2 \frac{1}{F(x,v)}\, dv \, dx \right)^{1/2}, \\
& {\mathcal{L}}^\infty_{bgk} : = \Big\{f \in L^1_{loc}({{\mathbb T}}^d \times {{\mathbb R}}^d), \, \sup_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f| \frac{1}{F(x,v)} < + \infty \Big\} ,
\quad \|f\|_{{\mathcal{L}}^\infty} = \sup_{{{\mathbb T}}^d \times {{\mathbb R}}^d} | f| \frac{1}{F(x,v)}
\end{aligned}$$ The space ${\mathcal{L}}^2_{bgk}$ is a Hilbert space endowed with the inner product $$\langle f, g \rangle_{{\mathcal{L}}^2_{bgk}} := \int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} \frac{f \, g}{F(x,v)} \, dv \, dx.$$
\[prop:WP-BGK\] Assume that $f_0 \in {\mathcal{L}}^2_{bgk}$. Then there exists a unique $f\in C^0({{\mathbb R}};{\mathcal{L}}^2_{bgk})$ solution of satisfying $f|_{t = 0} =f_0$, and we have $$\text{ for all } t \geq 0, \quad \frac{d}{dt} \| f(t)\|_{{\mathcal{L}}^2_{bgk}}^2 = - {D}_{bgk}(f(t)),$$ where $${D}_{bgk}(f)= \int_{{{\mathbb T}}^d} e^V\sigma(x) \int_{{{\mathbb R}}^d} \int_{{{\mathbb R}}^d} \frac{F(x,v) F(x,v')}{\rho_F(x)} \left( \frac{f(v)}{F(x,v)} - \frac{f(v')}{F(x,v')}\right)^2 \, dv' \, dv \, dx.$$ If moreover $f_0 \geq 0$ a.e., then for all $t \in {{\mathbb R}}$ we have $f(t, \cdot,\cdot)\geq 0$ a.e. (Maximum principle).
With the same geometric definitions of Section \[main\], we have the following results. Note that the set $\omega$ where the collisions are effective is equal to $\omega_x \times {{\mathbb R}}^d$, where $$\omega_x := \{x \in {{\mathbb T}}^d, \, \sigma(x)>0\}.$$
\[thmconv-BGK\] The following statements are equivalent.
1. The set $\omega$ satisfies the Unique Continuation Property.
2. The set $\omega$ satisfies the a.e.i.t. GCC and $\bigcup_{s\in {{\mathbb R}}^+}\phi_{-s}(\omega)$ is connected.
3. For all $f_0 \in {\mathcal{L}}^2_{bgk}$, denote by $f(t)$ the unique solution to with initial datum $f_0$. We have $$\label{convergeto0-BGK}
\left\|f(t)-\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx\right)\frac{F(x,v)}{\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} F(x,v) \, dv \,dx}\right\|_{{\mathcal{L}}^2_{bgk}} \to_{t \to +\infty} 0,$$
\[thmexpo-BGK\] The two following statements are equivalent:
1. $C^-(\infty) > 0$.
2. There exists $C>0, \gamma>0$ such that for any $f_0 \in {\mathcal{L}}^2({{\mathbb T}}^d \times {{\mathbb R}}^d)$, the unique solution to with initial datum $f_0$ satisfies $$\begin{gathered}
\label{decexpo-BGK}
\left\|f(t)-\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \right) \frac{F(x,v)}{\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} F(x,v) \, dv \,dx} \right\|_{{\mathcal{L}}^2} \\
\leq C e^{-\gamma t} \left\|f_0-\left(\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f_0 \, dv \,dx \right) \frac{F(x,v)}{\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} F(x,v) \, dv \,dx}\right\|_{{\mathcal{L}}^2}.\end{gathered}$$
We shall not dwell on the proofs of Theorems \[thmconv-BGK\] and \[thmexpo-BGK\], since they are very similar to those of Theorems \[thmconv-intro\] and \[thmexpo-intro\]. Indeed, we note that the structure of the equation is similar to that of , in the sense that the “degenerate dissipative” part is still made of a dissipative term plus a relatively compact term. This compactness, as usual, comes from averaging lemmas.
Let us just underline a crucial point in the proof $(ii.)$ implies $(i.)$ of Theorem \[thmconv-BGK\]. This comes from the fact that $F(x,v)$ does not separate the $x$ and $v$ variables, contrary to the Maxwellian equilibrium of and thus we have to be careful. Let us check that the proof we gave in the Boltzmann case is still relevant (see the proof of $(ii.) \implies (i.)$ of Theorem \[thmconv-intro\]).
Let $f \in C^0_t({\mathcal{L}}^2_{bgk})$ be a solution to $$\begin{aligned}
\label{eqhypo1-BGK} \partial_t f + v \cdot \nabla_x f - \nabla_x V \cdot \nabla_v f= 0, \\
\label{eqhypo2-BGK} f = \rho(t,x) F(x,v) \text{ on } {{\mathbb R}}^+ \times \omega.\end{aligned}$$ Assume that $\int_{{{\mathbb T}}^d \times {{\mathbb R}}^d} f \, dv \,dx=0$. The goal is to show that $f=0$.
To this purpose, as before, consider for $(t,x, v) \in {{\mathbb R}}^+ \times \omega$, $g(t,x) := \frac{1}{F(x,v)} \, f$ (note that by , $g$ does not depend on $v$). We have, for $(t,x, v) \in {{\mathbb R}}^+ \times \omega$: $$\partial_t g + v \cdot \nabla_x g = \frac{1}{F(x,v)} \left[\partial_t f + v \cdot \nabla_x f - v\cdot \nabla_x F \, \frac{f}{F}\right].$$ Since $f$ satisfies and , $$\partial_t f + v \cdot \nabla_x f = \nabla_x V \cdot \nabla_v f =\nabla_v F \cdot \nabla_x V \, \frac{f}{F}.$$ By definition of $F$, we have $$\nabla_x V \cdot \nabla_v F = v \cdot \nabla_x F,$$ from which we deduce that $g$ satisfies the free transport equation on $\omega$: $$\label{transg-BGK}
\text{ for all } (t,x, v) \in {{\mathbb R}}^+ \times \omega, \quad \partial_t g + v \cdot \nabla_x g =0.$$ We then conclude as in the proof of $(ii.) \implies (i.)$ of Theorem \[thmconv-intro\], *mutatis mutandis*.
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[[ <span style="font-variant:small-caps;"></span> Centre for the Studies of the Glass Bead Game ]{}]{}\
\[1\]
title: '\'
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\[1996/06/01\]
.
`dedicated to the Journeyers to the East`
`in the tradition of Hermann Hesse’s The Glass Bead Game`
Ł
Preface {#preface .unnumbered}
=======
- *We report an unexpected theoretical discovery of a spin one half matter field with mass dimension one.*
-
<!-- -->
- *We provide the first details on the unexpected theoretical discovery of a spin-one-half matter field with mass dimension one.*
-
With these opening lines Daniel Grumiller and I introduced an entirely new class of fermions. They carry mass dimension one. That is, they do not satisfy Dirac equation, but only the spinorial Klein-Gordon equation for spin one half. In the intervening decade and a half, the issues of non-locality, and Lorentz symmetry violation, have been completely resolved. But a self contained and an *ab initio* treatment of such an unexpected theoretical discovery is missing. It is therefore necessary to lift a logical version of this development from the pages of various journals to a monograph.
I present here what we know of the subject at the present moment (late 2018). In making this selection I have strictly confined, with a minor exception, to that part of the existing literature which has passed through my own pen and paper. This is not to negate the contributions of my collaborators, and many others who have worked on the subject, but to take full personal responsibility for the presented formalism.[^1]
Not unexpectedly, there is a group of physicists who have siezed upon the new construct and based much of their careers on exploiting the physical consequences and studying the underlying mathematical structure of the new theoretical discovery. This is evident from some one hundred papers, and several doctoral thesis, that are entirely devoted to the new spinors and the associated fermions.
Then there is a group of physicists who simply dismiss the subject as an impossibility. For the latter, I can only suggest that they first construct the eigenspinors of the $(1/2,0)\oplus(0,1/2)$ charge conjugation operator – with eigenvalues $\pm 1$ – and show that neither $\left(\gamma_\mu p^\mu + m\I_4\right)$ nor $\left(\gamma_\mu p^\mu - m\I_4\right)$ annihilates these spinors. Having done this preliminary exercise carefully, without falling into the temptation of Grassmann-isaton of the new spinors, they may start with Chapter \[ch5\] – returning to the earlier chapters only for notational details – and come to the end of chapter \[ch11\]. At that stage, they would have enough information to develop their own theory and to see if their calculations produce something similar to what follows in the remainder of the monograph.
The rest of the Preface provides a brief scientific journey of the author. It provides a context in which the reported results were obtained. What follows may thus be seen as a brief scientific autobiography that excludes, with the exception of the next paragraph, large parts of my work on the interface of the gravitational and quantum realms, and on neutrino oscillations.
————–
Sometime in the early 1980’s, on the banks of Charles river in Boston, my quantum mechanics teacher was scheduled to give three fifty-minutes lectures, thrice a week. Instead, to my pleasure, I was exposed to five lectures a week, each of three hours duration with a five minutes water break half the way through each of the lectures. And these continued for three trimesters. I learned many things, among them, significance of phases in the quantum description of reality. Years later, it was, in part, for that reason that a day after Christmas of 1995, seeing falling snow flakes on a road trip by car from the Maulbronn Monastery, Germany, to the French border, that I asked myself as to what is the difference between classical snow and quantum snow. I realised that each of the snow flakes had a different mass. Each flake thus picked up a different gravitationally induced phase.[^2] Soon, within minutes, I was thinking of solar neutrinos instead of snow flakes and a back of the envelope calculation rolled through me. It became clear that neutrino oscillations provide a set of flavour oscillation clocks and that these clocks redshift according to the general relativistic expectation, and suffer Zeeman like splitting in the oscillation frequencies for generalised flavour oscillations clocks. In the process I came to realise that there are instances when the gravitationally induced forces may be zero, but not the gravitationally induced phases. All this, in collaboration with Christoph Burgard, led to a shared 1996 First Prize from the Gravity Research Foundation (GRF), a Fourth Prize for 1997, and a series of other publications that inspired a few hundred papers devoted to the interface of the gravitational and quantum realms and won the third, in 2004, and the fifth in 2000, prizes from GRF.
When my quantum mechanics teacher moved from the east coast to the mid west, I returned with him to my original host university (which has been utterly flexible and graceful to me), I considered him as a natural advisor for my doctoral thesis. Gradually it dawned on me that my questions in Physics were different from his. For my teacher one must start a conversation with a Lagrangian, while for me I needed a more systematic approach to arrive at Lagrangians – be they be for the Maxwell field, or the Dirac, or any other matter or gauge field. In fact one cannot even fully formulate gauge covariance without knowing the kinematical description of the matter fields.
I now know that once the Lagrangian is given then the principles of quantum mechanics and inhomogeneous Lorentz symmetries – coupled with the operations of parity, time reversal, and charge conjugation – intermingle to make a theory predictable in the resulting S-matrix formalism.[^3] My aim was to understand the opening chapters of any quantum field theory text better. Given a representation space associated with the Lorentz algebra, and the behaviour under discrete symmetries, my desire was to derive Lagrangians for the objects inhabiting these spaces. I expected nothing more than to arrive at the standard results, but in my own way. This is already done by Steven Weinberg in his classic on quantum fields [@Weinberg:1995mt; @Weinberg:1996kr]
During the year I started working on my doctoral thesis, Lewis Ryder’s book on quantum field theory arrived on the scene. It provided a derivation of Dirac equation.[^4] I could easily extend his derivation to obtain Maxwell equations. So, I was happy for sometime that I could make progress in obtaining the kinematical structure of matters fields, and understand gauge fields from my own perspective. At this juncture, I arranged for a series of breakfast/lunch conversations with a nuclear physicist who had been my teacher for a wonderfully-taught course from Jackson’s electrodynamics. As a result of these conversations, I realised that I could be useful to the nuclear physics community by formulating a pragmatic approach to dealing with higher spin baryonic and mesonic resonances. These were copiously produced at the then new Continuous Electron Beam Accelerator Facility in Virginia.[^5]
As soon I submitted my doctoral work to the thesis clerk, came Christoph Burgard, and declared to me that Ryder’s derivation of Dirac equation is ‘all wrong.’[^6] And it turned out that Christoph was correct, as always. This is now explained in my review of Ryder’s book (written a few years later) and by Gaioli and Garcia Alvarez in their *Am. J. Phys.* paper.[^7] To me, it is again a story that weaves important phases in the analysis: for Dirac spinors the right and left transforming components of the particle and antiparticle rest-spinors have opposite relative phases. This important fact can be read off from Steven Weinberg’s analysis of the Dirac field. The result follows without invoking Dirac equation.
Soon after my arrival at the Los Alamos Meson Physics Facility[^8] in the July of 1992, I started to look at Majorana neutrinos. There was a significant element of confusion on the subject and I requested LAMPF to obtain for me an English translation of the 1937 paper of Majorana. I thus found out that in the 1937 paper there was no notion of Majorana spinors: The Majorana field was still the Dirac field, expanded in terms of the Dirac spinors, but with particle and antiparticle creation operators identified with each other.
On the other hand, I found in Pierre Ramond’s primer a systematic development of Majorana spinors: their origin resided in the fact that if $\phi$ transformed as a left-handed Weyl spinor then $\sigma_2 \phi^\ast$ transformed as a right-handed Weyl spinor ($\sigma_2 = \mbox{`second' Pauli matix}$). But $\phi$, and consequently the Majorana spinor, had to be treated as a Grassmann variable. Furthermore, Majorana spinors were looked upon as Weyl spinors in disguise.
I was uncomfortable with both of the mentioned elements. For the Dirac spinors, understood as a direct sum of the right and left transforming Weyl spinors, no such Grassmann-isation was necessary – at least in the operator formalism of quantum field theory. Another matter that concerned me was that for higher spin generalisation of the $\phi$-$\sigma_2\phi^\ast$ argument a replacement of $\sigma_2$ by its higher spin counterparts failed to do the magic of Pauli matrices, as Ramond had called it. For a while, this seemed to make a higher spin generalisation of Majorana spinors untenable.
I resolved these problems by taking note of the fact that for spin one half the charge conjugation operator has four, rather than two, independent eigenspinors. And that the $\phi$-$\sigma_2\phi^\ast$ argument works magically well for all spins if $\sigma_2$ was instead recognised, up to a phase, as Wigner’s time reversal operator, $\Theta$, for spin one half: if $\phi$ transforms as an $(0,j)$ object then $\Theta \phi^\ast$ transforms as a $(j,0)$ object – with $\Theta$ now a spin-$j$ Wigner time reversal operator. As a consequence the $(j,0)\oplus(0,j)$ object $$\left(
\begin{array}{c}
\mbox{a phase} \times \Theta \phi^\ast\\
\phi
\end{array}
\right)$$ becomes an eigenvector of the spin-$j$ charge conjugation operator if the indicated phases are chosen correctly to satisfy the self/anti-self conjugacy condition. This generalised Majorana spinors to all spins. And such new objects were not $(j,0)$, or $(0,j)$ objects in disguise: there were $2j$, and not $j$, independent $(j,0)\oplus(0,j)$ vectors. Half of them were self-conjugate under charge conjugation, and other half were anti self-conjugate. Later new names were to be invented to avoid confusion with the newly constructed $(j,0)\oplus(0,j)$ vectors.
During this phase, I also began to suspect that the dynamics associated with these new objects would carry unexpected features. It was as true for spin one half, as for higher spins.
All these observations were essentially a mathematical science fiction. In addition, I encountered the same problem as Aitchison and Hey did in their attempt to construct a Lagrangian for the c-number Majorana spinors. With some caveats, without the Lagrangian the story cannot unfold, cannot come to fruition. Bringing back the focus to spin one half, I expressed some of my frustration in an unpublished e-print where, towards a resolution of the problem, I initiated a work to construct dual space for the new set of four-component spinors. It allowed to define an adjoint for the quantum field with the new spinors as expansion coefficients. At this stage Daniel Grumiller joined my efforts and we calculated the vacuum expectation value of the time ordered product of the field with the newly defined adjoint – that is, the Feynman-Dyson propagator. This gave us a most startling result: the new spin one half fermionic field carried mass dimension one. This we presented as an “unexpected theoretical discovery” in JCAP and PRD – the two 2004 e-prints, were published in 2005, almost back to back due to a long, but constructive, refereeing process.
Soon after the publication of these papers, I moved from Zacatecas in Mexico to Canterbury in New Zealand. There, I formed a very active research group till it dismantled in the aftermath of the Christchurch earthquakes. The most active members of this group were my doctoral students: Cheng-Yang Lee, Sebastian Horvath, Dimitri Schritt, and Tom Watson. Though Tom stayed in the group only for a short time, he made an interesting contribution. The most important of these was that Ryder’s definition of the Dirac quantum field, as was the case with many other authors, was not consistent with the construction of quantum fields formulated by Steven Weinberg. This fact, along with what I’ll later describe as the IUCAA breakthrough in Section \[Sec:IUCAA\] of this monograph led to evaporating the problems of non-locality and Lorentz-symmetry violation.
After the Canterbury earthquakes, I took a two-year detour to Campinas in Brasil, and returned to India more or less permanently. Thus by 2017, in India I had on my hands a spin one half fermionic field that was local and did not suffer from the violation of the Lorentz symmetry.
————–
With this background, this monograph presents the new theory at a level that should be easily accessible to any good graduate student. An outstanding question that still remains is to reformulate Weinberg’s construction of quantum fields so as to accommodate this new field. My preliminary thoughts on evading the no-go result of Weinberg can be found in my latest paper in Europhysics Letters (EPL) written under the title, “Evading Weinberg’s no-go theorem to construct mass dimension one fermions: Constructing darkness.”
Acknowledgements {#ch16 .unnumbered}
================
Projects like these often evolve over years if not decades. In the process one’s scientific style takes birth, often in a merging of one’s own genius and an inspiration owed to least one great teacher. The latter for me was Dick Arnowitt. I am immensely grateful to him for teaching me many things of the quantum realm and for his absolute accessibility. Steven Weinberg’s books are another source of my inspiration, as is Dirac’s classic on quantum mechanics. While for Arnowitt a story begins with the Lagrangian density, for me once the Lagragian density is given one gives essentially the whole story. And so my quest was for the logical path that leads to Lagrangian densities. This monograph is a reflection of that quest, and that path taken. Arnowitt, in the very first lecture I attended by him, told us all that Dirac’s classic was the best book written in one hundred years [@Dirac:1930pam]. Weinberg, in my opinion does for quantum field theory what Dirac did for quantum mechanics. I am grateful to these scholars. I took Arnowitt’s emphasis on the importance of phases in quantum mechanics to heart and there is perhaps not a single publication of mine where this is not apparent.
My gratefulness also includes numerous referees and one in particular. He is Louis Michel. I urge the reader to read my indebtedness to him published as an acknowledgement [@Ahluwalia:1995ur]. I am thankful to Peter Herczeg for bringing certain sentiments of Louis Michel to me and for his friendship and scholarship during my 1992-1998 stay at Los Alamos.
Very special thanks go to Daniel Grumiller for joining my seed efforts in a 2003 preprint [@Ahluwalia:2003jt] and evolving them collaboratively into an ‘unexpected theoretical discovery’ reported in [@Ahluwalia:2004sz; @Ahluwalia:2004ab]. My students, Cheng-Yang Lee, Sebastian Horvath, and Dimitri Schritt, became my close friends and collaborators and contributed immensely in creating a warm scholarly ambiance in our research group at the University of Canterbury (Christchurch, New Zealand) and in developing the formalism of mass dimension one fermions. I am grateful to them, and to numerous other students who either attended my lectures at Canterbury and Zacatecas (Mexico) or/and worked under my supervision for projects or thesis.
For securing a continuing academic position at the University of Canterbury I am grateful to Matt Visser and to David Wiltshire and equally thankful for making that decade a very productive and pleasant one. For my two-year long detour to Brasil I am grateful to Marco Dias, Saulo Pereira, Julio Marny Hoff da Silva, Alberto Saa, and to Roldão da Rocha for their friendship and many insightful discussions on subject of this monograph.
Zacatecas is a beautiful small city in northern Mexico at roughly two thousand and five hundred meters. I very much enjoyed my tenure at Universidad Autónoma de Zacatecas and the city. The papers with Daniel Grumiller were published from there. Gema Mercado, then Director of the Department of Mathematics, not only invited me back from India to Zacatecas but she also provided an inspired scholarly ambiance, and a friendship and leadership of unprecedented selflessness. I am utterly grateful to Gema for that and for allowing me to pursue my work without hindrance and with encouragement and support.
I thank Llohann Sperança for discussions in the initial stages of this manuscript at Unicamp (São Paulo, Brasil). The breakthrough on the Lorentz symmetry and locality presented here began in late 2015 during a three-month long visit to the Inter-University Centre for Astronomy and Astrophysics (IUCAA, Pune) where I gave a series of lectures on mass dimension one fermions. For their insightful questions and the ensuing discussions, I thank the participants of those lectures and in particular Sourav Bhattachaya, Sumanta Chakraborty, Swagat Mishra, Karthik Rajeev, and Krishnamohan Parattu and my host Thanu Padmanabhan. Raghu Rangarajan (Physical Research Laboratory, Allahabad) carefully read the entire first draft of an important manuscript [@Ahluwalia:2016rwl] and provided many insightful suggestions. I am grateful to him for his generosity and for engaging in long insightful discussions. The calculations that led to the reported results were done at Centre for the Studies of the Glass Bead Game and Physical Research Laboratory.
Chia-Ren Hu and George Kattawar brought me to Texas A&M University (College Station, USA) for me to pursue my doctoral degree and supported me through my entire 1983-1991 stay there. I am immensely indebted to them for their conviction that a man could enter a Ph.D. program at age 31 and take his time reflecting to secure a Ph.D. at just a little shy of his 39th birthday. My gratefulness also goes to my supervisor for the Ph.D. degree, Dave Ernst. My tenure at Texas A&M was made particularly meaningful by the inspired friendship and collaboration with Christoph Burgard. I treasured and treasure his warmth, his insights, and many things *zimpoic*.
At the Los Alamos National Laboratory, Terry Goldman, Peter Herczeg, Mikkel Johnson, Hewyl White, among so many other friends like Cy Hoffman, George Glass and Nu Xu, kept their faith in my studies and supported my independence with warmth and scholarship. I am grateful to them, and those others who know who they are. Who sat down on mesas or walked with me, and shared their insights and wisdom.
My return to India has been warmly supported by Pankaj Jain through an Institute Fellowship at the Indian Institute of Technology Kanpur, and through similar grants and invitations by T. P. Singh (Tata Institute of Fundamental Research, Mumbai), Sudhakar Panda (Institute of Physics, Bhubaneshwar), Thanu Padmanbhan (IUCAA), Mohammad Sami (Jamia Milia Islamia, New Delhi). I thank them for welcoming me back and for opening doors for me. The scene where the reported work took place changed from Los Alamos National Laboratory in the States, to Universidad Autónoma de Zacatecas (UAZ) in Mexico, to the University of Canterbury in New Zealand, to Universidade Estadual de Campinas (Unicamp) in Brasil, to Inter-University Centre for Astronomy and Astrophysics (IUCAA) and to the Centre for the Studies of the Glass Bead Game in Bir, Himachal Pradesh. I am indebted to these institutions for reasons too many to enumerate.
Yeluripati Rohin and Suresh Chand read the final draft of the manuscript and caught several typos, and helped me improve the presentation. I thank them both. I thank Shreyas Tiruvaskar for his suggestions on an earlier draft.
I am utterly thankful to my editor, Simon Capelin, and Roisin Munnelly, and Sarah Lambert in the role of of Editorial Assistants. Without their patience and advice, and without the personal invitation of Simon, this monograph would simply not have come to exist.
Last, but not least my warm thanks go to my children Jugnu, Vikram, Shanti, and Wellner, and to my father Bikram Singh Ahluwalia. They all have been sages of wisdom and affection to me. Karan and Bobby, my brothers and their families, provided a home when I had none, for that and their warmth I am grateful. Sangeetha Siddheswaran is a constant source of affection and encouragement, and I am equally grateful to her for helping me bring this monograph to completion.
While writing this monograph works of J. M. Coetzee, Hermann Hesse and Carl Jung provided the poetic and moral background. The monograph is dedicated to Hermann Hesse’s Journeyers to the East.
During writing of this work Sweta Sarmah became an inspired inspiration in the ways of the Golu Molu. Many butter scotch ice creams to her.
Introduction
============
In a broad brush, the grand metamorphosis that has created the astrophysical and cosmic structures arise from an interplay of (a) Wigner’s work on unitary representations of the inhomogeneous Lorentz group including reflections, (b) Yang-Mills-Higgs framework for understanding interactions, and (c) an expanding universe governed by Einstein’s theory of general relativity. The wood is provided by the spacetime symmetries. These tell us whatever matter exists, it must be one representation or the other of the extended Lorentz symmetries [@Wigner:1939cj; @Wigner:1962ep]. The fire and the glue is provided by the principle of local gauge symmetries *à la* Yang and Mills [@Yang:1954ek] and by general relativistic gravity. The Wigner-Yang-Mills framework, as implemented in the standard model of the high energy physics, then provides the right hand side of the Einstein’s field equations to determine the evolution of that very spacetime which these matter and gauge fields give birth to in a mutuality yet not completely formulated to its quantum completion. Dark energy, thought to be needed for the accelerated expansion of the universe, and dark matter, required by data on the velocities of the stars in galaxies, the motion of galaxies in galactic clusters, and cosmic structure formation, keeps roughly ninety five percent of the universe dark, and of yet to be understood origin.
It is nothing less than the most sublime poetry and primal magic that this picture can explain the rise of mountains and flow of water in the rivers, and go as deep as to invoke metamorphosis of light into the water and the mountains, the stars and galaxies. Where a scientist knows where and how the water first came to be [@Hogerheijde:2011pq; @Podio:2013wnh], and a poet asks if there was thirst when the water first rose. The origin of biological structures remains an inspiring open subject.
It is within this framework, that we wish to add a new chapter and show how to construct dark matter and understand its darkness from first principles. It is thus, in this monograph we present an unexpected theoretical discovery of new fermions of spin one half. The fermions of the standard model, be they leptons or quarks, carry mass dimension three halves. The mass dimension of the new fermions is one. Their quartic self interaction, despite being fermions, is a mass dimension four operator as is their interaction with the Higgs. Their interaction with the standard model fermions is suppressed by one power of the Planck scale and because they couple to Higgs, quantum corrections can bring about tiny magnetic moments for the new fermions. These aspects make them natural dark matter candidate and can provide tiny interaction between matter and gauge fields of the standard model – something that is already suggested observationally [@Barkana:2018lgd]. Studies in cosmology hint that the quantum field associated with the new particles may also play an important role in inflation and accelerated expansion of the universe [@Boehmer:2007dh; @Boehmer:2010ma; @Basak:2012sn; @Basak:2014qea; @Pereira:2014wta; @Pereira:2017bvq; @Pereira:2016emd; @Rogerio:2017gvr].
We thus weave a story of how the non-locality of the first effort evaporated [@Ahluwalia:2004sz; @Ahluwalia:2004ab; @Ahluwalia:2016jwz; @Ahluwalia:2016rwl]. We tell of the evaporation of the violation of Lorentz symmetry. In the process, we construct a quantum field that is local and fermionic. It finds not its description in the Dirac formalism, but in a new formalism appropriate for its own nature. Beyond the immediate focus it makes explicit many insights, otherwise hidden in the work of Weinberg [@Weinberg:1995mt].
The meandering path from non-locality to locality, from Lorentz symmetry violation to preserving Lorentz symmetry, owes its existence to certain wide-spread errors and misconceptions in most textbook presentations of quantum field theory (and we had to learn, and correct these), and the eventual breakthrough to certain phases that affect locality and to a construction of a theory of duals and adjoints.
In the first volume, in chapters 2 to 5 of [@Weinberg:1995mt] Steven Weinberg proves what may be called a no-go theorem: a Lorentz and parity covariant local theory of spin half fermions must be based on a field expanded in terms of the eigenspinors of the parity operator, that is Dirac spinors – and nothing else. Furthermore, these expansion coefficients must come with certain relative phases. And in addition, there must be a specific pairing between the expansion coefficients and the annihilation and creation operators satisfying fermionic statistics.
A reader who finds these remarks mysterious, may undertake the exercise of comparing “coefficient functions at zero momentum” which Weinberg arrives at in his equations (5.5.35) and (5.5.36) with their counterparts written by some of the other popular authors, for example [@Ryder:1985wq; @Folland:2008zz; @Schwartz:2014md]. The book by Srednicki avoids these errors with profound consequences for the consistency of the theory with Lorentz symmetry and locality [@Srednicki:2007qs].
In arriving at the canonical spin one half fermionic field, Weinberg does not use or invoke Dirac equation, or the Dirac Lagrangian density. These follow by evaluating the vacuum expectation value of the time ordered product of the field and its adjoint at two spacetime points $(x,x^\prime)$. The resulting Feynman-Dyson propagator determines the mass dimensionality of the field to be three halves.[^9]
Thus, information about the mass dimensionality of a quantum field is spread over two objects: the field, and its adjoint. Weinberg first derives the field from general quantum mechanical considerations consistent with spacetime symmetries, cluster decomposition principle, and then as just indicated, uses this field to arrive at the Lagrangian density through evaluating the vacuum expectation value of the time ordered product of the field and its adjoint at two spacetime points $(x,x^\prime)$. The powers of spacetime derivatives that enter the Lagrangian density is not assumed, but it is determined by the representation space, and the mentioned formalism, in which the field resides. The broad brush lesson is: Given a spin, it is naive to propose a Lorentz covariant Lagrangian density. It must be derived *à la* Weinberg. The expansion coefficient, $f_\alpha$ and $f^\prime_\alpha$, of a quantum field, $\psi$, are determined by an appropriate finite dimensional representation of the Lorentz algebra and the symmetry of spacetime translation $$\psi = \sum_\alpha \big[f_\alpha \textbf{a}_\alpha + f^\prime_\alpha \textbf{b}^\dagger_\alpha\big]\nonumber$$ where $\textbf{a}_\alpha$ and $\textbf{b}_\alpha$ satisfy canonical fermionic or bosonic commutators or anticommutators. For simplicity of our argument we have suppressed the usual integration on four momentum, and it may be considered absorbed in the summation sign. As is clear from Weinberg’s work, though not explicitly stated by him, if $f_\alpha$ and $f^\prime_\alpha$ satisfy a wave equation, so do $ u_\alpha = e^{i \zeta_\alpha} f_\alpha$ and $v_\alpha = e^{i \xi_\alpha} f^\prime_\alpha$, with $\zeta_\alpha,\xi_\alpha\in \Re$. If the field $\psi$ has to respect Lorentz covariance, locality, and certain discrete symmetries then the phases, $ e^{i \zeta_\alpha}$ and $ e^{i \xi_\alpha}$ cannot be arbitrary, but must acquire certain values. Up to an overall phase factor, these are determined uniquely in the Weinberg formalism. Furthermore, the pairing of the $u_\alpha$ and $v_\alpha$ with the annihilation and creation operators is also not arbitrary. A concrete example of all this can be found in [@Ahluwalia:2016jwz]. The second subtle element is: how to define dual of $u_\alpha$ and $v_\alpha$, and the adjoint of $\psi$ (see below). We develop a general theory of these elements in this monograph suspecting that mathematicians may have already addressed this issue in one form or another – that said, a tourist guide by a mathematician has missed the issues that we point out [@Folland:2008zz].
Once these observations are taken into account, if one were to envisage a new fermionic field of spin one half and evade Weinberg’s no-go theorem then something non-trivial has to be done. Our approach would be to combine elements of Weinberg’s approach and that of a naive one indicated above. We shall take the $f_\alpha$ and $f^\prime_\alpha$ not to be complete set of eigenspinors of the spinorial parity operator but that of the spin one half charge conjugation operator. We shall fix the phases $e^{i \zeta_\alpha}$ and $e^{i \xi_\alpha}$ to control the covariance under various symmetries, and to satisfy locality. We will find that each of the eigenspinors of the charge conjugation operator has a zero norm under the canonical Dirac dual. This would lead us to an *ab initio* analysis of constructing duals and adjoints. In the process, we find that if the eigenspinors of the parity operators in the Dirac field are replaced by a complete set of the eigenspinors of the charge conjugation operator, and one chooses appropriate relative phases between the “coefficient functions at zero momentum,” and follows a Weinberg analogue of pairing of the expansion coefficients with the annihilation and creation operators, then the resulting field on evaluating the vacuum expectation value of the time ordered product of the field and its adjoint at two spacetime points $(x,x^\prime)$ is found to be endowed with mass dimension one. Thus, giving a fundamentally new fermionic field of spin one half.
————–
One of my younger friends, and a physicist in his own right, explains to me the new fermions with the following wisdom [@Swagat:2017pc], “Why should Parity get all the privilege? Charge Conjugation has equal rights." We will see here that he captures the essence of one of the main results of this monograph.
The monograph may also be seen as chapters envisaged by a referee of a 2006 Marsden Funding Application to the Royal Society of New Zealand. The referee report read, in part [@RefereeMarsden:2006nz]:
-
Thus this monograph presents the first long chapter envisaged by the referee and contains much that has been discovered since.
From time to time, a junior reader would come across a remark that is not immediately obvious. For example, after equation (\[eq:primordial-generators\]), there suddenly appears a paragraph reading, “Without the existence of two, rather than one, representations for each $\bmfj$ one would not be able to respect causality in quantum field theoretic formalism respecting Poincaré symmetries, or have antiparticles required to avoid causal paradoxes.” In such an instance our reader may simply go past such matters and continue. The chances are in the course of her studies, she will come to appreciate the insight, or perhaps disagree with it. I hope such liberties shall serve their purpose in the spirit of Hermann Hesse’s journeyers to the east, to whom this monograph is dedicated.
A trinity of duplexities
========================
A view is presented in which dark matter is seen as a continuation of historical emergence of spin and antiparticles.
From emergence of spin, to antiparticles, to dark matter
--------------------------------------------------------
Arguing for “some incompleteness” in the earlier works of Darwin and Pauli, Dirac confronts a “duplexity” phenomena: a discrepancy that the observed number of stationary states of an electron in an atom being twice the number given by the then-existing theory [@Dirac:1928hu; @Darwin:1927du; @Pauli:1927; @Uhlenbeck:1925ge; @Uhlenbeck:1926ge].[^10] The solution he proposed, with the subsequent development of the theory of quantum fields, not only resolved the discrepancy but it also introduced a new unexpected duplexity [@Tomonaga:1946zz; @PhysRev.76.749; @PhysRev.75.1736; @PhysRev.82.914; @Weinberg:1964cn; @tHooft:1973mfk; @Weinberg:1995mt]: For each spin one half particle, the Dirac theory predicted an antiparticle. Associated with this prediction was the charge conjugation symmetry – a notion that soon afterwards was generalised to all spins. This symmetry shall play a pivotal role in this monograph.
The doubling of the degrees of freedom, for spin one half fermions of Dirac, can be traced to the parity covariance built into the formalism. This symmetry requires not only the left-handed Weyl spinors but also the right-handed Weyl spinors. In the process for a spin one half particle we are forced to deal with four, rather than two, degrees of freedom. The antiparticles of the Dirac formalism may be interpreted as a consequence of this doubling.
Parenthetic remarks:
- The existence of antiparticles is not confined to spin one half as is beautifully argued by Feynman in his 1986 Dirac memorial lecture delivered under the title, “The reason for antiparticles” [@Feynman:1987gs]. It is based on a calculation of amplitudes for sources with strictly positive energy superpositions. A related argument by Weinberg emphasises that antiparticles are required to avoid causal paradoxes [@Weinberg:1972gc chap. 2, Sec. 13].
- For each spin, Lorentz algebra provides two separate representation spaces. These transmute into each other under the operation of parity. The existence of two separate representation spaces is important for the existence of antiparticles. It allows the doubling in the degrees of freedom required by the existence of antiparticles. That in turn allows to build a causal theory.[^11]
Fast forward a few decades, with the intervening years placing Dirac’s work on a more systematic footing, the new astrophysical and cosmological observations have now introduced a new duplexity. With the exception of interaction with gravity, these observations strongly hint that there exists a new form of matter which carries no, or limited, interactions with the matter and gauge fields of the standard model of high energy physics [@Bertone:2016nfn]. To distinguish it from the matter fields of the standard model of high energy physics the new form of matter has come to be called dark matter.
For some decades now supersymmetry was thought to provide precisely such a duplexity in a natural manner by introducing a symmetry that transmuted mass dimensionality *and* statistics of particles [@Coleman:1967ad; @Haag:1974qh]. However, at the date of this writing, despite intense searches there is no observational evidence for its existence.
Here I suggest that its origin instead lies in a new duplexity. And that dark matter is simply not yet another familiar particle of the types found in the standard model of the high energy physics and the standard general relativistic cosmology. The new duplexity, I suggest, is provided by mass dimension one fermions. Unlike supersymmetry I suspect that there exists a new symmetry that transmutes only the mass dimension of the fermions, and *not* the statistics.
\[tab:trinity\]
\[table2\]
[@c@lll@]{} Duplexity & Phenomena & Consequence\
1 & Doubling of states of an electron & Spin\
& in an atom &\
\
2 & Doubling of degrees of freedom & Antiparticles\
& for spin $\frac12$ particles ($m\ne 0$) &\
\
3[^12]& Doubling of types of matter fields by & Dark matter\
& introducing fermions of mass dimension one &\
The new fermions carry a foundational importance for the representations of the Lorentz symmetries and the particle content contained in them. The limited interactions of the new fermions with the standard model particles is an ineluctable consequence of its nature. The unexpected theoretical discovery of the new fermions thus provides a natural dark matter candidate. We outline the logical thread leading to the new proposal in Table 1.1 [@Ahluwalia:2016jwz].
The identification of dark matter with the mass dimension one fermions is consistent with the conjecture that whatever dark matter it must still be one representation or the other of the Lorentz[^13] symmetries [@Wigner:1939cj]. It also suggests a possible existence of a new symmetry yet to be discovered that mutates the mass dimensionality of fermions, without affecting the statistics. In that event the no-go theorems resulting from the works of Wigner, Weinberg, Lee and Wick may no longer apply [@Wigner:1962ep; @Weinberg:1964cn; @Weinberg:1964ev; @Weinberg:1995mt; @PhysRev.148.1385] and open up truly new physics systematically constructed on well-known and experimentally verified symmetries.
The new fermions do not allow the usual local gauge symmetries of the standard model. Concurrently, their mass dimension is in mismatch with mass dimension of the standard model fermions. It prevents them from entering the standard model doublets. The new fermions have a natural quartic self interaction with a dimensionless coupling constant, something that cannot occur for the Dirac/Majorana fermions. They also have a natural coupling with the Higgs and gravity. Additional interactions arise from quantum corrections.
As this monograph was composed a new unexpected aspect of the new fermions under rotation came to attention. It has important cosmological consequences. This is now the subject of Chapter \[ch10\].
Like the Majorana fermions the symmetry of charge conjugation plays a central role for the new fermions: while for the Majorana field the coefficient functions are eigenspinors of the parity operator, the field itself equals its charge conjugate. For the new fermions the field is expanded in terms of the eigenspinors of the charge conjugation operator. Once that is done, one may choose to impose the Majorana condition, but it is not mandated.
Thus the new formalism, in a parallel with the Dirac formalism, allows for darkly-charged fermions, and Majorana-like neutral fields.
To avoid possible confusion we remind the reader that both the Dirac and Majorana quantum field are expanded in terms of Dirac spinors. These are eigenspinors of the parity operator: $m^{-1} \gamma_\mu p^\mu$, see Chapter \[ch5\] below, and [@Speranca:2013hqa]. Eigenspinors of the charge conjugation operator are thought to provide no Lagrangian description in a quantum field theoretic construct [@Aitchison:2004cs App. P]. Thus placing the parity and charge conjugation symmetries on an asymmetric footing that is, as far as their roles in constructing spin one half quantum fields are concerned. Here I show that the Aitchison and Hey claim is in error (it seems to be edited out in later editions). It has remained hidden in a lack of full appreciation as to how one is to construct duals for spinors, and the associated adjoints that is, in the mathematics underlying the definition of the dual spinors via $\overline\psi(\p) = \psi(\p)^\dagger \gamma_0$ [@Ahluwalia:2016jwz; @Ahluwalia:2016rwl]. The resolution occurs through a generalisation of the Dirac dual presented here in chapter \[ch12\].
To develop the physics hinted above we are forced to complete the development of this mathematics. Taken to its logical conclusion it leads to the doubling of the fundamental form of matter fields. One form of matter is described by the Dirac formalism, while the other, that of the dark sector, by the new fermions reported here. For each sector the needed matter fields require a complete set of four, four-component spinors. For the former these are eigenspinors of the parity operator while for the latter these are eigenspinors of the charge conjugation operator (Elko).[^14]
Global phases associated with these eigenspinors, and the pairing of these eigenspinors with the creation and annihilation operators influence the Lorentz covariance and locality of the fields (as already discussed at some length in the Preface). This last observation, often ignored in textbooks, when coupled with the discovery of a freedom in defining spinorial duals accounts for the removal of the non-locality and restores the Lorentz symmetry for the mass dimension one fermions [@Ahluwalia:2016rwl; @Ahluwalia:2016jwz].
From elements of Lie symmetries to Lorentz algebra {#ch3}
==================================================
Our first exposure to Lorentz algebra often happens in the context of some course on the theory of special relativity. Because of historical reasons one often thinks of the two in the same breath. On some planet endowed with individuals who reflect on their origins one may arrive at Lorentz algebra by looking at the spectrum of the hydrogen atom. At another planet, observations on light may lead thinking beings to arrive at Maxwell equations, and then Lorentz algebra – and may be even at conformal algebra. Thus, Lorentz algebra is a unifying theme underlying all attempts to understand associated matter and gauge fields, and the very spacetime in which the associated quanta propagate. Somewhat poetically, that which walks and that in which it walks are determined by each other (this thread continues further in chapter \[ch15\]). With this being so, we here take the view that Lorentz algebra is deeper than the symmetries of the Minkowski space (where the additional symmetry of spacetime translations exist). Its different solutions furnish different representation spaces. Minkowski spacetime being just one of them.
This chapter develops the needed concepts in a pedagogic manner. Its pace is to the point, and leisurely, and exploits the opportunity to present a point of view that to my knowledge contains several novel points of view. This chapter is written for an advanced undergraduate student to provide her a simple entry into the subject at hand.
Introduction
------------
At Texas A&M University, I did not begin work on my doctoral thesis till such time I came to know of Eugene Wigner’s 1939 work and of Steven Weinberg’s 1964 papers [@Wigner:1939cj; @Weinberg:1964cn; @Weinberg:1964ev]. Not that I instantly understood them, or even appreciated their depth in the first encounter. But I realised that the Dirac equation in its 1928 form and Maxwell equations of classical electrodynamics are so utterly simple to arrive at. Modulo a careful handling of the discrete symmetries, all one needs is the unifying theme provided by the Loretnz algebra.[^15] And if the latter were to change, say at the Planck scale, so would these equations. Assertions such as ‘let us take a Lagrangian density that depends only on one or two spacetime derivatives of the fields,’ as I was to learn in the process, often led to mysterious pathologies for once the representation space to which a field belonged to was specified, one lost the freedom to invoke simplicity and convenience to choose how many spacetime derivatives entered the Lagrangian density. Many, if not all, such pathologies disappear if one is careful at the starting point: the Lagrangian density.
In other words, and as already alluded to above, my trouble with the quantum field theory courses as a doctoral student was that one had to provide a Lagrangian density, and even at the free field level it amounted to invoking the genius of Dirac or that of Maxwell coupled with a set of associated experiments and data. But theoretical physics to me was to be a logical exercise which depended on as few experiments as possible and explained all the relevant phenomena. This approach, if it existed, would then provide the unifying thread from which wave equations and Lagrangian densities would emerge. The hosts of experimental results would then simply be a logical consequence of this unifying theme and these Lagrangian densities. Additional formal structure would then be built from additional principles, such as that of local gauge invariance and the technology of the $S$-matrix theory would help extract many of the observables.[^16]
Roughly two decades later I have a better understanding as to from where do the Lagrangian densities come from and how one may evade certain no-go theorems of Weinberg [@Ahluwalia:2016jwz]. This is the story I want to develop. In the process we shall arrive at several known, and some totally unexpected, results. Very often the manner in which we arrive at the known results shall be significantly different from their presentation elsewhere in the physics literature. The role played by various symmetries shall be transparent, or so I hope. I make significant effort to keep the presentation at a level that makes the presented material easily accessible to the advanced undergraduate students and the beginning doctoral students of Physics. Some mathematicians may frown upon it but even for them it is hoped that there is, at times, new mathematics, and a lesson in the importance of phase factors in physics.
The first thing, then, is to introduce the notion of symmetry generators for a Lie algebra towards the eventual aim to facilitate bringing Lorentz algebra on the scene. To introduce the said notion it suffices to begin with the rotational symmetry in the familiar landscape of three spatial dimensions. With rotational symmetry in our focus we would not forgo the opportunity to emphasise the unavoidable inevitability of the quantum structure of the physical reality – or at least, make it more plausible.
Generator of a Lie symmetry
---------------------------
By a Lie symmetry we mean a symmetry that depends on a continuous parameter. These are to be distinguished from the discrete symmetries, such as that of parity. The familiar rotational symmetry in the ordinary space is one simple and important example. We will use it to introduce the notion of a generator and make quantum aspect of reality essentially unavoidable.
Consider a rotation of a frame of reference, or the vector itself (keeping the frame of reference fixed), with the following effect on a vector $\x =(x,y,z)$ $$\left(
\begin{array}{c}
x^\prime\\
y^\prime\\
z^\prime
\end{array}
\right) = \left(\begin{array}{ccc}
\cos\vartheta & \sin\vartheta & 0\\
-\sin\vartheta & \cos\vartheta & 0 \\
0 & 0 & 1
\end{array}
\right) \left(\begin{array}{c}
x\\
y\\
z
\end{array}\right)$$ with $0\le \vartheta\le 2 \pi$. Denoting the $3\times 3$ matrix that appears above by $R_z(\vartheta)$, we define a generator of rotation $J_z$ through the relation $$R_z(\theta) = {\e}^{i J_z \vartheta} \label{eq:def-of-Jz}$$ with $J_z$ a $3\times3$ matrix. The knowledge of $R_z(\vartheta)$ uniquely determines $$J_z =\frac{1}{i} \frac{\partial R_z(\vartheta)}{\partial\vartheta}\bigg\vert_{\vartheta = 0} = i \left(
\begin{array}{ccc}
0 & -1 & 0\\
1 &0 &0\\
0 & 0 & 0
\end{array}
\right). \label{eq:jz}$$
The factor of $i$ in the exponential of the definition (\[eq:def-of-Jz\]) is often omitted by mathematicians. We shall keep the factor of $i$. It makes $J_z$ hermitian. In the process it becomes a candidate for an observable in the quantum formalism.
The interval $dr^2 = dx^2+dy^2+dz^2$, besides other symmetry transformations of the Galilean group, is invariant not only under the transformation $R_z(\theta)$ but also under two additional transformations associated with rotations $$R_y(\psi) =
\left(
\begin{array}{ccc}
\cos\psi & 0 & -\sin\psi \\
0 & 1 & 0\\
\sin\psi & 0 & \cos\psi
\end{array}
\right) \stackrel{\textrm{\textrm{def} \textrm{$J_y$}}}{=} {\e}^{i J_y\psi}$$ and $$R_x(\phi) =
\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 &\cos\phi &\sin\phi \\
0 & -\sin\phi & \cos\phi
\end{array}
\right) \stackrel{\textrm{def $J_x$}}{=} {\e}^{i J_x \phi}$$ leading to the associated generators of the rotations $$J_y = \frac{1}{i} \frac{\partial R_y(\psi)}{\partial\psi}\bigg\vert_{\psi= 0} = i \left(
\begin{array}{ccc}
0 & 0 & 1\\
0 &0 &0\\
-1 & 0 & 0
\end{array}
\right) \label{eq:jy}$$ and $$J_x =\frac{1}{i} \frac{\partial R_x(\phi)}{\partial\phi}\bigg\vert_{\phi = 0} = i \left(
\begin{array}{ccc}
0 & 0 & 0\\
0 &0 &-1\\
0 & 1 & 0
\end{array}
\right). \label{eq:jx}$$ This is very elementary. But it allows us to introduce the important notion of generators of Lie symmetries in a familiar landscape. For the moment we refrain from studying boosts and spacetime translations.
A beauty of abstraction and a hint for the quantum nature of reality
--------------------------------------------------------------------
\[sec:quantum\]
Now comes an unreasonable beauty of abstraction. Since the matrices do not commute, the generators of rotations given by equations (\[eq:jz\]), (\[eq:jy\]), and (\[eq:jx\]) satisfy the algebraic relationship, or simply the algebra (or, Lie algebra) $$[J_x,J_y] = i J_z,\quad\mbox{and cyclic permutations}.\label{angular-momentum-algebra-in-3D}$$ The abstraction, well known to physicists and mathematicians, consists of the following. Let us momentarily forget how this algebra has arisen, and instead marvel at the infinitely many solutions that exist for this algebra. Each of these solutions is called a representation of the algebra, and the spaces on which the elements of the representation act are called representation spaces.
All of us know that (\[angular-momentum-algebra-in-3D\]) also follows from the fundamental commutator, but we now ask: Does the fundamental commutator follow from rotational symmetry?
Continuing the thread, not only are there finite-dimensional matrix representations of (\[angular-momentum-algebra-in-3D\]), infinitely many of them (this thought continues in chapter \[ch4\]), but there is also an infinite-dimensional representation that is made of differential operators, with $$\begin{aligned}
&J_x = \frac{1}{i}\left( y\frac{\partial}{\partial{z}} - z\frac{\partial}{\partial{y}} \right),\quad
J_y = \frac{1}{i}\left( z\frac{\partial}{\partial{x}} - x\frac{\partial}{\partial{z}} \right),\label{eq:jxjy}\\
&J_z = \frac{1}{i}\left( x\frac{\partial}{\partial{y}} - y\frac{\partial}{\partial{x}} \right).\label{eq:jzz}\end{aligned}$$ Many students of Physics encounter this result in the context of their first quantum mechanics course. Starting with the Heisenberg fundamental commutator, they are taught that taking the classical definition of the angular momentum and replacing the position and momentum by their quantum counterparts, one obtains (\[angular-momentum-algebra-in-3D\]), followed by (\[eq:jxjy\]) and (\[eq:jzz\]).
Given our discussion above, we ask the question: Does quantum aspect of reality spring from rotational symmetry and the implicit assumption of a continuous spacetime? The answer we adopt would then tell us if the quantum gravity spacetime would be discrete, not covered by Lorentz algebra and what modifications would the Heisenberg algebra and the deBroglie wave particle duality suffer. This view is supported by the Kempf, Mangano, and Mann [@Kempf:1994su] and by my own work [@Ahluwalia:2000iw; @Ahluwalia:1993dd].
Taking this view has the consequence that quantum mechanical foundations are seen as an inevitable consequence of the rotational symmetry. The primary result is the Heisenberg algebra, and from that follows the secondary result: the de Broglie wave particle duality We thus rewrite (\[eq:jxjy\]) and (\[eq:jzz\]) as $$\begin{aligned}
&J_x = \frac{1}{\hbar}\times\frac{\hbar}{i}\left( y\frac{\partial}{\partial{z}} - z\frac{\partial}{\partial{y}} \right),\quad
J_y = \frac{1}{\hbar}\times\frac{\hbar}{i}\left( z\frac{\partial}{\partial{x}} - x\frac{\partial}{\partial{z}} \right),\\
&J_z = \frac{1}{\hbar}\times\frac{\hbar}{i}\left( x\frac{\partial}{\partial{y}} - y\frac{\partial}{\partial{x}} \right).\end{aligned}$$ with $\hbar = h/2\pi$ as the reduced Planck’s constant. Identifying $$x\quad \mbox{and} \quad \frac{\hbar}{i}\frac{\partial}{\partial x},\quad
y\quad \mbox{and} \quad \frac{\hbar}{i}\frac{\partial}{\partial y},\quad
z\quad \mbox{and} \quad \frac{\hbar}{i}\frac{\partial}{\partial z}\quad$$ as position and momentum operators naturally leads to the Heisenberg’s fundamental commutators $$\left[x,p_x\right] = i \hbar,\quad \left[y,p_y\right] = i \hbar,\quad \left[z,p_z\right] = i \hbar
\label{eq:fundamentalC}$$ with $[x,y]=0$, etc. Now the eigenfunctions of the momentum operator $$\p = \frac{\hbar}{i}\,\nabla$$ have the form $$\exp\left(i\, \frac{\p^\prime\cdot \x^\prime}{\hbar}\right)\label{eq:eigenf}$$ where $\p^\prime$ and $\x^\prime$ denote eigenvalues of the operators $\p$ and $\x$. We find this Dirac-Schwinger notation useful but only invoke it when an ambiguity is likely to arise. The eigenfunctions (\[eq:eigenf\]) have the spatial periodicity $$\lambda =\frac{h}{p}$$ with $p=\vert\p^\prime\vert$. We are thus required that we associate a wave length $\lambda$ with momentum $p$ à la de Broglie.[^17]
In this interpretation the foundations of classical mechanics are at odds with the rotational symmetry and echo considerations related to the lack of stability of the algebra underling classical mechanics [@Flato:1982yu; @Faddeev:1989LD; @VilelaMendes:1994zg; @Chryssomalakos:2004gk].
To emphasise we repeat that the argument of the conventional courses on quantum mechanics begins with the fundamental commutator and results in the ‘angular momentum commutators.’ It is generally not realised that the de Broglie’s $\lambda = h/p$ is a direct consequence of the Heisenberg’s fundamental commutator. Whereas here we reverse that argument and see the fundamental commutator, and hence the wave particle duality, as a consequence of rotational symmetry. It makes it clear, or opens a discussion, that the classical description of reality is incompatible with the rotational symmetry. Quantum aspect of realty thus seem to spring from the rotational symmetry of the space in which events occur.
The presence of the momentum operator in the Heisenberg algebra introduces kinetic energy in the measurement process. It induces inevitable gravitational effects through the modification of the local curvature. These effects make position measurements non-commutative [@Ahluwalia:1993dd; @Doplicher:1994zv]. The implicit assumption of the commuting position operators of the arrived at quantum nature of reality must, therefore, undergo a modification at the Planck scale where gravitational effects become important. But rotational symmetry seems to make the fundamental commutator inevitable. This paradoxical circumstance can be averted if we entertain the possibility that quantum-gravity requires a non-commutative spacetime accompanied by a modified Heisenberg algebra.
In the argument above we can easily take $\hbar$ as some unknown constant with the dimension of angular momentum and then obtain its chosen identification by, say, looking at the spectrum of a system governed by the hamiltonian $
H = {\p^2}/{2 m} + \left({1}/{2}\right) m \omega^2 \x^2
$. The fundamental commutator yields the energy spectrum of such systems to be equidistant lines with separation $\hbar\omega$, and a zero point energy of $\frac{1}{2} \hbar\omega$.
A unification of the microscopic and the macroscopic
----------------------------------------------------
All this is not mathematical science fiction. Because of its simplicity and its manifest importance, it is good to recall that the stability of the Earth beneath us and Avogadro number of primal entities in a palmful of water speak of it. Take the simplest of the simple systems, the hydrogen atom. Classically, if one minimises the energy, $E=\left(p^2/2m\right) - e^2/r$, of the hydrogen atom then one immediately sees that the energy is minimised at $r=0$. The atom collapses to size zero. In contrast the fundamental commutator in (\[eq:fundamentalC\]) requires that $E$ be minimised to the constraint $r\, p\sim\hbar$. To implement this constraint one may set $r \sim \hbar/p$ and get $E = \left(p^2/2m\right) - e^2 p/\hbar$. Setting, $\partial{E}/{\partial p} =0$, then gives the $E$-minimising $p$, $p_0 \sim m e^2/\hbar$, for which $E$ takes its minimum value $E_0 \sim - m e^4/2 \hbar^2 \approx -13.6 ~\mbox{eV}$. The $r$ instead of collapsing to zero, is now constrained to $
r_0 =\hbar^2/{m e^2} = 0.5 \times 10^{-8} ~\textrm{cm}
$. It is a good measure of the size of most atoms. The reason being that for heavier nuclei with atomic number $Z$, for the outermost electron the $(Z-1)$ electrons screen the nucleus in such a way that the effective nuclear charge remains $e$ [@Weinberg:2012qm]. In this simple manner we not only understand the origin of the ionisation energy of the hydrogen atom but we also obtain the order of magnitude for the Avagadro’s number ($\sim (1/r_0)^3$).
These simple argument unify the microscopic, the hydrogen atom, with the macroscopic, the Earth. Beyond the stability of the planets, and burning of the stars, quantum nature of reality leaves its imprints in the entire cosmos and to possible physics beyond. Beyond, where rotational symmetry either ceases to be, or takes a new – not yet known – form in discreteness of spacetime [@Padmanabhan:2016eld].
Lorentz algebra {#sec:LorentzAlgebra}
---------------
On the planet Earth, the Lorentz algebra is usually arrived at by considering the transformations of the spatial and temporal specifications of events. Elsewhere in the cosmos one may arrive at the very same algebra by studying the spectrum of hydrogen atom, instead through the null result of the terrestrially famous 1887 experiment of Michelson and Morley [@Michelson:1887zz].
The folklore that the 1887 experiment requires Lorentz symmetries is, strictly speaking, not true [@Cohen:2006ky]. The Lorentz symmetries follow only if one requires in addition any of the four discrete symmetries. One of these discrete symmetries is Parity a symmetry known to be violated in the electroweak interactions [@Lee:1956qn; @Wu:1957my]. Remaining three are: time reversal, and charge conjugation conjugated with parity, or time reversal. The most familiar way to arrive at the Lorentz algebra is to simply note that absolute space and absolute time are now empirically untenable. While a rotation, say about the $z$-axis, *does not* mix time and space (units: speed of light is now taken as unity) $$\left(
\begin{array}{c}
t^\prime\\
x^\prime\\
y^\prime\\
z^\prime
\end{array}
\right) = \left(\begin{array}{cccc}
1& 0 & 0 & 0\\
0& \cos\vartheta & \sin\vartheta & 0\\
0& -\sin\vartheta & \cos\vartheta & 0 \\
0& 0 & 0 & 1
\end{array}
\right) \left(\begin{array}{c}
t\\
x\\
y\\
z
\end{array}\right).$$ a boost, say along the $x$-axis *does* $$\left(
\begin{array}{c}
t^\prime\\
x^\prime\\
y^\prime\\
z^\prime
\end{array}
\right) = \left(\begin{array}{cccc}
\cosh\varphi&\sinh\varphi& 0 & 0\\
\sinh\varphi& \cosh\varphi & 0 & 0\\
0 & 0 & 1&0 \\
0& 0 & 0 & 1
\end{array}
\right) \left(\begin{array}{c}
t\\
x\\
y\\
z
\end{array}\right)\label{eq:boost-x}$$ where the rapidity parameter, $\vp = \varphi\,\widehat \p$, is defined as $$\begin{aligned}
&\cosh\varphi = E/m = \gamma = 1/\sqrt{1-v^2}\nonumber\\
&\sinh\varphi = p/m = \gamma v\label{eq:rapidity-parameter}
\end{aligned}$$ with all symbols carrying their usual meaning. Denoting the $4\times 4$ boost matrix in (\[eq:boost-x\]) by $B_x(\varphi)$. The generator of the boost along the $x$-axis is thus $$K_x= \frac{1}{i}\frac{\partial B_x(\varphi)}{\partial\varphi}\bigg\vert_{\varphi=0} =
- i \left(
\begin{array}{cccc}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 &0\\
0 & 0 & 0 & 0
\end{array}
\right).\label{eq:kx}$$ This is complemented by the remaining two generators of the boosts $$K_y = -i \left(
\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 0 &0\\
0 & 0 & 0 & 0
\end{array}
\right),\quad K_z = -i \left(
\begin{array}{cccc}
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 &0\\
1 & 0 & 0 & 0
\end{array}
\right)\label{eq:kykz}$$ and the three generators of the rotations. These are directly read off from our work earlier with the added observation that under rotations $t$ and $t^\prime$ are identical $$J_x= - i \left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 &1\\
0 & 0 & -1 & 0
\end{array}
\right),\label{eq:jx-new}$$ $$J_y = -i \left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & -1\\
0 & 0 & 0 &0\\
0 & 1 & 0 & 0
\end{array}
\right),\quad J_z = -i \left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & -1 & 0 &0\\
0 & 0 & 0 & 0
\end{array}
\right).\label{eq:jyjz-new}$$
The six generators satisfy the following algebra $$\begin{aligned}
& \left[J_x,J_y\right] = i J_z\quad\textrm{and~cyclic~permutations}\label{eq:rotational-sub-algebra}\\
& \left[K_x,K_y \right] = - i J_z,\quad\textrm{and~cyclic~permutations}\\
& \left[J_x, K_x\right] = 0,\quad\textrm{etc.} \\
&\left[J_x,K_y\right] = i K_z,\quad\textrm{and~cyclic~permutations}.\end{aligned}$$ It is named Lorentz algebra.
Further abstraction: Un-hinging the Lorentz algebra from its association with Minkowski spacetime
-------------------------------------------------------------------------------------------------
To underline the mysteries of nature coded in the algebra just arrived at let us take the liberty of imagining that we forget how we arrived at this algebra. Each civilisation in the cosmos, sooner or later, is likely to arrive at this truth, this reflection of low-energy reality, in one way or another. Some in a way similar to ours, others in ways different, even ways we have not yet dreamed of. Lorentz algebra is a powerful unifying element in unearthing the nature of reality. Its various aspects thread through this monograph, with Physics as our primary focus.
We thus symbolically unhinge the Lorentz algebra from spacetime symmetries and express it as an abstract reality expressed through the following abstraction ,\[eq:abstractionJ\] such that $\mathfrak{J}$ and $\mathfrak{K}$ are no longer confined to being identified with the $4\times 4$ matrices given in (\[eq:kx\]) to (\[eq:jyjz-new\]), or with their unitarily transformed expressions, but still satisfy $$\begin{aligned}
&\left[\mathfrak{J}_x,\mathfrak{J}_y \right]= i \mathfrak{J}_z,\quad\mbox{and cyclic permutations}\label{eq:sub-new}\\
&\left[\mathfrak{K}_x,\mathfrak{K}_y\right] = - i\mathfrak{J}_z,\quad\mbox{and cyclic permutations}\\
&\left[\mathfrak{J}_x,\mathfrak{K}_x \right]=0, \quad\mbox{etc.}\\
&\left[\mathfrak{J}_x,\mathfrak{K}_y \right]= i \mathfrak{K}_z,\quad\mbox{and cyclic permutations}.
\label{eq:la-new}\end{aligned}$$ The ${\mathfrak{J}}_i$ and $\mathfrak{K}_j$, $i,j=x,y,z$, represent generators of rotations and boosts – in an abstract space. Their exponentiations $$\exp\left(i\bmfj\cdot\bm{\theta}\right), \quad \exp\left(i\bmfk\cdot\vp\right)
\label{eq:exponentiation}$$ give the group transformations under rotations and boosts for the ‘vectors’ spanning the associated representation space. While the underlying algebraic structure remains the same, each of the group transformations – chosen by a specific choice of $\bmfj$ and $\bmfk$ in (\[eq:exponentiation\]) – defines a new group. These group transformations act on vectors that inhabit the associated representation spaces. Upto a convention related freedom of a unitary transformation, the transformations of four vectors, to transformations of Dirac spinors, to ‘other vectors’ in infinitely many other representation spaces, are all obtained from (\[eq:exponentiation\]).
To cast the Lorentz algebra in a manifestly covariant form, we recall the definition of the three-dimensional Levi-Civita symbol \_[ijk]{} =
+1 &\
-1 &\
0 &
That is, if any two indices are equal the $\epsilon$ symbol vanishes. If all the indices are unequal, we have \_[ijk]{} = (-)\^p \_[123]{} where p, known as the parity of the permutation, is the number of interchanges of indices necessary to transmute $ijk$ into the order $123$. The factor $(-1)^p$ is called the signature of the permutation. Equipped with the Levi-Civita symbol we can define a completely antisymmetric operator \_ =
\_[ij]{} = - \_[ji]{} = \_[ijk]{} \_k\
\_[i0]{} = -\_[0i]{} = - \_i
with the greek indices $\mu$ and $\nu$ taking the values $0,1,2,3$, and the latin indices confined to the values $1,2,3$. We follow the Einstein convention. It assumes repeated indices are summed.
These definitions can be used to cast the Lorentz algebra in the following two equivalent form: = i(\_ \_ - \_ \_ + \_ \_ -\_ \_ ) where $\eta_{\mu\nu} = \textrm{diag}(1,-1,-1,-1)$ is the spacetime metric, and = i \_[ijk]{} \_k,= i \_[ijk]{} \_k,= -i \_[ijk]{} \_k \[eq:LorentzAlgebra\]
The $\J$ and $\K$ of section \[sec:LorentzAlgebra\] constitute the most familiar representation. The associated representation space for historical reasons is called Minkowski spacetime (or, generally the space of four vectors).
Representations of Lorentz Algebra {#ch4}
==================================
Poincaré algebra, mass, and spin
--------------------------------
For the purposes of a physicist a ‘representation of an algebra’ is simply a specific solution to the symmetry algebra under consideration.
Thus the six $4\times 4$ matrices given in equations (\[eq:kx\]) to (\[eq:jyjz-new\]) form a representation of the Lorentz algebra. It is a finite dimensional representation, and there are infinitely many of them. We will start considering them in the next section.
On the other hand, the set of generators $$\begin{aligned}
&J_x = \frac{1}{i}\left( y\frac{\partial}{\partial{z}} - z\frac{\partial}{\partial{y}} \right),\quad
J_y = \frac{1}{i}\left( z\frac{\partial}{\partial{x}} - x\frac{\partial}{\partial{z}} \right),\label{eq:diffa}\\
&J_z = \frac{1}{i}\left( x\frac{\partial}{\partial{y}} - y\frac{\partial}{\partial{x}} \right)\label{eq:diffb}\end{aligned}$$ coupled with $$\begin{aligned}
&K_x = {i}\left( t\frac{\partial}{\partial{x}} + x\frac{\partial}{\partial{t}} \right),\quad
K_y = {i}\left( t\frac{\partial}{\partial{y}} + y\frac{\partial}{\partial{t}} \right),\label{eq:diffc}\\
&K_z = {i}\left( t\frac{\partial}{\partial{z}} + z\frac{\partial}{\partial{t}} \right).\label{eq:diffd}\end{aligned}$$ provide an infinite dimensional representation of the same very algebra – the Lorentz albegra.
In the context of Minkowski space, Nature also supports the symmetry induced by the spacetime translations $$x^\mu\rightarrow x^{\prime\mu} = {\Lambda^\mu}_\nu x^\nu + a^\mu$$ generated by $$P_\mu = i\frac{\partial}{\partial x^\mu} \label{eq:sttr}$$ In the above we have defined ${\Lambda^\mu}_\nu$, the transformations of the Lorentz group, as follows $$\begin{aligned}
{\Lambda^\mu}_\nu =
\begin{cases}
{\Big[\exp(i\J\cdot \bm{\vt})\Big]^\mu}_\nu, & ~~~~~ \mbox{for rotations}\\ \\
{\Big[\exp(i\K\cdot \bm{\varphi})\Big]^\mu}_\nu, & ~~~~~\mbox{for boosts}
\end{cases}\end{aligned}$$ with $\J$ and $\K$ given by equations (\[eq:kx\]-\[eq:jyjz-new\]) and take $a^\mu$ as a constant four vector.
When one adjoins these four generators, encoded in (\[eq:sttr\]), to the six generators of rotations and boosts (\[eq:diffa\])-(\[eq:diffd\]) one obtains the 10 generators of the Poincaré algebra: $$\begin{aligned}
& \left[{J}_{\mu\nu},{J}_{\rho\sigma}\right] = i\left(\eta_{\nu\rho} {J}_{\mu\sigma}
- \eta_{\mu\rho} {J}_{\nu\sigma} + \eta_{\mu\sigma} {J}_{\nu\rho}
-\eta_{\nu\sigma} {J}_{\mu\rho}
\right) \\
& \left[P_\mu, J_{\rho\sigma}\right] = i\left( \eta_{\mu\rho} P_\sigma - \eta_{\mu\sigma} P_\rho\right) \\
& \left[P_\mu,P_\nu\right] =0 .\end{aligned}$$ It is remarkable that the fundamental notions of mass and spin arise from these symmetries. To see this, we introduce two ‘Casimir’ operators $$C_1 = P_\mu P^\mu, \quad C_2 = W_\mu W^\mu$$ with the Pauli Lubański pseudovector defined as [@Lubanski:1942jk] $$W_\mu \stackrel{\text{def}}{=} \frac{1}{2 m}\epsilon_{\mu\nu\rho\sigma}J^{\nu\rho} P^\sigma$$ The ‘extra’ factor of $1/m$ in the above definition is consistent with Lubański’s original paper. In considering the massless case we may introduce a related operator that is more in keeping with its later re-defintion. Now because the commutator $\left[P^\sigma,
P^\mu \right]$ vanishes, the projection of $W_\mu$ on the generators of spacetime translations vanishes $$W_\mu P^\mu =0$$ because the $\epsilon$ symbol is antisymmetric under the interchange of the indices $\sigma,\mu$ and $P^\sigma P^\mu$ is symmetric under the interchange of the same indices. It is then a few lines of exercise to show that (see, for example, [@Tung:1985na] or [@Ryder:1985wq]) $$\left[W^\mu,P^\mu\right] = 0$$ with the consequence that not only $W_\mu W^\mu$ is invariant under the Lorentz transformation but also under spacetime translations. And thus it commutes with all the ten generators of the Poincaré algebra. A parallel of the same argument tells us that $P_\mu P^\mu$ also commutes with all the ten generators of the Poincar’e algebra. The eigenvalues of the latter are identified with the square of the mass parameter $m$, and modulo a sign the eigenvalues of the former can be easily identified with the eigenvalues of the square of the generators of rotations: $- s(s+1)$.[^18] It is for these reasons that mass and spin arise in the description of the physical reality. Once one inertial observer measures them, all inertial observers related by Poincaré symmetries measure the same value.
We generally assume that electron has the same ‘mass’ and same ‘spin,’ here in our solar system as in a far away distant galaxy going back to the dawn of our universe – despite the fact that in earlier epochs Poincaré symmetries may have suffered a modification. But perhaps where and when these departures occur we enter another realm, the realm of massless particles, with hypothetical massless observers. These observers have no rest frame, and to us they lie in a realm where spacetime dimensionality changes, and time as we generally define does not exist. Such a dramatic change arises from singularities of length contraction and time dilation for truly massless particles.
### A cautionary remark
Except in the rest frame, the $s(s+1)$ encountered above is not an eigenvalue of $\bmfj^2$. It is $C_2$ whose eigenvalues remain invariant under Poincaré transformations and not the eigenvalues of $\bmfj^2$ except in the rest frame – or, in an accidental situation to be discussed below.
Representations of Lorentz algebra\[sec:representations-of-lorentz-algebra\]
----------------------------------------------------------------------------
To identify various representations of Lorentz algebra one often starts with introducing two $\mathfrak{su\left(2\right)}$ generators: $$\bmfa =\frac{1}{2}\left(\bmfj+ i \bmfk\right),\quad \bmfb =\frac{1}{2}\left(\bmfj- i \bmfk\right)$$ and studies representations of $\mathfrak{su\left(2\right)}\otimes\mathfrak{su\left(2\right)}$, keeping in mind that $$\begin{aligned}
&\left[\mathfrak{A}_x, \mathfrak{A}_y\right] = i \mathfrak{A}_z,\quad\textrm{and~cyclic~permutations}\\
&\left[\mathfrak{B}_x, \mathfrak{B}_y\right] = i \mathfrak{B}_z,\quad\textrm{and~cyclic~permutations}\\
&\left[\mathfrak{A}_i,\mathfrak{B}_j\right] = 0,\quad i,j=x,y,z. \label{eq:two-su(2)s}\end{aligned}$$ In this way one labels the resulting representation space by two labels $(a,b)$, with $a(a+1)$ and $b(b+1)$ being eigenvalues of $\bmfa^2$ and $\bmfb^2$, respectively. We find this complexification of the generators unnecessary. I mention it here because our reader may encounter it in her/his studies often. If due interpretational caution is not exercised all this can cause confusion as $\bmfj^2$ too is not a Casimir invariant of the Lorentz algebra (as just cautioned above), but only of the rotational subalgebra represented by the first of the equations in (\[eq:LorentzAlgebra\]). Introducing two $\mathfrak{su\left(2\right)}$ algebras through complexifications of the generators when one is permitted only real linear combination strictly speaking, takes us away from the Lorentz algebra and it is avoidable.
————–
To avoid conceptual confusion we do not follow this potentially misleading traditional approach but instead follow a straight forward method. For this we note that once a representation of $\bmfj$ is found[^19] that satisfies the first part of (\[eq:LorentzAlgebra\]) an inspection of the remainder of the Lorentz algebra shows that for each representation of $\bmfj$ there exist *two* independent primordial representations for the boost generators $$\bmfk = - i \bmfj,\quad \bmfk= + i\bmfj$$ The sets $$\underbrace{\bmfj, \bmfk= - i\bmfj}_{\mathcal{R}~\textrm{type}}\quad{\textrm{and}}\quad\underbrace{ \bmfj,\bmfk= i\bmfj}_{\mathcal{L}~\textrm{type}} \label{eq:primordial-generators}$$ provide two independent representations of the Lorentz algebra. As indicated, we call these representaions as $\mathcal{R}$ type and $\mathcal{\mathcal L}$ type, respectively. The reason for this nomenclature shall become apparent in section \[sec:RL\] below.
Without the existence of two, in contrast to one, representations for each $\bmfj$ one would not be able to respect causality in quantum field theoretic formalism respecting Poincaré symmetries, or have massive antiparticles required to avoid causal paradoxes. And thus this doubling of representations is more than an oddity, or a coincidence. It is a reflection a deep underlying thread of symmetries, causality, and degrees of freedom encountered in quantum fields.
In accordance with (\[eq:exponentiation\]), the associated symmetry transformations are implemented as follows $$\mathcal{R:}
\begin{cases}
\textrm{Rotations~by}: & \exp[i\bmfj\cdot{\boldmath\boldsymbol{\vt}}] \label{eq:R-type-rb} \\
\textrm{Boosts~by}: & \exp[ i (-i\bmfj)\cdot{\boldmath\boldsymbol{\varphi}}] = \exp[\bmfj\cdot{\boldmath\boldsymbol{\varphi}}]
\end{cases}$$ and $$\mathcal{L:}
\begin{cases}
\textrm{Rotations~by}: & \exp[i\bmfj\cdot{\boldmath\boldsymbol{\vartheta}}] \\
\textrm{Boosts~by}: & \exp[ i (i\bmfj)\cdot{\boldmath\boldsymbol{\varphi}}] = \exp[-\bmfj\cdot{\boldmath\boldsymbol{\varphi}}] \label{eq:L-type-rb}
\end{cases}$$ For these primordial representations the knowledge of $\bmfj$ completely determines the rotation, parameterised by ${\boldmath\boldsymbol{\vartheta}}$, and the boost, parameterised by ${\boldmath\boldsymbol{\varphi}}$, transformations.
The boosts do not carry the imaginary $i$ in the exponentiation. It is not that we have suddenly changed to the convention of the mathematics literature but because the $i$ in the exponentiation for the boost operator when encountered with the $\pm i$ in the expression for $\mathfrak{K}$ in (\[eq:primordial-generators\]) results in $\mp 1$. I make this parenthetic note because again and again in my seminars I encounter a question to this effect.
### Notational remark
In the language of the traditional approach of labelling representation spaces, the $\mathcal{R}$ and $\mathcal{L}$ stand for the $(j,0)$ and $(0,j)$ representation spaces, respectively: $\mathcal{R}\leftrightarrow (j,0),
\mathcal{L}\leftrightarrow (0,j).$
### Accidental Casimir
For the primordial representations introduced above despite the general remark made above about $\bmfj^2$ after equations (\[eq:two-su(2)s\]) $\bmfj^2$ does commute with all the generators given in (\[eq:primordial-generators\]). For this reason, in the restricted context of the $\mathcal{R}$ and $\mathcal{L}$ representation spaces, we introduce the term ‘accidental Casimir operator’ for $\bmfj^2$. Its invariant eigenvalue $s(s+1)$ coincides with that of the $C_2$ in all inertial frames.
Simplest representations of Lorentz algebra\[sec:RL\]
-----------------------------------------------------
With our focus on (\[eq:R-type-rb\]) and (\[eq:L-type-rb\]), the simplest non trivial $\bmfj$ that satisfies the Lorentz algebra (\[eq:LorentzAlgebra\]) is formed from the Pauli matrices $$\sigma_x=\left(
\begin{array}{cc}
0 & 1 \\
1 &0
\end{array}
\right),\quad
\sigma_y=\left(
\begin{array}{cc}
0 & -i \\
i &0
\end{array}
\right),\quad
\sigma_z=\left(
\begin{array}{cc}
1 & 0 \\
0 &-1
\end{array}
\right)\label{eq:Pauli-Matrices}$$ and is given by $$\bmfj = \s/2.$$ Through (\[eq:primordial-generators\]), it introduces two independent representations spaces of spin one half.
Referring to equations (\[eq:R-type-rb\]) and (\[eq:L-type-rb\]) the group transformations for rotations and boosts for these representation spaces are given by $$\begin{aligned}
& R_{\mathcal{R},\mathcal{L}}({\boldmath\boldsymbol{\vartheta}}) = \exp\left[{i \frac{{\boldmath\boldsymbol{\sigma}}}{2}\cdot{\boldmath\boldsymbol{\vartheta}} }\right] \label{eq:rotations-for-spin-1/2} \\
& B_\mathcal{R,L} (\vp) =\exp\left[\pm {\frac{{\boldmath\boldsymbol{\sigma}}}{2}\cdot{\boldmath\boldsymbol{\varphi}}} \right]
\label{eq:boosts-for-spin-1/2}\end{aligned}$$ where the upper sign is for the $\mathcal{R}$ type boosts, while the lower sign is for the $\mathcal{L}$ type boosts. The expression for the transformation for rotation is the same for both type of primordial representations.
Writing ${\boldmath\boldsymbol{\vartheta}} = \vartheta \,{\boldmath\boldsymbol{\widehat{n}}}$, with $\vartheta$ representing the angle of rotation and ${\boldmath\boldsymbol{\widehat{n}}}$ denoting a unit vector along the axis of rotation, the transformation for rotations (\[eq:rotations-for-spin-1/2\]) takes the form $$\begin{aligned}
R_{\mathcal{R},\mathcal{L}}({\boldmath\boldsymbol{\vartheta}}) & = \I + i \s\cdot{\boldmath\boldsymbol{\widehat{n}}}\frac{\vartheta}{2} -\frac{1}{2!}
\left(\s\cdot{\boldmath\boldsymbol{\widehat{n}}}\right)^2\left(\frac{\vartheta}{2}\right)^2
- i \frac{1}{3!}
\left(\s\cdot{\boldmath\boldsymbol{\widehat{n}}}\right)^3\left(\frac{\vartheta}{2}\right)^3 \nonumber\\
& \hspace{21pt} +\frac{1}{4!}
\left(\s\cdot{\boldmath\boldsymbol{\widehat{n}}}\right)^4\left(\frac{\vartheta}{2}\right)^4
+ i\frac{1}{5!}
\left(\s\cdot{\boldmath\boldsymbol{\widehat{n}}}\right)^5\left(\frac{\vartheta}{2}\right)^5 + \ldots \end{aligned}$$ Using the identity $\left(\s\cdot{\boldmath\boldsymbol{\widehat{n}}}\right)^2 = \I$, the above expansion reduces to $$\begin{aligned}
& R_{\mathcal{R},\mathcal{L}}({\boldmath\boldsymbol{\vartheta}}) =\bigg[1- \frac{1}{2!} \left(\frac{\vartheta}{2}\right)^2
+\frac{1}{4!} \left(\frac{\vartheta}{2}\right)^4 -\ldots
\bigg]\I \nonumber\\
&\hspace{51pt}+ i \s\cdot{\boldmath\boldsymbol{\hat{n}}}
\bigg[ \frac{\vartheta}{2} -\frac{1}{3!}\left( \frac{\vartheta}{2} \right)^3
+
\frac{1}{5!}
\left( \frac{\vartheta}{2} \right)^5-\ldots
\bigg]\end{aligned}$$ and simplifies to yield $$R_{\mathcal{R},\mathcal{L}}({\boldmath\boldsymbol{\vartheta}}) =
\cos\left(\frac{\vartheta}{2}\right)\I + i \s\cdot{\boldmath\boldsymbol{\widehat{n}}} \sin\left(\frac{\vartheta}{2}\right)\label{eq:rotation-for-Weyl-spinors}$$
An exactly similar calculation as above yields the boosts noted in (\[eq:boosts-for-spin-1/2\]) to be given by $$\begin{aligned}
B_\mathcal{R,L} (\vp) &= \cosh\left(\frac{\varphi}{2}\right)\I
\pm \s\cdot{\boldmath\boldsymbol{\widehat{p}}}\,
\sinh\left(\frac{\varphi}{2}\right) \\
&= \cosh\left(\frac{\varphi}{2}\right)\left[\I\pm\s\cdot {\boldmath\boldsymbol{\widehat{p}}}
\,\tanh\left(\frac{\varphi}{2}\right)\right] \label{eq:R-L-boost}\end{aligned}$$ where the the upper sign on the righthand side of the above result holds for the $\mathcal{R}$ type representations, and the lower sign holds for the $\mathcal{L}$ type representations. To further simplify (\[eq:R-L-boost\]) we use the identities $$\begin{aligned}
&\cosh\left(\frac{\varphi}{2}\right) = \sqrt{ \frac{\cosh\varphi+1}{2}} =\sqrt{\frac{E+m}{2 m}}\\
&\tanh\left(\frac{\varphi}{2}\right) =\frac{\sinh\varphi}{\cosh\varphi+1} = \frac{p}{E+m}\end{aligned}$$\[eq:boosts-for-R-L-spinors\] for the rapidity parameter defined in (\[eq:rapidity-parameter\]) and rewrite (\[eq:R-L-boost\]) as $$\begin{aligned}
B_\mathcal{R,L} (\vp) = \sqrt{\frac{E+m}{2 m}}\left[\I \pm \frac{\s\cdot\p}{E+m} \right]\label{eq:Weyl-boosts}\end{aligned}$$ Now note that the spin-$1/2$ helicity operator is defined as \[eq:helicity\] and thus $\s\cdot\p$ that appears in $B_\mathcal{R,L} (\vp)$ can be written as $ 2 p \mathfrak{h}$. This allows us to re-write (\[eq:Weyl-boosts\]) in the form B\_ () = \[eq:Weyl-boosts-h\]
The transformations $R_{\mathcal{R},\mathcal{L}}({\boldmath\boldsymbol{\theta}})$ and $B_{\mathcal{R},\mathcal{L}}
({\boldmath\boldsymbol{\varphi}})$ act on the two-dimensional representation spaces. For historical reasons the ‘vectors’ in these spaces are called $\mathcal{R}$-type and $\mathcal{L}$-type Weyl spinors. To gain insights into these representation spaces we work out the effect of the boosts and rotations on the eigenspinors of $\mathfrak{h}$. For an arbitrary four-momentum $p^\mu$ these are defined as $$\mathfrak{h} \,\phi_\pm^{\mathcal{R},\mathcal{L}}(p^\mu) = \pm \frac{1}{2} \phi_\pm^{\mathcal{R},\mathcal{L}}(p^\mu)\label{eq:Weyl-spinors-h}$$ Introducing the standard four momentum vector for massive particles $$k^\mu = \left(
\begin{array}{c}
m\\
\bm{0}
\end{array}
\right),\quad \bm{0}\stackrel{\textrm{def}}{=}\p \big\vert_{p\to 0}
\label{eq:standard-four-vector}$$ the Weyl spinors for a particle at rest $\phi_\pm^{\mathcal{R},\mathcal{L}}(k^\mu)$ also satisfy (\[eq:Weyl-spinors-h\]) $$\mathfrak{h} \,\phi_\pm^{\mathcal{R},\mathcal{L}}(k^\mu) = \pm \frac{1}{2} \phi_\pm^{\mathcal{R},\mathcal{L}}(k^\mu)$$ Once $\phi_\pm^{\mathcal{R},\mathcal{L}}(k^\mu)$ are chosen, the boosted $\phi_\pm^{\mathcal{R},\mathcal{L}}(p^\mu)$ follow immediately by the application of the boost operators (\[eq:Weyl-boosts-h\]). To avoid keeping track of two many $\pm$ and $\mp$ signs, consider $\mathcal{R}$ type Weyl spinors first $$\begin{aligned}
\phi_\pm^{\mathcal{R}}(p^\mu) & = B_\mathcal{R} (\vp)
\phi_\pm^{\mathcal{R}}(k^\mu) \\
& =\sqrt{\frac{E+m}{2 m}}\left[\I \pm \frac{p}{E+m} \I\right] \phi_\pm^{\mathcal{R}}(k^\mu) \end{aligned}$$ To explore the high-energy limit we note that for $p \gg m$, $$\begin{aligned}
\frac{p}{E+m} \approx 1-\frac{m}{p}
\end{aligned}$$ As a result we have $$B_\mathcal{R} \,(\vp) \phi_\pm^{\mathcal{R}}(k^\mu)
\approx \sqrt{\frac{E+m}{2 m}}\left[1\pm\left(1-\frac{m}{p}\right) \right] \phi_\pm^{\mathcal{R}}(k^\mu)
\label{eq:interpretation-R}$$ Implementing the high-energy limit by taking the massless limit one thus arrives at the result that for the $\mathcal{R}$ type Weyl spinors only the positive helicity degree of freedom survives. While for the $\mathcal{L}$ type Weyl spinors the counterpart of (\[eq:interpretation-R\]) reads $$B_\mathcal{L} \,(\vp) \phi_\pm^{\mathcal{L}}(k^\mu)
\approx \sqrt{\frac{E+m}{2 m}}\left[1\mp\left(1-\frac{m}{p}\right) \right] \phi_\pm^{\mathcal{L}}(k^\mu)
\label{eq:interpretation-L}$$ As a consequence, in the high energy limit only the negative helicity spinors survive.
The nomenclature of $\mathcal{R}$ and $\mathcal{L}$ type Weyl spinors arises from these high energy limits. For the $m=0$ case the helicity is an invariant under boosts, and from there arose the nomenclature of the right-handed ($\mathcal{R}$ type) and left-handed ($\mathcal{L}$ type) Weyl spinors. But for the massive case helicity is not an invariant and thus one must be careful. The $\mathcal{R}$ and $\mathcal{L}$ must be taken as labels for the representation spaces under consideration unless one wishes to restrict one’s calculations to a preferred frame for convenience. Both the $\mathcal{R}$ type and the $\mathcal{L}$ type Weyl spinors support positive helicity and the negative helicity spinors. A similar exercise for higher spins tells us that in the high energy limit only the maximal helicity $(\pm j)$ associated with the $j$-vectors survives. By $j$-vectors we mean $(2j+1)$ column vectors with $(2j+1)$ complex entries either in the $\mathcal{R}$ or $\mathcal{L}$ representation space. In this language massive Weyl spinors are $1/2$-vectors.
Equation (\[eq:rotation-for-Weyl-spinors\]) implies that the action of $R_{\mathcal{R},\mathcal{L}}({\boldmath\boldsymbol{\vartheta}}) $ on Weyl spinors for $\vartheta=2\pi$, irrespective of the axis of rotation ${\boldmath\boldsymbol{\widehat{n}}}$, introduces a phase factor equal to $-1$. It requires a $4\pi$ rotation for a spinor to return to itself. In general Weyl spinors pick up a helicity-dependent phase factor $$\phi(p^\mu) \to \begin{cases}\e^{+i\vartheta/2} \phi(p^\mu),\hspace{3pt}\mbox{for the positive helicity Weyl spinor}\\
\e^{-i\vartheta/2} \phi(p^\mu),\hspace{3pt}\mbox{for the negative helicity Weyl spinor}
\end{cases}\label{eq:phasefactor}$$ We shall discover in Chapter \[ch10\] that these phase factors play a dramatic role for Elko, the eigenspinors of the charge conjugation operator.
Spacetime: Its construction from the simplest representations of Lorentz algebra {#sec:constructing-xt}
--------------------------------------------------------------------------------
The simplest representations of the Lorentz algebra provide the basic elements to describe the matter fields that we encounter around us, the basic blocks from which we are constructed from. The gauge fields that keep them interacting, glue them and add fire, carry a ‘vector’ spacetime index. This section is devoted to construct this vector index, in essence by constructing spacetime itself.
Towards this end, using the the $\mathcal{R}$ and $\mathcal{L}$ type generators for spin one half {\_ = /2,\_=-i/2}, {\_ = /2,\_=i/2} \[eq:rl-generators\] we introduce the following six generators for the $\mathcal{R}\otimes\mathcal{L}$ representation space $$\begin{aligned}
&{\mathfrak{J}}_i = \big[(\zeta_{\mathcal{R}})_i \otimes\I \big] + \big[\I\otimes(\zeta_{\mathcal{L}})_i\big] \label{eq:zeta-i}\\
& {\mathfrak{K}}_i = \big[(\kappa_{\mathcal{R}})_i \otimes\I \big]
+ \big[\I\otimes(\kappa_{\mathcal{L}})_i\big] \label{eq:kappa-i}\end{aligned}$$ with $i=x,y,z$. These generators correspond to rotation and boost transformations given by R()R\_ ()R\_ (),B()B\_ () B\_ ().\[eq:RB\]
The definitions (\[eq:RB\]), and (\[eq:zeta-i\]) and (\[eq:kappa-i\]) mutually define each other
\_i= R()\_[0]{} Each of the ${\mathfrak{J}}_i$ corresponds to taking ${\boldmath\boldsymbol{\widehat{n}}}$ in the defintion ${\boldmath\boldsymbol{\vartheta}} = \vartheta \,{\boldmath\boldsymbol{\widehat{n}}}$ along each of the three spatial axes. Similarly, \_i= B()\_[0]{} Each of the $\mathfrak{K}_i$ corresponds to taking ${\boldmath\boldsymbol{\widehat{\p}}}$ in the defintion ${\boldmath\boldsymbol{\varphi}} = \varphi \,{\boldmath\boldsymbol{\widehat{\p}}}$ along each of the three spatial axes.
We now show that modulo a convention-related unitary transformation $U$ given below in equation (\[eq:Umatrix\]), the $R({\boldmath\boldsymbol{\vartheta}})$ and $B({\boldmath\boldsymbol{\varphi}})$ are precisely the special relativistic transformations of the rotation and boosts.
To establish this claim all we have to do is to obtain $\mathfrak{J}_i$ and $\mathfrak{K}_i$ defined in (\[eq:zeta-i\]) and (\[eq:kappa-i\]) and show that a unitary transformation $U$ exists such that the $U$-transformed $\mathfrak{J}_i$ and $\mathfrak{K}_i$ reduce to $J_i$ and $K_i$ of special relativity. The $U$ simply implements the convention that the temporal coordinate is placed on top of the spatial co-ordinates in the standard sequence so that $x^\mu$ equals $(t,\x)$.
Implementing the definitions (\[eq:rl-generators\])-(\[eq:kappa-i\]) explicitly, and using Pauli matrices in the representation (\[eq:Pauli-Matrices\]) we obtain $$\mathfrak{J}_x= \frac{1}{2}\left(\begin{array}{cccc}
0 &1 & 1 &0 \\
1 & 0 & 0 & 1\\
1 & 0 & 0 &1\\
0 & 1 & 1 &0
\end{array}\right),$$ $$\mathfrak{J}_y= \frac{1}{2}\left(\begin{array}{cccc}
0 &- i & -i &0 \\
i & 0 & 0 & -i\\
i & 0 & 0 &-i\\
0 & i & i &0
\end{array}\right),\quad
\mathfrak{J}_z= \left(\begin{array}{cccc}
1 &0 & 0 &0 \\
0 & 0 & 0 & 0\\
0 & 0 & 0 &0\\
0 & 0 & 0 &-1
\end{array}\right)$$ and $$\mathfrak{K}_x= \frac{1}{2} \left(\begin{array}{cccc}
0 & i & -i &0 \\
i & 0 & 0 & -i \\
-i & 0 & 0 &i\\
0 & -i & i &0
\end{array}\right),$$ $$\mathfrak{K}_y= \frac{1}{2}\left(\begin{array}{cccc}
0 & 1& - 1 &0 \\
-1 & 0 & 0 & -1\\
1& 0 & 0 &1\\
0 & 1 & -1 &0
\end{array}\right),\quad
\mathfrak{K}_z= \left(\begin{array}{cccc}
0 &0 & 0 &0 \\
0 & - i & 0 & 0\\
0 & 0 & i &0\\
0 & 0 & 0 & 0
\end{array}\right)$$ A straight forward calculation now shows that with the $U$ defined as U (
[cccc]{} 0 & i & -i &0\
-i & 0 & 0 & i\
1 & 0 & 0 & 1\
0 & i & i &0
) \[eq:Umatrix\] we have the required result $$\begin{aligned}
& U \mathfrak{J}_i U^{-1} \to J_i \\ & U\mathfrak{K}_i U^{-1} \to K_i\end{aligned}$$ with $J_i$ and $K_i$ of special relativity given in (\[eq:kx\]) to (\[eq:jyjz-new\]). The usual $4\times 4$ transformation matrices of the special relativity are nothing but one or the other of the following $$\left[{\Lambda^\mu}_\nu \right]= U \left\{
\begin{array}{l}
R({\boldmath\boldsymbol{\vartheta}}) \\
B({\boldmath\boldsymbol{\varphi}})
\end{array}\right\} U^{-1}$$ The square brackets on the left hand side indicates that the expression is to be understood as the matrix $\Lambda$ defined by the elements enclosed.
The construction presented here may be interpreted as a parallel to Atiyah’s suggestion of a spinor being a square root of geometry [@Atiyah2013:ma].
A few philosophic remarks
-------------------------
It would become apparent as we open deeper into our discourse that the $\mathcal{R}\oplus\mathcal{L}$ representation space, in contrast to the $\mathcal{R}\otimes\mathcal{L}$ representation space just explored, supports all the fermionic matter fields of the standard model of the high energy physics. So that which walks the matter fields and that in which it walks, that is spacetime, are determined by each other. The unifying theme being the underlying Lorentz algebra. From a philosophic point of view the theory that deals with the measurement of spatial and temporal distances through the proverbial rods and clocks in fact explores the universality of symmetries that determine the very substance those very rods and clocks are made of. I believe that this point of view is very similar to the one adopted by Harvey Brown in his monograph [@Brown:2005hr].
It also assures us that the Planck-scale expected departures from the Lorentz algebra are likely to alter foundational principles that spring from the low-energy phenomena that we human are able to observe and study so far. Some of these principles and notions are those of Heisenberg algebra, wave particle duality, causality, particle-antiparticle symmetry, and the Lagrangians that govern matter and gauge fields in a Lorentz covariant framework. Any such departure may also be able to incorporate such notions as consciousness, and phenomena that are presently considered outside the realm of physics proper.
We make this observation because one can easily argue that non-commutativity of spacetime measurements already follows from the merger of quantum mechanics and the theory of general relativity [@Ahluwalia:1993dd; @Doplicher:1994zv; @Kempf:1994su; @Ahluwalia:2000iw]. In reference to the beautiful analysis of [@Chryssomalakos:2004gk] we note that in their analysis relativistic symmetries and Heisenberg algebra are considered as two separate entities to be merged together whereas in the approach taken in this monograph Heisenberg algebra is seen as implicitly contained in the Lorentz algebra.
However, incorporating the still deeper issues into the physics proper may require a unification of physics-related phenomena and consciousness-endowed biological systems [@Ahluwalia:2017owu].
Discrete symmetries: Part 1 (Parity) {#ch5}
====================================
Adapting the approach of [@Speranca:2013hqa], this chapter presents a first-principle discussion of the parity operator. We will find that the 1928-Dirac equation expresses the fact that the associated spinors are the eigenspinors of the parity operator, and that $m^{-1}\gamma_\mu p^\mu$ is the covariant parity operator. We will show that for the Dirac spinors the right- and left- transforming components necessarily carry the same helicity, and bring to attention a phase factor that determines whether a spinor is a particle spinor, or an antiparticle spinor. Our results coincide with that of [@Weinberg:1995mt], while pointing out a flaw in the standard treatment of the subject in books such as [@Ryder:1985wq; @Hladik:1999tt].
Discrete symmetries
-------------------
Our description of the nature of reality divides one unified whole into parts. These are connected with each other by being one representation or the other of the Lorentz algebra.
An example of this broadbrush observation is that each of the transformation for rotations and boosts (\[eq:R-type-rb\]) and (\[eq:L-type-rb\]) contains in it a reference to two type of spaces: one from the familiar four-vector space of Minkowski through ${\boldmath\boldsymbol{\theta}}$ and ${\boldmath\boldsymbol{\varphi}}$ (see section \[sec:constructing-xt\]), and second to the spaces on which the generators of rotations and boosts act (this time, Minkowski being just one of them). The latter are bifurcated in two types, the $\mathcal{R}$ and $\mathcal{L}$ type. These can transmute into each other if there exists a discrete transformation such that $$\Big\{ \exp[i\bmfj\cdot{\boldmath\boldsymbol{\vartheta}}] , \exp[ \pm \bmfj\cdot{\boldmath\boldsymbol{\varphi}}] \Big\} \leftrightarrows
\Big\{ \exp[i\bmfj\cdot{\boldmath\boldsymbol{\vartheta}}] , \exp[ \mp \bmfj\cdot{\boldmath\boldsymbol{\varphi}}] \Big\} \label{eq:rl}$$ Or, equivalently $$\mathcal{R} \leftrightarrows \mathcal{L}$$ There are two known ways in which this can be done. Each of these introduce a discrete symmetry. These merge seamlessly with the continuous kinematical symmetries and provide additional needed structure and understanding of reality. These are the symmetries of parity and charge conjugation:
- In the Minkowski four-vector space, parity is defined as the map: $$P: x^\mu = (t,\x) \to x^{\prime\mu}=(t,-\x). \label{eq:Definition-of-Parity-in-Four-Vectors-Space}$$ Under it the $\bmfj$ and ${\boldmath\boldsymbol{\vartheta}}$ remain unaltered but the rapidity parameter $\vp$ changes sign. As a consequence parity interchanges the two representation spaces and implements (\[eq:rl\]). In this chapter I start a discussion of the image of (\[eq:Definition-of-Parity-in-Four-Vectors-Space\]) in the $\mathcal{R}\oplus\mathcal{L}$ representation space for $s=1/2$ and develop the associated mathematical structure that gives us two identities, leading to the 1928 Dirac equation in the momentum space.[^20]
- A distinct discrete symmetry arises if the transmutation (\[eq:rl\]) is implemented by complex conjugating a spin $s$ representation space and acting it with the Wigner time reversal operator $\Theta$ defined as \^[-1]{} = - \^ It introduces a new discrete symmetry, that of charge conjugation. Again, taking $s=1/2$ as an example, I develop this discrete symmetry in detail in chapter \[ch6\].
Weyl spinors\[sec:notation\]
----------------------------
To bring a sharper focus to spin one half we first gather together what we have learned so far in a compact form. Under the Lorentz boosts the right- and left-handed Weyl spinors transform as[^21] $$\phi_{\mathcal{R}}(p^\mu) = \exp\left(+\frac{\s}{2}\cdot\vp\right)
\phi_\mathcal{R}(k^\mu),\quad
\phi_\mathcal{L}(p^\mu) = \exp\left(-\frac{\s}{2}\cdot\vp\right) \phi_\mathcal{L}(k^\mu)\label{eq:w}$$ where $\s$ represents the set of Pauli matrices $\{\sigma_x,\sigma_y,\sigma_z\}$ in their standard representation given in (\[eq:Pauli-Matrices\]); and the boost parameter $\vp$ is defined so that $\exp\left(i \K\cdot\vp\right)$ acting on the standard four momentum equals the general four momentum $
p^\mu = (E, p \sin\theta\cos\phi,p\sin\theta\sin\phi,p\cos\theta).
$ This yields $\cosh\varphi = E/m$, $\sinh\varphi= p/m$ with $\widehat{\vp} = \widehat {\p}$ \[in agreement with (\[eq:rapidity-parameter\])\], while $\K$ are the $4\times 4$ matrices for the generators of boosts in Minkowski space as per their definition in equations (\[eq:kx\]) and (\[eq:kykz\]).
The reader may recall that (\[eq:w\]) follow from the fact that $- i \s/2$ are the generators of the boosts for the right-handed Weyl representation space, while $+ i \s/2$ are for the left-handed Weyl representation space. For the direct sum of the right- and left-Weyl representation spaces, to be motivated below, the boost and rotation generators thus read $$\kb = \left[
\begin{array}{cc}
- i \s/2 & \0\\
\0 & + i \s/2
\end{array}
\right], \quad
\bz = \left[
\begin{array}{cc}
\s/2 & \0\\
\0 & \s/2
\end{array}
\right]. \label{eq:pi}$$
The set of generators $\{\K,\J\}$ and $\{\kb,\bz\}$, separately, satisfy the same unifying algebra, the Lorentz algebra given in equations (\[eq:LorentzAlgebra\]) and are simply its different representations.
The $\left[{\Lambda^\mu}_\nu\right] \stackrel{\textrm{def}}{=}\Lambda$ representing boosts and rotations in Minkowski space is thus given by $$\Lambda \stackrel{\textrm{def}}{=}\left\{
\begin{array}{cl}
\exp\left(i\K\cdot\vp\right) & \mbox{for Lorentz boosts} \\
\exp\left(i\J\cdot\vt\right) & \mbox{for rotations}
\end{array}.\right.\label{eq:Minkowski-boost-rotation}$$ If $\{\K,\J\}$ are in their canonical form the elements of $\Lambda$, $\K$, and $\J$ are of the form ${{a}^\mu}_\nu$, where the $4\times 4$ matrix $a$ stands generically for either one of them. Their indices may then be raised and lowered by the spacetime metric $\eta_{\mu\nu} =\textrm{diag}\{1,-1,-1,-1\}$.
Parity operator for the general four-component spinors \[sec:parity\]
---------------------------------------------------------------------
To construct the image of parity operator for $s=1/2$ the discussion above suggests to define a four-component spinor in the $\mathcal{R}\oplus\mathcal{L}$ representation space $$\psi(p^\mu) = \left( \begin{array}{c}
\phi_\mathcal{R}(p^\mu)\\
\phi_\mathcal{L}(p^\mu)
\end{array}
\right) \label{eq:gen-4-comp-psi}$$ with (p\^) = (i ) (k\^). \[eq:psi\] where $p^\mu = {\left[ \exp\left(i\K\cdot\vp\right)\right]^\mu}_\nu k^\nu$.
The $\psi(k^\mu)$ are generally called “rest spinors,” while the $\psi(p^\mu)$ are often named “boosted spinors.” However, since no frame is a preferred frame, the $\psi(k^\mu)$ and the infinitely many $\psi(p^\mu)$ reside in every frame.
These $\psi(p^\mu)$ spinor may be eigenspinors of the parity operator, or that of the charge conjugation operator, or any other operator in the $\left[\mathcal{R}\oplus\mathcal{L}\right]_{s=1/2}$ representation space. A physically important classification of spinors is by Lounesto [@Lounesto:2001zz Chapter 12]. For ready reference we remind that the rotation on $\psi(p^\mu)$ is implemented by $$\psi\left(p^{\prime\mu}\right) = \exp\left(i\bz\cdot\bm{\vartheta}\right) \psi\left(p^{\mu}\right)\label{eq:rotation-on-spinors}$$ where $p^{\prime\mu} = {\left[\exp\left(i\J\cdot\bm{\vartheta}\right)\right]^\mu}_\nu p^\nu$.
Thus for massive particles incorporating parity covariance doubles the degrees of freedom for a spin one half representation space from $2$ to $4$. This doubling brings in a new degree of freedom, that of antiparticles. The argument has a natural extension for all spins.
*A parenthetic remark*
Historically, antiparticles entered the theoretical physics scene through this doubling in the degrees of freedom. Later, it was realised that in a merger of quantum mechanical formalism and the theory of special relativity, causality could only be preserved if one introduced antiparticles, for all spins. For massive particles of spin one half, it is done by doubling of the degrees of freedom inherent in the definition of $\psi(p^\mu)$, while for the massless particles the doubling happens through incorporating both helicities degrees of freedom one from the massless $\mathcal{R}$ and the opposite from the massless $\mathcal{L}$ representation space. In a sense, the parity eigenvalues act as a charge under the charge conjugation operator, while helicity takes over the role of charge for massless particles and it gets interchanged by the charge conjugation operator.
Parity need not be invoked to incorporate antiparticles in the massless case, helicity picks up that role in the $m\ne 0$ to the $m=0$ transition. In fact the $\mathcal{R}$ and $\mathcal{L}$ representation spaces get decoupled in the indicated transition. It is natural to conjecture if helicity of massless particles, irrespective of spin, from neutrinos to photons to gravitons acts as a new type of charge. If so, does it couple to torsion, or a fundamentally new field? In a cosmological setting, it may suggest tiny difference in the cosmic background temperatures associated with the different helicities and it could also lead to matter-antimatter asymmetry.
An examination of the question, “How does $P$ affect $\psi(p^\mu)$?” yields the Dirac operator [@Speranca:2013hqa]. The argument is as follows.
In accordance with the opening discussion of this chapter, the parity operator interchanges the right- and left- handed Weyl representation spaces. The effect of $P$ on $\psi(p^\mu)$ is thus realised by a $4\times 4$ matrix $\mathcal{P}$:
- which up to a global phase must contain purely off-diagonal $2\times 2$ identity matrices $\openone$, and
- which in addition implements the action of $P$ on $p^\mu$.
Up to a global phase, chosen to be $1$ (unless needed otherwise), the effect of $\mathcal{P}$ on $\psi(p^\mu)$ is therefore given by (p\^) = \_[ [\_0]{}]{}(p\^) = \_0 (p\^). \[eq:psi2\] Here, $p^{\prime\mu}$ is the $P$ transformed $p^\mu$ while $\0$ and $\I$ represent $2\times 2$ null and identity matrices, respectively. This is where the general textbook considerations on $\mathcal{P}$ stop. To be precise, the usual treatments arrive at $\mathcal{P}$ only after they introduce Dirac equation, and not at the primitive level we are pursuing. We will shortly see that the 1928 Dirac equation lies a short way ahead on the track.
For a general spinor, Sperança has noted that it provides a better understanding of $\mathcal{P}$ if in (\[eq:psi2\]) we note that $ \psi(p^{\prime\mu}) $ may be related to $ \psi(p^\mu)$ as follows [@Speranca:2013hqa] $$\psi\left(p^{\prime\mu}\right) = \exp\big[i \kb\cdot(-\vp)\big]\psi(k^\mu)$$ with $\kb$ defined in (\[eq:pi\]). But since from (\[eq:psi\]), $\psi(k^\mu) = \exp\left(-i\kb\cdot\vp\right)\psi(p^\mu)$ the above equation can be re-written as $$\psi\left(p^{\prime\mu}\right)\, =
\exp\left(-i \kb\cdot\vp\right)\exp\left(-i \kb\cdot\vp\right) \psi(p^\mu)
= \exp(- 2 i \kb\cdot\vp) \psi(p^\mu). \label{eq:zimpok167}$$ Substituting $\psi\left(p^{\prime\mu}\right)$ from the above equation in the $\mathcal{P}$-defining equation (\[eq:psi2\]), and on using the anti-commutativity of $\gamma_0$ with each of the generators of the boost ${\kb}$, $$\{\gamma_0,\kb_i\} = 0,\qquad \mbox{with}\,\, i=x,y,z$$ equation (\[eq:psi2\]) becomes $$\mathcal{P} \psi(p^\mu) = \exp(2 i \kb\cdot\vp) \gamma_0 \psi(p^\mu).\label{eq:psi3}$$ Using the definition of $\kb$ given in equation (\[eq:pi\]) and recalling from (\[eq:rapidity-parameter\]) that $\vp = \varphi\,\widehat \p$, the $\exp(2 i \kb\cdot\vp) \gamma_0$ factor in the above equation can be re-written as $$\exp(2 i \kb\cdot\vp) \gamma_0 =
\exp\left[\left(
\begin{array}{cc}
{{\boldmath\boldsymbol{\sigma}}}\cdot\widehat{\p} &\0 \\
\0 & - {{\boldmath\boldsymbol{\sigma}}\cdot\widehat{\p}}
\end{array}\right) \varphi
\right]\gamma_0 \label{eq:expansion}$$ Taking note of the identity (
[cc]{} &\
& -
)\^2 = (
[cc]{} &\
&
) we immediately obtain $$\begin{aligned}
\exp & \left[\left(
\begin{array}{cc}
{{\boldmath\boldsymbol{\sigma}}}\cdot\widehat{\p} &\0 \\
\0 & - {{\boldmath\boldsymbol{\sigma}}\cdot\widehat{\p}}
\end{array}\right) \varphi
\right] \nonumber\\ & \hspace{29pt} =
\left(\begin{array}{cc}
\I \cosh\varphi + {\boldmath\boldsymbol{\sigma}}\cdot\widehat\p \sinh\varphi & \0\\
\0 & \I \cosh\varphi - {\boldmath\boldsymbol{\sigma}}\cdot\widehat\p\sinh\varphi
\end{array}\right) \end{aligned}$$ With $\gamma_0$ defined in (\[eq:psi2\]), this result transforms (\[eq:expansion\]) to $$\exp(2 i \kb\cdot\vp) \gamma_0 = \left(\begin{array}{cc}
\0 & \I \cosh\varphi + {\boldmath\boldsymbol{\sigma}}\cdot\widehat\p \sinh\varphi\\
\I \cosh\varphi - {\boldmath\boldsymbol{\sigma}}\cdot\widehat\p\sinh\varphi &\0
\end{array}\right)$$ Making the substitutions $$\begin{aligned}
\cosh\varphi \to \frac{p^0}{m}\nonumber\\
\sinh\varphi \to \frac{p}{m}\end{aligned}$$ and using the identifications $$\gamma_0 \stackrel{\textrm{def}}{=} \left(\begin{array}{cc}
\0 & \openone \\
\openone & \0\end{array}\right),\quad
\bm{\gamma} \stackrel{\textrm{def}}{=} \left(\begin{array}{cc}
\0 &\s \\
-\s & \0
\end{array}\right).\label{eq:diracgamma-lower}$$ where $\gamma_\mu=(\gamma_0,\bm{\gamma})$, $\bm{\gamma} \stackrel{\textrm{def}}{=}\gamma_1,\gamma_2,\gamma_3$, are the canonical Dirac matrices in the Weyl representation, yields $$\exp\left( 2 i \kb\cdot\vp\right) \gamma_0 = m^{-1} \gamma_\mu p^\mu \label{eq:zimpok177}$$ Thus (\[eq:psi3\]) now takes the form $$\mathcal{P} \, \psi(p^\mu) = m^{-1}\gamma_\mu p^\mu \, \psi(p^\mu).\label{eq:psi4}$$
Up to a global phase this exercise yields the parity operator for the $\left[\mathcal{R}\oplus\mathcal{L}\right]_{s=1/2}$ representation space to be $$\mathcal{P} = m^{-1} \gamma_\mu p^\mu. \label{eq:P}$$ In this manner the ‘recipe’ contained in the discussion surrounding equation (\[eq:psi2\]) transforms into a clear-cut mathematical operator and makes the physical content of the 1928 Dirac equation transparent. The $\mathcal{P}$ applies to all the $\left[\mathcal{R}\oplus\mathcal{L}\right]_{s=1/2}$ 4-component spinors of the type $\psi(p^\mu)$. Only its eigenspinors, despite wide spread misconceptions to the contrary, are Dirac spinors for, they alone satisfy the Dirac equation.
The eigenvalues of $\mathcal{P}$ are $\pm 1$. Each of these has a two fold degeneracy $$\mathcal{P}\, \psi^S_\sigma(p^\mu) = + \psi^S_\sigma(p^\mu),\quad
\mathcal{P}\, \psi^A_\sigma(p^\mu) = - \psi^A_\sigma(p^\mu). \label{eq:Dirac-p-new}$$ The subscript $\sigma$ is the degeneracy index, while the superscripts refer to self and anti-self conjugacy of $\psi(p^\mu)$ under $\mathcal{P}$. With the help of Eq. (\[eq:P\]), Eq. (\[eq:Dirac-p-new\]) translates to $$\left(\gamma_\mu p^\mu -m \openone\right) \psi^S_\sigma(p^\mu) = 0,\quad
\left(\gamma_\mu p^\mu +m\openone\right) \psi^A_\sigma(p^\mu) = 0.\label{eq:Dirac-pd}$$
*Notational remark*
The Dirac’s [@Dirac:1928hu] $u_\sigma(p^\mu)$ and $v_\sigma(p^\mu)$ spinors are thus seen as the eigenspinors of the parity operator, $\mathcal{P}$, with eigenvalues $+1$ and $-1$, respectively: $$\psi^S_\sigma(p^\mu) \to u_\sigma(p^\mu),\quad \psi^A_\sigma(p^\mu)\to v_\sigma(p^\mu).
\label{eq:dirac-sa-uv}$$
At this stage equations (\[eq:Dirac-pd\]) should not be given a higher status than that of identities. This is not to undermine their eventual importance but that importance, history aside, resides in a quantum field theoretic context with $\psi^S_\sigma(p^\mu)$ and $\psi^A_\sigma(p^\mu)$ serving as expansion coefficients of the Dirac field [@Weinberg:1995mt] and *not* as wave functions in momentum space.
We end this part of the discussion on the spinorial parity operator by noting that $
\mathcal{P}^2 = \I _4$, and that the form of equation (\[eq:Dirac-pd\]) is preserved under a global transformation $$\psi_\sigma(p^\mu) \to \psi_\sigma^\prime(p^\mu)= \exp( i \alpha) \psi_\sigma(p^\mu),\quad
\forall \sigma
\label{eq:remarks}$$ with $\alpha\in\mathfrak{R}$.
The parity constraint on spinors, locality phases, and constructing the Dirac spinors
-------------------------------------------------------------------------------------
To obtain the explicit form of the Dirac spinors we use the following strategy
(a) first we obtain the constraint put on a general four-component spinor for $p^\mu=k^\mu$, and construct $\psi(k^\mu)$
(b) then act the boost operator (i ) = \[eq:sb-again\] on $\psi(k^\mu)$.
To accomplish this calculation we re-write equations (\[eq:Dirac-p-new\]) into the form
$$\begin{aligned}
m^{-1}\left[
\left(
\begin{array}{cc}
\0 &\I\\
\I & \0
\end{array}
\right) p_0 + \left(
\begin{array}{cc}
\0 &\s\cdot\widehat\p\\
- \s\cdot\widehat\p & \0
\end{array}
\right) p
\right]& \left( \begin{array}{c}
\phi_\mathcal{R}(p^\mu)\\
\phi_\mathcal{L}(p^\mu)
\end{array}
\right) \nonumber\\
&= \pm \left( \begin{array}{c}
\phi_\mathcal{R}(p^\mu)\\
\phi_\mathcal{L}(p^\mu)
\end{array}
\right) \end{aligned}$$
For $p^\mu = k^\mu$ this reduces to the constraint (
[c]{} \_(k\^)\
\_(k\^)
) = (
[c]{} \_(k\^)\
\_(k\^)
) Thus, keeping the notation introduced in (\[eq:dirac-sa-uv\]) in mind, we have $$\phi_\mathcal{R}(k^\mu) = \begin{cases} + \phi_\mathcal{L}(k^\mu), & \mbox{for}~u_\sigma(k^\mu)~ \mbox{spinors}\\
- \phi_\mathcal{L}(k^\mu), & \mbox{for}~v_\sigma(k^\mu)~\mbox{spinors}\end{cases}\label{eq:constraint12}$$ That is, for a four-component spinor $\psi(p^\mu)$ to be an eigenspinor of the $\mathcal{P}$ the relative phase between the left- and right- transforming Weyl spinors at rest must be the same for the self-conjugate spinors and opposite for the anti-self conjugate spinors.
This result stands in conflict with the well-known textbook treatments of the subject [@Ryder:1985wq; @Hladik:1999tt] and must be corrected [@Gaioli:1998ra; @Ahluwalia:1998dv]. Christoph Burgard arrived at this result simply by examining the form of the rest spinors as found in most textbooks [@Burgard:1989tamu]. Specifically, Ryder and Hladik assume that $\phi_\mathcal{R}(k^\mu) = \phi_\mathcal{L}(k^\mu)$ and claim that this in conjunction with the results (\[eq:w\]) yields Dirac equation: $\left(i \gamma_\mu\partial^\mu - m\right) \psi(x) =0$. Their construct yields $\left(\gamma_\mu p^\mu -m \right)\psi(p^\mu) =0$, but misses $\left(\gamma_\mu p^\mu +m \right)\psi(p^\mu) =0$. With the latter missing one must make two mistakes to arrive at the Dirac equation – this thread continues in reference [@Ahluwalia:2016jwz].
The constraint (\[eq:constraint12\]) is consistent with Weinberg’s analysis [@Weinberg:1995mt]. His analysis contains two additional elements:
- one, in each of the $\psi(k^\mu)$ the spin projections – or helicities – for $\phi_\mathcal{R}(k^\mu)$ and $\phi_\mathcal{L}(k^\mu)$ must be the same; and
- second, when these are used to construct the expansion coefficients of a quantum field the $\psi_\sigma(k^\mu)$ cannot have arbitrary global phases but must have specific values.[^22]
The first of the mentioned results, in conjunction with (\[eq:phasefactor\]), implies that under rotation by an angle $\vartheta$ the Dirac spinors pick up a global phase $e^{\pm i\vartheta/2}$ depending on the helicity of the Weyl components involved. It ought to be emphasised that this result is specific to Dirac spinors. It does not hold for general four-component spinors. An example of this assertion is found in Chapter \[ch10\].
To incorporate the phase factors constraints coded in (\[eq:constraint12\]) into the eigenspinors of the $\mathcal{P}$ we define the Weyl spinors at rest $\phi(k^\mu)$ $$\s\cdot\widehat \p\, \phi_\pm(k^\mu) = \pm \phi_\pm(k^\mu)\label{eq:hd}$$ and take $$\begin{aligned}
\phi_+(k^\mu) & = \sqrt{m} \textrm{e}^{i\vartheta_1} \left(
\begin{array}{c}
\cos(\theta/2)\exp(- i \phi/2)\\
\sin(\theta/2)\exp(+i \phi/2)
\end{array}
\right)
\label{eq:zimpok52-new} \\
\phi_-(k^\mu) & = \sqrt{m} \textrm{e}^{i\vartheta_2} \left(
\begin{array}{c}
\sin(\theta/2)\exp(- i \phi/2)\\
- \cos(\theta/2)\exp(+i \phi/2)
\end{array}
\right) \label{eq:zimpok52-new-new}\end{aligned}$$ with $\vartheta_1, \vartheta_2 \in \Re$. In contrast to our 2005 work [@Ahluwalia:2004sz; @Ahluwalia:2004ab] where we set $\vartheta_1 = \vartheta_2 = 0$, we now take $\vartheta_1 = 0, \vartheta_2 = \pi$. These are part of the locality phase factors, the phase factors responsible for removing non-locality of the cited earlier works. With this choice of phase factors the $\phi(k^\mu)$ read $$\begin{aligned}
\phi_+(k^\mu) & = \sqrt{m} \left(
\begin{array}{c}
\cos(\theta/2)\exp(- i \phi/2)\\
\sin(\theta/2)\exp(+i \phi/2)
\end{array}
\right)
\label{eq:zimpok52-new-new2} \\
\phi_-(k^\mu) & = \sqrt{m} \left(
\begin{array}{c}
- \sin(\theta/2)\exp(- i \phi/2)\\
\cos(\theta/2)\exp(+i \phi/2)
\end{array}
\right).
\label{eq:zimpok52-new-new-new}\end{aligned}$$ In terms of these two-component rest spinors, we define the four-component rest spinors $$\begin{aligned}
& u_+(k^\mu) = \textrm{e}^{i \lambda_1}\left(
\begin{array}{c}
\phi_+(k^\mu) \\
\phi_+(k^\mu)
\end{array}
\right),\quad
u_-(k^\mu) = \textrm{e}^{i \lambda_2}\left(
\begin{array}{l}
\phi_-(k^\mu) \\
\phi_-(k^\mu)
\end{array}
\right)\\
& v_+(k^\mu) = \textrm{e}^{i \lambda_3}\left(
\begin{array}{c}
\phi_-(k^\mu) \\
- \phi_-(k^\mu)
\end{array}
\right),\quad
v_-(k^\mu)
= \textrm{e}^{i \lambda_4} \left(\begin{array}{c}
\phi_+(k^\mu) \\
-\phi_+(k^\mu)
\end{array}
\right)\end{aligned}$$ with $\lambda_1,\lambda_2,\lambda_3,\lambda_4\in \Re$. We set $\lambda_1=\lambda_2=\lambda_3 =0$ and $\lambda_4=\pi$, $$\begin{aligned}
& u_+(k^\mu) = \left(
\begin{array}{c}
\phi_+(k^\mu) \\
\phi_+(k^\mu)
\end{array}
\right),\quad
u_-(k^\mu) = \left(
\begin{array}{c}
\phi_-(k^\mu) \\
\phi_-(k^\mu)
\end{array}
\right)\label{eq:Diracuk}\\
& v_+(k^\mu) = \left(
\begin{array}{c}
\phi_-(k^\mu) \\
- \phi_-(k^\mu)
\end{array}
\right),\quad
v_-(k^\mu)
= \left(\begin{array}{c}
- \phi_+(k^\mu) \\
\phi_+(k^\mu)
\end{array}
\right).\label{eq:Diracvk}\end{aligned}$$ to be consistent with equations (5.5.35) and (5.5.36) of Weinberg’s analysis [@Weinberg:1995mt]. That analysis ensures correct incorporation of Lorentz symmetries, parity covariance, and locality. Apart from the chosen phases, it is $\phi_+(k^\mu) $ which enters the definition of $v_-(k^\mu) $, and $\phi_-(k^\mu) $ which enters the definition of $v_+(k^\mu) $. In contrast, it is $\phi_-(k^\mu) $ which enters the definition of $u_-(k^\mu) $, and $\phi_+(k^\mu) $ which enters the definition of $u_+(k^\mu) $. This is important for the correct pairing of the creation and annihilation operators with $u_\pm(k^\mu)$ and $v_\pm(k^\mu)$ when constructing the Dirac quantum field.
Following item (b) of the strategy that opened this section, the $u_\sigma(p^\mu)$ and $v_\sigma(p^\mu)$ for an arbitrary $p^\mu$ follow by acting the $\left[\mathcal{R}\oplus\mathcal{L}\right]_{s=1/2}$ boost given in equation (\[eq:sb-again\]) on the rest spinors just enumerated. This exercise yields $$\begin{aligned}
u_+(p^\mu) & = \alpha \left(
\begin{array}{l}
\alpha_+ \,\phi_+(k^\mu)\\
\alpha_-\,\phi_+(k^\mu)\\
\end{array}
\right) \label{eq:upp}\\
u_-(p^\mu) & = \alpha
\left(
\begin{array}{l}
\alpha_-\,\phi_-(k^\mu)\\
\alpha_+\,\phi_-(k^\mu)\
\end{array}
\right) \label{eq:uspinors}\end{aligned}$$ and $$\begin{aligned}
v_+(p^\mu) & = \alpha
\left(
\begin{array}{r}
\alpha_-\,\phi_-(k^\mu)\\
- \,\alpha_+\,\phi_-(k^\mu)\
\end{array}
\right) \label{eq:vp}\\
v_-(p^\mu) &= \alpha \left(
\begin{array}{r}
-\,\alpha_+ \,\phi_+(k^\mu)\\
\alpha_-\,\phi_+(k^\mu)\
\end{array}
\right) \label{eq:vm}\end{aligned}$$ where $$\begin{aligned}
\alpha = \sqrt{\frac{E+m}{2 m}},\quad \alpha_\pm = \left(1\pm\frac{p}{E+m}\right)\label{eq:apm}\end{aligned}$$
The task is done: On the one hand we have constructed the constraint that a $\mathcal{R}\oplus\mathcal{L}$ spinor must satisfy for it to be an eigenspinor of the parity operator $m^{-1}\gamma_\mu p^\mu$ and unearthed how Dirac spinors follow when working through this approach. On the other hand we have also introduced certain phase factors that affect the Lorentz and parity covariance, and locality, of the quantum field these spinors allow to be constructed.
Dirac spinors are known since 1928. What we have added to the subject is its simple underlying structure, an act that leads us to look at the charge conjugation operator and its eigenspinors in the next chapter. It is there that a reader to whom all this is fairly well known would encounter her folkloric wisdom challenged, unexpectedly.
Discrete symmetries: Part 2 (Charge conjugation) {#ch6}
================================================
We now embark on developing a second discrete symmetry implied by the transmutation (\[eq:rl\]). It is the symmetry of charge conjugation. The reader is likely to have encountered it before, in particular, in the context of the 1928-Dirac equation. Here, we will see it arise in a manner that unifies its origin to the transmutation (\[eq:rl\]) and the symmetry of parity discussed in the previous chapter. It spawns by complex conjugating $\mathcal{R}$, or $\mathcal{L}$, representation spaces for spin one half followed by the operation of the Wigner time reversal operator. The extension of the argument to all spins – or more precisely, to all representation spaces of the type $\left[\mathcal{R}\oplus\mathcal{L}\right]_{j=\textrm{any}}$ – follows in a parallel manner. We show that the the charge conjugation operator transmutes the parity eigenvalues. This in turn leads to the anti-commutativity of the parity and charge conjugation operators.
Magic of Wigner time reversal operator
--------------------------------------
For spin $s$, given a $\bmfj$ the Wigner time reversal operator is defined as: $$\Theta_{[s]} \bmfj \Theta_{[s]}^{-1} \stackrel{\textrm{def}}{=} - \bmfj^\ast$$ with $\Theta_{[s]} = (-1)^{s+\sigma}\delta_{\sigma^\prime,-\sigma}$ and $\Theta_{[s]}^\ast \Theta_{[s]} = (-1)^{2 s}$. We abbreviate $\Theta_{\left[1/2\right]}$ to $\Theta$. In $\Theta$ we mark the rows and columns in the order $\{-1/2,1/2\}$.
For spin one half $
\bmfj = \s/2$. As such, the Wigner time reversal operator acts on the Pauli matrices as follows $$\Theta \s\Theta^{-1} = - \s^\ast\label{eq:wt}$$ with $$\Theta= \left(\begin{array}{cc} 0 & -1 \\ 1 & 0\end{array}\right), \quad
\Theta^{-1} = - \Theta.$$
It allows the following ‘magic’ to happen:
> - If $\phi_R(p^\mu)$ transforms as a right-handed Weyl spinor then $\zeta_\rho \Theta \phi^\ast_R(p^\mu)$ – with $\zeta_\rho$ an arbitrary phase factor – transforms as a left-handed Weyl spinor.
>
> - If $\phi_L(p^\mu)$ transforms as a left-handed Weyl spinor then $\zeta_\lambda \Theta \phi^\ast_L(p^\mu)$ – with $\zeta_\lambda$ an arbitrary phase factor – transforms as a right-handed Weyl spinor.
>
The way this comes about is as follows. First complex conjugate both the equations in (\[eq:w\]), then multiply from the left by $\Theta$, and use the above defining feature of the Wigner time reversal operator. This sequence of manipulations after using the freedom to multiply these equations by phase factors $\zeta_\rho$ and $\zeta_\lambda$ respectively ends up with the result $$\Big[\zeta_\rho\Theta \phi^\ast_R(p^\mu)\Big] = \exp\left( - \frac{\s}{2}\cdot\vp\right)\Big[\zeta_\rho\Theta \phi^\ast_R(k^\mu)\Big]$$ and $$\Big[\zeta_\lambda\Theta \phi^\ast_L(p^\mu)\Big] = \exp\left( + \frac{\s}{2}\cdot\vp\right)\Big[\zeta_\lambda\Theta \phi^\ast_L(k^\mu)\Big]
\label{eq:boostL}$$ yielding the claimed magic of the Wigner time reversal operator. This crucial observation motivates the introduction of two sets of four-component spinors [@Ahluwalia:1994uy; @Ahluwalia:2004ab] $$\rho(p^\mu) = \left(\begin{array}{c}
\phi_R(p^\mu)\\
\zeta_\rho \Theta \phi^\ast_R(p^\mu)
\end{array}
\right) , \quad
\lambda(p^\mu) = \left(\begin{array}{c}
\zeta_\lambda \Theta \phi^\ast_L(p^\mu)\\
\phi_L(p^\mu)
\end{array}
\right).
\label{eq:lambda}$$ The $\rho(p^\mu)$ do not provide an additional independent set of spinors. For that reason we do not consider them further and confine our attention to $\lambda(p^\mu)$ only [@Ahluwalia:1994uy].
Confining to spin one half, Ramond [@Ramond:1981pw] introduces this result as ‘magic of Pauli matrices’ where $ \Theta$ gets concealed in Pauli’s $\sigma_y$, which equals $i \Theta$. More importantly, his analysis misses the full multiplicity of the phase factor $\zeta_\lambda$. Our argument in terms of the Wigner time reversal operator $\Theta$ has the advantage that it immediately generalises to higher spin $\mathcal{R}\oplus\mathcal{L}$ representation spaces.
In $\lambda(p^\mu)$ we have four, rather than two, independent four-component spinors. The helicities of the right and left transforming components of $\lambda(p^\mu)$ are opposite (see Section \[Sec:sevenpointone\]) – in sharp contrast to the eigenspinors of the parity operator. This circumstance allows the $\lambda(p^\mu)$ to escape their interpretation as Weyl spinors in a four-component disguise. This doubling is more than of an academic interest. It is required, as we shall see later in this monograph, to incorporate antiparticles when the $\lambda(p^\mu)$ are used as expansion coefficients of a quantum field.
Charge conjugation operator for the general four-component spinors
------------------------------------------------------------------
With the just made observations at hand we are led to entertain the possibility that in addition to the symmetry operator $\mathcal{P}$ there exists a second symmetry operator from (\[eq:rl\]), which up to a global phase factor, has the form $$\mathcal{C} \stackrel{\textrm{def}}{=} \left(\begin{array}{cc}
\0 & \alpha \Theta \\
\beta\Theta & \0
\end{array}
\right) K \label{eq:cc}$$ where $K$ complex conjugates to its right. The arguments that lead to (\[eq:cc\]) are similar to the ones that give (\[eq:psi2\]). Requiring $\mathcal{C}^2$ to be an identity operator determines $\alpha = i, \beta= -i$ (where we have used $K^2=1$). It results in $$\mathcal{C} = \left(\begin{array}{cc}
\0 & i\Theta \\
-i \Theta & \0
\end{array}
\right) K = \gamma_2 K\label{eq:gamma2K}$$ with $$\gamma_2 =
\left(\begin{array}{cc}
\0 &\sigma_y \\
-\sigma_y & \0
\end{array}\right).$$ There also exists a second solution with $\alpha = - i, \beta= i$. But this does not result in a physically different operator and in any case the additional minus sign can be absorbed in the indicated global phase. This is the same operator that appears in the particle-antiparticle symmetry associated with the 1928 Dirac equation.
We have thus arrived at the charge conjugation operator from the analysis of the symmetries of the $4$-component representation space of spinors. This perspective has the advantage of immediate generalisation to any spin: if $\Theta_{[s]}$ is taken as Wigner time reversal operator for spin $s$ then the spin-$s$ charge conjugation operator in the $2(2s+1)$ dimensional representation space becomes [@Lee:2012td equation (A10)] $$\mathcal{C} = \left[\begin{array}{cc}
\0 & -i \Theta_{[s]}^{-1} \\
-i \Theta_{[s]} & \0
\end{array}
\right] K. \label{eq:c}$$For spin one half, $\Theta^{-1} = - \Theta$; consequently, the above expression coincides with the result for spin one half given in equation (\[eq:gamma2K\]).
Both $\mathcal{P}$ and $\mathcal{C}$ arise without reference to any wave equation, or equivalently without assuming a Lagrangian density. In fact, it will be apparent from our presentation that Lagrangian densities should be derived rather than assumed.
Transmutation of $\mathcal{P}$ eigenvalues by $\mathcal{C}$, and related results
---------------------------------------------------------------------------------
We immediately note that $\mathcal{C}$ defined above transmutes the eigenvalues of the $\mathcal{P}$ operator. As a consequence they anti-commute. To see this we apply $\mathcal{C}$ from the left on (\[eq:P\]) $$\begin{aligned}
\mathcal{C}{\mathcal{P}} & =
\gamma_2 K \big[ m^{-1} \gamma_\mu p^\mu\big] \nonumber \\
& = \gamma_2 K \big[m^{-1} \gamma_{(\mu\ne 2)} p^{(\mu\ne 2)} +
m^{-1} \gamma_2 p^2 \big] \nonumber\\
& =
\gamma_2 \big[m^{-1} \gamma_{(\mu\ne 2)} p^{(\mu\ne 2)} -
m^{-1} \gamma_2 p^2 \big] K \nonumber\\
& = \big[- m^{-1} \gamma_{(\mu\ne 2)} p^{(\mu\ne 2)} -
m^{-1} \gamma_2 p^2 \big] \gamma_2 K \nonumber\\
& = - \big[ m^{-1} \gamma_\mu p^\mu\big] \gamma_2 K \nonumber\\
& = -\mathcal{P}\mathcal{C}.\end{aligned}$$ Where we have successively used the facts that in Weyl representation $\gamma_2$ is imaginary (see (\[eq:Pauli-Matrices\]) and (\[eq:diracgamma-lower\])), and that $\gamma_2$ anti-commutes with $\gamma_{(\mu\ne 2)}$. This anti-commutativity of the parity and charge conjugation operators { , } = 0\[eq:CP-formal\] immediately yields the result that if $\psi(p^\mu)$ are eigenspinors of the parity operator (p\^) = (p\^) then the $\mathcal{C}$ transformed spinors $\mathcal{C} \psi(p^\mu)$ carry opposite parity eigenvalue = . Stated differently, (\[eq:Dirac-pd\]) changes as follows $$\begin{aligned}
\underbrace{\left(\gamma_\mu p^\mu -m \openone\right) \psi^S_\sigma(p^\mu) = 0,\quad
\left(\gamma_\mu p^\mu +m\openone\right) \psi^A_\sigma(p^\mu) = 0}_\downarrow~~~~~\nonumber\\
\overbrace{\left(\gamma_\mu p^\mu + m \openone\right)\left[\mathcal{C} \psi^S_\sigma(p^\mu)\right] = 0,\quad
\left(\gamma_\mu p^\mu - m\openone\right) \left[\mathcal{C}\psi^A_\sigma(p^\mu)\right] = 0}.\label{eq:Dirac-pd-C}\end{aligned}$$ In this sense, the relative eigenvalues of the $\mathcal{P}$ may be identified as a charge under the charge conjugation operator.
The result on the anti-commutativity of the $\mathcal{C}$ and $\mathcal{P}$ operators arrived at (\[eq:CP-formal\]) may be obtained in another manner and it checks the internal consistency of various choices of phase factors. For this analysis we note that (\[eq:dirac-sa-uv\]) together with (\[eq:Dirac-pd-C\]) imply v\_(p\^) u\_(p\^), u\_(p\^) v\_(p\^) where the proportionality constants may depend on $\sigma$. We wish to find the proportionality constant(s), so we act $\mathcal{C}$ on each of the $u_\sigma(p^\mu)$ and $v_\sigma(p^\mu)$. With expression for $\mathcal{C}$ given by (\[eq:gamma2K\]) and for $u_+(p^\mu) $ given by (\[eq:upp\]) we work through first of the calculations as follows u\_+(p\^) = i (
[c]{} (1-)\_+\^(k\^)\
- (1+) \_+\^(k\^)
) and since $ \Theta\phi^\ast_+ (k^\mu) = \phi_-(k^\mu) $ u\_+(p\^) = i (
[c]{} (1-)\_-(k\^)\
- (1+) \_-(k\^).
) Use of (\[eq:vp\]) then yields u\_+(p\^) = i v\_+(p\^). The use of another identity $ \Theta\phi^\ast_- (k^\mu) = - \, \phi_+(k^\mu) $ and calculations similar to as above then tells us that u\_(p\^) = i v\_(p\^), v\_(p\^) = i u\_(p\^).\[eq:cuv\] As a consequence, \^2 = \_4,{,} = 0\[eq:CPanticommutator-P\] for the eigenspinors of the parity operator. This is so because $$\begin{aligned}
& \mathcal{C}\mathcal{P} \, u_-(p^\mu) = \mathcal{C} u_-(p^\mu) = i v_-(p^\mu)\\
& \mathcal{P}\mathcal{C} \, u_-(p^\mu) = \mathcal{P} \left(i v_-(p^\mu)\right) = - i v_-(p^\mu).\end{aligned}$$ Together, they yield the claimed anti-commutativity of the $\mathcal{C}$ and $\mathcal{P}$ for the eigenspinors of the parity operator $\mathcal{P}$.
Introducing the time reversal operator $\mathcal{T} = i \gamma \mathcal{C}$, with $\gamma$ given by equation (\[eq:gamma5\]) below, an explicit calculation gives the result $$\begin{aligned}
\mathcal{T} u_+(p^\mu) &= - u_-(p^\mu),\quad \mathcal{T} u_-(p^\mu) =
u_+(p^\mu)\label{eq:tr12}\\
\mathcal{T} v_+(p^\mu) &= v_-(p^\mu),\quad \mathcal{T} v_-(p^\mu) =
- v_+(p^\mu).\label{eq:tr34}\end{aligned}$$ The introduced time reversal operator commutes with the charge conjugation operator and also with the parity operator $$\left[\mathcal{C},\mathcal{T}\right]=0,\quad \left[\mathcal{P},\mathcal{T}\right]=0$$ In conjunction with the facts that $$\mathcal{P}^2 = \I_4,\quad \mathcal{T}^2=-\I_4$$ all the results obtained above combine to yield $$\left(\mathcal{C}\mathcal{P}\mathcal{T}\right)^2 = \I_4$$ for the Dirac spinors.
Eigenspinors of charge conjugation operator, Elko {#ch7}
=================================================
In the usual language, the 1937 Majorana field starts with the Dirac field and identifies the $b^\dagger_\sigma(\p)$ with the $a^\dagger_\sigma(\p)$. In the process Majorana constructed a fundamentally neutral field. Later Majorana spinors were introduced as eigenspinors of the charge conjugation operator, see for example Ramond’s primer [@Ramond:1981pw]. But as soon these were motivated by the ‘magic of Pauli matrices,’ they were elevated to Grassmann variables. Furthermore, they were simply considered as Weyl spinors in a four component form – giving rise to the ‘disguise’ argument. Nothing similar is required of the Dirac spinors – at least, in the operator formalism of quantum field theory, which in our opinion is an unambiguous conceptual continuation of the quantum formalism [@Dirac:1930pam]. Finding the grassmann-isation of doubtful validity, at least for the reasons given in [@Ramond:1981pw] and elsewhere [@Aitchison:2004cs], we here construct eigenspinors of the charge conjugation operator. We not only avoid the ‘disguise’ argument, but we also refrain from grassmann-isation. The result is a set of four four-component spinors which stand at par with the Dirac spinors. Taken to their logical consequence the new spinors lead to mass dimension one fermions. To avoid a possible confusion from arising we call the new spinors as Elko (igenspinoren des adungsonjugationsperators) – or, simply as eigenspinors of the charge conjugation operator.
Elko {#Sec:sevenpointone}
----
We have at our disposal the charge conjugation operator from equation (\[eq:gamma2K\]) and the new spinors suggested by the magic of the Wigner time reversal operator from equation (\[eq:lambda\]): $$\begin{aligned}
& \mathcal{C} = \left(\begin{array}{cc}
\0 & i \Theta \\
-i \Theta & \0
\end{array}
\right) K, \label{eq:ccRepeat}\\
& \lambda(p^\mu) = \left(\begin{array}{c}
\zeta_\lambda \Theta \phi^\ast_L(p^\mu)\\
\phi_L(p^\mu)
\end{array}
\right).
\label{eq:lambdaRepeat}
\end{aligned}$$ We now establish the following two results:
- The $\lambda(p^\mu)$ become eigenspinors of the charge conjugation operator with doubly degenerate eigenvalues, $\pm 1$, if the phase $\zeta_\lambda$ that appears in $\lambda(p^\mu)$ is set to $\pm i$.
- In contrast to the eigenspinors of the parity operator, the right- and left- transforming components of the new spinors have opposite helicities.
To establish the first of the two enumerated results we act the charge conjugation operator on the new spinors. After the action of $K$ in $\mathcal{C}$, the result reads $$\mathcal{C} \lambda(p^\mu) =
\left(\begin{array}{cc}
\0 & i\Theta \\
-i \Theta & \0
\end{array}
\right)
\left(\begin{array}{c}
\zeta^\ast_\lambda \Theta \phi_L(p^\mu)\\
\phi^\ast_L(p^\mu)
\end{array}
\right)$$ Exploiting the property $\Theta^2 = -\I$ simplifies the above expression into $$\mathcal{C} \lambda(p^\mu) =
\left(\begin{array}{cc}
i \Theta \phi^\ast_L(p^\mu)\\
i \zeta^\ast_\lambda \phi_L(p^\mu)
\end{array} \right).
\label{eq:Cev}$$ The choice $\zeta_\lambda = \pm i$ makes $\lambda(p^\mu)$ become eigenspinors of $\mathcal{C}$ with doubly degenerate eigenvalues $\pm 1$: $$\mathcal{C} \lambda^S(p^\mu) = + \lambda^S(p^\mu),\quad
\mathcal{C} \lambda^A(p^\mu) = - \lambda^A(p^\mu), \label{eq:elko-cc}$$ where $$\lambda(p^\mu) = \left\{
\begin{array}{ll}
\lambda^S(p^\mu) =
\left(\begin{array}{c} i \Theta \phi_L^\ast(p^\mu)\\
\phi_L(p^\mu)
\end{array} \right)
& \mbox{for} ~ \zeta_\lambda = + i \\ \\
\lambda^A(p^\mu) =
\left(\begin{array}{c}- i \Theta \phi_L^\ast(p^\mu)\\
\phi_L(p^\mu)
\end{array} \right) & \mbox{for} ~ \zeta_\lambda = - i
\end{array}\right.
\label{eq:lsa}$$ To establish the second of the two enumerated results we proceed as follows. With the structure $\zeta_\lambda \Theta \phi^\ast_L(p^\mu)$ in mind, we complex conjugate (\[eq:hd\]) to get (suppressing the subscript $\mathcal{L}$) $$\s^\ast \cdot\widehat{\p}\, \phi^\ast_\pm(k^\mu) = \pm \phi^\ast_\pm(k^\mu)\label{eq:hdd}$$ and then replace $\s^\ast$ by $- \Theta\s\Theta^{-1}$ in accordance with (\[eq:wt\]) \^[-1]{} \^\_(k\^) = \^\_(k\^) Next we use the fact that $\Theta^{-1} \leftrightharpoons -\Theta$ to replace $\Theta\s\Theta^{-1}$ by $ \Theta^{-1}\s\Theta$ \^[-1]{} \^\_(k\^) = \^\_(k\^) A left multiplication by $\Theta$ furnishes us the result = \[eq:zimpok3\] The result (\[eq:boostL\]) immediately translates the validity of the above expression for all $p^\mu$ = \[eq:zimpok3new\] Thus, the right- and left- transforming components of $\lambda(k^\mu)$ are constrained to have opposite helicities – in sharp contrast to the constraints (\[eq:constraint12\]) for the eigenspinors of the parity operator $\mathcal{P}$. We will see in Chapter \[ch10\] that this fact endows Elko with an unusual property under rotation. Exploiting (\[eq:phasefactor\]) it results in an unexpected cosmological effect.
Restriction on local gauge symmetries {#sec:Restriction}
-------------------------------------
Because of the presence of the operator $K$ in Eq. (\[eq:gamma2K\]) a global transformation of the type $$\lambda(p^\mu) \to \lambda^\prime(p^\mu)= \exp( i \mathfrak{a} \alpha)
\lambda(p^\mu) \label{eq:counterpart}$$ with $\mathfrak{a}^\dagger = \mathfrak{a}$, a $4\times 4$ matrix and $\alpha\in\mathfrak{R}$, does not preserve the self/anti-self conjugacy of $\lambda(p^\mu)$ under $\mathcal{C}$ unless the matrix $\mathfrak{a}$ satisfies the condition $$\gamma_2 \mathfrak{a}^\ast +\mathfrak{a} \gamma_2 = 0$$ The general form of $\mathfrak{a}$ satisfying this requirements is $$\begin{aligned}
\mathfrak{a} & =
\left[
\begin{array}{cccc}
\epsilon & \beta & \lambda & 0 \\
\beta^\ast & \delta & 0 & \lambda \\
\lambda^\ast & 0 & -\delta & \beta \\
0 & \lambda^\ast & \beta^\ast & -\epsilon \\
\end{array}
\right]
\label{eq:mathfraka}\end{aligned}$$ with $ \epsilon,\delta \in \mathfrak{R}$ and $\beta,\lambda \in\mathfrak{C}$ (with no association with the same symbols used elsewhere in this work).
For a field constructed with the eigenspinors of $\mathcal{C}$ as expansion coefficients, the usual local $U(1)$ interaction is ruled out as the form of $\mathfrak{a}$ given by (\[eq:mathfraka\]) does not allow a solution with $\mathfrak{a}$ proportional to an identity matrix. As remarked around equation (\[eq:remarks\]), for the eigenspinors of the $\mathcal{P}$, defined by (\[eq:Dirac-p-new\]), the counterpart of (\[eq:counterpart\] ) is trivially satisfied. And thus the two fields, one based on $\mathcal{P}$ eigenspinors and the other constructed from the $\mathcal{C}$ eigenspinors, carry intrinsically different possibility for their interaction through local gauge fields. The simplest non-trivial choice consistent with (\[eq:mathfraka\]) is given by $$\mathfrak{a} = \gamma = \frac{i}{4!} \epsilon_{\mu\nu\lambda\sigma}
\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\sigma = \left(\begin{array}{cc}
\I &\0\\
\0 & -\I
\end{array}\right)\label{eq:gamma5}$$ where $\epsilon_{\mu\nu\lambda\sigma}$ is defined as $$\epsilon_{\mu\nu\lambda\sigma} = \left\{
\begin{array}{cl}
+1, & \mbox{for $\mu\nu\lambda\sigma$ even permutation of 0123}\\
- 1,& \mbox{for $\mu\nu\lambda\sigma$ odd permutation of 0123}\\\
0, & \mbox{if any two of the $\mu\nu\lambda\sigma$ are same}
\end{array}\right.$$
Construction of Elko {#ch8}
====================
With Elko defined in the previous chapter we now provide explicit construction of the new spinors and discuss the subtle departure from our 2005 publications [@Ahluwalia:2004sz; @Ahluwalia:2004ab]. These departure, when coupled with the new dual introduced in Chapter \[ch11\], lie at the heart of evaporating away the non-locality and a lack of full Lorentz covariance of our earlier works. The arguments that follow are adapted from [@Ahluwalia:2016rwl; @Ahluwalia:2016jwz].
Elko at rest
------------
To obtain an explicit form of Elko requires the ‘rest’ spinors $\lambda(k^\mu)$. That done, we then have for an arbitrary $p^\mu$ $$\lambda(p^\mu)
= \sqrt{\frac{E + m }{2 m}}
\left[
\begin{array}{cc}
\I + \frac{\boldsymbol{\sigma}\cdot\mathbf{p}}{E +m} & \0 \\
\0 & \I - \frac{\boldsymbol{\sigma}\cdot\mathbf{p}}{E +m}
\end{array}
\right] \lambda(k^\mu)
\label{eq:elkoboost-again}$$ with the boost operator above the same as in (\[eq:sb-again\]).
To construct $\lambda(k^\mu)$ we have to choose global phases for each of the two $\phi_\pm(k^\mu)$ as discussed after equations (\[eq:zimpok52-new\]) and (\[eq:zimpok52-new-new\]). With that choice made we are still left with the additional phase freedom $$\lambda^S_+(k^\mu) = e^{i\xi_1} \left[
\begin{array}{c}
i \Theta\left[\phi_+(k^\mu)\right]^\ast\\
\phi_+(k^\mu)
\end{array}
\right],\hspace{7pt}
\lambda^S_-(k^\mu) = e^{i\xi_2} \left[
\begin{array}{c}
i \Theta\left[\phi_-(k^\mu)\right]^\ast\\
\phi_-(k^\mu)
\end{array}
\right] \label{eq:zimpok0}$$ and $$\lambda^A_+(k^\mu) = e^{i\xi_3} \left[
\begin{array}{c}
- i \Theta\left[\phi_-(k^\mu)\right]^\ast\\
\phi_-(k^\mu)
\end{array}
\right],
\hspace{7pt}
\lambda^A_-(k^\mu) = e^{i\xi_4} \left[
\begin{array}{c}
- i \Theta\left[\phi_+(k^\mu)\right]^\ast\\
\phi_+(k^\mu)
\end{array}
\right]
\label{eq:zimpok91}$$ with $\xi_1,\xi_2,\xi_3,\xi_4\in \Re$. We set $\xi_1=\xi_2=\xi_3 =0$ and $\xi_4=\pi$ $$\begin{aligned}
\lambda^S_+(k^\mu) & = \left[
\begin{array}{c}
i \Theta\left[\phi_+(k^\mu)\right]^\ast\\
\phi_+(k^\mu)
\end{array}
\right],\hspace{7pt}
\lambda^S_-(k^\mu) = \left[
\begin{array}{c}
i \Theta\left[\phi_-(k^\mu)\right]^\ast\\
\phi_-(k^\mu)
\end{array}
\right] \label{eq:zimpok0new}\\
\lambda^A_+(k^\mu) &= \left[
\begin{array}{c}
- i \Theta\left[\phi_-(k^\mu)\right]^\ast\\
\phi_-(k^\mu)
\end{array}
\right],
\hspace{7pt}
\lambda^A_-(k^\mu) = \left[
\begin{array}{c}
i \Theta\left[\phi_+(k^\mu)\right]^\ast\\
- \phi_+(k^\mu)
\end{array}
\right]
\label{eq:zimpok91new}
\end{aligned}$$
This choice of phases, coupled with advances in understanding duals and adjoints [@Ahluwalia:2016rwl; @Ahluwalia:2016jwz], completely resolves the lingering problems with locality and Lorentz covariance encountered in [@Ahluwalia:2004sz; @Ahluwalia:2004ab; @Ahluwalia:2008xi; @Ahluwalia:2009rh; @Ahluwalia:2010zn]. Table 7.1 tabulates the differences just mentioned explicitly. For the Dirac field, the counterpart of these observations follow seamlessly in the Weinberg’s formalism. Since Weinberg does not note these details explicitly they seem to have escaped many of the recent textbook expositions on the theory of quantum fields. This has the consequence that the Dirac field, as presented, for example, in [@Ryder:1985wq; @Folland:2008zz; @Schwartz:2014md], hides violation of locality and Lorentz symmetry. The former can be seen by locality analysis of the those fields on Majorana-isation, while the latter only becomes apparent on comparing the rest spinors with those of Weinberg’s analysis. One way of understanding part of this story is to realise that in a quantum field the complete set of spinors enter as expansion coefficients and the freedom of a global phase that each of these carried before – by, say satisfying the Dirac equation – is lost under the usual summation on the helicity degrees of freedom.
\[tab:comparison\]
-----------------------------------------------------------------------------------------------------
Here In reference [@Ahluwalia:2004sz; @Ahluwalia:2004ab][^23]
----------------------- ----------------------------------------------------------------------- -- --
$ \lambda^S_+(k^\mu)$ $\lambda^S_{\{-,+\}}(k^\mu)$
$\lambda^S_-(k^\mu)$ $-\lambda^S_{\{+,-\}}(k^\mu) $
$ \lambda^A_+(k^\mu)$ $- \lambda^A_{\{+,-\}}(k^\mu)$, ${\textrm {and\; not\;}
- \lambda^A_{\{-,+\}}(k^\mu)} $
$\lambda^A_-(k^\mu)$ $-\lambda^A_{\{-,+\}}(k^\mu)$, ${\textrm{ and\; not}\;}
- \lambda^A_{\{+,-\}}(k^\mu)$
-----------------------------------------------------------------------------------------------------
: A comparison of $\lambda(p^\mu)$ with those used in earlier work.[]{data-label="table2new"}
Elko are not Grassmann nor are they Weyl in disguise
----------------------------------------------------
Under the Dirac dual, as would be seen in Chapters \[ch11\] and \[ch12\], the naive mass term for Elko identically vanishes. So one would be tempted to treat Elko as grassmann numbers – see, for example, [@Aitchison:2004cs]. As already remarked in Chapter \[ch7\], we find this transition from the complex valued Elko to their grassmann-isation as mathematically untenable. We would treat Elko as one treated the Dirac spinors in the operator formalism of quantum field theory.
Similarly, one should not be tempted to consider Elko as Weyl spinors in disguise. This is because I include the neglected $\zeta_\lambda = -1$ in our formalism. It would be seen as we proceed that just as the Dirac spinors span a four dimensional representation space, the Elko too are endowed with a completeness relation in the same four dimensional representation space (see equation (\[eq:completeness-li\])). Furthermore, we can exploit the algebraic identity $$\Theta\phi^\ast_\pm (k^\mu) = \pm \, \phi_\mp(k^\mu) \label{eq:algebraic-identity}$$ to rewrite the Elko at rest, given by equations (\[eq:zimpok0\]) and (\[eq:zimpok91\]), into $$\begin{aligned}
\lambda^S_+(k^\mu) = \left(\begin{array}{cc}
i\phi_-(k^\mu) \\
\phi_+(k^\mu)
\end{array}\right),\quad \lambda^S_-(k^\mu) = \left(\begin{array}{cc}
- i\phi_+(k^\mu) \\
\phi_-(k^\mu)
\end{array}\right) \\
\lambda^A_+(k^\mu) = \left(\begin{array}{cc}
i\phi_+(k^\mu) \\
\phi_-(k^\mu)
\end{array}\right),\quad \lambda^A_-(k^\mu) = \left(\begin{array}{cc}
i\phi_-(k^\mu) \\
-\phi_+(k^\mu)
\end{array}\right)\end{aligned}$$ and compare these with the Dirac spinors at rest given in equations (\[eq:Diracuk\]) and (\[eq:Diracvk\]) in the same degrees of freedom: $\phi_\pm(k^\mu)$. Written in this form all the phase choices, relative between the right- and left- transforming components, and global for each of the spinors, along with their helicities become manifest and stand in contrast to their Dirac counterpart.
Elko for any momentum
---------------------
The interplay of the result (\[eq:zimpok3\]) with the boost (\[eq:sb-again\]) and the chosen form of $\lambda(k^\mu)$ in (\[eq:zimpok0new\]) to (\[eq:zimpok91new\]) results in the following form for $\lambda(p^\mu)$ $$\begin{aligned}
\lambda^S_+(p^\mu) &= \sqrt{\frac{E+m}{2 m} }\left( 1-\frac{p}{E+m}\right)\lambda^S_+(k^\mu),\label{eq:lsp}\\
\lambda^S_-(p^\mu) &= \sqrt{\frac{E+m}{2 m} }\left( 1+\frac{p}{E+m}\right)\lambda^S_-(k^\mu)\label{eq:name}
\end{aligned}$$ and $$\begin{aligned}
\lambda^A_+(p^\mu) &= \sqrt{\frac{E+m}{2 m} }\left( 1+\frac{p}{E+m}\right)\lambda^A_+(k^\mu),\label{eq:lap}\\
\lambda^A_-(p^\mu) &= \sqrt{\frac{E+m}{2 m} }\left( 1-\frac{p}{E+m}\right)\lambda^A_-(k^\mu).
\label{eq:lam}\end{aligned}$$ Or, in a more compact form $$\begin{aligned}
\lambda^S_+(p^\mu) &= \beta_-\lambda^S_+(k^\mu),
\quad
\lambda^S_-(p^\mu) = \beta_+\lambda^S_-(k^\mu)\\
\lambda^A_+(p^\mu) &= \beta_+\lambda^A_+(k^\mu),
\quad
\lambda^A_-(p^\mu) = \beta_-\lambda^A_-(k^\mu). \end{aligned}$$ where $$\beta_\pm = \alpha \alpha_\pm$$ with $\alpha$ and $\alpha_\pm$ defined in equation (\[eq:apm\]).
In a sharp contrast to the eigenspinors of the parity operator the here-considered eigenspinors of the charge conjugation operator, $\lambda^{S,A}_\pm(p^\mu)$, are simply the rest spinors $\lambda(k^\mu)$ scaled by the indicated energy-dependent factors. In particular, for the eigenspinors of $\mathcal{C}$ the boost does not mix various components of the ‘rest-frame’ spinors.
An inspection of equations (\[eq:lsp\]) to (\[eq:lam\]) suggests that for massless $\lambda(p^\mu)$ the number of degrees of freedom reduces to two, that is those associated with $\lambda^S_-(p^\mu)$ and $\lambda^A_+(p^\mu)$ while the $\lambda^S_+(p^\mu)$ and $\lambda^A_-(p^\mu)$ vanish identically.
We will see below that the parity operator takes $\lambda_-^{S}(p^\mu) \to \lambda_+^{S}(p^\mu)$ and $\lambda_+^{A}(p^\mu) \to \lambda_-^{A}(p^\mu)$. Combining these two observations we conclude that in the massless limit $\lambda(p^\mu)$ have no reflection.
Strictly speaking for massless particles there is no rest frame, or ‘rest-frame’ spinors. The theory must be constructed *ab initio* except that the massless limit of certain massive representation spaces yields the massless theory. The $\mathcal{R}$ and $\mathcal{L}$ representation spaces belong to that class. We refer the reader to Weinberg’s 1964 work on the subject [@Weinberg:1964ev]. That such a limit may be taken for the $\mathcal{R}\oplus\mathcal{L}$ representation space is apparent from Weinberg’s analysis.
The four four-component Elko, $\lambda^{S,A}_\pm(p^\mu)$, are the expansion coefficients of the new quantum field to be introduced below.
We parenthetically note that as soon as the first papers introducing Elko and mass dimension one fermions were published da Rocha and Rodrigues Jr. noted that Elko belong to class 5 spinors [@daRocha:2005ti] in the Lounesto classification [@Lounesto:2001zz Chapter 12].
A hint for mass dimension one fermions {#ch9}
======================================
\[sec:elko-do-not-satisfy-Dirac-equation\]
A first hint towards mass dimension one fermions arises from the observation that the momentum space Dirac operators $(\gamma_\mu p^\mu \pm m \I)$ do not annihilate Elko $$\left(\gamma_\mu p^\mu \pm m \I\right)\lambda(p^\mu) \ne 0\label{eq:DiracNot}$$
To prove this we act $\gamma_\mu p^\mu$ on each of the Elko enumerated in equations (\[eq:lsp\])-(\[eq:lam\]). We begin this exercise with $ \lambda^S_+(p^\mu)$ $$\begin{aligned}
\gamma_\mu p^\mu \lambda^S_+(p^\mu) = \sqrt{\frac{E+m}{2 m}} &\left[ 1 - \frac{p}{E+m}\right]\nonumber \\
& \times \underbrace{ \left[
E \gamma_0 + p \left(\begin{array}{cc}
0 & \s\cdot\widehat{\p} \\
-\s\cdot\widehat{\p} & 0
\end{array}\right)
\right] }_{\gamma_\mu p^\mu}
\lambda^S_+(k^\mu)\label{eq:zimpok4}
\end{aligned}$$ and notice that $$\left(\begin{array}{cc}
0 & \s\cdot\widehat{\p} \\
-\s\cdot\widehat{\p} & 0
\end{array}\right) \lambda^S_+(k^\mu) = \left(\begin{array}{cc}
0 & \s\cdot\widehat{\p} \\
-\s\cdot\widehat{\p} & 0
\end{array}\right)
\left(
\begin{array}{c}
i \Theta\left[\phi_+(k^\mu)\right]^\ast\\
\phi_+(k^\mu)
\end{array}
\right).$$ But \_+(k\^) = \_+(k\^) while according to (\[eq:zimpok3\]) $$\s\cdot\widehat{\p} \,\Big[\Theta\left[\phi_+(k^\mu)\right]^\ast\Big]
= - \Theta\left[\phi_+(k^\mu)\right]^\ast$$ Therefore, we have the result $$\begin{aligned}
\left(\begin{array}{cc}
0 & \s\cdot\hat{\p} \\
-\s\cdot\hat{\p} & 0
\end{array}\right)& \lambda^S_+(k^\mu) =
\left(\begin{array}{c}
\phi_+(k^\mu)\\
i \Theta\left[\phi_+(k^\mu)\right]^\ast
\end{array}
\right) \nonumber \\
& = \left(\begin{array}{cc}
\0 & \openone \\
\openone & \0\end{array}\right)
\left(\begin{array}{c}
i \Theta\left[\phi_+(k^\mu)\right]^\ast\\
\phi_+(k^\mu)
\end{array}
\right) \end{aligned}$$ That is $$\left(\begin{array}{cc}
0 & \s\cdot\hat{\p} \\
-\s\cdot\hat{\p} & 0
\end{array}\right) \lambda^S_+(k^\mu) =
\gamma_0 \lambda^S_+(k^\mu).$$
As a consequence (\[eq:zimpok4\]) simplifies to $$\gamma_\mu p^\mu \lambda^S_+(p^\mu) = \sqrt{\frac{E+m}{2 m}} \left(1 - \frac{p}{E+m}\right)
\left(
E + p
\right) \gamma_0\lambda^S_+(k^\mu).\label{eq:zimpok5}$$ The standard dispersion relation allows for the replacement $$\left(1 - \frac{p}{E+m}\right)(E+p)
\rightarrow m \left(1 + \frac{p}{E+m}\right)$$ To evaluate $\gamma_0 \lambda^S_+(k^\mu) $ we first expand it as $$\gamma_0 \lambda^S_+(k^\mu) =
\left(
\begin{array}{c}
\phi_+(k^\mu) \\
i\Theta\phi^\ast_+(k^\mu)
\end{array}
\right)$$ and exploit the algebraic result (\[eq:algebraic-identity\]) to make the following substitutions $$\phi_+(k^\mu) \to i \left(i \Theta\phi^\ast_-(k^\mu) \right),\quad
i \Theta\phi^\ast_+(k^\mu) \to i\phi_-(k^\mu)$$ to obtain the identity $$\gamma_0 \lambda^S_+(k^\mu) = i \lambda^S_-(k^\mu)$$ Combined, these two observations reduce (\[eq:zimpok5\]) to $$\gamma_\mu p^\mu \lambda^S_+(p^\mu) = i m \sqrt{\frac{E+m}{2 m}} \left(1 + \frac{p}{E+m}\right) \lambda^S_-(k^\mu).\label{eq:zimpok6}$$ Using (\[eq:name\]) in the right-hand side of (\[eq:zimpok6\]) gives $$\gamma_\mu p^\mu \lambda^S_+(p^\mu) = i m \lambda^S_-(p^\mu). \label{eq:er-a1}$$ An exactly similar exercise complements (\[eq:er-a1\]) with $$\begin{aligned}
\gamma_\mu p^\mu \lambda^S_-(p^\mu) & = - i m \lambda^S_+(p^\mu) \label{eq:er-a2}\\
\gamma_\mu p^\mu \lambda^A_-(p^\mu) & = i m \lambda^A_+(p^\mu) \label{eq:er-b1}\\
\gamma_\mu p^\mu \lambda^A_+(p^\mu) & = - i m \lambda^A_-(p^\mu) . \label{eq:er-b2}\end{aligned}$$ These results are consistent with those of [@Dvoeglazov:1995eg; @Dvoeglazov:1995kn]. Translated in words these equations combine to yield the following pivotal result: $(\gamma_\mu p^\mu \pm m)$ do not annihilate $\lambda(p^\mu)$.
Equations (\[eq:er-a1\]) to (\[eq:er-b2\]), coupled with the discussion surrounding (\[eq:counterpart\]), contain the rudimentary seeds for the kinematical and dynamical content of the quantum field built upon $\lambda(p^\mu)$ as its expansion coefficients: first, $\lambda(p^\mu)$ are annihilated by the spinorial Klein-Gordon operator (and not by the Dirac operator), and second, the resulting kinematic structure cannot support the usual gauge symmetries of the standard model of the high energy physics. To prove the the former claim, we multiply (\[eq:er-a1\]) from the left by $\gamma_\nu p^\nu $ and use (\[eq:er-a1\]) and (\[eq:er-a2\]) in succession \_p\^ \_p\^\^S\_+(p\^) = i m \_p\^\^S\_-(p\^) = im ( - i m \^S\_+(p\^)) = m\^2 \^S\_+(p\^) \[eq:zimpok1952\] and then utilise the fact that the left hand side of the above equation can be rewritten exploiting $\{\gamma_\mu,\gamma_\nu\}= 2 \eta_{\mu\nu} \I_4$ – where $\eta_{\mu\nu}$ is the space-time metric with signature $(+1,-1,-1,-1)$) – as \_p\^ \_p\^= {\_,\_} p\^p\^ = \_p\^p\^\_4 . Substituting this result in (\[eq:zimpok1952\]), and rearranging gives $$(\eta_{\mu\nu}p^\mu p^\nu \I_4 - m^2\I_4)\lambda^{S}_+(p^\mu) = 0.$$ Repeating the same exercise with (\[eq:er-a2\]) to (\[eq:er-b2\]) as the starting point, yields $$(\eta_{\mu\nu}p^\mu p^\nu \I_4 - m^2\I_4)\lambda(p^\mu)= 0. \label{eq:skg}$$ where $\lambda(p^\mu)$ stands for any of the four eigenspinors of the charge conjugation operator, $\lambda^{S,A}_\pm(p^\mu)$.
Lest the appearances betray, the $\lambda^{S,A}_\pm(p^\mu)$ are not a set of four scalars, disguised in four-component form repeated four times, but spinors in the $\mathcal{R}\oplus\mathcal{L}$ representation space of spin one half. Under boosts these spinors contract or dilate by factors of $$\beta_\pm = \sqrt{\frac{E+m}{2 m} }\left[ 1\pm \frac{p}{E+m}\right]$$ in accord with equations (\[eq:lsp\]) to (\[eq:lam\]) and pick up a minus sign under $2\pi$ rotation as dictated by (\[eq:rotation-on-spinors\]).[^24]
CPT for Elko
============
Since $\mathcal{P} = m^{-1}\gamma_\mu p^\mu$ the results obtained in the previous chapter also translate to the action of $\mathcal{P}$ on the eigenspinors of the charge conjugation operator $\mathcal{C}$ $$\begin{aligned}
\mathcal{P} \lambda^S_+(p^\mu) &= i \lambda^S_-(p^\mu) \label{eq:er-a1P}\\
\mathcal{P} \lambda^S_-(p^\mu) & = - i \lambda^S_+(p^\mu) \label{eq:er-a2P}\\
\mathcal{P} \lambda^A_-(p^\mu) & = i \lambda^A_+(p^\mu) \label{eq:er-b1P}\\
\mathcal{P} \lambda^A_+(p^\mu) & = - i \lambda^A_-(p^\mu) . \label{eq:er-b2P}\end{aligned}$$ The above can be compacted into the following $$\mathcal{P} \lambda^S_\pm (p^\mu) = \pm i \lambda^S_\mp(p^\mu),\quad
\mathcal{P}\lambda^A_\pm (p^\mu) = \mp i \lambda^A_\mp(p^\mu) \label{eq:parity-b2}$$ and lead to $$\mathcal{P}^2=\I_4\label{eq:psq}$$ Acting $\mathcal{C}$ from the left on the first of the above equations gives $$\mathcal{C} \mathcal{P} \lambda^S_+(p^\mu) = - i \mathcal{C} \lambda^S_-(p^\mu) = -i \lambda^S_-(p^\mu) . \label{eq:zimpok100}$$ On the other hand $$\mathcal{P} \mathcal{C} \lambda^S_+(p^\mu) =
\mathcal{P} \lambda^S_+(p^\mu) = i \lambda^S_-(p^\mu) .$$ Adding the above two results leads to anti-commutativity for the $\mathcal{C}$ and $\mathcal{P}$ for $ \lambda^S_+(p^\mu) $. Repeating the same exercise for $\lambda^S_-(p^\mu)$ and $\lambda^A_\pm(p^\mu) $ establishes that $\mathcal{C}$ and $\mathcal{P}$ anticommute for all the eigenspinors of the charge conjugation operator $$\{\mathcal{C}, \mathcal{P}\} = 0\label{eq:cp-z}$$ just as seen before in (\[eq:CPanticommutator-P\]) for the eigenspinors of the parity operator $\mathcal{P}$.
The action of the time reversal operator on $\lambda(p^\mu)$ is as follows (as an explicit calculation shows): $$\begin{aligned}
\mathcal{T}\lambda^S_+(p^\mu) = i \lambda^A_-(p^\mu), \quad
\mathcal{T}\lambda^S_-(p^\mu) = - i \lambda^A_+(p^\mu) \label{eq:timereversalS}\\
\mathcal{T}\lambda^A_+(p^\mu) = i \lambda^S_-(p^\mu), \quad
\mathcal{T}\lambda^A_-(p^\mu) = - i \lambda^S_+(p^\mu)
\label{eq:timereversalA}
\end{aligned}$$ with the square of $\mathcal{T}$ acting on Elko giving, $-\I_4$ $$\mathcal{T}^2 = - \I_4\label{eq:tsq}$$ In addition, we find that $$\left[\mathcal{C},\mathcal{T}\right]=0,\quad \left[\mathcal{P},\mathcal{T}\right]=0\label{eq:pt}$$ hold for Elko, as for the Dirac spinors. In consequence, for Elko $\left(\mathcal{C}\mathcal{P}\mathcal{T}\right)^2=\I_4$. This contrasts with contrary results reported in our own earlier work, and those of several other authors. The difference can be traced to an unambiguous treatment $\mathcal{P}$ for Elko, and to the differences tabulated in Table 7.1.
Elko in Shirokov-Trautman, Wigner, and Lounesto classifications
===============================================================
In the Shirokov-Trautman classification [@Shirokov:1960ym; @Trautman:2005qx] a spinor is classified by the possible choices of signs $\lambda,\mu,\nu\in \{+,-\}$ in the relations $$\mathcal{P} \mathcal{T} = \lambda\, \mathcal{T} \mathcal{P},\quad
\mathcal{P}^2=\mu\, \I_4,\quad\mathcal{T}^2 = \nu\, \I_4$$ Commutativity of $\mathcal{P}$ and $\mathcal{T}$ as given in equation (\[eq:pt\]) gives $\lambda= +$, while results (\[eq:psq\]) and (\[eq:tsq\]) give $\mu=+$ and $\nu=-$. Thus Elko belongs to $(\lambda,\mu,\nu)$ class identified as $(+,+,-)$.
In the Wigner classification one may go beyond the relative intrinsic parity of the particles and antiparticles to be same for bosons, and opposite for fermions [@Wigner:1962ep]. The anti-commutativity of $\mathcal{C}$ and $\mathcal{P}$ reached in equation (\[eq:cp-z\]) places Elko in the same class as the Dirac spinors. This should not make the reader infer that all other properties of Elko and Dirac spinors are the same but only that both of these spinors have particles and antiparticles with the opposite relative intrinsic parity.
Lounesto classification is based on the bilinear invariants associated with the spinors under the standard Dirac dual [@Lounesto:2001zz]. An early analysis given in reference [@daRocha:2005ti] established that Elko belong to class 5 in this classification. The cited work has been extended in a series of later papers, see for example in [@HoffdaSilva:2017waf; @HoffdaSilva:2016ffx], with additional new structure unearthed by the use of the dual appropriate for Elko. This classification may be particularly helpful for studying currents associated with Dirac spinors and Elko, and the gauge symmetries they arise from. However, Lounesto did not consider the issue of associated Lagrangian densities for each of the classes that he proposed.
Rotation induced effects on Elko {#ch10}
================================
Despite its simple roots in the spin one half representation of the Lorentz algebra the celebrated minus sign that a $ 2 \pi$ rotation induces on Dirac spinors has intrigued the physicist and the lay scholar alike [@Aharonov:1967zz; @Dowker:1969ia; @RAUCH1974369; @Werner:1975wf; @Silverman:1980mp; @Horvathy:1984sm; @Klein:1976qc]. For a general rotation about an axis, say $\widehat{\p}$, by an angle $\vartheta$ the effect of rotation is simply a multiplication of the original spinor by a phase factor $\exp(\pm i \vartheta/2)$ – the sign determined by the helicity of the spinor.
In sharp contrast to the Dirac spinors, Elko not only acquire a minus sign for the same $2 \pi$ rotation but the effect of a general rotation is to create a specific admixture of the self-conjugate and anti-self conjugate spinors – the resulting spinors are still eigenspinors of the charge conjugation operator. This apparent paradox is resolved. This property is a direct consequence of the fact that the right and left transforming components of Elko are endowed with opposite helicities (see equation (\[eq:zimpok3new\]), and each of these picks up an opposite phase in accordance with the result (\[eq:phasefactor\]).[^25]
Setting up an orthonormal cartesian coordinate system with $\widehat{\textbf{p}}$ as one of its axis
----------------------------------------------------------------------------------------------------
Understanding the new spinors vastly simplifies in the basis we have chosen for them. Their right- and left- transforming components have their spin projections assigned not with respect an external $\widehat{z}$-axis but to a self referential unit vector associated with their motion: that is $\widehat{\textbf{p}}$. To establish the result outlined above we find it convenient to erect an orthonormal cartesian coordinate system with $\widehat{\textbf{p}}$ as one of its axis. With $\widehat{\textbf{p}}$ chosen as[^26] $$\left(s_\theta c_\phi, s_\theta s_\phi,c_\theta\right)$$ we introduce two more unit vectors $$\begin{aligned}
\widehat{\eta}_+ & \stackrel{\textrm{def}}{=}\frac{1}{\sqrt{2+a^2}}\left(1,1,a\right), \quad a \in \Re \\
\widehat{\eta}_- & \stackrel{\textrm{def}}{=}
\frac{\widehat{\textbf{p}}\times\widehat{\eta}_+ }
{\sqrt{\left(\widehat{\textbf{p}}\times\widehat{\eta}_+ \right)
\cdot
\left(\widehat{\textbf{p}}\times\widehat{\eta}_+ \right)}}
\label{eq:etam}\end{aligned}$$ and impose the requirement $$\begin{aligned}
& \widehat{\eta}_+\cdot \widehat{\eta}_- = 0,\quad
\widehat{\eta}_+\cdot \widehat{\textbf{p}} = 0,\quad
\widehat{\eta}_-\cdot \widehat{\textbf{p}} = 0\\
& \widehat{\eta}_+\cdot \widehat{\eta}_+ = 1,\quad
\widehat{\eta}_-\cdot \widehat{\eta}_- = 1.\end{aligned}$$ Requiring $\widehat{\eta}_+\cdot \widehat{\textbf{p}} $ to vanish reduces $\widehat{\eta}_+ $ to $$\widehat{\eta}_+ = \frac{1}{\sqrt{2 + (1+s_{2\phi})t^2_\theta}}\Big( 1,1,-(c_\phi + s_\phi)t_\theta\Big).$$ Definition (\[eq:etam\]) then immediately yields $$\begin{aligned}
\widehat{\eta}_- = \frac{1}{\sqrt{2 + (1+s_{2\phi})t^2_\theta}}\Big( -c_\theta - s_\theta s_\phi(c_\phi+s_\phi)t_\theta, \\
c_\theta + s_\theta c_\phi(c_\phi+s_\phi)t_\theta,
s_\theta(c_\phi-s_\phi) \Big).\end{aligned}$$
In the limit when both the $\theta$ and $\phi$ tend to zero, the above-defined unit vectors take the form $$\begin{aligned}
\widehat{\eta}_+\big\vert_{\theta\to 0,\phi\to 0} &= \frac{1}{\sqrt{2}}\left(1,1,0\right) \label{eq:etaplus}\\
\widehat{\eta}_-\big\vert_{\theta\to 0,\phi\to 0} &= \frac{1}{\sqrt{2}}\left(-1,1,0\right)\label{eq:etaminus}\end{aligned}$$ and orthonormal system $\left(\widehat{\eta}_-,\widehat{\eta}_+,\widehat{\textbf{p}}\right)$ does not reduce to the standard cartesian coordinate system $(\widehat{\textbf{x}},\widehat{\textbf{y}},\widehat{\textbf{z}})$. For this reason, without destroying the orthonormality of the introduced unit vectors and guided by (\[eq:etaplus\]) and (\[eq:etaminus\]), we exploit the freedom of a rotation about the $ \widehat{\textbf{p}}$ axis (in the plane defined by $\widehat{\eta}_+$ and $\widehat{\eta}_-$) and introduce $$\widehat{\textbf{p}}_\pm \stackrel{\textrm{def}}{=} \frac{1}{\sqrt{2}}\left(\widehat{\eta}_+ \pm \widehat{\eta}_-\right)$$ The set of axes $\left(\widehat{\textbf{p}}_-,\widehat{\textbf{p}}_+,\widehat{\textbf{p}}\right)$ do indeed form a right-handed coordinate system that reduces to the standard cartesian system in the limit both the $\theta$ and the $\phi$ tend to zero.
Generators of the rotation in the new coordinate system
-------------------------------------------------------
We now introduce three generators of rotations about each of the three axes $$\texttt{J}_- \stackrel{\textrm{def}}{=} \frac{\s}{2}\cdot \widehat{\textbf{p}}_-
,\quad
\texttt{J}_+ \stackrel{\textrm{def}}{=} \frac{\s}{2}\cdot \widehat{\textbf{p}}_+
,\quad
\texttt{J}_p \stackrel{\textrm{def}}{=} \frac{\s}{2}\cdot \widehat{\textbf{p}}$$ $\texttt{J}_p $ coincides with the helicity operator $\mathfrak{h} $ defined in (\[eq:helicity\]). As a check, a straightforward exercise shows that the three $\texttt{J}'s$ satisfy the $\mathfrak{s}\mathfrak{u}(2)$ algebra needed for generators of rotation $$\left[\texttt{J}_-,\texttt{J}_+\right] = i \texttt{J}_p,\quad
\left[\texttt{J}_p,\texttt{J}_-\right] = i \texttt{J}_+,\quad
\left[\texttt{J}_+,\texttt{J}_p\right] = i \texttt{J}_-$$ Since Elko reside in the $\mathcal{R}\oplus\mathcal{L}$ representation space and as far as rotations are concerned both the $\mathcal{R}$ and $\mathcal{L}$ spaces are served by the same generators of rotations. We therefore introduce $$\texttt{h}_ - =\left(\begin{array}{cc}
{\mathtt{J}_-} & \texttt{0}_2\\
\texttt{0}_2 &{\mathtt{J}_-}
\end{array}
\right),\quad
\texttt{h}_ + =\left(\begin{array}{cc}
{\mathtt{J}_+} & \texttt{0}_2\\
\texttt{0}_2 &{\mathtt{J}_+}
\end{array}
\right),\quad
\texttt{h}_ p =\left(\begin{array}{cc}
{\mathtt{J}_p} & \texttt{0}_2\\
\texttt{0}_2 &{\mathtt{J}_p}
\end{array}
\right)$$ where $\texttt{0}_2$ is a $2\times 2$ null matrix. A straight forward calculation then yields the result $$\begin{aligned}
\textrm{h}_p \,\lambda^S_+(p^\mu) &= -\frac{1}{2}\lambda^A_-(p^\mu),\quad
\textrm{h}_p \,\lambda^S_-(p^\mu) = -\frac{1}{2}\lambda^A_+(p^\mu)\label{eq:hplamdaS}\\
\textrm{h}_p \,\lambda^A_+(p^\mu) & = -\frac{1}{2}\lambda^S_-(p^\mu),\quad
\textrm{h}_p \,\lambda^A_-(p^\mu) = -\frac{1}{2}\lambda^S_+(p^\mu)\label{eq:hplamdaA}\end{aligned}$$ with the consequence that each of the Elko is an eigenspinor of $\texttt{h}_p^2$ $$\textrm{h}_p^2 \,\lambda^{S,A}_\pm(p^\mu) = \frac{1}{4}\lambda^{S,A}_\pm(p^\mu)$$ The action of $\texttt{h}_-$ and $\texttt{h}_+$ on Elko is more involved, for instance $$\texttt{h}_+\,\lambda^S_+(p^\mu) = - \frac{1}{2}\left(
\alpha \lambda^S_-(p^\mu) + \beta\lambda^A_+(p^\mu)\right)$$ with $$\begin{aligned}
\alpha = - \frac{im (m+E)\big(\cos\phi(1+\sec\theta)+(-1+\sec\theta)\sin\phi\big)}
{\sqrt{4+2(1+\sin 2\phi)\tan^2\theta}}\\
\beta= - \frac{m (m+E)\big(\cos\phi(-1+\sec\theta)+(1+\sec\theta)\sin\phi\big)}
{\sqrt{4+2(1+\sin 2\phi)\tan^2\theta}}\end{aligned}$$ but the action of their squares is much simpler and reads $$\textrm{h}_-^2 \,\lambda^{S,A}_\pm(p^\mu) = \frac{1}{4}\lambda^{S,A}_\pm(p^\mu),\quad
\textrm{h}_+^2 \,\lambda^{S,A}_\pm(p^\mu) = \frac{1}{4}
\lambda^{S,A}_\pm(p^\mu).$$ This exercise thus establishes that $$\texttt{h}^2 \stackrel{\textrm{def}}{=} \textrm{h}_- ^2+ \textrm{h}_+^2 + \textrm{h}_p^2$$ while acting on each of the Elko yields $$\texttt{h}^2 \lambda^{S,A}_\pm = \frac{3}{4} \lambda^{S,A}_\pm =\frac{1}{2}
\left(1+ \frac{1}{2}\right) \lambda^{S,A}_\pm$$ and confirms spin one half for Elko.
For ready reference, we note the counterpart of (\[eq:hplamdaS\]) and (\[eq:hplamdaA\]) for the Dirac spinors: $$\begin{aligned}
\textrm{h}_p \, u_+(p^\mu) &= \frac{1}{2} u_+(p^\mu),\quad
\textrm{h}_p \,u_-(p^\mu) = -\frac{1}{2}u_-(p^\mu)
\label{eq:hplamdau}\\
\textrm{h}_p \,v_+(p^\mu) & = -\frac{1}{2}v_+(p^\mu),\quad
\textrm{h}_p \,v_-(p^\mu) = \frac{1}{2}v_-(p^\mu).\label{eq:hplamdav}\end{aligned}$$
The new effect
--------------
The formalism developed in this chapter so far immediately establishes the opening claim of this chapter. For simplicity we consider a rotation by $\vartheta$ about $\widehat{\p}$ axis and find that a $2\pi$ rotation induces the expected minus sign for Elko, but for a general rotation it mixes the self and antiself conjugate spinors: $$\exp\left({i \texttt{h}_p \vartheta}\right)\, \lambda^S_\pm(p^\mu) = \cos\left({\vartheta}/{2}\right) \lambda^S_\pm(p^\mu) - i \sin\left({\vartheta}/{2}\right)
\lambda^A_\mp(p^\mu)\label{eq:22}$$ and $$\exp\left({i \texttt{h}_p \vartheta}\right)\, \lambda^A_\pm(p^\mu) = \cos\left({\vartheta}/{2}\right) \lambda^A_\pm(p^\mu) - i \sin\left({\vartheta}/{2}\right)
\lambda^S_\mp(p^\mu).\label{eq:23}$$ In contrast, for the Dirac spinors we have the well-known result $$\exp\left({i \texttt{h}_p \vartheta}\right) \psi_\pm(p^\mu) = \exp(\pm i\vartheta/2) \psi_\pm(p^\mu)$$ where $\psi_\pm(p^\mu)$ stands for any one of the four $u$ and $v$ spinors of Dirac given in equations (\[eq:upp\]) to (\[eq:vm\]).
`Do the results ([eq:22]) and ([eq:23]) imply that rotation induces loss of self/anti-self conjugacy under charge conjugation \mathcal{C}?` The answer is: no. To see this we observe that $$\begin{aligned}
\mathcal{C} \left[ i \lambda^A_{\pm}(\p)\right] &=& (-i)(-\lambda^A_{\pm}(\p))
=+ \left[ i \lambda^A_{\pm}(\p)\right] \label{eq:obs1}\\
\mathcal{C} \left[ i \lambda^S_{\pm}(\p)\right] &=& (-i)(\lambda^S_{\pm}(\p))
=- \left[ i \lambda^S_{\pm}(\p)\right].\label{eq:obs2}\end{aligned}$$ We thus define a set of new self and anti-self conjugate spinors (see (\[eq:22\]) and (\[eq:23\])) $$\begin{aligned}
\lambda^s(\p) &\stackrel{\text{def}}{=}& \cos\left({\vartheta}/{2}\right) \lambda^S_\pm(\p) - i \sin\left({\vartheta}/{2}\right) \lambda^A_\mp(\p) \\
\lambda^a(\p)& \stackrel{\text{def}}{=} &\cos\left({\vartheta}/{2}\right) \lambda^A_\pm(\p) - i \sin\left({\vartheta}/{2}\right) \lambda^S_\mp(\p)\end{aligned}$$ and verify that $$\mathcal{C}\lambda^{s}_\pm (\p) = + \lambda^{s}_\pm (\p), \quad\mathcal{C}\lambda^{a}_\pm (\p) = - \lambda^{a}_\pm (\p).$$ As a result (\[eq:22\]) and (\[eq:23\]) reduce to $$\begin{aligned}
\exp\left({i \texttt{h}_p \vartheta}\right)\, \lambda^S_\pm(\p) = \lambda^s_\pm(\p) \\
\exp\left({i \texttt{h}_p \vartheta}\right)\, \lambda^A_\pm(\p) = \lambda^a_\pm(\p)\end{aligned}$$ and confirm that rotation preserves $\mathcal{C}$-self/anti-self conjugacy of Elko. The explicit expressions for the new set of Elko are now readily obtained, and read: $$\lambda^s_\pm(\p) = \varrho_\pm\lambda^S_\pm(\p),\quad
\lambda^a_\pm(\p) = \varrho_\mp\lambda^A_\pm(\p)$$ where the $4\times 4$ matrices $\varrho_\pm$ are defined as $$\varrho_\pm =\left(
\begin{array}{cc}
e^{\mp i\vartheta/2} \mathbb{I} & \mathbb{O}\\
\mathbb{O} & e^{\pm i\vartheta/2} \mathbb{I}
\end{array}
\right).$$ In the above expression, $\mathbb{I}$ and $\mathbb{O}$ are $2\times2$ identity and null matrices respectively. Apart from the special values of $\vartheta$ for which $$\varrho_\pm =\bigg\{\begin{array}{ll}
- 1, & \mbox{for $\theta = 2\pi$}\\
+ 1,& \mbox{for $\theta = 4\pi$}.
\end{array}$$ Elko and Dirac spinors behave differently.
The next question thus arises: Since in all physical observables Elko appear as bilinears, do the sets $\{\lambda^S_\pm(\p),
\lambda^A_\pm(\p)\}$ and $\{\lambda^s_\pm(\p),
\lambda^a_\pm(\p)\}$ yield identical results? The answer is: yes. It comes about because $$(\varrho_\pm)^\dagger \gamma_0 \varrho_\mp = \gamma_0.$$ So from the orthonormality relations, to the completeness relations, to the spin sums, the two sets $\{\lambda^S_\pm(\p),
\lambda^A_\pm(\p)\}$ and $\{\lambda^s_\pm(\p),
\lambda^a_\pm(\p)\}$ carry identical results.
Elko-Dirac interplay, a temptation and a departure {#ch11}
==================================================
Null norm of massive Elko and Elko-Dirac interplay
--------------------------------------------------
The result that the spin one-half eigenspinors of the charge conjugation operator satisfy only the Klein Gordon equation would suggest that at the ‘classical level’ the Lagrangian density for the $\lambda(x)$ would simply be $$\mathfrak{L}(x) = \partial^\mu{\overline{\lambda}(x)}\,\partial_\mu {{\lambda(x)}} - m^2 {\overline{\lambda}}(x) \lambda(x)\label{eq:wrong}$$ where $\lambda(x)$ is a classical field with $\lambda^{S,A}(p^\mu)$ as its Fourier coefficients. This apparently natural choice is too naive. Its validity is challenged by an explicit calculation which shows that under the Dirac dual $$\overline{\lambda}^{S,A}_\alpha(p^\mu)\stackrel{\rm def}{=}\left[\lambda^{S,A}_\alpha(p^\mu)\right]^\dagger \gamma_0$$ the norm of each of the four $\lambda^{S,A}_\alpha(p^\mu)$ identically vanishes [@Ahluwalia:1994uy] $$\overline{\lambda}^S_\alpha(p^\mu) \lambda^S_\alpha(p^\mu) = 0,\quad
\overline{\lambda}^A_\alpha(p^\mu) \lambda^A_\alpha(p^\mu) = 0 .\label{eq:nullnorm}$$ At the simplest, one may use the $\lambda(p^\mu)$ enumerated in equations (\[eq:lsp\]) to (\[eq:lam\]) and verify null norm (\[eq:nullnorm\]) by a brute force algebraic calculation. Here we follow a more detailed route to arrive at the null norm of massive Elko. In the process we would not only see how precisely this comes about but also gain deeper insight into the relation between the Dirac spinors and Elko.
Since both the Dirac spinors and Elko reside in the same $\mathcal{R}\oplus\mathcal{L}\vert_{s=1/2}$ representation space, we can expand Elko in terms of the Dirac spinors as [@Ahluwalia:1993xa; @Ahluwalia:2004ab] $$e_a = \sum_{b=1}^{4} \Omega_{ab} d_b,\quad a,b = 1,2,3,4$$ where we have abbreviated Elko and Dirac spinors as $$\left\{
\begin{array}{ll}
e_1 \\
e_2\\
e_3\\
e_4
\end{array}\right\}
\stackrel{\textrm{def}}{=}
\left\{
\begin{array}{ll}
\lambda^S_+(p^\mu) \\
\lambda^S_-(p^\mu)\\
\lambda^A_+(p^\mu)\\
\lambda^A_-(p^\mu)
\end{array}\right\},\quad
\left\{
\begin{array}{ll}
d_1 \\
d_2\\
d_3\\
d_4
\end{array}\right\}
\stackrel{\textrm{def}}{=}
\left\{
\begin{array}{ll}
u_+(p^\mu) \\
u_-(p^\mu)\\
v_+(p^\mu)\\
v_-(p^\mu)
\end{array}\right\}$$ and $$\Omega_{ab} = \begin{cases}
+ (1/2 m)\overline{d}_b e_a,\quad \mbox{\textrm{for b=1,2}}\\
- (1/2 m)\overline{d}_b e_a,\quad \mbox{\textrm{for b=3,4}}
\end{cases}$$ An explicit calculation using equations (\[eq:lsp\]) to (\[eq:lam\]) for the Elko and equations (\[eq:upp\]) to (\[eq:vm\]) for the Dirac spinors yields $$\Omega = \frac{1}{2} \left(
\begin{array}{cccc}
1 & i & i & 1\\
-i & 1 & -1 & i\\
i & 1 & -1 & -i\\
-1 & i & i & -1
\end{array}
\right)$$where rows are enumerated as $a=1,2,3,4$ and columns as $b=1,2,3,4$. With this formulation at hand we immediately see that we may explicitly write any of the Elko in terms of Dirac spinors. For instance $$\lambda^S_+(p^\mu) = \frac{1}{2} \Big( u_+(p^\mu) + i u_-(p^\mu) + i v_+(p^\mu) + v_-(p^\mu\Big)\label{eq:lambdasa}$$ which yields under the Dirac dual reads $$\overline{\lambda}^S_+(p^\mu) = \frac{1}{2} \Big( \overline{u}_+(p^\mu) - i \overline{u}_-(p^\mu) -i \overline{v}_+(p^\mu) +\overline{v}_-(p^\mu)\Big).$$ Thus, using the orthonormality of the Dirac spinors under the Dirac dual, we get two positive and two negative contributions – each equaling $2 m$ in magnitude – to the norm of $\lambda^S_+(p^\mu)$ under the Dirac dual. With the result that $\overline{\lambda}^S(p^\mu) \lambda^S(p^\mu)$ vanishes. Exactly the same pattern and result holds for the remaining three Elko.
So the mass matrix in the Lagrangian density (\[eq:wrong\]) would have vanishing diagonal elements. The off diagonal elements would either vanish identically, or would be pure imaginary because, for instance $$\overline{\lambda}^S_-(p^\mu) = \frac{1}{2} \Big( i \overline{u}_+(p^\mu)
+ \overline{u}_-(p^\mu)
-\overline{v}_+(p^\mu) -i \overline{v}_-(p^\mu)\Big)$$ and it results in $$\overline{\lambda}^S_-(p^\mu) \lambda^S_+(p^\mu) = i 2m\label{eq:i2m}$$
Further on Elko-Dirac interplay
-------------------------------
The above discussed Elko-Dirac structure makes it clear that the inclusion of anti self conjugate spinors is necessary for a proper understanding of the eigenspinors of the charge conjugation operator.
The result that Elko cannot satisfy Dirac equation, as encoded in equations (\[eq:er-a1P\]) to (\[eq:er-b2P\]), can now be derived in a way which makes transparent inevitability of the conclusion. To see this one may act the parity operator $\mathcal{P}$, $m^{-1}\gamma_\mu p^\mu$, on equation (\[eq:lambdasa\]) to the effect $$\begin{aligned}
m^{-1}\gamma_\mu p^\mu \,\lambda^S_+(p^\mu)
&= \frac{1}{2}
\Big( \underbrace{ m^{-1} \gamma_\mu p^\mu \,u_+(p^\mu)}_{ u_+(p^\mu)} + i \underbrace{m^{-1}\gamma_\mu p^\mu \,u_-(p^\mu) }_{u_-(p^\mu) }
\nonumber \\
& + i \underbrace{m^{-1}\gamma_\mu p^\mu \,v_+(p^\mu)}_{-v_+(p^\mu)} + \underbrace{m^{-1}\gamma_\mu p^\mu\, v_-(p^\mu)}_{-v_-(p^\mu)}\Big)\nonumber\\
&= \frac{1}{2}\Big(u_+(p^\mu)+i u_-(p^\mu) - i v_+(p^\mu) - v_-(p^\mu)) \Big)\nonumber\\
& =i \underbrace{\frac{1}{2}\Big(-i u_+(p^\mu)+ u_-(p^\mu) - v_+(p^\mu) + i v_-(p^\mu) }_{\lambda^S_-(p^\mu)} \Big)\nonumber\\
& = i \lambda^S_-(p^\mu).\end{aligned}$$ That is $\mathcal{P} \lambda^S_+(p^\mu) = i \lambda^S_-(p^\mu)$. Results (\[eq:er-a2P\]) to (\[eq:er-b2P\]) follow similarly.
Finally, it is insightful to check how the results (\[eq:timereversalS\]) and (\[eq:timereversalA\]) on time reversal come from the Elko-Dirac relationship. Towards this end one may act the time reversal operator $\mathcal{T}$, $i\gamma\mathcal{C}$, on $\lambda^S_+(p^\mu)$ expanded in terms of Dirac spinors in (\[eq:lambdasa\]) to the effect $$\begin{aligned}
\mathcal{T}\,\lambda^S_+(p^\mu) &= \frac{1}{2} i \gamma \mathcal{C} \Big( u_+(p^\mu) + i u_-(p^\mu) + i v_+(p^\mu) + v_-(p^\mu)\Big)\nonumber\\
&= \frac{1}{2} i \gamma \Big( \mathcal{C} u_+(p^\mu) - i \mathcal{C} u_-(p^\mu) - i \mathcal{C} v_+(p^\mu) + \mathcal{C} v_-(p^\mu)\Big).
\end{aligned}$$ The action of the charge conjugation operator $\mathcal{C}$ can now be implemented through equations (\[eq:cuv\]) to obtain $$\mathcal{T}\,\lambda^S_+(p^\mu) =
\frac{1}{2} i \gamma \Big( i v_+(p^\mu) + v_-(p^\mu) + u_+ (p^\mu) + i u_-(p^\mu)\Big).$$ This requires us to note that $\gamma$ acts on the Dirac spinors as follows $$\gamma\, u_\pm(p^\mu) = \mp\, v_\mp(p^\mu),\quad
\gamma\, v_\pm(p^\mu) = \pm\, u_\mp(p^\mu).$$ As a consequence we have $$\begin{aligned}
\mathcal{T}\,\lambda^S_+(p^\mu) & =
\frac{1}{2} i \Big( i u_-(p^\mu) - u_+(p^\mu) - v_- (p^\mu)
+ i v_+(p^\mu\Big)\nonumber\\
& = i \underbrace{\frac{1}{2} \Big( - u_+(p^\mu) + i u_-(p^\mu) + i v_+(p^\mu) - v_- (p^\mu)
\Big)}_{\lambda^A_-(p^\mu)}\end{aligned}$$ That is $$\mathcal{T}\,\lambda^S_+(p^\mu) = i \lambda^A_-(p^\mu)$$ in agreement with (\[eq:timereversalS\]). The check for the remainder of the results in (\[eq:timereversalS\]) and (\[eq:timereversalA\]) follows exactly the same calculational flow.
These calculations provide an intimate dependence of Elko on Dirac spinors, and of Dirac spinors on Elko. Since Elko are superposition of the particle *and* antiparticle Dirac spinors, which in configuration space have different time evolutions, Elko and Dirac spinors cannot be governed by the same Lagrangian density.
A temptation, and a departure\[sec:temptation\]
-----------------------------------------------
One may thus be tempted to suggest that we introduce, instead, a Majorana mass term and treat the components of $\lambda(x)$ as anticommuting numbers (that is, as Grassmann numbers). This, in effect – but with doubtful mathematical justification – would immediately promote a c-number classical field to the quantum field of Majorana and demand that the kinetic term be restored to that of Dirac. In the process we will be forced to abandon Elko as possible expansion coefficients of a quantum field, and return to a field with Dirac spinors as its expansion coefficients $$\begin{aligned}
\psi(x) = & \int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{\sqrt{2 E(\p)}} \nonumber\\ &\times\sum_\sigma \Big[ a_\sigma(\p) u_\sigma(\p) \exp(- i p_\mu x^\mu)
+\, b^\dagger_\sigma(\p) v_\sigma(\p) \exp(i p_\mu x^\mu){\Big]}
\label{eq:DiracField}\end{aligned}$$ which on setting $$b_\sigma^\dagger(\p) = a_\sigma^\dagger(\p),\qquad
{
\mbox{\textrm{\citep{Majorana:1937vz}}}}\label{eq:Majorana1937}$$ and introducing $\mathcal{R}$ and $\mathcal{L}$ transforming components, $\psi_R(x)$ and $\psi_L(x)$, can be written as $$\psi(x) = \left(
\begin{array}{c}
\psi_R(x)\\
\psi_L(x)
\end{array}
\right)$$ with[$$\begin{aligned}
\psi_R(x) = & \int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{\sqrt{2 E(\p)}} \nonumber\\ &\times\sum_\sigma \Big[ a_\sigma(\p) u^R_\sigma(\p) \exp(- i p_\mu x^\mu)
+\, a^\dagger_\sigma(\p) v^R_\sigma(\p) \exp(i p_\mu x^\mu){\Big]}\\
\psi_L(x) = & \int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{\sqrt{2 E(\p)}} \nonumber\\ &\times\sum_\sigma \Big[ a_\sigma(\p) u^L_\sigma(\p) \exp(- i p_\mu x^\mu)
+\, a^\dagger_\sigma(\p) v^L_\sigma(\p) \exp(i p_\mu x^\mu){\Big].}\end{aligned}$$ ]{} On using the identities[$$u^R_\sigma(\p) = \Theta \left[ v^L_\sigma(\p) \right]^\ast,\quad
v^R_\sigma(\p) = \Theta \left[ u^L_\sigma(\p) \right]^\ast$$ ]{} the resulting field would then have the form of a Majorana field $$\psi^M(x) =
\left(
\begin{array}{c}
\Theta \psi^\ast_L(x)\\
\psi_L(x)
\end{array}
\right)$$ where the superscript $M$ symbolises the Majorana condition (\[eq:Majorana1937\]), and the $\ast$ symbol on $\psi^\ast_L(x)$ complex conjugates without transposing, and takes $a_\sigma(\p)$ to $a^\dagger_\sigma(\p)$, and vice versa. While it has a striking formal resemblance to Elko its physical and mathematical content is very different. Elko is a complex number valued four component spinor. It has certain transformation properties. It has null norm under the Dirac dual, and this fact is not enough to warrant to endow it with a Grassmann character. The $\psi^M(x)$ is a quantum field with complex number valued four component Dirac spinors as its expansion coeeficients, it has its own transformation properties. Its Grassmann character arises from interpreting the creation and annihilation operators as anticommuting fermionic operators. For Elko the $\mathcal{R}$ and $\mathcal{L}$ transforming components have opposite helicities, no such interpretation can be associated to $\mathcal{R}$ and $\mathcal{L}$ transforming components of $\psi^M(x)$.
If we do not fall into this temptation for Elko an unexpected theoretical result follows that naturally leads us to a new class of fermions of spin one half. The question on the path of departure is deceptively simple: If the Dirac dual $$\overline{\psi}(p^\mu)=\left[\psi(p^\mu)\right]^\dagger\gamma_0 \label{eq:Dirac-dual}$$ was not given how shall we go about deciphering it? And, is this a unique dual, or is there a freedom in its definition?, and what physics does it encode?
These questions are rarely asked in the physics literature. An exception for the definition, called a “convenience” by Weinberg, is to note that the counterpart of $\Lambda$ in (\[eq:Minkowski-boost-rotation\]) for the $\mathcal{R}\oplus\mathcal{L}\vert_{s=1/2}$ representation space $$D(\Lambda) \stackrel{\textrm{def}}{=}\left\{
\begin{array}{cl}
\exp\left(i\kb\cdot\vp\right) & \mbox{for Lorentz boosts} \\
\exp\left(i\bz\cdot\vt\right) & \mbox{for rotations}
\end{array}\right.$$ is not unitary, but pseudounitary[^27] $$\gamma_0 D(\Lambda)^\dagger \gamma_0 = D(\Lambda)^{-1}.\label{eq:pseudounitarity}$$ Weinberg uses this observation to motivate the definition (\[eq:Dirac-dual\]).
For Elko the completion of (\[eq:nullnorm\]) and (\[eq:i2m\]) is given by $$\begin{aligned}
&\overline\lambda^S_\pm(p^\mu) \lambda^S_\pm(p^\mu) = 0, \quad \overline\lambda^S_\pm(p^\mu) \lambda^A_\pm(p^\mu) = 0,\quad
\overline\lambda^S_\pm(p^\mu) \lambda^A_\mp(p^\mu) =0 \label{eq:norm-a}\\
&\overline\lambda^A_\pm(p^\mu) \lambda^A_\pm(p^\mu) = 0,\quad \bar\lambda^A_\pm(p^\mu) \lambda^S_\pm(p^\mu) = 0,\quad
\overline\lambda^A_\pm(p^\mu) \lambda^S_\mp(p^\mu) =0 \label{eq:norm-b} \end{aligned}$$ and $$\overline\lambda^S_\pm(p^\mu) \lambda^S_\mp(p^\mu) = \mp \,2 i m,\quad
\overline\lambda^A_\pm(p^\mu) \lambda^A_\mp(p^\mu) = \pm \,2 i m. \label{eq:norm-c}$$ Because of (\[eq:norm-c\]), we can define – see below – a new dual and make it convenient to formulate and calculate the physics of quantum fields with Elko as their expansion coefficients. Once a lack of uniqueness of the Dirac dual is discovered, the new way to accommodate the pseudounitarity captured by (\[eq:pseudounitarity\]) is introduced. It opens up concrete new possibilities to go beyond the Dirac and Majorana fields for spin one half fermions without violating Lorentz covariance and without introducing non-locality. The programme now is as follows:
- Introduce a new dual so that not only the norm of Elko is Lorentz invariant (and $\in \Re$), but also the spins sums.
- Define a quantum field in terms of Elko, with the creation and annihilation operators satisfying the usual fermionic anticommutators.
- Using the new dual, define a new adjoint for the quantum field. Calculate the vacuum expectation value of the time ordered product of the field and its adjoint.
- From the Feynman-Dyson propagator thus obtained, decipher the mass dimensionality of the new field, and check the Schwinger locality by calculating the relevant anticommutators.
With the exception of this chapter, we consciously refrained from introducing a dual of the spinors. The reason was simple. The Elko had to be obtained in a linear fashion. Its full poetry and beauty revealed without the shadows of the intricacies of the dual space. The inevitability of the final result hinted, and exposed, in its utter simplicity. That done we could then take our reader through the more amorphous cloud of duals and adjoints, to establishing the hint in its full maturing, knowing that others may refine it further but not alter its essence.
This shared with the reader, we proceed to developing the theory of dual spaces. Its flow holds as good for the representation space of our immediate interest as for any other representation space. But by confining to the indicated space we are able to make our argument less heavy, notationally. Beginning in the next chapter the reader would learn the deeper structure from which (\[eq:Dirac-dual\]) arises, and why there is no proper metric for Weyl spinors, and for $\mathcal{R}$ and $\mathcal{L}$ representation spaces, separately, in general. We would also learn to construct the metric for many other representation spaces. In particular, we would easily see how the Lorentz algebra – without reference to any other fact – gives the metric for the Minkowski space, with the twist of a multiplicative phase factor. This phase may help us to understand four-vector spaces at a deeper quantum level. In the process we will discover how to construct the needed metric, with symmetries as the starting point.
An *ab initio* journey into duals {#ch12}
=================================
Motivation and a brief outline
------------------------------
A $n$ dimensional representation space may be spanned by $n$ independent vectors $\{\zeta_1,\zeta_2,\ldots\zeta_n\}$. These could be Weyl spinors, four-component spinors, four-vectors, or ‘vectors’ spanning any representation. Our task is to introduce a unified approach to constructing dual spaces, and a metric that helps define bi-linear invariants.
Our starting point is Lorentz algebra, and the representation spaces on which the symmetry transformations act through exponentiation of the generators in the sense defined in the opening chapters. Our motivation to look at the duals *ab initio* resides in the null norm of Elko under the Dirac dual.
While considering a representation space, we look at each of the basis vectors $\zeta_\alpha$, $\alpha=1,2, \ldots n$, as a complex number valued column vector. Our task is to define a dual that allows us to construct scalars (or more generally bilinears), that are invariant (covariant) under a set of physically interesting symmetry transformations. These, besides the Lorentz transformations, would include the discrete transformations, of parity, of time reversal, and of charge conjugation. For this task we generate a mechanism that transforms each of the $\zeta_\alpha$ into a row. We identify this mechanism, in part, with the complex conjugation of each of the $\zeta_\alpha$ and transposing it. In addition, motivated by the null norm of Elko under the Dirac dual, we introduce a pairing operator that uniquely pairs (with the freedom to multiply with an $\alpha$ dependent phase factor) each of these newly generated rows in an invertible manner with each of the $\zeta_\alpha$. Say, the $\alpha$th column with the $\beta$th row and generate the set of row vectors, $\{{\zeta}^\prime_1,{\zeta}^\prime_2,\ldots{\zeta}^\prime_n\}$ $${\zeta}^\prime_\alpha \stackrel{\textrm{def}}{=} \left(\Xi \,\zeta_\alpha\right)^\dagger\label{eq:dualdef}$$ where we have taken the liberty of first pairing and then implementing the complex conjugation and transposition. Of $\Xi$ we require that its inverse exists, and that its square is an identity operator, $\Xi^2 = \I_n$.
To construct scalars from the ${\zeta}^\prime_\alpha$ and $\zeta_\alpha$ we introduce a $n\times n$ matrix $\eta$, and define a dual vector, or just a dual $${\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\zeta_\alpha}}}\stackrel{\textrm{def}}{=} {\zeta}^\prime_\alpha \eta$$ We call $\eta$ the metric for the representation space under consideration. It is constrained to keep the bi-linear product $$\chi_\alpha= {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\zeta_\alpha}}}\,\zeta_\alpha$$ invariant under a chosen set of transformation, or symmetries. These constraints, as we will discover, generally take the form of commutators/anticommutators of $\eta$, with the relevant generators of the symmetry algebra, to vanish.
*Additional constraints *
It turns out that the dual as defined above still has an element of freedom. This freedom we shall discuss in detail.
It is this general procedure that we now implement for the spinor spaces.
The dual of spinors: constraints from the scalar invariants \[sec:constraint-from-scalar-invariants\]
-----------------------------------------------------------------------------------------------------
Consider a general set of $4$-component massive spinors $\varrho(p^\mu)$. We would like these to be orthonormal under the dual we are seeking. These do not have to be eigenspinors of $\mathcal{P}$, that is, Dirac spinors, or the eigenspinors of $\mathcal{C}$ – that is, Elko. As a specific implementation of the preceding discussion, we examine a general form of the dual defined as $${\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\varrho}}}_\alpha(p^\mu) \stackrel{\rm def}{=}{\big[}\Xi(p^\mu) \, \varrho_\alpha(p^\mu){\big]}^\dagger \eta
\label{eq:gendual}$$ where $\eta$ is a $4\times 4$ matrix, with its elements $\in \mathbb{C}$. The task of $\Xi(p^\mu)$ is to take any one of the $\varrho_\alpha(p^\mu)$ and transform it, up to a phase, into one of the spinors $\varrho_{\alpha^\prime}(p^\mu)$ from the same set. It is not necessary that the indices $\alpha^\prime$ and $\alpha$ be the same. We require $\Xi(p^\mu)$ to define an invertible map, with $\Xi^2 = \I_4$ (possibly, up to a phase).
The Dirac and Elko dual: a preview
----------------------------------
The Dirac dual arises when we identify the pairing matrix with the identity matrix $$\Xi(p^\mu) =\I_4\label{eq:Dirac-Xi}$$ for which $\beta =\alpha$ $$\varrho_\alpha(p^\mu)\to\varrho_\alpha(p^\mu),\quad\forall\alpha$$ and identifying $\varrho_\alpha(p^\mu)$ with the eigenspinors of the parity operator, that is with the $u_\sigma(p^\mu)$ and $v_\sigma(p^\mu)$.
The starting point for the Elko dual goes back to an early unpublished preprint before it was fully realised that these provide expansion coefficients for a mass dimension one fermionic field of spin one half [@Ahluwalia:2003jt]. The hint arose from the results found here in equations (\[eq:norm-a\]) to (\[eq:norm-c\]). That early choice was shown by Sperança to arise from the pairing matrix [@Speranca:2013hqa] $$\begin{aligned}
\Xi(p^\mu) \stackrel{\rm def}{=} \frac{1}{2 m} \sum_{\alpha=\pm}
\Big[\lambda^S_\alpha(p^\mu)\bar\lambda^S_\alpha(p^\mu) - \lambda^A_\alpha(p^\mu)\bar\lambda^A_\alpha(p^\mu) \Big].\label{eq:Xi}
\end{aligned}$$ It induces the needed map $$\begin{aligned}
\lambda^S_+(p^\mu)& \to i \lambda^S_-(p^\mu)\label{eq:map1}\\
\lambda^S_-(p^\mu)& \to- i \lambda^S_+(p^\mu)\label{eq:map2}\\
\lambda^A_+(p^\mu)& \to - i \lambda^A_-(p^\mu)\label{eq:map3}\\
\lambda^A_-(p^\mu)& \to+ i \lambda^S_+(p^\mu).\label{eq:map4}\end{aligned}$$ if $\varrho_\alpha(p^\mu)$ are identified with Elko, $\lambda_\alpha(p^\mu)$. That is $$\begin{aligned}
\Xi(p^\mu) \lambda^S_\pm(p^\mu) &= \pm i \lambda^S_\mp(p^\mu)\label{eq:mapS}\\
\Xi(p^\mu) \lambda^A_\pm(p^\mu) &= \mp i \lambda^A_\mp(p^\mu). \label{eq:mapA}\\end{aligned}$$
Constraints on the metric from Lorentz, and discrete, symmetries
----------------------------------------------------------------
We now wish to determine the metric $\eta$, and $\Xi(p^\mu)$ explicitly. We will see that the choices (\[eq:Dirac-Xi\]), (\[eq:mapS\]) and (\[eq:mapA\]), are indeed allowed. For the boosts, the requirement of a Lorentz invariant norm translates to $$\underbrace{\Big[\Xi(k^\mu)\varrho(k^\mu)\Big]^\dagger \eta \,\varrho(k^\mu) }_{{\chi(k^\mu)}}
= \underbrace{\Big[\Xi(p^\mu)\varrho(p^\mu)\Big]^\dagger \eta \,\varrho(p^\mu)}_{\chi(p^\mu)}
\label{eq:boostdemand}$$ with a similar expression for the rotations. Expressing $\varrho(p^\mu)$ as $\exp(i \kb\cdot\vp) \varrho(k^\mu)$, and using $\kb^\dagger = -\kb$ (for an explicit form of $\bm{\kappa}$ see Eq. (\[eq:pi\])), the right-hand side of the above expression can be re-written as $$\Big[\Xi(p^\mu)\varrho(p^\mu)\Big]^\dagger \eta \,\varrho(p^\mu)
= \varrho^\dagger(k^\mu)\, e^{i\boldsymbol{\kappa\cdot\varphi}} \, \Xi^\dagger(p^\mu)\, \eta\, e^{i\boldsymbol{\kappa\cdot\varphi}}\, \varrho(k^\mu).\label{eq:breq}$$ Since $$\Xi(p^\mu) = e^{i\boldsymbol{\kappa\cdot\varphi}}\, \Xi(k^\mu)\, e^{-i\boldsymbol{\kappa\cdot\varphi}} \label{eq:XiTransformation-b}$$ the $\Xi^\dagger(p^\mu)$ evaluates to $$\Xi^\dagger(p^\mu) \rightarrow e^{i\boldsymbol{\kappa^\dagger\cdot\varphi}}\, \Xi^\dagger(k^\mu)\, e^{-i\boldsymbol{\kappa^\dagger\cdot\varphi}}
= e^{- i\boldsymbol{\kappa\cdot\varphi}}\, \Xi^\dagger(k^\mu)\, e^{i \boldsymbol{\kappa\cdot\varphi}} .$$ Using this result in (\[eq:breq\]) yields $$\begin{aligned}
\Big[\Xi(p^\mu)\varrho(p^\mu)\Big]^\dagger \eta \,\varrho(p^\mu)
&=
\varrho^\dagger(k^\mu) \Xi^\dagger(k^\mu) e^{i\boldsymbol{\kappa\cdot\varphi}} \eta \,e^{i\boldsymbol{\kappa\cdot\varphi}}\varrho(k^\mu) \nonumber \\
&=
\Big[\Xi(k^\mu) \varrho(k^\mu)\Big]^\dagger e^{i\boldsymbol{\kappa\cdot\varphi}} \eta \,e^{i\boldsymbol{\kappa\cdot\varphi}}\varrho(k^\mu).
\end{aligned}$$ Using the just derived result into the right hand side of (\[eq:boostdemand\]) gives $$\Big[\Xi(k^\mu)\varrho(k^\mu)\Big]^\dagger \eta \,\varrho(k^\mu)
=
\Big[\Xi(k^\mu) \varrho(k^\mu)\Big]^\dagger e^{i\boldsymbol{\kappa\cdot\varphi}} \eta \,e^{i\boldsymbol{\kappa\cdot\varphi}}\varrho(k^\mu).$$ Up to a caveat to be noted below in section \[sec:Aremark\], it gives the constraint $$\eta = e^{i\boldsymbol{\kappa\cdot\varphi}} \eta \,e^{i\boldsymbol{\kappa\cdot\varphi}} \label{eq:bc}$$ That is, the metric $\eta$ must anticommute with each of boost generators $\kb$ $$\{\kappa_i,\eta\}= 0,\qquad i = x,y,z. \label{eq:eta-boost}$$ Implementing the above constraint with $i=z,y,z$ in succession reduces $\eta$ to have the form $$\left(\begin{array}{cccc}
0 & 0 & a+i b & 0\\
0 & 0& 0 & a+ i b \\
c + i d & 0 & 0 & 0\\
0 & c+i d & 0 & 0
\end{array}\right)\label{eq:etaafterboost}$$ where $a,b,c,d \in \Re$. This is valid for all four-component spinors.
### A freedom in the definition of the metric {#sec:Aremark}
Strictly speaking the equality in (\[eq:bc\]) is up to a freedom of an operator which carries the $\varrho(p^\mu)$ as its eigenspinor with eigenvalue unity: $$\eta = e^{i\boldsymbol{\kappa\cdot\varphi}} \eta \,e^{i\boldsymbol{\kappa\cdot\varphi}} \Gamma$$ with $\Gamma\varrho(p^\mu) = \varrho(p^\mu)$. Existence of $\Gamma$ operator(s) is an important freedom as it may reduce the symmetries under consideration, possibly to a subgroup of Lorentz.
Apart from our Elko experience [@Ahluwalia:2010zn] that we shall further share with our reader in this chapter, the above remark arises from a 2006 Cohen and Glashow observation [@Cohen:2006ky] that, “Indeed, invariance under HOM(2), rather than (as is often taught) the Lorentz group, is both necessary and sufficient to ensure that the speed of light is the same for all observers, and inter alia, to explain the null result to the Michelson-Morley experiment and its more sensitive successors.” These subgroups are defined by the four algebras summarised in Table 11.1. The generators $$\mathfrak{T}_1\stackrel{\textrm{def}}{=}\mathfrak{K}_x + \mathfrak{J}_y, \quad \mathfrak{T}_2\stackrel{\textrm{def}}{=}
\mathfrak{K}_y - \mathfrak{J}_x$$ form the group T(2) – a group isomorphic to the group of translations in a plane.
Following Cohen and Glashow a relativity formed by adjoining four spacetime translations with any of these groups is termed Very Special Relativity (VSR). While each of the four VSRs have distinct physical characters they all share the remarkable defining property that incorporation of either $P$, $T$, $CP$, or $CT$ enlarges these subgroups to the full Lorentz group. At present, VSR does not violate any of the existing tests of Special Relativity and provides an unexpected realm for the investigation of spacetime symmetries including violation of discrete symmetries, and isotropy of space.
\[tab:VSR\]
Name Generators Sub algebra
--------------------- --------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
$ \mathfrak{t}(2)$ $\mathfrak{T}_1,\mathfrak{T}_2$ $\left[\mathfrak{T}_1,\mathfrak{T}_2\right]=0$
$\mathfrak{e}(2)$ $\mathfrak{T}_1,\mathfrak{T}_2,\mathfrak{J}_z$ $\left[\mathfrak{T}_1,\mathfrak{T}_2\right]=0$, $\left[\mathfrak{T}_1,\mathfrak{J}_z\right]= -i \mathfrak{T}_2$, $\left[\mathfrak{T}_2,\mathfrak{J}_z\right]= i \mathfrak{T}_1$
$\mathfrak{hom}(2)$ $\mathfrak{T}_1,\mathfrak{T}_2,\mathfrak{K}_z $ $\left[\mathfrak{T}_1,\mathfrak{T}_2\right]=0$, $\left[\mathfrak{T}_1,\mathfrak{K}_z\right]= i \mathfrak{T}_1$, $\left[\mathfrak{T}_2,\mathfrak{K}_z\right]= i \mathfrak{T}_2$
$\mathfrak{sim}(2)$ $\mathfrak{T}_1,\mathfrak{T}_2,\mathfrak{J}_z,\mathfrak{K}_z$ $\left[\mathfrak{T}_1,\mathfrak{T}_2\right]=0$, $\left[\mathfrak{T}_1,\mathfrak{K}_z\right]= i \mathfrak{T}_1$, $\left[\mathfrak{T}_2,\mathfrak{K}_z\right]= i \mathfrak{T}_2$
$\left[\mathfrak{T}_1,\mathfrak{J}_z\right]= -i \mathfrak{T}_2$, $\left[\mathfrak{T}_2,\mathfrak{J}_z\right]= i \mathfrak{T}_1$, $\left[\mathfrak{J}_z,\mathfrak{K}_z\right]=0$
: Algebras associated with the four VSR subgroups[]{data-label="table2new"}
A detailed discussion of VSR can be found in a Masters thesis by Gustavo Salinas De Souza [@Gustavo:Thesis:2005] and in the context of Elko in [@Ahluwalia:2010zn]. Our analysis of Elko provides a concrete example of a VSR quantum field that exhibits non-locality governed by a VSR operator $$\mathcal{G}(p^\mu) \stackrel{\textrm{def}}{=}
\left(\begin{array}{cccc}
0 & 0 & 0 & -i e^{-i\phi} \\
0 & 0 & i e^{i \phi} & 0 \\
0 & -i e^{-i \phi} & 0 & 0\\
i e^{i\phi} & 0 & 0 & 0
\end{array}\right). \label{eq:Gdef}$$ This result is consistent with the claim of Cohen and Glashow that a departure from SR to VSR introduces non-locality. In its character $\mathcal{G}(p^\mu) $ is similar to $\Gamma$ becuase $
\mathcal{G}(p^\mu)\, \lambda^S(p_\mu) = \, \lambda^S(p_\mu), \mbox{and}~
\mathcal{G}(p^\mu)\, \lambda^A(p_\mu) = - \, \lambda^A(p_\mu)
$.
As a parenthetic remark we note that Nakayama shows a way to circumvent the no-go locality result of VSR [@Nakayama:2018fib; @Nakayama:2017eof] while Ilderton shows how non-locality and VSR symmetries appear on averaging observables over rapid field oscillations of a Lorentz covariant theory [@Ilderton:2016rqk]. Cheng-Yang Lee circumvents the non-locality problem by introducing a configuration space $\mathcal{G}(p^\mu)$ at the cost of introducing fractional derivatives [@Lee:2014opa].
Given our experience with Elko we conjecture that $\Gamma$ encodes a theoretical mechanism of Lorentz symmetry breaking in a generic manner to the subgroups summarised in Table 11.1. After we have developed the notion of Elko dual further, we shall pick up this thread in section \[Sec:IUCAA\]
### The Dirac dual\[sec:Dirac-dual\]
The simplest choice for $\Xi(p^\mu)$ is the identity operator as in (\[eq:Dirac-Xi\]). We will now show that the well-known Dirac dual corresponds to this choice. With $\Xi(p^\mu) = \openone_4$, the counterpart of (\[eq:boostdemand\]) for rotations reads $$\varrho^\dagger(p_\mu) \, \eta \,\varrho(p_\mu)
= \varrho^\dagger(p^\prime_\mu)\, \eta \,\varrho(p^\prime_\mu)
\label{eq:rotationdemand}$$ where $\varrho(p^\prime_\mu) = e^{i\bm{\zeta}\cdot\bm{\theta}}\varrho(p_\mu)$. Using $\bm{\zeta}^\dagger
=\bm{\zeta}$ (for an explicit form of $\bm{\zeta}$ see Eq. (\[eq:pi\])) translates (\[eq:rotationdemand\]) to $$\varrho^\dagger(p_\mu) \, \eta \,\varrho(p_\mu)
=
\varrho^\dagger(p_\mu) \, e^{-i \bm{\zeta}\cdot{\bm{\theta}}}\, \eta
e^{i \bm{\zeta}\cdot{\bm{\theta}}}\,\varrho(p_\mu).$$ Modulo a freedom noted in section \[sec:Aremark\], it gives the constraint $$\eta=
e^{-i \bm{\zeta}\cdot{\bm{\theta}}}\, \eta
e^{i \bm{\zeta}\cdot{\bm{\theta}}} .$$ That is, the metric $\eta$ must commute with each of rotation generators $\bm{\zeta}$ $$\left[{\zeta}_i,\eta\right] =0,\qquad i = x,y,z. \label{eq:eta-rotation}$$ The constraint that $\eta$ must commute with $\bm{\zeta}_i$ does not reduce $\eta$ constrained by its anticommutativity with each of the boost generators, $\kappa_i$.
However, following an analysis paralleling the two previous calculations the demand for the norm to be invariant under the parity transformation $\mathcal{P}$ is readily obtained to be $$\begin{aligned}
\eta = m^{-2} \left[ \gamma_\mu p^\mu\right]^\dagger \eta \, \gamma_\mu p^\mu
\label{eq:parityconstraint}\end{aligned}$$ A direct evaluation of the right hand side of the parity constraint (\[eq:parityconstraint\]) with $\eta$ given by (\[eq:etaafterboost\]) gives the result $$m^{-2} \left[ \gamma_\mu p^\mu\right]^\dagger \eta \, \gamma_\mu p^\mu
= \left(\begin{array}{cccc}
0 & 0 & c+i d & 0\\
0 & 0& 0 & c+ i d \\
a + i b & 0 & 0 & 0\\
0 & a+i b & 0 & 0
\end{array}\right)$$ thus requiring $c=a$, and $d=b$. In consequence, the parity constraint further reduces the metric $\eta$ to $$\eta= \left(\begin{array}{cccc}
0 & 0 & a+i b & 0\\
0 & 0& 0 & a+ i b\\
a + i b & 0 & 0 & 0\\
0 & a+i b & 0 & 0
\end{array}\right)$$ Requiring the norm of the Dirac spinors to be real forces the choice $b=0$, and then $a$ is simply a scale factor. It may be chosen to be unity $$\eta= \left(\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0& 0 & 1\\
1& 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{array}\right)$$ \[eq:etacanonicalZimpok\] We thus reproduce the canonical Dirac dual: it is defined by the choice of $\Xi(p^\mu)=\openone_4$, and the constraints on $\eta$ given by (\[eq:eta-boost\]), (\[eq:eta-rotation\]) and (\[eq:parityconstraint\]).
On the path of our departure we learn that the Dirac dual has additional underlying structure. In particular, it gives us a choice to violate parity, or to preserve it, depending on whether we choose $a/c$ in $\eta$ of (\[eq:etaafterboost\]) to be unity, or different from unity. With the former choice we reproduce the standard result (\[eq:Dirac-dual\]), while the latter choice gives us a first-principle control on the extent to which parity may be violated in nature, or in a given physical process.
The Elko dual\[sec:elko-dual\]
------------------------------
The dual for $\lambda(p^\mu)$ first introduced in [@Ahluwalia:2003jt], and refined in [@Ahluwalia:2008xi; @Ahluwalia:2009rh] can now be more systematically understood by observing that those results correspond to the $\Xi(p^\mu)$ of the second example considered above (and given in equation (\[eq:Xi\])). Expression (\[eq:Xi\]) can be evaluated to yield a compact form $$\Xi(p^\mu) =
m^{-1} \mathcal{G}(p^\mu) \gamma_\mu p^\mu.
\label{eq:XizZmpok}$$ where $\mathcal{G}(p^\mu)$ is defined in equation (\[eq:Gdef\]). Out of the four variables $m$, $p$, $\theta$ and $\phi$ that enter the definition $$p^\mu = (E, p \sin\theta\cos\phi,p\sin\theta\sin\phi,p\cos\theta)$$ $\mathcal{G}(p^\mu) $ depends only on $\phi$. The analysis of the boost constraint remains unaltered, with the result that we still have $$\{\kb_i,\eta\}= 0,\quad i = x,y,z . \label{eq:eta-boost-b}$$ The analysis for the rotation constraint changes. It begins as $$\Big[m^{-1} \mathcal{G}(p^\mu) \gamma^\mu p_\mu \, \lambda(p_\mu)\Big]^\dagger \,\eta\, \lambda(p_\mu)
=
\Big[m^{-1} \mathcal{G}(p^{\prime\mu}) \gamma^\mu p^\prime_\mu \, \lambda(p^\prime_\mu)\Big]^\dagger \,\eta\, \lambda(p^\prime_\mu)$$ where $\lambda(p_\mu)$ represents any of the four $\lambda^{S,A}_\pm(p^\mu)$, and the primed quantities refer to their rotation-induced counterparts. The above expression simplifies on using the following identities $$\mathcal{G}(p^\mu)\, \lambda(p_\mu) = \pm \, \lambda(p_\mu),\qquad [\mathcal{G}(p^\mu),\,\gamma^\mu p_\mu ]=0,\label{eq:important}$$ where the upper sign in the first equation above holds for $\lambda^S(p_\mu)$ and the lower sign is for $\lambda^A(p_\mu)$. The result is $$\Big[ \gamma^\mu p_\mu \, \lambda(p_\mu)\Big]^\dagger \,\eta\, \lambda(p_\mu)
=
\Big[\gamma^\mu p^\prime_\mu \, \lambda(p^\prime_\mu)\Big]^\dagger \,\eta\, \lambda(p^\prime_\mu).\label{eq:rotation-constraint-b}$$ Expressing $\lambda(p^{\prime\mu})$ as $e^{i \bm{\zeta}\cdot\bm{\theta}} \lambda(p^\mu)$, and using $\bm{\zeta}^\dagger
=\bm{\zeta}$, the right-hand side of the above expression can be written as $$\Big[\gamma^\mu p^\prime_\mu \, \lambda(p^\prime_\mu)\Big]^\dagger \,\eta\, \lambda(p^\prime_\mu) =
\lambda^\dagger(p_\mu)\,
e^{-i\bm{\zeta}\cdot\bm{\theta}}
\left(\gamma^\mu p^\prime_\mu \right)^\dagger\,\eta\,
e^{i\bm{\zeta}\cdot\bm{\theta}} \,\lambda(p_\mu). \label{eq:extra}$$ On taking note that $$\gamma^\mu p^\prime_\mu = e^{i\bm{\zeta}\cdot\bm{\theta}} \gamma^\mu p_\mu
e^{-i\bm{\zeta}\cdot\bm{\theta}}$$ equation (\[eq:extra\]) becomes $$\Big[\gamma^\mu p^\prime_\mu \, \lambda(p^\prime_\mu)\Big]^\dagger \,\eta\, \lambda(p^\prime_\mu) =
\Big[\gamma^\mu p_\mu\,\lambda(p_\mu)\Big]^\dagger\,
e^{-i\bm{\zeta}\cdot\bm{\theta}}
\eta\,
e^{i\bm{\zeta}\cdot\bm{\theta}} \,\lambda(p_\mu)$$ Comparing the above expression with (\[eq:rotation-constraint-b\]) gives the constraint $$\eta = e^{-i\bm{\zeta}\cdot\bm{\theta}}
\eta\,
e^{i\bm{\zeta}\cdot\bm{\theta}}$$ That is, the metric $\eta$ must commute with each of rotation generators $\bm{\zeta}$ $$\left[\zeta_i,\eta\right] = 0 ,\quad i = x,y,z \label{eq:eta-rotation-c}$$ Despite a non-trivial $\Xi(p^\mu)$, this is the same result as before.
It is readily seen that $\left[\Xi(p^\mu)\right]^2=\I$ and $\left[\Xi(p^\mu)\right]^{-1}$ indeed exists and equals $\Xi(p^\mu)$ itself. Thus, the dual for $\lambda(p^\mu)$ is defined by the choice of $\Xi(p^\mu)$ given by (\[eq:XizZmpok\]).
So far the constraints on $\eta$ turn out to be same as for the Dirac dual. To distinguish it from other possibilities the new dual at the intermediate state of our calculations is represented by $${\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}_\alpha(p^\mu) \stackrel{\rm def}{=} {\big[}\Xi(p^\mu)\, \lambda_\alpha(p^\mu){\big]}^\dagger \eta
\label{eq:dual-b}$$ with $\eta$ given by (\[eq:etacanonicalZimpok\]). This choice of $\eta$ is purely for convenience at the moment. If the new particles to be introduced here are indeed an element of the physical reality then the ratio $a/c$ must not be set to unity but determined by appropriate observations/experiments.
Using (\[eq:mapS\]) and (\[eq:mapA\]), (\[eq:dual-b\]) translates to $$\begin{aligned}
{\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_+(p^\mu) &= - i \left[\lambda^S_-(p^\mu)\right]^\dagger \eta \nonumber\\
{\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_-(p^\mu) &= i \left[\lambda^S_+(p^\mu)\right]^\dagger\eta\nonumber\\
{\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_+(p^\mu) &= i \left[\lambda^A_-(p^\mu)\right]^\dagger\eta\nonumber \\
{\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_-(p^\mu) &= - i \left[\lambda^A_+(p^\mu)\right]^\dagger\eta
\label{eq:otherd-3}\end{aligned}$$
We can thus rewrite the results (\[eq:norm-a\]), (\[eq:norm-b\]), and (\[eq:norm-c\]) into the following orthonormality relations $$\begin{aligned}
& {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \lambda^S_{\alpha^\prime}(p^\mu)
= 2 m \delta_{\alpha\alpha^\prime}\nonumber \\
& {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \lambda^A_{\alpha^\prime}(p^\mu)
= - 2 m \delta_{\alpha\alpha^\prime} \nonumber \\
& {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \lambda^A_{\alpha^\prime}(p^\mu) = 0 =
{\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \lambda^S_{\alpha^\prime}(p^\mu)\label{eq:zimpokJ9c}\end{aligned}$$ with $\alpha$ and $\alpha^\prime = \pm$.
The dual of spinors: constraint from the invariance of the Elko spin sums \[sec:constraints-from-the-spin-sums\]
----------------------------------------------------------------------------------------------------------------
The Dirac spin sums $$\begin{aligned}
\sum_{\sigma=+,-} u_\sigma(p^\mu)\overline{u}_\sigma(p^\mu) & = \gamma_\mu p^\mu + m \I_4 \label{eq:spinsumsDiracu}\\
\sum_{\sigma=+,-} v_\sigma(p^\mu)\overline{v}_\sigma(p^\mu) & = \gamma_\mu p^\mu - m \I_4 \label{eq:spinsumsDiracv}\end{aligned}$$ immediately follow from the definition of the Dirac dual and expressions for the $u_\pm(p^\mu)$ and $v_\pm (p^\mu)$ given in equations (\[eq:upp\]) to (\[eq:vm\]). These are covariant under Lorentz symmetries and as such require no further scrutiny of the Dirac dual – that is, as far as the construction of the dual is concerned.
The spin sums (\[eq:spinsumsDiracu\]) and (\[eq:spinsumsDiracv\]) yield the completeness relations for the Dirac spinors $$\frac{1}{2 m}\sum_{\sigma=+,-}
\Big(
u_\sigma(p^\mu)\overline{u}_\sigma(p^\mu) -
v_\sigma(p^\mu)\overline{u}_\sigma(p^\mu) \Big) = \I_4$$
With $\lambda^S_\alpha(p^\mu)$ and $\lambda^A_\alpha(p^\mu)$ given by equations (\[eq:lsp\]) to \[eq:lam\]), and their respective duals defined by equations $(\ref{eq:otherd-3})$, the spin sums for Elko $$\sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \;\;{\rm and} \;\;
\sum_{\alpha}\lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu)$$ are completely defined. The first of the two spin sums evaluates to $$\begin{aligned}
& i\; \underbrace{ \left[ \frac{E + m}{2 m}
\left(1 - \frac{{p^2}}{(E+m)^2}\right)\right]}_{=1} \nonumber\\ &\hspace{51pt}\times\underbrace{\bigg( - \lambda^S_+(k^\mu) \left[\lambda^S_-(k^\mu) \right]^\dagger
+ \lambda^S_-(k^\mu) \left[\lambda^S_+(k^\mu)\right]^\dagger
\bigg)\eta}_{= -i m \left[ \I_4 + \mathcal{G}(p^\mu) \right]}. \end{aligned}$$ The second of the spin sums can be evaluated in exactly the same manner. The combined result is $$\begin{aligned}
\index{Spin sums!Elko}
\sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) & = +
m \big[\I_4 + \mathcal{G}(p^\mu) \big] \label{eq:sss}\\
\sum_{\alpha} \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) & =
- m \big[\I_4 - \mathcal{G}(p^\mu)\big] \label{eq:ssa}\end{aligned}$$ with $\mathcal{G}(p^\mu)$ as in Eq. (\[eq:Gdef\]). These spin sums have the eigenvalues $\{0,0,2m,2m\}$, and $\{0,0,-2m,-2m\}$, respectively. Since eigenvalues of projectors must be either zero or one [@Weinberg:2012qm Section 3.3], we define $$\begin{aligned}
P_S &\stackrel{\textrm{def}}{=}\frac{1}{2 m} \sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu)= \frac{1}{2} \big[\I_4 + \mathcal{G}(p^\mu) \big] \label{eq:s}
\\
P_{A}&\stackrel{\textrm{def}}{=} -\frac{1}{2 m} \sum_{\alpha} \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) = \frac{1}{2} \big[\openone_4 -\mathcal{G}(p^\mu)\big] \label{eq:a}\end{aligned}$$ and confirm that indeed they are projectors and furnish the completeness relation $$P_S^2 =P_S,\quad P_A^2 = P_A,\quad P_S+P_A = \openone_4. \label{eq:compl}$$
Because $\mathcal{G}(p^\mu)$ is not Lorentz covariant its appearance in the spin sums violates Lorentz symmetry.
Till late 2015, all efforts to circumvent this problem had failed and gave rise to a suggestion that the formalism can only be covariant under a subgroup of the Lorentz group suggested by Cohen and Glashow [@Cohen:2006ky; @Ahluwalia:2010zn].
The IUCAA breakthrough
----------------------
\[Sec:IUCAA\]
However, during a set of winter-2015 lectures I gave on mass dimension one fermions at the Inter-University Centre for Astronomy and Astrophysics (IUCAA) it became apparent that there is a freedom in the definition of the dual.[^28] It allows a re-definition of the dual in such a way that the Lorentz invariance of the orthonormality relations remains intact, but it restores the Lorentz covariance of the spin sums. In fact it makes them invariant [@Ahluwalia:2016rwl].
The basic idea is already discussed briefly in section \[sec:Aremark\]. Since the construction of Elko dual was not sufficiently developed yet, our discussion, of necessity, was framed in terms of Elko, rather than its dual. Now, it is possible to bypass that limitation and explore the solution to Lorentz symmetry breaking by considering the following re-definition of the Elko dual $${\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \to {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) = {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \mathcal{A},\quad
{\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \to {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) = {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \mathcal{B}\label{eq:ab}$$ with $\mathcal{A}$ and $\mathcal{B}$ constrained to have the following non-trivial properties: The $\lambda^S_\alpha(p^\mu)$ must be eigenspinors of $\mathcal{A}$ with eigenvalue unity, and similarly $\lambda^A_\alpha(p^\mu)$ must be eigenspinors of $\mathcal{B}$ with eigenvalue unity$$\mathcal{A} \lambda^S_\alpha(p^\mu) = \lambda^S_\alpha(p^\mu),\quad
\mathcal{B} \lambda^A_\alpha(p^\mu) = \lambda^A_\alpha(p^\mu),\label{eq:4jan-a}\\$$ and additionally $\mathcal{A}$ and $\mathcal{B}$ must be such that $${\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu)\mathcal{A} \lambda^A_{\alpha^\prime}(p^\mu)=0,\quad {\overset{{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu)\mathcal{B} \lambda^S_{\alpha^\prime}(p^\mu) = 0.
\label{eq:4jan-b}$$ Under the new dual while the orthonormality relations (\[eq:zimpokJ9c\]) remain unaltered in form $$\begin{aligned}
& {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \lambda^S_{\alpha^\prime}(p^\mu)
= 2 m \delta_{\alpha\alpha^\prime}\label{eq:zimpokJ9an}\\
& {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \lambda^A_{\alpha^\prime}(p^\mu)
= - 2 m \delta_{\alpha\alpha^\prime} \label{eq:zimpokJ9bn} \\
& {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) \lambda^A_{\alpha^\prime}(p^\mu) = 0 =
{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \lambda^S_{\alpha^\prime}(p^\mu)\label{eq:zimpokJ9cn}\end{aligned}$$ the same very re-definition alters the spin sums to $$\begin{aligned}
\sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) & =
m \big[\I_4 + \mathcal{G}(p^\mu) \big] \mathcal{A} \label{eq:sss-new}\\
\sum_{\alpha} \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) & =
- m \big[\I_4 - \mathcal{G}(p^\mu)\big] \mathcal{B} \label{eq:ssa-new}\end{aligned}$$ In what follow we will show that $\mathcal{A}$ and $\mathcal{B}$ exist that satisfy the dual set of requirements encoded in (\[eq:4jan-a\]) and (\[eq:4jan-b\]) and at the same time find that a specific form of $\mathcal{A}$ and $\mathcal{B}$ exists that renders the spin sums Lorentz invariant.
The spin sums determine the mass dimensionality of the quantum field that we will build from the here-constructed $\lambda(p^\mu)$ as its expansion coefficients. They, along with their duals, enter the evaluation of the Feynman-Dyson propagator. For consistency with (\[eq:er-a1\])-(\[eq:er-b2\]) and (\[eq:skg\]) this mass dimensionality must be one. This can be achieved in the formalism we are developing if the spin sums are Lorentz invariant, and are proportional to the identity.
Thus, up to a constant to be taken as $2$ to preserve orthonormality relations $\mathcal{A}$ and $\mathcal{B}$ must be inverses of $\big[\I_4 + \mathcal{G}(p^\mu) \big]$ and $\big[\I_4 - \mathcal{G}(p^\mu) \big]$ respectively. But since the determinants of $\big[\I_4 \pm \mathcal{G}(p^\mu)\big]$ identically vanish we proceed in a manner akin to that of Penrose [@PSP:2043984] and Lee [@Lee:2014opa], and with $\tau \in \Re$ we introduce a $\tau$ deformation of the spin sums (\[eq:sss-new\]) and (\[eq:ssa-new\])[^29] $$\begin{aligned}
\sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) & =
m \big[\I_4 + \tau\mathcal{G}(p^\mu) \big] \mathcal{A}\Big\vert_{\tau \to 1} \label{eq:sss-newnew}\\
\sum_{\alpha} \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) & =
- m \big[\I_4 - \tau\mathcal{G}(p^\mu)\big] \mathcal{B}\Big\vert_{\tau \to 1}. \label{eq:ssa-newnew}\end{aligned}$$ We will see that the $\tau \to 1$ limit is non pathological in the infinitesimal small neighbourhood of $\tau = 1$ in the sense we shall make explicit. We choose $\mathcal{A}$ and $\mathcal{B}$ to be $$\begin{aligned}
& \mathcal{A} = 2 \big[I_4 + \tau \mathcal{G}(p^\mu)\big]^{-1} = 2 \left(\frac{\I_4 - \tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \\
& \mathcal{B} = 2 \big[I_4 - \tau \mathcal{G}(p^\mu)\big]^{-1} = 2 \left(\frac{\I_4 + \tau \mathcal{G}(p^\mu)}{1-\tau^2}\right).\end{aligned}$$ Making use of the identity $\mathcal{G}^2(p^\mu) = \I_4$, Eqs. (\[eq:sss-newnew\]) and (\[eq:ssa-newnew\]) simplify to: $$\begin{aligned}
\sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) & =
2 m \big[\I_4 + \tau\mathcal{G}(p^\mu) \big] \left(\frac{\I_4 - \tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \bigg\vert_{\tau \to 1} \nonumber \\
&=2 m \I_4 \left(\frac{1-\tau^2}{1-\tau^2}\right)\bigg\vert_{\tau\to 1} =
2m \I_4\label{eq:sss-newnew-a}\\
\sum_{\alpha} \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) & =
2 m \big[\I_4 - \tau\mathcal{G}(p^\mu) \big] \left(\frac{\I_4 + \tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \bigg\vert_{\tau \to 1} \nonumber \\
&=2 m \I_4 \left(\frac{1-\tau^2}{1-\tau^2}\right)\bigg\vert_{\tau\to 1} =
- 2m \I_4
\label{eq:ssa-newnew-b}\end{aligned}$$ We now return to the orthonormality relations. Since from the first equation in (\[eq:important\]), $\mathcal{G}(p^\mu) \lambda^S(p^\mu) =
\lambda^S(p^\mu)$ while $\mathcal{G}(p^\mu) \lambda^A(p^\mu) = -
\lambda^A(p^\mu)$, we have the result demanded by the requirement (\[eq:4jan-a\]) $$\begin{aligned}
\mathcal{A} \lambda^S_\alpha(p^\mu) & = 2 \left(\frac{\I_4-\tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \lambda^S_\alpha(p^\mu)
= 2 \left(\frac{1-\tau}{1-\tau^2}\right) \lambda^S_\alpha(p^\mu)
\nonumber\\
&= \left(\frac{2}{1+\tau}\right)\bigg\vert_{\tau\to 1} \lambda^S_\alpha(p^\mu) \nonumber\\
& = \lambda^S_\alpha(p^\mu)
\end{aligned}$$ and $$\begin{aligned}
\mathcal{B} \lambda^A_\alpha(p^\mu) &= 2 \left(\frac{\I_4+\tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \lambda^A_\alpha(p^\mu)
= 2 \left(\frac{1-\tau}{1-\tau^2}\right) \lambda^A_\alpha(p^\mu) \nonumber\\
& = \left(\frac{2}{1+\tau}\right)\bigg\vert_{\tau\to 1} \lambda^A_\alpha(p^\mu)\nonumber\\
& = \lambda^A_\alpha(p^\mu)\end{aligned}$$ where in the first two terms on the right hand side of each of the above equations the ${\tau\to 1}$ limit has been suppressed.
To examine the fulfilment of requirement (\[eq:4jan-b\]) we note that $$\begin{aligned}
{\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu)\mathcal{A} \lambda^A_{\alpha^\prime}(p^\mu) =
2 {\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu)\left(\frac{\I_4 - \tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \lambda^A_{\alpha^\prime}(p^\mu) \nonumber \\
= 2 \left(\frac{1}{1-\tau}\right)\bigg\vert_{\tau\to 1}
\underbrace{{\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu)\lambda^A_{\alpha^\prime}(p^\mu)}_{=\;0 ~\mbox{\small{(see eq. \ref{eq:zimpokJ9c})}}}
= \;0\\
{\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu)\mathcal{B} \lambda^S_{\alpha^\prime}\alpha(p^\mu) =
2 {\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) \left(\frac{\I_4 + \tau \mathcal{G}(p^\mu)}{1-\tau^2}\right) \lambda^S_{\alpha^\prime}(p^\mu) \nonumber \\
= 2 \left(\frac{1}{1-\tau}\right)\bigg\vert_{\tau\to 1}
\underbrace{{\overset{\:{}^{{}^{\boldsymbol{\sim}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu)\lambda^S_{\alpha^\prime}(p^\mu)}
_{=\;0 ~\mbox{\small{(see eq. \ref{eq:zimpokJ9c})}}}
= 0\end{aligned}$$ where the final equalities are to be understood as ‘in the infinitesimally close neighbourhood of $\tau =1$, but not at $\tau=1$.’ We will accept it as physically acceptable cost to be paid for the $\tau$ deformation forced upon us by the non-invertibility of $\big[\I_4 \pm \mathcal{G}(p^\mu)\big]$. With this caveat, constraints (\[eq:4jan-a\]) and (\[eq:4jan-b\]) on $\mathcal{A}$ and $\mathcal{B}$ are satisfied resulting in the Lorentz invariant spin sums $$\begin{aligned}
&\sum_{\alpha} \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu) = 2m \I_4\label{eq:sss-newnew-a-new}\\
&\sum_{\alpha} \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu) = - 2m \I_4
\label{eq:ssa-newnew-b-new}\end{aligned}$$ without affecting the Lorentz invariance of the orthonormality relations (\[eq:zimpokJ9an\])-(\[eq:zimpokJ9cn\]).[^30]
The completeness relation that follows from the Lorentz invariant spin sums (\[eq:sss-newnew-a-new\]) and (\[eq:ssa-newnew-b-new\]) takes the form $$\begin{aligned}
\frac{1}{4 m} \sum_{\alpha} \left( \lambda^S_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(p^\mu)
- \lambda^A_\alpha(p^\mu) {\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(p^\mu)\right) = \I_4
\label{eq:completeness-li}\end{aligned}$$
We thus conclude that a systematic analysis of spinorial duals we have resolved a long-standing problem on the construction of a Lagrangian density for the c-number Majorana spinors, extended here into Elko. The sought after Lagrangian density is not as given in equation (\[eq:wrong\]), or as conjectured and found not to exist in [@Aitchison:2004cs App. P], but $$\mathfrak{L}(x) = \partial^\mu{{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}(x)}\,\partial_\mu {{\lambda(x)}} - m^2 {{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}}(x) \lambda(x).\label{eq:correct}$$ Strictly speaking, this result should be taken as suggestive till it is fully established in its quantum field theoretic incarnation in the next chapter.
Mass dimension one fermions {#ch13}
===========================
A quantum field with Elko as its expansion coefficient
------------------------------------------------------
We now use the eigenspinors of the charge conjugation operator, Elko: $\lambda^S_\alpha(p^\mu)$ and $\lambda^A_\alpha(p^\mu)$, as expansion coefficients to define a new quantum field of spin one half $$\begin{aligned}
\mathfrak{f}(x) \stackrel{\textrm{def}}{=} & \int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{\sqrt{2 m E(\p)}} \nonumber \\ & \times \sum_\alpha \Big[ a_\alpha(\p)\lambda^S_\alpha(\p) e^{- i p\cdot x}
+\, b^\dagger_\alpha(\p)\lambda^A_\alpha(\p)e^{ i p\cdot x}
{\Big]}
\label{eq:newqf}\end{aligned}$$ where we have taken the liberty to notationally replace the $\lambda(p^\mu)$ by $\lambda(\p)$. To decipher the mass dimensionality of $\mathfrak{f}(x)$ and to develop a quantum field theoretic formalism for the new field we define its adjoint $$\begin{aligned}
{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \stackrel{\textrm{def}}{=} & \int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{ \sqrt{2 m E(\p)}} \nonumber\\ &\times
\sum_\alpha \Big[ a^\dagger_\alpha(\p){\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(\p) e^{ i p\cdot x}
+ b_\alpha(\p){\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(\p) e^{-i p\cdot x}{\Big]}\label{eq:newadjoint}\end{aligned}$$
The creation and annihilation operators, at this stage, are left free to obey fermionic $$\begin{aligned}
& \left\{a_\alpha(\p),a^\dagger_{\alpha^\prime}(\p^\prime)\right\} = \left(2 \pi \right)^3 \delta^3\hspace{-2pt}\left(\p-\p^\prime\right) \delta_{\alpha\alpha^\prime} \label{eq:a-ad-zimpok}\\
& \left\{a_\alpha(\p),a_{\alpha^\prime}(\p^\prime)\right\} = 0,\quad \left\{a^\dagger_\alpha(\p),a^\dagger_{\alpha^\prime}(\p^\prime)\right\} =0\label{eq:aa-adad-zimpok}\end{aligned}$$ or bosonic $$\begin{aligned}
& \left[a_\alpha(\p),a^\dagger_{\alpha^\prime}(\p^\prime)\right] = \left(2 \pi \right)^3 \delta^3\hspace{-2pt}\left(\p-\p^\prime\right) \delta_{\alpha\alpha^\prime} \label{eq:a-ad-zimpok}\\
& \left[a_\alpha(\p),a_{\alpha^\prime}(\p^\prime)\right] = 0,\quad \left[a^\dagger_\alpha(\p),a^\dagger_{\alpha^\prime}(\p^\prime)\right] =0\label{eq:aa-adad-zimpok-zimpok}\end{aligned}$$ statistics. We assume similar anti-commutativity/commutativity for $b_\alpha(\p)$ and $b^\dagger_\alpha(\p)$. Under the assumption that the vacuum state $\vert\hspace{3pt}\rangle$ is normalised to unity, they fix the normalisation of one particle states $$\vert\p,\alpha,a\rangle \stackrel{\textrm{def}}{=}a^\dagger_\alpha(\p)\vert\hspace{3pt}\rangle,\quad \vert\p,\alpha,b\rangle \stackrel{\textrm{def}}{=}b^\dagger_\alpha(\p)\vert\hspace{3pt}\rangle$$ to be $$\begin{aligned}
\langle\p^{\prime},\alpha^\prime,a\vert\p,\alpha,a\rangle = \left(2 \pi \right)^3 \delta^3\hspace{-2pt}\left(\p-\p^\prime\right) \delta_{\alpha\alpha^\prime} \nonumber\\
\langle\p^{\prime},\alpha^\prime,b\vert\p,\alpha,b\rangle = \left(2 \pi \right)^3 \delta^3\hspace{-2pt}\left(\p-\p^\prime\right) \delta_{\alpha\alpha^\prime}
\label{eq:orthonormality-zimpok}\end{aligned}$$ In the above expressions, we use the key $$a = \textrm{particle}, \quad b=\textrm{antiparticle}$$
A hint that the new field is fermionic
--------------------------------------
A Lie algebraically stable theoretical framework requires that position and momentum measurements do not commute: theories with special relativistic symmetries must be quantum in nature [@Flato:1982yu; @Faddeev:1989LD; @VilelaMendes:1994zg; @Chryssomalakos:2004gk].[^31] With the ensuing irreducible product of $\hbar/2$ between the uncertainties in the position and the momentum measurements this necessity forces non-vanishing amplitude for the propagation of a particle from an emission event $x$ to a space-like separated absorption event $y$ – that is to a classically forbidden region. But since time ordering of events is not preserved for space-like separations, causal paradoxes come to exist unless the same very process that was interpreted as a particle propagation for the set of observers with $y_0 > x_0$ is re-interpreted as propagation of an antiparticle for observers for whom $y_0 < x_0$ [@Weinberg:1972gc Ch. 2, Sec.13]. In the quantum field theoretic framework – only known way to merge quantum and relativistic realms – the processes that connect space-like separated events are mediated by virtual particles, particles that are off shell, that is, with energy, momentum, and mass that deviate from: $E^2 = \p^2 +m^2$. The time-energy uncertainty relation intervenes to protect the energy-momentum conservation for fluctuations off the mass shell.
With this background, to decipher the statistics for the $\mathfrak{f}(x)$ and ${\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)$ system we now consider two space-like separated events, $x$ and $y$, along the lines of [@Ahluwalia:2015vea]. Referring to the observation on the lack of time ordering preservation for such a set-up we recognise the existence of two sets of inertial frames, ones in which $y_0 > x_0$ and the ones in which the reverse is true, $x_0 > y_0$. We call these sets of inertial frame as $\mathcal{O}$ and $\mathcal{O}^\prime$ respectively. In $\mathcal{O}$, we calculate the amplitude for a particle to propagate from $x$ to $y$ and in $\mathcal{O}^\prime$ the amplitude for an antiparticle to propagate from $y$ to $x$. Causality requires that these two amplitudes may only differ, at most, by a phase factor:[^32] $$\textrm{Amp}(x\to y, \textrm{particle})\big\vert_\mathcal{O}
=e^{i \theta} \textrm{Amp}(y\to x, \textrm{antiparticle})\big\vert_{\mathcal{O}^\prime}\label{eq:phase}$$ with $\theta\in\R$. The definition of $\mathfrak{f}(x)$ and its adjoint ${\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)$ given in equations (\[eq:newqf\]) and (\[eq:newadjoint\]), respectively, allow us to obtain the following concrete results for the needed amplitudes: $$\begin{aligned}
\textrm{Amp}&(x\to y, \textrm{particle})\big\vert_\mathcal{O} =
\langle\hspace{3pt}\vert \mathfrak{f}(y)
{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)\vert\hspace{3pt}\rangle\nonumber\\
& =\int\frac{\text{d}^3p}{(2 \pi)^3}\left(\frac{1}{2 m E(\p)}\right)
e^{-ip\cdot(y-x)}
\sum_\alpha\lambda^S_\alpha(\p)
{\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(\p) \label{eq:amplitudeP}\end{aligned}$$ and $$\begin{aligned}
\textrm{Amp}(y\to x, \textrm{antiparticle})\big\vert_{\mathcal{O}^\prime}
&=\langle\hspace{3pt}\vert {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(y)\vert\hspace{3pt}\rangle\big\vert_{\mathcal{O}^\prime}\nonumber\\
& = \left[\langle\hspace{3pt}\vert {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(y)\vert\hspace{3pt}\rangle\big\vert_{\mathcal{O}}\right]_{(x-y)\to (y-x)}\nonumber\\
=\int\frac{\text{d}^3p}{(2 \pi)^3}\left(\frac{1}{2 m E(\p)}\right)
& e^{-ip\cdot(y-x)}
\sum_\alpha\lambda^A_\alpha(\p)
{\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(\p) \end{aligned}$$ The Elko spin sums (\[eq:sss-newnew-a-new\]) and (\[eq:ssa-newnew-b-new\]) when substituted in the above calculated amplitudes, yields the result that the phase factor in (\[eq:phase\]) is minus one: $e^{i \theta} = -1$. Furthermore, a direct calculation shows that $$\textrm{Amp}(y\to x, \textrm{antiparticle})\big\vert_{\mathcal{O}^\prime}
= \textrm{Amp}(y\to x, \textrm{antiparticle})\big\vert_{\mathcal{O}}$$ Combined, the above two results translate the relation (\[eq:phase\]) to $$\textrm{Amp}(x\to y, \textrm{particle})\big\vert_\mathcal{O}
= - \textrm{Amp}(y\to x, \textrm{antiparticle})\big\vert_{\mathcal{O}}\label{eq:phase-b}$$ That is $$\left\{{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x),\mathfrak{f}(y)\right\}=0$$
This is a hint, a strong hint, that the new field must be fermionic. We thus take the choice (\[eq:aa-adad-zimpok\]) as part of the definition of the $\mathfrak{f}(x)$.
In the standard S-matrix theory the events that we detect at ‘spatial infinity’ – that is, at distances far away from the interaction region – contain contribution from virtual particles as well as from on-sell particles. This amplitude is the Feynman-Dyson propagator. It is considered in the next section.
However, before we undertake that study we explicitly substitute the spin sum from (\[eq:sss-newnew-a-new\]) into (\[eq:amplitudeP\]), do the resulting integration, and introduce the normalisation constant alluded to above[^33] $$\begin{aligned}
\textrm{Amp}(x\to y, \textrm{particle})\big\vert_\mathcal{O} &= \frac{i}{2} m^2
\int\frac{\text{d}^3p}{(2 \pi)^3}\left(\frac{1}{ E(\p)}\right)
e^{-ip\cdot(y-x)}\,
\I_4 \nonumber\\
& =\frac{i}{4 \pi^2} \frac{m^3}{\sqrt{\epsilon^2-\tau^2} }K_1(m
\sqrt{\epsilon^2-\tau^2}) \label{eq:normalisationAdded}\end{aligned}$$ where $\epsilon\stackrel{\textrm{def}}{=}\vert\x^\prime - \x\vert$, $\tau\stackrel{\textrm{def}}{=}\vert\x^{0\prime} - x^0\vert$, $\epsilon > \tau$ so that $y$ and $x$ represent events separated by space-like interval, and $K_\nu(\ldots)$ is the modified Bessel function of the second kind of order one, that is with $\nu=1$.
For a historical thread (in the context of Dirac fermions), involving Dirac, Pauli, and Feynman, we refer our reader to the discussion following equation (2.13) of [@Ahluwalia:2011rg].
Amplitude for propagation
-------------------------
We now study amplitude of propagation from $x$ to $x^\prime$ without spacetime interval being restricted to space-like separations. It would reveal the mass dimensionality of the new field to be one. We would be mindful that causal paradoxes can only be avoided for contributions from space-like separations if we allow particles to be replaced by antiparticles whenever time ordering of the events is reversed. In the interaction region we do not measure spacetime coordinates, $x$ and $x^\prime$ – all our measurements in the scattering/collision processes take place in ‘far away regions.’ This creates an unavoidable ambiguity in time ordering of the absorption and emission of the virtual particles. This is incorporated in our calculations, as in the standard quantum field theoretic formalism, by adding all possible amplitudes that connect $x$ and $x^\prime$, at the same time we keep in mind the time ordering.
With this background, we note that:
- The fermionic statistics requires the amplitude for the $x \to x^\prime$ propagation to be antisymmetric under the exchange $x\leftrightarrow x^\prime$.
- Causality requires that a particle, or an antiparticle, cannot be absorbed before it is emitted, and vice versa.
In the S-matrix formulation of quantum field theory, amplitude for a particle to propagate from $x$ to $x^\prime$ incorporates all these facts by giving it the general form $$\begin{aligned}
\mathcal{A}_{x\to x^\prime} &=
\textrm{Amp} (x\to x^\prime, \textrm{particle})\big\vert_{t^\prime>t}
- \textrm{Amp}(x^\prime\to x, \textrm{antiparticle})\big\vert_{t>t^\prime}
\nonumber\\
& = \xi \Big(\underbrace{\langle\hspace{3pt}\vert
\mathfrak{f}(x^\prime){\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)\vert\hspace{3pt}\rangle \theta(t^\prime-t)
- \langle\hspace{3pt}\vert
{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(x^\prime)\vert\hspace{3pt}\rangle \theta(t-t^\prime)}_{\langle\hspace{4pt}\vert \mathfrak{T} ( \mathfrak{f}(x^\prime) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{f}}}(x)\vert\hspace{4pt}\rangle}\Big)\end{aligned}$$ where $\xi\in\C$ is to be determined from the normalisation condition that $
\mathcal{A}_{x\to x^\prime}
$ integrated over all possible separations $x-x^\prime$ be unity,[^34] and $\mathfrak{T}$ is the time ordering operator. The two vacuum expectation values that appear in $\mathcal{A}_{x\to x^\prime}$ can be evaluated as before but with the care that $(x-x^\prime)$ is no longer restricted to a space-like separation, the result is $$\begin{aligned}
\langle\hspace{3pt}\vert
\mathfrak{f}(x^\prime){\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)\vert\hspace{3pt}\rangle & =\int\frac{\text{d}^3p}{(2 \pi)^3}\left(\frac{1}{2 m E(\p)}\right)
e^{-ip\cdot(x^\prime-x)}
\sum_\alpha\lambda^S_\alpha(\p)
{\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(\p) \label{eq:amplitudeP-newS}
\\
\langle\hspace{3pt}\vert
{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(x^\prime)\vert\hspace{3pt}\rangle
& = - \int\frac{\text{d}^3p}{(2 \pi)^3}\left(\frac{1}{2 m E(\p)}\right)
e^{ip\cdot(x^\prime-x)}
\sum_\alpha\lambda^A_\alpha(\p)
{\overset{{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(\p) .\label{eq:amplitudeP-newA}\end{aligned}$$ The two Heaviside step function can be replaced by their integral representations $$\begin{aligned}
\theta(t^\prime-t) &= \lim_{\epsilon\to 0^+} \int\frac{\text{d}\omega}{2\pi i}
\frac{e^{i \omega (t^\prime-t)}}{\omega- i \epsilon} \\
\theta(t-t^\prime) &= \lim_{\epsilon\to 0^+} \int\frac{\text{d}\omega}{2\pi i}
\frac{e^{i \omega (t-t^\prime)}}{\omega- i \epsilon}\end{aligned}$$ where $\epsilon,\omega\in\R$. Using these results, and
- substituting $\omega \to p_0 = -\omega+E(\p)$ in the first term and $\omega \to p_0 = \omega- E(\p)$ in the second term
- and using the results (\[eq:sss-newnew-a-new\]) and (\[eq:ssa-newnew-b-new\]) for the spin sums
we get $$\mathcal{A}_{x\to x^\prime} = i \,2 \xi \int\frac{\text{d}^4 p}{(2 \pi)^4}\,
e^{-i p_\mu(x^{\prime\mu}-x^\mu)}
\frac{\I_4}{p_\mu p^\mu -m^2 + i\epsilon}\label{eq:AmplitudeWithXi}$$ We fix $\xi$ by the requirement that this amplitude when integrated over all possible separations $x^\prime-x$ yields unity. Towards that we take note of the integral representation of the delta function $$\delta(x-a) = \frac{1}{2 \pi} \int_{-\infty}^{\infty}e^{i(x-a) t} \text{d}t$$ and its symmetric aspect $\delta(x) = \delta(-x)$, and obtain $$i \,2 \xi \int\frac{\text{d}^4 p}{(2 \pi)^4}\,
\delta(p_\mu)
\frac{\I_4}{p_\mu p^\mu -m^2 + i\epsilon} =
\I_4$$ As such we have $$\frac{ i \,2 \xi }{-m^2+i\epsilon} =1$$ In the limit $\epsilon\to 0$ $$\xi = \frac{i m^2}{2}$$ Up to a possible global phase $e^{i\gamma}$ mentioned earlier, the amplitude (\[eq:AmplitudeWithXi\]) becomes $$\mathcal{A}_{x\to x^\prime} = - m^2 \int\frac{\text{d}^4 p}{(2 \pi)^4}\,
e^{-i p_\mu(x^{\prime\mu}-x^\mu)}
\frac{\I_4}{p_\mu p^\mu -m^2 + i\epsilon}$$ with $\epsilon\to 0^+$.
The $\xi$ differs from $\varpi$ of [@Ahluwalia:2004ab] by a factor of half: $\xi = (1/2) \varpi $. The origin of this difference resides in the new spin sums (\[eq:sss-newnew-a-new\]) and (\[eq:ssa-newnew-b-new\]).
Mass dimension one fermions {#mass-dimension-one-fermions}
---------------------------
To decipher the mass dimension of the new fermions we must know the Feynman-Dyson propagator associated with the set $\mathfrak{f}(x)$ and ${\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)$.
The Feynman-Dyson propagator is defined to be proportional to $\mathcal{A}_{x\to x^\prime} $ in such a way that the proportionality constant is adjusted to make the Feynman-Dyson propagator coincide with the Green function associated with the equation of motion for the field $\mathfrak{f}(x)$. To determine this proportionality constant we act the spinorial Klein-Gordon operator on the amplitude $\mathcal{A}_{x\to x^\prime}$ $$\left(\partial_{\mu^\prime} \partial^{\mu^\prime} \I_4 + m^2\I_4\right)
\mathcal{A}_{x\to x^\prime} = m^2 \delta^4(x^\prime - x)$$ So we define the Feynman-Dyson propagator $$\begin{aligned}
S_{\textrm{FD}}(x^\prime-x) & \stackrel{\textrm{def}}{=} - \frac{1}{m^2}
\mathcal{A}_{x\to x^\prime}\nonumber\\
&= \int\frac{\text{d}^4 p}{(2 \pi)^4}\,
e^{-i p_\mu(x^{\prime\mu}-x^\mu)}
\frac{\I_4}{p_\mu p^\mu -m^2 + i\epsilon}\label{eq:FD-prop-b}\end{aligned}$$ With this definition $$\left(\partial_{\mu^\prime} \partial^{\mu^\prime} \I_4 + m^2\I_4\right)
S_{\textrm{FD}}(x^\prime-x) = - \delta^4(x^\prime - x)\label{eq:KGDiracDelta}$$ and the Feynman-Dyson propagator in terms of the new field and its adjoint takes the form $$S_{\textrm{FD}}(x^\prime-x)= -\frac{i}{2}\left\langle\hspace{4pt}\left\vert \mathfrak{T} ( \mathfrak{f}(x^\prime) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{f}}}(x)\right\vert\hspace{4pt}\right\rangle$$
Following the canonical discussion on the mass dimensionality of quantum fields given in [@Weinberg:1995mt Section 12.1] we find that mass dimension of the field $\mathfrak{f}(x)$ is one $$\mathfrak{D}_{\mathfrak{f}} = 1 \label{eq:df1}$$ and not three-half, as is the case for the Dirac field: for large momenta $p$, $S_{\textrm{FD}} (x^\prime-x) \propto p^{-1}$ for the latter, while for the new field $S_{\textrm{FD}} (x^\prime-x) \propto p^{-2}$. This result is precisely what was hinted at by our discussion in section \[sec:elko-do-not-satisfy-Dirac-equation\].
The discussion of section \[sec:elko-do-not-satisfy-Dirac-equation\], when coupled with the results encoded in equations (\[eq:FD-prop-b\]) and (\[eq:KGDiracDelta\]), suggests the following free field Lagrangian density for the new field $$\mathfrak{L}_0(x) = \frac{1}{2}\left(\partial^a{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}\,\partial_a {\mathfrak{f}}(x) - m^2 {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(x)\right) \label{eq:fieldlagrangian}$$ where we have taken liberty to let $a,b,\ldots$ take the flat space values $(0,1,2,3)$, and reserve $\mu,\nu,
\mbox{etc}$ to take on the values for the names of the co-ordinates in the gravitational context, for example $(t,x,y,z)$. The motivation to introduce the factor of half resides in our desire to obtain canonical form of the locality anticommutators. It does not alter the equation of motion.
Like the Dirac fermions, the new fermions, are of spin one half. Both satisfy Klein-Gordon equation. For the former the “factorisation” of the Klein-Gordon operator occurs through the Dirac operator, while for the latter it occurs through equations (\[eq:er-a1\]) to (\[eq:er-b2\]). The expansion coefficients of the former are a complete set of eigenspinors of the parity operator, while for the latter the expansion coefficients are a complete set of eigenspinors of the charge conjugation operator. Both the fields are local as we shall soon discover in section \[sec:locality\].
Gravity enters the realm of Elko and mass dimension one fermions by introducing tetrads $e^a_\mu$ through $e^a_\mu e^b_\nu \eta_{ab} = g_{\mu\nu}$. Here $g_{\mu\nu}$ is the spacetime metric and $\eta_{ab}=\mbox{diag}(1,-1,-1,-1)$. Thus, for example, space-time Dirac matrices are connected with their flat space counterpart by $\gamma^\mu = e^\mu _a \gamma^a$, which consequently satisfy $$\left\{\gamma^\mu,\gamma^\nu\right \} = 2 g^{\mu\nu}$$ The Latin indices – $a,b,\ldots = 0,1,\ldots$, called non-holonomic indices – refer to a local inertial frame. Greek indices – $\mu,\nu,\ldots = t,\ldots$, called holonomic indices – refer to a generic non-inertial frame.
The Lagrangian density (\[eq:fieldlagrangian\]) for the mass dimension one fermions in a torsion-free gravitational field follows the prescription $$\begin{aligned}
\partial_a \mathfrak{f}(x) & \to \nabla_\mu \mathfrak{f}(x) \stackrel{\textrm{def}}{=} \partial_\mu \mathfrak{f}(x)- \Gamma_\mu \mathfrak{f}(x)\\
\partial_a{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) & \to \nabla_\mu{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}} (x)\stackrel{\textrm{def}}{=} \partial_\mu {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}} +
{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \Gamma_\mu\end{aligned}$$ where $$\Gamma_\mu = \frac{i}{2}\omega^{ab}_\mu\left(
\mathfrak{J}_{ab} \Big\vert_{(\mathcal{R}\oplus\mathcal{L})_{s=1/2}} \right)$$ and the spin connection is defined as $$\omega^{ab}_\mu = e^a_\nu\left(\partial_\mu e^{\nu b}
+ e^{\sigma b}\Gamma^\nu_{\mu\sigma}\right)$$ with $\Gamma^\nu_{\mu\sigma}$ denoting the Christoffel symbol associated with $g_{\mu\nu}$. In addition \_[ab]{}\_[()\_[s=1/2]{}]{} =
\_[ij]{} = - \_[ji]{} = \_[ijk]{} \_k\
\_[i0]{} = -\_[0i]{} = - \_i
Equation (\[eq:pi\]) provides explicit expressions for $\kb$ and $\bz$, respectively the boost and rotation generators in the $\mathcal{R}\oplus\mathcal{L}$ representations space for spin one half.
This prescription is as valid for Elko as for the Dirac spinors because the spinorial covariant derivative depends on the generators of the Lorentz algebra for spin one half in the indicated fashion. Böhmer has verified this observation independently, without invoking our argument, in [@Boehmer:2007dh]. The physics of Elko and torsion is discussed at length in [@Fabbri:2010ws; @KOUWN:2013wza; @Fabbri:2014foa; @Pereira:2017efk; @Pereira:2017bvq]. In the presence of torsion, denoted by a ‘tilde’ above the symbols, the spinorial covariant derivatives no longer commute $$\left[\tilde{\nabla}_\mu,\tilde{\nabla}_\nu \right]\ne \0$$ and (\[eq:fieldlagrangian\]) is modified to $$\tilde{\mathfrak{L}}(x) = \frac{1}{2}\left(\tilde\nabla^\mu{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}\,\tilde{\nabla}_\mu {\mathfrak{f}}(x) - m^2 {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(x)\right) \label{eq:fieldlagrangianB}$$ allowing for such additional interactions as $$\frac{1}{2} \tilde\nabla_\mu{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)\sigma^{\mu\nu}\tilde\nabla_\nu\mathfrak{f}(x)$$ with $$\sigma^{\mu\nu}
\stackrel{\textrm{def}}{=}
\frac{1}{4}\Big[\gamma_\mu,\gamma_\nu\Big]$$ This was first noted by Luca Fabbri in [@Fabbri:2010ws].
Despite formal similarity, Elko and Dirac spinors experience gravity differently. From a physical point of view the $\mathcal{R}$ and $\mathcal{L}$ transforming components of Elko pick up different, and in certain circumstances opposite, gravitationally-induced phase factors. This is apparent from their helicity structure [@Ahluwalia:2004kv].
Another distinguishing feature of Elko arises from the contribution to the energy-momentum tensor from the variation of spin connection. For the Dirac spinors this contribution identically vanishes. In contrast, for Elko it idoes not. This was first noted by Christian Böhmer in [@Boehmer:2010ma].
Locality structure of the new field\[sec:locality\]
---------------------------------------------------
Now that we have $\mathfrak{L}_0(x)$ we can calculate the momentum conjugate to $\mathfrak{f}(x)$ $$\mathfrak{p}(x) = \frac{\partial \mathfrak{L}_0(x)}
{\partial {\dot{\mathfrak{f}}(x)}} = \frac{1}{2} \frac{\partial}{\partial t}{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x).$$ To establish that the new field is local we calculate the standard equal-time anti-commutators. The first of the three anti-commutators we calculate is the ‘$\mathfrak{f}$-$\mathfrak{p}$’ anti-commutator $$\left\{ \mathfrak{f}(t,\x),\;\mathfrak{p}(t,\x^\prime) \right\}.$$ With the defintion $$\sum_{\alpha,\bf{p}} \stackrel{\textrm{def}}{=}
\int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{\sqrt{2 m E(\p)}}
\sum_\alpha$$ it expands to $$\begin{aligned}
\frac{1}{2} \sum_{\alpha,\bf{p}}\sum_{\alpha',\bf{p'}}
\left(iE(\p'\right) {\Big[}
& \left\{a_\alpha(\p),a^\dagger_{\alpha'} (\p')\right\}
\lambda^S_\alpha(\p) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_{\alpha'}(\p')
e^{-i p\cdot x+i p'\cdot x'}\nonumber\\
& - \left\{b_\alpha(\p),b^\dagger_{\alpha'} (\p')\right\}
\lambda^A_\alpha(\p) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_{\alpha'}(\p')
e^{i p\cdot x- i p'\cdot x'}
{\Big]}\end{aligned}$$ Replacing each of the anticommutators by $\left(2 \pi \right)^3 \delta^3\hspace{-2pt}\left(\p-\p^\prime\right)
\delta_{\alpha\alpha^\prime}$ and performing the $\p'$ integration, and $\alpha'$ summation, followed by (a) change of variables $\p\to -\p$ in the second integration, and (b) noting that the $t$ dependence in the exponentials cancels out, we get $$\frac{i}{4 m} \int \frac{\text{d}^3p}{(2\pi)^3}
e^{i\bf{p}\cdot(\bf{x}-\bf{x'})}
\sum_\alpha\left(
\lambda^S_\alpha(\p) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_{\alpha}(\p)
- \lambda^A_\alpha(- \p) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_{\alpha}(-\p)
\right)$$ The first spin sum evaluates to $2 m\I$, while the second equals $-2 m\I$, giving the result $4 m \I$ for the summation over $\alpha$. This gives us $$\left\{ \mathfrak{f}(t,\x),\;\mathfrak{p}(t,\x^\prime) \right\} = i \delta^3\left(\x-\x^\prime\right) \openone_4 .\label{eq:lac-1}$$ Had we not included a factor $1/2$ in the definition of the Lagrangian density in (\[eq:fieldlagrangian\]) we would have gotten an additional factor of $2$ multiplying the delta function in the above result.
A still simpler calculation shows that the remaining two, that is, ‘$\mathfrak{f}$-$\mathfrak{f}$’ and ‘$\mathfrak{p}$-$\mathfrak{p}$’, equal time anti-commutators vanish $$\left\{ \mathfrak{f}(t,\x),\;\mathfrak{f}(t,\x^\prime) \right\} = 0, \quad
\left\{ \mathfrak{p}(t,\x),\;\mathfrak{p}(t,\x^\prime) \right\} = 0.\label{eq:lac-2and3}$$ The field $\mathfrak{f}(x)$ is thus local in the sense of Schwinger [@PhysRev.82.914 Sec. II, Eqs. 2.82]. It is a much stronger condition of locality than that adopted by Schwartz [@Schwartz:2013pla Sec. 24.4].
### Majorana-isation of the new field\[sec:Majorana-isation\]
Even though field $\mathfrak{f}(x)$ is uncharged under local U(1) supported by the Dirac fields of the standard model of high energy physics, it may carry a charge under a different local U(1) gauge symmetry such as the one suggested in the discussion around (\[eq:mathfraka\]). This gives rise to the possibility of having a fundamentally neutral field in the sense of Majorana [@Majorana:1937vz] $$\begin{aligned}
\mathfrak{m}(x) &= \int \frac{\text{d}^3p}{(2\pi)^3} \frac{1}{ \sqrt{2 m E(\p)}} \nonumber\\ &\times \sum_\alpha \Big[ a_\alpha(\p)\lambda^S(\p) \exp(- i p_\mu x^\mu)
+ \, a^\dagger_\alpha(\p){\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A(\p) \exp(i p_\mu x^\mu){\Big]} \end{aligned}$$ with momentum conjugate $$\mathfrak{q} = \frac{1}{2} \frac{\partial}{\partial t}{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{m}}}}(x).$$ The calculation for the ‘$\mathfrak{m}$-$\mathfrak{q}$’ equal time anti-commutators goes through exactly as before and one gets $$\left\{ \mathfrak{m}(t,\x),\;\mathfrak{q}(t,\x^\prime) \right\} = i \delta^3\left(\x-\x^\prime\right) \openone_4 .$$ The calculation of the remaining two anti-commutators requires knowledge of the following ‘twisted’ spin sums $$\begin{aligned}
&\sum_\alpha\left[ \lambda^S_\alpha(\p)\left[ {\lambda}^A_\alpha(\p)\right]^T +
\lambda^A_\alpha(-\p) \left[{\lambda}^S_\alpha(-\p)\right]^T \right]\\
& \sum_\alpha\left[ \left[{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(\p)\right]^T {{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}}^A_\alpha(\p) +
\left[{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^A_\alpha(-\p)\right]^T {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}}^S_\alpha(-\p)\right].\end{aligned}$$ One finds that each of these vanishes. With this result at hand, we immediately decipher vanishing of the ‘$\mathfrak{m}$-$\mathfrak{m}$’ and ‘$\mathfrak{q}$-$\mathfrak{q}$’, equal time anti-commutators $$\left\{ \mathfrak{m}(t,\x),\;\mathfrak{m}(t,\x^\prime) \right\} = 0, \quad
\left\{ \mathfrak{q}(t,\x),\;\mathfrak{q}(t,\x^\prime) \right\} = 0.$$ The field $\mathfrak{m}(x)$, like $\mathfrak{f}(x)$, is thus local in the sense of Schwinger [@PhysRev.82.914 Sec. II, Eqs. 2.82].
Mass dimension one fermions as a first principle dark matter {#ch14}
============================================================
Mass dimension one fermions as dark matter
------------------------------------------
- A mass dimension mismatch between the new fermions and the standard model matter fields – one versus three halves – forbids them to enter the standard model doublets. In the process their interaction with the standard model fields is severely restricted.
One exception is the dimension four operators $$\lambda_{\textrm\small \mathfrak{f}\phi} {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(x) \,\phi^\dagger(x)\phi(x),
\quad
\lambda_{\textrm\small \mathfrak{m}\phi} \,{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{m}}}}(x) \mathfrak{m}(x) \,\phi^\dagger(x)\phi(x)$$ where $ \lambda_{\textrm\small \mathfrak{f}\phi}$ and $ \lambda_{\textrm\small \mathfrak{m}\phi}$ are dimensionless coupling constant, and $\phi(x)$ is the Higgs doublet. For the Dirac field, a similar interaction is a dimension five operator. It is thus suppressed by one power of the unification/Planck scale.
Therefore, mass dimension one fermions are natural dark matter candidate. Compared to bosonic dark matter a fermionic dark matter brings with it Pauli exclusion principle to affect structure formation.
- The system of mass dimension one fermions further supports perturbatively re-normalisable dimension-four quartic self interactions $$\lambda_\mathfrak{f} \left({\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x) \mathfrak{f}(x)\right)^2,\quad\
\lambda_\mathfrak{m} \,\left({\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{m}}}}(x) \mathfrak{m}(x)\right)^2$$ where $\lambda_\mathfrak{f}$ and $\lambda_\mathfrak{m}$ are dimensionless coupling constants. A similar quartic self interaction for the Dirac field, with or without Majorana-isation, is a dimension six operator. It is thus suppressed by two power of the unification/Planck scale.
Observational evidence that favours self interacting dark matter is reviewed in [@Tulin:2017ara] and in a Ph.D. thesis by Robertson [@Robertson:2017igt]. An early reference is [@Spergel:1999mh]. Following these references we simply note that self interacting dark matter provides a heat transport mechanism from the outer hotter to the cooler inner region of dark matter halos. It thermalises the inner halo and leads to a uniform velocity distribution as one moves outward in the halo.
- The very definition of Elko does not allow covariance under local phase transformations of the standard model (see, section \[sec:Restriction\] for local U(1)). This endows the new fermions with a natural darkness with respect to the standard model fields while leaving open the possibility of new gauge symmetries that apply to mass dimension one fermions.
Beyond the dimension-four interactions mentioned above one may also introduce the following Yukawa couplings of dimension three and half [@ArnabDasgupta] $$\begin{aligned}
&\lambda_1\,
\varphi(x) \,\overline{\psi}(x)\mathfrak{f}(x),\quad
\lambda_2\,\varphi(x) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}} (x) {\psi}(x)\\
&\lambda_3 \,
\varphi(x) \,\overline{\psi}(x)\mathfrak{m}(x),\quad
\lambda_4\,\varphi(x) {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{m}}}} (x) {\psi}(x)\label{eq:Yukawa}\end{aligned}$$ where $\psi(x)$ is a Dirac/Majorana field, $\varphi(x)$ is a scalar field, and $\lambda's$ are dimensionfull coupling constants. These may be used to violate conservation of all three of the following: lepton number, electric charge, and dark charges[^35]
The interactions of the form $$\epsilon {\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{f}}}}(x)\left[\gamma^a,\gamma^b\right]
\mathfrak{f}(x) F_{ab}(x),\quad
\varepsilon \,{\overset{\:{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{m}}}}(x)\left[\gamma^a,\gamma^b\right]
\mathfrak{m}(x) F_{ab}(x)$$ with dimensionless couplings $\epsilon$ and $\varepsilon$ are severely restricted due to stringent limits on photon mass [@PhysRevD.69.107501; @PhysRevLett.90.081801; @Bonetti:2016cpo]. Nevertheless, they could have significant astrophysical and cosmological implications where the smallness of the couplings may be compensated by huge dark matter densities. One crucial arena where such couplings may manifest are in 21-cm cosmology [@Barkana:2018qrx].
A conjecture on a mass dimension transmuting symmetry {#Sec:conjecture}
-----------------------------------------------------
A single component dark matter sector is likely to be an oversimplified view of the cosmological reality with the possibility of distorting our intuitions. We conjecture that the realm of dark matter is populated by a set of fields of the $\mathfrak{f}(x)$ type and its Majorana-ised form $\mathfrak{m}(x)$. We envisage the possibility that a mass dimension transmuting symmetry exists. It takes mass dimension one fermions to mass dimension three halves fermions, and vice versa. Unlike supersymmetry the conjectured new symmetry does not alter the statistics, but only the mass dimensionality.
In this possibility for every standard model fermion there exists a mass dimension one fermion, and vice versa. The possibility that Higgs is shared by both the sectors is the simplest unifying theme we can envisage. To complete our conjecture we suggest that dark gauge fields mirror the standard model gauge fields. The coupling constants and masses we leave as free parameters, but we would be surprised if they were not related to those of the standard model. Echoing the arguments of [@Hardy:2014mqa; @McDermott:2017vyk]: Combined, the dark sector may support some sort of dark fusion leading to dark nucleosynthesis.
*A parenthetic remark *
For reasons to be soon encountered the dark sector supported by Elko has distinctive gravitational signatures with torsion revealing aspects that are absent with the Dirac spinors. Elko cosmology is still in its infancy, and given its first principle origins it holds promise to give us observational signatures not yet anticipated by cosmologists. Observations can be easily misinterpreted in the absence of a systematic development of a cosmology that fully incorporates the new fermions. Such a process was begun by Christian Böhmer, and is now an active realm of researches by a new generation of physicists mostly from South America, Europe, and Asia.
Elko inflation and Elko dark energy
-----------------------------------
Echoing the closing remarks of the last section, Christian Böhmer was the first cosmologist to realise that:
- Elko not only help formulate mass dimension one fermions for dark matter, as Daniel Grumiller and I had argued, but also that Elko could serve as as a source of inflation and could drive the accelerated expansion of the universe [@Boehmer:2006qq; @Boehmer:2007dh; @Boehmer:2008rz].
- In a stark contrast to Dirac spinors, Elko energy momentum tensor $T_{\mu\nu}$ has an important non-vanishing contribution from the variation of spin connection [@Boehmer:2010ma].
These early works inspired a series of efforts to explore Elko as a source inflation, dark matter, and dark energy. Christian Böhmer was also the first to note that the spin angular momentum tensor associated with Elko cannot be entirely expressed as an axial torsion vector [@Boehmer:2006qq]. He emphasised that this important difference from the Dirac spinors arises due to different helicity structures of the Elko and Dirac spinors. His groundbreaking paper also put forward a tiny coupling of Elko to Yang-Mills fields and discussed its implications for consistently coupling massive spin one field to the Einstein-Cartan theory. Restricting to the Einstein-Elko system he constructed analytical ghost Elko solutions with the property of a vanishing energy-momentum tensor [@Boehmer:2006qq]. This was done to make the analytical calculations possible[^36] and he showed that, “the Elko …are not only prime dark matter candidates but also prime candidates for inflation.” With his collaborators Böhmer has placed Elko cosmology on a firm footing with an eye on the available data. We refer the reader to references [@Boehmer:2007ut; @Boehmer:2008rz; @Boehmer:2008ah; @Boehmer:2009aw; @Boehmer:2010ma] for details. While building Elko cosmology he has coined the term “dark spinors” for Elko.
The group of Julio Hoff da Silva and Saulo Pereira, focusing on exact analytical solutions, have taken Elko cosmology significantly beyond Böhmer’s initial pioneering efforts. We refer the reader to their publications [@daSilva:2014kfa; @Pereira:2014wta; @S:2014dja; @Pereira:2014pqa]. Concurrently extending the work of Böhmer, Gredat and Shankaranarayanan have considered an Elko-condensate driven inflation and shown that it is favoured by existing observational data [@Gredat:2008qf]. This work has been followed by Basak and Shankaranarayanan to prove that, “Elko driven inflation can generate growing vector modes even in the first order." This allows them to generate vorticity during inflation to produce primordial magnetic field [@Basak:2014qea].
Basak et al. show that Elko cosmology provides two sets of attractor points. These correspond to slow and fast-roll inflation. The latter being unique to Elko [@Basak:2012sn]. For earlier contribution to Elko cosmology from this group we refer the reader to [@Shankaranarayanan:2010st; @Shankaranarayanan:2009sz]. The cosmological coincidence problem in the context of Elko is discussed by Hao Wei in reference [@Wei:2010ad], and has been revisited in [@Bahamonde:2017ize Sec. 7.1]. One of the early papers on stability of de Sitter solution in the context of Elko is [@Chee:2010ju].
Elko cosmology has gained a significant and independent boost through a recent study of phantom dark-energy Elko/dark-spinors undertaken by Yu-Chiao Chang *et al.* In the context of Einstein-Cartan theory it resolves a host of problems with phantom dark energy models and predicts a final de Sitter phase for our universe at late time with or without dark matter [@Chang:2015ufa]. Their work not only makes Elko and mass dimension one fermions more physically viable but it also lends concrete physicality to torsion as an important possible element of reality.
Darkness is relative, not self referential
------------------------------------------
To avoid confusion, a clarifying remark seems necessary: The dark sector need not be self referentially dark. To dark matter, and to the all pervading field – dark energy – responsible for the accelerated expansion of the universe, we are dark. But, darkness is relative, and the dark sector may not be self referentially dark. It may support its own gauge fields, and carry its own luminosity. Our luminosity arises from the gauge fields supported by the matter fields which use eigenspinors of the parity operator as its expansion coefficients. A self referentially luminous dark sector may arise from the gauge fields that matter fields based on Elko support.
Continuing the story {#ch15}
====================
In this closing chapter, I take the liberty of suggesting what in essence are research projects that a beginning students may pursue. Most of these have been developed in detail in my private notes. I would be happy to share the details with anyone interested in pursuing them further.
Constructing the spacetime metric from Lorentz algebra
------------------------------------------------------
Returning to section \[sec:constructing-xt\] we continue an *ab initio* examination of four vectors spanning the $\mathcal{R}\otimes\mathcal{L}\vert_{s=1/2}$ representation space. We denote these vectors by $\chi$, and take them in the form of four-component columns with their elements in $\C$. We shall assume that we have implemented the rotation defined by $U$ of equation (\[eq:Umatrix\]), then if $\chi$ represents the position of an event then the elements take values in $\Re$ but otherwise this restriction is not necessary.
Following the discussion in chapter \[ch12\] the simplest dual for $\chi$ is $$\overline{\chi} \stackrel{\textrm{def}}{=} \chi^\dagger\eta$$ In order that $\overline{\chi} \chi$ is an invariant for observers connected by Lorentz transformations, the metric $\eta$ must satisfy $$\{K_i,\eta\}=0,\quad \left[J_i,\eta\right]=0,\qquad i=x,y.z \label{eq:constraints}$$ with the boost and rotation generators, $K_i$ and $J_i$, defined by equations (\[eq:kx\]) to (\[eq:jyjz-new\]).
We give $\eta$ the form $$\eta = \left(
\begin{array}{cccc}
a+i \alpha & b+i \beta & c + i \sigma & d + i\delta\\
e +i \epsilon & f + i\phi & g + i\gamma & h + i \lambda\\
j + i \zeta & k + i \kappa & m + i \mu & n + i\nu\\
p + i \omega & q + i\rho & r + i\chi & s + i \psi
\end{array}
\right)$$ with $a \ldots s, \alpha\ldots\psi \in \Re$ and implement the constraints (\[eq:constraints\]). Its anti-commutativity with $K_z$ reduces it to the form $$\eta = \left(
\begin{array}{cccc}
a+i \alpha & 0 & 0 & d + i\delta\\
0 & f + i\phi & g + i\gamma & 0\\
0 & k + i \kappa & m + i \mu & 0\\
-d - i \delta & 0 & 0 & -a - i\alpha
\end{array}
\right)$$ while anti-commutativity with $K_y$ restricts it further to $$\eta = \left(
\begin{array}{cccc}
a+i \alpha & 0 & 0 & 0\\
0 & f + i\phi & 0 & 0\\
0 & 0 & -a - i \alpha& 0\\
0 & 0 & 0 & -a - i\alpha
\end{array}
\right)$$ Its final form is reached by implementing its anti-commutativity with $K_x$, and reads $$\eta = \delta e^{i\xi}\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & -1& 0\\
0 & 0 & 0 & -1
\end{array}
\right)$$ where we have defined $a + i \alpha = \delta e^{i\xi}$, with $\delta,\xi\in\Re$. Since $\delta$ only sets the scale of the norm we may set it to unity, to the effect that $$\eta = e^{i\xi}\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & -1& 0\\
0 & 0 & 0 & -1
\end{array}
\right)\label{eq:SpacetimeMetricWithPhase}$$ The commutativity of $\eta$ with the generators of rotations places no further restrictions on the metric.
This exercise does two things for us. It algebraically constructs the metric for $\mathcal{R}\otimes\mathcal{L}\vert_{s=1/2}$ representation space, and unearths a phase that the Lorentz invariant norms allow. Second, it provides a unified way of looking at spinors, spacetime, and four vectors.
If we require a reality of the norm for $\chi$, with all its four elements $\in\Re$, the phase angle $\xi$ can be restricted as follows $$\xi=
\begin{cases}
0, & \textrm{to yield $\eta$ in the West coast form}\\
\pi, & \textrm{to give the East coast version of $\eta$}
\end{cases}$$ This is useful for spacetime vectors. On the other hand, in a quantum context where $\chi$ may be a vector field, a non-vanishing $\xi$, $0 \le \xi \le \pi$, opens a discussion which was not possible hitherto: that is, before the discovery of the phase $e^{i\xi}$ in (\[eq:SpacetimeMetricWithPhase\]).
The $\mathbf {\left[\mathcal{R}\otimes\mathcal{L}\right]_{s=1/2}}$ representation space
---------------------------------------------------------------------------------------
A field transforming according to $\left[\mathcal{R}\otimes\mathcal{L}\right]_{s=1/2}$ representations can be bifurcated into two Casimir sectors: one, with eigenvalue $2=1(1+1)$, and the other with eigenvalue $0=0(0+1)$. The eigenvalue $2$ sector is three fold degenerate, which *in the rest frame* can be distinguished with eigenvalues $+1,0,-1$ of spin one $J_z$. The Casimir sector with eigenvalue $0$ is non-degenerate.
A quantum field defined with the three degrees of freedom associated with the Casimir sector of eigenvalue $2$ is known to violate unitarity at high energies. The unitarity is restored to construct a renormalisable theory in the form of the standard model of high energy physics at the cost of introducing Higgs through spontaneous symmetry breaking. Historically, it served as a departure to develop gauge theories.
It is and was conventional to project out the Casimir sector with the vanishing eigenvalue. But, can one project out degrees of freedom from a representation space without violating something deep about the symmetries that gives birth to the very representation space on which a quantum field is built upon?
To cosnider this question we backtrack to our discussion in Section 2.2 of [@Ahluwalia:2004ab] that a similar projecting out from the $\left[\mathcal{R}\oplus\mathcal{L}\right]_{s=1/2}$ representation space would have excluded antiparticles from the Dirac’s theory. Mass dimension one fermions come to exist because we do not ignore the two degrees of freedom associated with the anti-self conjugate spinors. Incorporating them in the theory brings about dark matter, or at least a first principle candidate for dark matter with quartic self interaction, and an unusual property associated with rotation of dark matter clouds based on Elko.
With that background we suspect that something similar may be happening in the conventional treatment of the $\left[\mathcal{R}\otimes\mathcal{L}\right]_{s=1/2}$ representation space. Inadvertently, we may be projecting out something that could morph into a Higgs.
One can now define a field that contains expansion coefficients corresponding to the three degrees of freedom associated with the Casimir invariant $2$ sector, and one from the Casimir invariant $0$ sector. The dual of the expansion coefficients is so defined that the two Casimir sectors have their own phase angle $\xi$. This modifies the adjoint of the field. As a result the the phase angles end up showing in the vacuum expectation value of the calculated Feynman-Dyson propagator and can be so adjusted as to obtain a well-behaved propagator that does not lead to unitarity violation at high energies, or to the negative norm states. The latter aspect is related to the fact that the creation and destruction operators satisfy the bosonic commutators.
The result is that the resulting field contains, what in the rest frame, can be called spin one and spin zero. Such a scenario can serve as an *ab initio* starting point to better understand origin of the Higgs mechanism and spontaneous symmetry breaking.[^37] Extended to ‘spin $2$’, the argument suggests ‘spin $1$’ and ‘spin $0$’ Higgs like bosons.
Maxwell equations and beyond
----------------------------
When one extends the work of Chapter \[ch5\] to the $\mathcal{R}\oplus\mathcal{L}$ representation space of spin one and two, and takes the massless limit, one reaches a deeper understanding of Maxwell equations and Einstein’s gravity in the weak field limit. A helpful reference is [@Weinberg:1964ev], but care must be taken to check the invertibility of $\mathfrak{J}\cdot \nabla$ operator [@Ahluwalia:1993pj].
Appendix: Further reading {#ch17 .unnumbered}
=========================
Recently Saulo Pereira and R. C. Lima have shown that an asymptotically expanding universe creates low-mass mass dimension one fermions much more copiously than Dirac fermions (of the same mass) [@Pereira:2016eez]. If their preliminary results remain essentially unaffected by the new results presented here it would significantly help us to develop a first-principle cosmology based on Elko and mass dimension one fermions.
Various Brazilian-Italian group of physicists have examined such important topics as Hawking radiation of mass dimension one fermions [@daRocha:2014dla], and continue to develop mathematical physics underlying Elko [@daRocha:2007sd; @daRocha:2007pz; @daRocha:2008we; @HoffdaSilva:2009is; @daRocha:2011xb; @daRocha:2011yr; @Bernardini:2012sc; @daSilva:2012wp; @daRocha:2013qhu; @Cavalcanti:2014uta; @daRocha:2014dla; @Bonora:2015ppa; @daRocha:2016bil; @Rogerio:2016grn; @daSilva:2016htz; @HoffdaSilva:2017vic; @Neto:2017vgx]. Of these we draw particular attention to mass-dimension transmuting operators considered in [@daRocha:2007pz; @HoffdaSilva:2009is]. It would help define a new symmetry between the Dirac field and the field associated with mass dimension one fermions if mass-dimension transmuting operators could be placed on a rigorous footing after incorporating locality and Lorentz covariance.[^38]
Max Chaves and Doug Singleton have suggested that mass dimension one fermions of spin one half may have a possible connection with mass-dimension-one vector particles with fermionic statistics [@Chaves:2008gd]. It may be worth examining if a new fundamental symmetry may be constructed that relates the works of [@daRocha:2007pz; @HoffdaSilva:2009is] with those of Chaves and Singleton.
Localisation of Elko in the brane has been considered in references [@Liu:2011nb; @Jardim:2014xla; @Zhou:2017bbj]. Elko in the presence of torsion has been a subject of several insightful papers by Luca Fabbri. We refer the reader to these and related publications [@Fabbri:2009ka; @Fabbri:2010qv; @Fabbri:2010ws; @Fabbri:2010va; @Fabbri:2011mi; @Fabbri:2012yg; @Fabbri:2014foa]. Cosmological solutions of 5D Einstein equations with Elko condensates were obtained by Tae Hoon Lee where it was found that there exist exponentially expanding cosmological solution even in the absence of a cosmological constant [@Lee:2012zze].
We parenthetically note that all the works discussed so far remain essentially unchanged with the new developments reported here. In view of the new results on locality and Lorentz covariance it is important to revisit the analysis and claims of [@Basak:2011wp] and also those calculations that use full apparatus of the theory of quantum fields, and not merely Elko. In the same thread, given the interest in mass dimension one fermions a number of $S$-matrix calculations were done and published [@Dias:2010aa; @Lee:2015sqj; @Alves:2014qua; @Alves:2014kta; @Agarwal:2014oaa; @Lee:2015jpa; @Alves:2017joy]. These need to be revisited also.
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[^1]: Of the reader it is assumed that she is at home with the theory of special relativity, and first few chapters of books on the theory of quantum fields. With that background she would be ready for the journey through this monograph
[^2]: I now realise that this inference requires a revision due to the inevitable modification of the wave particle duality in the Planck realm [@Kempf:1994su; @Ahluwalia:2000iw]: going from a snow flake to the neutrino mass eigenstates one goes from the Planck-scale induced gravitational modifications of the wave particle duality to the low energy realm of quantum mechanics where the de Broglie’s wave particle duality holds to a great accuracy [@PhysRevLett.91.090408]. For $C_{60} F_{48}$ molecule the experiment finds fringe visibility lower than expected. It may be indicative that the modification to the de Broglie wave particle duality may become significant at a much lower energy.
[^3]: This is the quantum field theoretic formalism presented in [@Weinberg:1995mt; @Weinberg:1996kr]. Its origins go back to [@Weinberg:1964cn; @Weinberg:1964ev; @Weinberg:1969di]. Its most celebrated offspring is the standard model of high energy physics.
[^4]: Dirac equation, as originally introduced and as now understood are two very different things. The original acted $(i\gamma_\mu\partial^\mu - m\I_4)$ on a spinor, while that of the standard model of high energy physics the same very operator acts on a spinorial quantum field.
[^5]: Now called Thomas Jefferson National Accelerator Facility.
[^6]: To be fair to Lewis Ryder, his derivation, in essence reproduced the then-existing literature on the subject. For a parallel treatment as Ryder’s our reader may consult [@Hladik:1999tt]. It apparently began in the Istanbul lectures in early 1960’s with Feza Gürsey hosting the theoretical physics school.
[^7]: At the time I wrote my book review I was not aware of the analysis of Gaioli and Garcia Alvarez. I only learned of their work when, unexpectedly one day, they walked into my office at the Los Alamos National Laboratory (as by then I was a Director’s Fellow there) and told me about their publication. Lewis Ryder in the second edition of his book does cite Gaioli and Garcia Alvarez, but without attending to the raised concerns.
[^8]: Now named Los Alamos Neutron Science Center.
[^9]: See chapter 12 of Weinberg’s cited monograph for a rigorous definition of mass dimensionality of a quantum field.
[^10]: For the contribution of Otto Stern to this story, see [@Pakvasa:2018xlz].
[^11]: A departure from this remark must be made when dealing with massless particles, and parity violation.
[^12]: Here, perhaps interchanging the entries in the second and third columns may better reflect the logical order. As we proceed the reader would discover why mass dimension one fermions are a first principle candidate for dark matter.
[^13]: Note, we say Lorentz symmetry rather than spacetime symmetry. For the latter is just one representation arising out of the Lorentz algebra. See, section \[sec:constructing-xt\] for further remarks.
[^14]: As noted earlier: Elko is a German acronym for **E**igenspinoren des **L**adungs**k**onjugations**o**perators introduced in [@Ahluwalia:2004sz; @Ahluwalia:2004ab]. In English, it translates to eigenspinors of the charge conjugation operator.
[^15]: In a quantum field theoretic context significantly more structure needs to be introduced to arrive at the Dirac and Maxwell equation for the fields [@Weinberg:1995mt].
[^16]: Unknown to me for sometime, similar questions were being asked by Steven Weinberg in 1960’s. In 1980’s when I was to take quantum field theory courses, not too far from Texas A&M, he was offering quantum field theory courses at the University of Texas at Austin. Had I attended those courses this monograph would not have come to be written. The beauty of his formalism, how he viewed and views quantum field theory, and its seduction was too intense to think that a couple places needed to be explored deeper leading to the results presented in this monograph.
[^17]: We shall often set $\hbar$ and the speed of light $c$ to be unity.
[^18]: In making this identification with $\J^2$ we implicitly assume the abstraction explicitly noted in equation (\[eq:abstractionJ\]) that allows $s$ to take half integral and integral values.
[^19]: To find explicit form of $\bmfj$ a junior reader may consult any good book on quantum mechanics ranging from Dirac’s classic [@Dirac:1930pam], to recent lectures by Weinberg [@Weinberg:2012qm], or to a very pedagogically written two volume set by Cohen-Tannoudji et al. [@Cohen-Tannoudji:1977qm].
[^20]: The genesis of this chapter goes back to a pizza and hand written explanation by Llohann Sperança on a paper napkin one evening in Barão Geraldo in Campinas. This was during the two years I spent at IMECC, Unicamp, on a long term visit.
[^21]: The placing of $\mathcal{R}$ and $\mathcal{L}$ as a superscript or as a subscript is purely for convenience in a given context.
[^22]: These observations, though not explicitly stated, can be easily read off from the mentioned analysis.
[^23]: The ${\bf 0}$ there carries the same meaning as $k^\mu$ here.
[^24]: See, Chapter \[ch10\] for departures from Dirac spinors.
[^25]: This chapter is an adapted version of [@Ahluwalia:2018hfm]
[^26]: Where $\sin[\theta]$ is abbreviated as $s_\theta$ with obvious extension to other trigonometric function.
[^27]: \[fn:8\]Strictly speaking, Weinberg’s discussion is for a spin one half quantum field but it readily adapts to the c-number spinors, $\psi(p^\mu)$, in the $\mathcal{R}\oplus\mathcal{L}\vert_{s=1/2}$ representation space.
[^28]: The concrete outline of the breakthrough passed through me at a conversation with Krishnamohan Parattu in the parking lot of Akashganga guest house at IUCAA just as I was leaving for the Centre for the Studies of the Glass Bead Game that I was setting up in Bir, Himachal Pradesh.
[^29]: Unlike Lee [@Lee:2014opa] we refrain from introducing configuration counterpart of $\mathcal{G}(p^\mu)$ to avoid brining in the theory fractional derivatives.
[^30]: In [@Rogerio:2016mxi] Rogerio et al. provide additional support for the new dual introduced here.
[^31]: A weaker version of this argument based solely on the rotational symmetry is given in section \[sec:quantum\].
[^32]: For the smoothness of the discussion, we suppress a normalisation factor with the dimensions of inverse length squared till equation (\[eq:normalisationAdded\]).
[^33]: The far from trivial integration was done by Sebastian Horvath in our collaborative work [@Ahluwalia:2011rg].
[^34]: Or, more precisely $e^{i\gamma}$, with $\gamma\in \R$.
[^35]: A similar mass dimension four coupling, without a quartic self interaction term, of Dirac/Majorana fermions to a scalar can be found in [@Kainulainen:2015sva].
[^36]: This assumption was later placed on a more natural footing by [@Chang:2015ufa] *et al.*
[^37]: The just outlined calculation for the new field has been a subject of various discussions with Sebastian Horvath, Cheng-Yang Lee, and Dimitri Schritt as is the extension to higher spins. It must be considered preliminary.
[^38]: The need for a mass-dimension transmuting symmetry was first suggested by the present author to Roldão da Rocha several years ago and is briefly mentioned in section \[Sec:conjecture\].
|
---
abstract: 'We consider a near-critical binary mixture with addition of antagonistic salt confined between weakly charged and selective surfaces. A mesoscopic functional for this system is developed from a microscopic description by a systematic coarse-graining procedure. The functional reduces to the Landau-Brazovskii functional for amphiphilic systems for sufficiently large ratio between the correlation length in the critical binary mixture and the screening length. Our theoretical result agrees with the experimental observation \[Sadakane et.al. J. Chem. Phys. [**139**]{}, 234905 (2013)\] that the antagonistic salt and surfactant both lead to a similar mesoscopic structure. For very small salt concentration $\rho_{ion}$ the Casimir potential is the same as in a presence of inorganic salt. For larger $\rho_{ion}$ the Casimir potential takes a minimum followed by a maximum for separations of order of tens of nanometers, and exhibits an oscillatory decay very close to the critical point. For separations of tens of nanometers the potential between surfaces with a linear size of hundreds of nanometers can be of order of $k_BT$. We have verified that in the experimentally studied samples \[Sadakane et.al. J. Chem. Phys. [**139**]{}, 234905 (2013), Leys et.al. Soft Matter [**9**]{}, 9326 (2013)\] the decay length is too small compared to the period of oscillations of the Casimir potential, but the oscillatory force could be observed closer to the critical point.'
author:
- Faezeh Pousaneh
- Alina Ciach
title: ' Effect of antagonistic salt on confined near-critical mixture '
---
Introduction
============
Addition of small amount of salt to a binary mixture near a demixing critical point can significantly change its properties. When inorganic salt is added to water and organic liquid, the two-phase region enlarges [@fuess:99:0]. In contrast, antagonistic (or amphiphilic) salt added to such mixture leads to shrinking of the two phase region; it can even disappear when the amount of salt is large enough [@sadakane:13:0]. In addition, in the case of the antagonistic salt a peak in the structure factor for the wavenumber $k>0$ was observed in the one-phase region [@sadakane:09:0; @sadakane:11:0; @leys:13:0; @sadakane:13:0]. The peak indicates thermodynamically stable inhomogeneities on the length scale $\sim 10nm$ [@sadakane:13:0; @leys:13:0]. Moreover, in a few cases a lamellar phase was observed for some region of the phase diagram [@sadakane:09:0; @sadakane:13:0; @leys:13:0]. For low salt concentration the shape of the structure factor was described with a good accuracy by the formula obtained by Onuki and Kitamura [@onuki:04:0]. For larger amount of salt (when the two-phase region disappears) the experimental structure factor was described with better accuracy by the formula derived earlier for bicontinuous microemulsion or sponge phases [@sadakane:13:0; @lei:97:0; @porcar:03:0]. In addition, the structure factor of the lamellar phase was fitted with a good accuracy [@sadakane:13:0] by the formula developed for a stack of membranes [@sadakane:13:0; @nallet:93:0]. These observations strongly suggest that the key features of the mesoscopic structure do not depend on whether antagonistic salt or surfactant is added to a mixture of inorganic and organic solvents. In this work we address the question of the above similarity on a theoretical level by comparing the Landau-type functionals for the two systems. For this purpose we first develop a Landau-type functional starting from a microscopic density functional theory and using the same coarse-graining procedure as used earlier for the hydrophilic salt [@ciach:10:0; @pousaneh:14:0]. In the next step we verify under what conditions the Brazovskii functional of the solvent concentration, developed earlier for amphiphilic systems [@gompper:94:0; @ciach:01:2; @brazovskii:75:0] can be obtained.
The effects of confinement on the near-critical mixture with antagonistic salt were not investigated experimentally yet. Theoretical investigations based on the theory of Onuki and Kitamura [@onuki:04:0] focused very briefly on colloid particles immersed in the critical mixture with salt [@okamoto:11:0]. In contrast, the effective potentials between a flat substrate and a colloid particle immersed in the critical mixture with hydrophilic salt, and between two particles in such mixture were measured in several impressive experiments [@hertlein:08:0; @nellen:11:0; @gambassi:09:0; @bonn:09:0; @nguyen:13:0]. Similarity between the structure factors in the investigated system and sponge phases suggests that an oscillatory force between confining surfaces, observed experimentally for surfactant mixtures [@antelmi:99:0], could occur in a presence of antagonistic salt too.
The effective potential between surfaces confining the critical mixture with ions is interesting from both the fundamental and the practical point of view. Surfaces with similar adsorption preferences attract each other, while surfaces with different adsorption preferences repel each other [@gambassi:09:0; @vasiliev:11:0; @krech:96:0; @krech:94:0] when the confined fluid is near its critical point. The range of this so called thermodynamic Casimir potential [@krech:94:0; @gambassi:09:0] is equal to the bulk correlation length $\xi\propto |T-T_c|^{-\nu}$ with $\nu\approx 0.63$, and becomes macroscopic when the critical temperature $T_c$ is approached. When the surfaces are charged, then depending on the surface charges electrostatic repulsion or attraction is added to the Casimir potential. The resulting potential can have a minimum or a maximum, depending on the ratio between $\xi$ and the screening length $\lambda_D$. By changing the salt concentration and the temperature, one can change $\lambda_D$ and $\xi$, and tune the shape of the potential between the surfaces, e.g. the surfaces of particles [@hertlein:08:0; @nellen:11:0; @gambassi:09:0; @nguyen:13:0]. Temperature can be changed in a reversible manner, and reversible structural changes can be induced [@bonn:09:0; @nguyen:13:0; @dang:13:0]. In particular, when the effective potential has a minimum, then analogs of the gas-liquid and liquid-solid transitions between the particles were observed [@nguyen:13:0].
The experimental results for inorganic salt show that the effective potential cannot be described by just a sum of the Casimir and the electrostatic potential [@gambassi:09:0; @pousaneh:14:1]. In particular, attraction was measured for some temperature range between like charge hydrophilic and hydrophobic surfaces [@nellen:11:0], whereas both the Casimir and the electrostatic potentials are repulsive in this case. Both potentials, however, can be modified because of different solubility of the ions in the two components of the solvent. For this reason quite different behavior of the effective potential between charged and selective surfaces can be expected for the antagonistic salt [@okamoto:11:0], and we address this question here.
Several groups tried to explain the experiment in Ref. [@nellen:11:0]. The attraction for a range of temperatures was obtained in two different approaches. The first one is based on the theory of Onuki and Kitamura [@onuki:04:0]. The attraction appears when the solubility in water of the anion and the cation differ significantly from each other and the hydrophobic surface is neutral [@bier:11:0; @samin:12:0]. On the other hand, in the theory developed in Ref.[@ciach:10:0] for hydrophilic ions, the concentration profile near the charged hydrophobic surface can be nonmonotonic, because the attraction of the organic particles competes with the attraction of the hydrated ions [@pousaneh:11:0; @pousaneh:14:0]. When $\lambda_D>\xi$, excess of water may appear at some distance from the surface, therefore it may act as a hydrophilic one. The shapes of the effective potential for different temperatures in this theory are quite similar to the experimental curves, but the fitting was not attempted yet [@pousaneh:14:1]. Importantly, good quantitative agreement between this theory and the experiments conducted for $\lambda_D<\xi$ [@hertlein:08:0; @gambassi:09:0] was obtained in Ref.[@pousaneh:12:0], in contrast to the theories based on the Onuki and Kitamura functional. In this work we develop a Landau-type theory for the antagonistic salt following the same strategy as in Ref.[@ciach:10:0; @pousaneh:14:0].
In the Onuki and Kitamura model the critical binary mixture is described by the phenomoenological Landau functional of the concentration $\phi$. The ions are treated as a two-component ideal gas whose particles interact with the Coulomb potential, and the corresponding entropy and electrostatic energy are added to the Landau functional. Finally, the coupling between the critical mixture and the ions of the form $\phi(w_+n_++w_-n_-)$ is added, where $n_i$ and $w_i$ denote the density and the preferential salvation of the $i$-th ion. Note that in this model the van der Waals interactions between the ions are neglected, and the entropy of the four-component mixture is approximated by a sum of the entropy of the two two-component subsystems. For this reason the coupling between the two subsystems beyond the above bilinear term is not taken into account.
The full microscopic density-functional theory of a four-component mixture is more accurate and justified than a phenomenological model, but in practice it is too difficult. Moreover, such a detailed description is not necessary when the characteristic length scales are mesoscopic. On the other hand, a microscopic theory is a good starting point for a derivation of the Landau-type functional by a coarse-graining procedure. Such a strategy was successful in the case of the hydrophilic salt [@pousaneh:14:1; @ciach:10:0]. In the next section we develop in the same way a Landau-type functional for the antagonistic salt added to the near-critical binary solvent. In sec.3 we focus on the disordered phase in the bulk. We calculate the correlation function, and compare our functional with the Landau-Brazovskii functional for microemulsions. In sec.4 we consider the concentration and charge profiles in a semiinfinite system and in a slit, and calculate the effective potential between the surfaces. In sec.5 a comparison of the structure factor with the experimental results reported in Ref.[@sadakane:13:0; @leys:13:0] is performed. For the model parameters obtained from the fitting of the structure factor to the experimental results the concentration profiles and the effective potential are computed. Sec.6 contains summary and discussion. In particular, we compare the effective potential obtained in the models for the hydrophilic and the antagonistic salt.
The model {#generic}
=========
We consider a four-component mixture containing water, organic liquid and antagonistic salt which has hydrophilic cations and hydrophobic anions. The fluid is in contact with a reservoir with fixed temperature and chemical potential of all the components. In equilibrium the distribution of the components corresponds to the minimum of the grand potential $$\begin{aligned}
\label{OmegaL}
\Omega= -PV \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;
\\
\nonumber
= U_{vdW}+U_{el}-TS-\int_Vd{\bf r}^*\mu_i\rho_i^*({\bf r}),\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\end{aligned}$$ where $P$, $V$ and $S$ are pressure, volume and entropy respectively, and $U_{el}$, $U_{vdW}$ are the electrostatic and the van der Waals contributions to the internal energy. $\rho_i ({\bf r})$ and $\mu _i$ are the local number density and the chemical potential of the $i$-th component respectively, with $i=w,l$ corresponding to water and organic solvent (for example lutidine or methylpyridine), and $i=+,-$ corresponding to the cations and the anions. We consider dimensionless distance $r^*=r/a$ and dimensionless densities, i.e. the length is measured in units of $a$, and $\rho_i^*=a^3 N_i/V$, where $a^3$ is the average volume per particle in the liquid phase, and $N_i$ denotes the number of the $i$-th kind particles in the volume $V$. We neglect compressibility of the liquid and assume that the total density is fixed, $$\label{fixrho}
\sum_{i=\{w,l,+,-\}}\rho_i^*=1.$$ The microscopic details, in particular different sizes of molecules are disregarded, since we are interested in the local densities in the regions much larger than $a$. To simplify the notation we shall omit the asterisk for the dimensionless distance.
The electrostatic energy is given by [@barrat:03:0] $$\label{el}
U_{el}=\int_V d{\bf r} \left[ -\frac{\epsilon}{8\pi}(\bigtriangledown{\psi_{el}({\bf r})})^2
+e{\rho_q^*({\bf r})} {\psi_{el}({\bf r})} \right],\\$$ where ${\psi_{el}({\bf r})}$ is the electrostatic potential which satisfies the Poisson equation (note that the length $|{\bf r}|$ and density are dimensionless) $$\begin{aligned}
\label{Poisson}
\bigtriangledown ^2 \psi_{el} ({\bf r})=-\frac {4\pi e} {a\epsilon}\rho_q^*({\bf r}).\end{aligned}$$ We neglect the dependence of the dielectric constant $\epsilon$ on the solvent concentration ([@ciach:10:0; @pousaneh:14:0]). $\rho_q^*({\bf r})$ is the local dimensionless charge density $$\begin{aligned}
\label{rhoqdef}
\rho_q^*({\bf r})=\rho_+^*({\bf r})-\rho_-^*({\bf r}),\end{aligned}$$ and $e$ is the elementary charge.\
We assume the usual form of the internal energy ${\cal U}_{vdW}$, $$\begin{aligned}
\label{vW}
{\cal U}_{vdW}
=\int_{V}d{\bf r}\int_{V} d{\bf r}' \frac{1}{2}\rho_i^*({\bf r})V_{ij}({\bf r}-{\bf r}') g_{ij}({\bf r}-{\bf r}')
\rho_j^*({\bf r}'),\end{aligned}$$ where $V_{ij}$ and $g_{ij}$ are the vdW interaction and the pair correlation function between the corresponding components respectively. The very complex expression for ${\cal U}_{vdW}$ can be approximated by a much simpler form in systems with particular properties of $V_{ij}$, and in special thermodynamic states, such as the neighborhood of the critical point. Close to the critical point and for small amount of ions the correlation and screening lengths are large, and the local densities vary slowly on the microscopic length scale. Thus, the density $\rho_j^*({\bf r'})$ can be Taylor expanded about ${\bf r}$. In the case of attractive interactions the expansion can be truncated at the second-order therm [@ciach:10:0; @ciach:13:03]. As a result, ${\cal U}_{vdW}$ is given by a single integral over ${\bf r}$ with the integrand depending on the densities and their gradients at ${\bf r}$, and on the zeroth, $J_{ij}=-\int_{V}d{\bf r}V_{ij}({\bf r})g_{ij}({\bf r})$, and the second, $\bar J_{ij}=-\int_{V}d{\bf r}V_{ij}({\bf r})g_{ij}({\bf r})r^2/6$, moments of $V_{ij}g_{ij}$ [@ciach:10:0; @ciach:13:03]. Since we consider two inorganic (water and cation) and two organic (metylpyridine and anion) components, we can further simplify the expression for $U_{vdW}$ by assuming appropriate relations between $J_{ij}$. We assume that the differences between the interactions of all the inorganic components are negligible; likewise, we neglect the differences between the interactions of all the organic components. Accordingly, we assume
$$\begin{aligned}
\label{J1}
J_{ww}\approx J_{++} \approx J_{w+}\\\nonumber
J_{ll} \approx J_{--} \approx J_{l-}\\\nonumber
J_{wl}\approx J_{w-} \approx J_{l+} \approx J_{+-}.\end{aligned}$$
with analogous assumptions for $\bar J_{ij}$. After some algebra we obtain $$\begin{aligned}
\label{uvdW0}
{\cal U}_{vdW}=U_{vdW}[\Phi]
+J_1\int_V \Phi({\bf r})d{\bf r} +J_0V ,\end{aligned}$$ where $$\begin{aligned}
\label{uvdW}
U_{vdW}[\Phi]=\frac{J}{2}\int_V \Big[-b{\Phi({\bf r})}^2 + (\nabla {\Phi({\bf r})})^2 \Big]d{\bf r}\end{aligned}$$ and $$\begin{aligned}
\label{OP}
\Phi({\bf r})= \rho_w^*({\bf r})-\rho_l^*({\bf r}) +\rho_q^*({\bf r})\end{aligned}$$ is the difference between the local dimensionless density of the inorganic and organic components (see (\[rhoqdef\])), $J_1=\frac{1}{4}(J_{ww}-J_{ll})$, $J_0=\frac{1}{8}(J_{ww}+J_{ll}+2J_{wl})$, and $$\begin{aligned}
J=\frac{1}{4}(\bar J_{ww}+\bar J_{ll}-2\bar J_{wl})
\end{aligned}$$ is the energy parameter relevant for the phase separation. The parameter $b$ is associated with the difference between the second and the zeroth moments of the interactions. In the case of the lattice model with nearest-neighbor interactions $b=2d$ in a $d$-dimensional system. We neglect the constant term in (\[uvdW0\]), and add the linear term in (\[uvdW0\]) to $-\mu_i\sum_i\rho_i^*({\bf r})$ in Eq.(\[OmegaL\]). As a consequence, the internal energy is given in Eq.(\[uvdW\]) and the chemical potentials are modified.
Finally, we postulate the lattice gas or ideal-mixing form for the entropy $$\label{S}
-TS=k_BT\int_V d{\bf r}\sum_{\{i= w, l, +, -\}}\rho_i^*({\bf r})\ln\rho_i^*({\bf r}).$$ Eq. (\[S\]) should be expressed in terms of the new variables $\Phi$, $\rho_q^*$ and the dimensionless ion density $$\begin{aligned}
\label{rhoion}
\rho_{ion}^*({\bf r})=\rho_+^*({\bf r})+\rho_-^*({\bf r}),\end{aligned}$$ since there are 3 independent variables when (\[fixrho\]) holds.
In this theory the thermal equilibrium for fixed temperature, total density and chemical potentials corresponds to the global minimum of the grand potential (\[OmegaL\]) with the electrostatic and the vdW contributions to the internal energy given in Eqs.(\[el\]) and (\[uvdW\]) respectively, and the entropy given in (\[S\]). The equilibrium forms of $\Phi({\bf r})$ and $\rho_{ion}^*({\bf r})$ are the solutions of the Euler-Lagrange (EL) equations obtained from the extremum conditions $\delta \Omega/\delta {\rho_{ion}^*({\bf r})}=0=\delta \Omega/\delta \Phi({\bf r})=0=\delta \Omega/\delta \rho_q^*({\bf r})$. Note that neither $U_{el}$ nor $U_{vdW}$ depends on $\rho_{ion}^*({\bf r})$. For this reason $\rho_{ion}^*({\bf r})$ can be easily expressed in terms of $\Phi({\bf r})$ and $\rho_q^*({\bf r})$ with the help of the equation $\delta \Omega/\delta {\rho_{ion}^*({\bf r})}=0$ (see Appendix A), and the grand potential becomes a functional of two fields, $\Phi({\bf r})$ and $\rho_{q}^*({\bf r})$.
properties of the disordered phase {#corr}
==================================
Landau-type functional in the Gaussian approximation
----------------------------------------------------
In thermodynamic conditions corresponding to stability of a uniform fluid $\rho_q^*({\bf r})=0$ and both $\Phi({\bf r})$ and $\rho_{ion}^*({\bf r})$ are independent of the space position, $\Phi({\bf r})=\bar \Phi$ and $\rho_{ion}^*({\bf r})=\bar \rho_{ion}^*$. For the stability analysis and in order to calculate the correlation functions we introduce the Landau-Ginzburg (LG) functional $$\label{L0}
{\cal L}[\varPhi({\bf r}), \rho_q^*({\bf r})]=\beta \Omega[ \Phi({\bf r}),\rho_{ion}^*({\bf r}),\rho_q^*({\bf r})]-
\beta\Omega[\bar\Phi,\bar\rho_{ion}^*,0],$$ where $\rho_{ion}^*({\bf r})$ is expressed in terms of $\Phi({\bf r})$ and $\rho_q^*({\bf r})$ (Appendix A) and $$\begin{aligned}
\label{fields}
\varPhi({\bf r})=\Phi({\bf r})-\bar \Phi.\end{aligned}$$
We write the internal energy in the Fourier representation (Appendix B). Next we Taylor-expand the entropy minimized with respect to $\rho_{ion}^*$ and approximate it by a second-order polynomial in $\varPhi$ and $\rho_q^*$. This way we obtain the LG functional in the Gaussian approximation, $$\label{calL}
{\cal L}_G[\varPhi({\bf k}),\rho({\bf k})]=\frac{1}{2}\int_V \frac{d{\bf k}}{(2\pi)^d}
\bigg \{ \tilde C_{\Phi\Phi}( k) \tilde \varPhi({\bf k})
\tilde \varPhi(-{\bf k})+2 \tilde C_{\Phi q}( k) \tilde \varPhi({\bf k})
\tilde \rho_q^*(-{\bf k})
+ \tilde C_{q q}( k) \tilde \rho_q^*({\bf k})\tilde \rho_q^*(-{\bf k}) \bigg\},
\hskip2cm$$ where $$\label{Ck}
\left\{
\begin{array}{l }
\tilde C_{\Phi\Phi}( k)= \beta^* (\xi_0^{-2}+k^2),
\vspace{0.7cm}\\
\tilde C_{\Phi q}( k)= -\frac{1}{(1-\bar \rho_{ion}^*-\bar\Phi^2)},
\vspace{0.7cm}\\
\tilde C_{q q}( k)= \frac{1}{\bar \rho_{ion}^*}+\frac{1}{(1-\bar \rho_{ion}^*-\bar\Phi^2)}
+\frac{1}{\bar \rho_{ion}^*\lambda_D^{*2}k^2 } ,
\end{array} \right.$$\
with $1/\beta^*=T^*=k_BT/J$, the dimensionless screening length $\lambda_D^*$ given by $$\label{lambdaD}
\lambda_D^{*2}=\lambda_D^2/a^2=\frac{a\epsilon}{4\pi e^2 \beta \bar\rho_{ion}^*},$$ and with $$\label{xi}
\xi_0= \bigg( \frac{T^*}{ (1-\bar \rho_{ion}^*-\bar\Phi^2)}-b\bigg)^{-1/2}.$$
For $ \tilde \rho_q^*({\bf k})=0$ Eq.(\[calL\]) reduces to the standard Landau functional for the upper critical point. $\xi_0$ is the dimensionless correlation length in the case of $\tilde \rho_q^*({\bf k})=0$ and $T^*>T^*_c$, where $T^*_c=b(1-\bar \rho_{ion}^*)$ is the critical temperature in the reduced units. $\bar\Phi=0$ at the critical point. The temperature of the upper critical point decreases after addition of the solute, because the entropy increases, and the energy is not changed in this model (see (\[J1\])). In the case of the lower critical point $T^*-T^*_c$ in (\[xi\]) should be replaced by $|T^*-T^*_c|$.
The structure and the boundary of stability of the disordered phase
-------------------------------------------------------------------
According to the density functional theory [@evans:79:0] the correlation function for concentration fluctuations, $\tilde G( k)$ is given by $$\label{G0}
\tilde G( k)= \frac{\tilde C_{q q }( k)}{\tilde C_{\Phi \Phi}( k)\tilde C_{q q }( k)
-\tilde C_{\Phi q}( k)^2},$$ and its explicit form can be easily obtained from Eq. (\[Ck\]), $$\label{G}
\tilde G( k)= \xi_0^{2}T^* \bigg\{ 1+\xi_0^2k^2
-\frac{ a_N\xi_0^2 k^2}{\lambda_D^{*2} k^2 +a_D}\bigg\}^{-1}.$$ We have introduced $$\begin{aligned}
\label{aN}
a_N= \frac{T^*\bar \rho_{ion}^* \lambda_D^{*2}}{ (1-\bar \rho_{ion}^*-\bar\Phi^2)(1-\bar\Phi^2)}
% \\ =\frac{ T^*}{4\pi \ell_B^*(1-\bar \rho_{ion}^*-\bar\Phi^2)(1-\bar\Phi^2)}\end{aligned}$$ and $$\label{aD}
a_D=\frac{(1-\bar \rho_{ion}^*-\bar\Phi^2)}{(1-\bar\Phi^2)}.$$ Our formula (\[G\]) is very similar to the one obtained in Ref.[@onuki:04:0]. The $k$-dependence is the same, but in Ref.[@onuki:04:0] $a_D=1$, and our expression for $a_N$ is somewhat different. We should stress that in our theory $\xi_0$ is equal to the correlation length in the system with suppressed charge waves.
Note that $ {\cal L}_G[\varPhi,\rho_q]$ in Eq.(\[calL\]) can be minimized with respect to $\rho_q^*$, because $\tilde C_{qq}(k)>0$. At the minimum $\tilde\rho_q^*({\bf k})=-\frac{\tilde C_{\Phi q}(k)}{\tilde C_{qq}(k)}\tilde \varPhi({\bf k})$ , and the functional takes the form $$\label{LC}
{\cal L}_G[\varPhi]=\frac{1}{2}\int_V \frac{d{\bf k}}{(2\pi)^d} \tilde C(k) \tilde \varPhi({\bf k})
\tilde \varPhi(-{\bf k})$$ where (see (\[G\])) $$\tilde C(k) = \tilde G^{-1}(k).$$ $\tilde C(k)$ takes the minimum $$\label{Ck0}
\tilde C(k_0)=\beta^*\Big[
\big(\frac{\lambda_D^*}{\xi_0}\Big)^{2}-(\sqrt a_N-\sqrt a_D)^2\Big]\lambda_D^{*-2}$$ for $k=k_0$ with $$\label{k0}
k_0^2\lambda_D^{*2}=\sqrt{a_D}
(\sqrt{a_N} -\sqrt{a_D})
%=\frac{\Big[\lambda_D
%\sqrt{ T^*\bar\rho_{ion}} -(1-\bar\rho_{ion}-\bar\Phi^2)
%\Big]}{(1-\bar\Phi^2)\lambda_D^2}$$ where $a_N$ and $a_D$ are given in (\[aN\]) and (\[aD\]). When $a_N<a_D$, then $\tilde C(k)$ takes the minimum at $k=0$. The minimum of $\tilde C(k)$ (or the maximum of $\tilde G(k)$) with $k_0>0$ occurs only if $a_N>a_D$, or explicitly $$(k_BT)^2>J\frac{4\pi e^2}{a\epsilon} (1-\bar\rho_{ion}^*-\bar\Phi^2)^2.$$ In our theory developed for an upper critical point the inhomogeneous structure in the disordered phase can occur when the thermal energy is larger than the geometric mean of the energy cost of the salt molecule dissociation and the vdW energy gain of phase-separated system, $J$. At the same time $J$ is the energy cost of a local interface (see the term $\frac{J}{2}\int d{\bf r}(\nabla \Phi)^2$ in Eq.(\[uvdW\])), and $4\pi e^2/(a\epsilon)$ is the energy gain when the two ions approach each other from the two sides of the interface.
The correlation function in the real-space representation can be obtained by the pole analysis of $\tilde G(k)$. We have found that the damped oscillatory and the monotonic decay of correlations occur for $(\sqrt a_N-\sqrt a_D)^2<(\lambda_D^{*}/\xi_0)^2<(\sqrt a_N+\sqrt a_D)^2$, and $(\lambda_D^{*}/\xi_0)>(\sqrt a_N+\sqrt a_D)$ respectively. The boundary between these two cases, $(\lambda_D^{*}/\xi_0)=(\sqrt a_N+\sqrt a_D)$ is called the disorder surface in the $(T^*,\bar\rho_{ion}^*,\bar\Phi)$ phase diagram, as in the amphiphilic systems [@ciach:01:2].
When $(\sqrt a_N-\sqrt a_D)=(\lambda_D^{*}/\xi_0)$ then $\tilde C(k_0)=0$ (i.e. $\tilde G(k_0)\to \infty$), and the homogeneous phase is at the boundary of stability in the mean-field approximation (MF). At the corresponding $\lambda$- surface in the $(T^*,\bar\rho_{ion}^*,\bar\Phi)$ phase space (we use this name by analogy with the $\lambda$-line [@ciach:00:0] in the two-dimensional phase space) $k_0$ is given by $$k_0^2\lambda_D^*\xi_0=\sqrt a_D\approx 1.$$ The wavelength of the concentration wave at the$\lambda$- surface is equal to a geometric mean of the inverse correlation length and the inverse screening length. This relation allows for a quick verification if mesoscopic inhomogeneities ($k_0\sim 2\pi/10 nm^{-1}$) can occur in the investigated experimental system.
Comparison with microemulsions
------------------------------
When we limit ourselves to the most probable concentration waves, with the wavenumbers near the maximum of the structure factor, $k\simeq k_0>0$, then we can make the approximation $$\label{C}
\tilde C(k) \simeq \tilde C(k_0)+ c(k^2-k_0^2)^2.$$ We took into account that $\tilde C(k)$ is a function of $k^2$, and truncated the Taylor expansion in $k^2$ about $k_0^2$ at the second-order term. After inserting (\[C\]) in (\[LC\]) we obtain the Landau-Brazovskii functional [@brazovskii:75:0] in the Gaussian approximation $$\label{LB}
{\cal L}_G[\varPhi]\approx\frac{c}{2}\int_V \frac{d{\bf k}}{(2\pi)^d}
\tilde \varPhi({\bf k}) [t_0+(k^2-k_0^2)^2]
\tilde \varPhi(-{\bf k})$$ where $k_0$ is given in (\[k0\]), $$\label{t0}
t_0=\tilde C(k_0)/c,$$ $\tilde C(k_0)$ is given in (\[Ck0\]) and $c=\beta^*\lambda_D^{*2}/\sqrt{a_Na_D}$. The Landau-Brazovskii functional was successfully used for a description of the structure of block copolymers [@leibler:80:0], binary or ternary surfactant mixtures [@gompper:94:0; @gozdz:96:1] and recently for the colloidal self-assembly [@ciach:13:03]. In each case the physical interpretation of the order-parameter $\varPhi$ is different. In particular, in the case of ternary surfactant mixtures $\varPhi$ is interpreted as a concentration difference between the polar and nonpolar components, in close analogy with the present case where $\varPhi$ is a concentration difference between the inorganic and organic components.
From Eq.(\[C\]) we can easily obtain the approximate expression for the correlation function in the real-space representation [@ciach:01:2], $$\label{Cr}
G(r)=\frac{\lambda}{2\pi r}e^{-r/\xi}\sin\big(\frac{2\pi r}{\lambda}\big),$$ where the dimensionless correlation length and period of the exponentially damped oscillatory decay, $\xi$ and $\lambda$, are given by $$\xi^2=\frac{2}{\sqrt{t_0+k_0^4}-k_0^2}$$ and $$\Big(
\frac{\lambda}{2\pi}
\Big)^2=\frac{2}{\sqrt{t_0+k_0^4}+k_0^2}.$$ Note that $\xi$ differs from $\xi_0$ and diverges at the boundary of stability of the disordered phase, $t_0=0$. Moreover, $\lambda\le 2\pi/k_0$; the equality holds when $t_0=0$.
We should mention that in the Brazovskii theory the dominant fluctuations ($k\simeq k_0$) lead to fluctuation-induced first-order phase transition to the lamellar phase beyond the Gaussian approximation [@brazovskii:75:0], or as shown recently to a transition to a nematic phase [@barci:13:0]. The $k$-dependence of $\tilde G(k)$ in the disordered phase remains almost unchanged beyond the Gaussian approximation, however, but $t_0$, and hence the model parameters, become renormalized [@brazovskii:75:0]. For this reason Eq.(\[G\]) can reasonably well describe the experimental results, but the MF theories cannot correctly predict the microscopic expressions for the parameters present in Eq.(\[G\]).
The similarity between our system and ternary surfactant mixtures was confirmed by the agreement of the experimental results for the former and the formulas developed for the latter case [@sadakane:13:0]. Here we have shown that starting from a simple microscopic density functional theory we can obtain by a systematic coarse-graining procedure the same Landau-Brazovskii functional (\[LB\]) that describes the amphiphilic systems.
Effects of confinement
======================
The Euler-Lagrange equations
----------------------------
In order to obtain the form of $\Phi$ and $\rho_q^*$ it is necessary to solve the EL equations for $\Phi$, $\psi$, $\rho_{ion}^*$ and $\rho_q^*$. The expressions corresponding to the full functional (\[OmegaL\]) with the electrostatic and the vdW contributions to the internal energy given in Eqs.(\[el\]) and (\[uvdW\]) respectively, and the entropy given in (\[S\]) are rather long, and are given in Appendix A. Here we discuss the linearized EL equations for $\bar\Phi=0$ (see (\[EL1\]) and (\[EL2\]) in the Appendix A) that can be written in the form $$\label{Philin}
\nabla^2\nabla^2\Phi({\bf r})+a_2\nabla^2\Phi({\bf r})+a_0\Phi({\bf r})=0$$ and $$\label{EL1l}
\rho_q^*(z)=\frac{(1-\bar \rho_{ion}^*)}{T^*}\Bigg[\xi_0^{-2} \Phi(z)-\nabla^2\Phi(z)\Bigg] ,$$ where $\nabla^2$ denotes the laplacian, $$\label{a2}
a_2
%=\frac{\bar\rho_{ion}T^*}{1-\bar\rho_{ion}}-\kappa^2 (1-\bar\rho_{ion})-\xi_0^{-2}
=[(a_N-a_D)-\lambda_D^{*2}\xi_0^{-2}]\lambda_D^{*-2},
% =-\Big( \frac{\lambda_D^2}{\sqrt{a_Na_D}}t_0-2k_0^2 \Big)$$ and $$a_0
%=\lambda_D^{-2}\xi_0^{-2}(1-\bar\rho_{ion})
=a_D\xi_0^{-2}\lambda_D^{*-2}.$$
Let us focus for simplicity on one-dimensional concentration profiles. The solution of (\[Philin\]) for $\varPhi$ depending only on $z$ is a linear combination of the exponential terms $\exp(\pm\lambda_iz)$ with $$\label{delta12}
\lambda_{1,2}^2=\frac{-a_2\pm\sqrt\Delta}{2}$$ where $\Delta$ can be written in the form $$\label{Delta}
\Delta =\Big[
\lambda_D^{*2}\xi_0^{-2}-(\sqrt a_N-\sqrt a_D)^2
\Big]\Big[
\lambda_D^{*2}\xi_0^{-2}-(\sqrt a_N+\sqrt a_D)^2
\Big]\lambda_D^{*-4}.$$ Note that $\Delta=0$ when either the first or the second factor in (\[Delta\]) vanishes. The first factor in (\[Delta\]) is proportional to $t_0$ (see (\[t0\])), hence $\Delta$ vanishes at the $\lambda$-surface $t_0=0$. The second factor vanishes at the disorder surface $\lambda_D^{*}/\xi_0=(\sqrt a_N+ \sqrt a_D) $. For $(\sqrt a_N-\sqrt a_D)^2<(\lambda_D^{*}/\xi_0)^2<(\sqrt a_N+\sqrt a_D)^2$ the $\lambda_i$ are complex conjugate numbers, because $\Delta<0$. For $\lambda_D^{*}/\xi_0>(\sqrt a_N+ \sqrt a_D) $ both $\lambda_i$ are real, while for $(\lambda_D^{*}/\xi_0)^2<(\sqrt a_N-\sqrt a_D)^2$ (i.e. $t_0<0$) both $\lambda_i$ are imaginary. Note that for $t_0=0$ we obtain an oscillatory function with the wavenumber of oscillations equal to $k_0$, in consistency with the results of the previous section.
The boundary conditions for the EL equations
--------------------------------------------
When the system is in contact with a selective and charged surface at $z=0$, then there is additional contribution to the internal energy [@ciach:10:0; @pousaneh:12:0; @pousaneh:14:0] $$U_s=\Big[J\Big(\frac{\Phi^2(0)}{2}-h_0\Phi(0)
\Big) + e\sigma_0\psi(0)\Big] A^*,$$ where $A^*$ is the dimensionless area of the confining surface, $\sigma_0$ is the dimensionless surface charge (the charge per area $a^2$), $h_0$ is the dimensionless surface field describing the preferential adsorption of the inorganic (for $h_0>0$) or organic (for $h_0<0$) components, and the first term follows from the missing fluid neighbors for $z\le 0$. In the case of a slit with another surface at $z=L$ there is analogous contribution to the internal energy. The above surface terms lead to the boundary conditions for the EL equations [@ciach:10:0; @pousaneh:14:0] $$\label{BCel}
\nabla\Phi(0)-\Phi(0)=-h_0, \hskip2cm \nabla\Phi(L)+\Phi(L)=h_L$$ and $$\label{BCel1}
\nabla \psi (0)=-\frac{4\pi e}{\epsilon}\sigma_0,\hskip2cm \nabla \psi (L)=\frac{4\pi e}{\epsilon}\sigma_L.$$ In the linearized theory $$\psi({\bf r})=\frac{k_BT}{e(1-\bar\rho_{ion}^*)}\Big[\Phi({\bf r})-\frac{\rho_q^*({\bf r})}{\bar\rho_{ion}^*}\Big]$$ when $\bar\Phi=0$, and from (\[EL1l\]) we obtain using (\[BCel1\]) the second boundary conditions for the equation (\[Philin\]) $$\label{BC2}
\Phi^{'''}(0)-\zeta^{-2}\Phi'(0)
=-\frac{T^*}{\lambda_D^{*2}}\sigma_0,\\ \hskip2cm
\Phi^{'''}(L)-\zeta^{-2}\Phi'(L)
=\frac{T^*}{\lambda_D^{*2}}\sigma_L$$ where $$\label{zeta}
\zeta^{-2}=\xi_0^{-2}-\frac{T^*\bar\rho_{ion}^*}{1-\bar\rho_{ion}^*}$$
The local concentration and the local charge density in a semiinfinite system
-----------------------------------------------------------------------------
We shall limit ourselves to a near-surface structure in the disordered phase and to the critical composition $\bar\Phi=0$. The composition of the near-surface layer depends on both the wall-fluid van der Waals interactions $h_0$ and on the sign and value of the surface charge $\sigma_0$. A hydrophilic surface attracts water, and a negatively charged surface attracts water-soluble ions. In contrast, the positively charged surface attracts ions soluble in the organic solvent, and in the case of the positively charged hydrophilic surface the composition in its vicinity depends on the ratio $h_0/\sigma_0$. We shall limit ourselves to $h_0>0$.
Let us first focus on the case of $\lambda_D^{*}/\xi_0>(\sqrt a_N+ \sqrt a_D) $, where both $\lambda_i$ are real numbers. From (\[Philin\]) and (\[EL1l\]) we have $$\label{Phi(z)}
\Phi(z)=A_1e^{-\lambda_1 z}+A_2e^{-\lambda_2 z}$$ and $$\label{rhoq(z)}
\rho_q^*(z)=\frac{(1-\bar \rho_{ion}^*)}{T^*}\Bigg[
(\xi_0^{-2}-\lambda_1^2)A_1e^{-\lambda_1 z}+(\xi_0^{-2}-\lambda_2^2)A_2e^{-\lambda_2 z}\Bigg]$$ where $A_i$ are determined by the boundary conditions and depend on $\sigma_0$ and $h_0$ (see Appendix C). Note that the asymptotic decay at large separation of both the excess concentration and the charge density is given by the inverse decay length $\min(\lambda_1,\lambda_2)$, while in the case of the hydrophilic salt the decay length of the excess concentration is $\xi_0$, and the decay length of the charge is $\lambda_D^*$. This difference follows from the coupling of the concentration and charge fluctuations already in the Gaussian approximation in the case of the antagonistic salt.
The two decay lengths approach $\xi_0$ and $\lambda_D^*$ only far away from the disorder line, i.e. for $\lambda_D^{*}/\xi_0\gg(\sqrt a_N+ \sqrt a_D) $ (low ionic strength, away from the critical point). In the limit of $\xi_0/\lambda_D^{*}\to 0$ Eqs.(\[Phi(z)\]) and (\[rhoq(z)\]) take the forms $$\Phi(z)\simeq h_0e^{-z/\xi_0}$$ and $$\rho_q^*(z)\simeq -\frac{\sigma_0}{\lambda_D^*}e^{-z/\lambda_D^*},$$ where we took into account that $a_D\approx 1$.
The difference between the two decay lengths decreases when the disorder line ($\lambda_D^{*}/\xi_0=(\sqrt a_N+ \sqrt a_D) $) is approached, and at the disorder line they become identical, $$\lambda_1^2=\lambda_2^2
%=\lambda_D^{*-2}(a_D+\sqrt{a_Na_D})
=\frac{\sqrt a_D}{\lambda_D^*\xi_0}.$$
For $(\sqrt a_N- \sqrt a_D)<\lambda_D^{*}/\xi_0<(\sqrt a_N+ \sqrt a_D) $ Eq.(\[Phi(z)\]) can be written inthe form $$\Phi(z)=A\cos(\lambda_{im}z+\theta)e^{-\lambda_{re}z}$$ with similar damped oscillatory decay of $\rho_q^*(z)$, because $\lambda_1=\lambda_{re}+i\lambda_{im}$ and $\lambda_2=\lambda_{re}-i\lambda_{im}$ are complex conjugate numbers.
The structure and effective potential between parallel surfaces
---------------------------------------------------------------
We consider two selective and charged surfaces ($h_{0}, \sigma_0$ and $h_L,\sigma_L$) which are separated by the distance L. We limit ourselves to identical surfaces, with $h=h_0=h_L$ and $\sigma=\sigma_0=\sigma_L$. The local concentration has the form $$\label{OPcon}
\Phi(z)={\cal A}_1[e^{-\lambda_1 z}+e^{-\lambda_1 (L-z)}]+{\cal A}_2[e^{-\lambda_2 z}+e^{-\lambda_2 (L-z)}]$$ in the case of the structureless fluid, and $$\label{OPcon1}
\Phi(z)={\cal A}[\cos(\lambda_{im}z+\vartheta)e^{-\lambda_{re}z}+\cos(\lambda_{im}(L-z)+\vartheta)e^{-\lambda_{re}(L-z)}]$$ in the presence of mesoscopic inhomogeneities. The expressions for the amplitudes and the phase are rather long and will not be given here. The results of the linear theory for the excess concentration profiles, Eqs.(\[OPcon\]) and (\[OPcon1\]), are compared with the numerical solutions of the full EL equations, Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]) for $T^*=6$ in Figs. \[OP1\]-\[OP4\].
As Fig. \[OP1\] shows, for small surface charges, surface fields and ion concentrations, and for temperatures far from the critical point, the linear theory agrees very well with the numerical solutions of the EL equations. Figs. \[OP2\] and \[OP3\] show the appearance of the periodic structure upon approaching the critical point and upon increasing $\lambda_D^*$. Finally, Fig. \[OP4\] indicates that for bigger surface charges and surface fields the linear theory differs significantly from the numerical results, however the qualitative agreement is preserved.\
![ The dimensionless concentration profile defined in Eq. \[OP\] (left) and the dimensionless charge density (right). Solid lines correspond to numerical results of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), while dash lines show analytical results (Eqs. (\[Phi(z)\]), (\[rhoq(z)\])). Dimensionless parameters are $\lambda_D^*= 5$, $|T/T_c-1|= 0.1$, $\bar \rho_{ion}^* = 0.001$, $h= 0.001$, $\sigma= -0.001$ for top curves and $\sigma= 0.001$ for bottom curves. The distance from the left wall $z$ is in units of the microscopic length $a\approx 0.4 nm$. []{data-label="OP1"}](./Figures/Fig1_a.eps "fig:") ![ The dimensionless concentration profile defined in Eq. \[OP\] (left) and the dimensionless charge density (right). Solid lines correspond to numerical results of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), while dash lines show analytical results (Eqs. (\[Phi(z)\]), (\[rhoq(z)\])). Dimensionless parameters are $\lambda_D^*= 5$, $|T/T_c-1|= 0.1$, $\bar \rho_{ion}^* = 0.001$, $h= 0.001$, $\sigma= -0.001$ for top curves and $\sigma= 0.001$ for bottom curves. The distance from the left wall $z$ is in units of the microscopic length $a\approx 0.4 nm$. []{data-label="OP1"}](./Figures/Fig1_b.eps "fig:")
![ The dimensionless concentration profile. Solid lines correspond to numerical results of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), while dash lines show analytical results, Eq. \[Phi(z)\]. The dimensionless parameters are $h= 0.001$, $\sigma= -0.001$, $\lambda_D^*= 10$, $\bar \rho_{ion}^* = 0.005$, (thin curves) $ |T/T_c-1|= 0.002$, and (thick curves) $|T/T_c-1|= 0.005$. The distance from the left wall $z$ is in units of the microscopic length $a\approx 0.4 nm$. []{data-label="OP2"}](Figures/Fig2.eps)
![The dimensionless concentration profile. Solid lines correspond to numerical results of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), while dash lines show analytical results, Eq. \[Phi(z)\]. The dimensionless parameters are $h= 0.001$, $\sigma= -0.001$, $ |T/T_c-1|= 0.001$, (thick curves) $\lambda_D^*= 20$, $\bar \rho_{ion}^* = 0.00125$, and (thin curves) $\lambda_D^*= 10$, $\bar \rho_{ion}^* = 0.005$. The distance from the left wall $z$ is in units of the microscopic length $a\approx 0.4 nm$. []{data-label="OP3"}](Figures/Fig3.eps)
![ The dimensionless concentration profile (left) and the dimensionless charge density (right). The dimensionless parameters are $h= 0.03$, $\sigma= -0.01$, $\lambda_D^*= 10$, $\bar \rho_{ion}^* = 0.005$ and $| T/T_c-1|= 0.002$. The distance from the left wall $z$ is in units of the microscopic length $a\approx 0.4 nm$.[]{data-label="OP4"}](./Figures/Fig4_a.eps "fig:") ![ The dimensionless concentration profile (left) and the dimensionless charge density (right). The dimensionless parameters are $h= 0.03$, $\sigma= -0.01$, $\lambda_D^*= 10$, $\bar \rho_{ion}^* = 0.005$ and $| T/T_c-1|= 0.002$. The distance from the left wall $z$ is in units of the microscopic length $a\approx 0.4 nm$.[]{data-label="OP4"}](./Figures/Fig4_b.eps "fig:")
The excess grand potential (\[L0\]) consists of the surface tension contribution, $(\gamma_0+\gamma_L)A^*$, and the effective potential between the confining surfaces of the dimensionless area $A^*$, $A^*\Psi(L)$. When the concentration $\Phi$ and the charge density $\rho_q^*$ satisfy the EL equations (\[EL2\]) and (\[EL1\]), then $$\begin{aligned}
\label{pot}
\gamma_0+\gamma_L+\Psi(L)=-\frac{J}{2}\Bigg[\Big(h_0 -\frac{\sigma_0T^*}{1-\bar\rho_{ion}^*}\Big)\Phi(0)+
\Big( h_L-\frac{\sigma_LT^*}{1-\bar\rho_{ion}^*}\Big)\Phi(L)\\
\nonumber
+\frac{T^*}{\bar\rho_{ion}^*(1-\bar\rho_{ion}^*)}\Big(\sigma_0\rho_q(0)+\sigma_L\rho_q(L)\Big)\Bigg].
\end{aligned}$$ The results of the previous section give us for $\lambda_D^{*}/\xi_0>(\sqrt a_N+ \sqrt a_D) $ $$\label{Psi(L)}
\Psi(L)=C_1e^{-\lambda_1 L}+C_2e^{-\lambda_2 L}$$ and for $(\sqrt a_N- \sqrt a_D)<\lambda_D^{*}/\xi_0<(\sqrt a_N+ \sqrt a_D) $ the above can be written in the form $$\Psi(L)=C\cos(\lambda_{im}L+\theta)e^{-\lambda_{re}L}$$ where the expressions for the amplitudes are too long to be given here.
In Figs. \[pot.1\]-\[pot.2\] the effective potential per microscopic area $a^2$ between identical surfaces, obtained in the linearized and the nonlinear theory is presented for different $\lambda_D^*$, $T^*$ and different surface charge and selectivity of the surfaces. Fig. \[pot.1\] presents a very good agreement at large distances between the linear theory and the full EL equations for small surface fields and surface charge, and for temperatures far from the critical point. When the critical point of the binary mixture is approached and the disorder surface is crossed, an oscillatory force between the surfaces is seen (Fig. \[pot.2\] (right)). Note that for all the considered cases there is qualitative agreement between the linear theory and the numerical results.
![The effective potential per microscopic area $a^2$ between two identical surfaces. Solid line corresponds to numerical solutions of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]) and dash line corresponds to Eq.(\[pot\]). The dimensionless parameters are $h= 0.001$, $\sigma= 0.001$, $\lambda_D^*= 5$, $ |T/T_c-1|= 0.1$ and $\bar \rho_{ion}^* = 0.001$. []{data-label="pot.1"}](Figures/Fig5.eps)
![ The effective potential per microscopic area $a^2$ between two identical surfaces. Solid line corresponds to numerical solutions of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]) and dash line corresponds to Eq.(\[pot\]). The dimensionless parameters are $h= 0.001$, $\sigma= -0.001$, $\lambda_D^*= 15$, $\bar \rho_{ion}^* = 0.005$ and (Left) $|T/T_c-1|= 0.005$ and (right) $ |T/T_c-1|= 0.002$.[]{data-label="pot.2"}](Figures/Fig6_a.eps "fig:") ![ The effective potential per microscopic area $a^2$ between two identical surfaces. Solid line corresponds to numerical solutions of the full EL equations Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]) and dash line corresponds to Eq.(\[pot\]). The dimensionless parameters are $h= 0.001$, $\sigma= -0.001$, $\lambda_D^*= 15$, $\bar \rho_{ion}^* = 0.005$ and (Left) $|T/T_c-1|= 0.005$ and (right) $ |T/T_c-1|= 0.002$.[]{data-label="pot.2"}](Figures/Fig6_b.eps "fig:")
The potential between surfaces of area $400nm\times400nm \approx 10^6a^2 $ is $10^6\Psi(L)$. Note that for such a mesoscopic surface the first two extrema in Fig.\[pot.2\] are both of order of $k_BT$.
the case of experimentally studied samples {#cexp}
==========================================
In this section we apply our theory to the systems studied experimentally in Ref.[@sadakane:13:0; @sadakane:09:0; @leys:13:0]. In the experiments the antagonistic salt sodium tetraphenylborate (NaBPh$_4$) was added to the $3-$methylpyridine (3MP) and heavy water mixture near its lower critical point (LCP). Small-angle neutron scattering (SANS) was performed and a periodic structure with a periodicity of about $10 nm$ was reported in the experiment. The structure factor of the ternary mixture was determined for a few salt concentrations and a few temperatures. The structure factor was fitted to the expression obtained by Onuki and Kitamura [@onuki:04:0] that has the same $k$ dependence as our Eq.(\[G\]).
Our theory has been developed for the upper critical point. In order to apply it to the system that phase separates for increasing temperature, we assume that in the coarse-grained description of the considered mixture the interaction parameter $J$ depends on $T$ in such a way that near the LCP $T^*=k_BT/J(T)$ decreases for increasing $T$. We shall not attempt to reproduce the phase diagram. Our purpose is a calculation of the effective potential between the confining surfaces for the samples studied experimentally in Refs.[@sadakane:13:0; @sadakane:09:0; @leys:13:0]. In order to model the particular samples, we assume that $a=0.4 nm$, $\bar\Phi=0$ for the samples with near-critical composition, $\lambda_D^*$ is given in Eq.(\[lambdaD\]) and we fit Eqs.(\[G\])-(\[aD\]) to the measured scattering intensity $S(k)$, assuming that $S(k)=S_0\tilde G(k)$. The remaining parameters are taken from Ref.[@sadakane:13:0; @leys:13:0]. The selected samples and the parameters obtained from the fitting are given in Table I, and the fitting of the formulas (\[G\])-(\[aD\]) to the experimental curves is shown in Fig.\[fitLey\].
![ SANS intensity measurements for near-critical composition of the 3MP, D$_2$O and NaBPh$_4$ mixture. Top: $T= 293K$ (bottom curve), $T= 313 K$ (top curve) and $\rho_{ion}=7.2mM/L$ [@leys:13:0]. Bottom: $T=280K$, $\rho_{ion}= 300mM/L$ (top curve), and $\bar\rho_{ion}= 1mM/L$ (bottom curve)[@sadakane:13:0]. The remaining parameters are shown in Table \[tab1\]. The solid lines are the theoretical prediction of the correlation function, Eq.(\[G\]). The fit for $\bar\rho_{ion}= 300mM/L$ is better than in Ref.[@sadakane:13:0] because we take into account the dependence of $a_D$ on $\bar\rho_{ion}$.[]{data-label="fitLey"}](Figures/Fig7_a.eps "fig:") ![ SANS intensity measurements for near-critical composition of the 3MP, D$_2$O and NaBPh$_4$ mixture. Top: $T= 293K$ (bottom curve), $T= 313 K$ (top curve) and $\rho_{ion}=7.2mM/L$ [@leys:13:0]. Bottom: $T=280K$, $\rho_{ion}= 300mM/L$ (top curve), and $\bar\rho_{ion}= 1mM/L$ (bottom curve)[@sadakane:13:0]. The remaining parameters are shown in Table \[tab1\]. The solid lines are the theoretical prediction of the correlation function, Eq.(\[G\]). The fit for $\bar\rho_{ion}= 300mM/L$ is better than in Ref.[@sadakane:13:0] because we take into account the dependence of $a_D$ on $\bar\rho_{ion}$.[]{data-label="fitLey"}](Figures/Fig7_b.eps "fig:")
$T (K) $ $\rho_{ion}(mM/L)$ $\lambda_D(nm)$ $\xi_0 (nm)$ $ a_N$ $S_0\tilde G_{\Phi\Phi}(0)$ $T^*$ $\Lambda_1 (nm)$ $\Lambda_2 (nm)$
--------------- -------------------- ----------------- -------------- ---------- ----------------------------- ------- ------------------- -------------------
313 7.2 2.6 5.82 $ 1.20$ $1.42\cdot 10^{-8}$ $106$ $5.5$ $ 34.6$
293 7.2 2.6 2.16 $ 1.73 $ $1.55\cdot 10^{-8}$ $152$ $2.94$ $ 25.1$
280 300 0.3 $3.89$ $ 1.05$ $0.03875$ 160 1.90 8.26
280 1 5.2 $0.75$ $ 1.03 $ $7.59\cdot 10^{-8}$ 160 0.76 5.16
\[fitLeytab\]
: Parameters characterizing the samples studied in Ref.[@leys:13:0] (the first two rows) and in Ref.[@sadakane:13:0] (the last two rows). The data in the first three columns are taken or computed from the data given in Refs.[@leys:13:0; @sadakane:13:0]. The data in the columns 4-6 are obtained from the fitting of the SANS intensity to the correlation function $\tilde G$ given in Eq.(\[G\])-(\[aD\]). The data in column 7 are computed from Eq.(\[aN\]). For the first three rows the columns 8 and 9 show the decay length and the period of the damped oscillatory decay, i.e. $\Lambda_1=1/\lambda_{re}$ and $\Lambda_2=2\pi/\lambda_{im}$ (Eq.\[lambdare\]). In the last row (1 mM/L sample) the two decay lengths of the double exponential decay are shown, i.e. $\Lambda_i=1/\lambda_i$ (see Eq.(\[delta12\])). []{data-label="tab1"}
The excess concentration profiles and the effective potential for the samples studied in Ref.[@leys:13:0] and in Ref.[@sadakane:13:0] are shown in Figs. \[OPLey\] and \[OPSadakane\] respectively. We plot the results of the full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), between two identical weakly selective and weakly charged surfaces, for the parameters given in Table. \[tab1\]. To obtain $ |T/T_c-1|$ from $\xi_0$ we use the critical exponent $\nu=0.63$ instead of $1/2$ in Eq.(\[xi\]). The structure factor takes a maximum for $k=k_0>0$ for three of the selected samples, and for $k_0=0$ for the low salt concentration, $\rho_{ion}=1 mmol/L$. For $k_0>0$ the exponentially damped oscillatory decay of correlations occurs, but the decay length is short compared to the period of oscillations (table I). For this reason the second extremum is much smaller than the first one, and is not seen on the plot. In the $k_0=0$ case the decay of correlations is given by two exponential functions.
![ (Left) Excess concentration profile (dimensionless) and (right) the effective potential per microscopic area $a^2$ (with $a\approx 0.4nm$) resulted from full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), as a function of the distance between two identical surfaces. The parameters are selected according to experiment [@leys:13:0] for $7.12 mM/L$ of $NaBPh_4$ added to $D_2O$ and metylpyridine at $313 K$ (dash line) and $293 K$ (solid line) given in Table. \[tab1\]. The dimensionless selectivity and surface charge are $h=0.001$ and $\sigma=0.001$ respectively. []{data-label="OPLey"}](Figures/Fig8_a.eps "fig:") ![ (Left) Excess concentration profile (dimensionless) and (right) the effective potential per microscopic area $a^2$ (with $a\approx 0.4nm$) resulted from full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), as a function of the distance between two identical surfaces. The parameters are selected according to experiment [@leys:13:0] for $7.12 mM/L$ of $NaBPh_4$ added to $D_2O$ and metylpyridine at $313 K$ (dash line) and $293 K$ (solid line) given in Table. \[tab1\]. The dimensionless selectivity and surface charge are $h=0.001$ and $\sigma=0.001$ respectively. []{data-label="OPLey"}](Figures/Fig8_b.eps "fig:")
![ (Top row) Excess concentration profiles (dimensionless) and (bottom row) the effective potential per microscopic area $a^2$ (with $a\approx 0.4nm$) resulted from full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), as a function of the distance between two identical surfaces. The parameters are selected according to experiment [@sadakane:13:0] for the $300 mM/L$ (solid lines) and $1 mM/L$ (dash lines) of $NaBPh_4$ added to $D_2O$ and metylpyridine sample at $T=280K$ [@sadakane:13:0], given in Table. \[tab1\]. The dimensionless selectivity and surface charge are $h=0.001$ and $\sigma=0.001$ respectively.[]{data-label="OPSadakane"}](./Figures/Fig9_a.eps "fig:") ![ (Top row) Excess concentration profiles (dimensionless) and (bottom row) the effective potential per microscopic area $a^2$ (with $a\approx 0.4nm$) resulted from full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), as a function of the distance between two identical surfaces. The parameters are selected according to experiment [@sadakane:13:0] for the $300 mM/L$ (solid lines) and $1 mM/L$ (dash lines) of $NaBPh_4$ added to $D_2O$ and metylpyridine sample at $T=280K$ [@sadakane:13:0], given in Table. \[tab1\]. The dimensionless selectivity and surface charge are $h=0.001$ and $\sigma=0.001$ respectively.[]{data-label="OPSadakane"}](./Figures/Fig9_b.eps "fig:") ![ (Top row) Excess concentration profiles (dimensionless) and (bottom row) the effective potential per microscopic area $a^2$ (with $a\approx 0.4nm$) resulted from full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), as a function of the distance between two identical surfaces. The parameters are selected according to experiment [@sadakane:13:0] for the $300 mM/L$ (solid lines) and $1 mM/L$ (dash lines) of $NaBPh_4$ added to $D_2O$ and metylpyridine sample at $T=280K$ [@sadakane:13:0], given in Table. \[tab1\]. The dimensionless selectivity and surface charge are $h=0.001$ and $\sigma=0.001$ respectively.[]{data-label="OPSadakane"}](./Figures/Fig9_c.eps "fig:") ![ (Top row) Excess concentration profiles (dimensionless) and (bottom row) the effective potential per microscopic area $a^2$ (with $a\approx 0.4nm$) resulted from full EL equations, (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]), as a function of the distance between two identical surfaces. The parameters are selected according to experiment [@sadakane:13:0] for the $300 mM/L$ (solid lines) and $1 mM/L$ (dash lines) of $NaBPh_4$ added to $D_2O$ and metylpyridine sample at $T=280K$ [@sadakane:13:0], given in Table. \[tab1\]. The dimensionless selectivity and surface charge are $h=0.001$ and $\sigma=0.001$ respectively.[]{data-label="OPSadakane"}](./Figures/Fig9_d.eps "fig:")
Note the change of the shape of $\Psi(L)$ with increased amount of salt (Fig.\[OPSadakane\]). Because of the very short screening length the electrostatic repulsion that dominates in the $1mM/L$ sample is suppressed. In the $300mM/L$ sample a repulsion barrier $\sim 0.6 k_BT$ for surfaces of area $1.6\cdot 10^5nm^2$ at the separation $L\approx 5 nm$ occurs, and the barrier is followed by a strong attraction for $L<3nm$. For the intermediate salt concentration, $\bar\rho_{ion}=7.2 mM/L$ a minimum of the potential of the depth $\sim 0.5 k_BT$ for surfaces of area $1.6\cdot 10^5nm^2$ occurs for $L\simeq 20nm$.
Discussion and summary
======================
We have developed a Landau-type functional for a near-critical mixture with addition of antagonistic salt. We used the same coarse-graining procedure as in the case of the inorganic salt [@ciach:10:0; @pousaneh:12:0; @pousaneh:14:0]. In both cases we have postulated that the OP for the phase separation, $\Phi$, is the concentration difference between the inorganic and organic components. While in the case of the inorganic salt $\Phi=\rho_w^*-\rho_l^*+\rho_{ion}^*$, in the case of the antagonistic salt $\Phi=\rho_w^*-\rho_l^*+\rho_q^*$. Since there are two types of ions, we have postulated that the entropy has a form of the ideal entropy of mixing of a four-component mixture. The entropy of mixing has quite different form for the two different OP when expressed in terms of $\Phi$, $\rho_{ion}^*$ and $\rho_q^*$. As a consequence, the linearized EL equations for $\Phi$ and $\rho_q^*$ are coupled in the case of the antagonistic salt, and decoupled in the case of the inorganic salt. In the absence of the coupling of the linearized EL equations the decay lengths of $\Phi$ and $\rho_q^*$ are $\xi_0$ and $\lambda_D^*$ respectively. In the presence of the coupling the decay of both $\Phi(z)$ and $\rho_q^*(z)$ is given by the decay lengths $1/\lambda_1,1/\lambda_2$ that differ from $\xi_0$ and $\lambda_D^*$. Another important consequence of the above coupling between $\Phi$ and $\rho_q^*$ is the lack of a qualitative difference between the solutions of the linearized and nonlinear EL equations. On the other hand, in our theory for the inorganic salt qualitatively different results in the linearized and nonlinear theories have been obtained [@pousaneh:11:0; @pousaneh:14:0].
We should note that our theory is similar to the theory developed by Onuki and Kitamura [@onuki:04:0] and studied in Ref.[@okamoto:11:0]. The main difference between the two theories is the OP of the phase separation. In Refs.[@onuki:04:0; @okamoto:11:0] the OP is identified with the concentration in the binary mixture, $\rho_w^*-\rho_l^*$, while we define the OP in Eq.(\[OP\]). We take into account the van der Waals interactions between the ions, and consider the entropy of the four-component mixture, instead of a sum of the entropies of the two 2-component subsystems. Important advantage of our approach is the link between the mesoscopic and the microscopic description that was a starting point of our derivation. Thanks to this link our theory is justified on a more fundamental level. However, in the case of the antagonistic salt the difference between our theory and the theory of Onuki and Kitamura is less significant than in the case of the inorganic salt. In the latter case the nonlinear coupling between $\Phi$ and $\rho_q^*$ plays a crucial role in our theory, and is ignored in Ref.[@onuki:04:0; @okamoto:11:0].
The effect of very small amount of solute should be independent of its kind. Our functionals for the inorganic and antagonistic salt, however, differ significantly from each other. Nevertheless, we have verified that for small amount of ions and not very close to the critical point (for example for $\rho_{ion}^*=0.001$ and $|T/T_c-1|=0.005$) the effective potential between parallel external surfaces has essentially the same form for the inorganic and antagonistic salt. For larger $\rho_{ion}^*$ a quantitative difference between $\Psi(L)$ in the presence of the inorganic or the antagonistic salt can be seen (Fig.\[pot.11\]). Qualitatively different shapes of $\Psi(L)$ are obtained if the correlation length is sufficiently large compared to the period of the concentration oscillations. The same amount of antagonistic salt leads to a deeper minimum of the effective potential at shorter separation than in the case of the inorganic salt, and the repulsion barrier occurs. This shape of the potential may occur when $\rho_{ion}^*$ is big enough, and the critical point is approached (Fig.\[pot.11\]).
We have obtained a mesoscopic functional of the Landau-Brazovskii form (\[LB\]) from a microscopic density-functional theory for a four-component mixture by a systematic coarse-graining procedure. A functional of the same form was successfully applied to amphiphilic systems [@gompper:94:0; @ciach:01:2]. Thus, our theoretical result and the experimental observations of similarity between the mesoscopic structure induced by amphiphiles and antagonistic salt [@sadakane:13:0] are complementary. We have limited to the Gaussian approximation here. It is well known that beyond the Gaussian approximation the fluctuations with the wavelength $k\simeq k_0$ yield a significant contribution to the correlation function, especially close to $t_0=0$. The $k$-dependence, however, is not changed up to a small correction; only $t_0$ is renormalized in the 1-loop approximation. We fitted our expression for $\tilde G(k)$ to the experimental curves. Since the decay of the OP and the shape of the effective potential are determined by the correlation function, in our results for $\Psi$ the renormalization of the model parameters is partially taken into account. This way we avoid the overestimation of the mesoscopic structure typical for the men-field results.
We have calculated the effective potential for 4 samples investigated experimentally in Ref.[@leys:13:0; @sadakane:13:0]. The maximum of the structure factor occurs for $k_0>0$ for three of these samples. The oscillatory concentration and effective potential can be seen only in the sample with the largest concentration of ions, however. This is because the period of oscillations, $2\pi/\lambda_{im}$ is large compared to the decay length $1/\lambda_{re}$ (Table I), whereas a detectable oscillatory potential can be expected for $1/\lambda_{re}>2\pi/\lambda_{im}$. Based on our results (Fig.\[OPSadakane\]) and on the analogy with surfactant mixtures where such forces were measured [@antelmi:99:0], we may expect oscillatory effective potential between the confining surfaces closer to the critical point or near the transition to the lamellar phase.
The potential in Figs.\[OPSadakane\] and \[pot.11\] is attractive at short separations and repulsive at larger separations (SALR). For the SALR-type potentials between colloid particles finite clusters are expected, because the barrier prevents the clusters from further growth [@stradner:04:0; @ciach:13:03].
We conclude that effective interactions between weakly charged colloid particles immersed in a near-critical mixture with antagonistic salt can exhibit very rich behavior. As a result, the particles can form different types of structures. Small changes of the amount of ions or temperature can lead to qualitative changes of the interactions between the particles. The sensitivity of the effective interactions to the thermodynamic state is a property that may allow for manipulating with the structure of colloids.
![ Effective potential obtained from numerical solutions of the full EL equations. Solid lines correspond to the current theory while the dash lines show the results of a theory of hydrophilic ions [@pousaneh:12:0; @pousaneh:14:0]. The parameters are $h=0.001$, $\sigma=-0.001$, $\lambda_D^*=10$, $\bar \rho_{ion}^* = 0.005$, (left) $| T/T_c-1|= 0.005$, (right) $| T/T_c-1|= 0.002$. $L$ is in units of the microscopic distance $a=0.4 nm$. []{data-label="pot.11"}](./Figures/Fig10_a.eps "fig:") ![ Effective potential obtained from numerical solutions of the full EL equations. Solid lines correspond to the current theory while the dash lines show the results of a theory of hydrophilic ions [@pousaneh:12:0; @pousaneh:14:0]. The parameters are $h=0.001$, $\sigma=-0.001$, $\lambda_D^*=10$, $\bar \rho_{ion}^* = 0.005$, (left) $| T/T_c-1|= 0.005$, (right) $| T/T_c-1|= 0.002$. $L$ is in units of the microscopic distance $a=0.4 nm$. []{data-label="pot.11"}](./Figures/Fig10_b.eps "fig:")
We thank Dr. Koichiro Sadakane and Dr. Jan Leys for providing us the data with their experimental structure factor. The work of FP was realized within the International PhD Projects Programme of the Foundation for Polish Science, cofinanced from European Regional Development Fund within Innovative Economy Operational Programme “Grants for innovation”. AC acknowledges the financial support by the NCN grant 2012/05/B/ST3/03302.
Appendices
==========
The Euler-Lagrange equations
-----------------------------
From $\delta \Omega(\Phi({\bf r}),\rho_q^*({\bf r}),\rho_{ion}({\bf r}))/\delta {\rho_{ion}^*({\bf r})}=0$ we obtain $$\label{mu2}
\mu_{ion}=\frac{T^*}{2}\ln\Bigg(
\frac{\rho_{ion}^{*2}({\bf r})-\rho_q^{*2}({\bf r})}
{(1-\rho_{ion}^*({\bf r}))^2-(\Phi({\bf r})-\rho_q^*({\bf r}))^2}
\Bigg) .$$ In a uniform fluid Eq.(\[mu2\]) takes the form $$\label{mu1}
\mu_{ion}=T^*\ln R,$$ where $$R^2=\frac{\bar\rho_{ion}^{*2}}{(1-\bar\rho_{ion})^{*2} -\bar\Phi^2}.$$ By equating RHS of Eqs. (\[mu1\]) and (\[mu2\]) we obtain $$\label{rhoelim}
\rho_{ion}^*({\bf r})=\frac{-R^2+\sqrt{R^2-(1-R^2)\Big[ R^2(\Phi({\bf r})-\rho_q^*({\bf r}))^2-\rho_q^*({\bf r})^2\Big] }}
{1-R^2}.$$ With the help of Eq. (\[rhoelim\]) we can eliminate $\rho_{ion}^*({\bf r})$ from Eq. (\[S\]). The remaining EL equations are obtained in a similar way, and have the forms $$\label{epsi}
e\beta\psi_{el}({\bf r})+\frac{1}{2}\ln\Bigg[
\frac{\Big(\rho_{ion}^*({\bf r})+\rho_q^*({\bf r})\Big)\Big(1-\Phi({\bf r})\Big) +
\Big(\rho_q^*({\bf r})^2 -\rho_{ion}^*({\bf r})^2\Big)}{\Big(\rho_{ion}^*({\bf r})-
\rho_q^*({\bf r})\Big)\Big(1+\Phi({\bf r})\Big) + \Big(\rho_q^*({\bf r})^2 -\rho_{ion}^*({\bf r})^2\Big)}
\Bigg]=0$$ and $$\label{EL3}
\frac{d^2\Phi({\bf r})}{d {\bf r}^2}= -b{\Phi({\bf r})}+\frac{T^*}{2}\ln\Bigg[
\frac{1-\rho_{ion}^*({\bf r})+\Phi({\bf r})-\rho_q^*({\bf r})}{1-\rho_{ion}^*({\bf r})-\Phi({\bf r})+\rho_q^*({\bf r})}
\Bigg].$$ Eqs. (\[rhoelim\])-(\[EL3\]) and (\[Poisson\]) form a closed set of two differential and two algebraic equations.
When $\rho_q^*({\bf r})$, $\varPhi$ and $$\begin{aligned}
\vartheta({\bf r})=\rho_{ion}^*({\bf r})-\bar\rho_{ion}^*\end{aligned}$$ are small, then we can solve analytically the linearized EL equations. Here we focus on the critical composition, $\bar\Phi=0$. Let us limit ourselves to the functions that depend only on $z$. From (\[EL3\]) we obtain $$\label{EL1}
\frac{d^2\Phi(z)}{dz^2}= \xi_0^{-2} \Phi(z) -\frac{T^*}{(1-\bar \rho_{ion}^*)}\rho_q^*(z),$$ and from (\[epsi\]) and (\[Poisson\]) we have $$\label{EL2}
\frac{d^2\rho_q^*(z)}{dz^2}=\kappa^2(1- \bar\rho_{ion}^*)\rho_q^*(z)+\bar \rho_{ion}^* \frac{d^2\Phi(z)}{dz^2}.$$
Internal energy in Fourier representation
-----------------------------------------
The electrostatic energy Eq. (\[el\]) in the Fourier representation is given by $$\label{elfou}
U_{el}=\frac{1}{2}\int_V \frac{d{\bf k}}{(2\pi)^d} \bigg(\frac{4\pi e^2}{a\epsilon k^2}\bigg )
\tilde \rho_q^*({\bf k})\tilde \rho_q^*({\bf k}),\\$$ where the wavenumber $k$ is dimensionless (in $a^{-1}$ -units) and we used the Poisson equation in Fourier representation $$\label{elfouk}
k^2 \tilde \psi_{el}({\bf k}) =\frac{4\pi e}{a\epsilon} \tilde \rho_q^* ({\bf k}).$$ The van der Waals contribution Eq. (\[uvdW\]) in the Fourier representation takes the form $$\label{vWfou}
U_{vdW}=\frac{J}{2}\int_V \frac{d{\bf k}}{(2\pi)^d} (-b+k^2) \tilde \Phi({\bf k}) \tilde \Phi(-{\bf k}).\\$$
Parameters in the solutions of the EL equations
-----------------------------------------------
The amplitudes in Eq.(\[Phi(z)\]) are $$\left\{
\begin{array}{l }
A_1=\frac{h_0(\lambda_2^2-\zeta^{-2})\lambda_2-
\frac{T^*\sigma_0}{\lambda_D^{*2}}(1+\lambda_2)}{D},
\vspace{0.4cm}\\
A_2=-\frac{h_0(\lambda_1^2-\zeta^{-2})\lambda_1-\frac{T^*\sigma_0}{\lambda_D^{*2}}(1+\lambda_1)}{D}.
\end{array} \right.$$ with $$D=(\lambda_1-\lambda_2)[\zeta^{-2}-\lambda_1^2-\lambda_2^2-\lambda_1\lambda_2(1+\lambda_1+\lambda_2)]$$
where $\lambda_i$ and $\zeta$ are given in Eq.(\[delta12\]) and (\[zeta\]) respectively. The real and imaginary parts of the inverse decay lengths $\lambda_i$ are $$\label{lambdare}
\left\{
\begin{array}{l }
\lambda_{re}^2=\frac{1}{4\lambda_D^{*2}}\Bigg[
\Big(
\sqrt{a_D} +\frac{\lambda_D^*}{\xi_0}
\Big)^2-a_N
\Bigg],
\vspace{0.4cm}\\
\lambda_{im}^2=\frac{1}{4\lambda_D^{*2}}\Bigg[a_N-
\Big(\frac{\lambda_D^*}{\xi_0}-
\sqrt{a_D}
\Big)^2
\Bigg] .
\end{array} \right.$$
[36]{}
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|
---
abstract: 'We calculate accurate bound states and resonances of two interesting perturbed Coulomb models by means of the Riccati-Padé method. This approach is based on a rational approximation to a modified logarithmic derivative of the eigenfunction and produces sequences of roots of Hankel determinants that converge towards the eigenvalues of the equation.'
author:
- |
Francisco M. Fernández\
INIFTA (Conicet, UNLP), División Química Teórica,\
Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16,\
1900 La Plata, Argentina\
E–mail: [email protected]
title: Calculation of bound states and resonances in perturbed Coulomb models
---
Introduction \[sec:intro\]
==========================
In a most interesting series of papers Killingeck et al [@KGJ04; @KGJ05; @KGJ06] and Killingbeck [@K07; @K07b] have shown that perturbation theory and the Hill–series method are suitable tools for the calculation of bound states and resonances of simple quantum–mechanical models. In order to obtain the complex eigenvalues that correspond to unstable states they resort to a complex parametrization of the methods that they call “complexification ”.
Another approach that proves useful for the accurate calculation of bound states and resonances is the Riccati–Padé method (RPM) based on a rational approximation to a modified (or regularized) logarithmic derivative of the eigenfunction [@FMT89a; @FMT89b; @F92; @FG93; @F95; @F95b; @F95c; @F96; @F96b; @F97]. In this paper we apply the RPM to the interesting perturbed Coulomb problems discussed recently by Killingeck [@K07b] with the purpose of challenging the recently developed asymptotic iteration method [@CHS03; @F04; @CHS05; @CHS05b; @F05; @B05; @B05b].
In Section \[sec:RPM\] we outline the main features of the RPM. In Section \[sec:boundstates\] we discuss a perturbed Coulomb model with interesting bound states. In Section \[sec:resonances\] we calculate the resonances for a slightly modified model with continuum states. Finally, in Section \[sec:conclusions\] we draw conclusions on the performance of the RPM.
The Riccati–Padé method (RPM)\[sec:RPM\]
========================================
Suppose that we want to obtain sufficiently accurate solutions to the eigenvalue equation $$\psi ^{\prime \prime }(x)+Q(x)\psi (x)=0,\;Q(x)=E-V(x)
\label{eq:Schrodinger}$$ where $Q(x)$ can be expanded as $$Q(x)=\sum_{j=0}^{\infty }Q_{j-2}x^{\beta j-2} \label{eq:Q_series}$$ about $x=0$. We transform the linear differential equation (\[eq:Schrodinger\]) into a Riccati one for the modified logarithmic derivative of the eigenfunction: $$f(x)=\frac{s}{x}-\frac{\psi ^{\prime }(x)}{\psi (x)} \label{eq:f(x)}$$ On substituting (\[eq:f(x)\]) into (\[eq:Schrodinger\]) we obtain $$f^{\prime }(x)+\frac{2s}{x}f(x)-f(x)^{2}-Q(x)-\frac{s(s-1)}{x^{2}}=0
\label{eq:Riccati}$$ We choose $s(s-1)=-Q_{-2}$ in order to remove the pole at origin, and, as a result, the expansion $$f(x)=x^{\beta -1}\sum_{j=0}^{\infty }f_{j}x^{\beta j} \label{eq:f_series}$$ for the solution to the Riccati equation (\[eq:Riccati\]) converges in a neighbourhood of $x=0$. Notice that if we substitute the expansions (\[eq:Q\_series\]) and (\[eq:f\_series\]) into the Riccati equation (\[eq:Riccati\]) we easily obtain the series coefficients $f_{j}$ as a function of $E$ and the known potential parameters $Q_{j}$.
We rewrite the partial sums of the expansion (\[eq:f\_series\]) as rational approximations $x^{\beta -1}[N+d/N](z)$, where $z=x^{\beta }$, in such a way that $$\lbrack N+d/N](z)=\frac{\sum_{j=0}^{N+d}a_{j}z^{j}}{\sum_{j=0}^{N}b_{j}z^{j}}=\sum_{j=0}^{2N+d+1}f_{j}z^{j}+\mathcal{O}(z^{2N+d+2}) \label{eq:Pade}$$ In order to satisfy this condition the Hankel determinant $H_{D}^{d}$, with matrix elements $f_{i+j+d-1}$, $i,j=1,2,\ldots ,D$, vanishes. Here, $D=N+1=2,3,\ldots $ is the determinant dimension and $d=0,1,\ldots $. The main assumption of the Riccati–Padé method (RPM) is that there is a sequence of roots $E^{[D,d]}$ of $H_{D}^{d}(E)=0$ for $D=2,3,\ldots $ that converges towards a given eigenvalue of equation (\[eq:Schrodinger\]). Comparison of sequences with different $d$ values is useful to estimate the accuracy of the converged results.
Notice that we do not have to take the boundary conditions explicitly into account in order to apply the RPM; the approach selects them automatically. In addition to the answers expected from physical considerations, the RPM also yields unwanted solutions as shown below.
Model with bound states \[sec:boundstates\]
===========================================
From the ansatz $\varphi (r,\lambda )=r\exp \left( -r-\lambda r^{2}\right) $ and the equation $\varphi ^{\prime \prime }(r,\lambda )/\left[ 2\varphi
(r,\lambda )\right] =V(r,\lambda )-E(\lambda )$ we derive a potential–energy function $V(r,\lambda )=-1/r+2\lambda r+2\lambda ^{2}r^{2}$ if $E(\lambda )=-1/2+3\lambda $. For $\lambda >0$ $\varphi (r,\lambda )$ and $E(\lambda )$ are a pair of eigenfunction and eigenvalue of the Schrödinger equation with the potential $V(r,\lambda )$. For $\lambda <0$ $E(\lambda )=-1/2+3\lambda $ is not an eigenvalue of the Schrödinger equation because the corresponding eigenfunction $\varphi (r,\lambda )$ is not square integrable. Curiously enough, $e(\lambda )=-1/2-3\lambda $ is close to the ground–state eigenvalue of the Schrödinger equation $$\psi ^{\prime \prime }(r)+2\left[ E-V_{1}(r)\right] \psi (r)=0,\;V_{1}(r)=-\frac{1}{r}-2\lambda r+2\lambda ^{2}r^{2},\;\lambda >0 \label{eq:V1(r)}$$ when $\lambda $ is sufficiently small. Killingbeck [@K07b] calculated the energy shift $\Delta (\lambda )=E(\lambda )-e(\lambda )$ very accurately for several values of $\lambda $ by means of the Hill–series method.
Our interest in this model stems from the fact that $1/r-\varphi ^{\prime
}(r,-\lambda )/$ $\varphi (r,-\lambda )=1-2\lambda r$ is an exact rational function and therefore $e(\lambda )$ will always be a root of the Hankel determinants even though it does not correspond to a square–integrable eigenfunction if $\lambda >0$. This unwanted solution will appear as an exact multiple root of the Hankel determinant, very close to the physical one when $\lambda $ is close to zero.
If $\lambda <2/27$ the potential–energy function (\[eq:V1(r)\]) exhibits three stationary points: a minimum at $r_{1}<0$, a maximum at $r_{2}\geq 4$ and a shallow minimum at $r_{3}>4$. On the other hand, there is only a minimum at $r_{1}<0$ when $\lambda >2/27$. Obviously, only the stationary points at $r>0$ make sense from a physical point of view, and we expect the RPM to yield better results in the latter case. The expansion of the solution to the Riccati equation about $r=0$ will require many terms in order to take into account the shallow minimum that will move away from origin as $\lambda $ decreases. In this case we expect to face the necessity of Hankel determinants of greater dimension in order to obtain the shift to a given accuracy as $\lambda $ decreases. This unfavourable situation is an interesting test for the RPM that has not been applied to this kind of problems before.
The Hankel determinants are polynomial functions of $\Delta $ and $\lambda $. For example, $H_{D}^{0}(\Delta ,\lambda )=\Delta ^{D-1}P_{D}(\Delta
,\lambda )$ and, therefore, the approximation to the energy shift is given by a root of $P_{D}(\Delta ,\lambda )=0$ that approaches the multiple root $\Delta =0$ as $\lambda $ decreases. Table \[tab:V1\] shows $\Delta
(\lambda )$ for some values of $\lambda $ calculated with determinants of dimension $D\leq 20$. In order to estimate the last stable digit we compared the sequences of roots with $d=0$ and $d=1$. As expected from the argument above, the accuracy decreases as $\lambda $ decreases if we do not increase the maximum value of $D$ consistently, but in all cases we have verified that there is a sequence of roots of the Hankel determinants that converge towards the ground–state eigenvalue. Present results agree with those calculated by Killingbeck by means of the Hill–series method[@K07b].
Model with no bound states \[sec:resonances\]
=============================================
It has already been shown that the RPM is a most efficient tool for the calculation of resonances in the continuum of simple quantum–mechanical models[@F95; @F96b; @F97]. However, for completeness in what follows we consider the potential–energy function $$V_{2}(r)=-\frac{1}{r}+2\lambda r-2\lambda ^{2}r^{2} \label{eq:V2(r)}$$ that is closely related to the preceding one but does not support bound states because it is unbounded from below as $r\rightarrow \infty $. In this case we expect unstable or resonant states with complex eigenvalues that correspond to tunnelling from the Coulomb well.
Table \[tab:V2\] shows present results obtained from Hankel sequences with $D\leq 20$. As in the preceding example we compared sequences with $d=0$ and $d=1$ in order to estimate the last stable digit. Our results agree with those reported by Killingbeck[@K07b], except for $\lambda =0.08$. While the first digits of the imaginary part of our eigenvalue agree with those in Killingbeck’s Table 3[@K07b], the real part is completely different. The disagreement is due to a misprint in Killingbeck’s Table 3 for that particular entry. In fact, we have found that the real part of the eigenvalue reported by Killingbeck for $\lambda =0.08$ corresponds to $\lambda =0.05$ instead, as shown in present Table \[tab:V2\].
Conclusions \[sec:conclusions\]
===============================
We have shown that the RPM is suitable for the accurate calculation of bound states and resonances of perturbed Coulomb problems. The first model, equation (\[eq:V1(r)\]), considered in this paper exhibits interesting features that were not faced in previous applications of the RMP[@FMT89a; @FMT89b; @F92; @FG93; @F95; @F95b; @F95c; @F96; @F96b; @F97]. A shallow minimum that moves forward from origin as the potential parameter $\lambda $ decreases makes it necessary to resort to Hankel determinants of increasing dimension in order to obtain eigenvalues of a given accuracy. On the other hand, we had already proved that the RPM is suitable for the calculation of resonances in the continuum, and we simply confirmed this strength of the approach by means of the second model chosen above.
The applicability of the RPM is not restricted to eigenvalue equations. We have recently applied it to several ordinary nonlinear differential equations[@AF07]. Since most of them are not Riccati equations we called this variant of the method Padé–Hankel, but the strategy is basically the same outlined in this paper.
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------ ------------------------------
0.10 $3.41730960373299\ 10^{-2} $
0.09 $2.31341988422733\ 10^{-2} $
0.08 $1.4212168993068\ 10{-2} $
0.07 $7.546639486534\ 10^{-3} $
0.06 $3.1738752354\ 10^{-3} $
0.05 $8.93101948\ 10^{-4} $
0.04 $1.1819718\ 10^{-4} $
------ ------------------------------
: Energy shift $\Delta(\lambda)$ for the ground–state energy of the perturbed Coulomb model (\[eq:V1(r)\])[]{data-label="tab:V1"}
------ --------------------------- ----------------------------
0.10 $-0.27519233330828482428$ $1.3918964850900\ 10^{-8}$
0.09 $-0.29265795893536614770$ $7.9213310722\ 10^{-10}$
0.08 $-0.31105186469292522577$ $2.094858859\ 10^{-11}$
0.05 $-0.372260539194895485$
------ --------------------------- ----------------------------
: Resonance for the $1s$ state of the perturbed Coulomb model (\[eq:V2(r)\])[]{data-label="tab:V2"}
|
---
abstract: 'We consider the nonlinear transverse magnetic moment that arises in the Meissner state of superconductors with a strongly anisotropic order parameter. We compute this magnetic moment as a function of applied field and geometry, assuming d-wave pairing, for realistic samples, finite in all three dimensions, of high temperature superconducting materials. Return currents, shape effects, and the anisotropy of the penetration depth tensor are all included. We numerically solve the nonlinear Maxwell-London equations for a finite system. Results are discussed in terms of the relevant parameters. The effect, which is a probe of the order parameter symmetry in the bulk, not just the surface, of the sample should be readily measurable if pairing is in a d-wave state. Failure to observe it would set a lower bound to the s-wave component.'
address: |
School of Physics and Astronomy and Minnesota Supercomputer Institute\
University of Minnesota\
Minneapolis, Minnesota 55455-0149
author:
- 'Igor Žutić and Oriol T. Valls'
title: Nonlinear transverse magnetic moment in anisotropic superconductors
---
Introduction
============
The question of the pairing state of high temperature superconductors (HTSC’s) continues to baffle researchers in the field. A recent review [@agl] of many of the experimental results closes after over eighty pages with the tentative conclusion that the only state compatible with all experiments would have to exhibit two separate transitions, in contrast with the single transition invariably observed in HTSC’s. Worse, even different experiments performed on the very same single crystal sample seem to lead to contradictory conclusions.[@buanyang] This situation makes it particularly desirable to find good probes that may as unambiguously as possible discern the structure of the order parameter (OP) function.
One such probe is afforded by the nonlinear Maxwell-London electrodynamics of exotic pairing states in superconductors in the Meissner state. It was pointed out four years ago[@ysprl] in the context of a two-dimensional model (HTSC’s are nearly all highly anisotropic, layered materials) that the presence of order parameter lines of nodes on a cylindrical Fermi surface would lead to observable nonlinear effects in the electrodynamic properties. These effects were shown to be[@ysprl; @sv; @ys] potentially capable of yielding a signature for the structure of the order parameter function, and therefore for the nodal structure of the gap itself. The most obvious of them is the presence of a transverse magnetic moment ${\bf m}_\perp$ perpendicular to the direction of an applied magnetic field ${\bf H}$ lying in the $x-y$ ($a-b$) plane (the $z$ axis is taken to be along the $c$ crystallographic direction, perpendicular to the stack of planes in the lattice structure). This transverse magnetic moment has an angular dependence, as the sample is rotated about the $z$ axis, reflecting directly the periodicity of the energy spectrum, that is, of the square of the order parameter, on the azimuthal angle $\phi$. This signal would be particularly prominent if the order parameter has nodes or very deep minima as a function of angle. The transverse moment can be measured directly or inferred from the torque it produces.
Electrodynamic effects are of particular interest because they probe the superconducting properties over a scale given by the field penetration depth, while many other experimental methods, such as tunneling and Josephson junctions, probe only over the scale of the correlation length, which in HTSC’s is over two orders of magnitude smaller. Thus, electrodynamics probes the bulk properties of the superconducting material, not merely the surface. Some of the apparent conflicts among experimental results may be resolved[@bahcall] if the order parameter symmetries in the bulk and at the surface differ.
For these reasons, investigations of the transverse magnetic moment effect have continued. For the model of a “slab” shaped sample, infinite in the $x-y$ directions and of fixed thickness, and a cylindrical Fermi surface, the equations involved in the nonlinear Maxwell-London electrodynamics have been solved numerically[@sv] for the experimentally relevant ranges of applied magnetic field and temperature. At $T=0$, for the same geometry and assumptions, a perturbative, but accurate, analytic solution has been obtained[@ys]. In Ref. results were obtained for both a pure d-wave state and a mixture of $s$ and $d$ waves. The amplitude of the transverse magnetization at low temperatures was found to increase[@sv] as approximately the second power of the applied field, and, since one is dealing with an effect arising from a finite penetration depth, as the surface area of the sample. The results[@sv; @ys] indicated that, for a pure d-wave state, ${\bf m_\perp}$ should be detectable by a SQUID magnetometer in single crystal samples, even though the applied field is limited, by the need to keep the sample in the Meissner regime, to the field of first flux penetration.
The results of only one experiment seeking to find the transverse magnetization[@confusion] effect have been published[@buan]. No evidence of a transverse magnetic moment having the proper periodicity was obtained, but the nature of the data and of its analysis was such that the absence of evidence could not be conclusively taken to be evidence of absence. The resolution of the measurements was, over most of the field range studied, comparable to the signal measured. At the higher fields used the signal was below, but only by a factor of two, the theoretically expected results for a pure d-wave state, based on the infinite slab assumptions[@sv] discussed in the above paragraph, scaled to the finite size of the sample. Although a subsequent analysis[@buanyang] of the field dependence of the experimental signal [@buan] was also supportive of the inference that no nodes were observed, a completely definitive statement about the existence of nodes could not be made because of theoretical and experimental uncertainties.
Although this paper deals with the theoretical questions it is useful to briefly mention here the main difficulty associated with the experiments: except in the ideal world of theorists where all samples, even when finite, are spheres or other highly symmetric bodies, there [*always is*]{} in any superconductor, a transverse magnetic moment because of demagnetization effects[@orlando]. For the rectangular samples used in the experiment[@buan] these effects are much larger than the nonlinear contribution. Of course, their main periodicity, $\pi$, is twice that associated ($\pi/2$) with the d-wave nonlinear effects. Fourier analysis of the data, as done in Ref. does filter out the main spurious demagnetization signal, but its $\pi/2$ harmonic still will be confounded with the nonlinear signal and, although it can partly be sorted out[@buanyang] because of its different, linear, magnetic field dependence, the noise introduced in the original signal by the Fourier transforms and subtractions takes a heavy toll. The work in this paper is motivated in large part by experiments now being planned[@amgpc] on samples having approximately a flat disk shape that minimizes the demagnetization effect for an applied field lying in the $a-b$ plane . It seems to us that reaching conclusions from experimental data taken on samples lacking rotational symmetry would be difficult.
The theoretical uncertainties are related to the use in the work discussed above of two approximations: first, the use of an infinite slab geometry neglects the contribution to ${\bf m}_\bot$ of return currents flowing along the sides of the sample, parallel to the z-axis. Furthermore, the penetration depth for those currents, $\lambda_c$ is much larger than that in the a-b plane, $\lambda_{ab}$. Since ${\bf m}_\perp$ increases with penetration depth, the question of how the larger $\lambda_c$ must be included in the analysis arises. The effect of $\lambda_c$ was included in the analysis of the data[@buan] but in a purely heuristic way, which assumed that its influence was rather strong. Although this assumption was relaxed in a subsequent reanalysis[@buanyang], the whole question remains a major obstacle in reaching conclusions from experimental evidence. A finite $\lambda_c$ implies also abandoning the notion of a cylindrical Fermi surface. Further, the very neglect of finite size effects is suspect: they have been shown[@lin] to affect the analysis of penetration depth experiments in rather large samples. The uncertainties related to these complications in analyzing experimental data in the context of the geometric approximations of Ref. are about as large as the experimental uncertainties and it might be argued that, in combination, are about as large as the discrepancy between theory and experiment.
In this paper we therefore undertake the examination of Maxwell-London electrodynamics, including nonlinear contributions, for finite samples. We include in the study the effects of $\lambda_c$, which affect the electromagnetic response even at linear order in the field, and we specifically consider geometries relevant to the experimental systems currently been studied. Because of these realistic assumptions, not only is an analytic solution unattainable even at zero temperature, but the numerical work involved is quite considerable. Our results confirm that a transverse magnetic moment should be observable in obtainable single crystal samples, if the bulk pairing is in a d-wave state. The effect discussed in this paper is a fingerprint for the existence of an OP with d-wave symmetry. Its experimental observation in suitable samples, would constitute very hard to refute evidence for d-wave pairing in the bulk. Failure to observe this effect would at the very least put a lower bound on the existence of an s-wave component.
In the next Section we present the equations solved and we discuss the methods we use. In Section III, we present our results and predictions for experimental outcomes, assuming a d-wave order parameter is present. Our conclusions are given in the last Section.
Methods {#sec:methods}
=======
Maxwell-London electrodynamics
------------------------------
The equations of superconducting electrodynamics are in textbooks [@orlando; @degennes] and need no rederivation. We will merely introduce here our notation, and briefly recall a couple of easy to overlook points. For the linear case, a particularly clear discussion of the complications that occur when the penetration depth tensor is anisotropic is in Chapter 3 of Ref.().
Outside the sample the current is ${\bf j}=0$ and therefore in the steady state the field ${\bf H}$ satisfies the Maxwell equations $\nabla \cdot {\bf H}=0$ and $\nabla \times {\bf H}=0$. One can therefore write ${\bf H}$ in terms of a scalar potential $\Phi$ satisfying the Laplace equation:
\[out\] $${\bf H}=-\nabla \Phi \label{equationa}$$ $$\nabla^2 \Phi =0 \label{equationb}$$
In the interior of the sample one must consider three fields: ${\bf H}$, ${\bf j}$ and the “superfluid velocity field” ${\bf v}$ defined as: $${\bf v}=\frac{\nabla \chi}{2} + \frac{e}{c} {\bf A},
\label{vdef}$$ where $\chi$ is the phase of the superconducting order parameter, ${\bf A}$ the vector potential, and $e$ the proton charge. We set $\hbar=k_B=1$. Eq.(\[vdef\]) is equivalent to Eq. (5.99) in Ref.. The field ${\bf v}$ has dimensions of momentum, not velocity, but the factors associated with the effective mass and its anisotropies are more conveniently dealt with by placing them elsewhere. The three equations that one requires for these three fields are first, the second London equation obtained by taking the curl of (\[vdef\]): $$\nabla \times {\bf v}=\frac{e}{c}{\bf H}
\label{londoneq}$$ second, Ampère’s law: $$\nabla\times {\bf H}=\frac {4 \pi}{c}{\bf j}
\label{ampere}$$ and finally, a constitutive equation relating ${\bf j}$ and ${\bf v}$ which is discussed in general in the next subsection. For the usual linear case this relation is of course simply: $${\bf j}=-e\tilde{\rho}{\bf v}
\label{linear}$$ where $\tilde{\rho}$ is the superfluid density tensor. The nonlinear contribution is discussed below (see Eq. (\[jtzero\])). $\tilde{\rho}$ is related to the penetration depth tensor $\tilde{\Lambda}$, whose components in the diagonal representation are the square of the London penetration depths, by the relation: $$\tilde{\rho}=\frac{c^2}{4 \pi e^2} \tilde{\Lambda}^{-1}
\label{rholam}$$ When these tensors are proportional to the identity, then one can combine Eqns.(\[londoneq\]) and (\[ampere\]) and find that any one of the three fields considered satisfies the vector Helmholtz equation. But this is not true[@orlando] when $\tilde{\rho}$ is anisotropic. One still has, however, the completely general equation: $$\nabla\times\nabla\times{\bf v}=\frac{4 \pi e}{c^2}{\bf j(v)}
\label {maxlon}$$ which is valid whatever the relation ${\bf j(v)}$ might be. Eq. (\[maxlon\]) will be the basic equation we will consider here.
These equations must be solved with appropriate boundary conditions. These are: first, at infinity (that is, very far away from the finite sample) ${\bf H}$ must reduce to the applied field. Second, deep inside the sample all fields must vanish. Finally, at the interface ${\bf H}$ must be continuous[@degennes; @reitz] and the component of ${\bf j}$ normal to the interface must vanish.
Once the currents ${\bf j}$ inside the sample are known, the magnetic moment can be obtained by integration:[@jackson] $${\bf m}=\frac{1}{2 c}\int d{\bf r} \: {\bf r}\times {\bf j(v(r))}
\label{magmom}$$
It is convenient to rewrite this equation in terms of surface integrals. Using Eq. (\[ampere\]) and formulas for vector calculus [@jackson2] we derive: $$\begin{aligned}
{\bf m} &=\frac{1}{8 \pi}\int_{s} d^2S \: {\bf n} \: ({\bf r}\cdot {\bf H})
+\frac{1}{8 \pi}\int_{s} d^2S \: {\bf r} \: ({\bf H}\cdot {\bf n}) \nonumber \\
&- \frac{1}{8 \pi}\int_{s} d^2S \: {\bf H} \: ({\bf n}\cdot {\bf r})
+\frac{c}{8 e\pi}\int_{s} d^2S \: {\bf n} \times {\bf v}
\label{eq:magmoms}\end{aligned}$$ Integration is performed over the sample surface S, [**r**]{} is the position vector for a point on the surface S, and [**n**]{} is the unit normal pointing outwards.
In the linear regime, and for the case which is experimentally relevant here where one has a small but finite value of the ratio $\lambda/d$ between a typical penetration depth $\lambda$ and a characteristic sample dimension $d$, the components of ${\bf m}$ can generically be written in the form $m=m_0(1-\alpha(\lambda/d)+{\it O}(\lambda/d)^2)$. Examples for values of the positive constant $\alpha$ are given in textbooks[@fetwal; @london]. One has, then, a reduction in the magnetic moment due to current penetration in the material. The nonlinear effects may be conveniently viewed for our purposes as anisotropic, field-dependent corrections to the values of the $\alpha$ coefficients.
The supercurrent
----------------
To obtain the supercurrent response one has to specify the full constitutive relation ${\bf j(v)}$ as discussed above. The linear contribution has been given in Eqns. (\[linear\]) and (\[rholam\]). The anisotropy in $\tilde{\Lambda}$ is related to that of the Fermi surface and can be related to the effective mass tensor[@orlando]. The anisotropy of the order parameter considered in this work would not, if the Fermi surface were isotropic, lead to any anisotropies in the penetration depth[@sv], at linear order in the magnetic field. Thus we have: $$\tilde{\Lambda}=\frac{c^2 \tilde{m}}{4 \pi n e^2}
\label{lamm}$$ where $\tilde{m}$ is the effective mass tensor and $n$ the carrier density.
The nonlinear corrections to ${\bf j(v)}$ were first discussed by Bardeen[@bardeen] in the context of an s-wave superconductor. In that case, at $T=0$, the current-velocity relation is linear for velocities less than a critical velocity $v_c$ defined below. Nonlinear corrections to the current-velocity relation arise from the thermal population of quasiparticles. These corrections are small[@ys] (cubic in the ratio $v/v_c$) and in the regime of larger flow velocities vortex nucleation may occur before these effects become important[@degennes]. On the other hand, in unconventional superconductors, particularly those with nodes in the OP, nonlinear corrections are significantly larger and qualitatively different than in a superconductor with an s-wave excitation gap. The region on the Fermi surface (FS) close to the nodes in the gap provides enhanced nonlinear response due to the higher population of quasiparticles. There the energy required to break electron pairs will be reduced and will vanish along the direction of the lines of nodes in the gap. Thus nonlinear corrections in the presence of nodes in the OP are nonvanishing even at $T=0$ and the nonlinear behavior exhibits anisotropy with respect to the relative angle between applied magnetic field and the lines of nodes of the gap function.
For the purpose of computing ${\bf m_\perp}$ for a crystallographycally strongly anisotropic HTSC, in the geometry considered here, we are concerned with the angular dependence of the OP in the plane of the applied field, the $a-b$ plane. In most of the numerical calculations performed in this work, we consider an order parameter of the pure d-wave form: $$\Delta=\Delta_0 \sin(2 \phi)
\label{op}$$ where $\phi$ is the azimuthal angle referred to a node and $\Delta_0$ the amplitude. The periodicity of ${\bf m_\perp}$, which equals that of the energy, is therefore $\pi/2$. It would be possible, as we shall see, to generalize the calculations to other $\phi$ dependence for the OP if necessary. Possible dependence of $\Delta$ on the $c$ direction is very uncertain because of multilayering effects[@agl], but we believe it is unlikely to be important. This follows from the above physical considerations, and from the brief discussion of results for alternative forms in the next Section and Appendix A. On the other hand, we must properly take into account the strong anisotropy of $\tilde{\Lambda}$ in the $c$ direction, which is related to that of the Fermi surface.
The assumption of isotropy in the $a-b$ plane deserves further discussion, since anisotropies in the in-plane penetration depth in HTSC’s are known to exist.[@kim] Anisotropy in $\lambda_{ab}$ will lead to a contribution to ${\bf m_\perp}$ of periodicity $\pi$ (instead of $\pi/2$ for the nonlinear signal), and (see the paragraph below (\[eq:magmoms\])) down by a factor of $\sim \lambda/d$ (the ratio of a typical penetration depth to a typical sample dimension) from the longitudinal, linear magnetic moment. This contribution will be linear, not quadratic, in the field. Hence, from an experimental point of view, it is in effect a small correction to the effective demagnetization factor which is, although to a small extent, present even in macroscopically symmetric samples. Thus, Fourier analysis of the experimental signal and examination of the field dependence of the harmonics would separate this from the nonlinear effect. As for the theoretical results discussed here it is sufficient to interpret our symbol $\lambda_{ab}$ for the $a-b$ plane penetration depth ($\lambda_c$ is that along the c-direction) as the geometric mean of $\lambda_a$ and $\lambda_b$. Finally, if because of orthorhombic anisotropy the nodes are not precisely at $\pi/2$ angles, this can be accounted for by adding a small constant[@agl] to (\[op\]) and therefore the $\pi/2$ Fourier component of the response would be little affected.
For an order parameter with nodes, it is known[@sv; @ys] that the details of the FS shape are not important for the nonlinear properties we deal with here: only the nodes and their symmetry matter. Thus, our chief concern about the FS is to describe the anisotropy in $\tilde{\Lambda}$. For this purpose, we have used in our calculations an ellipsoidal (of revolution) Fermi surface characterized by effective masses $m_{ab}$ (in the $a-b$ plane) and $m_c$, as in Eq. (\[ef\]). We introduce the ratio $\delta\equiv (m_c/m_{ab})$. We define a speed $v_f$ in terms of the Fermi energy $\epsilon_f$ as $v_{f}^{2}=2 \epsilon_f /m_{ab}$ which facilitates comparison with two-dimensional results.
The general expression [@sv; @ys] for ${\bf j(v)}$, including nonlinear terms, can be written as: $$j=-eN_f\int d^2s \: n(s) {\bf v}_f [({\bf v}_f \cdot {\bf v})
+2\int^{\infty}_0 d\xi \: f(E(\xi)+{\bf v}_f\cdot {\bf v})]
\label{jtzero}$$ where $N_f$ is the total density of states at the Fermi Level, $n(s)$ is the density of states at point $s$ at the Fermi surface, normalized to unity, ${\bf v}_f(s)$ is the $s$-dependent Fermi velocity, $f$ the Fermi function, and $E=\sqrt{\xi^2+\left| \Delta(s) \right|^2}$. In general, the integrals in Eq. (\[jtzero\]) can be evaluated only numerically, as was done in 2-d in Ref.. However it was shown there that at the temperatures (about two Kelvin) where the experiments are performed, one is very near the zero temperature limit. Therefore we confine ourselves in this work to this limit, where, within the assumptions discussed above, it is possible to derive an analytic expression for ${\bf j(v)}$, valid in the limit where $\delta$ is large. Having an analytic expression for ${\bf j(v)}$ makes the subsequent substantial numerical work easier. In a magnetic field (hence ${\bf v}\neq 0$) even at $T=0$ there exists a region with $\left|\Delta(s) \right|+{\bf v}_f \cdot {\bf v}<0$ where it is possible to have a quasiparticle population. One can then perform, as in Ref. an approximate but accurate integration of the general formula Eq. (\[jtzero\]), to obtain the nonlinear corrections at zero temperature (details are given in Appendix A). The resulting ${\bf j(v)}$ relation is valid under the assumptions for the FS and the OP discussed above, and for any three dimensional strongly anisotropic superconductor (independent of sample geometry). Introducing cartesian axes x’-y’ fixed to the crystal along the nodal directions the expression we obtain is:
\[jnl\] $$j_{x',y'} =-e\rho_{ab} v_{x',y'} (1-\frac{9\pi}{64 v_c}\left|v_{x',y'}
\right|)$$ $$j_z =-e\rho_{c} v_{z} (1-\frac{3\pi}{32 v_c}\frac{v_{x'}^2+v_{y'}^2}
{\left|v_{x'}\right|+\left|v_{y'}\right|})$$
where $v_c\equiv \Delta_0/ v_f$. The components of the superfluid density tensor are $\rho_{ab}=\frac{1}{3}N_f v_f^2$, $\:$ $\rho_{c}=\rho_{ab}/\delta$, in the $a-b$ plane and along the $c$ crystallographic axis respectively. Even for YBCO reported[@erange] measured values for $\delta=m_c/m_{ab}=(\lambda_c/\lambda_{ab})^2$ are quite large, between 15 and 50. Omitted terms in the above relation are those involving higher powers of $v/v_c \ll$ 1 (for the regime of experimental interest in the Meissner state) or supressed by the small parameter $\delta^{-1}$.
Geometry
--------
In this subsection we discuss the sample geometry considered in the calculation of the supercurrent response in the Meissner state. The dependence of measured quantities such as the local magnetic field, actual current distribution and magnetic moment on sample shape is one of the issues of particular interest in this work, as explained in the Introduction.
The experimentally relevant regime is that of small but finite ratio of penetration depths to typical sample dimension. As mentioned below Eqn.(\[eq:magmoms\]), the geometric dependence of the effect under investigation in this regime[@film] is proportional to these ratios. Thus, the coefficient of this finite size term must be accurately calculated. A “locally flat” approximation (assuming a purely exponential decay for the fields away from the surface) is not sufficient for this purpose, as one can check even in the linear case from the exact solution for a sphere[@reitz]. It is therefore necessary to solve the complete boundary value problem,that is, to find the solution to the equations (\[out\]) and (\[maxlon\]) outside and inside of the sample. This is in principle a numerically difficult undertaking, since [*a priori*]{} it involves solving a system of partial differential equations with nonlinear terms in the entire three dimensional space, but with currents being confined to a very small region where great precision is required. Although sophisticated variable grid methods could perhaps be found to take care of these complications, the relatively symmetric shape of the required experimental samples allows for a simpler approach.
Single crystals of HTSC materials are typically flat, much thinner in the c-direction than in the $a-b$ plane. The magnetic field, we recall, is applied parallel to this plane along the $x$ direction. The geometry and coordinate system we consider are sketched in Fig. \[fig1\]. Because of the complications associated with demagnetization factors, as described in the Introduction, experiments must be performed on single crystals of a highly symmetric shape. This is achieved by laser cutting, or shaving[@bgg] the crystals so that their cross section in the direction perpendicular to the $z$ axis is circular. Their shape is, therefore, roughly a disk, thinning towards the edges because of mechanical disintegration associated with the cutting. They have smooth edges and can therefore be described as ellipsoids of revolution. This geometry has also a considerable computational advantage: the form of the solution outside the sample in the limit $\tilde{\Lambda}=0$ is known exactly since the potential $\Phi$ (see (\[out\])) satisfies trivial Neumann boundary conditions at the interface and can be found by electrostatic analogy. The solution contains a single parameter which is simply related to $m_0$, the value of the longitudinal magnetic moment in the zero penetration limit. When the penetration depth is finite, the longitudinal moment does change, but its correct value can be determined from the boundary conditions and the solution inside through an iteration process as described in the next Section. But, as important as these simplifications are, they only go so far: the fundamental equation (\[maxlon\]) is not separable in spheroidal coordinates.
We therefore consider a supercondutor in the shape of a flat ellipsoid of revolution (an oblate spheroid) with the axis of revolution along the z-axis. Its major and minor semiaxes are denoted by $A$ and $C$ respectively, and we have $A > C$ for actual samples. We take (see Fig. \[fig1\]) a coordinate system fixed to the direction of the magnetic field, with its z-axis parallel to the $c$ crystalographic direction of the superconductor (and parallel to the $C$ semiaxis of an ellipsoid). The field is applied along the x-axis, and we picture the experiment as being performed by rotating the crystal about the z-axis. As the crystal is rotated the axes $x-y$ remain fixed in space, and should not be confused with the previously introduced $x'-y'$ coordinates, affixed to the crystal structure.
Even with the nonseparability of the equations, it is still convenient from the point of view of fulfilling the interface boundary conditions, to introduce oblate spheroidal coordinates $\alpha, \beta, \varphi$. In the definition we use[@arfken], they are related to cartesian coordinates by the transformation:
\[eq:coor\] $$x = f \cosh \alpha \sin \beta \cos \varphi \\
\label{coora}$$ $$y = f \cosh \alpha \sin \beta \sin \varphi \\
\label{coorb}$$ $$z = f \sinh \alpha \cos \beta
\label{coorc}$$
where $0\leq \alpha<\infty,
0\leq\beta\leq\pi, 0\leq\varphi\leq 2\pi$, and $f$ is a focal length scale factor. The surface of an ellipsoid corresponds to a given value of $\alpha$.
For an ellipsoid of revolution (about its $z$ axis) in the presence of uniform magnetic field $\bf{ H}_a$ applied along the direction that we denote as the $x$ axis the magnetic potential in the outside region for $\lambda=0$ has the form [@russian] $\Phi=-H_a x(1+g(\sinh\alpha))$, where the first term is the potential for the uniform applied field ${\bf H}_a$ and the gradient of the second term is the field generated by the superconducting ellipsoid. Writing out in detail the function $g$ one has: $$\Phi=-H_a f P^1_1(i\sinh\alpha) P^1_1(\cos\beta)\cos\varphi+
A_1 f Q^1_1(i\sinh\alpha) P^1_1(\cos\beta) \cos\varphi
\label{eq:phiout}$$ where $P^{1}_{1}$ and $Q^{1}_{1}$ are associated Legendre functions of the first and second kind respectively. The parameter $A_1$ is determined from the boundary conditions. It is proportional to $m_{\|}$, the longitudinal magnetic moment: $$m_{\|}=\frac{2}{3}f^3A_1
\label{longmag}$$ Its value at $\lambda=0$ is: $$A_1(0)=\frac{- H_a\sinh\alpha_0}{1+1/\cosh^2\alpha_0-\sinh\alpha_0
\arctan(1/\sinh\alpha_0)}
\label{a1}$$ where $\alpha=\alpha_0$ is the value at the surface of the ellipsoid. Eqns.(\[a1\]) and (\[longmag\]) can be combined to yield the usual expression for $m_0$, the zero penetration depth, purely longitudinal, magnetic moment, in terms of the ellipsoidal demagnetization factors.[@ll] At finite $\tilde{\Lambda}$ the value for $A_1$ is obtained from the iterative procedure discussed in the next subsection. In the spherical limit $\sinh\alpha\rightarrow\infty$ we also recover the standard result[@reitz; @london].
Results
=======
Numerical procedure
-------------------
Let us discuss now the iterative procedure we use to solve (\[maxlon\]), while avoiding to have to numerically solve the field equations in all space. For clarity, let us focus first on the case where the nonlinear effects are neglected but the penetration depths are finite. This has two effects on the solution outside: first, the value of $A_1$ (or of $m_{\|}$) deviates from the value given by (\[a1\]). Secondly, the field acquires higher order multipoles, i.e. the field potential acquires additional terms: $$\Phi = \Phi_a +A_1 f Q^1_1(i\sinh\alpha) P^1_1(\cos\beta) \cos\varphi
+\sum_{n\geq 1}A_{n} f Q^1_n(i\sinh\alpha) P^1_n(\cos\beta) \cos\varphi
\label{figen}$$ where $\Phi_a$ is the potential corresponding to the applied field (the first term on the right side of (\[eq:phiout\])) and the gradient of the rest is the field created outside the sample by the current distribution in it. The terms with $n>1$ vanish at $\tilde{\Lambda}=0$ (and also in the spherical case if $\tilde{\Lambda}$ is isotropic.) The fields generated by $A_1$ are not purely dipolar, since they have ellipsoidal symmetry. However, the dipole moment is determined by this parameter only. At the first step of the iteration we solve the linear version of (\[maxlon\]) for the current field, assuming that the outside field is given by its $\lambda=0$ limit, i.e. the gradient of (\[eq:phiout\]) with (\[a1\]). Parameter counting shows that in order to do so we must give up one boundary condition, and accordingly we temporarily sacrifice the continuity of the “radial” field ($\hat{\alpha}$ component). From the resulting current distribution we compute ${\bf m}$ through (\[eq:magmoms\]). This is of course not the same as the input moment. We then replace this obtained value in the external potential (through (\[longmag\])) and repeat the procedure[@flat] until the moment generated by the computed currents equals the input value. The iteration is considerably simplified by observing that several terms in (\[eq:magmoms\]) vanish explicitly when $\tilde{\Lambda}=0$ and by making use of the fact that the penetration depths are small[@zv] compared to the sample dimensions. Once the iteration is concluded, one finds that $H_\alpha$ is continuous except in a very narrow band near the equator corresponding to a symmetry higher than dipolar. This can then be eliminated by adding small values for higher order $A_n$’s to the expansion (\[figen\]), but symmetry considerations and examination of Eq. (\[eq:magmoms\]) show that these additions do not affect the already determined value of the sample magnetic moment. It is easy and very instructive to verify analytically that this procedure recovers the known result[@reitz] for the $\lambda$ dependent magnetic moment of a sphere with an isotropic $\tilde{\Lambda}$.
The same procedure is used with the nonlinear terms, with only two important differences: first, to the field outside one must add a transverse dipole, i.e. a contribution of the form of the last term in (\[eq:phiout\]) rotated $90^\circ$: $$\Phi_{\bot}=A_{1\bot} f Q^1_1(i\sinh\alpha) P^1_1(\cos\beta) \sin\varphi
\label{eq:phiout2}$$ where $\frac{2}{3}f^3 A_{1\bot}=m_\bot$. Of course this term does not exist when the penetration depths vanish, since the nonlinear effects are absent unless the field can penetrate the sample. In principle one should also consider higher order multipole terms, as in (\[figen\]) but we have found that any such terms are below the precision level of our numerical results. Both components of the moment ${\bf m}$ are determined through the vector relation (\[eq:magmoms\]). The other important difference is the obvious one of using (\[jnl\]) for the ${\bf j(v)}$ relation. There are also practical differences, however, since in the linear equations the variable $\varphi$ can be separated out while in the full nonlinear case all three coordinates are coupled.
In the actual solution of the equations we use a relaxation method. We proceed in two steps: first we solve the linear problem, which involves only two variables, since then the $\varphi$ dependence of all quantities can be determined analytically. The iterated solution for that problem is then used as the initial guess in the full three-variable nonlinear problem.
We discretize the differential equations expressed in ellipsoidal (oblate spheroidal) coordinates, on a three dimenssional grid. An obvious advantage of the ellipsoidal grid is that it simplifies consideration of boundary conditions at the surface of the ellipsoid, given by equation (\[londoneq\]). Numerically, it is more intricate to consider boundary conditions that involve derivatives, and accuracy is increased if the grid points are also boundary points.
The discretization procedure we have used has an estimated error quadratic in the spacing between the grid points. All quantities involved in equations (\[maxlon\]) have definite parity with respect to the exchange $z$ $\rightarrow$$-z$ and it is sufficient to solve them only for one half of the sample. Accuracy is predominantly governed by the spacing between grid points along the $\hat{\alpha}$ direction. In the actual numerical solution (for half the sample), we consider ellipsoids of different shapes (different $C/A$) and sizes (different $A$). It is appropriate to increase $n_\alpha$ proportionally to $C/A$. Denoting by $n_{\alpha}$ the number of grid points along the $\hat{\alpha}$ direction, the smallest $n_{\alpha}$ used was 100, and the largest 800, spaced in the region of nonvanishing currents. The number of grid points along the $\hat{\beta}$ and $\hat{\varphi}$ directions was $n_{\beta}=50$ and $n_{\varphi}=30$ respectively. Increasing these numbers by a factor of two gave only effects below the numerical accuracy attained, which is about two significant figures, an error much smaller that the uncertainty arising from the imprecise knowledge of the experimental values of the input parameters.
As one of the checks on the accuracy of the algorithm we use, it is instructive to consider the case of an isotropic spherical sample, (both the sample and the Fermi surface are spheres, $\delta=1$) in the linear regime, where the analytical solution is known. To ensure that we tested the same algorithm, we treated the sphere as the the spherical limit of ellipsoidal coordinates, which corresponds to $\sinh \alpha$ $\rightarrow \infty$. We used $\sinh \alpha$ =1000, equivalent to eccentricity $e=\sqrt{1-(C/A)^2}$=0.001 with the grid given by $n_{\alpha}=200$, $n_{\beta}=50$ and $n_{\varphi}=30$ grid points. We solved equations (\[maxlon\]) in a spherical shell of thickness 10$\lambda$ and studied both the magnetic moment and the current distributions. The accuracy for the current on any grid point corresponded to four significant figures for the region where currents are important. The magnetic moment calculated both from (\[magmom\]) and from the surface integrals (\[eq:magmoms\]) agrees with the exact result, including the correct finite penetration depth correction.
We have also checked that our results for the longitudinal magnetic moment extrapolated to the zero penetration depth limit agree with the known analytic result for ellipsoids. We have also verified that the magnetic moment calculated from (\[magmom\]) agrees from that found from the surface integrals (\[eq:magmoms\]). The latter procedure is, however, much more convenient.
Numerical results and discussion
--------------------------------
In performing the calculations and describing the results, it is convenient to introduce dimensionless quantities. Because of the shape of the samples, we use $\lambda_{ab}$ as the unit of length. We then define the dimensionless fields: ${\bf V}$, ${\bf J}$, and ${\bf h}$: $${\bf V}=\frac{\bf v}{v_c},\qquad {\bf h}=\frac{{\bf H}}{H_0},\qquad
{\bf J}=\frac{c H_0}{4\pi \lambda_{ab}} \, {\bf j},
\label{eq:dimless}$$ where we have introduced a characteristic magnetic field $H_0$ as: $$H_0=\frac {\phi_0}{\pi ^2 \lambda_{ab} \xi_0}
\label{h0}$$ where $\phi_0$ is the flux quantum and $\xi_0=v_f/\pi \Delta_0$ is the in-plane superconducting coherence length. The definition of (\[h0\]) involves precisely the same numerical factors as that used in Ref.. The required equations are easily rewritten in terms of these quantities. The boundary conditions for the velocity field in (\[maxlon\]) are now: $$\nabla\times {\bf V}={\bf h} \mid _{\alpha=\alpha_0}
\label{eq:lon2}$$ where the right hand side is the external dimensionless field at the surface of the ellipsoid and from now on the derivatives are with respect to dimensionless length. The remaining boundary condition that there is no normal component of current at the surface is readily obtained from Eqs. (\[jnl\]). The relation between ${\bf j}$ and ${\bf v}$ in equations (\[jnl\]) then becomes:
\[maxlon2\] $$( \nabla\times\nabla\times {\bf V})_{x',y'} =- V_{x',y'}
(1-\frac{9\pi}{64}\left|V_{x',y'}
\right|)
\label{maxlon2a}$$ $$(\delta \nabla\times\nabla\times {\bf V})_{z} =
-V_z(1-\frac{3 \pi}{32}\frac{V_{x'}^2+V_{y'}^2}
{\left|V_{x'}\right|+\left|V_{y'}\right|})
\label{maxlon2b}$$
where we recall $\delta=(\lambda_c/\lambda_{ab})^2=m_c/m_{ab}$. Equations (\[maxlon2\]) are transformed to ellipsoidal coordinates, as defined above. Expressions for the superfluid density tensor and the differential operator $\nabla\times\nabla \times {\bf V}$, in ellipsoidal coordinates are included in Appendix B.
In Figures \[linearcur\] and \[nonlincur\] we show some of the results for the currents. These figures illustrate some of the physics, as well as the quality of the numerics. In Fig \[linearcur\] we show the current $j_z$ going along the $z$ direction in the $x-y$ plane as a function of distance from the surface of the sample starting at the point with cartesian coordinates $(0,A,0)$. This current is overwhelmingly determined by the usual linear response to the field. One can clearly see that its decay as a function of depth from the surface is governed by the $\lambda_c$ penetration depth, and not by $\lambda_{ab}$. The next Figure illustrates the difference between the linear and nonlinear components of the current. It shows the current, again as a function of depth, starting at the center of the top of the sample, i.e. a point with cartesian coordinates $(0,0,C)$. The component $j_y$, normal to the applied field, is very predominantly “linear” and one can see that this time the decay is governed by the in-plane penetration depth, as expected from the geometry. The component $j_x$ along the applied field, on the other hand, arises exclusively from nonlinear effects: symmetry considerations show that it vanishes in the linear limit. Its overall scale is down by a large factor (basically the ratio of longitudinal and transverse moments). One sees that even though the overall decay of $j_x$ is determined by the scale $\lambda_{ab}$, its behavior is very far from exponential: it changes sign as a function of depth. This can be readily understood: at the positions plotted, close to the center of the the top of the ellipsoid, $j_x$ is approximately proportional (from Ampère’s law) to the derivative with respect to $z$ of the anomalous component of the field, $H_y$. This field component nearly vanishes at the surface (it would vanish for a slab) and decreases exponentially at depths larger than $\lambda_{ab}$. Hence, at this position in the sample, $H_y$ has an extremum as a function of depth, and its derivative with respect to $z$ [*must*]{} change sign at some point, as we find. This shows the delicate intricacy of the nonlinear current patterns inside the material.
Before proceeding with the detailed discussion of the dependence of our results on the relevant physical quantitites (i.e. size, shape, applied field, and penetration depths) we illustrate their general scope by describing our prediction for the transverse magnetic moment of a possible HTSC superconducting sample, assumed to be in a pure d-wave pairing state. This is done in Fig. \[predict\]. We show there results for $m_\perp$ as a function of applied field. The quantity shown is the maximum value of $m_\perp$ as the crystal is rotated. It is assumed that the sample is an ellipsoid with $A=2$ mm and $C=0.1$ mm. Material parameters are taken to be $\xi_0=20\AA$, $\lambda_{ab}=1800\AA$, and $\lambda_c=9000\AA$. The characteristic field $H_0$ would be about 5800 gauss. One can see that the magnetic moments involved are readily accessible to measurement. Predictions for samples of other sizes, shapes and material parameters can be conveniently extracted from the information given below.
We have obtained results for the transverse magnetic moment for a wide range of the experimentally accessible values of the appropriate dimensionless parameters, which as we shall see, can be taken to be ${H_a}/{H_0}$, $\lambda_c/\lambda_{ab}$, ${\lambda_{ab}}/{A}$, and ${C}/{A}$, the aspect ratio of the ellipsoid of revolution. We did not consider in our study the “thin film” situation where the sample is so small or so thin that its relevant dimensions are comparable to or smaller than the corresponding penetration depth. This case would be of no interest since the nonlinear effects then are vanishingly small[@sv], and it is excluded from the analysis that follows.
We begin our general discussion of the results by performing some dimensional analysis. The quantity $4 \pi m_\bot$ has dimensions of magnetic field times volume. The expression $$Q=\frac{4 \pi m_{\bot}}{H_a V}
\label{Qeq}$$ where $V$ is the sample volume, is therefore dimensionless. Even though $Q$ is suitable for some purposes, it is more convenient to analyze the dependence of the results on sample size in a different way. The reason is that $4\pi m_\bot$ does not scale with the sample volume but[@sv] as its surface area. The coefficient of proportionality between transverse moment and area depends on the sample [*shape*]{} and one can express this dependence through the aspect ratio[@noecc] ${A}/{C}$. Since the area of an ellipsoid of revolution is $\pi A^2$ times a function of ${A}/{C}$, it is easier for the purpose of giving results in a form more accessible to experimentalists, to scale explicitly results for ellipsoids of the same aspect ratio (i.e. the same shape) by a factor of the “disk” area ${\cal S}\equiv \pi A^2$. The third length that goes into the volume factor in the units of $m_\bot$ is[@sv; @ys] a penetration depth. Since we are dealing with rather large $A/C$ ratios, with currents predominantly in the $a-b$ plane, it is advantageous for our purposes to take this length into accout by writing out a factor of $\lambda_{ab}$ explicitly. Thus we put, as a first step: $$4 \pi m_{\bot}(\psi)=
{\gamma}(\frac{A}{C},\frac{\lambda_c}{\lambda_{ab}},H)
{\cal S} \lambda_{ab} f(\psi)
\label{mfirst}$$ Here and hereafter we denote the magnitude of the applied field simply by $H$, rather than $H_a$, as there is no longer a possibility of confusion. It is not surprising in view of the above dimensional analysis that we find ${\gamma}$ to be a rather weak function of its first two arguments and independent[@scaling] of ${\cal S}$ and of $\lambda_{ab}/A$. The angular dependence of $m_{\bot}$ is given by the function $f(\psi$) in terms of the angle $\psi$ between the applied field and a node. We normalize $f(\psi)$ so that its value is unity at its maximum.
Our results for the angular dependence are shown in Figure \[shape\]. The points shown are the values obtained from our numerical calculation. The error bars indicate the numerical uncertainty. The solid line represents the analytic result for the two dimensional calculations in the slab case[@sv; @ys], normalized in the same way. We see from the Figure that the shape of $f(\psi)$ is, within numerical uncertainty, the same as for the flat case, where, with the same normalization, $f(\psi)=3\sqrt{3} \sin\psi\cos\psi(\cos\psi-\sin\psi)$, (for $0<\psi<\pi/2$). This is important, because[@sv] the $\pi/2$ Fourier coefficient of $f(\psi)$, as analytically calculated from the above expression, is unity to three significant figures. We can therefore identify here the coefficient of $f(\psi)$ in (\[mfirst\]) with the Fourier amplitude $m_{\bot}$ as introduced earlier in the paper. The results shown in this Figure were obtained at $H/H_0=0.1$, $\delta=16$, and $A/C=19$, but similar results are obtained in all other cases studied.
As can already be seen in Figure \[predict\], the coefficient ${\gamma}$ depends strongly on its third argument, the applied field. We have studied the field range $0<H/H_0<0.2$ at $0.05$ intervals. We expect that the nonlinear $m_{\bot}$ is proportional to $H^2$, as in the slab case at zero temperature. We therefore conclude that a very convenient way of writing our results for the amplitude is: $$4 \pi m_{\bot}= {\cal M}(\frac {A}{C},\frac{\lambda_c}{\lambda_{ab}})
\:\frac{H}{H_0}\:H\:\lambda_{ab}\: {\cal S} \:f(\psi).
\label{mtr2}$$ Equation (\[mtr2\]) implies explicitly that $m_{\bot}/ H$ is a function of field only through the ratio $H/H_0$. We have verified that, as expected, ${\cal M}$ is independent of the field. This can be seen in Fig. \[hsquare\], where we plot the quantity $G\equiv 10^2 4\pi m_\bot/(H\lambda_{ab}{\cal S})$ vs $H$. We see that our results, represented by the symbols with error bars, are on a straight line, which is indicated by a best fit. Except for the field, the parameter values are the same as in the previous Figure. As in the case of the angular dependence, this qualitative result holds in all cases studied. The slope of the best fit straight line in plots such as that in this figure is used to extract ${\cal M}$.
The geometric aspect ratios we have considered in the nonlinear case range from $A/C=$7 up to 19 (that is from $\sinh\alpha=$0.14433 to 0.05270). Comparison of the results with largest eccentricity to previous work on the slab case illustrates the effect of the return currents. This comparison cannot be made by taking $C\rightarrow 0$ because we must have $C>>\lambda_{ab}$. The central portion of a flat ellipsoid can be identified as a “slab”. It is simpler to make the comparison in terms of $Q$ defined in (\[Qeq\]). It follows from our dimensional analysis that: $$Q=\tilde{\gamma}\frac{H}{H_0}\frac{\lambda}{2C}f(\psi)
\label{qscale}$$ The corresponding result for the slab is of the same form, and the coefficient may be extracted from the $T=0$ results of Ref.. When all the relevant factors, such as the different definition of $H_0$, the normalization of $f(\psi)$, and the setting of the parameter[@ys] $\mu$ (slope of the OP near a node) to $\mu=2$ are taken into account, one finds that the slab value is $\tilde{\gamma}=0.056$. We obtain $\tilde{\gamma}=0.092$ for our flattest ellipsoid at $\delta=16$, a number only weakly dependent on $\delta$. Hence, the presence of return currents enhances the nonlinear effects but, as hinted by the recent reanalysis[@buanyang] of the experimental data, the enhancement is about three times smaller, in typical situations, than the $\delta$ dependent factor postulated in Ref. .
We next show the dependence of our results on the remaining parameters, $A/C$ and $\lambda_c/\lambda_{ab}$. As explained above, our results are most useful to experimentalists if given in terms of the scaled dimensionless amplitude ${\cal M}$. We therefore summarize the dependence of this quantity on the mentioned length ratios in Table I. This table can be used, in conjuction with Eq. (\[mtr2\]) to compute the theoretical predictions, when planning experiments or comparing data and theory. The rows are the relevant material parameter, given as the square of the penetration depth ratio, and the columns are the crystal shape. For crystals that are not quite ellipsoidal, one should choose a $C/A$ ratio in the table so that the surface to volume ratio of the ellipsoid agrees with that of the crystal. As the results vary slowly across the rows and columns of the Table, interpolation and reasonable extrapolation are obviously feasible. The value of ${\cal M}$ in the Table must be multiplied by the square of the applied field and by the cross sectional area of the sample in the $a-b$ plane, then divided by $H_0$ (which is again a material-dependent parameter), to obtain the expected value of the magnetic moment amplitude to be observed. The Table covers anisotropy parameter $\delta$ values up to 50.
The dependence of ${\cal M}$ on its arguments is weak, less important than the unavoidable uncertainty in the values of the experimental input parameters such as those that go into e.g. the determination of $H_0$. The slow dependence of ${\cal M}$ on $\delta$ can be roughly understood as follows: the portion of the current loop which is along the $c$ direction does of course contribute to the magnetic moment. However, provided, as in the Table, that $A>>\lambda_c>\lambda_{ab}$ the exact value of $\lambda_c$ does not matter very much, since (as seen in Fig. \[nonlincur\]) the decay of the nonlinear current is governed primarily by $\lambda_{ab}$. The nonlinear moment is then limited by the smaller of the penetration depths. The slight downtrend with $\delta$ may be due to changes in the complicated current patterns discussed in connection with Fig. \[nonlincur\]. The slight decrease with decreasing eccentricity is likely to be due to the influence of the $j_z$ component (dominated by linear effects) in a thicker crystal. The Table indicates that flatter crystals are actually preferable to harder to obtain thick ones. Again, however, a simple explanation in intuitive terms is hard to come by, since these results must be related to the intricacies of the nonlinear current patterns.
Finally we consider the sensitivity of our results to the unknown dependence of the order parameter on the crystallographic $c$ direction. In Appendix A, we have calculated the coefficients of the nonlinear terms in the current (as in Eq. (\[jnl\])) for the simplest form of the appropriate[@sigrist] 3d OP. We have verified that an increase occurs in the quantity ${\cal M}$ but it is negated by a compensating increase in $H_0$, where then one should replace $\xi_0$ by some average of in and out of plane correlation lengths. We conclude that the influence of any such dependence can be neglected as compared with the uncertainty in the material parameters.
conclusions
===========
The detailed calculations for realistic samples performed here strongly confirm that the transverse magnetization effect should be readily observed in available crystals if the order parameter has nodes of the form expected for d-wave OP symmetry. From our results, see e.g. Fig. \[predict\], it follows that the transverse magnetic moment generated in a typical sample of about 2 mm diameter and 0.1 mm thickness (the precise thickness not being very important) should be $4\pi m_\bot \approx 10^{-6} {\rm gauss\: cm^3}$. These numbers are above what can be readily measured by standard experimental techniques: the rather high uncertainty levels quoted in past experiments[@buan] arise from the geometrical problems described in the Introduction and also from easy to overcome uncertainties[@anandpc] in the angular positioning the sample. We have seen that the return currents in the z-direction produce an enhancement of the effect, although not nearly as large as that estimated in Ref.. The effect of the boundary conditions and the finite size on the result is somewhat intricate and, perhaps not surprisingly, we have not been able to find a clear physical picture in terms of some nonlinear generalization of the demagnetization factors. Although our results for the influence of sample geometry and anisotropy in penetration depth were not amenable to simple summing up either, the tabulated values of the quantity ${\cal M}$ should facilitate experimental design and interpretation.
As briefly indicated in Appendix A, (see below Eq.(\[phic\])) it is easy at $T=0$ to take into account different shapes of the d-wave order parameter function close to the nodes. Our calculations can also be extended to other symmetries of the OP, such as an $s+d$ or $s+id$ state, or to take into account dependence of the OP in the c-direction. In the worst case, even at finite temperatures, the functional ${\bf j(v)}$ could be, numerically evaluated and used in the calculations, although possibly at considerable cost in computer time, by means of a lookup table and interpolation scheme.
We have not considered in this work the effect of impurities. This is unnecessary as it was shown in Ref. that the impurity concentration required to noticeably decrease the nonlinear Meissner effect at low temperatures is such that it would clearly reduce the transition temperature of the material. This result will not depend on the geometry of the sample. On the other hand, we have considered here only the case where the order parameter has nodes, not just dips. The nonlinear effects are in fact extremely sensible to the presence of nodes and $m_\bot$ will decrease substantially, in finite as in infinite samples[@sv] if an s-wave component eliminates the nodes. The magnitude of this decrease can be gauged from the infinite slab work[@sv] since, again, it should not be excessively sensitive to return currents or sample shape. Therefore, a negative experimental result could be used to put a [*lower*]{} bound on the amount of s-wave component present.
The methods discussed here can be extended to other sample shapes, to finite temperatures, and to the computation of other measurable effects that arise from the same nonlinear phenomena.
We thank A. Bhattacharya, A.M. Goldman, J. Buan and D. Grupp for many enlightening conversations concerning the experimental implications of our work, and B.P. Stojković for reading a draft of this work. We also thank B.Bayman and J. Sauls for discussions. I. Ž. acknowledges support from the Foster Wheeler and Stanwood Johnston Memorial Fellowships.
currents
=========
We derive here the expression for ${\bf j(v)}$ at $T=0$ starting from equation (\[jtzero\]). Our procedure is a generalization, valid in the case of interest where the anisotropy $\delta$ is large, of the two dimesional derivation[@ys]. We consider an ellipsoidal Fermi surface: $$\epsilon _f=\frac{k_x^2+k_y^2}{2m_{ab}}+\frac{k_z^2}{2m_c}\equiv
\frac{1}{2} m_{ab} v_f^2
\label{ef}$$ where we introduce $v_f$ as in Section II. To establish our notation we briefly review the first, linear term in (\[jtzero\]): $$j=-eN_f\int d^2s \: n(s) {\bf v}_f ({\bf v}_f \cdot {\bf v})
\label{firstterm}$$ which is due to the condensate contribution to the current. For its evaluation one uses the convenient standard textbook method of rescaling $k_z$ by a factor of $\delta^{-1/2}$. The components of ${\bf v}_f$ are then given by:
\[vcomponents\] $$(v_f)_{x'}=v_f \sin \theta \cos \phi \label{vfa}$$ $$(v_f)_{y'}=v_f \sin \theta \sin \phi \label{vfb}$$ $$(v_f)_{z'}=\frac{v_f}{ \sqrt{\delta}} \cos \theta \label{vfc}$$
where $(\theta,\phi)$, ($\phi$ measured from the x’ axis) are the spherical angles of a vector with components $k_x, k_y, k_z/\delta^{1/2}$. The scalar product in (\[firstterm\]) can then be written as: $${\bf v}_f \cdot {\bf v}=v_f (v_{x'} \sin \theta \cos \phi
+v_{y'} \sin \theta \sin \phi +\frac{v_{z'}}{ \sqrt{\delta}} \cos \theta)
\label{vfv}$$ The integration can be performed by replacing $\int d^2 s \: n(s)$ by $\int _{\Omega} d \phi \: d \theta \sin \theta /(4 \pi)$ which yields $$j_{x',y'}=-e \rho_{ab} v_{x',y'} \label{jlina}$$ $$j_{z}=-e \rho_{c} v_{z} \label{jlinb}$$ With $\rho_{ab}= \frac {1}{3} N_f v_f^2$, $\rho_c=
\rho_{ab}/ \delta$, as quoted in Section II, (we recall that $N_f$ is the total density of states, for both spins). Thus, as is well known, the effect of the ellipsoidal Fermi surface [@muz] is to rescale $\rho_c$ by a factor of $1/ \delta$ relative to $\rho_{ab}$.
The nonlinear corrections are contained in the second term of (\[jtzero\]) which is due to the quasiparticle backflow. We denote this term by ${\bf j}_{qp}$. At T=0, one can perform the $\xi$ integration: $${\bf j}_{qp}=-2e N_f \int_{\Omega}d \phi \: d \theta \sin \theta /(4 \pi)
\:{\bf v}_f
\: \Theta (-{\bf v}_f \cdot {\bf v} - \left| \Delta (\phi) \right|)
\sqrt{({\bf v}_f \cdot {\bf v})^2- \left| \Delta (\phi) \right|^2}
\label{jqp}$$ where $\Delta(\phi)$ is given by (\[op\]). In the Meissner regime, $\left| v_f v \right|\ll \Delta_0$ and the contribution to the quasiparticles arises from narrow wedges along the nodal regions, approximately described by the azimuthal angle $\phi \lesssim (v/ v_c) \ll 1$. A superfluid velocity ${\bf v}$ will give rise to backflow currents because of its components along nodal regions, separated by an azimuthal angle $\pi /2$. In two dimensions, ${\bf v}$ can be[@ys] uniquely decomposed into two “jets” $v_1 \hat{x}'$ and $v_2 \hat{y}'$ along two nodal directions. In the present three-dimensional case, where the anisotropy $\delta$ is large, one can proceed in a similar way. We decompose ${\bf v}$ into the sum of two jets each directed along a nodal direction and tilted by the same angle $\omega$ with respect to the z-axis. Thus we write ${\bf v}=v_{x'} \hat{x}' +v_{y'} \hat{y}'+v_{z} \hat{z}={\bf v}_1+{\bf v}_2$. with: $${\bf v}_1=v_{x'} \hat{x}'+\cot \omega |v_{x'}| \hat{z} \label{v1a}$$ $${\bf v}_2=v_{y'} \hat{y}'+\cot \omega |v_{y'}| \hat{z} \label{v1b}$$ $$\cot \omega=\frac {v_z}{\left| v_{x'} \right|+ \left| v_{y'} \right|}
\label{v1c}$$ The decomposition thus specified is unique, and it ensures that the component $v_z$ is distributed along x’-z and y’-z planes proportionally to the corresponding projections of ${\bf v}$ along the $\hat{x}'$, and $\hat{y}'$ directions. It is easy to see however, that any other decomposition of $v_z$ would lead to the same results for the nonlinear currents derived below, except for higher order corrections in $\delta^{-1}$ which shall be neglected in any case.
The total phase space contributing to the quasiparticle part of the current can be obtained by considering the effects of ${\bf v}_1$ and ${\bf v}_2$ separately. Let us consider the effect of ${\bf v}_1$ in (\[jqp\]). Quasiparticle excitations are allowed in the region described by $\phi \leq \phi _c$, where $({\bf v}_f \cdot {\bf v}_1)^2= (\Delta_0 \sin \phi _c)^2$ and $$\phi_c^2= \frac{v_f^2 v_{x'}^2}{4 \Delta_0^2} (\sin ^2 \theta+
\frac{2}{ \sqrt{\delta}} \sin \theta \cos \theta \cot \omega
\frac{\left| v_{x'} \right|} {v_{x'}}+\frac{ 1} {\delta} \cos ^2
\theta \cot ^2 \omega)
\label{phic}$$ where we have approximated the order parameter in the nodal region by $\Delta(\phi) \approx \Delta _0 2 \phi$. It would be easy to write instead $\Delta(\phi)\approx\Delta_0\mu \phi$, with $\mu$ being a free parameter representing the slope of the OP function near the node, as was done in Ref.. This can be viewed as modifying the characteristic field $H_0$ (Eq. (\[h0\]) by a factor of $\mu/2$.
The integrals involved in the calculation of the nonlinear contribution to the relation ${\bf j(v)}$ due to the ${\bf v}_1$ are then of the form $$I_{x'}=2 \Delta_0 \int_0^{\pi} d \theta \sin^2 \theta \int_{-\phi_c}^{\phi_c}
d \phi \: v_f \sqrt{\phi_c^2 - \phi^2}$$ $$I_{z1}=2 \Delta_0 \int_0^{\pi} d \theta \sin \theta \cos \theta
\frac{1} {\sqrt{\delta}}
\int_{-\phi_c}^{\phi_c} d \phi \:
v_f \sqrt{\phi_c^2 - \phi^2}$$ After integration over the angles $(\theta, \phi)$ we get: $$I_{x'}= \frac {3 \pi ^2}{32} v_f^2 \frac {v_{x'}^2}{v_c}
(1+ \frac {\cot ^2 \omega}{3 \delta}) \label{ix}$$ $$I_{z1}= \frac {\pi ^2}{16 \: \delta} v_f^2 \frac {1}{v_c}
(v_{x'}^2 \cot \omega) \label{iz1}$$ The contribution to the quasiparicle current due to ${\bf v}_2$, is calculated in a precisely similar manner and it gives analogous expressions for $I_{y'}$ and $I_{z2}$. Combining the contributions due to ${\bf v}_1$, ${\bf v}_2$ and omitting, as indicated above, subleading terms of order $(1/ \delta)$ in Eq. (\[ix\]) we get: $$j_{qp \:x',y'}=e N_f v_f^2 \frac {3 \pi }{64} \frac {v_{x',y'}
\left|v_{x',y'}\right|}{v_c}
\label{jqpx}$$ $$j_{qp \:z}= e N_f v_f^2 \frac { \pi }{32 \: \delta} \frac {1}{v_c}
(v_{x'}^2 \cot \omega+v_{y'}^2 \cot \omega) \label{jqpz}$$ In the above expression we substitute $\cot \omega$ given in (\[v1c\]) and recover the nonlinear part of Eq. (\[jnl\]).
If, as an alternative, we consider an OP function of the naive 3-d form[@sigrist] $\propto (k_x^2-k_y^2)$: $$\Delta(\theta,\phi)=\Delta_1 \sin^2(\theta)\sin(2\phi)
\label{opalt}$$ then all integrals can still be done and one obtains instead: $$j_{qp \:x',y'}=e N_f v_f^2 \frac {\pi }{16} \frac {v_{x',y'}
\left|v_{x',y'}\right|}{v_c}
\label{jqpxa}$$ $$j_{qp \:z}= e N_f v_f^2 \frac { \pi }{8 \: \delta} \frac {1}{v_c}
(v_{x'}^2 \cot \omega+v_{y'}^2 \cot \omega) \label{jqpza}$$ where $v_c$ is now defined in terms of $\Delta_1$. The important nonlinear coefficient in (\[jqpxa\]) is a factor of $4/3$ larger than that in \[jqpx\]. The results for ${\cal M}$ increase by the same factor. However, in Eq. (\[h0\]) the quantity $\xi_0$ should be replaced by some average such as $(\xi_{ab}^2 \xi_c)^{1/3}$ which is smaller by about the same amount.
Quantities in spheroidal coordinates
=====================================
The superfluid density tensor, given in the obvious cartesian coordinates by a diagonal tensor with components $(\rho_{ab}, \rho_{ab}, \rho_c)$, is converted to oblate spheroidal coordinates by performing the appropriate transformation. One obtains: $$\tilde{\rho} = \rho_{ab}
\left[ \begin{array}{ccc} \frac{\sinh^2\alpha \sin^2\beta + \delta^{-1}
\cosh^2\alpha \cos^2\beta}{\sinh^2\alpha+\cos^2\beta} &
\frac{(1-\delta^{-1})\sinh\alpha \cosh\alpha
\sin\beta \cos\beta}{\sinh^2\alpha+\cos^2\beta} & 0 \\
\frac{(1-\delta^{-1})\sinh\alpha \cosh\alpha
\sin\beta \cos\beta}{\sinh^2\alpha+\cos^2\beta} &
\frac{\delta^{-1}\sinh^2\alpha \sin^2\beta +
\cosh^2\alpha \cos^2\beta}{\sinh^2\alpha+\cos^2\beta} & 0 \\
0 & 0 & 1\end{array} \right]$$
To solve equations (\[maxlon\]) the expression $\nabla\times\nabla\times {\bf v}$ should be transformed to oblate spheroidal coordinates: $$\begin{aligned}
f^2 (\nabla\times\nabla\times {\bf v})_{\alpha} &=
a_0\partial_{\beta\beta}v_{\alpha}+
a_1\partial_{\beta}v_{\alpha}+
a_2\partial_{\phi\phi}v_{\alpha}+
a_3v_{\alpha}+
a_4\partial_{\alpha\beta}v_{\beta}+
a_5\partial_{\alpha}v_{\beta}+
a_6\partial_{\beta}v_{\beta} \nonumber \\
&+a_7v_{\beta}
+a_8\partial_{\alpha\phi}v_{\phi}+a_9\partial_{\phi}v_{\phi} \end{aligned}$$
$$\begin{aligned}
f^2 (\nabla\times\nabla\times {\bf v})_{\beta} &=
b_0\partial_{\alpha\beta}v_{\alpha}+
b_1\partial_{\alpha}v_{\alpha}+
b_2\partial_{\beta}v_{\alpha}+b_3v_{\alpha}+
b_4\partial_{\alpha\alpha}v_{\beta}+
b_5\partial_{\alpha}v_{\beta}+
b_6\partial_{\phi\phi}v_{\beta} \nonumber \\
&+b_7v_{\beta}+
b_8\partial_{\beta\phi}v_{\phi}+b_9\partial_{\phi}v_{\phi} \end{aligned}$$
$$\begin{aligned}
f^2 (\nabla\times\nabla\times {\bf v})_{\phi} &=
p_0\partial_{\alpha\phi}v_{\alpha}+
p_1\partial_{\phi}v_{\alpha}+
p_2\partial_{\beta\phi}v_{\beta}+
p_3\partial_{\phi}v_{\beta}+
p_4\partial_{\alpha\alpha}v_{\phi}+
p_5\partial_{\beta\beta}v_{\phi} \nonumber \\
&+p_6\partial_{\alpha}v_{\phi}+
p_7\partial_{\beta}v_{\phi}+p_8 v_{\phi}\end{aligned}$$
we recall that f is a focal length scale factor. The coefficients $a_i,b_i,p_i$ are given by (using the abreviations $t\equiv\sinh\alpha$, $u\equiv\cosh\alpha$, $s\equiv\sin\beta$, $c\equiv\cos\beta$, $w\equiv(t^2+c^2)^{1/2}$): $$\begin{aligned}
&a_0=-a_4=-b_0=b_4=p_4=p_5=-\frac {1}{w^2} \nonumber \\
&a_1=-\frac{w^2}{u^2}a_5=p_7=\frac{ws}{u}p_3=-\frac{c}{w^2 s} \nonumber \\
&a_2=b_6=-p_8=-\frac{1}{u^2 s^2} \nonumber \\
&a_3=\frac{2 c^2-t^2+3 t^2 c^2}{w^6} \nonumber \\
&a_6=\frac{t u }{w^4} \nonumber \\
&a_7=\frac{tuc(3+t^2-2c^2)}{w^6 s} \nonumber \\
&a_8=b_8=p_0=p_2=\frac{u}{t}a_9=\frac{s}{c}b_9=\frac{1}{u w s} \nonumber \\
&b_1=-\frac{c s}{w^4} \nonumber \\
&b_2=\frac{u s}{w}p_1=-\frac{t s^2}{u w^4}\nonumber \\
&b_3=\frac{tcs(3+2 t^2-c^2)}{w^6 u} \nonumber \\
&b_5=p_6=-\frac{2t}{u w^2} \nonumber \\
&b_7=\frac{2t^2-c^2-3 t^2 c^2}{w^6} \nonumber\end{aligned}$$ The right hand side of (\[maxlon\]) has to be transformed to spheroidal coordinates and added to the expressions above.
J. Annett, N. Goldenfeld, and A. J. Leggett, to appear in [*Physical properties of High Temperature Superconductors*]{}, World Scientific, Singapore, (1996). J. Buan [*et al*]{} preprint. S.K. Yip and J.A. Sauls, Phys. Rev. Lett. [**69**]{}, 2264, (1992). B.P. Stojković and O.T. Valls, Phys. Rev. [**B51**]{}, 6049, (1995). D. Xu, S.K. Yip and J.A. Sauls, Phys Rev. [**B51**]{}, 16233, (1995). See for example, S.R. Bahcall, unpublished. Magnetic moment is magnetization times volume. As both quantites are given in “emu” in gaussian units, much confusion between them occurs in the literature. Here we will give results in terms of $4\pi m$, in gauss times volume, as it is in fact done in the experimental work cited here. J. Buan, B.P. Stojković, N.E. Israeloff, C.C. Huang, A.M. Goldman, and O.T. Valls, Phys. Rev. Lett. [**72**]{}, 2632, (1994). T.P. Orlando and K.A. Delin, [*Foundations of Applied Superconductivity*]{}, Addison-Wesley, Reading, (1991). A.M. Goldman, private communication. Z.H. Lin, G.C. Spalding, A.M. Goldman, B.F. Bayman, and O.T. Valls, Europhysics Lett. [**32**]{}, 573, (1995). P.G. DeGennes, [*Superconductivity of Metals and Alloys*]{} Addison-Wesley, reading, (1989). J.R. Reitz, F.J. Milford, and R.W. Christy, [*Foundations of Electromagnetic Theory*]{}, Addison-Wesley, Reading, (1989), page 329. J. D. Jackson, [*Classical Electrodynamics*]{}, John Wiley, New York, (1975), page 181. J. D. Jackson, [*op. cit.*]{} inside cover. A. Fetter and J.D. Walecka, [*Theory of Many Particle Systems*]{}, McGraw-Hill, New York, (1971). F. London, [*Superfluids*]{}, v 1, John Wiley, New York, (1950). J. Bardeen, Rev. Mod. Phys. [**34**]{}, 667, (1962). D.N. Basov [*et al*]{}, Phys. Rev. Lett. [**74**]{}, 598, (1995). K. Zhang [*et al*]{}, Phys. Rev. Lett. [**73**]{}, 2484, (1994). See e.g. N.M. Plakida, [*High Temperature Superconductivity*]{}, Springer, Berlin, (1995). In the limit $\lambda\sim d$ the nonlinear effects disappear very quickly, as the sample becomes essentially normal. A. Bhattacharya, D. Grupp, A.M. Goldman, and U. Welp, submitted to Applied Physics Letters. G. Arfken, [*Mathematical Methods for Physicists*]{}, Academic Press, New York, (1970), Chapter 2. This subject was dropped in later editions. N.N. Lebedev, I.P. Skalskaya, and Y.S. Uflyand, [*Worked Problems in Applied Mathematics*]{}, Dover, New York, (1979). L.D. Landau and E.M. Lifshitz, [*Electrodynamics of Continuous Media*]{}, Pergamon, Oxford, (1960). In the “slab” case this procedure would yield immediately the correct result, since then there is no “radial” field and no boundary condition is sacrificed. Additional mathematical details will be given elsewhere. One could use the eccentricity instead, but $C/A$ is more immediately accessible. The computational systems used, on which this scaling has been verified, range in size all the way up to the actual size of the physical samples. M. Sigrist and T.M. Rice, Zeitschrift fur Physik B (Condensed Matter) [**68**]{}, 9, (1987) A. Bhattacharya private communication. C.H. Choi and P. Muzikar, Phys. Rev. [**B39**]{}, 11296, (1989).
------------------------------------- -------- -------- -------
$\delta=(\lambda_c/\lambda_{ab})^2$ A/C=19 A/C=10 A/C=7
16 0.061 0.060 0.059
25 0.058 0.057 0.056
36 0.057 0.054 0.052
50 0.055 0.051 0.049
------------------------------------- -------- -------- -------
: The dimensionless quantity ${\cal M}$, defined by Eq. (), as a function of the material parameter $\delta$ and of sample shape.[]{data-label="table1"}
|
---
abstract: |
Accurate predictions of pollutant concentrations at new locations are often of interest in air pollution studies on fine particulate matters (PM$_{2.5}$), in which data is usually not measured at all study locations. PM$_{2.5}$ is also a mixture of many different chemical components. Principal component analysis (PCA) can be incorporated to obtain lower-dimensional representative scores of such multi-pollutant data. Spatial prediction can then be used to estimate these scores at new locations. Recently developed predictive PCA modifies the traditional PCA algorithm to obtain scores with spatial structures that can be well predicted at unmeasured locations. However, these approaches require complete data, whereas multi-pollutant data tends to have complex missing patterns in practice. We propose probabilistic versions of predictive PCA which allow for flexible model-based imputation that can account for spatial information and subsequently improve the overall predictive performance.
[**Keywords:** air pollution, multi-pollutant analysis, missing data, dimension reduction]{}
author:
- 'Phuong T. Vu'
- 'Timothy V. Larson'
- 'Adam A. Szpiro'
bibliography:
- 'VuPT-ProPrPCA.bib'
date: 'August 22, 2019'
title: 'Probabilistic Predictive Principal Component Analysis for Spatially-Misaligned and High-Dimensional Air Pollution Data with Missing Observations'
---
=1
Introduction \[sec-intro\]
==========================
In recent years, there has been a growing interest in studying the role and health impact of PM$_{2.5}$, which is fine particulate matter with aerodynamic diameter less than 2.5 $\mu$m [@brook2004air]. PM$_{2.5}$ is a complex mixture of many components, and its chemical profile may vary drastically across time and space [@brook2004air; @bell2007spatial; @dominici2010protecting]. Obtaining a lower-dimensional representation of PM$_{2.5}$ multi-pollutant data is often necessary, as including many correlated pollutants in a statistical model is problematic. Principal component analysis (PCA) [@jolliffe1986principal] is an unsupervised dimension reduction technique that has gained popularity in multi-pollutant analysis [@dominici2003health].
Examples of environmental studies utilizing PM$_{2.5}$ data include studies on the associations between various health outcomes and long-term [@pope2002lung; @kunzli2005ambient; @miller2007long; @chan2015long; @kaufman2016association] or short-term [@gold2000ambient; @tolbert2007multipollutant; @pascal2014short; @achilleos2017acute; @hsu2017ambient; @tian2017addressing] exposures to PM$_{2.5}$. Many studies have suggested that the associations between PM$_{2.5}$ total mass and various health outcomes can be modified by some specific constituents or the overall chemical composition [@franklin2008role; @bell2009hospital; @krall2013short; @zanobetti2014health; @dai2014associations; @kioumourtzoglou2015pm2; @wang2017long; @keller2018pollutant].
In the United States, PM$_{2.5}$ studies often rely on data collected from regulatory monitoring networks managed by the Environmental Protection Agency (EPA). Unfortunately, for many pollution-health association studies, these fixed monitoring sites are usually not at the same locations where health outcomes are available. Such *spatial misalignment* motivates an exposure modeling stage in which a spatial prediction model, such as land-use regression or universal kriging, is often used to estimate the exposure at unmeasured locations where pollutant data is not observed [@brauer2003estimating; @kunzli2005ambient; @crouse2010postmenopausal; @bergen2013national; @chan2015long].
Derivation of a lower-dimensional representation of PM$_{2.5}$ multivariate data prior to making these spatial predictions is necessary, as predicting chemically and spatially correlated pollutant surfaces is challenging and intractable in most cases. As PCA is capable of performing dimension reduction without meddling with the health outcomes, it can be easily integrated in the analysis of spatially-misaligned data. Using PCA, a lower-dimensional scores of the multi-pollutant data at monitoring locations can be obtained. These monitoring scores, along with geographic covariates, can then be used in a spatial prediction model to estimate the corresponding scores at unmeasured locations. However, PCA does not account for exogenous geographic information and spatial correlations across neighboring locations. Hence, PCA may produce scores that summarize the monitoring data well but are difficult to be predicted at unmeasured locations. A spatially predictive PCA algorithm [@jandarov2017novel] was developed to mitigate this issue by producing scores with spatial patterns that can be subsequently predicted well at new locations.
An additional challenge arises in practice where there is often a large amount of missing data, especially for multi-pollutant monitoring data. For example, not all PM$_{2.5}$ components are measured at all monitoring sites, either due to environmental considerations, logistic constraints or lack of resources. The missing patterns can sometimes be complex or spatially informative. Neither traditional PCA nor predictive PCA is well-equipped to deal with missing data, and thus a separate imputation step is required prior to dimension reduction. Existing non-parametric imputation schemes, ranging from simple mean imputation to sophisticated matrix completion, do not account for external spatial information. They may therefore distort the underlying spatial structure in the original data even before the dimension reduction stage, and thus negatively impact the predictive performance in the final stage.
In this paper, our goal is to enhance the dimension reduction procedure under the presence of missing data by proposing a probabilistic framework in place of the deterministic algorithm of predictive PCA. Similar to [@jandarov2017novel], our methods seek to produce principal components that can be well predicted at new locations. The added probabilistic assumptions allow for flexible model-based imputation that takes into account the embedded geographic and spatial information, and thus eliminates the need for a preprocessing stage with non-parametric imputation.
Motivating example \[sec-motivating\]
=====================================
To illustrate the merit of our proposed methods, we use data collected nationally by the Air Quality System (AQS) network of monitors managed by the EPA. Measurements of annually averaged PM$_{2.5}$ total mass and its components are only collected at a few subnetworks of AQS. For consistency with previous related work [@keller2017covariate; @jandarov2017novel], we choose to use the 2010 data from the Chemical Speciation Network (CSN), of which monitoring sites are located strategically in various urban areas. Data is available for 21 components of PM$_{2.5}$: elemental carbon (EC), organic carbon (OC), sulfate ion (SO$^{2-}_4$), nitrate ion (NO$^{-}_3$), aluminum (Al), arsenic (As), bromine (Br), cadmium (Cd), calcium (Ca), chromium (Cr), copper (Cu), iron (Fe), potassium (K), magnesium (MN), sodium (Na), sulfur (S), silicon (Si), selenium (Se), nickel (Ni), vanadium (V), and zinc (Zn).
Geographic covariates are obtained for all available sites through the Exposure Assessment Core Database by the MESA Air team at the University of Washington. Data on roughly 600 Geographic Information System (GIS) covariates are available, including distances from roads, distances from major pollution sources, land-use information, vegetation indices, etc. The specific sources and attributions of these geographic covariates are carefully described in [@bergen2013national].
Data for 2010 is available for 221 CSN sites, with only 130 of those sites having complete data on all 21 components. Overall the amount of missing data in 2010 is roughly 32.1%. Not only do we compare the predictive performances following the application of different PCA methods, but we also examine how different the chemical profiles are when considering only complete sites versus all available data. The data processing, analysis procedures, and results are discussed in Section \[sec-application\].
Review of PCA and predictive PCA \[sec-review\]
===============================================
We denote ${\mbox{\boldmath $X$}}\in \mathds{R}^{n \times p}$ as the exposure data with $p$ pollutants observed at $n$ monitoring sites with spatial coordinates ${\mbox{\boldmath $s$}}_1, ..., {\mbox{\boldmath $s$}}_n$. The exposure data ${\mbox{\boldmath $X$}}$ may contain missing elements as some pollutants are not measured at all monitoring site. Let ${\mbox{\boldmath $r$}}_i$ be a vector of $k$ geographic covariates pertaining to the $i$-th monitoring sites. Variables corresponding to locations where exposure data is of interest but not measured are distinguished by an asterisk, i.e. $n^*, {\mbox{\boldmath $X$}}^*, {\mbox{\boldmath $s$}}^*_1, ..., {\mbox{\boldmath $s$}}^*_{n^*}, {\mbox{\boldmath $r$}}^*_1, ..., {\mbox{\boldmath $r$}}^*_{n^*}$.
The data of interest, ${\mbox{\boldmath $X$}}^*$, is high-dimensional but inaccessible. If ${\mbox{\boldmath $X$}}^*$ were observed, dimension reduction could be applied directly to obtain a lower-dimensional representation ${\mbox{\boldmath $U$}}^* \in \mathds{R}^{n^* \times q}$ where $q < p$. Because of spatial misalignment, a spatial prediction model is required to estimate the unobserved exposures. Modeling highly correlated surfaces is challenging and inefficient given the final aim of recovering only the lower-dimensional ${\mbox{\boldmath $U$}}^*$. Thus, a sensible modeling procedure under the presence of spatially misaligned multi-pollutant data with missing observations may consist of several steps: (1) imputation for missing data, (2) dimension reduction to derive scores at monitoring sites, and (3) spatial prediction to estimate corresponding scores at new locations. In this paper, we focus on dimension reduction using PCA, an unsupervised technique that is suitable for handling spatially-misaligned data.
Traditional PCA provides a mapping from the original $p$-dimensional exposure surface to a corresponding $q$-dimensional representation where ${\mbox{\boldmath $X$}}\approx {\mbox{\boldmath $U$}}{\mbox{\boldmath $V$}}^{\mathsf{T}}$ for $q<p$. We refer to the orthogonal columns of ${\mbox{\boldmath $V$}}\in \mathds{R}^{p\times q}$ as the loadings or principal directions. The columns of ${\mbox{\boldmath $U$}}\in \mathds{R}^{n\times q}$, $\{{\mbox{\boldmath $u$}}_1, ..., {\mbox{\boldmath $u$}}_q \}$, are the principal component (PC) scores. These PC scores can be thought of as linear combinations of the original features of ${\mbox{\boldmath $X$}}$. These newly transformed variables are considered uncorrelated due to orthogonality of the loadings, which is an attractive feature of PCA. The PCA algorithm is also optimal in the sense that the derived PC scores are conveniently ordered by the amount of variability explained in ${\mbox{\boldmath $X$}}$.
While PCA provides a unique solution in the reduced dimensions, the algorithm can be reformulated into a series of biconvex optimization problems, in which the loading and corresponding score of each PC can be solved in an iterative fashion [@shen2008sparse], $$\min_{{\mbox{\boldmath $u$}}, {\mbox{\boldmath $v$}}} \Big\Vert {\mbox{\boldmath $X$}}- {\mbox{\boldmath $u$}}{\mbox{\boldmath $v$}}^{\mathsf{T}}\Big\Vert^2_F \hspace{0.25cm} \text{s.t.} \hspace{0.1cm} \Vert {\mbox{\boldmath $v$}}\Vert_2 = 1.$$ Utilizing such optimization framework, [@jandarov2017novel] develop a spatially predictive PCA algorithm (PredPCA hereafter) by directly incorporating spatial information in the objective function: $$\min_{{\mbox{\boldmath $\alpha$}}, {\mbox{\boldmath $v$}}} \bigg\Vert {\mbox{\boldmath $X$}}- \left( \frac{{\mbox{\boldmath $Z$}}{\mbox{\boldmath $\alpha$}}}{\Vert {\mbox{\boldmath $Z$}}{\mbox{\boldmath $\alpha$}}\Vert_2} \right) {\mbox{\boldmath $v$}}^{\mathsf{T}}\bigg\Vert^2_F ,$$ where ${\mbox{\boldmath $Z$}}= \begin{bmatrix}
{\mbox{\boldmath $R$}}& \tilde{{\mbox{\boldmath $R$}}}
\end{bmatrix}$, in which ${\mbox{\boldmath $R$}}\in \mathds{R}^{n \times k}$ contains $k$ GIS covariates, and $\tilde{{\mbox{\boldmath $R$}}} \in \mathds{R}^{n \times \tilde{k}}$ contains $\tilde{k}$ thin-plate spline basis functions. The induced PC score, ${\mbox{\boldmath $Z$}}{\mbox{\boldmath $\alpha$}}/ \Vert {\mbox{\boldmath $Z$}}{\mbox{\boldmath $\alpha$}}\Vert_2$, is constrained to have an underlying smooth spatial structure guided by geographic and spatial information encoded in ${\mbox{\boldmath $Z$}}$. An advantage of PredPCA over PCA is the capability to identify principal directions that lead to spatially predictable PC scores at unmeasured locations. Recent work by [@bose2018adaptive] further improves PredPCA by adaptively selecting information to be included in ${\mbox{\boldmath $Z$}}$ for each PC.
When monitoring data is incomplete, simply omitting locations with missing data may reduce the usable sample size substantially; thus, imputation is often required prior to dimension reduction. Non-parametric techniques, ranging from mean imputation to matrix completions, are based on observed pollutant values but not additional spatial information. When the missingness is spatially informative, such imputation schemes may heavily bias the results of these algorithms.
In the next section, we propose a probabilistic framework that aims to derive spatially predictive PC scores, with the ability to handle incomplete monitoring data and induce flexible model-based imputation that accounts for spatial and geographic information.
Probabilistic predictive PCA \[sec-proposed\]
=============================================
Probabilistic formulation with a latent variable model: the Krige algorithm
---------------------------------------------------------------------------
[@tipping1999probabilistic] proposed a probabilistic formulation of PCA based on a Gaussian latent variable model. Their model assumes ${\mbox{\boldmath $X$}}= {\mbox{\boldmath $u$}}{\mbox{\boldmath $v$}}^{\mathsf{T}}+ {\mbox{\boldmath $E$}}$, where ${\mbox{\boldmath $u$}}\sim \mathcal{N}({\mbox{\boldmath $0$}}, {\mbox{\boldmath $I$}}_n)$, ${\mbox{\boldmath $v$}}\in \mathds{R}^p$, $\Vert {\mbox{\boldmath $v$}}\Vert_2 = 1$, and the elements of ${\mbox{\boldmath $E$}}$ are independently and identically distributed (i.i.d.) with mean zero and variance $\gamma^2$. We extend this framework by directly imposing a spatial mean and covariance structure on the latent variable space. That is, given a desired number of PCs, $q$, our model assumes $$\begin{aligned}
{\mbox{\boldmath $X$}}&= \sum^q_{l = 1} \left( {\mbox{\boldmath $u$}}_l {\mbox{\boldmath $v$}}_l^{\mathsf{T}}+ {\mbox{\boldmath $E$}}_l \right),\\
{\mbox{\boldmath $u$}}_l &= {\mbox{\boldmath $R$}}{\mbox{\boldmath $\beta$}}_l + \boldsymbol{\eta}_l,\end{aligned}$$ where ${\mbox{\boldmath $\beta$}}_l \in \mathds{R}^k$ includes the coefficients corresponding to the geographic covariates in ${\mbox{\boldmath $R$}}$, while $\boldsymbol{\eta}_l \in \mathds{R}^n$ has zero mean and spatial covariance $\Sigma(\boldsymbol{\xi}_l)$, with $\boldsymbol{\xi}_l$ denoting the spatial covariance parameters of the latent space. We use similar constraint $\Vert {\mbox{\boldmath $v$}}_l \Vert_2 = 1$, and assume that $\Sigma(\boldsymbol{\xi}_l)$ has no nugget effect. The latent score ${\mbox{\boldmath $u$}}_l$ is stochastic with a full spatial distribution.
Let $\Theta_l$ be the collection of the model parameters, $\{{\mbox{\boldmath $v$}}_l, {\mbox{\boldmath $\beta$}}_l, \gamma_l^2, \boldsymbol{\xi}_l\}$, corresponding to the $l$-th PC. When the monitoring data is complete, estimate of the first loading, ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_1$, can be obtained using the original data matrix ${\mbox{\boldmath $X$}}$. The corresponding score ${\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_1$ at monitoring locations can then be calculated by projecting ${\mbox{\boldmath $X$}}$ onto the direction of ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_1$. In later steps, $\Theta_l$ can be estimated using ${\mbox{\boldmath $X$}}_l = {\mbox{\boldmath $X$}}_{l-1} - {\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_{l-1} {\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_{l-1}^{\mathsf{T}}$, where ${\mbox{\boldmath $X$}}_1 = {\mbox{\boldmath $X$}}$. The PC score ${\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_l$ can then be derived by projecting ${\mbox{\boldmath $X$}}_l$ onto ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_l$. Note that we use projection of the data matrix to obtain the PC score in each step instead of using model estimate of the latent mean ${\mbox{\boldmath $R$}}{\mbox{\boldmath $\beta$}}_l$. When some elements of ${\mbox{\boldmath $X$}}$ are missing, estimation of $\Theta_l$ is based only on the observed elements of ${\mbox{\boldmath $X$}}_l$. Estimated PC score ${\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_l$ can then be made by projecting the model-based imputed exposure data onto the direction of ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_l$.
Our approach to estimate $\Theta_l$ in each step is similar to the EM algorithm employed by [@tipping1999probabilistic]. We consider the latent variable ${\mbox{\boldmath $u$}}_l$ to be the “missing" portion, and thus the “complete" data consists of the observed ${\mbox{\boldmath $X$}}_l$ and the latent variable ${\mbox{\boldmath $u$}}_l$. The goal is then to maximize the joint likelihood of ${\mbox{\boldmath $X$}}_l$ and ${\mbox{\boldmath $u$}}_l$. The mathematical details and algorithms for both complete and missing monitoring data are described in the Supplemental Materials. We refer to this framework as the probabilistic predictive PCA, or ProPrPCA, hereafter. Specifically, we call this algorithm ProPrPCA-Krige due to the kriging formulation in the model assumptions.
Our ProPrPCA-Krige model is closely related to the SupSVD model recently proposed by [@li2016supervised]. The SupSVD model is expressed as ${\mbox{\boldmath $X$}}= {\mbox{\boldmath $U$}}{\mbox{\boldmath $V$}}^{\mathsf{T}}+ {\mbox{\boldmath $E$}}$ where ${\mbox{\boldmath $U$}}= {\mbox{\boldmath $Y$}}{\mbox{\boldmath $B$}}+ \mathbf{F}$. Here ${\mbox{\boldmath $U$}}$ is a the latent score matrix, ${\mbox{\boldmath $V$}}$ is a full-rank loading matrix, $\mathbf{F}$ and ${\mbox{\boldmath $E$}}$ are error matrices. [@li2016supervised] also propose an EM approach to estimate the model parameters. The ProPrPCA-Krige model is also related to the envelope model proposed in [@cook2010envelope], which is a more general version compared to SupSVD. As discussed in [@li2016supervised], the SupSVD model attempts to extract a low-rank representation of the original data based on some auxiliary data, while the envelope model aims to reduce variation in regression coefficient estimation. We note that our model is motivated by spatial misalignment where data are not observed at cohort locations, but some geographic information is available. The end goal is also different from the SupSVD and envelope models, as we seek to accurately predict a low-rank representation of the data at unmeasured locations. Thus, our model is designed such that patterns of available covariates and spatial structure are properly induced in the latent scores at locations where we have data, so that we can easily predict them at new locations. An additional contribution is that we develop EM algorithms for parameter estimation for both complete and missing data scenarios.
Probabilistic formulation with thin-plate spline basis: the Spline algorithm
----------------------------------------------------------------------------
While the ProPrPCA-Krige algorithm is cohesive with a prediction stage using universal kriging, the parameter estimation appears to be computational burdensome. In general, the EM algorithm is often computationally expensive and convergence is not always guaranteed. We propose a more simplified version of ProPrPCA, $${\mbox{\boldmath $X$}}= \sum^q_{l=1}\left( ({\mbox{\boldmath $Z$}}{\mbox{\boldmath $\beta$}}_l){\mbox{\boldmath $v$}}_l^{\mathsf{T}}+ {\mbox{\boldmath $E$}}_l \right),$$ where ${\mbox{\boldmath $Z$}}$ contains thin-plate spline functions similar to PredPCA. Compared to the ProPrPCA-Krige model, the latent score ${\mbox{\boldmath $u$}}_l$ no longer has a stochastic component. Instead, ${\mbox{\boldmath $u$}}_l$ is now a smooth structure enriched with spatial patterns included in ${\mbox{\boldmath $Z$}}$.
The overall procedure to obtain PC scores is similar to the Krige algorithm. The algorithm with complete monitoring data is shown in Table \[tab-spline-alg\]. When some elements of ${\mbox{\boldmath $X$}}_l$ are missing, estimation of $\hat{\Theta}_l = \{{\mbox{\boldmath $v$}}_l, {\mbox{\boldmath $\beta$}}_l, \gamma_l^2\}$ is based on the observed elements of ${\mbox{\boldmath $X$}}_l$, and estimated PC score $\hat{{\mbox{\boldmath $u$}}}_l$ can be derived by projecting the model-based imputed exposure matrix onto the direction of ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_l$. When the monitoring data is complete, the algorithm for parameter estimation at each step is straightforward. The mathematical derivations and the algorithm for missing data are described in the Supplemental Materials. We refer to this model as ProPrPCA-Spline due to the use of thin-plate spline basis functions.
Simulations \[sec-simulation\]
==============================
We conduct two sets of simulations to compare the different PCA approaches. The first set involves a low-dimensional setting with three-pollutant exposure surfaces. The second set illustrates a higher-dimensional setting with 15 generated pollutant surfaces. In both cases, the multi-pollutant data is generated on a $100 \times 100$ grid ($N = 10,000) $.
In each simulation, we randomly choose 400 training locations and 100 testing locations. We then apply the four competing methods (PCA, PredPCA, ProPrPCA-Krige, and ProPrPCA-Spline) to the training data, ${\mbox{\boldmath $X$}}^{train}$, to obtain the corresponding loading ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_l^{train}$ and score ${\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_l^{train}$, for $l = 1, ..., q$ where $q$ is a desired number of PCs. We then use ${\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_l^{train}$ and relevant covariate information to obtain $\hat{{\mbox{\boldmath $u$}}}_l^{test}$, predicted scores at testing locations, in a universal kriging model with an exponential covariance assumption. Finally, we compare the predicted scores to the known scores, ${\mbox{\boldmath $u$}}_l^{test}$, which is defined by projecting ${\mbox{\boldmath $X$}}^{test}$ onto the direction of ${\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_l^{train}$.
We also consider various scenarios in which some training data is missing. These scenarios include missing completely at random (MCAR) , with 30%, 35%, and 40% of missing data, and missing at random (MAR), in which the missing patterns are associated with the generated spatial covariates. When there is missing data, we apply low-rank matrix completion (LRMC) via the SoftImpute algorithm [@mazumder2010spectral] to fill in the missing entries prior to PCA and PredPCA.
There are several metrics to evaluate the predictive performance. The metric of interest is the prediction R$^2$ adapted from [@szpiro2011does], which reflects the correlation between ${\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_l^{test}$ and ${\mbox{\boldmath $u$}}_l^{test}$. We also look at the reconstruction error (RE), defined as $\Vert {\mbox{\boldmath $X$}}^{test} - \hat{{\mbox{\boldmath $X$}}}^{test} \Vert_F$ where $\hat{{\mbox{\boldmath $X$}}}^{test} = \hat{{\mbox{\boldmath $U$}}}^{test} (\hat{{\mbox{\boldmath $V$}}}^{train})^{{\mathsf{T}}}$, $\hat{{\mbox{\boldmath $U$}}}^{test} = \begin{bmatrix}
{\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_1^{test} & ... & {\mbox{$\hat{{\mbox{\boldmath $u$}}}$}}_q^{test}
\end{bmatrix}$, and $\hat{{\mbox{\boldmath $V$}}}^{train} = \begin{bmatrix}
{\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_1^{train} & ... & {\mbox{$\hat{{\mbox{\boldmath $v$}}}$}}_q^{train}
\end{bmatrix}$.
Three-dimensional exposure surfaces
-----------------------------------
We simulate three-dimensional surfaces with $\{{\mbox{\boldmath $x$}}_1, {\mbox{\boldmath $x$}}_2, {\mbox{\boldmath $x$}}_3\}$, and three independent covariates $\{{\mbox{\boldmath $r$}}_1, {\mbox{\boldmath $r$}}_2, {\mbox{\boldmath $r$}}_3\}$. Only ${\mbox{\boldmath $r$}}_1 \sim {\mbox{${\cal N}$}}({\mbox{\boldmath $0$}}, {\mbox{\boldmath $I$}}_N)$ is “observed" and thus used in the universal kriging model. Both ${\mbox{\boldmath $r$}}_2 \sim {\mbox{${\cal N}$}}({\mbox{\boldmath $0$}}, {\mbox{\boldmath $I$}}_N)$ and ${\mbox{\boldmath $r$}}_3 \sim {\mbox{${\cal N}$}}({\mbox{\boldmath $0$}}, {\mbox{\boldmath $I$}}_N)$ are unobserved and primarily used to induce correlations across $\{{\mbox{\boldmath $x$}}_1, {\mbox{\boldmath $x$}}_2, {\mbox{\boldmath $x$}}_3\}$. We generate data such that ${\mbox{\boldmath $x$}}_1 = 4{\mbox{\boldmath $r$}}_1 + 2{\mbox{\boldmath $r$}}_3 + {\mbox{\boldmath $\epsilon$}}_1$, ${\mbox{\boldmath $x$}}_2 = 3{\mbox{\boldmath $r$}}_2 + {\mbox{\boldmath $\epsilon$}}_2$, and ${\mbox{\boldmath $x$}}_3 = 2{\mbox{\boldmath $r$}}_1 + 4{\mbox{\boldmath $r$}}_2 + {\mbox{\boldmath $\epsilon$}}_3$, where ${\mbox{\boldmath $\epsilon$}}_1, {\mbox{\boldmath $\epsilon$}}_2, {\mbox{\boldmath $\epsilon$}}_3 \sim {\mbox{${\cal N}$}}({\mbox{\boldmath $0$}}, \Sigma)$, where $\Sigma$ has an exponential structure with partial sill $\sigma^2 = 3.5^2$, nugget $\tau^2 = 1$, and range $\phi = 50$. Under this setting, only ${\mbox{\boldmath $x$}}_1$ and ${\mbox{\boldmath $x$}}_3$ are predictable by ${\mbox{\boldmath $r$}}_1$. While not dependent on ${\mbox{\boldmath $r$}}_1$, ${\mbox{\boldmath $x$}}_2$ is moderately correlated with ${\mbox{\boldmath $x$}}_3$ via ${\mbox{\boldmath $r$}}_2$. We also generate a second set of data in which the errors ${\mbox{\boldmath $\epsilon$}}_1, {\mbox{\boldmath $\epsilon$}}_2, {\mbox{\boldmath $\epsilon$}}_3 \sim {\mbox{${\cal N}$}}({\mbox{\boldmath $0$}}, {\mbox{\boldmath $1$}})$ . For MAR scenarios, ${\mbox{\boldmath $x$}}_1$ is missing at training locations where ${\mbox{\boldmath $r$}}_1$ values are larger than its 80th sample percentile, while ${\mbox{\boldmath $x$}}_2$ and ${\mbox{\boldmath $x$}}_3$ have 20% MCAR. We look at the first PC for these simulations, i.e. $q = 1$.
Figure \[fig-toy-scen2\] shows the prediction R$^2$’s and REs across 1,000 simulations for data generated with spatially correlated noise. Table \[tab-loading-scen2\] displays the means and standard deviations of the estimated loadings from each method when the training data is complete. The principal direction produced by PCA is loaded heavily on ${\mbox{\boldmath $x$}}_3$ and only moderately on both ${\mbox{\boldmath $x$}}_1$ and ${\mbox{\boldmath $x$}}_2$. This leads to poor predictive performance for PCA (median R$^2 = 0.40$). Meanwhile, loadings from the other three methods put the most weight on ${\mbox{\boldmath $x$}}_1$ and some on ${\mbox{\boldmath $x$}}_3$, thus they have higher prediction R$^2$’s (median R$^2$’s are about 0.75) and lower REs.
Under MCAR scenarios, prediction R$^2$’s substantially decrease and REs increase for both PCA and PredPCA as the amount of missing data increases. Median R$^2$ of PredPCA drops to as low as 0.64 when training data is 35% MCAR. On the other hand, there are only some subtle reductions in the predictive performances of both ProPrPCA approaches. Under MAR, the performances of both PCA and PredPCA are significantly worse. While ProPrPCA-Krige performs better than PredPCA on average, the variability in performance is high across simulations. Despite not achieving the same level as when the data is complete, ProPrPCA-Spline has the highest predictive performance among the four competing methods.
Table \[tab-loading-scen1\] shows the estimated loadings with complete data, while Figure \[fig-toy-scen1\] shows the prediction R$^2$’s and REs across 1,000 simulations for data generated with independent noise. Similar trends, where ProPrPCA outperforms the rest when missing data is more severe, are also observed in this set of generated data.
High-dimensional exposure surfaces
----------------------------------
We also demonstrate the performance of ProPrPCA algorithms via simulations with 15 generated pollutants. The full setup is described in the Supplemental Materials. Overall, the high-dimensional exposure surfaces are generated from three underlying scores, ${\mbox{\boldmath $u$}}_1$, ${\mbox{\boldmath $u$}}_2$, and ${\mbox{\boldmath $u$}}_3$. The data generating mechanism is such that ${\mbox{\boldmath $u$}}_1$ is the most spatially predictable, ${\mbox{\boldmath $u$}}_2$ is moderately predictable, and ${\mbox{\boldmath $u$}}_3$ is not predictable by any covariates used in the universal kriging model. The loadings used to generate the data are sparse, in order to clearly identify the behaviors of the PCA methods. That is, the first five pollutants, $({\mbox{\boldmath $x$}}_1, {\mbox{\boldmath $x$}}_2, {\mbox{\boldmath $x$}}_3, {\mbox{\boldmath $x$}}_4, {\mbox{\boldmath $x$}}_5)$, are generated from ${\mbox{\boldmath $u$}}_1$. Meanwhile, $({\mbox{\boldmath $x$}}_6, {\mbox{\boldmath $x$}}_7, {\mbox{\boldmath $x$}}_8, {\mbox{\boldmath $x$}}_9, {\mbox{\boldmath $x$}}_{10})$ are generated from ${\mbox{\boldmath $u$}}_2$, and $({\mbox{\boldmath $x$}}_{11}, {\mbox{\boldmath $x$}}_{12}, {\mbox{\boldmath $x$}}_{13}, {\mbox{\boldmath $x$}}_{14}, {\mbox{\boldmath $x$}}_{15})$ are generated from ${\mbox{\boldmath $u$}}_3$. For MAR scenario, we induce a mild spatial pattern in the missing data for the first five pollutants. [In these simulations, we evaluate the predictive performance based on two PCs, i.e. $q = 2$.]{}
We create two scenarios: scenario 1 with $Var({\mbox{\boldmath $u$}}_1) = 10$, $Var({\mbox{\boldmath $u$}}_2) = 7.5$, and $Var({\mbox{\boldmath $u$}}_3) = 5$, and scenario 2 with $Var({\mbox{\boldmath $u$}}_3) = 10$, $Var({\mbox{\boldmath $u$}}_1) = 7.5$, and $Var({\mbox{\boldmath $u$}}_2) = 5$. In scenario 1, where the order of variance contribution is the same as the order of spatial predictability, we expect all methods to identify [linear combinations of ${\mbox{\boldmath $u$}}_1$ and ${\mbox{\boldmath $u$}}_2$ as the first two PCs]{} when training data is complete. In scenario 2, the non-predictable score ${\mbox{\boldmath $u$}}_3$ has the highest variance contribution. Thus we expect PCA to identify [linear combinations of ${\mbox{\boldmath $u$}}_3$ and ${\mbox{\boldmath $u$}}_1$ for the first two PCs, with a large contribution of ${\mbox{\boldmath $u$}}_3$ for the first PC. Meanwhile,]{} we anticipate the other predictive methods to still pick [linear combinations of]{} ${\mbox{\boldmath $u$}}_1$ and ${\mbox{\boldmath $u$}}_2$.
Table \[tab-scen123\] shows the results for the prediction R$^2$’s across 1,000 simulations under scenario 1. As expected under scenario 1, all methods perform comparably when the training data is complete. While the results for MCAR 30% and 40% are not shown in this chapter, we observed similar patterns to the three-dimensional simulations where the performance of PCA and PredPCA decreases steadily as the amount of MCAR missing data increases. Under MCAR 35% setting, ProPrPCA-Spline has the best median R$^2$’s for both PCs.
Under MAR, data among the first five pollutants are more likely to be missing at locations with extreme geographic covariate values. This setup effectively has an impact on the actual variance contributions of the underlying scores in a given sample, and particularly lowers the variability contributed by ${\mbox{\boldmath $u$}}_1$. As a result, for PC1, PCA is likely to produce loadings with higher contribution from ${\mbox{\boldmath $u$}}_2$ than before. As the predictive methods (PredPCA and ProPrPCA) attempt to balance out the trade-off between data representativeness and spatial predictability, these methods will also likely to obtain linear combinations with more weights from ${\mbox{\boldmath $u$}}_2$ for PC1 than before. Subsequently, linear combinations obtained for PC2 will have more weights from ${\mbox{\boldmath $u$}}_1$ than before. This explains the decreases in median R$^2$’s of PC1 for all methods but slight increases for PC2. ProPrPCA-Spline notably has the best median R$^2$ for PC1.
We further compare the differences in R$^2$ values between ProPrPCA-Spline and PredPCA in Figure \[fig-hd-s123\]. With complete training data, ProPrPCA-Spline outperforms PredPCA for only less than 60% of the simulations, and the magnitude of the difference between the two methods is rather negligible. Under MCAR 35%, ProPrPCA-Spline outperforms PredPCA for both PCs in 69.7% of the 1,000 simulations, and, for 28.5% of the time, ProPrPCA-Spline is better in one of the PCs. Finally, under MAR, there are only 2.5% of the simulations in which ProPrPCA-Spline is worse than PredPCA for both PCs. There are 38.7% of the simulations where ProPrPCA-Spline is better for only PC1 (blue top-left quadrant). Particularly for points lying in this quadrant, the greater spread along the y-axis implies that a higher increase in R$^2$ for PC1 is often accompanied by a smaller decrease in R$^2$ for PC2. Thus ProPrPCA-Spline shows more prominent benefits for PC1 without trading off too much in predictability of PC2.
Table \[tab-scen312\] and Figure \[fig-hd-s312\] show the corresponding results under scenario 2. In this scenario, as expected, PCA often identifies linear combinations of ${\mbox{\boldmath $u$}}_3$ and ${\mbox{\boldmath $u$}}_1$ as the first two PCs, and thus the predictive performance is generally poor, especially for PC1. ProPrPCA-Krige severely underperforms compared to PredPCA and ProPrPCA-Spline, even with complete data. Both PredPCA and ProPrPCA-Spline produce similar median R$^2$’s with complete data. Similar to scenario 1, ProPrPCA-Spline performs consistently well with an increasing amount of MCAR, while the performance of PredPCA deteriorates. ProPrPCA-Spline shows clear benefits under MAR, particularly for PC1 (0.72) compared to PredPCA (0.63). The visualization of the differences in prediction R$^2$’s between ProPrPCA-Spline and PredPCA in Figure \[fig-hd-s312\] further supports similar conclusions to those of scenario 1.
Data application \[sec-application\]
====================================
Methods
-------
In this section, we first compare the pollutant profiles obtained by different dimension reduction methods to the annual average 2010 CSN data. Prior to our analysis, we take a similar approach to [@keller2017covariate] and convert the mass concentrations of PM$_{2.5}$ components to proportions by dividing by the total mass of PM$_{2.5}$, and then log-transform these proportions. We also follow a similar preprocessing procedure as described in [@keller2017covariate] and [@jandarov2017novel] to the GIS covariates to be used in the predictive algorithms and spatial prediction model. That is, we remove covariates that are missing at all chosen sites, have the same values in at least 80% of the sites, or have at least 2% of their values being more than five standard deviations away from the sample mean. We also remove land-use covariates whose maximal value is only 10% among all chosen sites. Finally, we apply PCA on the processed GIS data and use the first five PCs in later stages.
After the preprocessing procedure, we end up with a total of 221 CSN sites, only 130 of which have complete data on all 21 PM$_{2.5}$ components. We first apply three methods, PCA, PredPCA, and ProPrPCA-Spline, on the 130 [ sites with complete data (the “complete" set)]{}. We then proceed to apply these methods on all 221 CSN sites [ (the “full" set)]{}, where LRMC is applied prior to PCA and PredPCA. The goal is to assess how the estimated loadings and PC scores change when using only sites with complete data compared with using all available sites. The design matrix, ${\mbox{\boldmath $Z$}}$, used in PredPCA and ProPrPCA-Spline includes the five PCs of GIS covariates and thin-plate spline basis functions generated from the spatial coordinates, similar to [@jandarov2017novel]. We do not use ProPrPCA-Krige in our comparison because of its computational cost and inferior performance compared to ProPrPCA-Spline in our previously described simulations.
We also conduct leave-one-site-out cross-validation to compare the predictive performances among these methods. In each round of cross-validation, we leave out one site among the complete sites as test data. We then perform dimension reduction and fit a universal kriging model on training data comprised of either only the remaining complete sites [ (the “complete" training data)]{}, or all remaining sites [ (the “full" training data), while the testing data in each round stays the same.]{} The goal is to assess the predictive performance of different methods with both complete and missing data.
Results
-------
### The multi-pollutant profile
Figure \[fig-real-feature1\] shows the estimated loadings and the spatial distributions of corresponding scores of the first PC for four combinations of method and dataset: PCA applied to [ the complete set]{}, PredPCA applied to [ the complete set]{}, imputation followed by PredPCA applied to [ the full set]{}, and ProPrPCA-Spline applied to [ the full set]{}. The results for ProPrPCA-Spline when using [ the complete set]{} (not shown here) are essentially identical to PredPCA results.
The estimated PC1 loadings are similar across PredPCA applied to either sets and to ProPrPCA-Spline, with highly positive weights on SO$^{2-}_4$ and S and highly negative weights on Al, Ca, Na, and Si. Highly positive scores are observed in the east and part of the Midwest, probably due to sulfur emissions from coal combustion [@thurston2011source; @hand2012seasonal]. Negative scores are observed in the west and southwest, and have a classic resuspended soil profile [@thurston2011source; @tong2012long; @clements2017source]. While the spatial distribution of PCA scores looks similar to other methods, loadings obtained by PCA applied to [ the complete set]{} are fundamentally different than the rest, with much weaker positive weights on SO$^{2-}_4$ and S, and strongly negative weights on many additional elements, including Cr, Cu, Fe, Mn, Ni, Zn.
Figure \[fig-real-feature2\] shows the estimated loadings and the score distributions for the PC that has a highly positive composition of Na, Ni, and V. This feature corresponds to PC3 obtained by [PCA or PredPCA applied to the complete set]{}, and PC2 obtained by [PredPCA or ProPrPCA-Spline applied to the full set]{}. ProPrPCA-Spline results in highly positive scores along the west coast, the east coast, and southeast region, possibly due to residual oil combustion [@thurston2011source], and marine aerosol [@thurston2011source; @kotchenruther2017effects]. ProPrPCA-Spline also identifies pronounced negative loadings on Zn and NO$^{-}_3$. The remaining three combinations of methods and datasets are able to produce fairly similar maps with strongly positive scores along the west coast and across the northern east coast, although they fail to highlight some relevant coastal locations in the southeast region.
Figure \[fig-real-feature3\] shows the results for features highly positive in NO$^{-}_3$ and Zn, which corresponds to PC2 obtained by [PCA or PredPCA applied to the complete set]{}, and PC3 obtained by [PredPCA or ProPrPCA-Spline applied to the full set]{}. For all methods, highly positive scores are observed in the northern Midwest, possibly due to nitrate hazes [@coutant2003compilation; @pitchford2009characterization; @hand2012seasonal]. Additionally, loadings produced by ProPrPCA-Spline are also strongly positive in Ni, V, and negative in Al, Si, with greater magnitude compared to other methods. Thus, moderately positive scores are also observed along the west coast. ProPrPCA-Spline also results in highly positive scores in the southeast region due to the calcium poor soils in that region compared to Al and Si content [@shacklette1984element].
### Cross-validation results
Finally, we look at the predictive performances in leave-one-site-out cross-validations, and the results are shown in Figure \[fig-cv\]. While having decent performance for PC2 and PC3 (R$^2 = 0.51$), using PCA applied to [ the complete training data]{} yields a poor result for PC1 (R$^2 = 0.24$). PredPCA has similar performances for PC1 with either [complete or full training data]{}. However, there is a substantial trade-off in performances between PC2 and PC3, which can potentially be explained by the switching between PC2 and PC3 observed in the pollutant profile. ProPrPCA-Spline applied on [ the full training data]{} shows the highest predictive performance for PC1 (R$^2 = 0.57$) and PC3 (R$^2 = 0.69$), but suffers from a decrease in the ability to predict PC2 well (R$^2 = 0.35$).
A possible explanation to the overall relatively low R$^2$’s for all methods is that we use the same pre-specified spatial information encoded in ${\mbox{\boldmath $Z$}}$ to characterize the spatial variability across all PCs, which may not be effective. A potential solution, which is beyond the scope of this paper, is adaptive selection of features to be included in ${\mbox{\boldmath $Z$}}$, which is proposed and discussed in [@bose2018adaptive].
Discussion
==========
In this chapter, we propose a probabilistic extension to the PredPCA algorithm developed by [@jandarov2017novel]. The proposed ProPrPCA algorithms can be applied to misaligned multi-pollutant data with missing observations. The ultimate goal is to improve the predictive performance of the exposure modeling stage that is often required in air pollution studies that rely on fixed site monitoring data. [ In spite of its simplicity, these probabilistic extensions are nontrivial and effective in mitigating the impact of missing data on the predictive performance of the exposure model. The proposed methods also eliminate the necessity of a separate imputation procedure prior to dimension reduction.]{} The scientific motivation, especially in health-pollution studies on PM$_{2.5}$ and its components, includes the ability to use estimated PC scores at study locations as effect modifiers for the main health associations of interest.
We have demonstrated via simulations that ProPrPCA-Spline consistently outperforms its competitors under various missing observation scenarios. Its computational speed is on par with both PCA and PredPCA, which are non likelihood-based methods. The complex version, ProPrPCA-Krige, assumes a universal kriging formulation for the latent variable, with the mean model enriched by spatial covariates, and spatial correlations among the residuals. ProPrPCA-Spline incorporates thin-plate spline basis functions, which can be regarded as an alternative to a fixed low-rank kriging model [@kammann2003geoadditive]. Intuitively, the latent specification of ProPrPCA-Krige would have been cohesive with the later prediction stage using universal kriging. Possible explanations for the inferior performance of the Krige algorithm in simulations include the difficult nature of the numerical optimization for spatial variance parameters, the number of parameters to estimate, and no guaranteed convergence to the global optima using the EM algorithm.
PCA is closely related to factor analysis [@harman1976modern], k-mean clustering [@macqueen1967some], or positive matrix factorization [@paatero1994positive], which have recently been used as source apportionment or dimension reduction for exposure data prior to health analyses [@sarnat2008fine; @ostro2011effects; @zanobetti2014health; @ljungman2016impact]. These applications, however, have been limited to time-series analysis in specific regions, without the challenge of spatial misalignment and severe missing data. Recent work by [@keller2017covariate] and [@jandarov2017novel] has modified the traditional clustering and PCA methods, respectively, to the setting of spatially-misaligned multi-pollutant data, where the products of the dimension reduction procedure are desired to be spatially predictable. We further extend these frameworks by considering the realistic challenge of missing monitoring data. Our proposed framework essentially performs model-based imputation, which is cohesive and complementary to the spatial prediction stage. While one can impute the original data with sophisticated low-rank matrix completion techniques, which also operate based on the assumption of a latent variable structure, such methods only rely on observed measures. Therefore, if the missing patterns depend on external geographic covariates, such imputation schemes cannot recover the correct data structure.
In the literature, spatial latent variable models have been explored under the Bayesian framework. For example, [@wang2003generalized] proposed a generalized common spatial factor model using MCMC techniques. [@hogan2004bayesian] formulated a Bayesian factor analysis model, which was later extended by [@liu2005generalized] to motivate a generalized spatial structural equations model, and by [@zhu2005generalized] to deal with spatiotemporal data. These rich modeling approaches have not been utilized in the setting of multi-pollutant analysis with spatial misalignment. The main goal of these models is often to explain the associations between the original variables and the underlying factors. Here the goal of an improved PCA algorithm is to obtain a lower-dimensional representation of the data in a spatially predictive way for subsequent use in spatial prediction and health regression.
The multi-stage procedure in analyzing health-pollution association under spatial misalignment is a common and pragmatic approach [@crouse2010postmenopausal; @bergen2013national; @chan2015long]. However, it is important to be mindful of the potential implications of measurement errors and model uncertainty of the spatial prediction stage on the health inference model, a topic which has been discussed extensively in [@szpiro2013measurement]. Additionally, these authors emphasized that the spatially structured components of the covariates used in the health model should be included in the exposure modeling stage to guarantee a consistent estimation of the health effects. In the multi-pollutant setting with missing observations, additional stages of imputation and dimension reduction lead to more complicated layers of uncertainty. Our proposed methods eliminate the need of a separate imputation step prior to dimension reduction, as these two steps are handled simultaneously using a model-based approach. A possible alternative to the multi-stage paradigm is a unified approach where both exposure and health data are considered simultaneously in a joint model, while leveraging the factor analysis framework to perform dimension reduction. [@szpiro2013measurement] point out several disadvantages of such joint model, including sensitivity to influential or outlying health data, vulnerability to model mis-specifications, and computational burden, especially with multi-pollutant data.
While we focus our discussion in this chapter exclusively on studies involving data on PM$_{2.5}$ and its components, our proposed method is both appropriate for other multi-pollutant studies and applicable to other fields in general where spatial misalignment necessitates an exposure modeling procedure. Future work includes further understanding and improvement of the ProPrPCA-Krige algorithm, and a possible extension to spatiotemporal data.
Supporting Information
======================
Data used in this paper are available upon request through the MESA Air team at the University of Washington. Supplemental materials can be provided upon email request.
[@\*[1]{}[p@]{}]{} **Input** ${\mbox{\boldmath $X$}}$, ${\mbox{\boldmath $Z$}}$, $q$, and $t_{max}$\
**for** $l$ in $\{1, ..., q \}$ **do**\
${\mbox{\boldmath $X$}}_l \leftarrow {\mbox{\boldmath $X$}}_{l-1} - \hat{{\mbox{\boldmath $u$}}}_{l-1}\hat{{\mbox{\boldmath $v$}}}^{\mathsf{T}}_{l-1} $ where ${\mbox{\boldmath $X$}}_0 = {\mbox{\boldmath $X$}}$, $\hat{{\mbox{\boldmath $u$}}}_0 = {\mbox{\boldmath $0$}}$, and $\hat{{\mbox{\boldmath $v$}}}_0 = {\mbox{\boldmath $0$}}$\
**Initialize** ${\mbox{\boldmath $v$}}^{(0)}_l$, $(\gamma^{(0)}_l)^2$, ${\mbox{\boldmath $\beta$}}^{(0)}_l$, and $t = 1$\
**while** not converged **or** $t < t_{max}$ **do**\
${\mbox{\boldmath $v$}}^{(t+1)}_l \leftarrow {\tilde{{\mbox{\boldmath $v$}}}_l}/{\Vert \tilde{{\mbox{\boldmath $v$}}}_l \Vert_2} $ where $\tilde{{\mbox{\boldmath $v$}}}_l \leftarrow {{\mbox{\boldmath $X$}}_l^{\mathsf{T}}{\mbox{\boldmath $Z$}}{\mbox{\boldmath $\beta$}}^{(t)}_l }/{\big\Vert {\mbox{\boldmath $Z$}}{\mbox{\boldmath $\beta$}}^{(t)}_l \big\Vert^2_2} $\
${\mbox{\boldmath $\beta$}}^{(t+1)}_l \leftarrow \left( {\mbox{\boldmath $Z$}}^{\mathsf{T}}{\mbox{\boldmath $Z$}}\right)^{-1} \left({\mbox{\boldmath $Z$}}\otimes {\mbox{\boldmath $v$}}^{(t+1)}_l \right)^{\mathsf{T}}\text{vec}({\mbox{\boldmath $X$}}_l) $\
$(\gamma^{(t+1)}_l)^2 \leftarrow (np)^{-1} \big\Vert \text{vec}({\mbox{\boldmath $X$}}_l) - ({\mbox{\boldmath $I$}}_n \otimes {\mbox{\boldmath $v$}}^{(t+1)}_l){\mbox{\boldmath $Z$}}{\mbox{\boldmath $\beta$}}^{(t+1)}_l \big\Vert^2_2$\
$t \leftarrow t+1$\
**end while**\
$\hat{{\mbox{\boldmath $v$}}}_l \leftarrow {\mbox{\boldmath $v$}}^{(t)}_l$, $\hat{\gamma}^2_l \leftarrow (\gamma^{(t)}_l)^2$, $\hat{{\mbox{\boldmath $\beta$}}}_l \leftarrow {\mbox{\boldmath $\beta$}}^{(t)}_l$\
$\hat{{\mbox{\boldmath $u$}}}_l = {\mbox{\boldmath $X$}}_l \hat{{\mbox{\boldmath $v$}}}_l$\
**end for**\
**Output** $\{ \hat{{\mbox{\boldmath $v$}}}_1, ...,\hat{{\mbox{\boldmath $v$}}}_q \}$, $\{ \hat{{\mbox{\boldmath $u$}}}_1, ...,\hat{{\mbox{\boldmath $u$}}}_q \}$, $\{ \hat{{\mbox{\boldmath $\beta$}}}_1, ...,\hat{{\mbox{\boldmath $\beta$}}}_q \}$, $\{ \hat{\gamma}^2_1, ...,\hat{\gamma}^2_q \}$\
![Prediction R$^2$’s and reconstruction errors across 1,000 replications with three-dimensional surface generated with spatially correlated noises. Under missing data scenarios, LRMC is used prior to the application of either PCA or PredPCA.[]{data-label="fig-toy-scen2"}](Figure_Simulations_3d_scenario2){width="6.5in"}
$X_1$ $X_2$ $X_3$
----------------- ------------- -------------- -------------
PCA 0.40 (0.11) 0.41 (0.09) 0.80 (0.07)
PredPCA 0.88 (0.04) -0.07 (0.04) 0.46 (0.09)
ProPrPCA-Krige 0.85 (0.04) -0.11 (0.08) 0.50 (0.08)
ProPrPCA-Spline 0.86 (0.03) -0.12 (0.07) 0.49 (0.07)
: Means (standard deviations) of estimated PC1 loadings across 1,000 replications with three-dimensional surface with spatially correlated noise and complete training data.[]{data-label="tab-loading-scen2"}
![Prediction R$^2$’s and reconstruction errors across 1,000 replications with three-dimensional surface generated with independent noises. Under missing data scenarios, LRMC is used prior to the application of either PCA or PredPCA.[]{data-label="fig-toy-scen1"}](Figure_Simulations_3d_scenario1){width="6.5in"}
$X_1$ $X_2$ $X_3$
----------------- ------------- ------------- -------------
PCA 0.53 (0.06) 0.39 (0.04) 0.75 (0.03)
PredPCA 0.89 (0.02) 0.01 (0.02) 0.45 (0.04)
ProPrPCA-Krige 0.88 (0.02) 0.03 (0.04) 0.47 (0.04)
ProPrPCA-Spline 0.89 (0.02) 0.01 (0.03) 0.46 (0.04)
: Means (standard deviations) of estimated PC1 loadings across 1,000 replications with three-dimensional surface with independent noise and complete training data.[]{data-label="tab-loading-scen1"}
![Differences in prediction R$^2$ values between ProPrPCA-Spline and PredPCA for high-dimensional scenario 1. Each dot represents result from one simulation. Percentages indicate the proportion out of 1,000 simulations.[]{data-label="fig-hd-s123"}](Figure_DifferenceInRsq_s123){width="6.35in"}
PC1 Complete MCAR 35% MAR
----------------- ---------- ---------- ------
PCA 0.01 0.01 0.00
PredPCA 0.81 0.78 0.63
ProPrPCA-Krige 0.70 0.66 0.41
ProPrPCA-Spline 0.81 0.80 0.72
: The median prediction R$^2$’s across 1,000 simulations for high-dimensional scenario 2. Under missing data scenarios, LRMC is used prior to either TradPCA or PredPCA.[]{data-label="tab-scen312"}
PC2 Complete MCAR 35% MAR
----------------- ---------- ---------- ------
PCA 0.78 0.74 0.60
PredPCA 0.56 0.54 0.62
ProPrPCA-Krige 0.30 0.26 0.23
ProPrPCA-Spline 0.56 0.56 0.59
: The median prediction R$^2$’s across 1,000 simulations for high-dimensional scenario 2. Under missing data scenarios, LRMC is used prior to either TradPCA or PredPCA.[]{data-label="tab-scen312"}
![Differences in prediction R$^2$ values between ProPrPCA-Spline and PredPCA for high-dimensional scenario 2. Each dot represents result from one simulation. Percentages indicate the proportion out of 1,000 simulations.[]{data-label="fig-hd-s312"}](Figure_DifferenceInRsq_s312){width="6.35in"}
![Estimated loadings for feature with highly positive weights on SO$^{2-}_4$ and S, and corresponding scores, obtained from different PCA algorithms applied to 2010 CSN data: PCA and PredPCA applied to the complete set (130 sites with complete data), PredPCA and ProPrPCA-Spline applied to the full set (all 221 available sites). []{data-label="fig-real-feature1"}](Figure_Feature1_CSN2010){width="6.5in"}
![Estimated loadings for feature with highly positive weights on Na, Ni, and V, and corresponding scores, obtained from different PCA algorithms applied to 2010 CSN data: PCA and PredPCA applied to the complete set (130 sites with complete data), PredPCA and ProPrPCA-Spline applied to the full set (all 221 available sites). []{data-label="fig-real-feature2"}](Figure_Feature2_CSN2010){width="6.5in"}
![Estimated loadings for feature with highly positive weights on NO$^{-}_3$ and Zn, and corresponding scores, obtained from different PCA algorithms applied to 2010 CSN data: PCA and PredPCA applied to the complete set (130 sites with complete data), PredPCA and ProPrPCA-Spline applied to the full set (all 221 available sites). []{data-label="fig-real-feature3"}](Figure_Feature3_CSN2010){width="6.5in"}
![Prediction R$^2$’s from leave-one-site-out cross-validation on 2010 CSN data. Sites with complete PM$_{2.5}$ component data are used as testing data. Training data may include only complete sites, or all available sites.[]{data-label="fig-cv"}](Figure_Crossvalidation_CSN2010){width="6.5in"}
|
---
abstract: |
In this paper we introduce a new type of Pascal’s pyramids. The new object is called hyperbolic Pascal pyramid since the mathematical background goes back to the regular cube mosaic (cubic honeycomb) in the hyperbolic space. The definition of the hyperbolic Pascal pyramid is a natural generalization of the definition of hyperbolic Pascal triangle ([@BNSz]) and Pascal’s arithmetic pyramid. We describe the growing of hyperbolic Pascal pyramid considering the numbers and the values of the elements. Further figures illustrate the stepping from a level to the next one.\
[*Key Words: Pascal pyramid, cubic honeycomb, regular cube mosaic in hyperbolic space.*]{}\
[*MSC code: 52C22, 05B45, 11B99.*]{}
author:
- 'László Németh[^1]'
title: '**Hyperbolic Pascal pyramid** '
---
Introduction {#sec:introduction}
============
There are several approaches to generalize the Pascal’s arithmetic triangle (see, for instance [@BSz]). A new type of variations of it is based on the hyperbolic regular mosaics denoted by Schläfli’s symbol $\{p,q\}$, where $(p-2)(q-2)>4$ ([@C]). Each regular mosaic induces a so called hyperbolic Pascal triangle (see [@BNSz; @NSz1]), following and generalizing the connection between the classical Pascal’s triangle and the Euclidean regular square mosaic $\{4,4\}$. For more details see [@BNSz], but here we also collect some necessary information.
The hyperbolic Pascal triangle based on the mosaic $\{p,q\}$ can be figured as a digraph, where the vertices and the edges are the vertices and the edges of a well defined part of lattice $\{p,q\}$, respectively, and the vertices possess a value that give the number of different shortest paths from the base vertex to the given vertex. Figure \[fig:Pascal\_layer6\] illustrates the hyperbolic Pascal triangle when $\{p,q\}=\{4,5\}$. Here the base vertex has two edges, the leftmost and the rightmost vertices have three, the others have five edges. The quadrilateral shape cells surrounded by the appropriate edges correspond to the squares in the mosaic. Apart from the winger elements, certain vertices (called “Type $A$”) have $2$ ascendants and $3$ descendants, while the others (“Type $B$”) have $1$ ascendant and $4$ descendants. In the figures we denote vertices type $A$ by red circles and vertices type $B$ by cyan diamonds, further the wingers by white diamonds (according to the denotations in [@BNSz]). The vertices which are $n$-edge-long far from the base vertex are in row $n$. The general method of preparing the graph is the following: we go along the vertices of the $j^{\text{th}}$ row, according to the type of the elements (winger, $A$, $B$), we draw the appropriate number of edges downwards ($2$, $3$, $4$, respectively). Neighbour edges of two neighbour vertices of the $j^{\text{th}}$ row meet in the $(j+1)^{\text{th}}$ row, constructing a new vertex type $A$. The other descendants of row $j$ have type $B$ in row $j+1$. In the sequel, $\binomh{n}{k}$ denotes the $k^\text{th}$ element in row $n$, which is either the sum of the values of its two ascendants or the value of its unique ascendant. We note, that the hyperbolic Pascal triangle has the property of vertical symmetry.
![Hyperbolic Pascal triangle linked to $\{4,5\}$ up to row 6[]{data-label="fig:Pascal_layer6"}](Pascal_layer6_e.jpg){width="0.99\linewidth"}
The 3-dimensional analogue of the original Pascal’s triangle is the well-known Pascal’s pyramid (or more precisely Pascal’s tetrahedron) (left part in Figure \[fig:Eulidean\_pyramid\]). Its levels are triangles and the numbers along the three edges of the $n^{\text{th}}$ level are the numbers of the $n^{\text{th}}$ line of Pascal’s triangle. Each number inside in any levels is the sum of the three adjacent numbers on the level above [@B; @har].
In the following we define a Pascal pyramid in the hyperbolic space based on the hyperbolic regular cube mosaic (cubic honeycomb) with Schläfli’s symbol $\{4,3,5\}$ as a generalisation of the hyperbolic Pascal triangle and the classical Pascal’s pyramid which are based on the hyperbolic planar mosaic $\{4,5\}$ and the Euclidean regular cube mosaic $\{4,3,4\}$, respectively. (We write the hyperbolic one without an “apostrophe", similarly to the writing of the classical Pascal’s triangle and the hyperbolic Pascal triangle.)
Construction of the hyperbolic Pascal pyramid
=============================================
In the hyperbolic space there are 7 regular mosaics and one of them is the regular cube mosaic $\{4,3,5\}$ (see [@C]), which is the hyperbolic analogue of the Euclidean regular cube mosaic.
First of all for the further examination we summarise some properties of the cubic honeycomb $\{4,3,5\}$, which is not as well-known as the Euclidean one. The vertex figures of the hyperbolic cube mosaic are icosahedra with Schläfli’s symbol $\{3,5\}$. Thus the nearest vertices to a certain vertex $V$ form an icosahedron in this mosaic. It means, considering an arbitrary vertex $V$ of the mosaic, that the number of cubes around $V$ is as many as the number of the faces of the icosahedron, namely 20 and the number of the mosaic edges from $V$ (degree of $V$) is as many as the number of vertices on an icosahedron, namely 12. There are 5 cubes around a mosaic edge as there are 5 faces around a vertex on the icosahedron. In Figure \[fig:belt0\] we can see a vertex figure with one cube and the twenty cubes around the centre $V$ of the icosahedron. Vertex $X$ and $W$ are two nearest vertices of the mosaic to $V$ and around the edge $V$-$W$ there are 5 cubes. We mention that the edges of the icosahedron are not the edges of the mosaic, they are the face diagonals of the cubes.
![Vertex figure of $V$ and cubes around $V$[]{data-label="fig:belt0"}](belt0)
Now, we consider the hyperbolic (and Euclidean) cubic honeycomb. We define the part ${\cal P}$ of the mosaic which can induce the hyperbolic Pascal pyramid (and the classical Pascal’s pyramid).
Take a cube of the mosaic as a base cell of ${\cal P}$ and let $V_0$ be a vertex of it. Take the three cubes of the mosaic which have common faces with the base cell but do not contain $V_0$. (In the right part of Figure \[fig:border3d\] we can see the construction of ${\cal P}$ and by comparison the left part shows the Euclidean one.) Reflect these cubes across their own faces which are opposite the touching faces with the base cube. Reflect again the new cubes across the faces which are opposite the previous cubes, and so on limitless. This way we give the “edge”s of the border of ${\cal P}$ (blue cubes in Figure \[fig:border3d\]) and the convex parts of the mosaic-levels defined by any two “edge"s give the border of ${\cal P}$. Finally, the convex part of the bordered parts of the mosaic is the well defined ${\cal P}$. The shape of this convex part of the mosaic resembles an infinite tetrahedron.
Let ${\cal G}_{\cal P}$ be the graph, in which the vertices and edges are the vertices and edges of ${\cal P}$. We label an arbitrary vertex $V$ of ${\cal G}_{\cal P}$ by the number of different shortest paths along the edges of ${\cal P}$ from $V_0$ to $V$. We mention that all the edges of the mosaic are equivalent. Some labelled vertices can be seen in Figure \[fig:border3d\]. Let the labelled ${\cal G}_{\cal P}$ be the hyperbolic Pascal pyramid (more precisely the hyperbolic Pascal tetrahedron), denoted by ${\cal HPP}$. Considering the Euclidean mosaic $\{4,3,4\}$ instead of the hyperbolic one in the definition above the classical Pascal’s pyramid returns.
![Construction of the border of $\cal{P}$[]{data-label="fig:border3d"}](border3d.jpg)
Let level 0 be the vertex $V_0$. Level $n$ consists of the vertices of whose edge-distances from $V_0$ are $n$-edge (the distance of the shortest path along the edges of ${\cal P}$ is $n$). It is clear, that the labelled graphs indicated by the outer boundaries of ${\cal P}$ are the hyperbolic Pascal triangles based on the regular hyperbolic planar mosaic $\{4,5\}$.
The right part of Figure \[fig:hyperbolic\_pyramid\] shows the hyperbolic Pascal pyramid up to level 4, when the digraph ${\cal G}_{\cal P}$ is directed from $V_0$ according to the growing distance from $V_0$ (compare Figures \[fig:border3d\] and \[fig:hyperbolic\_pyramid\]). Moreover, Figures \[fig:layer3to4\] and \[fig:layer4to5\] show the growing from a level to the next one in case of some lower levels. The colours and shapes of different types of the vertices are different. (See the definitions later.) The numbers without colouring and shapes refer to vertices in the lower level in every figure. The graphs growing from a level to the new one contain graph-cycle with six nodes. These graph-cycles figure the convex hulls of the parallel projections of the cubes from the mosaic, where the direction of the projection is not parallel to any edges of the cubes.
![Euclidean and hyperbolic Pascal pyramid[]{data-label="fig:hyperbolic_pyramid"}](Eulidean_pyramid.jpg "fig:"){width="48.00000%"} ![Euclidean and hyperbolic Pascal pyramid[]{data-label="fig:hyperbolic_pyramid"}](hyperbolic_pyramid.jpg "fig:"){width="48.00000%"}
\[fig:Eulidean\_pyramid\]
![Connection between levels two, three and four in []{data-label="fig:layer3to4"}](layer2to3.jpg "fig:") ![Connection between levels two, three and four in []{data-label="fig:layer3to4"}](layer3to4.jpg "fig:")
![Connection between levels four and five in []{data-label="fig:layer4to5"}](layer4to5.jpg){width="0.9\linewidth"}
In the following we describe the method of the growing of the hyperbolic Pascal pyramid and we give the sum of the paths connecting vertex $V_0$ and level $n$.
Growing of the hyperbolic Pascal pyramid
========================================
In the classical Pascal’s pyramid the number of the elements on level $n$ is $(n+1)(n+2)/2$ and its growing from level $n$ to $n+1$ is $n+2$, on the contrary in the hyperbolic Pascal pyramid it is more complex.
As the faces of are the hyperbolic Pascal triangles, then here are three types of vertices $A$, $B$ and $1$ corresponding to the Introduction and [@BNSz]. From all $A$ and $B$ start only one edge to the inside of the pyramid, because five cubes close around an edge of the mosaic (see Figure \[fig:border3d\]). The types of inside vertices of these edges differ from the types $A$ and $B$, denote them by type $C$ and type $D$, respectively. The left part of Figure \[fig:from\_border\] presents a cube, in which the upper face is on the border of and a vertex $A$ on level $i$ generates a vertex $C$ inside of with 3 incoming edges. The right part shows that all vertices $B$ imply a vertex $D$ with two incoming edges.
![Groing from border to inside[]{data-label="fig:from_border"}](from_border)
The growing methods of them are illustrated in Figure \[fig:gowing3d\_1\] (compare it with the growing method in [@BNSz]).
![Growing method in case of the faces[]{data-label="fig:gowing3d_1"}](gowing3d_1)
As a cube has three edges in all vertices, then during the growing (step from level $i-1$ to level $i$) an inner arbitrary vertex $V$ on level $i$ can be reached from level $i-1$ with three, two or just one edges. This fact allows us a classification of the inner vertices. Let the type of a vertex on level $i$ is $C$, $D$ or $E$, respectively, if it has three, two or one joining edges to level $i-1$ (as before). Figure \[fig:gowing3d\_icosa\] shows vertex figures of the inner vertices of . Vertices $W_{i-1}$ (small green circles) are on level $i-1$, we don’t know their types (or not important to know) and the centres are on level $i$. The other vertices of the icosahedron are on level $i+1$ and the classification of them gives their types. An edge of the icosahedron and its centre $V$ determine a square (a face of a cube) from the mosaic. (Recall, that an edge of the icosahedron is a diagonal of a face of a cube from the mosaic.) Since from a vertex of a square we can go to the opposite vertex two ways, then a vertex $X$ of the icosahedron, where $X$ and a $W_{i-1}$ are connected by an edge, can be reached with two paths from level $i-1$. (For example in Figure \[fig:belt0\], between vertex $W$ and $X$ there are the paths $W\!-\!V\!-\!X$ and $W\!-\!U\!-\!X$.) So, the type of the third vertex of the faces on the icosahedron whose other two vertices are $W_{i-1}$ are $C$. The types of the vertices which connect to only one $W_{i-1}$ with icosahedron-edge are $D$, the others are type $C$. See Figure \[fig:gowing3d\_icosa\]. In case of the vertex figure of $C$ or $D$ a vertex $W_{i-1}$ can be vertex $A$ or $B$, respectively. In the figures we denote vertices type $C$ by blue hexagons, vertices type $D$ by green pentagons and vertices type $E$ by yellow squares. The blue thick directed edges are mosaic-edges between levels $i-1$ and $i$, while the red thin ones are between levels $i$ and $i+1$. We mention that in case of Pascal’s pyramid there are only type $C$ inner vertices.
![Growing method in case of the inner vertices with icosahedra[]{data-label="fig:gowing3d_icosa"}](gowing3d_icosa){width="0.99\linewidth"}
In Figure \[fig:gowing3d\_2\] the growing method is presented in case of the inner vertices. They come from the centres and vertices from the icosahedra in Figure \[fig:gowing3d\_icosa\].
![Growing method in case of the inner vertices[]{data-label="fig:gowing3d_2"}](gowing3d_2)
Three new $C$, $D$ and $E$ connect for all vertices type $C$, but from the explanation above all new vertices $C$ and $D$ connect altogether three or two other vertices on level $i$, respectively. So, for the correct calculation we correspond just one third or half of them to the examined vertices $C$, respectively. All the new vertices $C$ connect to just one vertex on level $i$. By the help of the similar consideration in case of vertices $D$ and $E$, we can calculate the number of vertices on level $i+1$, recursively, without multiplicity.
Finally, we denote the sums of vertices types $A$, $B$, $C$, $D$ and $E$ on level $i$ by $a_i$, $b_i$, $c_i$, $d_i$ and $e_i$, respectively.
Summarising the details $(i\geq4)$ and calculating the numbers of vertices in some lower levels $(i<4)$ from Table \[table:typeof\_vertices\], we prove the Theorem \[th:growing\_type\].
\[th:growing\_type\] The growing of the numbers of the different types of the vertices are described by the system of linear inhomogeneous recurrence sequences $(n\geq1)$ $$\label{eq:seq01}
\begin{array}{ccl}
a_{n+1}&=& a_n+b_n+3,\\
b_{n+1}&=& a_n+2b_n,\\
c_{n+1}&=& \frac13 a_n+c_n+\frac23 d_n,\\
d_{n+1}&=& \frac12 b_n+\frac32c_n+2d_n+\frac52e_n,\\
e_{n+1}&=& 3 c_n+4d_n+6 e_n,
\end{array}$$ with zero initial values.
Moreover, let $s_n$ be the number of all the vertices on level $n$, so that $s_0=1$ and $$\label{eq:sn}
s_{n}= a_n+b_n+c_n+d_n+ e_n+3 \qquad (n\geq1).$$
Table \[table:typeof\_vertices\] shows the numbers of the vertices on levels up to 10.
$n$ 0 1 2 3 4 5 6 7 8 9 10
------- --- --- --- ---- ---- ----- ----- ------ ------- -------- ---------
$a_n$ 0 0 3 6 12 27 66 168 435 1134 2964
$b_n$ 0 0 0 3 12 36 99 264 696 1827 4788
$c_n$ 0 0 0 1 3 9 34 174 1128 8251 63315
$d_n$ 0 0 0 0 3 24 177 1347 10467 82029 644808
$e_n$ 0 0 0 0 3 39 357 2952 23622 186984 1474773
$s_n$ 1 3 6 13 36 138 736 4908 36351 280228 2190651
: *Number of types of vertices $(n\leq10)$*\[table:typeof\_vertices\]
\[theorem:numvertex4q\] The sequences $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{d_n\}$, $\{e_n\}$ and $\{s_n\}$ can be described by the same fifth order linear homogeneous recurrence sequence $$\label{recurcde}
x_n=12x_{n-1}-37x_{n-2}+37x_{n-3}-12x_{n-4}+x_{n-5} \qquad (n\ge6),$$ the initial values are in Table \[table:typeof\_vertices\]. The sequences $\{a_n\}$, $\{b_n\}$ can be also described by $$x_n=4x_{n-1}-4x_{n-2}+x_{n-3} \qquad (n\ge4).\label{recurab}$$ Moreover, the explicit formulae $$\begin{aligned}
a_n&=&\left(-\frac{9}{2}+\frac{21}{10}\sqrt{5}\right)\alpha_1^n+\left(-\frac{9}{2}-\frac{21}{10}\sqrt{5}\right)\alpha_2^n+3, \\
b_n&=&\left(3-\frac{6}{5}\sqrt{5}\right)\alpha_1^n+\left(3+\frac{6}{5}\sqrt{5}\right)\alpha_2^n-3,\\
c_n&=&\left(-\frac{33}{10}+\frac{3}{2}\sqrt{5}\right)\alpha_1^n+
\left(-\frac{33}{10}-\frac{3}{2}\sqrt{5}\right)\alpha_2^n+ \left(\frac{122}{15}-\frac{21}{10}\sqrt{15}\right)\alpha_3^n\\
& & \qquad
+\left(\frac{122}{15}+\frac{21}{10}\sqrt{15}\right)\alpha_4^n+\frac13, \\
d_n&=&\left(\frac{27}{5}-\frac{12}{5}\sqrt{5}\right)\alpha_1^n+
\left(\frac{27}{5}+\frac{12}{5}\sqrt{5}\right)\alpha_2^n+ \left(-\frac{213}{20}+\frac{11}{4}\sqrt{15}\right)\alpha_3^n\\
& & \qquad+
\left(-\frac{213}{20}-\frac{11}{4}\sqrt{15}\right)\alpha_4^n-\frac32, \\
e_n&=&\left(-\frac{21}{10}+\frac{9}{10}\sqrt{5}\right)\alpha_1^n+
\left(-\frac{21}{10}-\frac{9}{10}\sqrt{5}\right)\alpha_2^n+ \left(\frac{31}{10}-\frac{4}{5}\sqrt{15}\right)\alpha_3^n\\
& & \qquad+
\left(\frac{31}{10}+\frac{4}{5}\sqrt{15}\right)\alpha_4^n+1, \\
s_n&=&\left(-\frac{3}{2}+\frac{9}{10}\sqrt{5}\right)\alpha_1^n+
\left(-\frac{3}{2}-\frac{9}{10}\sqrt{5}\right)\alpha_2^n+ \left(\frac{7}{12}-\frac{3}{20}\sqrt{15}\right)\alpha_3^n\\
& & \qquad+
\left(\frac{7}{12}+\frac{3}{20}\sqrt{15}\right)\alpha_4^n+\frac{17}6 \\\end{aligned}$$ are valid, where $\alpha_1=(3+\sqrt{5})/2$, $\alpha_2=(3-\sqrt{5})/2$, $\alpha_3=4+\sqrt{15}$ and $\alpha_4=4-\sqrt{15}$.
For the proof of Theorem \[theorem:numvertex4q\] we apply Theorem \[th:recur\].
\[th:recur\] Let the real linear homogeneous recurrence sequences $a^{(j)}$ embedded in each other be given the following way ($k\geq 2,
\enskip i\geq 0,\enskip j=1,2,\dots ,k)$ $$\begin{aligned}
a^{(1)}_{i+1}&=&m_{1,1}a^{(1)}_i+m_{1,2}a^{(2)}_i+ \dots +m_{1,k}a^{(k)}_i \nonumber \\
a^{(2)}_{i+1}&=&m_{2,1}a^{(1)}_i+m_{2,2}a^{(2)}_i+ \dots +m_{2,k}a^{(k)}_i \nonumber \\
&\vdots & \\
a^{(k)}_{i+1}&=&m_{k,1}a^{(1)}_i+m_{k,2}a^{(2)}_i+ \dots +m_{k,k}a^{(k)}_i, \nonumber \\
\noalign{\noindent with initial values $a^{(j)}_0 \in \mathbb{R}$ and } \nonumber \\
r_{i+1}&=&{\alpha}_1a^{(1)}_i+{\alpha}_2a^{(2)}_i+ \dots +{\alpha}_{k} a^{(k)}_i, \qquad r_0 \in \mathbb{R}.\end{aligned}$$ In a shorter form $$\begin{aligned}
{\mathbf{a}}_{i+1}&=&{\mathbf{M}}{\mathbf{a}}_i,\label{eq:am}\\
r_{i+1}&=&\veca^T{\mathbf{a}}_i, \label{eq:ra}\end{aligned}$$ where ${\mathbf{M}}=\{m_{i,j}\}_{k\times k}$, ${\mathbf{a}}_j=[a^{(1)}_j\enskip a^{(2)}_j \enskip \dots \enskip a^{(k)}_j]^T$, $ \veca=[\alpha _1 \enskip \alpha _2 \enskip \dots \enskip \alpha _k]^T$ and $rank({\mathbf{M}})=k$.\
If ${\beta}_i \in \mathbb{R}$ and $$\begin{aligned}
r_i={\beta}_1r_{i-1}+{\beta}_2r_{i-2}+ \dots +{\beta}_kr_{i-k}, \qquad (i\geq k)\label{eq:rr}\end{aligned}$$ then ${\beta}_i$ are the coefficients of characteristic polynomial of matrix ${\mathbf{M}}$.
Moreover, if the matrix ${\mathbf{M}}$ has $\ell$ distinct eigenvalues with one algebraic multiplicity and ${\gamma}_j$ $(1\leq j\leq \ell)$ are the coefficients of minimal polynomial of ${\mathbf{M}}$, then $$\begin{aligned}
r_i={\gamma}_1r_{i-1}+{\gamma}_2r_{i-2}+ \dots +{\gamma}_{\ell}r_{i-\ell} \qquad (i\geq k \geq \ell).\label{eq:rrg}\end{aligned}$$
From and we obtain, that $$\begin{aligned}
r_i=\veca ^T{\mathbf{a}}_{i-1}=\veca ^T{\mathbf{M}}{\mathbf{a}}_{i-2}&=&\veca ^T{\mathbf{M}}^2{\mathbf{a}}_{i-3}=\dots =\veca ^T{\mathbf{M}}^{k-1}{\mathbf{a}}_{i-k},\\
r_{i-j}&=&\veca ^T{\mathbf{M}}^{k-(j+1)}{\mathbf{a}}_{i-k} \qquad (j=0,1,\dots ,k-1).\\[2mm]
\noalign{\noindent It follows from $rank({\mathbf{M}})=k$ (has inverse), that}\\[-3.5mm]
r_{i-k}&=&\veca ^T{\mathbf{M}}^{-1}{\mathbf{a}}_{i-k}.
\end{aligned}$$ We substitute the results into (\[eq:rr\]), $$\begin{aligned}
\veca ^T{\mathbf{M}}^{k-1}{\mathbf{a}}_{i-k}&=&{\beta}_1 \veca ^T{\mathbf{M}}^{k-2}{\mathbf{a}}_{i-k}+ \dots + {\beta}_j \veca ^T{\mathbf{M}}^{k-(j+1)}{\mathbf{a}}_{i-k}+\dots
+{\beta}_k \alpha ^T{\mathbf{M}}^{-1}{\mathbf{a}}_{i-k} \\
&=&\sum _{j=1}^k \Big( {\beta}_j \veca ^T{\mathbf{M}}^{k-(j+1)}{\mathbf{a}}_{i-k}\Big)=\veca ^T \Bigg( \sum _{j=1}^k \Big(
{\beta}_j {\mathbf{M}}^{k-(j+1)}\Big) \Bigg){\mathbf{a}}_{i-k} .\end{aligned}$$ We gain, that $$\begin{aligned}
\veca ^T \Bigg({\mathbf{M}}^{k-1}- \sum _{j=1}^k \Big(
{\beta}_j {\mathbf{M}}^{k-(j+1)}\Big) \Bigg) {\mathbf{a}}_{i-k}= 0 .\end{aligned}$$ As $\veca ^T$ and ${\mathbf{a}}_{i-k}$ can be any elements of the vector space $\mathbb{R}^k$, $$\begin{aligned}
{\mathbf{M}}^{k-1}- \sum _{j=1}^k \Big(
{\beta}_j {\mathbf{M}}^{k-(j+1)}\Big) = {\mathbf{0}},\end{aligned}$$ thus $$\begin{aligned}
\label{eq:charM}
{\mathbf{M}}^{k}&=&\sum _{j=1}^k {\beta}_j {\mathbf{M}}^{k-j}.\end{aligned}$$ Using the well-known *Cayley-Hamilton Theorem*, from the equation $$x^k={\beta}_1x^{k-1}+{\beta}_2x^{k-2}+ \dots +{\beta}_k$$ is the characteristic equation of matrix ${\mathbf{M}}$. If the matrix ${\mathbf{M}}$ has $\ell$ distinct eigenvalues with one algebraic multiplicity and ${\gamma}_j$ are the coefficients of the minimal polynomial of ${\mathbf{M}}$, then the method of the proof can be followed step by step for $\ell$ elements of $r_i$ and for ${\gamma}_j$ coefficients too, thus also holds.
Let $v_n=3$ $(n\geq1)$ be a constant sequence and $v_0=1$. The value $v_n$ gives the number of vertices type “1" on level $n$. Substitute $3=v_n$ into the first equation of and complete the equations system with $v_{n+1}=v_n$. Than we have the system of linear homogeneous recurrence sequences $(n\geq1)$ $$\label{eq:seq01v}
\begin{array}{ccl}
a_{n+1}&=& a_n+b_n+ v_n,\\
b_{n+1}&=& a_n+2b_n,\\
c_{n+1}&=& \frac13 a_n+c_n+\frac23 d_n,\\
d_{n+1}&=& \frac12 b_n+\frac32c_n+2d_n+\frac52e_n,\\
e_{n+1}&=& 3 c_n+4d_n+6 e_n,\\
v_{n+1}&=& v_n
\end{array}$$ and $$\label{eq:snv}
s_{n}= a_n+b_n+c_n+d_n+ e_n+v_n \qquad (n\geq0).$$
Using the results of Theorem \[th:recur\] when $${\mathbf{M}}=\begin{pmatrix}
1 & 1 & 0 & 0 & 0 & 1 \\
1 & 2 & 0 & 0 & 0 & 0 \\
\frac13 & 0 & 1 & \frac23 & 0 & 0 \\
0 & \frac12 & \frac32 & 2 & \frac52 & 0 \\
0 & 0 & 3 & 4 & 6 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix},$$ ${\mathbf{a}}_j=[a_j\enskip b_j\enskip c_j\enskip e_j\enskip v_j\enskip]^T$, and $rank({\mathbf{M}})=6$ we gain that the solutions of system of linear recurrence equations are $\beta_1=12-t$, $\beta_2=-37+12t$, $\beta_3=37-37t$, $\beta_4=-12+37t$, $\beta_5=1-12t$, $\beta_6=t$, where $t\in \mathbb{R}$. As $r_n$ was an arbitrary equation, $r_n$ can be $s_n$, $a_n$, … $e_n$ with $\veca=(1,1,1,1,1,1)$, $\veca=(1,0,0,0,0,0)$, …, $\veca=(0,0,0,0,1,0)$, respectively. Moreover, let $t=0$, then we obtain the (degenerate) recurrence sequence .
As $a_{n+1}$, $b_{n+1}$ and $v_{n+1}$ are independent from $c_n$, $d_n$ and $e_n$, they form a system of homogeneous recurrence equations again with matrix ${\mathbf{M}}_{ab}=\left( \begin{smallmatrix}
1 & 1 & 1 \\
1 & 2 & 0 \\
0 & 0 & 1 \\
\end{smallmatrix}\right)$. Using the results of Theorem \[th:recur\] again we gain $\beta_1=4$, $\beta_2=-4$, $\beta_3=1$, so the equation holds.
The characteristic equation of is $$\label{eq:mini}
x^5={12}x^{4}-37x^{3}+37x^2-12x +1$$ and its solutions are $\alpha_1=(3+\sqrt{5})/2$, $\alpha_2=(3-\sqrt{5})/2$, $\alpha_3=4+\sqrt{15}$ and $\alpha_4=4-\sqrt{15}$ and $\alpha_5=1$. We mention that equation is the minimal polynomial of the matrix ${\mathbf{M}}$. (The roots of the characteristic equation of , $x^3=4x^2-4x +1$ are also $\alpha_1$, $\alpha_2$ and $\alpha_5$.) A suitable linear combination of their $n^{\text{th}}$ power provide the explicit formulae ([@sho]).
In Pascal’s pyramid the equations system also holds with suitable initial values. In this case, there is no type vertices $B$, $D$ and $E$, so $b_i=d_i=e_i=0$ for any $i$. Thus the hyperbolic Pascal pyramid is not only the geometric but also the algebraic generalization of Pascal’s pyramid.
The ratios of numbers of vertices from level to level tend to the biggest eigenvalue of the matrix ${\mathbf{M}}$. So, the growing ratio of is $\alpha_3=4+\sqrt{15}\approx 7.873$, on the contrary it is $1$ in case of the Euclidean case.
Sum of the values on levels in the hyperbolic Pascal pyramid
============================================================
The sum of the values of the elements on level $n$ in the classical Pascal’s pyramid is $3^n$ ([@B]). In this section we determine it in case of the hyperbolic Pascal pyramid.
Denote respectively $\hat{a}_{n}$, $\hat{b}_{n}$, $\hat{c}_{n}$, $\hat{d}_{n}$ and $\hat{e}_{n}$ the sums of the values of vertices type $A$, $B$, $C$, $D$ and $E$ on level $n$, and let $\hat{s}_{n}$ be the sum of all the values. From Figures \[fig:gowing3d\_1\] and \[fig:gowing3d\_2\] the results of Theorem \[th:recursum\] can be read directly. For example for all vertices type $A$, $B$ and $1$ on level $i$ generate two vertices type $A$ on level $i+1$ and it follows the first equation of . Table \[table:sumof\_vertices\] shows the sum of the values of the vertices on levels up to 10.
\[th:recursum\] If $n\geq1$, then $$\label{eq:seq02}
\begin{array}{ccl}
\hat{a}_{n+1}&=& 2\hat{a}_n+2\hat{b}_n+6,\\
\hat{b}_{n+1}&=& \hat{a}_n+2\hat{b}_n,\\
\hat{c}_{n+1}&=& \hat{a}_n+3\hat{c}_n+2 d_n,\\
\hat{d}_{n+1}&=& \hat{b}_n+3c_n+4\hat{d}_n+5\hat{e}_n,\\
\hat{e}_{n+1}&=& 3 \hat{c}_n+4\hat{d}_n+6 \hat{e}_n
\end{array}$$ with zero initial values.
Table \[table:sumof\_vertices\] shows the sums of values on levels up to 10.
$n$ 0 1 2 3 4 5 6 7 8 9 10
------- --- --- --- ---- ----- ------ ------ ------- -------- --------- -----------
$a_n$ 0 0 6 18 54 174 582 1974 6726 22950 78342
$b_n$ 0 0 0 6 30 114 402 1386 4746 16218 55386
$c_n$ 0 0 0 6 36 210 1452 12138 114684 1147002 11729148
$d_n$ 0 0 0 0 24 324 3600 38148 398112 4132596 42818208
$e_n$ 0 0 0 0 18 312 3798 41544 438270 4566120 47368110
$s_n$ 1 3 9 33 165 1137 9837 95193 962541 9884889 102049197
: *Sum of values of vertices $(n\leq10)$*\[table:sumof\_vertices\]
Let $\hat{s}_n$ be the sum of the values of all the vertices on level $n$, then $\hat{s}_0=1$ and $$\hat{s}_n = \hat{a}_n+\hat{b}_n+\hat{c}_n+\hat{d}_n+\hat{e}_n+3 \qquad (n\geq1).$$
The sequences $\{\hat{a}_n\}$, $\{\hat{b}_n\}$, $\{\hat{c}_n\}$, $\{\hat{d}_n\}$, $\{\hat{e}_n\}$ and $\{\hat{s}_n\}$ can be described by the same sixth order linear homogeneous recurrence sequence $$\label{recurcdehat}
\hat{x}_n = 18 \hat{x}_{n-1}-99\hat{x}_{n-2}+226\hat{x}_{n-3}-224\hat{x}_{n-4}+92\hat{x}_{n-5}-12\hat{x}_{n-6} \qquad (n\ge7),$$ the initial values are in Table \[table:sumof\_vertices\]. The sequences $\{\hat{a}_n\}$, $\{\hat{b}_n\}$ can be also described by $$\label{recurabhat}
\hat{x}_n = 5 \hat{x}_{n-1}-6\hat{x}_{n-2}+2\hat{x}_{n-3} \qquad (n\ge4).$$ The explicit formulae $$\begin{aligned}
\hat{a}_n &=&\left(\frac92\,\sqrt {2}-6 \right) \left( 2+\sqrt {2} \right) ^{n}+ \left(\frac92\,\sqrt {2}-6 \right) \left( 2-\sqrt {2} \right) ^{n}+6,\\
\hat{b}_n&=& \left(\frac92 -3\,\sqrt {2} \right) \left( 2+\sqrt {2} \right)^{n} + \left( \frac92+3\,\sqrt {2} \right) \left( 2-\sqrt {2} \right) ^{n} -6 \end{aligned}$$ and $$\begin{gathered}
\hat{s}_n=3+ \frac32\left(\sqrt {2}-1 \right) \left( 2+\sqrt {2} \right) ^{n}-
\frac32\left(\sqrt {2}+1 \right) \left( 2-\sqrt {2} \right)^{n} +\delta_4\alpha_4+\delta_5\alpha_5+\delta_6\alpha_6,\end{gathered}$$ where if $\varphi=\arctan(9\sqrt{101}/128)/3$, then $\alpha_4=(-\sqrt{85}\cos(\varphi)-\sqrt{3}\sqrt{85}\sin(\varphi)+13)/3 \approx 0.240683$, $\alpha_5=(-\sqrt{85}\cos(\varphi)+\sqrt{3}\sqrt{85}\sin(\varphi)+13)/3 \approx 2.408387$ and $\alpha_6=(2\sqrt{85}\cos(\varphi)+13)/3 \approx 10.350930$ are the roots of the equation ${x}^{3}-13{x}^{2}+28x-6=0$, moreover $\delta_4\approx 1.137480 $, $\delta_5\approx -0.144699$ and $\delta_6\approx 0.007219$.
Follow the proof of Theorem \[theorem:numvertex4q\] step by step. Using the results of Theorem \[th:recursum\] when $${\mathbf{M}}=\begin{pmatrix}
2 & 2 & 0 & 0 & 0 & 2 \\
1 & 2 & 0 & 0 & 0 & 0 \\
1 & 0 & 3 & 2 & 0 & 0 \\
0 & 1 & 3 & 4 & 5 & 0 \\
0 & 0 & 3 & 4 & 6 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix},\qquad
{\mathbf{M}}_{ab}=\left( \begin{matrix}
2 & 2 & 2 \\
1 & 2 & 0 \\
0 & 0 & 1 \\
\end{matrix}\right),$$ we gain that $\beta_1 = 18$, $\beta_2 = -99$, $\beta_3 = 226$, $\beta_4 = -224$, $\beta_5 = 92$, $\beta_6 = -12$ and the characteristic polynomial of ${\mathbf{M}}$ is $\left( x-1 \right) \left( {x}^{2}-4\,x+2 \right) \left( {x}^{3}-13\,{x}^{2}+28\,x-6 \right)$ and its roots are $\alpha_i$ $(i=1, \dots , 6)$. As the exact values of coefficients $\delta_j$ $(j=4,\, 5,\, 6)$ are very complicated (for sequences $\hat{c}_n$, $\hat{d}_n$, $\hat{e}_n$ also), we give their numerical values in case sequence $\hat{s}_n$ by the help of MAPLE software. But from the characteristics polynomial of ${\mathbf{M}}_{ab}$ the sequences $\hat{a}_n$ and $\hat{b}_n$ are given in exact explicit forms.
The growing ratio of values of is $\approx\!10.351$, while it is $3$ in case of the Euclidean case.
[2015]{}
Ahmia, M. – Szalay, L., On the weighted sums associated to rays in generalized Pascal triangle, submitted.
Belbachir, H., Németh, L., Szalay, L., Hyperbolic Pascal triangles, Applied Mathematics and Computation, to appear (arXiv:1503.02569).
Belbachir, H. – Szalay, L., On the arithmetic triangles, Siauliai Math. Sem., [**9**]{} (17) (2014), 15-26.
Bondarenko, B. A., Generalized Pascal triangles and pyramids, their fractals, graphs, and applications. Translated from the Russian by Bollinger, R. C. (English) Santa Clara, CA: The Fibonacci Association, vii, 253 p. (1993). www.fq.math.ca/pascal.html
Coxeter, H. S. M., Regular honeycombs in hyperbolic space, Proc. Int. Congress Math., Amsterdam, Vol. III. (1954), 155-169.
Harris, J, M., Hirst, J. L., Mossinghoff, M. J, Combinatorics and Graph Theory, Springer, (2008).
Németh, L., Szalay, L., Alternating sums in hyperbolic Pascal triangles, submitted.
Shorey, T. N., – Tijdeman, R., Exponential diophantine equation, Cambridge University Press, 1986, p. 33.
[^1]: University of West Hungary, Institute of Mathematics, Hungary. *[email protected]*
|
---
abstract: 'Simultaneous observation of characteristic 3-dimensional (3D) signatures in the electron velocity distribution function (VDF) and intense quasi-monochromatic waves by the Magnetospheric Multiscale (MMS) spacecraft in the terrestrial magnetosheath are investigated. The intense wave packets are characterised and modeled analytically as quasi-parallel circularly-polarized whistler waves and applied to a test-particle simulation in view of gaining insight into the signature of the wave-particle resonances in velocity space. Both the Landau and the cyclotron resonances were evidenced in the test-particle simulations. The location and general shape of the test-particle signatures do account for the observations, but the finer details, such as the symmetry of the observed signatures are not matched, indicating either the limits of the test-particle approach, or a more fundamental physical mechanism not yet grasped. Finally, it is shown that the energisation of the electrons in this precise resonance case cannot be diagnosed using the moments of the distribution function, as done with the classical ${\bf E}.{\bf J}$ “dissipation” estimate.'
bibliography:
- 'libRes.bib'
title: 'Resonant whistler-electron interactions: MMS observations vs. test-particle simulation'
---
Characteristic double-branch signatures in the electron Velocity Distribution Function (VDF) are observed simultaneously with a whistler wave.
The wave, applied to test-particles, produces signatures in the VDF through Landau and cyclotron resonances.
This resonant wave-particle interaction cannot be diagnosed in the Magnetospheric MultiScale (MMS) observations through the dissipative $\mathbf{E}\cdot\mathbf{J}$ term.
Introduction
============
Resonant wave-particle interactions are one of the few mechanisms in collisionless plasmas that enable a net transfer of energy from oscillating electromagnetic fields to moving charged particles. They play a fundamental role in various regions of the near-Earth plasma environment, such as the bowshock, the radiation belts, the polar cusp or the magneto-tail [@Mazelle00; @Grison05; @thorne2010grl; @fujimoto2011ssr; @krasnoselskikh2013ssr]. In (fully developed) plasma turbulence, wave-particle interactions are also thought to play a leading role in dissipating energy as the turbulent cascade proceeds from large (fluid) to small (kinetic) scales [@bruno2013lrsp; @Sahraoui20]. In the solar wind (and to some extent the magnetosheath), the most debated dissipation processes are the Landau damping [@Landau46; @Howes08; @Schekochihin09; @Gary09; @Sahraoui10; @Podesta10; @Sulem15; @Kobayashi17], cyclotron damping [@Leamon98; @Kasper08; @Cranmer14; @he2015apj] and stochastic heating [@chandran10], which all would involve different spatial or temporal scales. Often magnetic reconnection is also evoked as a potential dissipation process in localized current sheets that self-consistently form in turbulence plasmas [@Matthaeus84; @Retino07; @Sundkvist07; @Chasapis15]. However, even within such localized coherent structures, Landau damping is shown to be very effective in numerical simulations of collisionless magnetic reconnection [@tenbarge13; @Loureiro13; @numata15].\
Despite their role in energy dissipation, a [*direct*]{} diagnosis of wave-particle resonances in numerical simulations and in-situ data remains elusive. The difficulty to approach these processes stems from the need to measure [*simultaneously*]{} the 7D VDF (3 spatial dimensions, 3 velocity dimensions, and time) with high temporal and velocity space resolutions to access the kinetic scales, and the 4D structure of the electric and magnetic fields. While the latter could have been achieved at the magnetohydrodynamic (MHD) and ion scales using the Cluster data and appropriate data analysis techniques [@Sahraoui06; @Narita10; @Sahraoui10], the former became possible only in recent years thanks to the MMS mission [@burch2016ssr]. MMS indeed provides us with the highest ever achieved resolution of the particle VDFs, both in time and velocity space [@pollock2016ssr]. Furthermore, thanks to its small inter-spacecraft separations ($\sim 10$ km) MMS allows us to probe in 3D kinetic spatial scales of the fluctuations fields. On the other hand, the increasing computer capabilities allows achieving Vlasov simulations with high-enough phase space resolution to unravel the complex nature of the kinetic dissipation in turbulent collisionless plasmas [@Cerri18]. The present article is part of the ongoing efforts in this direction.\
A few observational approaches on signatures of such mechanisms were previously proposed, and here we focus on studies manipulating 3-dimensional VDFs. reported the unambiguous observation of wave-ion resonances leading to particle acceleration, associated with an ion cyclotron wave, in the Earth magnetosphere, displaying clear agyrotropic signatures (phase bunching) and their time evolution. In the work of , at the magnetopause, the energy exchange between electrons and kinetic Alfvén waves was studied in terms of the dissipative $\mathbf{E}.\mathbf{J}$ term, and trapped electrons were found in the wave minima. In the radiation belts, proposed empirical indications for Landau resonance signatures in the electron VDFs, as local minima in their velocity derivatives close to the parallel resonant velocity, in the presence of Chorus waves. and show indications in the solar wind of resonances in the proton VDFs, composed of a diffused, anisotropic core and secondary beam, which were linked to kinetic waves. In the Earth magnetosheath and using a field-particle correlation technique, presented structures in the fluctuating electron VDF close to the electron therrmal speed, which the authors linked to electron Landau damping.\
The method followed in the present study is similar to the approach of and , in that we first identify a neat electromagnetic wave, as intense and monochromatic as possible, study its potential effect on particle Velocity Distribution Functions (VDF), and then compare the expected signatures with the VDF observed outside and inside the region where the wave is observed. This approach enables an unequivocal, 3-dimensional comparison of resonant signatures in the VDF between observations and the simulation.
The wave studied in this work is a high-frequency quasi-parallel whistler mode, ubiquitous in both magnetospheric and the solar wind plasmas [@tao2012grl; @lacombe2014apj; @stansby2016apj]. Electrons are therefore the species of interest, and the frequencies of both the wave and the electron motion (gyration) are much higher than the fastest particle instruments operating in space, though in the reach of wave sensors. Anisotropies in the electron distribution functions are fundamental for the generation of whistler waves, as shown in various contexts such as the solar wind [@tong2019apj], magnetic reconnection regions [@Huang16; @yoo2018grl; @yoo2019pp], or mirror mode magnetic holes or other coherent structures in the magnetosheath [@Huang17; @ahmadi2018jgr]. , and have explored the theoretical link between whistler waves and solar wind electrons, in either the formation or the scattering of strahl and halo electrons. In their numerical approach, show how oblique whistler mode chorus in the magnetosphere can lead to electron acceleration up to a few MeV, via Landau, cyclotron, and higher order resonances.
To go beyond the 1-dimensional description of wave-particle resonances, in which a resonance is reduced to its associated (scalar) parallel speed, we explore the possibilities offered by a test-particle approach for a more comprehensive description of the mechanism and a direct comparison with the observations. This approach also presents the great advantage of isolating the effect of the wave on the particles, with no feedback allowed. A succinct view on particle energisation is also proposed, in order to appreciate whether the energy gained by the resonant particles can be quantified in observations.
Particle data analysis {#sec:partAnalysis}
======================
The particle data used in this study were recorded by the Fast Plasma Investigation (FPI) of the Magnetospheric Multiscale (MMS) mission [@burch2016ssr; @pollock2016ssr]. We work in a reference frame in which the average plasma flow velocity ${\mathbf{u}}(t) = \frac{m_{\mathrm{e}}{\mathbf{u}}_{\mathrm{e}}(t) + m_{\mathrm{i}}{\mathbf{u}}_{\mathrm{i}}(t)}{m_{\mathrm{e}} + m_{\mathrm{i}}}$ is zero, and the [*local*]{} magnetic field ${\bf B}_0$ – averaged over each 30 ms FPI measurement – is aligned with the $z$-direction. We define a spherical grid-of-interest in this reference frame, of arbitrary extent and resolution. For each FPI measurement of the VDF, this grid-of-interest is rotated and shifted to the instrument reference frame (cf. \[app:binningInterpolation\]) using the measured background magnetic field ${\bf B}_0$ and flow velocity, as well as the probe motion. The VDF is then interpolated at each node of the transformed grid, using a tricubic interpolation scheme, documented and tested in \[app:interpolation\]. The use of a spherical grid allows us to represent an averaged VDF in the $(v_\perp,v_\parallel)$-plane without the need of a binning process, which is a source of systematic artifacts when working with multidimensional data. These aspects are illustrated in \[app:binningInterpolation\]. In Figure \[fig:VDFNorms\], the left-hand polar plot shows the result for 10 time-averaged VDFs, with the parallel velocity given along the vertical axis. A regular spherical grid-of-interest of 200x200x200 nodes was used with a maximum extent of $1.5 \cdot 10^7$ m/s. To further ease the reading of the plots, a filled-contour representation is used.
Our goal is to study how the shape of the VDF may be affected by the presence of a wave. We wish to go further than the reduced description of the VDF given by its first order moments, namely its number density, its number flux density ($\sim$ bulk velocity) and its momentum flux density tensor ($\sim$ temperature in the thermal equilibrium case). For this purpose, we need a process that enhances potential patterns which might be “hidden” by the order zero distribution, or *background* distribution (not necessarily Gaussian/Maxwellian). In the example used in Figure \[fig:VDFNorms\], the VDF stretches over more than 6 orders of magnitude in the covered velocity space. The most obvious, order-0 shape found in the raw, original VDF is a centered, somewhat isotropic peak. A closer look may reveal obvious departure from the isotropy, with noticeably straight isocontours for parallel velocities around 0 m/s. We want a process which highlights these characteristics. In a numerical context, such a background, or equilibrium distribution $f_0$ is usually subtracted from $f(t)$, with the similar goal of enhancing higher order features (in such a case, the departure from $f_0$). It is often defined as the initial VDF $f(t=0)$, and sometimes as the time averaged distribution $<f(t)>$. But the resulting *fluctuating* distribution $\delta f(t) = f(t) - f_0$ might not be a valid concept when dealing with observations, as the *background*, order-0 distribution itself is generally varying – slightly or significantly – during the time interval of interest. In other words, as the plasma flows and the spacecraft moves, we never probe plasmas with the same parameters, which may be the case in a controlled simulation box. Using such a $f_0$ with observations results in significant patterns in velocity space, which should be avoided for studying instantaneous, fine details of the VDF.
With this motivation, we propose two different treatments of the VDF which do not rely on any other information than the instantaneous distribution itself. These two treatments provide two complementary views of the VDF, with different advantages and drawbacks discussed in the following sections. The first treatment is a *scaling*, during which we consider each energy level of the spherical grid-of-interest separately. For each of these spherical shells, the minimum value of the interpolated VDF is set to 0 and the maximum value set to 1. Values are then averaged over the gyro-angle (angle around the background magnetic field). This scaling is closely related to classical pitch-angle distributions, in which the VDF is shown for a few selected energy ranges, with the color-plot dynamics ranging from the lowest to the highest distribution value of each energy range (see also \[app:interpolation\]). In the proposed scaling, we virtually display 200 concentric, time-averaged pitch-angle distributions. In the next section, we will use a *scaled* pitch-angle distribution, by selecting only one limited energy range of the scaled VDF and plotting it over time (Figure \[fig:spectroVDF\]).
In the second treatment, each VDF values is *normalized* to a reference value, which we choose to be the VDF value at the same energy for $v_\parallel=0$ (i.e., a pitch-angle of $90^\circ$). All these central (equatorial) values are therefore set to one, appearing in white tones, while higher values appear as red tones and lower values as blue. A decimal logarithm is applied to the normalized values when plotted.
These two treatments do not have a physical motivation, they are only arbitrarily chosen to highlight VDF features.
![An example of an original VDF to the left, integrated over 300 ms, its scaled view in the middle, and its normalized view to the right.[]{data-label="fig:VDFNorms"}](normedDists_2.pdf){width="\textwidth"}
In Figure \[fig:VDFNorms\], the effect of the scaling on the original distribution is remarkable, with two parallel branches of values larger than 0.5 (i.e. red tones) stretching at constant parallel velocities. Properly speaking, these two structures are thick disks in the 3-dimensional velocity space. Another structure is found along the anti-parallel direction, a strahl-like structure greatly highlighted in comparison with the original VDF. The circular features of constant speed are unavoidable artifacts of this scaling, in which all energy levels are treated regardless of the others: the continuity of the VDF across energy is lost, and some energy shells are scaled differently compared to the neighbouring shells, resulting in circular visual artifacts. The normalization we introduced here-above was applied on the original VDF, in order to conserve this continuity across energy, for a complementary representation given in the right-most plot of Figure \[fig:VDFNorms\]. The circular artifacts vanished from the two branches – or disks – of higher density, which are now reaching to even higher perpendicular velocities. In this view, the strahl-like component is perceived as broader than in the scaled VDF. The normalization puts different weights on details compared to the scaling, and though being much less sharp, it appears in this case to be more sensitive to details at high velocities.
We now have a comprehensive and constraining way of examining an instantaneous VDF, with two different views of it. The 0-to-1 scaling and the normalization have different properties, already illustrated above and further appreciated in the following test-particle approach, which makes them a great tool for comparing numerical and observational results. These two methods are easy to implement and come with barely any computational cost. Together with the interpolation approach, they may offer interesting applications for characterising the multi-dimensional VDF in other heliophysics and planetary physics contexts.
Wave analysis and theoretical linear model {#sec:wave}
==========================================
![(a)-(b) B-field and E-field power density spectrograms. The electron gyrofrequency is indicated by the solid lines ($f_{ce}$ and $0.4 \ f_{ce}$). (c) Filtered E- and B-field wave forms. (d) Original B-field Power Density Spectra and its Butterworth filtered copy.[]{data-label="fig:waveForm"}](waveForm.pdf){width="\textwidth"}
In order to study wave-particle resonances in observational data, we have isolated one case displaying a clear wave activity within an otherwise “quiet” magnetosheath environment. Captured on the 8th of March 2019, the 3 minute-long interval starting at 13:56:10 displays contrasted, strong wave packets seen in the power density spectrograms of Figure \[fig:waveForm\] (a-b). The central frequency of these packets is about 250 Hz, getting slightly higher on the second half of the observation. It is comprised between 0.3 and 0.5 electron gyrofrequency $f_{ce}$.
We focus on the central event of constant frequency, selected in Figure \[fig:waveForm\] (a-b). The fields components are band-pass filtered using a Butterworth filter. The result is displayed in Figure \[fig:waveForm\] (d). The reference frame is aligned with the background magnetic field $\mathbf{B}_0$ averaged over the duration of the selected interval. We find that the parallel component of both fields is significantly smaller than the perpendicular components, however not null. In this frequency range, magnetic fluctuations up to 0.5 nT are observed around an average background magnetic field 25 nT strong (Figure \[fig:waveForm\] (c)). The electric field fluctuations are seen sometimes surpassing 1 mV/m.
![ One wave packet, indicated in Figure \[fig:waveForm\] (c) with a grey background, its analytical model, at position (0,0,0), and its hodogram in the minimum variance analysis frame. The first data point of the series is indicated by a cross.[]{data-label="fig:waveModel"}](waveModel.pdf){width="\textwidth"}
We zoom-in once more to study a single wave packet, highlighted with a grey background in the time series of Figure \[fig:waveForm\] (c) and expanded in Figure \[fig:waveModel\]. The background magnetic field $\mathbf{B}_0$ is now calculated over this shorter period. The fields components exhibit remarkably clean sinusoids within the envelope defining the packet. The two perpendicular ($x$ and $y$) components of both fields are almost equal in magnitude and separated in time by a phase of $\pi/2$, corresponding to an almost perfect right-handed circular polarisation. The polarisation is also nicely seen in the hodogram of Figure \[fig:waveModel\], given in the local minimum variance reference frame (the frame in which the variance along an axis – here the $z$-axis – of the vector time series is the smallest, 65 times smaller than the two other variances, almost equal to each other).
Whereas the four perpendicular components are defined by a similar envelope in the fields $\mathbf{E}$ and $\mathbf{B}$, the parallel (blue) component, also sinusoidal, follows a different time variation in $\mathbf{E}$ and $\mathbf{B}$, and has a slightly higher frequency compared to perpendicular components, in both fields (not shown). Given its right-handed circular polarisation and its non-zero parallel component, we identify the wave as a quasi-parallel whistler mode wave. Studying the same mode, and report its observations in a range from 0.1 to 0.6 gyro-frequency, a range in which the present case falls perfectly. Because of the wave right-handed circular polarity and its frequency, electrons are the species of interest for this study.
We use the following dispersion relation of the whistler mode, derived from the Appleton-Hartree relation [@hsieh2017jgr] in the limit of $\omega \ll \omega_{pe}$, valid in our case with $\omega<10^{-2}\omega_{pe}$:
$$\label{eq:dispRelation}
Y = \frac{X^2}{X^2+1}\cos(\theta)$$
with the normalized spatial frequency $X=k\cdot d_e$, the normalized angular frequency $Y=\omega/\omega_{ce}$, $k$ the amplitude of the wave vector, $\omega$ the wave angular frequency, $d_e$ the electron inertial length, $\omega_{pe}$ the electron plasma frequency, $\omega_{ce}$ the electron (angular) gyrofrequency, and $\theta$ the angle between the background magnetic field and the wave vector.
Because of apparent monochromatic nature of the observed waveforms, we can determine the wave vector angle $\theta$ with regard to the magnetic field using a minimum variance analysis over a sliding window as wide as the wave temporal period. In other words, we consider one pseudo-circle at a time, described by the wave vector, and find the orientation of the plane containing this circle. The result of this sliding minimum variance analysis is shown in the fourth row of Figure \[fig:waveModel\] for the magnetic field. We find that the wave is propagating with angles between 0 and 20 degrees from the background magnetic field. Finally, to determine if this quasi-parallel wave is propagating along or against the background magnetic field, we calculated the components of the Poynting vector $\mathbf{E} \times \mathbf{B}$ , which is parallel to the wave vector (see for instance the comprehensive work of on single-spacecraft estimation of whistler mode wave properties). The Poynting vector $\mathbf{S}$ (Figure \[fig:waveModel\] third row) is mostly parallel to the background magnetic field, as evidenced by its largest and positive parallel component: the wave propagates in a mostly parallel direction. We now have the frequency and the wave vector direction of the wave, and an estimation of the wave vector amplitude. It was also verified that the probe velocity in the plasma frame (in which $\mathbf{u}=0$, see Section \[sec:partAnalysis\]) is not significant with regard to the phase speed of the wave, and the observed wave frequency found in the spectrograms is not Doppler affected.
Despite how clean the observed waves are, they are still limited to one point in space. If we are to study the possible resonant interactions between the particles and this precise type of wave, we need its temporal *and* spatial description, or model. developed an analytical expression for the components of a quasi-parallel whistler mode wave, based on the electric field linear system proposed by and obeying the Faraday’s law. With the $k$-vector lying in the $(x,z)$-plane, the wave fields are given by the authors as
$$\begin{split}
\mathbf{B}_w & = \mathbf{e}_x B_x^w \cos(\Psi) + \mathbf{e}_y B_y^w \sin(\Psi) - \mathbf{e}_z B_z^w \cos(\Psi) \\
\mathbf{E}_w & = \mathbf{e}_x E_x^w \sin(\Psi) - \mathbf{e}_y E_y^w \cos(\Psi) + \mathbf{e}_z E_z^w \sin(\Psi) \\
\Psi & = \omega t - k_x x - k_z z
\end{split}$$
Following in the limit $\omega \ll \omega_{pe}$, we get the following polarizations
$$A_s = Y + \frac{Y^2-1}{\cos(\theta)-Y} \quad ; \ \
A_p = \frac{\sin(\theta)\cos(\theta)}{\sin^2(\theta)-1+\cos(\theta)/Y}$$
Finally, the wave fields components are expressed as
$$\label{eq:wave1}
\begin{gathered}
B_y^w = A_s(1-A_p \tan(\theta)) B_x^w \quad ; \ \ B_z^w = \tan(\theta) B_x^w , \\
E_x^w = A_s v_{p \parallel} B_x^w \quad ; \ \ E_y^w = v_{p \parallel} B_x^w \quad ; \ \ E_x^w = A_s A_p v_{p \parallel} B_x^w ,\\
B_x^w = \frac{B_w}{\sqrt{\cos^2(\Psi) + A_s^2(1-A_p \tan(\theta))^2 \sin^2(\Psi) + \tan^2(\theta)\cos^2(\Psi)}} ,
\end{gathered}$$
with the parallel phase speed $v_{p \parallel} = \omega/k_\parallel$ .
The observed and modeled wave and plasma parameters for the packet shown in Figure \[fig:waveModel\] are gathered in Table \[tab:param\]. Using these parameters, we have estimated the wave vector amplitude using Equation \[eq:dispRelation\], also given in the same table. We can now obtain the fully analytical, temporal and spatial model of the wave, using Equation \[eq:wave1\]. We note that no information about the electric field is fed to the analytical model, this field is thus completely constrained by the model.
The result for a fixed point in space is given in the lower-right panels of Figure \[fig:waveForm\] over a few wave periods. The analytical model results in a slightly higher $E_w / B_w$ ratio, but provides satisfactory wave forms and Poynting vector form and amplitude, additionally obeying Maxwell’s equations. We can now use this model to investigate resonant wave-particle interactions.
Observed modeled
--------------------------------- --------------------------- ----------------------------------
$B_0$ 25.4 nT 25.4 nT
$n_e$ 16.3 cm$^{-3}$ 16.3 cm$^{-3}$
$B_w$ \[0.2, 0.7\] nT 0.5 nT
$E_w$ \[0.5, 1.4\] mV/m 1.5 mV/m
$\omega \ (\omega/\omega_{ce})$ 1571 rad/s (0.35) 1571 rad/s (0.35)
$k$ ($k\cdot d_e$) $\emptyset \ (\emptyset)$ $5.9 \cdot 10^{-4}$ rad/m (0.77)
$\theta$ $[0^\circ, 20^\circ]$ $20^\circ $
$T_\parallel$ 45 eV 45 eV
$T_\perp$ 41 eV 45 eV
$\beta_e$ 0.46 $\emptyset$
: Wave parameters for the observation and the model, and additional plasma parameters.[]{data-label="tab:param"}
\
Resonant test-particles vs. observed VDF {#sec:testPart}
========================================
We will now examine the potential effect of the modeled wave on the electron dynamics, and see whether or not signatures in the electron VDF can be found. Before considering simulating the situation with a self-consistent model (fields and particles feedback on each other according to Maxwell’s equations), we explore the possibilities offered by a test-particle approach. The fields and waves are analytically described with constant frequency and wave-vector through time: there is no feedback from the particles on the fields. Thus, the presence of other particles, or the shape of the distribution, does not affect a single particle dynamics, which can be solved on its own. This allows for a very cheap, flexible first study of the resonant dynamics in the case of our simplified wave. A considerable advantage is that by doing so, we isolate the effects *of* the wave *on* the particles, and not the other way around, which is not directly possible in a self-consistent description.
In the absence of collisions and gravity, and using the analytical wave discussed in the previous section, we know the force experienced by a particle at any time and any position. The particles dynamics are solved according to this force using the classical Boris scheme [@boris1970], widely used by the Particle-In-Cell community because of its straightforward implementation and its great accuracy for this precise problem – charged particles moving in electromagnetic fields. We initialise 200 million particles following a 3-dimensional isotropic non-drifting Maxwellian distribution, characterised by the observed electron (parallel) temperature of 45 eV. These particles are homogeneously distributed in physical space, in a 2-dimensional periodic box, extending over one projected wavelength (parallel for one axis, perpendicular for the other). Results are given for a simulation time of 25 ms, corresponding to a bit more than 6 wave periods and about 15 electron gyrations.
![Test-particle results (top) compared to the observation (bottom), for an integration time of 60 ms. The Landau parallel resonant velocity is given by the horizontal dashed line and the cyclotron one by the solid lines. []{data-label="fig:comparison"}](normedDists_3.pdf){width="\textwidth"}
The Landau resonance is obtained for particles with a parallel velocity close to the wave parallel phase speed: seen from these particles’ perspective, the parallel component of the wave is almost static, and continuously accelerates them. This gives us a first resonant speed, $v_{Landau} = \omega/k_{\parallel}$. Because of their angle from the background magnetic field, the observed and modeled waves have a projected parallel component, which therefore enables this resonance. Particles with a parallel speed slightly slower than the parallel phase speed are being caught up by the wave and gain kinetic energy, whereas faster particles experience the opposite phenomenon. These resonant particles migrate in phase space, with a motion dependent on their initial phase space position. The wave we consider here has a dependence along the perpendicular direction: as a particle gyrates, its perpendicular position evolves, and so does the magnetic and electric wave components it experiences. A deeper, 3-dimensional description of these resonant motions, depending on $v_\parallel$ and $v_\perp$ is beyond the scope of this article, despite its great interest. These dynamics quickly result in a mixing of the particles in phase space around the parallel Landau speed. It is noteworthy that if one initialises the simulation with a flat velocity distribution (i.e. no density gradient in velocity space), no signature of this mixing can be found in velocity space. A density signature is indeed only visible if the resonant speed corresponds to high VDF gradients, which happens to be the case here. Very quickly, in just 6 wave periods, the resonant particles form an under/over-density centered on the Landau resonant speed, as seen in the scaled and normalized VDF of Figure \[fig:comparison\] (the Landau resonant speed is displayed by the dashed horizontal line).
The cyclotron resonance is only met in the presence of a circularly polarised wave. Particles with different parallel speeds experience a wave with a different frequency, a simple Doppler effect dependent on the particles velocity. For one particular parallel speed, given by $v_{Cyclo} = (\omega-\omega_{ce})/k_{\parallel}$, with $\omega_{ce}$ the electron gyrofrequency, the wave is seen with an angular frequency equal to the electron (angular) gyrofrequency: the particles gyrate synchronously with the rotation of the circularly polarised $\mathbf{B}_w$ and $\mathbf{E}_w$. Note that because $\omega<\omega_{ce}$, this resonant speed is negative (if $k_\parallel>0$): resonant particles are moving against the wave, which in turn significantly decreases the time during which they may interact with the wave. As discussed in the Landau case, such particles are continuously accelerated by the wave, and migrate in phase space, in an even more complex manner. The mixing of a high density gradient region again results in an under/over-density organised around the cyclotron speed given by the solid line in Figure \[fig:comparison\], with a sign opposite to the one of the Landau resonant speed. The over-density is found for lower parallel speeds but higher perpendicular speeds, corresponding to higher total speed for these particles.
The lower row of Figure \[fig:comparison\] gives a comparison to the VDF observed during the same wave packet. We use only two FPI observations, corresponding to an integration time of 60 ms. There as well, two over-densities are found just above the two resonant speeds, at the same location as for the test-particles. The scaled view of the test-particle VDF exhibits two strong, perfectly circular artefacts, which were discussed in the first section. The normalized view, conserving the continuity along the radial dimension, does not display such structures. An additional feature is also found along the anti-parallel direction. We will see in the next section that this beam does not correlate with the presence of the wave. The thermal parallel speed is indicated in the observed distributions. The speed is the same as the central speed of the over-densities: the resonant mixing of particles indeed happens where the VDF velocity gradient $\partial f/\partial v_\parallel$ is high, which could lead to an efficient damping of the wave. We note however that the relevance of this thermal speed is limited, when considering the large departures of the observed VDF from a Maxwellian distribution.\
The over densities are observed around similar absolute parallel velocities, despite the absolute value of the resonant parallel velocities being clearly different: the over-densities tend to be fairly symmetric with regard to the $(v_{\parallel}=0)$-plane. Note that this fact holds for the plasma and wave parameters analysed here. If the position of the two over-densities are matching nicely between the test-particles and the observed VDF, strong discrepancies in the shape of these signatures exist and are discussed below.
VDF time evolution
==================
To verify if this double branch signature indeed correlates with the presence of the wave over longer time scales, we show In Figure \[fig:spectroVDF\] (c-d) two additional electron VDFs integrated over a longer time of ten FPI measurements, corresponding to an integration time of 300 ms. In both cases, a strahl-like (beam) component is found along the anti-parallel direction, with a constant perpendicular width. In the absence of wave activity (last VDFs), we find a clearly anisotropic distribution, with a strong equatorial signature in the scaled view.\
When analysed during the maximum power of the wave activity, the VDF displays the double-branch signature. It was checked that during the entire time interval, no significant (i.e. higher than the measurement noise) agyrotropy are to be found. Ions (not shown here) present a similarly anisotropic distribution, virtually static over the time interval.\
Figure \[fig:spectroVDF\] (b) gives a view on the temporal evolution of the scaled VDF and its double-branch feature. For this representation, each 30 ms scaled VDF was averaged over a limited speed range between 5.8 and 6.3 $10^6$ m/s (indicated on the scaled view in panel (c)), and the result was plotted in the time $t$ and pitch-angle $\Theta$ dimensions in panel (b). Note that scaled pitch-angle distributions and classical pitch-angle distributions qualitatively converge as the energy range decreases. In this energy range, scaled values above 0.5 (red tones) form a striking feature centered on 90 degrees. When the waves are observed in the magnetic field spectrogram (panel (a)), the feature splits into two branches, corresponding to the double-branch pattern observed in velocity space. We note that such a pitch-angle distribution looks very similar to electron distributions in mirror modes, explored by , where the envelope of the pitch-angle distribution is shown to correlate with a critical pitch-angle, under which electrons get trapped in one mirror structure. We note, however, that in our case, the signature is ordered by (constant) parallel velocities, and not constant pitch-angles when all the energy range is considered.
![(a) B-field power density spectrogram. (b) scaled pitch-angle distribution for a speed range indicated in (c). (c)-(d) time-integrated VDFs and their scaled view, with the integration time indicated on the time series.[]{data-label="fig:spectroVDF"}](spectrogramsPitch.pdf){width="\textwidth"}
Particle energisation
=====================
We can follow the energisation of the test-particles either by summing up the total kinetic energy of all particles, or simply by taking the center of mass of their distribution (order 1 moment) and expressing it as a current density, with one parallel and one perpendicular component. The total kinetic energy and the two components of the current density are given in Figure \[fig:currentTestPart\]. We find that with the physical parameters of our problem, the maximum energisation happens very fast, in about 2 wave periods, or 5 gyroperiods. Therefore, the wave packets are sufficiently long to exchange energy with the particles. After about 5 wave periods, the overall energy gained by the test-particles, as well as the parallel current density, stabilise to a non-zero value. The test-particle approximation does not hold anymore, as this current would necessarily alter the electric and magnetic fields, which would act to decrease it back to zero. For this reason, it is likely that this estimated current for the present resonant wave-particle interaction is an upper bound for the self-consistent interaction. Barely any perpendicular current is found, indicating that the VDF remains gyrotropic at all times. The fact that the total kinetic energy of all particles and their parallel current have almost the same time evolution illustrates that these resonant interactions mostly result in an increase of the bulk velocity and only some of the energy is transformed into “temperature”, or change of the pressure tensor.
A crucial point is indicated by this time evolution and verified in the time evolution of the simulated VDF (not shown) is that the resonant signatures are not periodic: they remain at the same position (in velocity space) with the same over/under densities through time. This is a necessary condition for it to be observed by an instrument with an integration time longer than the wave period.
![Test-particle total kinetic energy (relative to its value at t=0) and current density, parallel and perpendicular. Time is given in unit of wave period $T_{wave}$.[]{data-label="fig:currentTestPart"}](current_testpart.pdf){width="\textwidth"}
In Figure \[fig:currentObs\], the observed electron current density (parallel and perpendicular) are displayed, together with the cropped spectrogram of the magnetic field. The parallel current is varying around 0, and no correlation with the wave activity can be found. Despite the strength and duration of the wave packets, the *background* current density turns out to be significantly larger than the estimated (upper bound) current rising from the resonant particle signatures. Alternative moments were also calculated ignoring the core of the distribution (using different speed threshold), showing barely any changes in the electron current. In turn, the dissipative term $\mathbf{E} \cdot \mathbf{J}_e$, using an electric field averaged over the electron integration time, shows no correlation with the wave activity.
![(a) Magnetic field spectrogram, (b) observed electron current and (c) dissipative term.[]{data-label="fig:currentObs"}](currentObs.pdf){width="\textwidth"}
Discussion & Conclusions
========================
We have first shown that VDF features originally hidden under high gradients of the background VDF can be highlighted using some scaling and normalization, without the use of a reference VDF $f_0$. We then isolated a strong, fairly narrow-band, quasi-parallel whistler mode wave, which could be analytically mimicked. We applied this analytical wave to a collection of test-particles to get a first sense of its potential effects on the electrons. We could map where and how particles resonate, and have found that the initial Maxwellian distribution is reshaped by Landau and cyclotron resonances. Test-particles and observed VDFs display two branches of higher density, around constant parallel velocities, in the scaled and normalized views of the VDF. The observed signatures were found to correlate nicely with the wave activity. Finally, this wave-particle interaction could not be detected in the observed current or the observed $\mathbf{E}.\mathbf{J}_e$ product, as both $\mathbf{E}$ and $\mathbf{J}_e$ (observed) present fluctuations of larger amplitude than the current and electric field used by or resulting from the model. Strong discrepancies between the simulation and the observations remain to be discussed.
Firstly, the strongest discrepancy between the observed VDF and the simulated one is the strong symmetry of the observed signatures around the $(v_\parallel=0)$-plane, further illustrated by the time evolution of the scaled pitch-angle distribution in Figure \[fig:spectroVDF\]. As such, it is difficult to prove that the Landau resonance, providing an already faint signature in the perfect test-particle set-up, may cause a resonant branch as strong as the observation, and so symmetric to the cyclotron branch. It actually *appears* as if a mirrored cyclotron branch shows up systematically with the wave activity, despite our wave analysis only bringing out one single mode, propagating in one well defined direction. Such symmetric signatures have been displayed in the works of (in the magnetosphere) and (in a nominal, or fully developed turbulence case), in which the authors did not isolate one single mode, and interpret the symmetric signatures as caused by waves propagating in both directions, parallel and anti-parallel. This symmetry remains the foremost open question of our study.
Secondly, the test-particle model cannot account for the transformation from the already strongly anisotropic VDF out of the wave activity (shown in panel (d) of Figure \[fig:spectroVDF\]) to the 2-branch signature. The disturbance of the VDF caused by resonances in the test-particle case is a local phenomenon in velocity space, it certainly cannot reshape macroscopically the distribution, make a large amount of the electron population migrate over large velocities. Were we able to easily initialise the particles according to the observed VDF outside the wave activity, the strong equatorial structure would remain there, with additional resonant signature expressed on the sides. While the test-particle simulation enlighten us on the micro-physics of the interaction during a very brief snapshot of the observations (when an intense wave activity is observed), the macroscopic configuration of the fields (magnetic gradients, mirrors, etc) should be considered to fully understand the time evolution of the electron VDF.
The limitations of the test-particle simulation are numerous, if one is to thoroughly compare its results and the observations. Instead, we suggest that this approach only points at the very first step of the full self-consistent wave-particle interaction. It shows us where and how in velocity space resonances should occur for a given a monochromatic wave, and gives a first hint on their phase density signatures. Most importantly, it also gives us a good sense of how fast these resonances can produce signatures in the VDF, and their efficiency. But it obviously cannot go further in the physics of the interaction, both an advantage and a drawback.\
This numerical approach is limited to the case of a purely monochromatic wave with a constant amplitude and a constant normal angle, whereas the observed wave packets exhibit some spectral breadth, a parallel component at a slightly higher central frequency, and additionally a propagation direction constantly evolving. Because of the periodic boundaries necessary for the simple test-particle approach, such a wave cannot be easily modeled. We refer to the work of , in which the authors specifically highlighted the effect of the amplitude modulation of chorus emissions in the magnetosphere, on particle acceleration. They, however, do not reconstruct distribution functions, and can therefore do without the periodicity constraint.\
The feedback of the particles on the wave is a different problem entirely, in terms of numerics at least. Thus the comparison should not be over-interpreted. We have shown that the observed wave is expected to have clear signatures on the electron distribution function, signatures matching well in position and somewhat in shape the observed signatures.\
\
The visualisations (scaling and normalization) developed for this study are a key element for a better, deeper characterisation of the particles distributions. It is in the numerical aspect of our work that these views reveal best the effect on the distribution, getting us rid of the need for an additional information (e.g. distributions at other times). They show great flexibility and versatility, which may be useful for other applications on observations and simulation data, and mostly their comparisons.\
Such an exploration of wave-particle resonances in 3 dimensions, and its direct comparison to observations, is – to the best of our knowledge – novel. It familiarises us with resonant dynamics and signatures, at a low cost and without the intrinsic complexity a self-consistent model adds to physical interpretations. This approach may present real promises for a more systematic recognition of resonant signatures in a more complex data, such as those of fully developed turbulence in the solar wind or the magnetosheath
E. Behar is funded through DIM-ACAV post-doctoral fellowship and from the European Unions Horizon 2020 research and innovation program under grant agreement No 776262 (AIDA, www.aida-space.eu). The authors also wish to thank Olivier Le Contel and Gérard Belmont for valuable discussions that have broadened the context of this study.
Binning versus interpolating {#app:binningInterpolation}
============================
![Illustration of a binning and an interpolation.[]{data-label="fig:interpolationSchem"}](interpSchem.pdf){width="\textwidth"}
In various publication, when handling multidimensional Velocity Distribution Functions (VDF), authors choose the somewhat more intuitive of rotating the data from the instrument frame to the frame of interest, to then *bin* the data within a defined grid, as illustrated in Figure \[fig:interpolationSchem\] in the first row. During this process, one measurement point ends up in one single cell of the grid-of-interest, and one cell of the grid-of-interest may receive zero one, or many measurements. Empty cells will appear on the representation, producing visual artifacts, and it is necessary to average the binned data over at least one dimension in order to *fill* as much as possible the grid-of-interest. We give such an example in a cylindrical representation of the VDF, right-most plot of the first row of Figure \[fig:interpolationSchem\], which shows about thirty electron distributions binned in a 2-dimensional grid defined, in the classical $(v_\parallel, v_\perp)$-plane. The binned data show strong artifact along the energy dimension, as the instrument energy levels are log-distributed: the higher the energy, or speed, the greater the energy steps. Therefore one regular Cartesian grid cannot be suitable for the entire energy range of the instrument: its resolution will be too coarse at low energies – where weighting of the binned data has to be taken into account properly – and too fine for high energies, leaving empty most of the grid-of-interest. The binning approach can only be used over a restricted range of energies, and is anyway not suitable for 3-dimensional analyses.\
The interpolation approach is illustrated in the second row of Figure \[fig:interpolationSchem\]. First, we define a 3-dimensional grid-of-interest – an array/set of coordinates – within the reference frame of our choice, with arbitrary extent and resolution. We apply the opposite rotation, shift, and scaling to a copy of the grid-of-interest, from the reference frame to the instrument frame, as illustrated in Figure \[fig:interpolationSchem\]. The values of the data are then interpolated at each node of the transformed grid (see next section): the data are continuously evaluated over the entire grid, leaving no room for artifacts, and avoiding additional weighing operation. All the results in the article use a spherical grid-of-interest, allowing at no additional cost the scaling and normalization of the VDF, and their straightforward representation in the $(v_\parallel, v_\perp)$-plane.
But interpolation with an order higher than 2 presents complications, and this approach is not universal and cannot be applied blindly.
Interpolating velocity space distributions {#app:interpolation}
==========================================
We have tested three different interpolation schemes, namely nearest-neighbour (order 0, only one point of the measured VDF is used), trilinear (order 1, 8 data points are used), and tricubic (order 2, 64 data points are used). The two first schemes are taken from the main scientific python library, SciPy, while the third was implemented by [@lekien2005nme], with one wrapper made available by the authors for a usage with Python.
![Interpolation.[]{data-label="fig:interpolation"}](interpMain.pdf){width="\textwidth"}
We now have a 3-dimensional array of interpolated VDF values, with two different sets of coordinates, one in the instrument frame (used for the interpolation), one in the reference frame of the study (the originally defined Grid-of-Interest). This array can be either analysed and visualised on its own, or added up to other arrays in order to average the data over a longer duration, all within the same frame. For instance, Figure \[fig:interpolation\] presents results using only one single electron VDF, which is the most challenging test for the methods, because of low detection rates for higher energies. This method has one invaluable interest: any resolution can be used without any of the risks inherent to a binning process, illustrated in Figure \[fig:interpolationSchem\]. Another great advantage when using a Cartesian grid-of-interest is that velocity derivatives of the data can be easily and straightforwardly computed. The advantage of using a spherical or cylindrical grid-of-interest was already illustrated. The method has one drawback, namely the computational cost. The interpolation in itself is more computationally demanding than a simple binning, and for most purposes we will have many more elements in the grid-of-interest than in the instrument grid, resulting in as many more calculations to compute.\
In Figure \[fig:interpolation\], a single electron distribution is used to illustrate the differences between the three interpolation schemes, using three different visualisations. In the first row, we give the profile (1-dimensional) of the interpolated values along the parallel direction. For this representation and in order to increase slightly the statistics, all values within an angular distance from the parallel axis ($+{\mathbf{B}}$) are selected and averaged, using a conic selection $\pi/6$-wide. For this purpose, a spherical grid-of-interest is a better choice, allowing one to simply average the interpolated data over one angular dimension, avoiding a binning process. The energy/speed levels of the instrument are also indicated as vertical lines, regularly spaced in this logarithmic representation. For this analysis, we only rotated the frame, without a shift, so the energy levels of the instruments remain centered on the origin of the Grid-of-Interest, making the results more readable.\
Another representation is proposed on the second row of Figure \[fig:interpolation\], in which we selected a specific speed range, averaged the data over this range, resulting in the given angular maps (2-dimensional). Here as well, using a spherical Grid-of-Interest makes this selection straightforward, selecting only one radial range in the array, again avoiding a binning process. The parallel axis is vertical and intersects these maps at their poles, and the equator correspond to a pitch angle of 90 degrees.\
In the parallel 1-dimensional profiles previously described, the nearest-neighbour interpolation results in steps centered around the instrument energy levels, as expected from such a scheme. This curve is reported in the two other VDF profiles for the trilinear and tricubic cases, in order to verify that all three interpolations indeed meet at the instrument speed levels, an important convergence test. These steps in the nearest-neighbour case directly correspond to the “pixels” found in the left-most angular map. This is the way particle data are often displayed, implying that within an instrument “pixel”, the observed flux is constant. This assumption is also widely used when integrating the plasma moments, which simply corresponds to a Riemann sum of the area under a function. The instrument poles are misaligned with the magnetic field direction, and can be seen close to the equator, 180 degrees apart, with triangular pixels meeting in one point.
The trilinear interpolation provides a smoother, more continuous profile. Thinking in one dimension, between two measurement points, interpolated values will follow a linear relationship. In this logarithmic representation, these linear segments show up as arc segments in-between each instrument energy/speed level. This is obviously an unwanted visual artifact, generated by the choice of the representation. Just as the 1-dimensional profile, the angular map shows a smoother result than in the nearest-neighbour case. In the same way that the nearest-neighbour can be linked to a Riemann sum, this linear interpolation converge to a trapezoidal rule in terms of VDF integral.
Finally, the tricubic interpolation provides the smoothest and most continuous curve for speeds up to 7000 km/s, for which the VDF is relatively high. At higher energies, strong artifacts are found, with oscillations between the last instrument speed levels. Two phenomena can account for this behaviour. The first could be the Runge’s phenomenon, namely oscillations at the edges of the interpolation interval when using polynomials of high degree (higher than two). We note however that we only use a third degree polynomial interpolation, which should limit this phenomenon, and these oscillations are only visually found for low fluxes, high speeds. The second phenomenon occurs when the VDF contains zero values. But most of all, this interpolation scheme, as derived and implemented by , can result in negative VDF values. This nonphysical result should be monitored, so these negative values remain small and insignificant for the analysis we want to perform.\
*Remark*: these interpolations and their unavoidable artifacts and drawbacks are not well suited for all purposes. For instance, the tricubic interpolation is valid only for strong signals, high VDF values, and should anyway be used with great caution, so these artifacts are not interpreted as physical features.\
As a first test, we have already verified that the three schemes converge at the instrument speed levels. A second important test is to make sure that the three interpolations do not show artifacts at the instrument poles. Indeed, the trilinear and tricubic schemes we use assume that the data are defined over a grid with cuboid elements: evenly spaced along each dimension, with possibly different spacings along each dimension. The poles of the instrument spherical grid should therefore present errors, as the array elements there strongly depart from cuboids. To test the overall error from each scheme, we have defined artificial, ideal instrument measurements, namely a drifting[^1] Maxwellian distribution. This way, we also know the real value of the distribution at the nodes of the rotated, shifted, scaled grid-of-interest, and can compare these analytical values to the interpolated ones. It allows us to creates error maps, presented in Figure \[fig:errorInterpolation\]. The first row shows one chosen Cartesian cut through the error distribution, while the second row gives one angular cut, or angular map. The nearest-neighbour interpolation results in the largest errors, just as expected from the steps seen in the 1-dimensional profiles of Figure \[fig:interpolation\]: the interpolation is alternately much smaller and much larger than the real analytical value of the distribution. These errors can largely surpass the 50 percent level, because of the steepness of the VDF. Since the Maxwellian distribution is convex for high energies/speed, the error given by the linear interpolation is positive-only for high speed values, as seen in the first row, middle panel. At these higher speeds, the error can reach up to 50 percent, whereas at lower speed values – of greater interest for us – the error becomes arbitrarily low. The tricubic interpolation shows the smallest error in this plane with mostly positive values. We note a slightly stronger error-ring at the lowest speeds covered by the instrument, which we link to the Runge’s phenomenon. At the high VDF values usually observed at these energies, this error may become significant for some purposes, though it has not been found to be the case for our analyses.
The instrument poles are not intersected by this plane, but we expect to find the greater error there. They however show up in any angular map, as seen in the second row of Figure \[fig:errorInterpolation\]. The poles still have the same position as previously, and can be easily spotted in each interpolation error map. The worst qualitative result is obtained for the tricubic interpolation, with relatively strong errors localised at the instrument poles, though the absolute error remains low (a few percents) and indeed extremely localised. We conclude that for our purpose, the three schemes are sound, resulting in acceptable errors where the observed VDF values are high, i.e. for speeds lower than $10^4$ m/s.
![Error interpolation.[]{data-label="fig:errorInterpolation"}](error_interpolation.pdf){width="\textwidth"}
As a third test, we have verified that the electron density, resulting from integrating the interpolated values using a simple Riemann integral, calculated for the three interpolations, matches the density provided by the instrument team. Their densities are also the plasma moment of order 0, but directly integrated in the instrument spherical coordinate system. The result is shown in the first row of Figure \[fig:momentsTest\], and one can find that the four curves are barely distinguishable.
![Moments tests.[]{data-label="fig:momentsTest"}](testDensity.pdf){width="\textwidth"}
[^1]: The drift is important for the significance of the test, as it misaligns the center of the artificial distribution, and the center of the instrument grid, testing the errors over all three dimensions.
|
---
abstract: |
Rare isotope beams of neutron-deficient $^{106,108,110}$Sn nuclei from the fragmentation of $^{124}$Xe were employed in an intermediate-energy Coulomb excitation experiment yielding $B(E2,
0^+_1 \rightarrow 2^+_1)$ transition strengths. The results indicate that these $B(E2,0^+_1 \rightarrow 2^+_1)$ values are much larger than predicted by current state-of-the-art shell model calculations. This discrepancy can be explained if protons from within the Z = 50 shell are contributing to the structure of low-energy excited states in this region. Such contributions imply a breaking of the doubly-magic $^{100}$Sn core in the light Sn isotopes.
author:
- 'C. Vaman$^1$'
- 'C. Andreoiu$^2$'
- 'D. Bazin$^1$'
- 'A. Becerril$^{1,3}$'
- 'B.A. Brown$^{1,3}$'
- 'C.M. Campbell$^{1,3}$'
- 'A. Chester$^{1,3}$'
- 'J.M. Cook$^{1,3}$'
- 'D.C. Dinca$^{1,3}$'
- 'A. Gade$^{1,3}$'
- 'D. Galaviz$^1$'
- 'T. Glasmacher$^{1,3}$'
- 'M. Hjorth-Jensen$^5$'
- 'M. Horoi$^6$'
- 'D. Miller$^{1,3}$'
- 'V. Moeller$^{1,3}$'
- 'W.F. Mueller$^{1}$'
- 'A. Schiller$^{1}$'
- 'K. Starosta$^{1,3}$'
- 'A. Stolz$^{1}$'
- 'J.R. Terry$^{1,3}$'
- 'A. Volya$^{4}$'
- 'V. Zelevinsky$^{1,3}$'
- 'H. Zwahlen$^{1,3}$'
title: 'Z=50 shell gap near $^{100}$Sn from intermediate-energy Coulomb excitations in even-mass $^{106 \mbox{--} 112}$Sn isotopes '
---
Numerous experimental and theoretical studies are currently focused on nuclear structure evolution far from the line of stability. In particular, the structure of neutron-deficient nuclei near the N=Z line is impacted by protons and neutrons occupying the same shell model orbitals. This letter reports observations which indicate that large spatial overlaps of valence orbitals in neutron-deficient, even-mass, tin isotopes, break the stability of the Z=50 shell gap near doubly-magic $^{100}$Sn.
$^{100}$Sn is the heaviest, doubly-magic, N=Z, particle-bound nucleus and therefore is of great interest for shell theory of heavy nuclei. However, it is very difficult to produce and experimentally study this nucleus. One way to approach $^{100}$Sn is to examine the evolution of nuclear properties along the Z = 50 chain of tin isotopes, which is the longest shell-to-shell chain of semi-magic nuclei investigated in nuclear structure to date. The nearly constant energy of the first excited 2$^+_1$ state between N=52 and N=80[@ram01], is one of the well known features of Sn isotopes, and it seems to indicate that effective nuclear interactions between nucleons of the same flavor outside a doubly-magic core do not affect the near-spherical nuclear shape [@cas01]. A probe of the stability of the Z=50 shell gap is provided by the electromagnetic transition rates between the $0^+_1$ ground and the first excited $2^+_1$ state, in even mass Sn isotopes. Even small admixtures of proton excitations across the Z=50 shell gap enhance significantly the electric quadrupole transition probability between the ground and the first excited states in contrast to the configurations with the closed Z=50 core and only neutrons in the valence space.
While experimental 2$^+_1$ state energies are well established, the reduced probability for the electric quadrupole transition from the ground state to the first excited state, $B(E2, 0^+_1 \rightarrow
2^+_1)$, has been sparsely known except for stable Sn isotopes. For neutron-rich tin nuclei, the measurements of these $B(E2)$ values have only recently been achieved due to progress in radioactive beam techniques [@rad05]. On the neutron-deficient side, the corresponding numbers are still unknown except for $^{108}$Sn measured recently in an intermediate-energy Coulomb excitation at GSI[@ban05]. The measurements on the neutron-deficient side of the Z=50 chain are hindered by the 6$^+_1$ isomeric state with a lifetime in the nanosecond range, while the expected lifetime for the 2$^+_1$ state is at least two orders of magnitude shorter. Therefore, for a measurement, the $2^+_1$ state must be populated from the ground state. Consequently, Coulomb excitation is the method of choice if beams of unstable nuclei are available, while other reactions, in particular fusion-evaporation, cannot be applied. This letter reports on the results of an intermediate energy Coulomb excitation experiment and the measurements of the corresponding $B(E2, 0^+_1\rightarrow
2^+_1 )$ strength of neutron-deficient $^{106-110}$Sn isotopes from the fragmentation of $^{124}$Xe. In addition, a measurement for $^{112}$Sn is reported as a check of consistency with existing experimental data.
Beams of rare isotopes are produced via projectile fragmentation at the National Superconducting Cyclotron Laboratory (NSCL) as documented in [@sto05]. In the current experiment a stable beam of $^{124}$Xe was accelerated by the Coupled Cyclotron Facility to 140 MeV/nucleon and fragmented on a 300 mg/cm$^2$ thick Be foil at the target position of the A1900 fragment separator [@mor03]. A combination of slits and a 165 mg/cm$^2$ Al wedge degrader were used at the A1900 to enhance the purity of the fragment of interest in the resulting cocktail beam. The properties of the Sn beams in this experiment are listed in Table \[A1900beams\].
------------ ----------- -------- ------------- --------------------
Isotope Energy Purity $\Delta$p/p Rate
\[MeV/u\] \[%\] \[%\] \[10$^3$ pps/pnA\]
$^{112}$Sn 80 50 0.10 19
$^{110}$Sn 79 50 0.10 21
$^{108}$Sn 78 17 0.34 17
$^{106}$Sn 81 2 0.34 0.7
------------ ----------- -------- ------------- --------------------
: Properties of the rare isotope Sn beams used in the current experiment.
\[A1900beams\]
Coulomb excitation of the above cocktail beams on a 212 mg/cm$^2$ thick $^{197}$Au target were studied using a combination of the Segmented Germanium Array (SeGA) [@mue01] for gamma-ray detection and the high resolution S800 spectrograph for particle identification and reconstruction of the reaction kinematics [@baz03]. For all four tin isotopes studied, a lithium-like and a beryllium-like charge state were delivered to the S800 focal plane and identified by their position on the Cathode Readout Drift Chamber (CRDC) detectors[@you99]. The mass and charge of the nuclei were extracted on an event-by-event basis from the time of flight and energy loss information.
The S800 CRDC detectors measure position and angle in dispersive and non-dispersive directions at the focal plane. This information can be used to reconstruct the trajectories of identified particles to the target position based on the knowledge of the magnetic field in the S800 [@baz03]. The proper trajectory reconstruction provides information on the scattering angle at the target, and, therefore, on the impact parameter in the Coulomb excitation process[@win79]. This information is crucial to relate the Coulomb excitation cross section at the intermediate energies to the reduced E2 transition probability. For the projectile excitation this relation is given by [@gla99]: $$\label{sigma}
\sigma_{proj}(E2,I_i\rightarrow I_f)\propto B(E2,I_i \rightarrow I_f)
Z^2_{tar}/b^2_{min},$$ where $b_{min}$ is the minimum impact parameter considered for the cross section measurement. The minimum impact parameter is chosen to be large enough to minimize the impact of nuclear force interference.
The procedure outlined above for a $B(E2, 0^+_1 \rightarrow 2^+_1)$ measurement from an angle-integrated Coulomb cross section has been applied in a number of successful experiments at the NSCL[@gla99; @cook01; @agade]. In the current study, however, the absolute Coulomb excitation cross section measurement was hindered by angular acceptance effects related to properties of the heavy mass and large charge beams. Thus, below, the experimental information on the transition rates was extracted from a relative measurement to excitations of the $^{197}$Au target.
Following Eq. \[sigma\] the ratio of the cross sections for the Sn projectile and Au target excitations in the current experiment is given by: $$\begin{aligned}
\label{sigmarat}
&&\frac{\sigma_{Sn}(E2,0^+_1\rightarrow 2^+_1)}
{\sigma_{Au}(E2,3/2^+_1\rightarrow 7/2^+_1)}=\;\;\;\;\;\;\;\;\;\;\;\nonumber\\&&
\frac{B_{Sn}(E2,0^+_1 \rightarrow 2^+_1)}{B_{Au}
(E2,3/2^+_1 \rightarrow 7/2^+_1)}\left( \frac{79}{50}\right)^2\end{aligned}$$ The dependence on $b_{min}$ and the reaction kinematics in this ratio is removed as long as safe Coulomb conditions are met. The ratio of the cross sections is measured from the ratio of gamma-ray intensities depopulating the $2^+_1$ state in Sn and the 7/2$^+_1$ state in the Au nuclei. Knowing the target $B(E2\uparrow)$ [@zho95] the corresponding transition rate for the projectile is extracted.
![\[spec1\] Gamma-ray spectra measured by the 90$^\circ$ ring of the SeGA for the $^{108 \mbox{--}112}$Sn projectiles (top) and the corresponding Au target (bottom) Coulomb excitations within the 45 mrad scattering angle in the laboratory reference frame.](fig1.eps){width="7cm"}
In view of the above, the analysis of the $^{108\mbox{--}112}$Sn data proceeded in the following way. A subset of particle-identified events with the impact parameter larger than 19.5 fm was selected; the corresponding scattering angle in the lab was 45 mrad. Next, the cross section ratio measurements were performed according to Eq.\[sigmarat\] for the downstream ring at 37$^\circ$ and the upstream ring at 90$^\circ$ separately, and the $B(E2\uparrow)$’s in Sn nuclei were extracted from these ratios. Spectra illustrating the quality of the data for the 90$^\circ$ ring are shown in Fig. \[spec1\]. The corresponding results are listed in Table \[res\]. It should be stressed that the precise value for the impact parameter is not crucial for this analysis.
Isotope $B(E2,0^+_1 \rightarrow 2^+_1)$ \[$e^2b^2$\] $\Delta_{stat}$ \[$e^2b^2$\] $\Delta_{sys}$ \[$e^2b^2$\]
------------ ---------------------------------------------- ------------------------------ -----------------------------
$^{112}$Sn 0.240 0.020 0.025
$^{110}$Sn 0.240 0.020 0.025
$^{108}$Sn 0.230 0.030 0.025
$^{106}$Sn 0.240 0.050 0.030
: Reduced E2 transition rates measured for $^{106
\mbox{--}112}$Sn isotopes. The results for $^{108 \mbox{--}112}$Sn correspond to the lab scattering angles smaller than 45 mrad, for the $^{106}$Sn the scattering angle limit was set by the S800 spectrograph acceptance [@baz03].
\[res\]
For the $B(E2,0^+_1 \rightarrow 2^+_1)$ measurement in $^{106}$Sn the off-line analysis requirement set on the impact parameter and the scattering angle was relaxed; however, the range of the scattering angles for detected events is still limited to 60 mrad by the angular acceptance of the S800 spectrograph. For all four isotopes the ratio of the projectile to the target Coulomb excitations was extracted using the data shown in Fig. \[spec2\]. A common scaling factor between these ratios and the measured $B(E2,0^+_1 \rightarrow 2^+_1)$ values was computed for $^{108 \mbox{--}112}$Sn and applied to the $^{106}$Sn; the resulting $B(E2)$ for $^{106}$Sn is reported in Table\[res\].\
![\[spec2\] Gamma-ray spectra measured with SeGA for the $^{106 \mbox{--}112}$Sn projectile (top) and the corresponding target (bottom) Coulomb excitations within the scattering angle limited by the S800 spectrograph acceptance.](fig2.eps){width="7cm"}
Experimental information on the $B(E2,0^+_1 \rightarrow 2^+_1)$ systematic in Sn isotopes based on the current measurement and Refs.[@ram01; @rad05; @ban05] is presented in Fig. \[resfig\]. The asymmetric behavior of the $B(E2\uparrow)$ with respect to the N=66 neutron mid-shell at A=116 is striking. This is in disagreement with several shell model $B(E2\uparrow)$ predictions including these from the Large Scale Shell Model calculations of Ref. [@ban05] performed with a $^{90}$Zr core, see Fig. \[smfig\] for the comparison. Shell model calculations consistently predict a $B(E2\uparrow)$ trend which is nearly parabolic and symmetric with respect to the midshell [@ban05; @vol05]. It reflects properties of the even-rank E2 tensor operator in the seniority scheme[@cas01]. In regard to other recently proposed theories, the experimental $B(E2\uparrow)$ strength is underpredicted by the Exact Pairing model of Ref. [@vol05]. It should also be pointed out that while the predictions of Relativistic Quasiparticle Random Phase Approximation of Ref. [@ans05] are consistent with the $B(E2\uparrow)$ values measured here for the most neutron-deficient Sn isotopes, the overall trend for the Sn isotopic chain in the middle of the shell is not well reproduced by these calculations.
![\[resfig\] Experimental data on $B(E2,0^+_1 \rightarrow
2^+_1)$ in Sn isotopic chain from the current results for $^{106\mbox{--}112}$Sn and from Refs. [@ram01; @rad05; @ban05]. The dotted line shows the predictions of the Large Scale Shell Model calculations of Ref. [@ban05] performed with $^{90}$Zr core. For $^{106\mbox{--}112}$Sn the error bars represent statistical errors; the corresponding systematic errors are marked by arrows. ](fig3.eps){width="8cm" height="6cm"}
An effect which can explain large $B(E2\uparrow)$ values in the neutron-deficient Sn isotopes may arise from correlation energy associated with nucleons occupying the same orbitals near N=Z line. An analogy can be drawn between the Sn and Ni isotopic chains. The $^{56-78}$Ni isotopes have valence neutron configurations, $
(f_{5/2},p_{3/2},p_{1/2},g_{9/2})^{A-56} $, similar in shell structure to those of $^{100-132}$Sn, $
(g_{7/2},d_{5/2},d_{3/2},s_{1/2},h_{11/2})^{A-100} $. Effective charges take into account coupling between the valence nucleons and the proton particle-hole excitations of the core not included in the model space. The empirical values of $ e_{p}=1.2 $ and $ e_{n}=0.8 $ [@ti52] apply to the full $ pf $ shell, and thus take into account coupling to the 2$\hbar\omega$ giant isoscalar and isovector quadrupole excitations [@brown]. The $ B(E2\uparrow) $ excitation strengths obtained in the $ (f_{5/2},p_{3/2},p_{1/2}) $ model with the GXPF1 interaction [@gx1] are 0.0126, 0.0249, 0.0264, 0.0243 and 0.0203 e$^{2}$ b$^{2}$ for $^{58,60,62,64,66}$Ni compared to experimental values [@ram01] of 0.0695(20), 0.0933(15), 0.0890(25), 0.0760(80) and 0.0620(90) e$^{2}$ b$^{2}$, respectively. The full $ pf $ shell results (including the $ f_{7/2} $ orbit) obtained with GXPF1 are $ B(E2)=$ 0.065, 0.106, 0.119, 0.082 and 0.047 e$^{2}$b$^{2}$, respectively[@ni54]. The coupling of valence neutrons to the low-lying $ 1p1h $ proton excitations \[$
(f_{5/2},p_{3/2},p_{1/2})(f_{7/2})^{-1} $\] of the Ni core could be taken into account by increasing the neutron effective charge from 0.8 to about 1.1 for all of the Ni isotopes leading to $ B(E2)=$ 0.024, 0.047, 0.050, 0.046 and 0.038 e$^{2}$b$^{2}$. Thus, in analogy, the effective charge of $ e_{n}=1 $ used for $^{112-130}$Sn in Ref.[@ban05] for calculations with the $^{100}$Sn core takes into account both the low-lying and high-lying (2$\hbar\omega$) quadrupole vibrations.
But effective charge is not enough to account for the large increase in the $ B(E2\uparrow) $ value for light Ni isotopes in the full $ pf
$ model space compared to that obtained in the $
f_{5/2},p_{3/2},p_{1/2} $ model space. To better understand the full $ pf $ model-space result for $^{58}$Ni we need to consider the type of two-proton excitations leading to the $4p2h$ configuration shown schematically in Fig. \[smfig\] for $^{102}$Sn. The low-lying spectrum of $^{58}$Ni can be described by mixing of $2p$ and $4p2h$ configurations (relative to $^{56}$Ni) with a collective band corresponding to the predominantly $4p2h$ state starting at 3.5 MeV [@gx1]. This mixing leads to an enhanced $ B(E2) $ for the ground state. The excitation energies of the multi-hole states [@hor] and the $B(E2)$ values [@now] slowly converge to their full-space values as a function of the number of nucleons excited from the $
f_{7/2} $ orbital. The $ 4p2h $ state is low in energy due to the alpha-correlation energy in the $ 4p $ structure as well as the pairing energy in the $ 2h $ structure. The alpha-correlation energy is particularly large near $ N=Z $ when protons and neutrons are in the same orbital, and when the valence configuration is “open" in the sense that many two-particle couplings are allowed. As neutrons are added to $^{56}$Ni, the alpha-correlation energy drops as the $f_{5/2},p_{3/2},p_{1/2} $ neutron orbitals become filled (and hence less “open"). To complete the analogy with Sn, improved results in comparison to experiment for the middle of the Ni isotopes require the addition of the $ g_{9/2} $ orbit [@lit].
![\[smfig\] Schematic representation of proton 2p2h excitations across the Z=50 shell gap in $^{102}$Sn. The occupation of the same proton and neutron orbitals above the Z=N=50 shell leads to $\alpha$-like correlations between the valence nucleons. ](fig4.eps){width="5cm"}
Thus, by considering these results for the Ni isotopes we can qualitatively understand (1) the origin of the large neutron effective charge and (2) a proposed origin for further $ B(E2\uparrow) $ enhancement towards $^{100}$Sn due to $ 2p2h $ proton excitations. In analogy to the $ pf $ calculations, we expect the full $ sdg $ model space results to converge slowly [@now] as a function of the number of nucleons excited out of the $ g_{9/2} $ orbit making the exact calculation difficult.
The $2p2h$ excitations across the Z=50 shell gap and $\alpha$-like correlations discussed above also influence observables other than $B(E2\uparrow)$’s. The correlations are likely to impact the $\alpha$-decay rates for nuclei above $^{100}$Sn. Next, low-lying 0$^+$ states in the light Sn isotopes built predominantly on $2p2h$ proton excitations are expected to exist close to the ground state with collective bands built on top of them. Last, a smooth band termination [@afa99] is expected for these bands due to the limited valence space. All these can be addressed experimentally.
In summary, the measured nearly constant $B(E2, 0^+_1 \rightarrow
2^+_1)$ strength of $\sim$0.24 $e^2b^2$ in $^{106\mbox{--} 110}$Sn isotopes is in disagreement with the current state-of-the-art shell model predictions. This discrepancy could be explained if protons from within the Z = 50 shell contribute to the structure of low-energy excited states in this region. Such contributions are favored and stabilized by the $\alpha$-like correlations for protons and neutrons occupying the same shell model orbitals. This result indicates breaking of the Z = 50 and N=50 gaps near the doubly-magic $^{100}$Sn.
This work is supported by the US National Science Foundation Grants No. PHY01-10253, PHY-0555366 and NSF-MRI PHY-0619407. One author (C.A.) would like to acknowledge the support received from the National Science and Engineering Research Council of Canada, the Swedish Foundation for Higher Education and Research and the Swedish Research Council. The authors would like to acknowledge computational resources provided by the MSU High Performance Computing Center and Center of High Performance Scientific Computing, at Central Michigan University.
[99]{} S. Raman, C.W. Nestor Jr and P. Tikannen, At. Data Nucl. Data Tables 78 (2001) 1. R. Casten, [*Nuclear Structure from a Simple Perspective*]{}, Oxford University Press, 2001. D.C. Radford [*et. al*]{},Nucl. Phys. [**A752**]{} (2005) 264c. A Banu[*et. al*]{}, Phys. Rev. C[**72**]{}, (2005) 061305(R). A. Stolz [*et. al*]{}, Nucl. Instr. &Methods [**B241**]{} (2005) 858. D.J.Morrissey [*et. al*]{}, Nucl. Instr.& Meth.[**B204**]{}(2003) 90. W.F.Mueller [*et. al*]{}, Nucl. Instr. & Meth. [**A466**]{} (2001) 492. D. Bazin [*et. al*]{}, Nucl. Instr. & Meth. Phys. Res. [**B204**]{} (2003) 629. J. Yurkon [*et. al*]{}, Nucl. Instr. & Meth.[**A422**]{} (1999) 291. A. Winther and K. Alder, Nucl. Phys. [**A319**]{} (1979) 518. T. Glasmacher, Ann. Rev. Nucl. Part. Sci. 48 (1998) 1. J. M. Cook,T. Glasmacher, A. Gade, Phys. Rev. C[**73**]{}, (2006) 024315. A. Gade [*et. al*]{}, Phys. Rev. C[**68**]{}, (2003) 014302. C. Zhou, Nucl. Data Sheets 76, 399 (1995). A. Volya and V. Zelevinsky, Nucl. Phys. [**A752**]{} (2005) 325. A. Ansari, Phys. Lett. [**B623**]{} (2005)37. D. C. Dinca,[*et al.*]{}, Phys. Rev. C[**71**]{}, (2005) 041302(R). B. A. Brown, A. Arima and J. B. McGrory, Nucl. Phys. A[**277**]{}, (1977) 77 . M. Honma, T. Otsuka, B. A. Brown and T. Mizusaki, Phys. Rev. C[**65**]{}, (2002) 061301(R); Phys. Rev. C[**69**]{}, (2004) 034335. The three $ f_{5/2},p_{3/2},p_{1/2} $ single-particle energies are readjusted to reproduce the spectrum of $^{57}$Ni. For the $B(E2)$ We use harmonic-oscillator radial wavefunctions with $b=1.01 A^{1/6}$). K. L. Yurkewicz, et al., Phys. Rev. C[**70**]{}, (2004) 054319. F. Nowacki, Nucl. Phys. [**A704**]{}, (2002) 223c. M. Horoi, B. A. Brown, T. Otsuka, M. Honma, T. Mizusaki, Phys. Rev. C[**73**]{}, (2006) 061305(R). A. F. Lisetskiy, B. A. Brown, M. Horoi and H. Grawe, Phys. Rev. C[**70**]{}, (2004) 044314. A.V. Afanasjev, D.B. Fossan, G.J. Lane and I. Ragnarsson, Phys. Rep. 322 (1999) 1.
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---
abstract: 'Let T be the standard torus of revolution in ${\mathbb{R}}^{3} $ with radii $b$ and $1$. Let $\alpha$ be a $(p,q)$ torus curve on T. We show that there are points of zero curvature on $\alpha$ for only one value of the variable radius of T, $\displaystyle b = \frac{p^2}{p^2 + q^2} $. The curve $\alpha$ has non-vanishing curvature for all other values of $b$. Moreover, for this value of $b$, there are exactly $q$ points of zero curvature on ${\alpha}$.'
address: 'University of Georgia Mathematics Department, Athens, GA 30602'
author:
- 'Edgar J. Fuller, Jr.'
bibliography:
- 'masterbib.bib'
title: Torus Curves With Vanishing Curvature
---
Introduction
============
In [@Costa:1988], Costa studies closed curves in ${\mathbb{R}}^n$ by examining their first $n$ derivatives. She exhibits $(p,q)$ curves on tori of revolution in ${\mathbb{R}}^3$ whose torsion is non-vanishing. These are parametric curves whose first three derivatives are linearly independent. She characterizes those tori for a given $(p,q)$ for which this occurs by analyzing the dimension of the span of these first three derivatives and determining when it drops below the maximum possible value of three.
A similar phenomenon can be observed for the curvature of $(p,q)$ torus curves. The tori for a given $(p,q)$ for which the curvature vanishes are much more restricted, however. In particular, for a fixed $(p,q)$ there is only one such torus in the one parameter family of homothety classes of tori such that the $(p,q)$-curve containing has of zero curvature (Theorem $\ref{main}$). This parameter value lies at the boundary of the range for non-vanishing torsion found by Costa (see Remark $\ref{rem1}$).
Torus Curves in ${\mathbb{R}}^3$
================================
The Curvature of a Torus Curve
------------------------------
The theory of curves and surfaces in ${\mathbb{R}}^3$ is a broad subject with many different approaches. A comprehensive exploration (with extensions to higher dimensions) is in Spivak [@Spivak:1978]. Struik [@Struik:1961] is a good reference for a classical perspective, while do Carmo [@doCarmo:1977] and Millman and Parker [@MilPark:1977] contain more modern developments. For the purpose of this note, our attention will be restricted to the case of curves on a torus of revolution, with the whole apparatus sitting in ${\mathbb{R}}^3$ so that the tools of extrinsic as well as intrinsic geometry can be brought to bear.
Consider the standard torus $T \subset {\mathbb{R}}^3$ as a circle of radius $b$ revolved about the $z$-axis with the center of the circle a distance of $1$ unit from the axis.
A parametrization of $T$ may be taken to be $$x(u,v) = ( (1+b\cos(v))\cos(u), (1+b\cos(v))\sin(u), b\sin(v) )$$ and on the surface $T$, we may take a (unitarized) moving frame corresponding to this parametrization:
$x_{u} = ( \sin(u), \cos(u), 0 )$\
$x_{v} = ( \sin(v)\cos(u), \sin(v)\sin(u), \cos(v) )$\
$n = ( \cos(u)\cos(v), \sin(u)\cos(v), \sin(v) )$\
We say that $\alpha$ is a $(p,q)$ torus curve in $T$ if the curve wraps around $T$ $p$ times in the horizontal direction and $q$ times in the vertical direction. A $(p,q)$ torus curve is parametrized by
$ \alpha (t) = ( (1+b\cos(qt))\cos(pt), (1+b\cos(qt))\sin(pt), b\sin(qt) )$
with the corresponding moving frame along $\alpha$ (inherited from the above moving frame by sending $u$ to $pt$ and $v$ to $qt$ ) given by:
$x_{u}(t) = ( \sin(pt), \cos(pt), 0 )$\
$x_{v}(t) = ( \sin(qt)\cos(pt), \sin(qt)\sin(pt), \cos(qt) ) $\
$n(t) = ( \cos(pt)\cos(qt), \sin(pt)\cos(qt), sin(pt) )$\
The curvature of $\alpha$ can be computed in the standard way as $$\kappa = \displaystyle \left\| \frac{d^2\alpha}{ds^2}\right\| = \left\| \frac{d^2\alpha}{dt^2}(\frac{dt}{ds})^2 +
\frac{d\alpha}{dt}\frac{d^2t}{ds^2} \right\| \label{curv}.$$
Direct computation of the curvature is hindered by the lack of a unit speed parametrization. As a partial remedy, we use the geometry of the curve $\alpha$ as a subset of $T$ to decompose the curvature of $\alpha$ into its normal and tangential components; i.e the normal curvature of $\alpha$, $\kappa_{n}$, and the geodesic curvature of $\alpha$, $\kappa_{g}$, respectively. The resulting relation
$$\kappa^2 = {\kappa_{n}}^2 + {\kappa_{g}}^2 \label{kdcom}$$
links the vanishing of the curvature $\kappa$ to the quantities ${\kappa}_g$ and ${\kappa}_n$ which are more tractable computationally because of the identities $$\displaystyle \kappa_{n} = {\operatorname{II}}(\alpha^{\prime},
\alpha^{\prime})\left(\frac{dt}{ds}\right)^2\hskip .5in
\displaystyle \kappa_{g} = [n, \alpha^{\prime}, \alpha^{\prime \prime}]\left(\frac{dt}{ds}\right)^3\label{kg}\medskip$$ where prime denotes differentiation with respect to t, \[ , , \] denotes the scalar triple product and ${\operatorname{II}}( , )$ is the second fundamental form for the torus, thought of as a symmetric bilinear form whose symmetric $2 \times 2$ matrix representation along ${\alpha}$ is $$\begin{pmatrix} (1+b\cos(qt))\cos(qt)&0\\0&b\\ \end{pmatrix}.$$ See, for example, [@MilPark:1977] for derivations of the results in $\eqref{kg}$.
The Main Result
---------------
With the above established we may state the following
\[main\] Let $\alpha$ be a $(p,q)$ torus curve with $p,q\neq 0$ on a standard torus of revolution with radii $b$ and $1$, $0<b<1$. Then $\alpha$ has points of zero curvature for only one value of $b$, namely $b= \frac{p^2}{p^2 + q^2}$. Moreover, for this value of $b$ there are precisely $q$ points of vanishing curvature on $\alpha$, all lying on the innermost longitudinal circle of the torus.
First of all, consider that by $\eqref{kdcom}$, the curvature will vanish if and only if the geodesic curvature and the normal curvature vanish simultaneously. Computing the geodesic curvature of $\alpha$ using $\eqref{kg}$ yields $$\kappa_{g}=\left[n,\alpha^{\prime},\alpha^{\prime\prime}\right]\left(\frac{dt}{ds}\right)^3$$ $$\begin{gathered}
= \biggl[ \begin{pmatrix} \cos(pt)\cos(qt)\\ \sin(pt)\cos(qt)\\ \sin(qt) \\ \end{pmatrix}, \begin{pmatrix}-bq\sin(qt)\cos(pt)-p(1+b\cos(qt))\sin(pt)\\ -bq\sin(qt)\sin(pt)+p(1+b\cos(qt))\cos(pt) \\ bq\cos(qt) \\ \end{pmatrix}, \\ \begin{pmatrix}-bq^2\cos(qt)\cos(pt)+2bqp\sin(qt)\sin(pt)-p^2(1+b\cos(qt))\cos(pt) \\ -bq^2\cos(qt)\sin(pt)-2bqp\sin(qt)\cos(pt)-p^2(1+b\cos(qt))\sin(pt) \\ -bq^2\sin(qt) \\ \end{pmatrix}\biggr] \\
\left( \frac{1}{(p^2(1+b\cos(qt))^2+q^2b^2))^{\frac{3}{2}}} \right) \end{gathered}$$ $$= \displaystyle \frac{(p\sin(qt))(p^2(1+b\cos(qt))+2q^2b^2)}{(p^2(1+b\cos(qt))^2+q^2b^2)^{\frac{3}{2}}}\label{kg_result}.$$ Since the denominator and the second factor of the numerator of $\eqref{kg_result}$ are never zero, in order to have this expression vanish we must have $\sin(qt) = 0$. The domain of ${\alpha}$ is $[0,2\pi]$ so we conclude that $t$ must be one of the values $$t=\frac{k\pi}{q}, k= 0 \ldots 2q \label{tsolv}$$ giving us $2q$ points on ${\alpha}$ to investigate.
Now, in order for the curvature of $\alpha$ to be zero, the normal curvature of ${\alpha}$ must also be zero. We compute that $$\begin{split}
\kappa_{n} &= {\operatorname{II}}(\alpha^{\prime}, \alpha^{\prime})(\frac{dt}{ds})^2 \\
&= ^{\begin{pmatrix} p&q\\ \end{pmatrix}} \begin{pmatrix} (1+b\cos(qt))\cos(qt)&0\\0&b\\ \end{pmatrix} \begin{pmatrix} p\\q\\ \end{pmatrix}\left(\frac{dt}{ds}\right)^2 \\
&= \frac{((1+b\cos(qt))\cos(qt))p^2 +
bq^2}{(p^2(1+\cos(qt))^2+q^2b^2)}.
\end{split}$$ The denominator of this last line is stricly positive so that all the zeroes of the normal curvature can be found by setting the numerator to zero. This yields a quadratic in $\cos(qt)$: $$bp^2(\cos(qt))^2 + p^2\cos(qt) + bq^2 = 0\label{kn_result}$$ Note that since $b$, $p$, and $q$ are all positive, as long as $\cos(qt) > 0$, $\eqref{kn_result}$ has no solutions. As a result, the values of $t$ from with $k$ even cause the normal curvature to be strictly positive (these are the points on the outside rim of the torus). Since the geodesic curvature must vanish simultaneously with the normal curvature, we disregard these values of $t$. Accordingly, the only remaining solutions from $\eqref{tsolv}$ (where $k$ is odd) make $\cos(qt)=-1$ and so $\eqref{kn_result}$ becomes $$\begin{split}
0 &= p^2b - p^2 + bq^2\\
&= b(p^2 + q^2) - p^2
\end{split}$$ which implies that $$b = \displaystyle \frac{p^2}{p^2 + q^2}.\label{b_exp}$$ Since $\cos(qt)=-1$ is forced by the simultaneous conditions ${\kappa}_g={\kappa}_n=0$, this value of $b$ is the only one for which the given $(p,q)$ torus curve may have points with vanishing curvature. Moreover, there are $q$ such points for this value of $b$, namely $$\{{\alpha}(t) | t=\frac{k\pi}{q}, k=1,3,\ldots, 2q-1\}$$
To see that these are indeed points of vanishing curvature for this value of $b$, assume $\eqref{b_exp}$ holds and compute $\kappa_{n}$ and $\kappa_{g}$ directly.
\[rem1\] Costa’s result states that for $$\frac{p^2}{p^2+q^2}<b<\frac{q^2-p^2}{2q^2+p^2},$$ the torsion of the $(p,q)$ curve ${\alpha}$ is nowhere-vanishing. Outside this range, the span of the first three derivatives of ${\alpha}$ has less than maximal dimension at some point of ${\alpha}$. Applying Theorem $\ref{main}$, we see that in fact, outside the range of non-vanishing torsion, ${\alpha}$ has points of vanishing curvature only at the left hand endpoint of this range and ${\alpha}$ has points of vanishing torsion for all others. Since the two events are mutually exclusive, this describes the singular properties of regular $(p,q)$ torus curves completely.
Planar Projections
==================
The following examples suggest that points of vanishing curvature on a torus curve ${\alpha}$ correspond to higher inflection points of the projection of ${\alpha}$ to the $(x,y)$-plane.
As a basic example, take $(p,q) = (2,3)$. In this case the curve ${\alpha}$ is the trefoil knot and $b = \frac{4}{13}$.
$b = .1$
$b = \frac{4}{13}$
$b = .5$
For $p=1$ or $q=1$, $\alpha$ is unknotted. For instance, $(p,q)=(1,4)$ yields projections that illustrate the phenomenon quite well. In this case $b=\frac{1}{17}$ and we have: .1in
$b = .01$
$b = \frac{1}{17}$
$b = .3$
The striking correlation between the higher order inflections seen in the planar projections and the points of zero curvature for the space curves above suggests a more geometric description of such points.
As in [@Costa:1988], define the $m$-tangent space to the regular curve ${\alpha}$ in ${\mathbb{R}}^n$ at ${\alpha}(t)$, denoted $\mathrm{T}_m ({\alpha},t)$, to be the span of the first $m$ derivatives of the parametrization of ${\alpha}$ with respect to $t$, thought of as vectors in ${\mathbb{R}}^{n}$. In the case that $\dim(\mathrm{T}_2 ({\alpha}, t))=1$, ${\alpha}$ has a point of zero curvature. An order $k$ inflection of a curve in ${\mathbb{R}}^n$ occurs when the first $k+1$ tangent spaces have rank 1 and the $k+2$ and higher tangent spaces have rank at least $2$. For space curves, a point of zero curvature is an inflection of order at least one. What we would like to show is that points of zero curvature in a $(p,q)$ torus curve are equivalent to inflections of order at least $2$ in an appropriately chosen planar projection of the curve.
Define the projection $\pi:{\mathbb{R}}^3\rightarrow{\mathbb{R}}^2$ to be the map projecting onto the plane perpendicular to the axis vector $\vec{u}=(0,0,1)$. Let $\beta(t) = \pi({\alpha}(t))$ be the planar projection of ${\alpha}$. Then ${\beta}$ is regular, and for $m=1,2,3$, ${\mathrm{T}}_m({\beta},t)=\pi({\mathrm{T}}_m({\alpha},t))$.
Now there is an inflection of order at least two at ${\beta}(t)$ when $$\dim({\mathrm{T}}_3({\beta},t))=1.$$ For this to happen, $\dim({\mathrm{T}}_3({\alpha},t))\leq 2$ since the projection $\pi$ decreases the dimension of this space by at most $1$. This yields a set of three possibilities for ${\mathrm{T}}_3({\alpha},t)$ and ${\mathrm{T}}_2({\alpha},t)$:
\[dim\] $$\dim({\mathrm{T}}_3({\alpha},t))=2 \;\mathrm{and}\; \dim({\mathrm{T}}_2({\alpha},t))=2\label{dima}$$ $$\dim({\mathrm{T}}_3({\alpha},t))=2 \;\mathrm{and}\; \dim({\mathrm{T}}_2({\alpha},t))=1 \label{dimb}$$ $$\dim({\mathrm{T}}_3({\alpha},t))=1 \;\mathrm{and}\; \dim({\mathrm{T}}_2({\alpha},t))=1 \label{dimc}$$
It is worth noting that none of these dimensions can be less than one since both ${\alpha}$ and ${\beta}$ are regular. Of these, case $\eqref{dimc}$ is the easiest to analyze, since here Theorem $\ref{main}$ applies. Computing ${\alpha}^{\prime} \;\mathrm{and} \; {\alpha}^{\prime\prime\prime}$ in the case that $b=\frac{p^2}{p^2+q^2}$ and $\cos(qt)=-1$ proves that these vectors cannot be linearly dependent at any of the zero curvature points and so $\dim({\mathrm{T}}_3({\alpha},t))=2$, a contradiction.
The two remaining cases are harder to discriminate. The second case $\eqref{dimb}$ is precisely the situation in Theorem $\ref{main}$. At points of zero curvature, direct computation of $$\det\begin{pmatrix} 0&0&1\\ &{\alpha}^{\prime} \\ &{\alpha}^{\prime\prime\prime} \\ \end{pmatrix}$$ shows that $(0,0,1) \in {\mathrm{T}}_3({\alpha},t)$ and so $\pi$ reduces its dimension by one. As a result, $\dim({\mathrm{T}}_3({\beta},t))=1$ and so for a $(p,q)$ torus curve, points of zero curvature show up in the projection to the plane of rotation as inflections of order at least two of the image plane curve.
Surprisingly, this is the only way in which higher order inflections can show up in the planar projection of ${\alpha}$ along $(0,0,1)$. In order for ${\beta}$ to have a higher order inflection at $t$ we must at least have $\dim({\mathrm{T}}_2({\beta},t))=1$ with the order of the inflection being determined by the first $m$-tangent space with $m > 2$ whose dimension is 2. The dimension of ${\mathrm{T}}_2({\beta},t)$ is the rank of the matrix whose rows consist of ${\beta}^{\prime}$ and ${\beta}^{\prime\prime}$. As a result, we have an inflection of ${\beta}$ at $t$ whenever the determinant of this matrix vanishes. In light of the dimensional analysis in $\eqref{dim}$, we have only to check that this condition is mutually exclusive to the last remaining case, $\eqref{dima}$. Now, $\dim({\mathrm{T}}_3({\alpha},t))\geq 1$ so we can detect the case where its dimension is $2$ by computing $$\det \begin{pmatrix} {\alpha}^{\prime}\\ {\alpha}^{\prime\prime}\\
{\alpha}^{\prime\prime\prime}\\ \end{pmatrix}.$$ If this determinant is zero, we have that either the torsion vanishes or the curvature vanishes so that, in fact, solving the equations $$\label{conds}
\det \begin{pmatrix}{\beta}^{\prime}\\ {\beta}^{\prime\prime}\\ \end{pmatrix} = 0
\hskip .3in \det \begin{pmatrix} {\alpha}^{\prime}\\ {\alpha}^{\prime\prime}\\
{\alpha}^{\prime\prime\prime} \\
\end{pmatrix} = 0$$ simultaneously yields all zero torsion or zero curvature points of ${\alpha}$ whose planar projections are inflections of ${\beta}$. A direct computation shows that these two equations in the two unknowns $b$ and $\cos(qt)$ yield only one solution, which is exactly the value of $b$ and $\cos(qt)$ given in Theorem $\ref{main}$. It was established above that these inflections are of order at least two. Consequently, we have proved
The planar projection of the $(p,q)$ torus curve ${\alpha}$ to the plane orthogonal to the axis of revolution of the torus has a higher order inflection if and only if the corresponding point on ${\alpha}$ has zero curvature.
acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank my advisor, Dr. Clint McCrory, for his support and infinite patience during the writing of this article.
|
---
abstract: |
This work is devoted to averaging principle of a two-time-scale stochastic partial differential equation on a bounded interval $[0,
l]$, where both the fast and slow components are directly perturbed by additive noises. Under some regular conditions on drift coefficients, it is proved that the rate of weak convergence for the slow variable to the averaged dynamics is of order $1-\varepsilon$ for arbitrarily small $\varepsilon>0$. The proof is based on an asymptotic expansion of solutions to Kolmogorov equations associated with the multiple-time-scale system.
address:
- 'College of Mathematics and Computer Science, Wuhan Textile University, Wuhan, 430073, PR China'
- 'School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China'
author:
- Hongbo Fu
- Li Wan
- Jicheng Liu
- Xianming Liu
title: 'Weak order in averaging principle for two-time-scale stochastic partial differential equations '
---
Stochastic partial differential equation; Averaging principle; Invariant measure; Weak convergence; Asymptotic expansion.
MSC: primary 60H15, secondary 70K70
Introduction
============
In a previous paper [@Brehier], Bréhier exhibited the strong and weak order of an averaging principle for the following class of slow-fast stochastic reaction-diffusion equation on a bounded interval $D=[0, l]$ of $\mathbb{R}$: $$\begin{aligned}
\begin{cases}
\frac{\partial }{\partial
t}x^\epsilon_t(\xi)= \Delta x^\epsilon_t(\xi)
+F(x^\epsilon_t(\xi),y^\epsilon_t(\xi)), \;\xi\in D,\;t>0,\\
\frac{\partial }{\partial
t}y^\epsilon_t(\xi)=\frac{1}{\epsilon} \Delta y^\epsilon_t(\xi)
+\frac{1}{\epsilon}G(x^\epsilon_t(\xi),y^\epsilon_t(\xi))+\frac{1}{\sqrt{\epsilon}}
\dot{W}_t(\xi),\;\xi\in D,\;t>0,\\
x_0^\epsilon(\xi)=x(\xi),\;y_0^\epsilon(\xi)=y(\xi), \;\xi\in D, \\
x^\epsilon_t(0)=x^\epsilon_t(l)=0,\; t\geq0,\\
y^\epsilon_t(0)=y^\epsilon_t(l)=0,\; t\geq0,
\end{cases}\label{eqation-orignal-0}\end{aligned}$$ where the leading linear operator $\Delta=\frac{\partial^2}{\partial\xi^2}$ is the Laplacian operator. The ratio of time-scale separation is described by the positive and small parameter $\epsilon$. With this time scale, the process $x_t^\epsilon(\xi)$ is always called as the slow component and $y^\epsilon_t(\xi)$ as the fast component. The drift coefficients $F$ and $G$ are suitable mappings from $ L^2(D)$ to itself. The stochastic perturbation $W_t$ is an $L^2(D)$-valued Wiener process with respect to a filtered probability space $(\Omega, \mathscr{F},
\mathscr{F}_t, \mathbb{P})$.
In many applications, it is of interest to describe dynamics of the slow variable. Since exact solution is hard to be known, a simple equation without fast component, which can capture the essential dynamics of slow variable, is highly desirable. The fundamental method to approximate slow solution $x^\epsilon_t(\xi)$ to equation (1.1) is the so-called averaging procedure. Under some conditions, it has been proven that the slow solution $x^\epsilon_t(\xi)$ to problem converges (as $\epsilon$ tends to $0$), in a suitable sense, to solution of the so-called averaged equation, obtained by eliminating fast component via taking the average of the coefficient $F$ over the slow equation.
Once one has proved the validity of averaging principle, a critical question arises as to how do we determine the rate of convergence for this procedure. In Bréhier [@Brehier], it has been proved that the strong convergence (approximation in pathwise) order is $\frac{1}{2}-\varepsilon$ while the weak convergence (approximation in law) order is $1-\varepsilon$, for arbitrarily small $\varepsilon>0$, on condition that the slow motion equation is a deterministic parabolic equation. Due to the arbitrariness of $\varepsilon$ we may say that strong (resp. weak) convergence order is $\frac{1}{2}^-$ (resp. $1^-$). If an additive noise is included in the slow motion equation, the strong convergence order will decrease to $\frac{1}{5}$. In this case, unfortunately, the methods used to prove the weak order will be invalid. The main difficulty is due to the fact that tactics depend on the time derivative of solution to the averaged equation, which does not exist in any general way with the case where the slow motion equation is perturbed with a noise. In a more recent work following the procedure inspired by Bréhier [@Brehier], Dong et al. [@Dong] establish weak order $1^-$ in stochastic averaging for one dimensional Burgers equation only in the particular case of an additive noise on the fast component.
Unlike in the above-mentioned papers, where the noise acts only in the fast motion, in the present paper we study a class of stochastic partial differential equations on the bounded interval $D=[0, l]$ of $\mathbb{R}$, involving two separated time scales, which can be written as: $$\begin{aligned}
\begin{cases}
\frac{\partial }{\partial
t}x^\epsilon_t(\xi)= \Delta x^\epsilon_t(\xi)
+F(x^\epsilon_t(\xi),y^\epsilon_t(\xi))+\sigma_1
\dot{W}^1_t(\xi), \;\xi\in D,\;t>0,\\
\frac{\partial }{\partial
t}y^\epsilon_t(\xi)=\frac{1}{\epsilon} \Delta y^\epsilon_t(\xi)
+\frac{1}{\epsilon}G(x^\epsilon_t(\xi),y^\epsilon_t(\xi))+\frac{\sigma_2}{\sqrt{\epsilon}}
\dot{W}^2_t(\xi),\;\xi\in D,\;t>0,\\
x_0^\epsilon(\xi)=x(\xi),\;y_0^\epsilon(\xi)=y(\xi), \;\xi\in D, \\
x^\epsilon_t(0)=x^\epsilon_t(l)=0,\; t\geq0,\\
y^\epsilon_t(0)=y^\epsilon_t(l)=0,\; t\geq0.
\end{cases}\label{eqation-orignal}\end{aligned}$$ Assumptions on regularity of the drift coefficients $F$ and $G$ will be given below. The noises $W^1_t(\xi)$ and $W^2_t(\xi)$ are independent Wiener processes which will be detailed in next section. The coefficients of noise strength $\sigma_1$ and $\sigma_2$ are positive constants. The coupled stochastic partial differential equation in form of arises from many physical systems when random spatio-temporal forcing is taken into account, such as diffusive phenomena in media, epidemic propagation and transport process of chemical species.
So far, the explicit order for weak convergence in averaging has not be extended to the general situation when both the fast and slow components are directly perturbed with some noises. In the current article, we will show that the [weak order]{} $1^-$ can be achieved even when there is a noise in the slow motion equation. More precisely, we prove that for any $T>0$ and a class of test functions $\phi: L^2(D)\rightarrow \mathbb{R}$, with continuous and bounded derivatives up to the third order, $$\begin{aligned}
|\mathbb{E}\phi( {x}^\epsilon_T)-\mathbb{E}\phi(\bar{X}_T)|\leq
C\epsilon^{1-r}\label{error}\end{aligned}$$ for any $r\in(0,1)$, where $C$ is a constant independent of $\epsilon$ (see Theorem \[theorem\]). In the estimate above, the averaged motion $\bar{X}_t$ solves the equation $$\begin{aligned}
\begin{cases}
\frac{\partial }{\partial
t }\bar{X}_t(\xi)=\Delta \bar{X}_t(\xi)+ \bar{F}(\bar{X}_t(\xi))+\sigma_1\dot{W}_t^{1}(\xi),\;\;\xi\in D,\;t>0,\\
\bar{X}_0(\xi)=x(\xi), \;\xi\in D, \\
\bar{X}_t(0)=\bar{X}_t(l)=0, \; t\geq 0,
\end{cases}\end{aligned}$$ with an averaged drift $\bar{F}(x):=\int_{L^2(D)}F(x,y)\mu^x(dy)$, where $\mu^x$ is the unique mixing invariant measure for fast variable with frozen slow component (see equation ).
In order to prove , we adopt an asymptotic expansion scheme as in [@Brehier] to decompose $\mathbb{E}\phi({x}^\epsilon_t)$ with respect to the scale parameter $\epsilon$ in form $$\mathbb{E}\phi({x}^\epsilon_t)=u_0+\epsilon u_1+r^\epsilon,$$ where the functions $u_0$, $u_1$ and $r^\epsilon$ are determined recursively and obey certain linear evolutionary equations. First of all, we identify leading term $u_0$ as $\mathbb{E}\phi(\bar{X}_t)$ by a uniqueness argument. To this purpose, we introduce the Kolmogorov operators with parameter to construct an evolutionary equation that describes both $u_0$ and $\mathbb{E}\phi(\bar{X}_t)$. Moreover, this also allows us to characterize the expansion coefficient $u_1$ by a Poisson equation associated with the generator of fast process so that we obtain an explicit expression of $u_1$. As a result, some a priori estimates guarantee the boundedness of function $u_1$.
The next key step consists in estimate for the remainder term described by a linear equation depending on $\mathcal{L}_2u_1$ and $\frac{\partial u_1}{\partial t}$, where $\mathcal{L}_2 $ is the Kolmogorov operator for slow motion equation with frozen fast component. Due to the presence of unbounded operator $\mathcal{L}_2$, we have to reduce the problem to its Galerkin finite dimensional version. Since the noise is included in the slow motion equation, the Itô formula is employed to derive an explicit expression for $\frac{\partial u_1}{\partial t}$, which is related to the [third derivative of]{} $\phi$. This is the reason why we have to require the test function to be $3$-times differentiable. After bounding the terms $\mathcal{L}_2u_1$ and $\frac{\partial u_1}{\partial t}$, the remainder $r^\epsilon$ in the expansions can be estimated by standard evolution equation method and the weak error with an explicit order is achieved, where Itô’s formula is used again to overcome the non-integrability of $r^\epsilon$ at zero point. We would like to stress that, due to the regular conditions imposed on noise in slow component (see and ), the solution process to slow equation enjoys values in the domain of dominating linear operator. This allows estimates using techniques similar to those in Bréhier [@Brehier].
Up to now, to our knowledge, this is the first to obtain the weak convergence order for averaging of stochastic partial differential equations in the case of a noise acting directly on the slow motion equation. It is certainly believable that our method can be applied to stochastic Burgers equation with regular noise such that weak order $1^-$ in averaging is obtained. This will extend work of Dong et al. [@Dong], as we do not require the slow motion equation is deterministic.
Averaging method plays a prominent role in the study of qualitative behavior of dynamical systems with two time scales and has a long and rich history. Their rigorous mathematical justification was due to Bogoliubov [@Bogoliubov] for the deterministic dynamical system. Further developments to ordinary differential equations of the averaging theory can be found in Volosov [@Volosov], Besjes [@Besjes] and Gikhman [@Gikhman]. The averaging result for stochastic differential equations of Itô type was firstly introduced in Khasminskii [@Khas], being an extension of the theory to stochastic case. Since then, much progress has been made for multiple-time-scale stochastic dynamical systems in finite dimensions, see for instance [@Freidlin-Wentzell1; @Freidlin-Wentzell2; @Givon1; @Khas2; @Khas3; @Kifer1; @Kifer2; @Kifer3; @LiuDi; @Vere1; @Vere2; @Wainrib; @Weinan] and the references therein. In particular, averaging for finite dimensional stochastic systems with non-Gaussian noise may be found in [@Xu; @Xu2; @Xu3; @Xu4]. In a series of recent papers, Cerrai and Freidlin [@Cerrai1] and Cerrai [@Cerrai2] studied an infinite dimensional version of averaging principle for partial differential equations of reaction-diffusion type with additive and multiplicative Wiener noise, respectively, where global Lipschitz conditions were imposed. In contrast to Lipschitz setting, averaging principle for parabolic equations with polynomial growth coefficients was explored in Cerrai [@Cerrai-Siam]. For the extensions to stochastic parabolic equations with non-Gaussian stable noise, we are referred to Bao et al. [@Bao]. For related works on averaging for infinite dimensional stochastic dynamical systems we refer the reader to [@wangwei; @Fu-Liu; @Fu-Liu-2; @Pei; @Thompson; @Hogele].
The rest of the paper is arranged as follows. Section 2 is devoted to the general notation and framework. The ergodicity of fast process and the main result are introduced in Section 3. Then some a priori estimates is presented in Section 4. In Section 5, we present an asymptotic expansion scheme. In the final section, we state and prove technical lemmas applied in the preceding section.
Throughout the paper, the letter $C$ below with or without subscripts will denote positive constants whose value may change in different occasions. We will write the dependence of constant on parameters explicitly if it is essential.
Notations and preliminary results {#notations}
=================================
To rewrite the system as an abstract evolutionary equation, we present some notations and recall some well-known facts for later use.
For a fixed domain $D=[0, l]$, let $H$ be the real, separable Hilbert space $L^2(D)$, endowed with the usual scalar product $\Big(\cdot, \cdot\Big)_H$. The corresponding norm is denoted by $\|\cdot\|$. Denote by $\mathcal{L}(H)$ the Banach space of linear and bounded operators from $H$ to itself, equipped with usual operator norm.
Let $\{ e_k(\xi)\}_{k\geq 1}$ denote the complete orthonormal system of eigenfunctions in $H$ such that, for $k = 1,2,\ldots$, $$\label{eigenfunction} -\Delta
e_k(\xi)=\alpha_ke_k(\xi),\;\;e_k(0)=e_k(l)=0 ,$$ with $0<\alpha_1\leq\alpha_2\leq\cdots\alpha_k\leq\cdots$. We would like to recall the fact that $e_k(\xi)=\sqrt{\frac{2}{l}}\sin\frac{k\pi\xi}{l}$ and $\alpha_k=-\frac{k^2\pi^2}{l^2}$ for $k = 1,2,\cdots$.
Let $A$ be the Laplacian operator $\Delta$ satisfying zero Dirichlet boundary condition, with domain $\mathscr{D}(A)=H^{ 1}_0(D)\cap
H^2(D)$, which generates a strongly continuous semigroup $\{S_t\}_{t\geq 0}$ on $H$, defined by, for any $h\in H$, $$\begin{aligned}
S_th=\sum\limits_{k\in \mathbb{N}} e^{-\alpha_kt}e_k\Big(e_k,
h\Big)_H.\end{aligned}$$ Here, for the sake of brevity, we omit to write the dependence of the spatial variable $\xi$. It is straightforward to check that $\{S_t\}_{t\geq0}$ is a contractive semigroup on $H$. For $\gamma\in [0,1]$ we defined the operator $(-A)^\gamma$ by $$\begin{aligned}
(-A)^\gamma x=\sum\limits_{k\in \mathbb{N}}\alpha_k^\gamma x_ke_k\in
H\end{aligned}$$ with domain $$\begin{aligned}
\mathscr{D}((-A)^\gamma)=\left\{x=\sum\limits_{k\in\mathbb{N}}x_ke_k\in H; \;
\|x\|^2_{(-A)^\gamma}:=\sum\limits_{k\in\mathbb{N}}\alpha_k^{2\gamma}x_k^2<\infty
\right\}.
\end{aligned}$$ Using the spectral decomposition of $A$, the semigroup $\{S_t\}_{t\geq 0}$ enjoys the following smooth property.
\[proposition\] For any $\gamma\in [0, 1]$ there exists a constant $C_\gamma>0$ such that $$\begin{aligned}
&&\!\!\!\!\!\!\|S_tx\|_{(-A)^\gamma}\leq C_\gamma
t^{-\gamma}e^{-\frac{\alpha_1}{2}t}\|x\|, \;t>0, x\in H,\label{propo-1}\\
&&\!\!\!\!\!\!\|S_tx-S_\tau x\|\leq C_\gamma
\frac{|t-\tau|^\gamma}{\tau^\gamma}e^{-\frac{\alpha_1}{2}\tau}\|x\|,\;t,
\tau>0,x \in H,\label{propo-2}\\
&&\!\!\!\!\!\!\|S_tx-S_\tau x\|\leq C_\gamma |t-\tau|^\gamma
e^{-\frac{\alpha_1}{2}\tau}\|x\|_{(-A)^\gamma},\;t,\tau>0, x\in
\mathscr{D}((-A)^\gamma).\label{propo-3}\end{aligned}$$
For the perturbation noises we suppose the following setting. For $i=1,2$, let $W_t^i$ be the Wiener process on a stochastic base $(\Omega, \mathscr{F}, \mathscr{F}_t, \mathbb{P})$ with a bounded covariance operator $Q_i: H\rightarrow H$ defined by $Q_ie_k=\lambda_{i,k}e_k$, where $\{\lambda_{i,k}\}_{k\in
\mathbb{N}}$ are nonnegative and $\{e_k\}_{k\in \mathbb{N}}$ is the complete orthonormal basis in $H$. Formally, for $i=1,2,$ Wiener processes $W^i_t$ can be written as the infinite sums (cf. Da Prato and Zabczyk [@Daprato]) $$\begin{aligned}
W_t^i=\sum\limits_{k\in\mathbb{N}}\sqrt{\lambda_{i,k}}B^{(i)}_{t,k}e_k,\end{aligned}$$ where $\{B^{(i)}_{t,k}\}_{k\in \mathbb{N}}$ are mutually independent real-valued Brownian motions on stochastic base $(\Omega,
\mathscr{F}, \mathscr{F}_t, \mathbb{P})$. For the sake of simplicity we prefer to assume that both $Q_1$ and $Q_2$ have finite trace, that is there exists a positive constant $C$ such that $$\begin{aligned}
Tr(Q_i)=\sum\limits_{k\in \mathbb{N}} {\lambda_{i,k}}\leq C,\;i=1,2.
\label{Trace}\end{aligned}$$ Moreover, we also assume $$\begin{aligned}
Tr\big((-A)Q_1\big)=\sum\limits_{k\in\mathbb{N}} \lambda_{1,k}
\alpha_k\leq C.\label{Tr-AQ}\end{aligned}$$ Concerning the drift coefficients $F$ and $G$ we shall impose the following conditions.\
(H.1) For each fixed $u\in H$, the mapping $F(u,\cdot):H\rightarrow
H$ is of a class $\mathcal {C}^3$, with bounded derivatives, uniformly with respect $u\in H$. Also suppose that for any $v\in H$, the mapping $F(\cdot,v): H\rightarrow H$ is of class $\mathcal
{C}^3$, with bounded derivatives, uniformly for $v\in H$.\
(H.2) For each fixed $u\in H$, the mapping $G(u,\cdot):H\rightarrow
H$ is of a class $\mathcal{C}^2$, with bounded derivatives, uniformly with respect $u\in H$. Also suppose that for any $v\in H$, the mapping $G(\cdot,v): H\rightarrow H$ is of class $\mathcal{C}^2$, with bounded derivatives, uniformly with respect $v\in H$. Moreover, we assume that $$\begin{aligned}
\sup\limits_{u\in H}\|G_v'(u,v)\|_{\mathcal{L}(H)}:=L_g<\alpha_1,\end{aligned}$$ where $G_v'$ denotes the derivative with respect to $v$ and $\|\cdot\|_{\mathcal{L}(H)}$ denotes the operator norm on $\mathcal{L}(H)$.
Under (H.1) and (H.1), it is not difficult to verify that there exist positive constants $K_F$ and $K_G$ such that $$\begin{aligned}
\|F(u_1,v_1)-F(u_2,v_2)\|\leq
K_F(\|u_1-u_2\|+\|v_1-v_2\|),\;u_1,u_2, v_1,v_2\in H,
\label{F-condi-1}\end{aligned}$$ and $$\begin{aligned}
\|G(u_1,v_1)-G(u_2,v_2)\|\leq
K_G(\|u_1-u_2\|+\|v_1-v_2\|),\;u_1,u_2, v_1,v_2\in H,\label{g-condi}\end{aligned}$$ which means $F, G: H\times H\rightarrow H$ are Lipschitz continuous.
Once introduced the main notations, system can be written as $$\begin{aligned}
\begin{cases}
dX_t^\epsilon=AX^\epsilon_tdt+F(X^\epsilon_t,Y^\epsilon_t)dt+\sigma_1dW_t^1,\;X_0^\epsilon=x, \\
dY_t^\epsilon=\frac{1}{\epsilon}AY^\epsilon_tdt+
\frac{1}{\epsilon}G(X^\epsilon_t,Y^\epsilon_t)dt+\frac{\sigma_2}{\sqrt{\epsilon}}dW_t^2,\;Y_0^\epsilon=y.
\end{cases}\label{abstr-equation}\end{aligned}$$ By virtue of conditions and , it is easy to check that system admits a unique mild solution, which, in order to emphasize the dependence on the initial data, is denoted by $(X_t^\epsilon(x,y),Y_t^\epsilon(x,y))$. This means that for any $t>0$, it holds $\mathbb{P}-a.s.$ that $$\begin{aligned}
\begin{cases}
X_t^\epsilon(x,y)=S_tx+\int_0^tS_{t-s}F(X^\epsilon_s(x,y),Y^\epsilon_s(x,y))ds
+\sigma_1\int_0^tS_{t-s}dW_s^1,\\
Y_t^\epsilon(x,y)=S_{\frac{t}{\epsilon}}y+\frac{1}{\epsilon}\int_0^tS_\frac{t-s}{\epsilon}G(X^\epsilon_s(x,y),Y^\epsilon_s(x,y))ds
+\frac{\sigma_2}{\sqrt{\epsilon}}\int_0^tS_\frac{t-s}{\epsilon}dW_s^2.\label{equation-mild}
\end{cases}\end{aligned}$$
Moreover, by using standard arguments, we have the following lemma.
\[moment-bound\] Under (H.1) and (H.2), for any $T>0$ and $x, y\in H $, there exists a positive constant $C_T$ such that for any $x, y\in H$ and $\epsilon\in (0,1]$, $$\begin{aligned}
&&\sup\limits_{t\in[0, T]}\mathbb{E}\|X^\epsilon_t(x,y)\|^2\leq
C_T(1+\|x\|^2+\|y\|^2),\\
&&\sup\limits_{t\in[0, T]}\mathbb{E}\|Y^\epsilon_t(x,y)\|^2\leq
C_T(1+\|x\|^2+\|y\|^2).\end{aligned}$$
To study weak convergence, we need to introduce some notations in connection with the test function. If $\mathcal {X}$ is a Hilbert space equipped with inner product $(\cdot,\cdot)_\mathcal {X}$, we denote by $\mathcal{C}^k(\mathcal {X},\mathbb{R})$ the space of all $k-$times continuously Fréchet differentiable functions $\phi:\mathcal {X}\rightarrow \mathbb{R}, x\mapsto \phi(x)$. By $\mathcal{C}_b^k(\mathcal {X},\mathbb{R})$ we denote the subspace of functions from $\mathcal{C}^k(\mathcal {X},\mathbb{R})$ which are bounded together with their derivatives. For $\phi\in
\mathcal{C}^m(\mathcal {X},\mathbb{R})$, we use the notation $D^m_{\underbrace{xx\cdots x}_{m -{times}}}\phi(x)$ for its $m$-th derivative at the point $x$.
Thanks to Riesz representation isomorphism, we may get the identity for $x,h\in \mathcal {X}$: $$D_x\phi(x)\cdot h=(D_x\phi(x), h)_\mathcal {X}.$$ For $\phi\in \mathcal{C}^2(\mathcal {X}, \mathbb{R})$, we will identify the second derivative $D^2_{xx}\phi(x)$ with a bilinear operator from $\mathcal {X}\times \mathcal {X}$ to $\mathbb{R}$ such that $$D^2_{xx}\phi(x)\cdot (h,k)=(D^2_{xx}\phi(x)h,k)_\mathcal {X},\;\; x,h,k\in \mathcal {X}.$$ On some occasions, we also use the notations $\phi',\phi''$ and $\phi'''$ instead of $D_{x}\phi$, $D_{xx}^2\phi$ and $D_{xxx}^3\phi$, respectively.
Ergodicity of $Y_t^x$ and averaging dynamics
============================================
For fixed $x\in H$ consider the problem associate to fast motion with frozen slow component $$\begin{aligned}
\begin{cases}
dY_t^x=AY_t^xdt+G(x, Y_t^x)dt+\sigma_2dW^2_t,\\
Y_0^x=y.\label{frozen}
\end{cases}\end{aligned}$$ Notice that the drift $G: H\times H\rightarrow H $ is Lipshcitz continuous. By arguing as before, for any fixed slow component $x\in
H$ and any initial data $y\in H$, problem has a unique mild solution denoted by $Y_t^{x}(y)$. Now, we consider the transition semigroup $P_t^x$ associated with process $Y_t^x(y)$, by setting for any $\psi \in \mathcal {B}_b(H)$ the space of bounded functions on $H$, $$P_t^x\psi(y)=\mathbb{E}\psi(Y_t^x(y)).$$ By adopting a similar approach used in [@Fu-Liu], we can show that $$\label{Fast-motion-energy-bound}
\mathbb{E}\|Y^x_t(y)\|^2\leq
C\left(e^{-(\alpha_1-L_g)t}\|y\|^2+\|x\|+1\right),\; t>0$$ for some constant $C>0$. This implies the existence of an invariant measure $\mu^x $ for the Markov semigroup $P^x_t$ associated with system on $H$ such that $$\int_HP^x_t \psi d\mu =\int_H\psi d\mu^x , \quad
t\geq 0$$ for any $\psi \in \mathcal {B}_b(H)$ (for a proof, see, e.g., [@Cerrai2], Section 2.1). We recall that in [@Cerrai-Siam] it is proved the invariant measure has finite $2-$moments: $$\label{mu-Momenent-bound}
\int_H\|y\|^2\mu^x(dy)\leq C(1+\|x\|^2).$$ Let $Y_t^x(y')$ be the solution of problem with initial value $Y_0=y'$, it is not difficult to show that for any $t\geq0$, $$\begin{aligned}
\label{initial-diff}
\mathbb{E}\|Y_t^x(y)-Y_t^x( y')\|^2\leq C\|y-y'\|^2e^{-\beta t}\end{aligned}$$ with $\beta=(\alpha_1-L_g)>0,$ which implies that $\mu^x$ is the unique invariant measure for $P^x_t$. This allows us to define an $H$-valued mapping $\bar{F}$ by averaging the coefficient $F$ with respect to the invariant measure $\mu^x$, that is, $$\bar{F}(x):=\int_HF(x,y)\mu^x(dy), x\in H, \label{aver-F}$$ and then, by using the condition , it is immediate to check that $$\begin{aligned}
\|\bar{F}(x_1)-\bar{F}(x_2)\|\leq K_F\|x_1-x_2\|, \; x_1, x_2\in H.
\label{barF-lip}\end{aligned}$$ According to the invariant property of $\mu^x$, and , we have $$\begin{aligned}
\nonumber \|\mathbb{E}F(x, Y_t^{x}( y))-\bar{F}(x) \|^2 &=&
\|\int_{H}\mathbb{E}\big(F(x, Y_t^{x}
(y)-F(x, Y_t^{x}(z)\big)\mu^x(dz) \|^2 \\
\nonumber&\leq&\int_{H}\mathbb{E}\left\|Y_t^{x}
(y)-Y_t^{x}(z)\right\|^2 \mu^x(dz)\\
\nonumber&\leq&e^{-\beta t}\int_{H}\|y-z\|^2 \mu^x(dz)\\
&\leq&Ce^{-\beta t}\big(1+\|x\|^2 +\|y\|^2
\big),\label{Averaging-Expectation}\end{aligned}$$ which means that $$\begin{aligned}
\bar{F}(x)=\lim\limits_{t\rightarrow +\infty}\mathbb{E}F(x,
Y^x_t(y)),\;x\in H. \label{bar-F-lim}\end{aligned}$$ Using this limit and Assumptions (H.1), it is possible to show that $$\begin{aligned}
\label{bar-F-derivative}
\|\bar{F}'(x)\cdot h\|\leq C\|h\|,\; x, h\in H.\end{aligned}$$ Now we introduce the effective dynamics: $$\begin{aligned}
\begin{cases}
\frac{\partial }{\partial
t }\bar{X}_t(\xi)=\Delta \bar{X}_t(\xi)+ \bar{F}(\bar{X}_t(\xi))+\sigma_1\dot{W}_t^{1}(\xi),\;\xi \in D, t> 0,\\
\bar{X}_t(0)=\bar{X}_t(l)=0, \;t\geq 0, \\
\bar{X}_0(\xi)=x(\xi), \;\xi\in D.
\end{cases}\end{aligned}$$ By using the notations introduced in Section \[notations\] it can be written as an abstract evolutionary equation in form $$\begin{aligned}
\label{Averaging-equation}
\begin{cases}
d\bar{X}_t=A\bar{X}_tdt+\bar{F}(\bar{X}_t)dt+\sigma_1dW^1_t,\;t>0,\\
\bar{X}_0=x.
\end{cases}\end{aligned}$$ For any initial datum $x\in H$, the equation admits a unique mild solution, which means that there exists a unique adapt process $\bar{X}_t(x)$ such that $$\begin{aligned}
\bar{X}_t(x)=S_tx+\int_0^tS_{t-s}\bar{F}(\bar{X}_s(x))ds+\sigma_1\int_0^tS_{t-s}dW^1_s,\;\mathbb{P}-a.s.,\;
t\geq0.\end{aligned}$$ Moreover, for any $T>0$ it can be easily proved that $$\begin{aligned}
\mathbb{E}\|\bar{X}_t(x)\|^2\leq C_T(1+\|x\|^2),\; t\in [0,
T].\label{bar-x-moment}\end{aligned}$$ Thanks to averaging principle (see Cerrai [@Cerrai2] for details), we have that the limit $$\begin{aligned}
\lim\limits_{\epsilon\rightarrow 0+}\sup\limits_{0\leq t\leq
T}\mathbb{E}\|\bar{X}_t(x)-{X}^{\epsilon
}_t(x,y)\|^2=0\label{aver-infinite}\end{aligned}$$ for any fixed $T>0.$ This means that the slow process $X^\epsilon_t(x,y)$ enjoys strong convergence to the averaging process $\bar{X}_t(x)$. Moreover, the strong order in averaging is $\frac{1}{5}^-$ (Bréhier [@Brehier]). The weak convergence using test functions is obvious. Our aim is to establish rigorously weak error bounds for the limit of slow process with respect to scale parameter $\epsilon$. The main result of this paper is the following, whose proof is postponed in the end of Section \[asym\].
\[theorem\] Assume that $x\in \mathscr{D}((-A)^\theta)$ for some $\theta\in (0,
1]$ and $y \in H.$ Then, under (H.1) and (H.2), for any $r\in (0, 1)$, $T>0$ and $\phi\in C_b^3(H, \mathbb{R})$, there exists a constant $C_{\theta,r,T,\phi,x,y}$ such that $$\begin{aligned}
\left|\mathbb{E}\phi(X^\epsilon_T(x,y))-\mathbb{E}\phi(\bar{X}_T(x))\right|\leq
C_{\theta,r,T,\phi,x,y}\epsilon^{1-r}.
\end{aligned}$$
Some a priori estimates
=======================
Before proving the main results, we need to state some technical lemmas used in subsequent section.
\[Xt-Xs\] Let the conditions (H.1) and (H.2) be satisfied and fix $x,y\in
H$ and $T>0$. Then for any $r\in (0,\frac{1}{2})$ there exists a constant $C_{r,T}>0$ such that for any $0< s\leq t\leq T$, we have $$\begin{aligned}
&&(\mathbb{E}\|X_t^\epsilon(x,y)-X^\epsilon_s(x,y)\|^2)^{\frac{1}{2}}\\
&&\leq
C_{r,T}\frac{|t-s|^{1-r}} {s^{1-r}}\|x\|\\
&&+ C_{r,T}(|t-s|^{\frac{1}{2}}+|t-s|^{1-r}+|t-s|^r)(1+\|x\|+\|y\|).\end{aligned}$$
We can write $$\begin{aligned}
X_t^\epsilon(x,y)-X_s^\epsilon(x,y)&=&(S_t -S_s)x+\int_s^tS_{t-\tau}F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\nonumber\\
&&+\int_0^s(S_{t-\tau}-S_{s-\tau})F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\nonumber\\
&&+\int_s^tS_{t-\tau}dW^1_\tau+\int_0^s(S_{t-\tau}-S_{s-\tau})dW^1_\tau.\label{Xt-Xs-0}\end{aligned}$$ In the next, we estimate separately the different terms in . By using , for the first term we have $$\begin{aligned}
\|(S_t-S_s)x\|\leq C_r\frac{|t-s|^{1-r}} {s^{1-r}}\|x\|.\label{Xt-Xs-1}
\end{aligned}$$ For the second term, by Lemma \[moment-bound\] and Hölder’s inequality, we obtain $$\begin{aligned}
&&\mathbb{E}\|\int_s^tS_{t-\tau}F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\|^2\nonumber\\
&&\leq
|t-s|\int_s^t\mathbb{E}\|S_{t-\tau}F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))\|^2d\tau\nonumber\\
&&\leq C|t-s|\int_s^t\mathbb{E}
(1+\|X_\tau^\epsilon(x,y)\|^2+\|Y^\epsilon_\tau(x,y)\|^2)d\tau\nonumber\\
&&\leq C_T|t-s|(1+\|x\|^2+\|y\|^2).\label{Xt-Xs-2}\end{aligned}$$ Concerning the third term, by using , we can deduce that $$\begin{aligned}
&&\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\|^2\\
&&\leq\mathbb{E}\left[\int_0^s\|(S_{t-\tau}-S_{s-\tau})F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))\|d\tau\right]^2\\
&&\leq C_r\mathbb{E}\left[\int_0^s\frac{(t-s)^{1-r}}{(s-\tau)^{1-r}}
{e^{-\frac{\alpha_1}{2}(s-\tau)}}\|F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))\|d\tau\right]^2.\\\end{aligned}$$ In view of Lemma \[moment-bound\] and Minkowski inequality, we get $$\begin{aligned}
&&\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\|^2\nonumber\\
&&\leq
C_r|t-s|^{2(1-r)}\left[\int_0^s\frac{e^{-\frac{\alpha_1}{2}(s-\tau)}}{(s-\tau)^{1-r}}
\big(\mathbb{E}\|F(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))\|^2\big)^{\frac{1}{2}}d\tau\right]^2\nonumber\\
&&\leq
C_r|t-s|^{2(1-r)}\left[\int_0^s\frac{e^{-\frac{\alpha_1}{2}(s-\tau)}}{(s-\tau)^{1-r}}
\mathbb{E}\big(1+\| X_\tau^\epsilon(x,y)\|
+\|Y^\epsilon_\tau(x,y)\|\big)
d\tau\right]^2\nonumber\\
&&\leq C_{r,T}|t-s|^{2(1-r)}(1+\|x\|^2+\|y\|^2),\label{Xt-Xs-3}\end{aligned}$$ here we have used fact $\int_0^{+\infty}\frac{e^{-\frac{\alpha_1}{2}s}}{s^{1-r}}ds<+\infty$ in the last step. For the forth term, we directly have $$\begin{aligned}
\mathbb{E}\|\int_s^tS_{t-\tau}dW^1_\tau\|^2&=&\sum\limits_{k=1}^\infty\lambda_{1,k}\int_s^t\|S_{t-\tau}e_k\|^2d\tau\nonumber\\
&\leq&Tr(Q_1)|t-s|.\label{Xt-Xs-4}\end{aligned}$$ The final term on the right-hand side of the can be treated as follows: $$\begin{aligned}
\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})dW^1_\tau\|^2&=&\sum\limits_{k=1}^\infty\lambda_{1,k}\int_0^s\|(S_{t-\tau}-S_{s-\tau})e_k\|^2d\tau\\
&=&\sum\limits_{k=1}^\infty\lambda_{1,k}\int_0^s\|\int_{s-\tau}^{t-\tau}AS_{\rho}e_kd\rho\|^2d\tau\\
&\leq&
C\sum\limits_{k=1}^\infty\lambda_{1,k}\int_0^s|\int_{s-\tau}^{t-\tau}\frac{1}{\rho}d\rho|^2d\tau,\end{aligned}$$ here the last inequality following from fact $\|AS_t\|_{\mathcal
{L}(H)}\leq Ct^{-1}$ for $t>0$. Note that for any $r\in (0,
\frac{1}{2})$, it holds $$\begin{aligned}
\int_0^s\left|\int_{s-\tau}^{t-\tau}\frac{1}{\rho}d\rho\right|^2d\tau&\leq&
\int_0^s(s-\tau)^{-2r}\left|\int_{s-\tau}^{t-\tau}\frac{1}{\rho^{1-r}}d\rho\right|^2d\tau\\
&=& r^{-2}\int_0^s(s-\tau)^{-2r}[(t-\tau)^r-(s-\tau)^r]^2d\tau\\
&\leq& r^{-2}\int_0^s(s-\tau)^{-2r}(t-s)^{2r}d\tau\\
&\leq&r^{-2}|t-s|^{2r}\frac{1}{1-2r}s^{1-2r}\\
&\leq&C_{r}|t-s|^{2r}T^{1-2r},\end{aligned}$$ which implies that $$\begin{aligned}
\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})dW^1_\tau\|^2\leq
C_{r,T}|t-s|^{ 2r}. \label{Xt-Xs-5}\end{aligned}$$ By taking - into account, we can deduce that$$\begin{aligned}
&&(\mathbb{E}\|X_t^\epsilon(x,y)-X^\epsilon_s(x,y)\|^2)^{\frac{1}{2}}\\
&&\leq
C_{r,T}\frac{|t-s|^{1-r}} {s^{1-r}}\|x\|\\
&&+ C_{r,T}(|t-s|^{\frac{1}{2}}+|t-s|^{1-r}+|t-s|^r)(1+\|x\|+\|y\|).\end{aligned}$$
\[Yt-Ys\] Let the conditions (H.1) and (H.2) be satisfied and fix $x,y\in
H$ and $T>0$. Then for any $r\in (0,\frac{1}{4})$ there exists a constant $C_{r, T}>0$ such that for any $0< s\leq t\leq T$, we have $$\begin{aligned}
(\mathbb{E}\|Y_t^\epsilon(x,y)-Y_s^\epsilon(x,y)\|^2)^{\frac{1}{2}}\leq
C_{r,T}(1+\|x\|+\|y\|)\left[\frac{|t-s|^{r}}{s^{r}}+\frac{|t-s|^{r}}{\epsilon^{r}}\right].\end{aligned}$$
We have the decomposition $$\begin{aligned}
Y_t^\epsilon(x,y)-Y_s^\epsilon(x,y)&=&[S_{\frac{t}{\epsilon}}y-S_{\frac{s}{\epsilon}}y]+
\frac{1}{\epsilon}\int_s^tS_{\frac{t-\tau}{\epsilon}}G(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\nonumber\\
&&+\frac{1}{\epsilon}\int_0^s(S_{\frac{t-\tau}{\epsilon}}-S_{\frac{s-\tau}{\epsilon}})G(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))d\tau\nonumber\\
&&+\frac{1}{\sqrt{\epsilon}}\int_s^tS_{\frac{t-\tau}{\epsilon}}
dW_\tau^2+\frac{1}{\sqrt{\epsilon}}\int_0^s(S_{\frac{t-\tau}{\epsilon}}-S_{\frac{s-\tau}{\epsilon}})
dW_\tau^2\nonumber\\
&:=&\sum\limits_{k=1}^5J^\epsilon_k(t,s).\label{Yt-Ys-0}\end{aligned}$$ By , it is immediate to check that $$\begin{aligned}
\|J^\epsilon_1(t,s)\|\leq
C_r\frac{|t-s|^r}{s^r}\|y\|.\label{Yt-Ys-1}\end{aligned}$$ By Minkowski inequality and Lemma \[moment-bound\], one can estimate $J^\epsilon_2(t,s)$ as follows: $$\begin{aligned}
\mathbb{E}\|J^\epsilon_2(t,s)\|^2&\leq&
\mathbb{E}\left(\frac{C}{\epsilon}\int_s^te^{-\alpha_1\frac{t-\tau}{\epsilon}}(1+
\|X_\tau^\epsilon(x,y)\|
+ \|Y^\epsilon_\tau(x,y)\|)d\tau\right)^2\nonumber\\
&\leq& C\mathbb{E}\left(\int_0^\frac{t-s}{\epsilon}e^{-\alpha_1\tau}
(1+\|X_{t-\epsilon\tau}^\epsilon\|+\|Y_{t-\epsilon\tau}^\epsilon\|)d\tau\right)^2\nonumber\\
&\leq& C\left(\int_0^\frac{t-s}{\epsilon}e^{-\alpha_1\tau}
\Big(\mathbb{E}(1+\|X_{t-\epsilon\tau}^\epsilon(x,y)\|+\|Y_{t-\epsilon\tau}^\epsilon(x,y)\|)^2\Big)^\frac{1}{2}d\tau\right)^2\nonumber\\
&=&C_T(1+\|x\|^2+\|y\|^2)(1-e^{-\alpha_1\frac{t-s}{\epsilon}})^2\nonumber\\
&\leq&C_{r, T}(1+\|x\|^2+\|y\|^2)\frac{|t-s|^{2r}}{\epsilon^{2r}},
\label{Yt-Ys-2}\end{aligned}$$ where, the last step is due to the inequality $1-e^{-a}\leq C_ra^r$ for $a>0, r\in (0, 1).$ Concerning $J^\epsilon_3(t,s)$, according to , Lemma \[moment-bound\] and Minkowski inequality, we get $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\mathbb{E}\|J^\epsilon_3(t,s)\|^2\nonumber\\
&\leq&\mathbb{E}\left(\frac{1}{\epsilon}\int_0^s\|S_{\frac{t-\tau}{\epsilon}}-S_{\frac{s-\tau}{\epsilon}}\|_{\mathcal{L}(H)}
\|G(X_\tau^\epsilon(x,y),Y^\epsilon_\tau(x,y))\|d\tau\right)^2\nonumber\\
&\leq&\mathbb{E}\left(\frac{C_r}{\epsilon}\int_0^s\frac{(t-s)^r}{(s-\tau)^r}e^\frac{-\alpha_1(s-\tau)}{2\epsilon}
(1+ \|X_\tau^\epsilon(x,y)\|+ \|Y_\tau^\epsilon(x,y)\|)d\tau\right)^2\nonumber\\
&\leq& C_r|t-s|^{2r} \left(\frac{1}{\epsilon}\int_0^s
\frac{e^\frac{-\alpha_1(s-\tau)}{2\epsilon}}{(s-\tau)^r}
\big(\mathbb{E}(1+ \|X_\tau^\epsilon(x,y)\|+ \|Y_\tau^\epsilon(x,y)\|)^2 \big)^{\frac{1}{2}} d\tau \right)^2 \nonumber\\
&\leq&C_r|t-s|^{2r}\frac{1}{\epsilon^{2r}}(1+\|x\|^2+\|y\|^2)\left(\int_0^{s/\epsilon}\frac{e^{-\frac{\alpha_1}{2}\tau}}{\tau^r}
d\tau\right)^2\nonumber\\
&\leq&C_r\frac{|t-s|^{2r}}{\epsilon^{2r}}(1+\|x\|^2+\|y\|^2)\left(\int_0^{+\infty}\frac{1}{\tau^r}e^{-\frac{\alpha_1}{2}\tau}d\tau\right)^2\nonumber\\
&\leq&C_r\frac{|t-s|^{2r}}{\epsilon^{2r}}(1+\|x\|^2+\|y\|^2).
\label{Yt-Ys-3}\end{aligned}$$ For $J^\epsilon_4(t,s)$ we have $$\begin{aligned}
\mathbb{E}\|J^\epsilon_4(t,s)\|^2&=&\frac{1}{\epsilon}\int_s^t\sum\limits_{k\in\mathbb{N}}
e^{-2(t-\tau)\alpha_k/\epsilon}
d\tau\nonumber\\
&=&\sum\limits_{k\in\mathbb{N}}\int_0^{(t-s)/\epsilon}e^{-2
\tau\alpha_k} d\tau\nonumber\\
&=&\sum\limits_{k\in\mathbb{N}}\frac{1}{2\alpha_k}(1-e^{-2\alpha_k(t-s)/\epsilon})\nonumber\\
&\leq& C_r\frac{|t-s|^{2r}}{\epsilon^{2r}}
\sum\limits_{k\in\mathbb{N}}\frac{1}{\alpha_k^{1-2r}}.\nonumber\end{aligned}$$ Recalling that we have assumed $r\in (0, \frac{1}{4})$, it follows that $\sum\limits_{k\in\mathbb{N}}\frac{1}{\alpha_k^{1-2r}}<+\infty.$ Therefore, we obtain $$\begin{aligned}
\mathbb{E}\|J^\epsilon_4(t,s)\|^2&\leq&
C_r\frac{|t-s|^{2r}}{\epsilon^{2r}}.\label{Yt-Ys-4}\end{aligned}$$ For $J^\epsilon_5(t,s)$ we have $$\begin{aligned}
\mathbb{E}\|J^\epsilon_5(t,s)\|&=&\frac{1}{\epsilon}\int_0^s\sum\limits_{k\in\mathbb{N}}e^{-2(s-\tau)\alpha_k/\epsilon}
(1-e^{-(t-s)\alpha_k/\epsilon})^2d\tau\nonumber\\
&\leq&\sum\limits_{k\in\mathbb{N}}(1-e^{-(t-s)\alpha_k/\epsilon})^2\frac{1}{2\alpha_k}(1-e^{-2s\alpha_k/\epsilon})\nonumber\\
&\leq&\sum\limits_{k\in\mathbb{N}}(1-e^{-(t-s)\alpha_k/\epsilon})^2\frac{1}{2\alpha_k}\nonumber\\
&\leq&C_r\sum\limits_{k\in\mathbb{N}}\frac{|t-s|^{2r}}{\epsilon^{2r}}\frac{1}{\alpha_k^{1-2r}}\nonumber\\
&\leq&C_r \frac{|t-s|^{2r}}{\epsilon^{2r}}. \label{Yt-Ys-5}\end{aligned}$$ Collecting together -, we obtain
$$\begin{aligned}
(\mathbb{E}\|Y_t^\epsilon(x,y)-Y_s^\epsilon(x,y)\|^2)^{\frac{1}{2}}\leq
C_{r,T}(1+\|x\|+\|y\|)\left[\frac{|t-s|^{r}}{s^{r}}+\frac{|t-s|^{r}}{\epsilon^{r}}\right].\end{aligned}$$
\[A-X\] Assume that $x\in \mathscr{D}((-A)^\theta)$ for some $\theta\in (0,
1]$. Then, under conditions (H.1) and (H.2), we have that $X_t^\epsilon(x,y)\in \mathscr{D}(-A)$, $\mathbb{P}-a.s.,$ for any $t>0$ and $\epsilon>0$. Moreover, for any $r\in (0, \frac{1}{4})$ it holds that $$\begin{aligned}
(\mathbb{E}\|AX_t^\epsilon(x,y)\|^2)^\frac{1}{2}\leq C_{r,T}
t^{\theta-1}\|x\|_{(-A)^\theta}+C_{r,T}(1+\|x\|+\|y\|)
(1+\frac{1}{\epsilon^{r}}),\;t\in [0, T].\end{aligned}$$
For any $t\in [0, T]$ we write $X_t^\epsilon(x,y)$ as $$\begin{aligned}
X_t^\epsilon(x,y)&=&\big[S_tx+\int_0^tS_{t-s}F(X_t^\epsilon(x,y),Y_t^\epsilon(x,y))ds\big]\\
&+&\int_0^tS_{t-s}[F(X_s^\epsilon(x,y),Y_s^\epsilon(x,y))-F(X_t^\epsilon(x,y),Y_t^\epsilon(x,y))]ds\\
&+&\int_0^tS_{t-s}dW_s^1\\
&:=&X^{\epsilon,1}_{t}(x,y)+X^{\epsilon,2}_{t}(x,y)+X^{\epsilon,3}_{t}(x,y).\end{aligned}$$ For $X^{\epsilon,1}_{t}(x,y)$ we have $$\begin{aligned}
\|AX^{\epsilon,1}_{t}(x,y)\|&=&\|AS_tx\|+\|(S_t-I)F(X_t^\epsilon(x,y),Y_t^\epsilon(x,y))\|\\
&\leq&
Ct^{\theta-1}\|x\|_{(-A)^\theta}+C(1+\|X_t^\epsilon(x,y)\|+\|Y_t^\epsilon(x,y)\|),\end{aligned}$$ so that, thanks to Lemma \[moment-bound\], we obtain $$\begin{aligned}
(\mathbb{E}\|AX^{\epsilon,1}_{t}(x,y)\|^2)^\frac{1}{2}&=&C_Tt^{\theta-1}\|x\|_{(-A)^\theta}+C_T(1+\|x\|+\|y\|).\label{A-X-1}\end{aligned}$$ From , we have $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\|AX^{\epsilon,2}_{t}(x,y)\|\\
&\leq& C\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{t-s}\|F(X_s^\epsilon(x,y),Y_s^\epsilon(x,y))-F(X_t^\epsilon(x,y),Y_t^\epsilon(x,y))\|ds\\
&\leq&C\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}[
\|X_t^\epsilon(x,y)-X_s^\epsilon(x,y)\|+
\|Y_t^\epsilon(x,y),Y_s^\epsilon(x,y)\|]ds,\end{aligned}$$ which implies $$\begin{aligned}
\mathbb{E}\|AX^{\epsilon,2}_{t}(x,y)\|^2&\leq&
C\Big[\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}
(\mathbb{E}\|X_t^\epsilon(x,y)-X_s^\epsilon(x,y)\|^2)^{\frac{1}{2}}
ds\Big]^2\nonumber\\
&+&C\Big[\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}
(\mathbb{E}\|Y_t^\epsilon(x,y)-Y_s^\epsilon(x,y)\|^2)^{\frac{1}{2}}
ds\Big]^2,\nonumber\end{aligned}$$ by making use of Minkowski inequality. If we take $r\in (0,
\frac{1}{4})$ as in Lemma \[Xt-Xs\], we get $$\begin{aligned}
&&\Big[\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}
(\mathbb{E}\|X_t^\epsilon(x,y)-X_s^\epsilon(x,y)\|^2)^{\frac{1}{2}}
ds\Big]^2\nonumber\\
&&\leq C_{r,
T}(1+\|x\|+\|y\|)^2\Big[\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^rs^{1-r}}ds+\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{\frac{1}{2}}
}ds\nonumber\\
&&\qquad+\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{(t-s)^{r}
}ds+\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{1-r }}ds\Big]^2\nonumber\\
&&\leq C_{r,
T}(1+\|x\|+\|y\|)^2\Big[\int_0^{t/2}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^rs^{1-r}}ds+\int_{t/2}^t\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^rs^{1-r}}ds\nonumber\\
&&\qquad+\int_0^{+\infty}\frac{e^{-\frac{\alpha_1}{2}s}
}{s^{\frac{1}{2} }
}ds+\int_0^{+\infty}\frac{e^{-\frac{\alpha_1}{2}s} }{s^{r }
}ds+\int_0^{+\infty}\frac{e^{-\frac{\alpha_1}{2}s}
}{s^{1-r } }ds\Big]^2\nonumber\\
&&\leq C_{r, T}(1+\|x\|+\|y\|)^2\Big[1+\int_0^{t/2}\frac{1}
{(t-s)^rs^{1-r}}ds+\int_{t/2}^t\frac{1}
{(t-s)^rs^{1-r}}ds\Big]^2\nonumber\\
&&\leq C_{r,
T}(1+\|x\|+\|y\|)^2\Big[1+\left(\frac{t}{2}\right)^{-r}\int_0^{t/2}\frac{1}
{ s^{1-r}}ds+\left(\frac{t}{2}\right)^{-(1-r)}\int_{t/2}^t\frac{1}
{(t-s)^r}ds\Big]^2\nonumber\\
&&=C_{r, T}(1+\|x\|+\|y\|)^2(1+\frac{1}{r}+\frac{1}{1-r})\nonumber\\
&&\leq C_{r, T}(1+\|x\|+\|y\|)^2. \label{A-X-1-1.6}\end{aligned}$$ By using a completely analogous way, due to Lemma \[Yt-Ys\], it is possible to show that $$\begin{aligned}
&&\Big[\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}
(\mathbb{E}\|Y_t^\epsilon(x,y)-Y_s^\epsilon(x,y)\|^2)^{\frac{1}{2}}
ds\Big]^2\\
&&\leq C_{r,
T}(1+\|x\|+\|y\|)^2\Big[\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{1-r}s^{r}}ds+\frac{1}{\epsilon^{r}}\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{1-r} }ds\Big]^2\\
&&\leq C_{r, T}(1+\|x\|+\|y\|)^2(1+\frac{1}{\epsilon^{2r}}),\end{aligned}$$ which, together with , allows us to get the estimate $$\begin{aligned}
\mathbb{E}\|AX^{\epsilon,2}_{t}(x,y)\|^2\leq C_{r,
T}(1+\|x\|+\|y\|)^2(1+\frac{1}{\epsilon^{2r}}).\label{A-X-1-2}\end{aligned}$$ Thus, it remains to estimate $AX^{\epsilon,3}_{t}(x,y)$. By straightforward computations and condition , we get $$\begin{aligned}
\mathbb{E}\|AX^{\epsilon,3}_{t}(x,y)\|^2
&=&\mathbb{E}\|\sum\limits_{k\in\mathbb{N}}\sqrt{\lambda_{1,k}}\alpha_ke_k\int_0^te^{-\alpha_k(t-s)}dB_{s,k}^{(1)}\|^2\nonumber\\
&=& \sum\limits_{k\in\mathbb{N}}\lambda_{1,k}
\alpha_k^2\int_0^te^{-2\alpha_{1,k}(t-s)}ds\nonumber\\
&\leq&C\sum\limits_{k\in\mathbb{N}}\lambda_{1,k} \alpha_k\nonumber\\
&\leq& C. \nonumber\end{aligned}$$ This, together with and , yields $$\begin{aligned}
(\mathbb{E}\|AX_t^\epsilon(x,y)\|^2)^\frac{1}{2}\leq C_{r, T}
t^{\theta-1}\|x\|_{(-A)^\theta}+C_{r, T}(1+\|x\|+\|y\|)
(1+\frac{1}{\epsilon^{r}}).\end{aligned}$$
\[bar-Xt-Xs\] Let the conditions (H.1) and (H.2) be satisfied and fix $x \in H$ and $T>0$. Then for any $r\in (0,\frac{1}{4})$ there exists a constant $C_{r, T}>0$ such that for any $0< s\leq t\leq T$, we have $$\begin{aligned}
(\mathbb{E}\| \bar{X}_t(x)-\bar{X}_s(x)\|^2)^\frac{1}{2} &\leq& C_{r, T}\frac{|t-s|^{1-r}} {s^{1-r}}\|x\|\\
&+&C_{r, T}(|t-s|^{\frac{1}{2} }+|t-s|^{1-r}+|t-s|^r)(1+\|x\|).\end{aligned}$$
It holds that $$\begin{aligned}
\bar{X}_t(x)
-\bar{X}_s(x)&=&S_tx-S_sx+\int_s^tS_{t-\tau}\bar{F}(\bar{X}_\tau(x))d\tau\nonumber\\
&+&\int_0^s[S_{t-\tau}\bar{F}(\bar{X}_\tau(x))-S_{s-\tau}\bar{F}(\bar{X}_\tau(x))]d\tau\nonumber\\
&+&\int_s^tS_{t-\tau}dW^1_\tau+\int_0^s(S_{t-\tau}-S_{s-\tau})dW^1_\tau.\label{bar-Xt-Xs-0}\end{aligned}$$ According to , we obtain $$\begin{aligned}
\|(S_t-S_s)x\|\leq C_r\frac{|t-s|^{1-r}} {s^{1-r}}\|x\|.
\end{aligned}$$ For the second term on the right-hand side of , by using we have $$\begin{aligned}
\mathbb{E}\|\int_s^tS_{t-\tau}\bar{F}(\bar{X}_\tau(x)
)d\tau\|^2&\leq&
|t-s|\int_s^t\mathbb{E}\|S_{t-\tau}\bar{F}(\bar{X}_\tau(x) )\|^2d\tau\\
&\leq&C|t-s|\int_s^t\mathbb{E}
(1+\|\bar{X}_\tau(x)\|^2 d\tau\\
&\leq&C_T|t-s|(1+\|x\|^2 ).\end{aligned}$$ Concerning the third term on the right-hand side of , we deduce $$\begin{aligned}
&&\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})\bar{F}(\bar{X}_\tau(x) )d\tau\|^2\\
&&\leq\mathbb{E}\left[\int_0^s\|(S_{t-\tau}-S_{s-\tau})\bar{F}(\bar{X}_\tau (x))\|d\tau\right]^2\\
&&\leq C_r\mathbb{E}\left[\int_0^s\frac{(t-s)^{1-r}}{(s-\tau)^{1-r}}
{e^{-\frac{\alpha_1}{2}(s-\tau)}}\|\bar{F}(\bar{X}_\tau(x) )\|d\tau\right]^2\\
&&\leq
C_r|t-s|^{2(1-r)}\left[\int_0^s\frac{e^{-\frac{\alpha_1}{2}(s-\tau)}}{(s-\tau)^{1-r}}
\big(\mathbb{E}\|\bar{F}(\bar{X}_\tau(x)
)\|^2\big)^{\frac{1}{2}}d\tau\right]^2,\end{aligned}$$ and then, by using once more , this yields $$\begin{aligned}
&&\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})\bar{F}(\bar{X}_\tau(x) )d\tau\|^2\\
&&\leq
C_r|t-s|^{2(1-r)}\left[\int_0^s\frac{e^{-\frac{\alpha_1}{2}(s-\tau)}}{(s-\tau)^{1-r}}
\mathbb{E}\big(1+\| \bar{X}_\tau (x)\| \big)
d\tau\right]^2\\
&&\leq C_{r,T}|t-s|^{2(1-r)}(1+\|x\|^2).\end{aligned}$$ By using arguments analogous to those used in Lemma \[Xt-Xs\], we have $$\begin{aligned}
\mathbb{E}\|\int_s^tS_{t-\tau}dW^1_\tau\|^2 \leq Tr(Q_1)|t-s|\end{aligned}$$ and $$\begin{aligned}
\mathbb{E}\|\int_0^s(S_{t-\tau}-S_{s-\tau})dW^1_\tau\|^2\leq
C_{r,T}|t-s|^{2r}.\end{aligned}$$ Therefore, collecting all estimate of terms appearing on the right-hand side of , we obtain $$\begin{aligned}
(\mathbb{E}\| \bar{X}_t(x)-\bar{X}_s(x)\|^2)^\frac{1}{2} &\leq& C_{r, T}\frac{|t-s|^{1-r}} {s^{1-r}}\|x\|\\
&+&C_{r, T}(|t-s|^{\frac{1}{2} }+|t-s|^{1-r}+|t-s|^r)(1+\|x\|).\end{aligned}$$
\[bar-X\] Assume that $x\in \mathscr{D}((-A)^\theta)$ for some $\theta\in (0,
1]$. Then, under conditions (H.1) and (H.2), we have that $\bar{X}_t \in \mathscr{D}((-A))$, $\mathbb{P}-a.s.,$ for any $t\in
[0, T]$ and $\epsilon>0$. Moreover, it holds that $$\begin{aligned}
(\mathbb{E}\|A\bar{X}_t(x) \|^2)^\frac{1}{2}\leq
C_Tt^{\theta-1}\|x\|_{(-A)^\theta}+C_T(1+\|x\|),\;t\in[0, T].\end{aligned}$$
The proof is analogous to that of the previous Lemma \[A-X\]. We write $$\begin{aligned}
\bar{X}_t (x)&=&\big[S_tx+\int_0^tS_{t-s}\bar{F}(\bar{X}_t(x) )ds\big]\nonumber\\
&&+\int_0^tS_{t-s}[\bar{F}(\bar{X}_s (x))-\bar{F}(\bar{X}_t^\epsilon(x) )]ds+\int_0^tS_{t-s}dW_s^1\nonumber\\
&:=&\bar{X}^{(1)}_{t}(x)+\bar{X}^{(2)}_{t}(x)+\bar{X}^{(3)}_{t}(x).\label{bar-X-0}\end{aligned}$$ For $\bar{X}^{(1)}_{t}(x)$, we have $$\begin{aligned}
\|A\bar{X}^{(1)}_{t}(x)\|&=&\|AS_tx\|+\|(S_t-I)\bar{F}(\bar{X}_t(x) )\|\\
&\leq& Ct^{\theta-1}\|x\|_{(-A)^\theta}+C(1+\|\bar{X}_t(x) \| ),\end{aligned}$$ and then, thanks to , we get $$\begin{aligned}
(\mathbb{E}\|A\bar{X}^{(1)}_{t}(x)\|^2)^\frac{1}{2}&=&C_Tt^{\theta-1}\|x\|_{(-A)^\theta}+C_T(1+\|x\|).\end{aligned}$$ Concerning $\bar{X}^{(2)}_{t}(x)$, we have $$\begin{aligned}
\|A\bar{X}^{(2)}_{t}(x)\|&\leq&
C\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{t-s}\|\bar{F}(\bar{X}_s(x))-\bar{F}(\bar{X}_t(x))\|ds\\
&\leq&C\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}
\|\bar{X}_t(x)-\bar{X}_s(x)\|ds,\end{aligned}$$ and then, according to Minkowski inequality and Lemma \[bar-Xt-Xs\], for a fixed $r_0\in (0, \frac{1}{4})$ we obtain $$\begin{aligned}
\mathbb{E}\|A\bar{X}^{(2)}_{t}(x)\|^2&\leq&
C\Big[\int_0^t\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{t-s}
(\mathbb{E}\|\bar{X}_t(x) -\bar{X}_s(x) \|^2)^{\frac{1}{2}} ds\Big]^2\\
&&\leq C_{r_0, T}(1+\|x\|
)^2\Big[\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{r_0}s^{1-r_0}}ds+\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{r_0}
}ds\\
&& +\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)} }{(t-s)^{
\frac{1}{2}} }ds+ +\int_0^{t}\frac{e^{-\frac{\alpha_1}{2}(t-s)}
}{(t-s)^{ 1-r_0} }ds \Big]^2\\
&\leq&C_{r_0, T}(1+\|x\| )^2.\end{aligned}$$ On the other hand, as shown in the Lemma \[A-X\], we have $$\begin{aligned}
\mathbb{E}\|A\bar{X}^{(3)}_{t}(x)\|^2\leq C.\end{aligned}$$ Therefore, collecting all estimates of terms appearing on the right-hand side of , we can conclude the proof.
Asymptotic expansions {#asym}
=====================
One of the main tools that we are using in order to prove the main result is Itô’s formula. On the other hand, here the operator $A$ is unbounded, and then we can not apply directly Itô’s formula. Therefore we have to proceed by Galerkin approximation procedure, to this purpose we need to introduce some notations. For arbitrary $n\in \mathbb{N}$, let $H^{(n)}$ denote the finite dimensional subspace of $H$, generated by the set of eigenvectors $\{e_1,e_2,\cdots, e_n\}$. Let $P_n : H \rightarrow H^{(n)}$ denote the orthogonal projection defined by $$\begin{aligned}
P_nh=\sum\limits_{k=1}^n\Big(h, e_k\Big)_H e_k,\; h\in H.\end{aligned}$$ We define $A_n:H^{(n)}\rightarrow H^{(n)}$ by $$\begin{aligned}
A_nh=AP_nh=P_nAh=\sum\limits_{k=1}^n(-\alpha_k)\Big(h,
e_k\Big)_He_k,\;h\in H^{(n)},\end{aligned}$$ which is the generator of a strongly semigroup $\{S_{t,n}\}_{t\geq
0}$ on $H^{(n)}$ taking the form $$\begin{aligned}
S_{t,n}h=\sum\limits_{k=1}^n e^{-\alpha_kt}\Big(e_k, h\Big)_He_k.\end{aligned}$$ Similarly, for arbitrary $n\in \mathbb{N}$ and $\gamma\in
\mathbb{R}$, one can define the $(-A_n)^\gamma:H^{(n)}\rightarrow
H^{(n)}$ as $$(-A_n)^\gamma h:=\sum\limits_{k=1}^n\alpha_k^{\gamma}\Big(e_k,
h\Big)_He_k,\;h\in H^{(n)}.$$ For each $n$ we consider the approximating problem of : $$\begin{aligned}
&&dX_t^{\epsilon,n}=A_nX^{\epsilon,n}_tdt+F_n(X^{\epsilon,n}_t,Y^{\epsilon,n}_t)dt+\sigma_1P_ndW_t^1,\label{abstr-slow-equation-finite}\\
&&dY_t^{\epsilon,n}=\frac{1}{\epsilon}A_nY^{\epsilon,n}_tdt+
\frac{1}{\epsilon}G_n(X^{\epsilon,n}_t,Y^{\epsilon,n}_t)dt+\frac{\sigma_2}{\sqrt{\epsilon}}P_ndW_t^2,\label{abstr-fast-equation-finite}\end{aligned}$$ with initial conditions $\;X_0^{\epsilon,n}:=x^{(n)}=P_nx,
Y_0^{\epsilon,n}:=y^{(n)}=P_ny$, where $F_n$ and $G_n$ are respectively defined by $$\begin{aligned}
&&F_n(u,v)=P_nF(u,v),\;u,v \in H^{(n)},\\
&&G_n(u,v)=P_nG(u,v),\;\;u,v \in H^{(n)}.\end{aligned}$$ Such a problem is the finite dimensional problem with Lipschitz coefficients. Under the assumption (H.1) and (H.2), it is easy to show that the problem - admits a unique *strong solution* taking values in $H^{(n)}\times H^{(n)}$, which is denoted by $(X_t^{\epsilon,n}(x^{(n)},y^{(n)}),Y_t^{\epsilon,n}(x^{(n)},y^{(n)}))$. Moreover, for any fixed $\epsilon>0$ and $x, y\in H$ it holds that $$\begin{aligned}
\lim\limits_{n\rightarrow
+\infty}\mathbb{E}\big(\|X_t^{\epsilon}(x,y)-X_t^{\epsilon,n}(x^{(n)},y^{(n)})\|^2
\big)=0\label{approx-infinite-1}\end{aligned}$$ and $$\begin{aligned}
\lim\limits_{n\rightarrow +\infty}\mathbb{E}\big(
\|Y_t^{\epsilon}(x,y)-Y_t^{\epsilon,n}(x^{(n)},y^{(n)})\|^2\big)=0,\label{approx-infinite-2}\end{aligned}$$ For any fixed $x\in H$, we consider frozen problem associate with equation in form $$\begin{aligned}
dY^{x,n}_t=A_nY^{x,n}_tdt+G_n(P_nx,Y^{x,n}_t)dt+\sigma_2P_ndW^2_t,\;Y^{x,n}_0=y^{(n)}.
\label{frozen-finite}\end{aligned}$$ Under (H.1) and (H.2), it is easy to check that such a problem admits a unique strong solution denoted by $Y^{x,n}_t(y^{(n)})$, which has a unique invariant measure $\mu^{x,n}$ on finite dimensional space $H^{(n)}$. The averaged equation for finite dimensional approximation problem can be defined as follows: $$\begin{aligned}
d\bar{X}^n_t(x^{(n)})=A_n\bar{X}^n_t(x^{(n)})dt+\bar{F}_n(\bar{X}^n_t(x^{(n)})dt+\sigma_1P_ndW^1_t,\;\bar{X}^n_0=x^{(n)},
\label{aver-finite-equation}\end{aligned}$$ with $$\begin{aligned}
\bar{F}_n(u)=\int_{H^{(n)}}F_n(u,v)\mu^{x,n}(dv),\;u\in H^{(n)}.\end{aligned}$$ The averaging principle guarantees $$\begin{aligned}
\lim\limits_{\epsilon\rightarrow0+}\left\{\mathbb{E}\|{X}^{\epsilon,n}_t(x^{(n)},y^{(n)})
-\bar{X}^n_t(x^{(n)})\|^2\right\}^\frac{1}{2}=0,\label{aver-finite}\end{aligned}$$ and the above limit is uniform with respect to $n\in \mathbb{N}$. By triangle inequality we obtain $$\begin{aligned}
\mathbb{E}\|\bar{X}_t(x)-\bar{X}^n_t(x^{(n)})\|&\leq&
\mathbb{E}\|\bar{X}_t(x)-{X}^{\epsilon
}_t(x,y)\|\\
&+&\mathbb{E}\|{X}^{\epsilon}_t(x,y)-{X}^{\epsilon,n}_t(x^{(n)},y^{(n)})\|\\
&+&\mathbb{E}\|{X}^{\epsilon,n}_t(x^{(n)},y^{(n)})-\bar{X}^n_t(x^{(n)})\|,\end{aligned}$$ which, together with and , yields $$\begin{aligned}
\lim\limits_{n\rightarrow\infty}\mathbb{E}\|\bar{X}_t(x)-\bar{X}^n_t(x^{(n)})\|=0.\label{approx-infinite-averaging}\end{aligned}$$
Note that for any $T>0$ and $\phi\in C_b^3(H, \mathbb{R})$ we have $$\begin{aligned}
\left|\mathbb{E}\phi(X^\epsilon_T(x,y))-\mathbb{E}\phi(\bar{X}_T(x))\right|
&\leq&\left|\mathbb{E}\phi(X^\epsilon_T(x,y))-\mathbb{E}\phi(X^{\epsilon,n}_T(x^{(n)},y^{(n)} ))\right|\\
&+&\left|\mathbb{E}\phi(X^{\epsilon,n}_T(x^{(n)},y^{(n)}))-\mathbb{E}\phi(\bar{X}^{n}_T(x^{(n)}
))\right|\\
&+&\left|\mathbb{E}\phi(\bar{X}^{n}_T(x^{(n)}
))-\mathbb{E}\phi(\bar{X}_T(x))\right|.\end{aligned}$$ According to the approximation results and the first and last terms above converge to zero as $n$ goes to infinity. In order to prove Theorem \[theorem\] we have only to show that for any $r\in (0, 1)$, it holds $$\begin{aligned}
\left|\mathbb{E}\phi(X^{\epsilon,n}_T(x^{(n)},y^{(n)}))-\mathbb{E}\phi(\bar{X}^{n}_T(x^{(n)}))\right|\leq
C_{\theta, r, T,\phi,x,y}\epsilon^{1-r} \label{finite-weak}\end{aligned}$$ for some constant $C_{\theta, r, T,\phi,x,y}$ independent of the dimension index $n$.
For all $n\in\mathbb{N}$, the regular conditions on drift coefficients $F$ and $G$ presented in (H.1) and (H.2) are still valid for $F_n$ and $G_n$, respectively, but replacing $H$ by $H^{(n)}$. In particular, the boundedness on derivatives associated with $F_n$ and $G_n$ are uniform with respect to dimension $n$. As a result, all properties satisfied by $(X_t^{\epsilon },
Y_t^{\epsilon })$, $Y_t^x$ and $P_t^x$ are still valid for $(X_t^{\epsilon,n},
Y_t^{\epsilon,n})$, $Y_t^{x,n}$ and for the transition semigroup $P_t^{x,n}$ corresponding to , respectively. Moreover, all estimates for $(X_t^{\epsilon,n}, Y_t^{\epsilon,n})$, $Y_t^{x,n}$ and $P_t^{x,n}$ are uniform with respect to $n\in
\mathbb{N}$. Similarly, $\bar{F}_n$ and $\bar{X}_t^{n}$ inherit all properties described for $\bar{F}$ and $\bar{X}_t$, respectively, with all estimates uniform with respect to $n\in \mathbb{N}$.
[In what follows]{}, the letter $C$ below with or without subscripts will denote generic positive constants independent of $\epsilon$ and dimension $n$, whose value may change from one line to another.
Let $\phi$ be the test function as in Theorem \[theorem\]. As usual, we use the notation $(X_t^{\epsilon,n}(x,y),
Y_t^{\epsilon,n}(x,y))$ to denote the solution to equation - with initial value $(X_0^{\epsilon,n}(x,y),
Y_0^{\epsilon,n}(x,y))=(x,y)\in H^{(n)}\times H^{(n)}$. For any $n\in\mathbb{N}$, we define a function $u_n^\epsilon: [0,
T]\times H^{(n)} \times H^{(n)}\rightarrow \mathbb{R}$ by $$u_n^\epsilon(t, x,y)=\mathbb{E}\phi(X_t^{\epsilon,n}(x,y)).$$
We now introduce two differential operators associated with the fast variable system and slow variable system in finite dimensional space, respectively: $$\begin{aligned}
\mathcal {L}_1^n\varphi(y)&=&\Big(A_ny+G_n(x,y),
D_y\varphi(y)\Big)_H\\
&&+\frac{1}{2}\sigma_2^2Tr(D^2_{yy}\varphi(y)Q_{2,n}^{\frac{1}{2}}(Q_{2,n}^{\frac{1}{2}})^*),\;
\varphi \in C_b^2(H^{(n)},\mathbb{R})\end{aligned}$$ and $$\begin{aligned}
\mathcal {L}_2^n\varphi(x)&=&\Big(A_nx+ F_n(x,y),
D_x\varphi(x)\Big)_{ {H}
}\\
&&+\frac{1}{2}\sigma_1^2Tr(D^2_{xx}\varphi(x)Q_{1,n}^{\frac{1}{2}}(Q_{1,n}^{\frac{1}{2}})^*),\;
\varphi \in C_b^2(H^{(n)},\mathbb{R} ),\end{aligned}$$ where $Q_{1,n}:=Q_1P_n$ and $Q_{2,n}:=Q_2P_n$ for any $n\in\mathbb{N}$. It is known that $u_n^\epsilon$ is a solution to the forward Kolmogorov equation: $$\begin{aligned}
\label{Kolm}
\begin{cases}
\frac{\partial}{\partial t}u_n^\epsilon(t, x, y)=\mathcal {L}^{\epsilon, n} u_n^\epsilon(t, x, y),\\
u_n^\epsilon(0, x,y)=\phi(x),
\end{cases}\end{aligned}$$ where $\mathcal {L}^{\epsilon, n}:=\frac{1}{\epsilon}\mathcal
{L}_1^n+\mathcal {L}_2^n.$
Also recall the Kolmogorov operator for the averaged system is defined as $$\begin{aligned}
\bar{\mathcal {L}^n}\varphi(x)&=&\Big(A_nx+\bar{F}_n(x),
D_x\varphi(x)\Big)_{ {H}
}\\
&&+\frac{1}{2}\sigma_1^2Tr(D^2_{xx}\varphi(x)Q_{1,n}^{\frac{1}{2}}(Q_{1,n}^{\frac{1}{2}})^*),\;
\varphi\in C_b^2(H^{(n)}, \mathbb{R} ).\end{aligned}$$ If we set $$\bar{u}_n(t, x)=\mathbb{E}\phi(\bar{X}_t^n(x)),$$ we have $$\begin{aligned}
\label{Kolm-Aver}
\begin{cases}
\frac{\partial }{\partial t}\bar{u}_n(t, x)=\bar{\mathcal {L}^n} \bar{u}_n(t, x),\\
\bar{u}_n(0, x)=\phi(x).
\end{cases}\end{aligned}$$ Then the weak difference at time $T$ can be rewritten as $$\begin{aligned}
\mathbb{E}\phi({X}^{\epsilon,n}_T)-\mathbb{E}\phi(\bar{X}^n_T)=u_n^\epsilon(T,
x,y)-\bar{u}_n(T,x).\end{aligned}$$ Henceforth, for the sake of brevity, we will omit to write the dependence of the temporal variable $t$ and spatial variables $x$ and $y$ in some occasion. For example, we often write $u^\epsilon$ instead of $u_n^\epsilon(t, x, y)$. Our aim is to seek an expansion for $u_n^\epsilon(T, x,y)$ with the form
$$\begin{aligned}
\label{asymp-expan}
u_n^\epsilon=u_{0,n}+\epsilon u_{1,n}+r^\epsilon_n,\end{aligned}$$
where $u_{0,n}$ and $u_{1,n}$ are smooth functions which will be constructed below, and $r^\epsilon_n$ is the remainder term.
The rest of this section is devoted to the proof of Theorem \[theorem\]. We will proceed in several steps, which have been structured as subsections.
**The leading term**
--------------------
Let us first determine the leading term. Now, substituting expansions into yields $$\begin{aligned}
\frac{\partial u_{0,n}}{\partial t}+\epsilon\frac{\partial
u_{1,n}}{\partial t}+\frac{\partial r^\epsilon_n}{\partial t}&=&
\frac{1}{\epsilon}\mathcal {L}_1^nu_{0,n}+\mathcal {\mathcal
{L}}_1^nu_{1,n}+\frac{1}{\epsilon}\mathcal {L}_1^nr^\epsilon_n\\
&&+\mathcal {L}_2^nu_{0,n}+\epsilon \mathcal {L}_2^nu_{1,n}+\mathcal
{L}_2^nr^\epsilon_n.\end{aligned}$$ By comparing orders of $\epsilon$, we obtain $$\begin{aligned}
&&\mathcal {L}_1^nu_{0,n}=0 \label{u-o-equ-1}\end{aligned}$$ and $$\begin{aligned}
&&\frac{\partial u_{0,n}}{\partial t}=\mathcal
{L}_1^nu_{1,n}+\mathcal {L}_2^nu_{0,n}.\label{u-0-equ-2}\end{aligned}$$ It follows from that $u_{0,n}$ is independent of $y$, which means $$u_{0,n}(t,x, y)=u_{0,n}(t,x).$$ We also impose the initial condition $u_{0,n}(0,x)=\phi(x).$ Since $\mu^{x,n}$ is the invariant measure of a Markov process with generator $\mathcal
{L}_1^n$, we have $$\begin{aligned}
\int_{H^{(n)}}\mathcal {L}_1^nu_{1,n}(t,x,y)\mu^{x,n}(dy)=0,\end{aligned}$$ which, by invoking , implies $$\begin{aligned}
\frac{\partial u_{0,n}}{\partial t}(t,x)&=&\int_{H^{(n)}}\frac{\partial u_{0,n}}{\partial t}(t,x)\mu^{x,n}(dy)\\
&=&\int_{H^{(n)}}\mathcal {L}_2^nu_{0,n}(t,x)\mu^{x,n}(dy)\\
&=&\left(A_nu_{0,n}(t,x)+\int_{H^{(n)}} F_n(x,y)\mu^{x,n}(dy),
D_xu_{0,n}(t,x)\right)_{ {H}}\nonumber\\
&&+\frac{1}{2}\sigma_1^2Tr(D^2_{xx}u_{0,n}(t,x)Q_{1,n}^{\frac{1}{2}}(Q_{1,n}^{\frac{1}{2}})^*)\\
&=&\bar{\mathcal {L}^n}u_{0,n}(t,x),\end{aligned}$$ so that $u_{0,n}$ and $\bar{u}_n$ satisfy the same evolution equation. By using a uniqueness argument, such $u_{0,n}$ has to coincide with the solution $\bar{u}_n$ and we have the following lemma:
\[u-0\] Assume (H.1) and (H.2). Then for any $x, y \in H^{(n)}$ and $T>0$, we have $u_{0,n}(T,x,y)=\bar{u}_n(T,x)$.
**Construction of** $ {u_{1,n}}$
--------------------------------
Let us proceed to carry out the construction of $u_{1,n}$. Thanks to Lemma \[u-0\] and , the equation can be rewritten as $$\begin{aligned}
\bar{\mathcal
{L}}^n\bar{u}_n=\mathcal{L}^n_1u_{1,n}+\mathcal{L}^n_2\bar{u}_n,\end{aligned}$$ and hence we get an elliptic equation for $u_{1,n}$ with form $$\begin{aligned}
\mathcal {L}_1^nu_{1,n}(t,x,y)=\Big(\bar{ F}_n(x)-F_n(x,y),
D_x\bar{u}_n(t,x)\Big)_{{H} }:=-\rho_n(t,x,y),\end{aligned}$$ where $\rho_n$ is of class $\mathcal{C}^2$ with respect to $y$, with uniformly bounded derivatives. Moreover, it satisfies for any $t\geq
0$ and $x\in H^{(n)}$, $$\begin{aligned}
\int_{H^{(n)}}\rho_n(t,x,y)\mu^{x, n}(dy)=0.\end{aligned}$$ [For any]{} $y\in H^{(n)}$ and $s>0$ we have $$\begin{aligned}
\frac{\partial}{\partial s}P_{s,n}^x\rho_n(t,x,y)&=&\Big(A_ny+G_n(x,y),D_y(P_{s,n}^x\rho_n(t,x,y))\Big)_H\nonumber\\
&+&\frac{1}{2}\sigma_2^2Tr[D^2_{yy}(P_{s,n}^x\rho_n(t,x,y))Q_{2,n}^{\frac{1}{2}}(Q_{2,n}^{\frac{1}{2}})^*],\end{aligned}$$ here $$P_{s,n}^x\rho_n(t,x,y)=\mathbb{E}\rho_n(t, x,Y_s^{x, n}(y))$$ satisfying $$\lim\limits_{s\rightarrow{+\infty}}\mathbb{E}\rho_n(t,
x,Y_s^{x,n}(y))=\int_{H^{(n)}}\rho_n(t,x,z)\mu^{x,n}(dz)=0.\label{rho-limits}$$ Indeed, by the invariant property of $\mu^{x,n}$ and Lemma \[ux\] in the next section, $$\begin{aligned}
&&\left|\mathbb{E}\rho_n(t,
x,Y_s^{x,n}(y))-\int_{H^{(n)}}\rho_n(t,x,z)\mu^{x,n}(dz)\right|\nonumber\\
&&=\left|\int_{H^{(n)}}\mathbb{E}[\rho_n(t, x,Y_s^{x,n}(y))-
\rho_n(t,
x,Y_s^{x,n}(z))\mu^{x,n}(dz)]\right|\nonumber\\
&&\leq\int_{H^{(n)}}\left|\mathbb{E}\Big( F_n(x,Y_s^{x,n}(z))-
F_n(x,Y_s^{x,n}(y)), D_x\bar{u}_n(t,x)\Big)_{{H}
}\right|\mu^{x,n}(dz)\nonumber\\
&&\leq C\int_{H^{(n)}} \mathbb{E} \|Y_s^{x,n}(z)- Y_s^{x,n}(y)\|
\mu^{x,n}(dz).\nonumber\\\end{aligned}$$ This, in view of and , yields $$\begin{aligned}
&&\left|\mathbb{E}\rho_n(t,
x,Y_s^{x,n}(y))-\int_{H^{(n)}}\rho_n(t,x,z)\mu^{x,n}(dz)\right|\nonumber\\
&&\leq Ce^{-\frac{\beta}{2}s}(1+\|x\|+\|y\|),\end{aligned}$$ which implies the equality . Therefore, we get $$\begin{aligned}
&&\Big(A_ny+G_n(x,y),D_y\int_0^{{+\infty}}P_{s, n}^x\rho_n(t,x,y)
ds\Big)_H\nonumber\\
&&+\frac{1}{2}\sigma_2^2Tr[D^2_{yy}\int_0^{{+\infty}}(P_{s,n}^x\rho_n(t,x,y))Q_{2, n}^{\frac{1}{2}}(Q_{2,n}^{\frac{1}{2}})^*]ds\nonumber\\
&&=\int_0^{{+\infty}}\frac{\partial}{\partial s}P_{s,n}^x\rho_n(t,x,y)ds\nonumber\\
&&=\lim\limits_{s\rightarrow{+\infty}}\mathbb{E}\rho_n(t,
x,Y_s^{x,n}(y))-\rho_n(t,x,y)\nonumber\\
&&=\int_{H^{(n)}}\rho_n(t,x,y)\mu^{x,n}(dy)-\rho_n(t,x,y)\nonumber\\
&&=-\rho_n(t,x,y),\nonumber\end{aligned}$$ which means $\mathcal{L}_1^n(\int_0^{{+\infty}}P_{s,n}^x\rho_n(t,x,y)
ds)=-\rho_n(t,x,y).$ Therefore, we can set $$\begin{aligned}
\label{u-1}
u_{1,n}(t,x,y)=\int_0^{+\infty}
\mathbb{E}\rho_n(t,x,Y^{x,n}_s(y))ds.\end{aligned}$$
\[u-1-abso-lemma\] Assume (H.1) and (H.2). [Then for any ]{}$x, y\in H^{(n)}$ and $T>0$, we have $$\begin{aligned}
|u_{1,n}(t,x,y)|\leq C_T(1 +\|x\|+\|y\|),\;t\in[0, T].
\label{u-1-abso}\end{aligned}$$
As known from , we have $$u_{1,n}(t,x,y)=\int_0^{{+\infty}}\mathbb{E}\Big(\bar{ F}_n(x)-
F_n(x,Y_s^{x,n}(y)), D_x\bar{u}_n(t,x)\Big)_{ {H} }ds.$$ This implies that $$\begin{aligned}
|u_{1,n}(t,x,y)|&\leq&\int_0^{{+\infty}}\|\bar{F}_n(x)- \mathbb{E}[
F_n(x,Y_s^{x,n}(y))]\|\cdot\|D_x\bar{u}_n(t,x)\| ds.\end{aligned}$$ Then, in view of Lemma \[ux\] and , this implies: $$\begin{aligned}
|u_{1,n}(t,x,y)|&\leq& C_T(1+\|x\|+\|y\|)\int_0^{{+\infty}}e^{-\frac{\beta}{2} s}ds\\
&\leq&C_T(1+\|x\|+\|y\|).\end{aligned}$$
**Determination of remainder** $ {r^\epsilon_n}$
------------------------------------------------
Once $u_{0,n}$ and $u_{1,n}$ have been determined, we can carry out the construction of the remainder $r^\epsilon_n$. It is known that $$\begin{aligned}
(\partial_t-\mathcal{L}^{\epsilon,n})u^\epsilon_n=0,\end{aligned}$$ which, together with and , implies $$\begin{aligned}
(\partial_t-\mathcal{L}^{\epsilon,n})r^\epsilon_n&=&-(\partial_t-\mathcal{L}^{\epsilon,n})
u_{0,n}-\epsilon(\partial_t-\mathcal{L}^{\epsilon,n})u_{1,n}\\
&=&-(\partial_t-\frac{1}{\epsilon}\mathcal{L}_1^n-\mathcal{L}_2^n)u_{0,n}-\epsilon(\partial_t
-\frac{1}{\epsilon}\mathcal{L}_1^n-\mathcal{L}_2^n)u_{1,n}\\
&=&\epsilon(\mathcal{L}_2^nu_{1,n}-\partial_tu_{1,n}).\end{aligned}$$ In order to estimate the remainder term $r^\epsilon_n$ we need the following crucial lemmas.
\[4.3\] [Assume that $x, y\in H^{(n)}$]{}. Then, under conditions (H.1) and (H.2), for any $T>0$ and $\theta\in(0, 1]$ we have $$\begin{aligned}
\left|\frac{\partial u_{1,n}}{\partial t}(t,x,y)\right|
&\leq&C_T(1+\frac{1}{t}+t^{\theta-1})(1+\|x\|+\|y\|+\|x\|_{(-A_n)^\theta})^2,\;
t\in [0, T].\end{aligned}$$
According to , we have $$\frac{\partial u_{1,n}}{\partial
t}(t,x,y)=\int_0^{{+\infty}}\mathbb{E}\left(\bar{F}_n(x)-
{F}_n(x,Y_s^{x,n}(y)), \frac{\partial }{\partial
t}D_x\bar{u}_n(t,x)\right)_{{H}}ds.\label{u1=deriv}$$ For any $h\in H^{(n)},$ $$\begin{aligned}
D_x\bar{u}_n(t,x)\cdot h&=&\mathbb{E}[\phi'(\bar{X}_t^n(x))\cdot
D_x\bar{X}_t^n(x)\cdot h]\nonumber\\
&=&\mathbb{E}\Big(\phi'(\bar{X}_t^n(x)),\eta^{h,x,n}_t\Big)_H,\label{u(t,x)h-deriv}\end{aligned}$$ here $\eta^{h,x,n}_t$ is the mild solution (also *strong solution*) of variation equation corresponding to the problem in form $$\begin{aligned}
\begin{cases}
d\eta ^{h,x,n}_t=\left(A_n\eta ^{h,x,n}_t+ \bar{{F}}_n'(\bar{X}_t^n(x))\cdot\eta ^{h,x,n}_t\right)dt,\\
\eta ^{h,x,n}_0=h.
\end{cases}\end{aligned}$$ Keep in mind that $\bar{X}_t^n(x)$ is the *strong solution* of equation with initial value $\bar{X}_0^n(x)=x$. By Itô’s formula in finite dimensional spaces, we get $$\begin{aligned}
\phi'(\bar{X}_t^n(x))&=&\phi'(x)+\int_0^t\phi''(\bar{X}_s^n(x))\cdot
[A_n\bar{X}_s^n(x)+\bar{F}_n(\bar{X}_s^n(x))]ds\\
&+&\int_0^t\phi''(\bar{X}_s^n(x))dW_s^{1,n}\\
&+&\frac{1}{2}\sum\limits_{k=1}^n\int_0^t\phi'''(\bar{X}_s^n(x))\cdot\Big(\sqrt{\lambda_{1,k}}e_k,\sqrt{\lambda_{1,k}}e_k\Big)ds,\end{aligned}$$ where $W_t^{1,n}:=\sum\limits_{k=1}^n\sqrt{\lambda_{i,k}}B^{(i)}_{t,k}e_k
$ denotes the $Q^{1,n}-$Wiener process in $H^{(n)}$. Then, by using again Itô’s formula, after taking the expectation we have $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\mathbb{E}\Big(\phi'(\bar{X}_t^n(x)),\eta^{h,x,n}_t\Big)_H\\
&=&\Big(\phi'(x),h\Big)_H\\
&+&\mathbb{E}\int_0^t\Big(\eta_s^{h,x,n},\phi''(\bar{X}_s^n(x))\cdot[A_n\bar{X}_s^n(x)+\bar{F}_n(\bar{X}_s^n(x))]\Big)_Hds\\
&+&\mathbb{E}\int_0^t\Big(\phi'(\bar{X}_s^n(x)),A_n\eta_s^{h,x,n}+\bar{F}'_n(\bar{X}_s^n(x))\cdot\eta_s^{h,x,n}\Big)_Hds\\
&+&\frac{1}{2}\mathbb{E}\sum\limits_{k=1}^n\int_0^t\Big(\phi'''(\bar{X}_s^n(x))\cdot
\big(\sqrt{\lambda_{1,k}}e_k,\sqrt{\lambda_{1,k}}e_k\big),\eta_s^{h,x,n}\Big)_Hds.\end{aligned}$$ Now, returning to and differentiating with respect to $t$, we obtain $$\begin{aligned}
\frac{\partial}{\partial t}(D_x\bar{u}_n(t,x)\cdot h)&=&\mathbb{E} \Big(\eta_t^{h,x,n},\phi''(\bar{X}_t^n(x))\cdot[A_n\bar{X}_t^n(x)+\bar{F}_n(\bar{X}_t^n(x))]\Big)_H\\
&+&\mathbb{E}\Big(\phi'(\bar{X}_t^n(x)),A_n\eta_t^{h,x,n}+\bar{F}'_n(\bar{X}_t^n(x))\cdot\eta_t^{h,x,n}\Big)_H\\
&+&\frac{1}{2}\mathbb{E}\sum\limits_{k=1}^n\Big(\phi'''(\bar{X}_t^n(x))\cdot\big(\sqrt{\lambda_{1,k}}e_k,\sqrt{\lambda_{1,k}}e_k\big),\eta_t^{h,x,n}\Big)_H,\end{aligned}$$ so that $$\begin{aligned}
\left|\frac{\partial}{\partial t}(D_x\bar{u}_n(t,x)\cdot
h)\right|&\leq&C\mathbb{E} \left[\|\eta_t^{h,x,n}\|
(\|A_n\bar{X}_t^n(x)\|+\|\bar{F}_n(\bar{X}_t^n(x))\|)\right]\\
&+&C\mathbb{E}
\|A_n\eta_t^{h,x,n}\|+C\mathbb{E}\|\bar{F}'_n(\bar{X}_t^n(x))\cdot\eta_t^{h,x,n}\|\\
&+& \mathbb{E}\sum\limits_{k=1}^\infty{\lambda_k} \|\eta_t^{h,x,n}\|.\\\end{aligned}$$ Then, as holds, by using Lemma \[bar-X\], Lemma \[eta-bound\] and Lemma \[eta-regularity\], it follows $$\begin{aligned}
\left|\frac{\partial}{\partial t}(D_x\bar{u}_n(t,x)\cdot
h)\right|&\leq&C_T\|h\|(t^{\theta-1}\|x\|_{(-A_n)^\theta}+1+\|x\|)\\
&&+C\|h\|(1+\frac{1}{t})(1+\|x\|)\\
&\leq&C\|h\|(1+\frac{1}{t}+t^{\theta-1}\|x\|_{(-A_n)^\theta}
+\frac{1}{t}\|x\|+\|x\|).\end{aligned}$$ Hence, as holds, from the above estimate and we get $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!\left|\frac{\partial u_{1,n}}{\partial
t}(t,x,y)\right| \leq
C_T(1+\frac{1}{t}+t^{\theta-1}\|x\|_{(-A_n)^\theta}
+\frac{1}{t}\|x\|+\|x\|)\\
&&\qquad\qquad\qquad\cdot\int_0^{{+\infty}}\mathbb{E}
\|\bar{ F}_n(x)- {F}_n(x,Y_s^{x,n}(y))\|ds\\
&\leq&C_T(1+\frac{1}{t}+t^{\theta-1}\|x\|_{(-A_n)^\theta}
+\frac{1}{t}\|x\|+\|x\|)(1+\|x\|+\|y\|)\int_0^{+\infty}e^{-\frac{\beta}{2}s}ds\\
&\leq&C_T(1+\frac{1}{t}+t^{\theta-1}\|x\|_{(-A_n)^\theta}
+\frac{1}{t}\|x\|+\|x\|)(1+\|x\|+\|y\|)\\
&\leq&C_T(1+\frac{1}{t}+t^{\theta-1})\left(1+\|x\|+\|y\|+\|x\|_{(-A_n)^\theta}\right)^2.\end{aligned}$$
\[4.4\] [Assume ]{}$x\in H^{(n)}$ and $y\in H^{(n)}$. Then, under conditions (H.1) and (H.2), for any $T>0$ we have $$\begin{aligned}
\left|\mathcal {L}_2^nu_{1,n}(t,x,y)\right|\leq
C_T\big(1+\|A_nx\|+\|x\|+\|y\| \big)\big(1+\|x\|+\|y\| \big), \;t\in
[0, T].\end{aligned}$$
As known, [for any]{} $x\in H^{(n)}$ it holds $$\begin{aligned}
\mathcal {L}_2^nu_{1,n}(t,x,y)&=&\Big(A_nx+ F_n(x, y),
D_xu_{1,n}(t,x,y)
\Big)_{H}\\
&+&\frac{1}{2}\sigma_2^2Tr\Big(D^2_{xx}u_{1,n}(t,x,y)Q_{1,n}^\frac{1}{2}(Q_{1,n}^\frac{1}{2})^*\Big).\end{aligned}$$ We will carry out the estimate of $\left|\mathcal {L}_2^nu_{1,n}(t,x,y)\right|$ in two steps.\
(**Step 1**) Estimate of $\Big( {A}_nx+ {F}_n(x, y),
D_xu_{1,n}(t,x,y) \Big)_{H}$.
For any $k\in {H}^{(n)}$, we have $$\begin{aligned}
& &D_xu_{1,n}(t,x,y)\cdot k\\
&&=\int_0^{{+\infty}}\Big(D_x(\bar{
F}_n(x)-\mathbb{E}{F}_n(x,Y_s^{x,n}(y)))\cdot k,D_x\bar{u}_n(t,x)\Big)_{{H}}ds\\
&&+\int_0^{{+\infty}}\Big(\bar{F}_n(x)-\mathbb{E} {F}_n(x,Y_s^{x,
n}(y)),D^2_{xx}\bar{u}_n(t,x)\cdot
k\Big)_{ {H} }ds\\
&&:=I_{1,n}(t,x,y,k)+I_{2,n}(t,x,y,k).\end{aligned}$$ Directly, we have $$\begin{aligned}
&&|I_{1,n}(t,x,y,k)|\nonumber\\
&&\leq \int_0^{+\infty} \left| \Big(D_x(\bar{F}_n(x)-
\mathbb{E}{F}_n(x,Y_s^{x,n}(y)))\cdot
k,D_x\bar{u}_n(t,x)\Big)_{{H}}\right| ds.\nonumber\end{aligned}$$ By making use of , the above yields $$\begin{aligned}
|I_{1,n}(t,x,y,k)|&\leq&C\|k\| \cdot\|D_x\bar{u}_n(t,x)\|
\cdot\int_0^{+\infty}
e^{-c s} (1+\|x\|+\|y\|)ds\nonumber\\
&\leq& C\|k\| \cdot\|D_x\bar{u}_n(t,x)\|(1+\|x\|+\|y\|)\nonumber \\
&\leq& C\|k\|(1+\|x\|+\|y\|), \label{I_1}\end{aligned}$$ where we used Lemma \[ux\] in the last step. By Lemma \[uxx\] and , we have $$\begin{aligned}
|I_{2,n}(t,x,y,k)|&\leq& \int_0^{{+\infty}}\left|\Big(\bar{F}_n(x)-
\mathbb{E} {F}_n(x,Y_s^{x,n}(y)),D^2_{xx}\bar{u}_n(t,x)\cdot k\Big)_{{H}}\right|ds\nonumber\\
&\leq& C\|k\|\int_0^{{+\infty}}\|\bar{F}_n(x)-
\mathbb{E}{F}_n(x,Y_s^{x,n}(y))\|ds\nonumber\\
&\leq&C\|k\|(1+\|x\| +\|y\|)\int_0^{+\infty} e^{-\frac{\beta}{2} s} ds\nonumber\\
&\leq&C\|k\|(1 +\|x\| +\|y\|).\nonumber\end{aligned}$$ Together with , this allows us to get $$\begin{aligned}
\left|D_xu_{1,n}(t,x,y)\cdot k\right|&\leq&C\|k\| (1+\|x\|+ \|y\|)\end{aligned}$$ which means $$\begin{aligned}
&&\left|\Big({A}_nx+ {F}_n(x,
y), D_xu_{1,n}(t,x,y) \Big)_{H}\right|\nonumber\\
&&\leq C\big(1+\|{A}_nx\|+\|x\|+\|y\| \big)\big(1+\|x\|+\|y\|
\big).\label{step1-estimate}\end{aligned}$$
(**Step 2**) Estimate of $Tr\Big(D^2_{xx}u_{1,n}(t,x,y)Q_{1,n}^\frac{1}{2}(Q_{1,n}^\frac{1}{2})^*\Big)$.\
By differentiating twice with respect to $x\in H^{(n)}$ in $u_{1,n}(t,x,y)$, for any $x, h, k\in H^{(n)}$ we have $$\begin{aligned}
&&D_{xx}u_{1,n}(t,x,y)\cdot (h,
k)\\
&&=\int_0^{{+\infty}}\Big(D^2_{xx}(\bar{{F}}_n(x)-
\mathbb{E}{F}_n(x,Y_s^{x,n}(y)))\cdot (h,k),D_x\bar{u}_n(t,x)\Big)_{{H}}ds\\
&&+\int_0^{{+\infty}}\Big(D_x(\bar{{F}}_n(x)-
\mathbb{E}{F}_n(x,Y_s^{x,n}(y)))\cdot h,D^2_{xx}\bar{u}_n(t,x)\cdot
k\Big)_{{H} }ds\\
&&+\int_0^{{+\infty}}\Big(D_x(\bar{{F}}_n(x)-
\mathbb{E}{F}_n(x,Y_s^{x,n}(y)))\cdot k,D^2_{xx}\bar{u}_n(t,x)\cdot
h\Big)_{{H} }ds\\
&&+\int_0^{{+\infty}}\Big(\bar{{F}}_n(x)-\mathbb{E}{F}_n(x,Y_s^{x,n}(y)),
D^3_{xxx}\bar{u}_n(t,x)\cdot (h,k)\Big)_{{H}
}ds\\
&&:=\sum\limits_{i=1}^4J_{i,n}(t,x,y,h,k).\end{aligned}$$ By taking Lemma \[mix-derivative-2\] and Lemma \[ux\] into account, we can deduce $$\begin{aligned}
&&|J_{1,n}(t,x,y,h,k)|\nonumber\\
&&\leq C\|h\|\cdot\|k\|\cdot\int_0^{+\infty}
e^{-c s} (1+\|x\|+\|y\|)ds\nonumber\\
&&\leq C\|h\|\cdot\|k\|(1+\|x\|+\|y\|). \label{J_1}\end{aligned}$$ Next, thanks to Lemma \[mix-derivative\] and Lemma \[uxx\] it holds $$\begin{aligned}
&&|J_{2,n}(t,x,y,h,k)|\nonumber\\
&&\leq C\|h\|\cdot\|k\| \cdot\int_0^{+\infty}
e^{-c s} (1+\|x\|+\|y\|)ds\nonumber\\
&&\leq C\|h\|\cdot\|k\|(1+\|x\|+\|y\|). \label{J_2}\end{aligned}$$ Parallel to , we can obtain the same estimate for $J_{3,n}(t,x,y, h,k)$, that is, $$\begin{aligned}
|J_{3,n}(t,x,y,h,k)| \leq C\|h\|\cdot\|k\|(1+\|x\|+\|y\|).
\label{J_3}\end{aligned}$$ Thanks to Lemma \[uxxx\] and , we get $$\begin{aligned}
&& |J_{4,n}(t,x,y,h,k)|\nonumber\\
&&\leq C\|h\|\cdot\|k\|
\cdot\int_0^{+\infty}e^{-\frac{\beta}{2} s} (1+\|x\|+\|y\|)ds\nonumber\\
&&\leq C\|h\|\cdot\|k\|(1+\|x\|+\|y\|). \label{J_4}\end{aligned}$$ Collecting together , , and , we obtain
$$\begin{aligned}
|D^2_{xx}u_{1,n}(t,x,y)\cdot (h, k)|\leq C
\|h\|\cdot\|k\|(1+\|x\|+\|y\|),\end{aligned}$$
so that, as the operator $Q_1$ has finite trace (see ), we get $$\begin{aligned}
&&\left|Tr\Big(D^2_{xx}u_{1,n}(t,x,y)Q_{1,n}^\frac{1}{2}(Q_{1,n}^\frac{1}{2})^*\Big)\right|\nonumber\\
&&=\sum\limits_{k=1}^n \left|D^2_{xx}u_{1,n}(t,x,y)\cdot\Big(\sqrt{\lambda_{1,k}}e_k, \sqrt{\lambda_{1,k}}e_k\Big)\right|\nonumber\\
&&\leq C(1+\|x\|+\|y\|). \label{Tr}\end{aligned}$$
Finally, by taking inequalities and into account, we can conclude the proof of the lemma.
As a consequence of Lemma \[4.3\] and \[4.4\], we have the following fact for the remainder term $r^\epsilon_n$.
\[4.5\] Under the conditions of Lemma \[4.3\], for any $r\in (0,
\frac{1}{4})$ we have $$\begin{aligned}
r_n^\epsilon(T,x,y)\leq C_{r,
T,\theta}\epsilon^{1-2r}(1+\|x\|^2+\|y\|^2+\|x\|^2_{(-A_n)^\theta}).\end{aligned}$$
By a variation of constant formula, we have $$\begin{aligned}
&&r_n^\epsilon(T,x,y)\nonumber\\
&&=\mathbb{E}[r_n^\epsilon(\delta_\epsilon,X^{\epsilon,n}_{T-\delta_\epsilon}(x,y),Y_{T-\delta_\epsilon}^{\epsilon,n}(x, y))]\nonumber\\
&&+\epsilon\mathbb{E}\left[\int^T_{\delta_\epsilon}(\mathcal{L}_2^nu_{1,n}-\frac{\partial
u_{1,n}}{\partial s})(s,
X^{\epsilon,n}_{T-s}(x,y),Y^{\epsilon,n}_{{T-s}}(x, y))
ds\right],\label{r-mild}\end{aligned}$$ where $\delta_\epsilon\in (0,\frac{T}{2})$ is a constant, only depending on $\epsilon>0$, to be chosen later. Now, we estimate the two terms in the right hand side of . Firstly, note that $u_n^\epsilon(0,x,y)=\bar{u}_n(0,x)$, it holds $$\begin{aligned}
r_n^\epsilon(\delta_\epsilon,
x,y)&=&u_n^\epsilon(\delta_\epsilon,x,y)-\bar{u}_n(\delta_\epsilon,x)-\epsilon
u_{1,n}(\delta_\epsilon,x,y)\\
&=&-\epsilon
u_{1,n}(\delta_\epsilon,x,y)+[u_n^\epsilon(\delta_\epsilon,x,y)-u_n^\epsilon(0,x,y)]\\
&&-[\bar{u}_n(\delta_\epsilon,x)-\bar{u}_n(0,x)].\end{aligned}$$ By lemma \[u-1-abso-lemma\], we have $$\begin{aligned}
|\epsilon u_{1,n}(\delta_\epsilon,x,y)|\leq C_T\epsilon
(1+\|x\|+\|y\|).\label{r-u1}\end{aligned}$$ By using Itô’s formula and taking the expectation we obtain $$\begin{aligned}
&&u_n^\epsilon(\delta_\epsilon,x,y)-u_n^\epsilon(0,x,y)\\
&&=\mathbb{E}\int_0^{\delta_\epsilon}\phi'(X_s^{\epsilon,n}(x,y))\cdot[A_nX_s^{\epsilon,n}(x,y)+F_n(X_s^{\epsilon,n}(x,y),Y_s^{\epsilon,n}(x,y))]
ds\\
&&\quad+\frac{1}{2}\mathbb{E}\sum\limits_{k=1}^n\int_0^{\delta_\epsilon}\phi''(X_s^\epsilon(x,y))\cdot\left(\sqrt{\lambda_{1,k}}e_k,
\sqrt{\lambda_{1,k}}e_k\right).\end{aligned}$$ Then, due to Lemma \[moment-bound\] and \[A-X\], for any $r\in
(0, \frac{1}{4})$ we have $$\begin{aligned}
&&\left|u_n^\epsilon(\delta_\epsilon,x,y)-u_n^\epsilon(0,x,y)\right|\nonumber\\
&&\leq C\int_0^{\delta_\epsilon}
\big[\mathbb{E}\|A_nX_s^{\epsilon,n}(x,y)\|+1+\mathbb{E}\|X_s^{\epsilon,n}(x,y)\|+\mathbb{E}\|Y_s^{\epsilon,n}(x,y)\|\big]
ds\nonumber\\
&&\quad+C Tr(Q_1)\delta_\epsilon\nonumber\\
&& \leq
C_{r,T}(\delta_\epsilon+\frac{\delta_\epsilon^\theta}{\theta}+\frac{\delta_\epsilon
}{\epsilon^r})(1+\|x\|_{(-A_n)^\theta}+\|x\|+\|y\|). \label{r-u}\end{aligned}$$ By using again Itô’s formula, we get $$\begin{aligned}
&&\bar{u}_n(\delta_\epsilon,x)-\bar{u}_n(0,x)\\
&&=\mathbb{E}\int_0^{\delta_\epsilon}\phi'(\bar{X}_s^n
(x))\cdot[A_n\bar{X}_s^n(x)+\bar{F}_n(\bar{X}_s^n(x))]
ds\\
&&\quad+\frac{1}{2}\mathbb{E}\sum\limits_{k=1}^n\int_0^{\delta_\epsilon}\phi''(\bar{X}_s^n(x))\cdot\left(\sqrt{\lambda_{1,k}}e_k,
\sqrt{\lambda_{1,k}}e_k\right).\end{aligned}$$ Then, thanks to Lemma \[bar-X\] and it holds $$\begin{aligned}
&&\left|\bar{u}_n(\delta_\epsilon,x)-\bar{u}_n(0,x)\right|\\
&&\leq C\int_0^{\delta_\epsilon}
\big[\mathbb{E}\|A_n\bar{X}_s^n(x)\|+1+\mathbb{E}\|\bar{X}_s^n(x)\|\big]
ds\\
&&\quad+C Tr(Q_1)\delta_\epsilon\\
&& \leq C_T(\delta_\epsilon+\frac{\delta_\epsilon^\theta}{\theta}
)(1+\|x\|_{(-A_n)^\theta}+\|x\|),\end{aligned}$$ which, in view of and , means that $$\begin{aligned}
|r_n^\epsilon(\delta_\epsilon, x,y)|\leq
C_{r,T}(\epsilon+\delta_\epsilon+\frac{\delta_\epsilon^\theta}{\theta}+\frac{\delta_\epsilon
}{\epsilon^r})(1+\|x\|_{(-A_n)^\theta}+\|x\|+\|y\|),\end{aligned}$$ so that, due to Lemma \[moment-bound\] and \[A-X\], [this easily]{} implies that $$\begin{aligned}
&&\mathbb{E}[r_n^\epsilon(\delta_\epsilon,X^{\epsilon,n}_{T-\delta_\epsilon}(x,y),Y_{T-\delta_\epsilon}^{\epsilon,n}(x,
y))]\\
&&\leq
C_{r,T}(\epsilon+\delta_\epsilon+\frac{\delta_\epsilon^\theta}{\theta}+\frac{\delta_\epsilon
}{\epsilon^r})(1+\mathbb{E}\|X^{\epsilon,n}_{T-\delta_\epsilon}(x,y)\|_{(-A_n)^\theta}+\mathbb{E}\|Y^{\epsilon,n}_{T-\delta_\epsilon}(x,y)\|)\\
&&\leq
C_{r,T}(\epsilon+\delta_\epsilon+\frac{\delta_\epsilon^\theta}{\theta}+\frac{\delta_\epsilon
}{\epsilon^r})\big(|T-\delta_\epsilon|^{\theta-1}\|x\|_{(-A)^\theta}+(1+\|x\|+\|y\|)
(1+\frac{1}{\epsilon^{r}})\big)\\
&&\leq C_{r, T,
\theta}(\epsilon+\delta_\epsilon+\frac{\delta_\epsilon^\theta}{\theta}+\frac{\delta_\epsilon
}{\epsilon^r})(1+\frac{1}{\epsilon^{r}})(\|x\|_{(-A)^\theta}+1+\|x\|+\|y\|).\end{aligned}$$ If we pick $\delta_\epsilon=\epsilon^{\frac{1}{\theta}}\leq
\epsilon$, we get $$\begin{aligned}
&&\mathbb{E}[r_n^\epsilon(\delta_\epsilon,X^{\epsilon,n}_{T-\delta_\epsilon}(x,y),Y_{T-\delta_\epsilon}^{\epsilon,n}(x,
y))]\nonumber\\
&&\leq C_{r, T,
\theta}\epsilon^{1-2r}(1+\|x\|_{(-A_n)^\theta}+\|x\|+\|y\|).\label{r-1-expectation}\end{aligned}$$ Next, we estimate the second term in the right hand side of . Thanks to Lemma \[4.3\] and Lemma \[4.4\], we have $$\begin{aligned}
&& \left|(\mathcal{L}_2^nu_{1,n}-\frac{\partial u_{1,n}}{\partial
s})(s,
X^{\epsilon,n}_{T-s}(x,y),Y_{T-s}^{\epsilon,n} (x,y))\right|\\
&&\leq C_T
(1+\frac{1}{s}+s^{\theta-1})\big[1+\|X^{\epsilon,n}_{T-s}(x,y)\|+\|Y^{\epsilon,n}_{T-s}(x,y)\|\big]^2\\
&&\quad+C_T\|{A}_nX^{\epsilon,n}_{T-s}(x,y)\|\left(1+\|X^{\epsilon,n}_{T-s}(x,y)\|+\|X^{\epsilon,n}_{T-s}(x,y)\|\right),\end{aligned}$$ and, according to the previous Lemma \[moment-bound\] and Lemma \[A-X\], this implies that
$$\begin{aligned}
&&\epsilon\left|\mathbb{E}\left[\int^T_{\delta_\epsilon}(\mathcal{L}_2^nu_{1,n}-\frac{\partial
u_{1,n}}{\partial s})(s,
X^{\epsilon,n}_{T-s}(x,y),Y^{\epsilon,n}_{{T-s}}(x, y))
ds\right]\right|\\
&&\leq C_T\epsilon
\int^T_{\delta_\epsilon}(1+\frac{1}{s}+s^{\theta-1})\mathbb{E}\big[1
+\|X^{\epsilon,n}_{T-s}(x,y)\|^2+\|Y^{\epsilon,n}_{T-s}(x,y)\|^2\big] ds \\
&&\quad+
C_{r,T}\epsilon\int_{\delta_\epsilon}^T\left[\mathbb{E}\|{A}_nX^{\epsilon,n}_{T-s}(x,y)\|^2\right]^\frac{1}{2}
\\
&&\qquad\qquad\qquad\cdot\left[\mathbb{E}(1+\|X^{\epsilon,n}_{T-s}(x,y)\|+\|X^{\epsilon,n}_{T-s}(x,y)\|)^2\right]^\frac{1}{2}ds\\
&&\leq C_{r,T}\epsilon(1+\|x\|^2+\|y\|^2+\|x\|^2_{(-A_n)^\theta})\\
&&\quad\quad\cdot \int^T_{\delta_\epsilon}(1+\frac{1}{\epsilon^r}+\frac{1}{s}+s^{\theta-1}+|T-s|^{\theta-1})ds\\
&&\leq C_{r,T} \epsilon(T+\frac{T^\theta}{\theta}+|\log T|
+|\log(\delta_\epsilon)|+\frac{T}{\epsilon^r})\\
&&\quad\quad\cdot(1+\|x\|^2+\|y\|^2+\|x\|^2_{(-A_n)^\theta})\\
&&\leq
C_{r,\theta,T}\epsilon(1+|\log\epsilon|+\frac{1}{\epsilon^r})(1+\|x\|^2+\|y\|^2+\|x\|^2_{(-A_n)^\theta})\\
&&\leq
C_{r,\theta,T}\epsilon^{1-r}(1+\|x\|^2+\|y\|^2+\|x\|^2_{(-A_n)^\theta}),\end{aligned}$$
which, together with , completes the proof.
Proof of [Theorem 3.1]{}
------------------------
Now we finish proof of main result introduced in Section 3
We stress that we need only to prove . With the notations introduced above, by Lemma \[u-0\], Lemma \[u-1-abso-lemma\] and Lemma \[4.5\], for any $r\in (0, 1)$ $x\in \mathscr{D}((-A)^{\theta})$ and $y\in H$ we have $$\begin{aligned}
&&\left|\mathbb{E}\phi(X^{\epsilon,n}_T(x^{(n)},y^{(n)}))-\mathbb{E}\phi(\bar{X}^{n}_T(x^{(n)}))\right|\\
&&=|u_n^\epsilon(T,x^{(n)},y^{(n)})-\bar{u}_n(T,x^{(n)}))|\\
&&=|u_{1,n}^\epsilon(T, x^{(n)},y^{(n)})|\epsilon+|r_n^\epsilon (T, x^{(n)},y^{(n)})|\\
&&\leq
C_{r,\theta,T}\epsilon^{1-r}(1+\|x^{(n)}\|^2+\|y^{(n)}\|^2+\|x^{(n)}\|^2_{(-A_n)^\theta})\\
&&\leq
C_{r,\theta,T}\epsilon^{1-r}(1+\|x\|^2+\|y\|^2+\|x\|^2_{(-A)^\theta}),\end{aligned}$$ where $C_{r,\theta,T}$ is a constant independent of the dimension $n$.\
The proof of Theorem \[theorem\] is completed.
Appendix
========
In this section, we state and prove some technical lemmas used in the previous sections. We first study the differential dependence on initial datum for the solution $\bar{X}_t^n(x)$ of the averaged system . In what follows we denote by $\eta ^{h,x,n}_t$ the derivative of $\bar{X}_t^n(x)$ with respect to $x$ along direction $h\in H^{(n)}$.
\[eta-bound\] Under (H.1) and (H.2), for any $x,h\in H^{(n)}$ and $T>0$ there exits a constant $C_T>0$ such that for any $x, h\in H^{(n)}$, $$\begin{aligned}
\|\eta ^{h,x,n}_t\| \leq C_T\|h\|, \;t\in [0, T].\end{aligned}$$
Note that $\eta^{h,x,n}_t$ is the mild solution of the first variation equation associated with the problem : $$\begin{aligned}
\begin{cases}
d\eta ^{h,x,n}_t=\left({A}_n\eta ^{h,x,n}_t+ \bar{{F}}'(\bar{X}_t^n(x))\cdot\eta ^{h,x,n}_t\right)dt,\\
\eta ^{h,x,n}_0=h.
\end{cases}\end{aligned}$$ This means that $\eta ^{h,x,n}_t$ is the solution of the integral equation $$\begin{aligned}
\eta^{h,x,n}_t={S}_{t,n}h+\int_0^t {S}_{t-s,n}[
\bar{{F}}'_n(\bar{X}_s(x))\cdot\eta ^{h,x,n}_s]ds,\end{aligned}$$ and then, due to and contractive property of $S_{t,n}$, we get $$\begin{aligned}
\|\eta ^{h,x,n}_t\| \leq \|h\| +C\int_0^t\|\eta ^{h,x,n}_s\|ds.\end{aligned}$$ Then by Gronwall lemma it follows that $$\begin{aligned}
\|\eta ^{h,x,n}_t\| \leq C_T\|h\|, \;t\in [0, T].\end{aligned}$$
\[eta-continuous\] Under the conditions of Lemma \[eta-bound\], for any $T>0$ and $r\in (0,1)$ there exists a constant $C_{r, T }>0$ such that for any $x,h\in H^{(n)}$ and $0<s\leq t\leq T$, $$\begin{aligned}
\|\eta_t^{h,x,n}-\eta_s^{h,x,n}\|\leq
C_{r,T}|t-s|^{1-r}(1+\frac{1}{s^{1-r}})\|h\|.\end{aligned}$$
See Proposition B.5 in [@Brehier].
\[eta-regularity\] Under the conditions of Lemma \[eta-bound\], for any $T>0$ there exists a constant $C_{T}>0$ such that for any $x, h\in H^{(n)}$, $$\begin{aligned}
\|A_n\eta^{h,x,n}_t\|\leq C_T (1+\frac{1}{t})(1+\|x\|)\|h\|,\;t\in
[0, T].\end{aligned}$$
See Proposition B.6 in [@Brehier].
After we have study the first order derivative of $\bar{X}_t^n(x)$, we introduce the second order derivative of $\bar{X}_t^n(x)$ with respect to $x$ in directions $h, k\in H^{(n)}$ denoted by $\zeta^{h,k,x,n}$, which is the solution of the second variation equation $$\begin{aligned}
\begin{cases}
d\zeta^{h,k,x,n}_t=\Big[{A}_n\zeta^{h,k,x,n}_t+
\bar{{F}}''_n(\bar{X}_t^n(x))\cdot(\eta^{h,x,n}_t,\eta^{k,x,n}_t)\\
\qquad\qquad\quad+\bar{{F}}'_n(\bar{X}_t^n(x))\cdot\zeta^{h,k,x,n}_t\Big]dt,\\
\zeta^{h,k,x}_0=0.
\end{cases}\label{variation-2}\end{aligned}$$
\[xi-2-bound\] Under the conditions of Lemma \[eta-bound\], for any $T>0$ there exists a constant $C_{T}>0$ such that for any $x, h, k\in H^{(n)}$, $$\|\zeta^{h,k,x,n}_t\|\leq C_T\|h\|\cdot \|k\|, \; t\in [0, T].$$
See Proposition B.7 in [@Brehier].
We now introduce the regular results for $\bar{u}_n(t,x)$ defined in Section \[asym\].
\[ux\] For any $T>0$, there exists a constant $C_T>0$ such that for any $x\in {H}^{(n)}$ and $t\in [0, T]$, we have $$\|D_{x}\bar{u}_n(t,x)\|\leq C_{T, \phi}.$$
Note that for any $t\in [0, T]$ and $h\in H^{(n)}$, $$\begin{aligned}
D_x\bar{u}_n(t,x)\cdot h
=\mathbb{E}\left(\phi'(\bar{X}_t^n(x)),\eta^{h,x,n}_t\right)_{H}.\end{aligned}$$ By Lemma \[eta-bound\], we have $$\begin{aligned}
|D_x\bar{u}_n(t,x)\cdot h|\leq C_T\sup\limits_{z\in H
}\|\phi'(z)\|\cdot\|h\|, \label{u-bar-deriv}\end{aligned}$$ so that $$\begin{aligned}
\|D_x\bar{u}_n(t,x)\| \leq C_{T, \phi}.\label{u-bar-deriv-1-1}\end{aligned}$$
\[uxx\] For any $T>0$, there exists a constant $C_{T, \phi}>0$ such that for any $x,h,k\in H^{(n)}$ and $t\in [0, T]$, we have $$\left|D^2_{xx}\bar{u}_n(t,x)\cdot(h,k)\right|\leq C_{T,\phi}\|h\|\cdot\|k\|.$$
For any $h, k \in H^{(n)}$, we have $$\begin{aligned}
D^2_{xx}\bar{u}_n(t,x)\cdot(h,k)&=&\mathbb{E}\big[\phi''(\bar{X}_t^n(x))\cdot(\eta^{h,x,n}_t,\eta^{k,x,n}_t)\nonumber\\
&&+\phi'(\bar{X}_t^n(x))\cdot \zeta^{h,k,x,n}_t\big], \label{5.1}\end{aligned}$$ where $\zeta^{h,k,x,n}$ is governed by variation equation . By invoking Lemma \[eta-bound\] and Lemma \[xi-2-bound\], we can get $$\begin{aligned}
|D^2_{xx}\bar{u}_n(t,x)\cdot(h,k)|\leq C_{T, \phi} \|h\|\cdot\|k\|.\end{aligned}$$
By proceeding again as in the proof of above lemma, we have the following result.
\[uxxx\] For any $T>0$, there exists $C_T>0$ such that for any $x,h,k,l\in
H^{(n)}$ and $t\in [0, T]$, we have $$D^3_{xxx}\bar{u}_n(t,x)\cdot(h,k,l)\leq C_{T,\phi}\|h\|\cdot\|k\|\cdot\|l\|.$$
Finally, we introduce some regular results which is crucial in order to prove some important estimates in Section \[asym\].
\[mix-derivative\] There exist constants $C, c>0$ such that for any $x, y, h\in
H^{(n)}$ and $t>0$ it holds $$\begin{aligned}
\|D_x (\bar{F}_n(x)-\mathbb{E}F_n(x, Y^x_t(y)))\cdot h\| \leq
Ce^{-ct}\|h\| \left(1+\|x\| +\|y\| \right).\end{aligned}$$
We shall follow the approach of [@Brehier Proposition C.2]. For any $t_0>0$, we set $$\begin{aligned}
\tilde{F}_{t_0,n}(x,y,t)=\hat{F}_n(x,y,t)-\hat{F}_n(x,y,t+t_0),\end{aligned}$$ where $$\begin{aligned}
\hat{F}_n(x,y,t):=\mathbb{E}F_n(x, Y^{x,n}_t(y)).\end{aligned}$$ Thanks to Markov property we may write that $$\begin{aligned}
\tilde{F}_{t_0,n}(x,y,t)&=&\hat{F}_n(x,y,t)-\mathbb{E}F_n(x,Y_{t+t_0}^{x,n}(y))\\
&=&\hat{F}_n(x,y,t)-\mathbb{E}\hat{F}_n(x, Y_{t_0}^{x,n}(y),t).\end{aligned}$$ In view of the assumption (H.1), $\hat{F}_n$ is Gâteaux-differentiable with respect to $x$ at $(x,y,t)$. Therefore, we have for any $h\in H^{(n)}$ that $$\begin{aligned}
D_x\tilde{F}_{t_0,n}(x,y,t)\cdot h&=&D_x\hat{F}_n(x,y,t)\cdot
h-\mathbb{E}D_x\left(\hat{F}_n(x, Y_{t_0}^{x,n}(y),t)\right)\cdot h\nonumber\\
&=&\hat{F}_{n,x}'(x,y,t)\cdot h-\mathbb{E}\hat{F}_{n,x}'(x,
Y_{t_0}^{x,n}(y),t)\cdot
h\nonumber\\
&&-\mathbb{E}\hat{F}_{n,y}'(x,
Y_{t_0}^{x,n}(y),t)\cdot\left(D_xY_{t_0}^{x,n}(y)\cdot
h\right),\label{7-1-1}\end{aligned}$$ where we use the symbol $\hat{F}_{n,x}'$ and $\hat{F}_{n,y}'$ to denote the derivative with respect to $x$ and $y$, respectively. Note that the first derivative $\varsigma_t^{x,y,
h,n}=D_xY_{t}^{x,n}(y)\cdot h$, at the point $x$ and along the direction $h\in H^{(n)}$, is the solution of variation equation $$\begin{aligned}
d\varsigma_t^{x, y, h,n}&=&\left(A_n\varsigma_t^{x,
y,h,n}+G_{n,x}'(x, Y_t^{x,n}(y))\cdot h+G_{n,y}'(x,
Y_t^{x,n}(y))\cdot\varsigma_t^{x, y,h,n}\right)dt\end{aligned}$$ with initial data $\varsigma_0^{x,y, h,n}=0$. Hence, thanks to [(H.2)]{}, it is immediate to check that for any $t\geq 0$, $$\begin{aligned}
\mathbb{E}\|\varsigma_t^{x,y, h,n}\|\leq C\|h\|.\label{7-2}\end{aligned}$$ Note that there exists a constant $c>0$, such that, for any $y_1,
y_2\in H^{(n)}$, it holds $$\begin{aligned}
\|\hat{F}_n(x,y_1, t)-\hat{F}_n(x,y_2,t)\|&=&\|\mathbb{E}F_n(x,
Y_t^{x,n}(y_1))-\mathbb{E}F_n(x,
Y_t^{x,n}(y_2))\|\nonumber\\
&\leq&C\mathbb{E}\|Y_t^{x,n}(y_1)-Y_t^{x}(y_2)\|\nonumber\\
&\leq& Ce^{-ct}\|y_1-y_2\|,\nonumber\end{aligned}$$ which implies $$\begin{aligned}
\|\hat{F}_{n,y}'(x, y,t)\cdot k\|\leq Ce^{-ct}\|k\|,\; k\in
H.\label{7-3}\end{aligned}$$ Therefore, thanks to and , we can conclude that $$\begin{aligned}
\|\mathbb{E}[\hat{F}_{n,y}'(x,
Y_{t_0}^{x,n}(y),t)\cdot\left(D_xY_{t_0}^{x,n}(y)\cdot
h\right)]\|\leq C e^{-ct}\|h\|.\label{7-4}\end{aligned}$$ Then, we directly have $$\begin{aligned}
&&\hat{F}_{n,x}'(x,y_1,t)\cdot h-\hat{F}_{n,x}'(x,y_2,t)\cdot h\nonumber\\
&&\quad=\mathbb{E}\left(F_{n,x}'(x, Y_t^{x,n}(y_1))\right)\cdot
h-\mathbb{E}\left(F_{n,x}'(x, Y_t^{x,n}(y_2))\right)\cdot h\nonumber\\
&&\quad\quad+\mathbb{E}\left(F_{n,y}'(x, Y_t^{x,n}(y_1))\cdot
\varsigma_t^{x,y_1,
h,n}-F_{n,y}'(x, Y_t^{x,n}(y_2))\cdot \varsigma_t^{x,y_2, h,n}\right)\nonumber\\
&&\quad= \mathbb{E}\left(F_{n,x}'(x, Y_t^{x,n}(y_1))\right)\cdot
h-\mathbb{E}\left(F_{n,x}'(x, Y_t^{x,n}(y_2))\right)\cdot h\nonumber\\
&&\quad\quad+\mathbb{E}\left([F_{n,y}'(x,
Y_t^{x,n}(y_1))-F_{n,y}'(x,
Y_t^{x,n}(y_2))]\cdot\varsigma_t^{x,y_1, h,n} \right)\nonumber\\
&&\quad\quad+\mathbb{E}\left(F_{n,y}'(x, Y_t^{x,n}(
y_2))\cdot(\varsigma_t^{x,y_1, h,n}-\varsigma_t^{x,y_2,
h,n})\right).\label{7-5}\end{aligned}$$ [First it is easy to show]{} $$\begin{aligned}
&&\|\mathbb{E}\left(F_{n,x}'(x, Y_t^{x,n}(y_1))\right)\cdot
h-\mathbb{E}\left(F_{n,x}'(x, Y_t^{x,n}(y_2))\right)\cdot h\|\nonumber\\
&&\quad\leq\mathbb{E}\|\left(F_{n,x}'(x, Y_t^{x,n}(y_1))\right)\cdot
h-\left(F_{n,x}'(x, Y_t^{x,n}(y_2))\right)\cdot h\|\nonumber\\
&&\quad\leq
C\mathbb{E}\|Y_t^{x,n}(y_1)-Y_t^{x,n}(y_2)\|\cdot\|h\|\nonumber\\
&&\quad\leq Ce^{-ct}\|y_1-y_2\|\cdot\|h\|.\label{7-6}\end{aligned}$$ Next, by Assumption (H.2) we have $$\begin{aligned}
&&\|\mathbb{E}\left([F_{n,y}'(x, Y_t^{x,n}(y_1))-F_{n,y}'(x,
Y_t^{x,n}(y_2))]\cdot\varsigma_t^{x,y_1, h,n} \right)\|\nonumber\\
&&\quad\leq\mathbb{E}\|[F_{n,y}'(x, Y_t^{x,n}(y_1))-F_{n,y}'(x,
Y_t^{x,n}(y_2))]\cdot\varsigma_t^{x,y_1, h,n}\|\nonumber\\
&&\quad\leq C\{\mathbb{E}\|\varsigma_t^{x,y_1,
h,n}\|^2\}^{\frac{1}{2}}\cdot\{\mathbb{E}\|Y_t^{x,n}(y_1)-Y_t^{x,n}(y_2)\|^2\}^{\frac{1}{2}}\nonumber\\
&&\quad\leq C e^{-ct}\|h\|\cdot\|y_1-y_2\|.\label{7-7}\end{aligned}$$ By making use of [Assumption (H.1)]{} again, we can show that there exists a constant $c'>0$ such that one has $$\begin{aligned}
&&\|\mathbb{E}\left(F_{n,y}'(x, Y_t^{x,n}
(y_2))\cdot(\varsigma_t^{x,y_1,
h,n}-\varsigma_t^{x,y_2, h,n})\right)\|\nonumber\\
&&\quad\leq\mathbb{E}\|\left(F_{n,y}'(x, Y_t^{x,n}(
y_2))\cdot(\varsigma_t^{x,y_1, h,n}-\varsigma_t^{x,y_2, h,n})\right)\|\nonumber\\
&&\quad\leq C\mathbb{E}\|\varsigma_t^{x,y_1, h,n}-\varsigma_t^{x,y_2, h,n}\|\nonumber\\
&&\quad\leq C e^{-c't}\|y_1-y_2\|\cdot\|h\|.\label{7-8}\end{aligned}$$ Collecting together , , and , we get $$\begin{aligned}
&&\|\hat{F}_{n,x}'(x,y_1,t)\cdot h-\hat{F}_{n,x}'(x,y_2,t)\cdot h\|\nonumber\\
&&\leq C e^{-c_0t}\|y_1-y_2\|\cdot\|h\|,\end{aligned}$$ which means $$\begin{aligned}
&&\|\hat{F}_{n,x}'(x,y,t)\cdot h-\mathbb{E}\hat{F}_{n,x}'(x,Y^{x,n}_{t_0}(y),t)\cdot h\|\nonumber\\
&&\leq C e^{-c_0t}(1+\|y\|)\cdot\|h\| \label{7-9}\end{aligned}$$ since $$\begin{aligned}
\mathbb{E}\|Y^{x,n}_{t_0}(y)\|\leq C(1+\|x\|+\|y\|).\end{aligned}$$ Returning to , by and we conclude that $$\begin{aligned}
\|D_x\tilde{F}_{t_0,n}(x,y,t)\cdot h\|\leq
Ce^{-ct}(1+\|x\|+\|y\|)\|h\|.\end{aligned}$$ By taking the limit as $t_0$ goes to infinity we obtain $$\begin{aligned}
\|D_x (\bar{F}_n(x)-\mathbb{E}F_n(x, Y^x_t(y)))\cdot h\| \leq
Ce^{-ct}\|h\| \left(1+\|x\| +\|y\| \right).\end{aligned}$$
Proceeding with similar arguments above we can obtain similar result concerning the second order differentiability.
\[mix-derivative-2\] There exist constants $C, c>0$ such that for any $x, y, h, k\in
H^{(n)}$ and $t>0$ it holds $$\begin{aligned}
\|D^2_{xx} (\bar{F}_n(x)-\mathbb{E}F_n(x, Y^{x,n}_t(y)))\cdot (h,
k)\| \leq Ce^{-ct}\|h\| \cdot \|k\|\left(1+\|x\| +\|y\| \right).\end{aligned}$$
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Professor Dirk Blömker for helpful discussions and comments. Hongbo Fu is supported by CSC scholarship (No. \[2015\]5104), NSF of China (Nos. 11301403) and Foundation of Wuhan Textile University 2013. Li Wan is supported by NSF of China (No. 61573011) and Science and Technology Research Projects of Hubei Provincial Department of Education (No. D20131602). Jicheng Liu is supported by NSF of China (No. 11271013). Xianming Liu is supported by NSF of China (No. 11301197)
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\[\]
|
---
abstract: 'A recent proof-of-principle study proposes an energy- and charge-conserving, nonlinearly implicit electrostatic particle-in-cell (PIC) algorithm in one dimension [\[]{}Chen et al, *J. Comput. Phys.*, **230** (2011) 7018[\]]{}. The algorithm in the reference employs an unpreconditioned Jacobian-free Newton-Krylov method, which ensures nonlinear convergence at every timestep (resolving the dynamical timescale of interest). Kinetic enslavement, which is one key component of the algorithm, not only enables fully implicit PIC a practical approach, but also allows preconditioning the kinetic solver with a fluid approximation. This study proposes such a preconditioner, in which the linearized moment equations are closed with moments computed from particles. Effective acceleration of the linear GMRES solve is demonstrated, on both uniform and non-uniform meshes. The algorithm performance is largely insensitive to the electron-ion mass ratio. Numerical experiments are performed on a 1D multi-scale ion acoustic wave test problem.'
address:
- 'Los Alamos National Laboratory, Los Alamos, NM 87545'
- 'University of Colorado Boulder, Boulder, CO 80309'
- 'University of New Mexico, Albuquerque, NM 87131'
author:
- 'G. Chen'
- 'L. Chacón'
- 'C. Leibs'
- 'D. Knoll'
- 'W. Taitano'
bibliography:
- 'kinetic.bib'
title: 'Fluid preconditioning for Newton-Krylov-based, fully implicit, electrostatic particle-in-cell simulations '
---
electrostatic particle-in-cell ,implicit methods ,direct implicit ,implicit moment ,energy conservation ,charge conservation ,physics based preconditioner ,JFNK solver
Introduction
============
The Particle-in-cell (PIC) method solves Vlasov-Maxwell’s equations for kinetic plasma simulations [@birdsall-langdon; @hockneyeastwood]. In the standard approach, Maxwell’s equations (or in the electrostatic limit, Poisson equation) are solved on a grid, and the Vlasov equation is solved by method of characteristics using a large number of particles, from which the evolution of the probability distribution function (PDF) is obtained. The field-PDF description is tightly coupled. Maxwells equations (or a subset thereof) are driven by moments of the PDF such as charge density and/or current density. The PDF, on the other hand, follows a hyperbolic equation in phase space, whose characteristics are self-consistently determined by the fields.
To date, most PIC methods employ explicit time-stepping (e.g. leapfrog scheme), which can be very inefficient for long-time, large spatial scale simulations. The algorithmic inefficiency of standard explicit PIC is rooted in the presence of numerical stability constraints, which force both a minimumtion of the smallest Debye length). Moreover, a fundamental issue with explicit schemes is numerical heating due to the lack of exact energy conservation in a discrete setting This problem is particularly evident for realistic ion-to-electron mass ratios.
Implicit methods hold the promise of nonlinear convergence, which leads to inconsistencies between fields and particle moments. As a result, significant numerical heating is often observed in long term simulations [@cohen-jcp-89-di_pic].
There has been significant recent work exploring fully implicit, fully nonlinear PIC algorithms, either Picard-based [@taitano-sisc-13-ipic] (following the implicit moment method school) or using Jacobian-Free Newton-Krylov (JFNK) methods [@chen-jcp-11-ipic; @markidis2011energy] (more aligned with the direct implicit school). In contrast to earlier studies, these nonlinear approaches enforce nonlinear convergence to a specified tolerance at every timestep. Their fully implicit character enables one to build in exact discrete conservation properties, such as energy and charge conservation [@chen-jcp-11-ipic; @taitano-sisc-13-ipic]. In these studies, particle orbit integration is sub-stepped for accuracy, and to ensure automatic charge conservation.
The purpose of this study is to demonstrate the effectiveness of fluid (moment) equations to accelerate a JFNK-based kinetic solver (moment acceleration in a Picard sense has already been demonstrated in Ref. [@taitano-sisc-13-ipic]). An enabling algorithmic component of the JFNK-based algorithm is the enslavement of particles to the fields, which removes particle quantities from the dependent variable list of the JFNK solver. With particle enslavement, memory requirements of the nonlinear solver are dramatically reduced. Particle equations of motion are orbit-averaged and evolve self-consistently with the field. The kinetic-enslaved JFNK not only makes the fully implicit PIC algorithm practical, but also makes the fluid preconditioning of the algorithm possible.
It is worth pointing out that the preconditioned JFNK approach proposed here can be conceptually viewed as an optimal combination of the direct implicit and moment implicit approaches. The fluid preconditioner is derived by taking the first two moments of the Vlasov equation, and then linearizing them into a so-called “delta-form” [@knoll2004jacobian]. Textbook linear analysis shows that such a system includes stiff electron modes in an electrostatic plasma. Although taking large timesteps for low-frequency field evolutions is desirable, previous work [@chen-jcp-11-ipic] indicates that the implicit CPU speedup over explicit PIC is largely insensitive to the timestep size for large enough timesteps owing to particle sub-cycling for orbit resolution. It is thus sufficient in this context to target the stiffest time scales supported, i.e., electron time scales. Therefore, we base our fluid preconditioner on electron moment equations only. The implicit timestep is chosen to resolve the ion plasma wave frequency. This is to resolve ion waves of all scales (including the Debye length scale which is physically relevant for some non-linear ion waves). For consistency with the orbit averaging of the kinetic solver, we take the time-average of the linearized moment equations in the preconditioner. We show that the fluid preconditioner is asymptotic preserving in the sense that it is well behaved in the quasineutral limit (as in Ref. [@degond-jcp-10-ap_pic]). However, beyond the study in Ref. [@degond-jcp-10-ap_pic], the algorithm proposed here is also well behaved for arbitrary electron-ion mass ratios.
The rest of the paper is organized as follows. Section \[sec:Kinetic-enslavement\] motivates and introduces the concept to kinetic enslavement in the implicit PIC formulation. Section \[sec:JFNK-method\] introduces the mechanics of the JFNK method and preconditioning. Section \[sec:formulation-pc\] formulates the fluid preconditioner of an electrostatic plasma system in detail, with an extension to 1D non-uniform meshes. Linear analysis of electron and ion waves, together with an asymptotic analysis of the preconditioner are also provided. Section \[sec:Numerical-experiments\] presents numerical parametric experiments to test the performance of the preconditioner. Finally, we conclude in Sec. \[sec:Conclusion\].
Kinetically Enslaved Implicit PIC \[sec:Kinetic-enslavement\]
=============================================================
We consider a collisionless electrostatic plasma system (without magnetic field) described by the Vlasov-Ampere equations in one dimension (1D) in both position ($x$) and velocity ($v$) [@chen-jcp-11-ipic]: $$\begin{aligned}
\frac{\partial f_{\alpha}}{\partial t}+v\frac{\partial f_{\alpha}}{\partial x}+\frac{q_{\alpha}}{m_{\alpha}}E\frac{\partial f_{\alpha}}{\partial v} & = & 0,\label{eq:vlasov}\\
\epsilon_{0}\frac{\partial E}{\partial t}+j & = & \left\langle j\right\rangle ,\label{eq:ampere}\end{aligned}$$ where $f_{\alpha}(x,v)$ is the particle distribution function of species $\alpha$ in phase space, $q_{\alpha}$ and $m_{\alpha}$ are the species charge and mass respectively, $E$ is the self-consistent electric field, $j$ is the current density, $\left\langle j\right\rangle =\int jdx/\int dx$, and $\epsilon_{0}$ is the vacuum permittivity. The evolution of Vlasov equation is solved by the method of characteristics, represented by particles evolving according to Newton’s equations of motion, $$\begin{aligned}
\frac{dx_{p}}{dt} & = & v_{p},\label{eq:dxpdt}\\
\frac{dv_{p}}{dt} & = & a_{p}.\label{eq:dvpdt}\end{aligned}$$ Here $x_{p}$, $v_{p}$, $a_{p}$ are the particle position, velocity, and acceleration, respectively, and $t$ denotes time. As a starting point, we may discretize Eqs. \[eq:ampere\], \[eq:dxpdt\], and \[eq:dvpdt\] by a time-centered finite-difference scheme, to find: $$\begin{aligned}
\epsilon_{0}\frac{E_{i}^{n+1}-E_{i}^{n}}{\Delta t}+j_{i}^{n+1/2} & = & \left\langle j\right\rangle ,\label{eq:discete-Ampere}\\
\frac{x_{p}^{n+1}-x_{p}^{n}}{\Delta t}-v_{p}^{n+1/2} & = & 0,\label{eq:discrete-xp}\\
\frac{v_{p}^{n+1}-v_{p}^{n}}{\Delta t}-\frac{q_{p}}{m_{p}}E(x_{p}^{n+1/2}) & = & 0,\label{eq:discrete-vp}\end{aligned}$$ where a variable at time level $n+\nicefrac{1}{2}$ is obtained by the arithmetic mean of the variable at $n$ and $n+1$, the subscript $i$ denotes grid-index and subscript $p$ denotes particle-index, $j_{i}=\sum_{p}q_{p}v_{p}S(x_{p}-x_{i})$, $E(x_{p})=\sum_{i}E_{i}S(x_{i}-x_{p})$, and $S$ is a B-spline shape function [@christensen2010functions].
It is critical to realize that solving the complete system of field-particle equations (i.e., with the field and particle position and velocity as unknowns) in a Newton-Krylov-based solver is impractical, due to the excessive memory requirements of building the required Krylov subspace. To overcome the memory challenges of JFNK for implicit PIC, the concept of kinetic enslavement has been introduced [@chen-jcp-11-ipic; @markidis2011energy]. With kinetic enslavement, the JFNK residual is formulated in terms of the field equation only, nonlinearly eliminating Eqs. \[eq:discrete-xp\] and \[eq:discrete-vp\] as auxiliary computations. The resulting JFNK implementation has memory requirements comparable to that of a fluid calculation. A single copy of particle quantities is still needed for the required particle computations.
One important implication of kinetic enslavement is that the enslaved particle pusher has the freedom of being adaptive in its implementation. This can be effectively exploited to overcome the accuracy shortcomings of using a fixed timestep $\Delta t$ to discretize the time derivatives of both field and particle equations [@Parker-jcp-93-bounded-multiscale-pic]. This is so because solving low-frequency field equations demands using large timesteps, but if particle orbits are computed with such timesteps, large plasma response errors result [@langdon-jcp-79-pic_ts]. In Ref. [@chen-jcp-11-ipic], a self-adaptive, charge-and-energy-conserving particle mover was developed that provided simultaneously accuracy and efficiency. Falgorithmic elements:
1. Estimate the sub-timestep $\Delta\tau$ using a second order estimator [@chen2013analytical].
2. Integrate the orbit over $\Delta\tau$ using a Crank-Nicolson scheme.
3. If a particle orbit crosses a cell boundary, make it land at the first encountered boundary.
4. Accumulate the particle moments to the grid points.
In the last step, the current density is orbit-averaged (over $\Delta t=\sum\Delta\tau$) to ensure global energy conservation. Additionally, binomial smoothing can be introduced without breaking energy or charge conservation. This is done in the particle pusher by using the binomially smoothed electric field, and the binomially smoothed orbit averaged current density in Ampere’s equation. The resulting Ampere’s equation reads: $$\epsilon_{0}\frac{E_{i}^{n+1}-E_{i}^{n}}{\Delta t}+SM(\bar{j})_{i}^{n+1/2}=\left\langle \overline{j}\right\rangle ^{n+1/2},\label{eq:bi-amperelaw}$$ where the orbit averaged current density is: $$\overline{j}_{i}^{n+1/2}=\frac{1}{\Delta t\Delta x}\sum_{p}\sum_{\nu=1}^{N_{\nu}}q_{p}S(x_{i}-x_{p}^{\nu+1/2})v_{p}^{\nu+1/2}\Delta\tau^{\nu}.\label{eq:javerage}$$ The binomial operator $SM$ is defined as $SM(Q)_{i}=\frac{Q_{i-1}+2Q_{i}+Q_{i+1}}{4}.$ A detailed description of the algorithm can be found in Ref. [@chen-jcp-11-ipic; @chen-jcp-12-ipic_gpu].
The kinetically enslaved JFNK residual is defined from Eq. \[eq:bi-amperelaw\] as: $$G_{i}(E^{n+1})=E_{i}^{n+1}-E_{i}^{n}+\frac{\Delta t}{\epsilon_{0}}\left(SM(\bar{j}[E^{n+1}])_{i}^{n+1/2}-\left\langle \overline{j}\right\rangle ^{n+1/2}\right).\label{eq:bi-amperelaw-residual}$$ The functional dependence of $\bar{j}$ with respect to $E^{n+1}$ has been made explicit. Evaluation of $\bar{j}[E^{n+1}]$ requires one particle integration step, and each linear and nonlinear iteration of the JFNK method requires one residual evaluation . We summarize the main elements of the JFNK nonlinear solver next.
The JFNK solver\[sec:JFNK-method\]
==================================
In its outer loop, JFNK employs Newton-Raphson’s method to solve a nonlinear system $\mathbf{G}(\mathbf{x})=0$, where $\mathbf{x}$ is the unknown, by linearizing the residual and inverting linear systems of the form: $$\left.\frac{\partial\mathbf{G}}{\partial\mathbf{x}}\right|^{(k)}\delta\mathbf{x}^{(k)}=-\mathbf{G}(\mathbf{x}^{(k)}),\label{eq:Newton-Raphson step}$$ with $\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\delta\mathbf{x}^{(k)}$, and $(k)$ denotes the nonlinear iteration number. Nonlinear convergence is reached when: $$\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}<\epsilon_{t}=\epsilon_{a}+\epsilon_{r}\left\Vert \mathbf{G}(\mathbf{x}^{(0)})\right\Vert _{2},\label{eq-Newton-conv-tol}$$ where $\left\Vert \cdot\right\Vert _{2}$ is the Euclidean norm, $\epsilon_{t}$ is the total tolerance, $\epsilon_{a}$ is an absolute tolerance, $\epsilon_{r}$ is the Newton relative convergence tolerance, and $\mathbf{G}(\mathbf{x}^{(0)})$ is the initial residual.
Such linear systems are solved iteratively with a Krylov subspace method (e.g. GMRES), which only requires matrix-vector products to proceed. Because the linear system matrix is a Jacobian matrix, matrix-vector products can be implemented Jacobian-free using the Gateaux derivative: $$\left.\frac{\partial\mathbf{G}}{\partial\mathbf{x}}\right|^{(k)}\mathbf{v}=\lim_{\epsilon\rightarrow0}\frac{\mathbf{G}(\mathbf{x}^{(k)}+\epsilon\mathbf{v})-\mathbf{G}(\mathbf{x}^{(k)})}{\epsilon},\label{eq:gateaux}$$ where $\mathbf{v}$ is a Krylov vector, and $\epsilon$ is in practice a small but finite number (p. 79 in [@kelley1987iterative]). Thus, the evaluation of the Jacobian-vector product only requires the function evaluation $\mathbf{G}(\mathbf{x}^{(k)}+\epsilon\mathbf{v})$, and there is no need to form or store the Jacobian matrix. This, in turn, allows for a memory-efficient implementation.
An inexact Newton method [@inexact-newton] is used to adjust the convergence tolerance of the Krylov method at every Newton iteration according to the size of the current Newton residual, as follows: $$\left\Vert J^{(k)}\delta\mathbf{x}^{(k)}+\mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}<\zeta^{(k)}\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}\label{eq-inexact-newton}$$ where $\zeta^{(k)}$ is the inexact Newton parameter and $J^{(k)}=\left.\frac{\partial\mathbf{G}}{\partial\mathbf{x}}\right|^{(k)}$ is the Jacobian matrix. Thus, the convergence tolerance of the Krylov method is loose when the Newton state vector $\mathbf{x}^{(k)}$ is far from the nonlinear solution, and tightens as $\mathbf{x}^{(k)}$ approaches the solution. Superlinear convergence rates of the inexact Newton method are possible if the sequence of $\zeta^{(k)}$ is chosen properly (p. 105 in [@kelley1987iterative]). Here, we employ the prescription: $$\begin{aligned}
\zeta^{A(k)} & = & \gamma\left(\frac{\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}}{\left\Vert \mathbf{G}(\mathbf{x}^{(k-1)})\right\Vert _{2}}\right)^{\alpha},\\
\zeta^{B(k)} & = & \min[\zeta_{max},\max(\zeta^{A(k)},\gamma\zeta^{\alpha(k-1)})],\\
\zeta^{(k)} & = & \min[\zeta_{max},\max(\zeta^{B(k)},\gamma\frac{\epsilon_{t}}{\left\Vert \mathbf{G}(\mathbf{x}^{(k)})\right\Vert _{2}})],\end{aligned}$$ with $\alpha=1.5$ , $\gamma=0.9$, and $\zeta_{max}=0.2$. The convergence tolerance $\epsilon_{t}$ is defined in Eq. \[eq-Newton-conv-tol\]. In this prescription, the first step ensures superlinear convergence (for $\alpha>1$), the second avoids volatile decreases in $\zeta_{k}$, and the last avoids oversolving in the last Newton iteration.
The Jacobian system Eq. \[eq:Newton-Raphson step\] must be preconditioned for efficiency. Here, we employ right preconditioning, which transforms the original system into the equivalent one: $$JP^{-1}\mathbf{y}=-\mathbf{G}(\mathbf{x})$$ where $J=\partial\mathbf{G}/\partial\mathbf{x}$ is the Jacobian matrix, $P$ is a preconditioner, and $\delta\mathbf{x}=P^{-1}\mathbf{y}$. The Jacobian-free preconditioned system employs $$\mathit{J}P^{-1}\mathbf{v}=\lim_{\epsilon\rightarrow0}\frac{\mathbf{G}(\mathbf{x}+\epsilon P^{-1}\mathbf{v})-\mathbf{G}(\mathbf{x})}{\epsilon}\label{eq:gateaux-prec}$$ for each Jacobian-vector product. An important feature of preconditioning is that, while it may substantially improve the convergence properties of the Krylov iteration (when $P$ approximates $J$ and is relatively easy to invert), it does not alter the solution of the system upon convergence.
The purpose of this study is to formulate an effective, fast preconditioner $P$ for the implicit PIC kinetic system. Before deriving the preconditioner, however, we review the fundamental CPU speedup limits of implicit vs. explicit PIC.
Performance limits of implicit PIC
==================================
As mentioned earlier, the ability of implicit PIC to take large timesteps without numerical instabilities does not necessarily translate into performance gains of implicit PIC over its explicit counterpart [@chen-jcp-11-ipic]. In this section, we summarize the back-of-envelope estimate for the CPU speedup introduced in the reference that supports this statement.
We begin by estimating the CPU cost for a given PIC solver to advance the solution for a given time span $\Delta T$ as: $$CPU=\frac{\Delta T}{\Delta t}N_{pc}\left(\frac{L}{\Delta x}\right)^{d}C,\label{eq:CPU-estimation-1}$$ where $N_{pc}$ is the number of particles per cell, ($L/\Delta x$) is the number of cells per dimension, $d$ is the number of physical dimensions, and $C$ is the computational complexity of the solver employed, measured in units of a standard explicit PIC Vlasov-Poisson leap-frogd timestep. Accordingly, the implicit-to-explicit speedup is given by: $$\frac{CPU_{ex}}{CPU_{im}}\sim\left(\frac{\Delta x_{im}}{\Delta x_{ex}}\right)^{d}\left(\frac{\Delta t}{\Delta t_{ex}}\right)\frac{1}{C_{im}},$$ where we denote $\Delta t$ to be the implicit timestep. Assuming that all particles take a fixed sub-timestep $\Delta\tau$ in the implicit scheme, and that the cost of one timestep with the explicit PIC solver is comparable to that of a single implicit sub-step, it follows that $C_{im}\sim N_{FE}\left(\Delta t/\Delta\tau_{im}\right)$, i.e., the cost of the implicit solver exceeds that of the explicit solver by the number of function evaluations ($N_{FE}$) per $\Delta t$ multiplied by the number of particle sub-steps $\left(\Delta t/\Delta\tau_{im}\right)$. Assuming typical values for $\Delta\tau_{im}\sim\min[0.1\Delta x/v_{th},\Delta t_{imp}]$, $\Delta t_{ex}\sim0.1\omega_{pe}^{-1}$, $\Delta x_{im}\sim0.2/k$, and $\Delta x_{ex}\sim\lambda_{D}$, we find that the CPU speedup scales as: $$\frac{CPU_{ex}}{CPU_{imp}}\sim\frac{0.2}{(5k\lambda_{D})^{d}}\min\left[\frac{1}{k\lambda_{D}},\sqrt{\frac{m_{i}}{m_{e}}}\right]\frac{1}{N_{FE}}.\label{eq:CPU-ex-im-1}$$ This result supports two important conclusions. Firstly, it predicts that the CPU speedup is asymptotically independent of the implicit time step $\Delta t$ for $\Delta t\gg\Delta\tau_{im}$. The effect of the implicit time step is captured in the extra power of one in the $(k\lambda_{D})$ term, once one accounts for sub-stepping, but that effect disappears when the mesh becomes coarse enough (i.e., $k\lambda_{D}<\sqrt{m_{e}/m_{i}}$). Also, it predicts that the speedup improves with larger ion-to-electron mass ratio, indicating that the approach is more advantageous when one employs realistic mass ratios. Because the CPU speedup is asymptotically independent of $\Delta t$, algorithmically it will be advantageous to use a time step that is large enough to be in the asymptotic regime, but no larger. This will motivate the choice in the preconditioner to include only electron stiff physics.
Secondly, Eq. \[eq:CPU-ex-im-1\] indicates that large CPU speedups are possible when $k\lambda_{D}\ll1$, particularly in multiple dimensions, but only if $N_{FE}$ is kept small and bounded. The latter point motivates the development of suitable preconditioning strategies. We focus on this in the next section.
Fluid preconditioning the electrostatic implicit PIC kinetic system\[sec:formulation-pc\]
=========================================================================================
The preconditioner of the nonlinear kinetic JFNK solver needs to return an approximation for the $E$-field update only. The approximate $E$-field update will be found from a linearized fluid model, consistently closed with particle moments. As will be shown, the fluid model provides an inexpensive approximation to the kinetic Jacobian. We demonstrate the concept in the 1D electrostatic, multispecies PIC model.
Formulation of the fluid preconditioner
---------------------------------------
Following standard procedure [@knoll2004jacobian], we work with the linearized form of the governing equations to derive a suitable preconditioner. The linearized, orbit-averaged, binomially smoothed 1D Ampere’s residual equation (Eq. \[eq:discete-Ampere\] with $E=E_{0}+\delta E$, and $\delta\bar{j}\equiv\int_{0}^{\Delta t}\delta jdt/\Delta t$) reads: $$\delta E=-\Delta t\left(G(E_{0})+\frac{1}{\varepsilon_{0}}SM(\delta\bar{j})\right),\label{eq:delta-Ampere-disc}$$ where $G(E_{0})=E_{0}-E^{n}+\frac{\Delta t}{\varepsilon_{0}}(SM(\bar{j}_{0}^{n+\nicefrac{1}{2}})-\left\langle \bar{j}_{0}\right\rangle )$ is the residual of Ampere’s law, the superscript $n$ denotes last timestep, and the subscript 0 of the $E$-field denotes the current Newton state. From the discussion in the previous section, for the purpose of preconditioning we consider only the linear response of electron contribution to the current ($\delta\bar{j}\simeq-e\delta\bar{\Gamma}$ where $\Gamma$ is the electron flux). Thus, the electric field update in the preconditioner will be found from: $$\delta E\approx-\Delta t\left(G(E_{0})-\frac{e}{\varepsilon_{0}}SM(\delta\bar{\Gamma})\right),\label{eq:delta-Ampere-e_only}$$ where $\delta\bar{\Gamma}=\frac{1}{\Delta t}\int_{0}^{\Delta t}dt\delta\Gamma(t)$, a time-average between timestep $n$ and $n+1$..
We approximate the linear response of the electron current via the continuity and momentum equations of electrons, closed with moments from particles (as in the implicit moment method [@mason-jcp-81-im_pic]). The continuity equation for electrons is $$\frac{\partial n}{\partial t}+\frac{\partial\Gamma}{\partial x}=0,\label{eq:continuity}$$ where where $n$ is electron number density. Linearizing, we obtain: $$\frac{\partial\delta n}{\partial t}=-\frac{\partial\delta\Gamma}{\partial x},\label{eq:delta-continuity}$$ where we have used particle conservation ($\partial n_{0}/\partial t+\partial\Gamma_{0}/\partial x=0$), which is satisfied at all iteration levels owing to exact charge conservation [@chen-jcp-11-ipic]. We then take the time-average ($\frac{1}{\Delta t}\int_{0}^{\Delta t}dt$, equivalently to the orbit average in Eq. \[eq:javerage\]) of Eq. \[eq:delta-continuity\] to obtain $$\delta n=-\Delta t\frac{\partial\delta\bar{\Gamma}}{\partial x}.\label{eq:delta-continuity-disc}$$ The update equation for $\delta\bar{\Gamma}$ is found from the electron momentum equation, which in conservative form reads $$m\left[\frac{\partial\Gamma}{\partial t}+\frac{\partial}{\partial x}\left(\frac{\Gamma\Gamma}{n}\right)\right]=-enE-\frac{\partial P}{\partial x}\label{eq:momentum}$$ where $m$ is the electron mass, $P\equiv nT$ is the electron pressure, and $T$ is the electron temperature. Linearizing it, we obtain: $$m\left[\frac{\partial\delta\Gamma}{\partial t}+\frac{\partial}{\partial x}\left(\frac{2\Gamma_{0}\delta\Gamma}{n_{0}}-\frac{\Gamma_{0}\Gamma_{0}}{n_{0}^{2}}\delta n\right)\right]+e(n_{0}\delta E+\delta nE_{0})+\frac{\partial(\delta nT_{0})}{\partial x}=0,\label{eq:delta-momentum}$$ where $T_{0}\equiv\int f(v)m(v-u)(v-u)dv/n_{0}$ is the current temperature (or normalized pressure). Closures for $\Gamma_{0}$, $n_{0}$ and $T_{0}$ are obtained from current particle information. In Eq. \[eq:delta-momentum\], we take $m\left[\partial\Gamma_{0}/\partial t+\partial(\Gamma_{0}\Gamma_{0}/n_{0})/\partial x\right]+en_{0}E_{0}+\partial(n_{0}T_{0})/\partial x=0$ by ansatz. To close the fluid model, we have neglected the linear temperature response $\delta T$.
To cast Eq. \[eq:delta-momentum\] in a useful form, we take its time-derivative to get (assuming that $n_{0}$, $E_{0}$, and $T_{0}$ do not vary with time): $$m\frac{\partial^{2}\delta\Gamma}{\partial t^{2}}+e(n_{0}\frac{\partial\delta E}{\partial t}+\frac{\partial\delta n}{\partial t}E_{0})+\frac{\partial}{\partial x}(T_{0}\frac{\partial\delta n}{\partial t})=0,\label{eq:delta-momentum-dt}$$ and then time-average the result to find (substituting Eqs. \[eq:delta-Ampere-disc\] and \[eq:delta-continuity\]): $$\frac{2m\delta\bar{\Gamma}}{\Delta t^{2}}+e^{2}n_{0}\delta\bar{\Gamma}-eE_{0}\frac{\partial\delta\bar{\Gamma}}{\partial x}-\frac{\partial}{\partial x}(T_{0}\frac{\partial\delta\bar{\Gamma}}{\partial x})=-n_{0}G(E_{0}).\label{eq:delta-momentum-avg}$$ Here, we have neglected the convective term for simplicity, and approximated the first time-derivative term as: $$\frac{\partial\delta\Gamma}{\partial t}\simeq\frac{2\delta\bar{\Gamma}}{\Delta t}\label{eq:ddGamma-dt}$$ (which is exact if $\delta\Gamma(t)$ is linear with $t$). We discretize Eq. \[eq:delta-momentum-avg\] with space-centered finite differences, resulting in a tridiagonal system, which we invert for $\delta\bar{\Gamma}$ using a direct solver. Finally, we substitute the solution of $\delta\bar{\Gamma}$ in Eq. \[eq:delta-Ampere-e\_only\] to find the $E$-field update.
Extension to curvilinear meshes
-------------------------------
The fully implicit PIC algorithm has been recently extended to curvilinear meshes [@chacon-jcp-13-curvpic]. In this section, we rewrite the above fluid model on a 1D non-uniform mesh using a map $x=x(\xi)$. In 1D, the curvilinear form of in Eqs. \[eq:continuity\] and \[eq:momentum\] can be derived straightforwardly by replacing every $dx$ with $\mathcal{J}d\xi$, where $\mathcal{J}\equiv dx/d\xi$ is the Jacobian. It follows that the continuity equation in logical space is written as: $$\frac{\partial n}{\partial t}+\frac{1}{\mathcal{J}}\frac{\partial\Gamma}{\partial\xi}=0.\label{eq:continuity-curv}$$ The transformed momentum equation is $$m\left[\frac{\partial\Gamma}{\partial t}+\frac{1}{\mathcal{J}}\frac{\partial}{\partial\xi}\left(\frac{\Gamma\Gamma}{n}\right)\right]=qnE-\frac{1}{\mathcal{J}}\frac{\partial P}{\partial\xi}.\label{eq:momentum-curv}$$ Similar to the procedure described above, linearizing and discretizing Eqs. \[eq:continuity-curv\] and \[eq:momentum-curv\] again results in a tridiagonal system.
Electrostatic wave dispersion relations
---------------------------------------
It is instructive to look at the dispersion relation of Eq. \[eq:delta-Ampere-disc\], \[eq:delta-continuity\] and \[eq:delta-momentum\], for both electrons and ions. Figure \[fig:eiwave-disper\] shows the dispersion relation of electron plasma waves and ion acoustic waves [@FChenbook], from which we make the following observations. The stiffest wave is the electron plasma wave, whose frequency $\omega_{pe}$ is essentially insensitive to the wave number $k$ for $k\lambda_{D}<1$. The wave frequency increases for $k\lambda_{D}>1$, but in that range the plasma wave is highly Landau-damped [@jackson1960longitudinal]. In contrast to the electron wave, the ion wave frequency increases with $k$ for $k\lambda_{D}<1$, but saturates at $\sim\omega_{pi}$ for $k\lambda_{D}>1$. In a propagating ion acoustic wave (IAW), nonlinear effects lead to wave steepening. Because of the wave dispersion, the IAW steepening stops when the high frequency waves propagate slower than the low frequency ones [@krall1997we]. Those high frequency ion waves are physically important, and therefore need to be resolved. For this reason, in our numerical experiments, we limit the implicit time step to $\Delta t\sim0.1\omega_{pi}^{-1}$. The frequency gap between the electron and ion waves is about a factor of $\sqrt{m_{i}/m_{e}}$, which provides enough room to place the algorithm in the large timestep asymptotic regime (Eq. \[eq:CPU-ex-im-1\]).
![\[fig:eiwave-disper\]Dispersion relations of electron and ion waves in an electrostatic plasma. The dispersions can be obtained by Fourier analysis of the fluid model of Eq. \[eq:delta-Ampere-disc\], \[eq:delta-continuity\] and \[eq:delta-momentum\], for both electrons and ions, assuming that $E_{0},\Gamma_{0}$, $n_{0},T_{0}=$const.](iaw-disper)
Asymptotic behavior of the implicit PIC formulation in the quasineutral limit.\[sub:Asymptotic-preservation-in\]
----------------------------------------------------------------------------------------------------------------
Since the implicit scheme is able to use large grid sizes and timesteps stably, it is important to ensure that the fluid preconditioner be able to capture relevant asymptotic regimes correctly [@degond-jcp-10-ap_pic]. In the context of electrostatic PIC, the relevant asymptotic regime is the quasineutral limit, which manifests when the domain length is much larger than the Debye length ($L\gg\lambda_{D}$) and when $m_{e}\ll m_{i}$. In this limit, the electric field must be found from the fluid equations [@fernsler2005quasineutral], and leads to the well know ambipolar electric field, $E=-\frac{1}{en}\partial_{x}P$.
In our context, the algorithm must be well behaved when $L$ varies from $\sim\lambda_{D}$ to $\gg\lambda_{D}$, and for arbitrary mass ratios. In particular, the fluid preconditioner must feature these properties to successfully accelerate the kinetic algorithm. To confirm that this is the case, following Ref. [@degond-jcp-10-ap_pic] we normalize the electron fluid equations to the following reference quantities: $$\hat{x}=\frac{x}{x_{0}},\:\hat{v}=\frac{v}{v_{0}},\:\hat{t}=\frac{tv_{0}}{x_{0}},\:\hat{n}=\frac{n}{n_{0}},\:\hat{q}=\frac{q}{q_{0}},\:\hat{m}=\frac{m}{m_{0}},\:\hat{E}=\frac{Eq_{0}x_{0}}{k_{B}T_{0}}.$$ We choose $x_{0}=L$, $v_{0}=\sqrt{k_{B}T_{0}/m_{0}}$, $q_{0}=e$, $m_{0}=m_{i}$. For electrons, $q=-e$, and hence $\hat{q}=-1$. The normalized preconditioning equations become: $$\begin{aligned}
\hat{\lambda}_{D}^{2}\frac{\partial\hat{E}}{\partial\hat{t}}-\hat{\Gamma} & = & 0,\label{eq:Ampere-law-norm}\\
\frac{\partial\hat{n}}{\partial\hat{t}}+\frac{\partial\hat{\Gamma}}{\partial\hat{x}} & = & 0,\label{eq:continuity-norm}\\
\hat{m}\frac{\partial\hat{\Gamma}}{\partial\hat{t}}+\hat{n}\hat{E}+\hat{T}\frac{\partial\hat{n}}{\partial\hat{x}} & = & 0,\label{eq:momentum-norm}\end{aligned}$$ where in Eq. \[eq:momentum-norm\] we have neglected the convective term. Substituting Eq. \[eq:Ampere-law-norm\] into Eq. \[eq:momentum-norm\], we find the equation for the electric field: $$\hat{m}\frac{\partial}{\partial\hat{t}}\left(\hat{\lambda}_{D}^{2}\frac{\partial\hat{E}}{\partial\hat{t}}\right)+\hat{n}\hat{E}+\hat{T}\frac{\partial\hat{n}}{\partial\hat{x}}=0,\label{eq:ap-momentum}$$ where $\lambda_{D}$ may change in time and space. The solution of $\hat{E}$ is well behaved as $\hat{m}\hat{\lambda}_{D}^{2}\rightarrow0$, where we indeed find that $\hat{n}\hat{E}=-\hat{T}\frac{\partial\hat{n}}{\partial\hat{x}}$, which is the correct (ambipolar) $E$-field. Our fluid preconditioner is based on the linearization of Eqs. \[eq:Ampere-law-norm\]-\[eq:momentum-norm\], and therefore inherits this asymptotic property. In what follows, we will demonstrate among other things the effectiveness of the preconditioner as we vary the domain size and the mass ratio.
Numerical experiments\[sec:Numerical-experiments\]
==================================================
We use the IAW problem for testing the performance of the fluid-based preconditioner. IAW propagation is a multi-scale problem determined by the coupling between electrons and ions. The 1D case used in Ref. [@chen-jcp-11-ipic] features large-amplitude IAWs in an unmagnetized, collisionless plasma without significant damping.
We initialize the calculation with the following ion distribution function: $$f(x,v,t=0)=f_{M}(v)\left[1+a\cos\left(\frac{2\pi}{L}x\right)\right]\label{eq:initf}$$ where $f_{M}(v)$ is the Maxwellian distribution, $a$ is the perturbation level, $L$ is the domain size. The spatial distribution is approximated by first putting ions randomly with a constant distribution, e.g. $x^{0}\in[0,L]$. The electrons are distributed in pairs with ions according to the Debye distribution[@williamson1971initial]. Specifically, in each $e$-$i$ pair, the electron is situated away from the ion by a small distance, $dx=\mathrm{ln}(R)$ where $R\in(0,1)$ is a uniform random number (note that we normalize all lengths with the electron Debye length). We then shift the particle position by a small amount such that $x=x^{0}+a\cos\left(\frac{2\pi}{L}x^{0}\right)$, with $a=0.2$.
For testing the solver performance with non-uniform meshes, the mesh adaptation in the periodic domain is provided by the map [@chacon-jcp-13-curvpic]: $$x(\xi)=\xi+\frac{L}{2\pi}(1-\frac{N\Delta x_{\nicefrac{L}{2}}}{L})\sin\left(\frac{2\pi\xi}{L}\right),\label{eq:map-xofxi}$$ which has the property that the Jacobian $J$ is also periodic. Here, $N$ is the number of mesh points, and $\Delta x_{\nicefrac{L}{2}}$ is the physical mesh resolution at $x=\xi=L/2$.
Before we begin the convergence studies, it is informative to look at the condition number of the Jacobian system, which can be estimated as the number of times we step over the explicit CFL: $$\sigma\propto\omega_{pe}\Delta t=0.1\frac{\omega_{pe}}{\omega_{pi}}=0.1\sqrt{\frac{m_{i}}{m_{e}}},\label{eq:condition_number}$$ where we have used that $\Delta t\sim0.1\omega_{pi}^{-1}$, and we have assumed $k\lambda_{D}<1$. The first important observation is that, as expected, the Jacobian system will become harder to solve as we increase the ion-to-electron mass ratio. Secondly, the condition number does not depend on $k\lambda_{D}$. The latter, while surprising, is a consequence of our chosen implicit time step upper bound. Dependence of $\sigma$ with $k\lambda_{D}$ is recovered for $k\lambda_{D}>1$, but in this regime Langmuir waves are highly Landau-damped [@jackson1960longitudinal], and do not survive in the system.
We demonstrate the performance of the fluid preconditioner by varying several relevant parameters, namely, the implicit timestep $\Delta t$, the mass ratio $m_{i}/m_{e}$, the domain length $L$, the mesh size $N_{x}$, and the number of particles per cell $N_{pc}$. We begin with the implicit timestep, which we vary from $0.01\omega_{pi}^{-1}$ to $0.25\omega_{pi}^{-1}$. For this test, we choose $L=100$, $N_{x}=128$, $N_{pc}=1000$, and $m_{i}/m_{e}=1836$. As shown in Figure \[fig:FE-dt\], the performance for preconditioned and unpreconditioned solvers is about the same for small time steps, where the Jacobian system is not stiff. However, significant differences in performance develop for larger timesteps, reaching a factor of 2 to 3 as the timestep approaches $0.2\omega_{pi}^{-1}$. Overall, the preconditioner is able to keep the linear and nonlinear iteration count fairly well bounded as the timestep increases.
![\[fig:FE-dt\]The performance of the JFNK solver against the timestep, with $L=100$, $N_{x}=128$, $N_{pc}=1000$, and $m_{i}/m_{e}=1836$. The number of function evaluations are well controlled by the preconditioner over a large range of $\Delta t$. ](./FE_dt-iaw)
For bounded $N_{FE}$, Eq. \[eq:CPU-ex-im-1\] predicts that the actual CPU time should be largely insensitive to the timestep size. This is confirmed in Fig. \[fig:CPU-dt\], which shows the CPU performance of a series of computations with a fixed simulation time-span. Clearly, the total CPU time is essentially independent of $\Delta t$ with preconditioning (but not without). Also, both with and without preconditioning, the average particle pushing time, which is the average CPU time used for all particle pushes during the simulation time-span, saturates for large enough time steps (e.g. $v_{the}\Delta t>1\sim10\Delta x$), indicating that we have reached an asymptotically large time step. Even though the CPU performance of the preconditioned solver is independent of $\Delta t$, the use of larger timesteps is beneficial for the following reasons. Firstly, the orbit-averaging performed to obtain the plasma current density helps with noise reduction, as it provides the time average of many samplings per particle [@cohen-jcp-82-orbit_averaging]. Secondly, the operational intensity (computations per memory operation) per particle orbit increases with the timestep, which helps enhance the computing performance (or efficiency) and offset communication latencies in the simulation [@chen-jcp-12-ipic_gpu].
![\[fig:CPU-dt\]Overall CPU performance as a function of timestep, comparing the unpreconditioned and preconditioned solvers in terms of the average particle pushing time (obtained by the total CPU time divided by the average number of iterations) (left) and wall clock CPU time (right). $L=100$, $N_{x}=128$, $m_{i}/m_{e}=1836$, and the time-span is fixed at 4.67$\omega_{pi}^{-1}$ for all computations.](./CPU_dt-iaw)
The performance of the preconditioner vs. the electron-ion mass ratio for both uniform and non-uniform meshes is shown in Tables \[tab:ES-prec-performance-iaw-uniform\] and \[tab:ES-prec-performance-IAW-non-uniform\]. To make a fair comparison, both uniform and non-uniform meshes have the same finest mesh resolution, which locally resolves the Debye length. From the tables it is clear that similar performance gains of the preconditioned solver vs. the unpreconditioned one are obtained for both uniform and non-uniform meshes. The dependence of the GMRES performance on the mass ratio is much weaker with the preconditioner: as the mass ratio increases by a factor of 100, the GMRES iteration count increases by a factor of 5 without the preconditioner, vs. a factor of 2 with the preconditioner. Although not completely independent of the mass ratio, the solver behavior is consistent with the asymptotic analysis made in Sec. \[sub:Asymptotic-preservation-in\].
------- -------- ------- -------- -------
Newton GMRES Newton GMRES
100 4 8 4 7
1600 5 21.2 4 10.1
10000 5.8 50.1 5.5 13.5
------- -------- ------- -------- -------
: \[tab:ES-prec-performance-iaw-uniform\]Solver performance with and without the fluid preconditioner for the IAW case with $L=100$, $N_{x}=512$, and $N_{pc}=1000$ on a uniform mesh. For all the test cases, $\Delta t=0.1\omega_{pi}^{-1}$. The Newton and GMRES iteration numbers are obtained by an average over 20 timesteps. For all the runs, we have kept the ion and electron temperature constant.
------- -------- ------- -------- -------
Newton GMRES Newton GMRES
100 4 7.6 4 7
1600 5 21.3 5.1 12.1
10000 5.8 48.6 5.3 16.5
------- -------- ------- -------- -------
: \[tab:ES-prec-performance-IAW-non-uniform\]Solver performance with and without the fluid preconditioner for the IAW case with the non-uniform mesh ($N_{x}=64$ and the smallest mesh size 0.2).
The impact of the domain length in the solver performance is shown in Fig. \[fig:FE\_L\_iaw\]. Clearly, the solver performance remains fairly insensitive to the domain length both with and without the preconditioner, even though the domain length varies from 10 to 1000 Debye lengths. This is consistent with the condition number analysis in Eq. \[eq:condition\_number\]. The impact of the preconditioner in the number of GMRES iterations is expected for the time step chosen.
![\[fig:FE\_L\_iaw\]Solver performance as a function of the domain size, with $N_{x}=64$, $N_{pc}=1000$, $\Delta t=0.1\omega_{pi}^{-1}$.](./FE_L-iaw)
![\[fig:FE-particle\]NFE of GMRES and Newton iterations as a function of average number of particles per cell for a domain size $L=100$ with $N_{x}=128$ uniformly distributed cells. ](./FE_np-iaw)
![\[fig:FE-particle\]NFE of GMRES and Newton iterations as a function of average number of particles per cell for a domain size $L=100$ with $N_{x}=128$ uniformly distributed cells. ](./FE_np-iaw-Newton)
The impact of the number of particles in the performance of the solver is shown in Fig. \[fig:FE-particle\], which depicts the iteration count of both Newton and GMRES vs. the number of particles. The timestep is varied by a factor of two, corresponding to about one-tenth and one-fifth of the inverse ion plasma frequency ($\omega_{pi}^{-1}$). As expected, the solver performs better with smaller timesteps and with larger number of particles. The number of linear and nonlinear iterations increases as the number of particles decreases. This behavior is likely caused by the increased interpolation noise associated with fewer particles: the noise in charge density results in fluctuations in the self-consistent electric field, making the Jacobian-related calculations less accurate, thus delaying convergence. The preconditioner seems to ameliorate the impact of having too few particles on the performance of the algorithm, thus robustifying the nonlinear solver.
![\[fig:FE-dx\]Solver performance vs. the number of grid-points $N_{x}$ for $\omega_{pi}\Delta t=0.093$, $L=100\lambda_{D}$, and $N_{pc}=1000$.](./FE_nx-iaw)
The impact of the number of grid points on the solver performance is shown in Fig. \[fig:FE-dx\] for $\omega_{pi}\Delta t=0.093$, $L=100\lambda_{D}$, and $N_{pc}=1000$. We see that the linear and nonlinear iteration count remains fairly constant with respect to $N_{x}$, with and without preconditioning. This is consistent with the condition number result in Eq. \[eq:condition\_number\] (which is independent of the wavenumber). However, despite the fact that the number of iterations is virtually independent of the number of grid points, the CPU time grows significantly with it. Figure \[fig:CPU-nx\] shows that the computational cost scales as $N_{x}^{2}$ for $N_{x}$ large enough. The reason is two-fold. On the one hand, since we keep the number of particles per cell fixed, the computational cost of pushing particles increases proportionally with the number of grid points. On the other hand, as we refine the grid, the cost per particle increases because particles have to cross more cells (for a given timestep). In multiple dimensions, the particle orbit will sample $N^{1/d}$ cells on average, for large enough $N$ (or $\Delta t$), with $N$ and $d$ denoting the total number of grid points and dimensions, respectively. Hence, the cost of particle crossing will scale as $N^{1/d}$, and the computational cost will scale as $N^{1+1/d}$. In this sense, the 1D configuration is the least favorable.
![\[fig:CPU-nx\]The performance of the JFNK solver against the number of grid-points, with $N_{pc}=1000$. The average particle pushing time is shown on the left and total CPU time for a total time-span $80$ is shown on the right. Both particle pushing time and total CPU time scale as $N_{x}^{2}$ for large enough $N_{x}$.](./CPU_nx-iaw)
The performance of the implicit PIC solver vs. the explicit PIC one is compared in Fig. \[fig:CPU-scaling\], which depicts the CPU speedup vs. $k\lambda_{D}$. For this test, we choose $m_{i}/m_{e}=1836$, $\Delta t=0.1\omega_{pi}^{-1}$, and $N_{pc}=1000$. In the implicit tests, the number of grid-points is kept fixed at $N_{x}=32$ as $L$ increases with $k=2\pi/L$. In the explicit computations, $\Delta x\simeq0.3\lambda_{D}$ is kept constant for stability, and therefore the number of grid-points increases with $L$. Both implicit and explicit tests employ a uniform mesh. We monitor the scaling power index of Eq. \[eq:CPU-ex-im-1\] with and without the preconditioner. We test the performance with a large implicit timestep (about 40 times larger than the explicit timestep). The scaling index is found to be $\sim1.86$ for small domain sizes, close to the expected value of 2. As $L$ increases, the scaling index becomes $\sim1$. The scaling index turns at $k\lambda_{D}\sim\sqrt{m_{e}/m_{i}}\sim0.025$, as predicted by Eq. \[eq:CPU-ex-im-1\]. The estimated scaling index of 2 would be recovered if one increased the timestep proportionally to $L$, but this would result in timesteps too large with respect to $\omega_{pi}^{-1}$. Overall, these results are in very good agreement with our simple estimates. The preconditioned solver gains about a factor of two compared to the un-preconditioned one, insensitively to $k\lambda_{D}$, which is consistent with the results depicted in Fig. \[fig:FE\_L\_iaw\]. We see that for $k\lambda_{D}<10^{-3}$, the implicit scheme delivers speedups of about three orders of magnitude vs. the explicit approach, while remaining exactly energy- and charge-conserving.
The setup in Fig. \[fig:CPU-scaling\] employs a uniform mesh. However, sometimes it is necessary to resolve the Debye length locally, e.g. at a shock front or a boundary layer near a wall. In this case, using a non-uniform mesh is advantageous [@chacon-jcp-13-curvpic]. We test the performance of the preconditioner on a non-uniform mesh for a nonlinear ion acoustic shock wave, as setup in Ref. [@chacon-jcp-13-curvpic]. Specifically, we use $L=100\lambda_{D}$, $N_{pc}=2000$, $N_{x}=64$, and $\Delta t=0.1\omega_{pi}^{-1}$ for 20 timesteps. The minimum resolution is $\Delta x=0.5\lambda_{D}$ at the shock location (as in [@chacon-jcp-13-curvpic], we perform the simulation in the reference frame of the shock). With a nonlinear tolerance $\epsilon_{r}=2\times10^{-4}$, we have found that, with preconditioning, the average number of Newton and GMRES iterations is 3 and 10.1, respectively, compared to 3.6 and 23 without preconditioning. The performance gain in the linear solve is about factor of two, comparable to that obtained for a uniform mesh with similar problem parameters. Similar performance gains are found with tighter nonlinear tolerances: for $\epsilon_{r}=10^{-8}$, we find 5.1 Newton and 46.6 GMRES iterations without preconditioning, vs. 5 and 20.5 with it.
![\[fig:CPU-scaling\]The implicit PIC solver performance compared with the explicit scheme. The performance gain increases with the domain size. For the parameters used, the performance gain of the implicit solver is enhanced by the preconditioner by about a factor of 2.](./scaling)
Conclusions\[sec:Conclusion\]
=============================
This study has focused on the development of a preconditioner for a recently proposed fully implicit, JFNK-based, charge- and energy-conserving particle-in-cell electrostatic kinetic model [@chen-jcp-11-ipic]. In the reference, it was found that, for large enough implicit time steps $\Delta t$, the potential implicit-to-explicit CPU speedup scaled as $\frac{1}{N_{FE}(k\lambda_{D})^{d}}$, with $N_{FE}$ the number of function evaluations per time step, and $k\lambda_{D}\propto\lambda_{D}/L$. Thus, large speedups are expected when $k\lambda_{D}\ll1$ provided that $N_{FE}$ is kept bounded. While the CPU speedup does not scale directly with $\Delta t$, the use of large $\Delta t$ is advantageous to maximize operational intensity [@chen-jcp-12-ipic_gpu] (i.e., to maximize floating point operations per byte communicated), and to control numerical noise via orbit averaging [@cohen-jcp-82-orbit_averaging].
We have targeted a preconditioner based on an electron fluid model, which is sufficient to capture the stiffest time scales, and thus enable the use of large implicit time steps while keeping the number of function evaluations bounded. The performance of the preconditioned kinetic JFNK solver has been analyzed with various parametric studies, including time step, mass ratio, domain length, number of particles, and mesh size. The number of function evaluations is found to be insensitive against changes in all of these, delivering a robust nonlinear solver. The CPU time of the implicit PIC solver is found to be insensitive to the time step (as expected), but to scale with the square of the number of mesh points in 1D. This scaling is due to the number of particles per cell being kept constant, and to the number of particle crossings increasing linearly with the mesh resolution. The latter scaling will be more benign in multiple dimensions, as particle orbits remain one-dimensional. Speedups of about three orders of magnitude vs. explicit PIC are demonstrated when $\lambda_{D}\ll L$ (i.e., in the quasineutral regime). Based on the speedup prediction in [@chen-jcp-11-ipic], more dramatic speedups are expected in multiple dimensions.
We conclude that the proposed algorithm shows much promise for extension to multiple dimensions and to electromagnetic simulations. This will be the subject of future work.
#### Acknowledgments {#acknowledgments .unnumbered}
This work was partially sponsored by the Office of Fusion Energy Sciences at Oak Ridge National Laboratory, and by the Los Alamos National Laboratory (LANL) Directed Research and Development Program. This work was performed under the auspices of the US Department of Energy at Oak Ridge National Laboratory, managed by UT-Battelle, LLC under contract DE-AC05-00OR22725, and the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory, managed by LANS, LLC under contract DE-AC52-06NA25396.
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abstract: 'A theory of quantum measurement introduced some time ago is modified so that it can be extended to a the general theory of state reduction in a way preserving the unitary-transformation symmetry of quantum mechanics. Moreover, it works also for meters that are not macroscopic. The examples of screening, of Stern-Gerlach measurement and of some experiments with superconducting rings illustrate the idea. The last example shows that the theory is falsifiable.'
---
[The phenomenon of state reduction]{}
P. Hájíček\
Institute for Theoretical Physics\
University of Bern\
Sidlerstrasse 5, CH-3012 Bern, Switzerland\
[email protected]
November 2014\
PACS number: 03.65.-w, 03.65.Ta, 07.07.Df, 85.25.Cp
Introduction
============
The standard quantum theory of measurement (for a recent review, see, e.g., [@WM]) assumes, following von Neumann [@JvN] and Heisenberg [@heisenb], that each individual measurement on a quantum system ${\mathcal S}$ gives a definite value, $r$, of the measured observable. The value can be [*read*]{} at the measuring apparatus (or meter) ${\mathcal M}$. The [*reading*]{} remains, however, a mysterious procedure. If the meter is considered as a quantum system then to observe it, another meter is needed, to observe this, still another is (the resulting series of measurements is called von-Neumann chain). At some (unknown) stage including the processes in the mind (brain?) of observer, there is the so-called Heisenberg cut at which the definite value emerges. Moreover, the final state associated with a definite value of $r$ cannot, in general, be a result of a unitary evolution. The standard theory can work by ignoring the mystery [@WM]. It is, so to speak, sufficient for all practical purposes (abbreviation FAPP introduced by John Bell) but it does not seem to be complete.
Some approaches to these questions make the assumption that the difference between the final state $|r\rangle$ associated with a unique value $r$ and state $$|\Psi\rangle = \sum_r c_r|r\rangle$$ describing a linear superposition of such states is not observable because the registration of observables, some correlations of which would reveal the difference, is either very difficult or because such observables do not exist. One can then deny that the transition from $|\Psi\rangle$ to $|r\rangle$ really takes place and so assume that the unique result is only apparent (no-collapse scenario). Other attempts (collapse scenario) do assume that the reduction is a real process and postulate a new dynamics that leads directly to $|r\rangle$. There are many examples of no-collapse approach, e.g., [@schloss; @Hepp; @bub].
An example of the collapse scenario is known as Dynamical Reduction Program (DRP) [@GRW; @pearle]. It postulates new universal dynamics that is non-linear and stochastic. The main idea, the so-called spontaneous localisation, is the assumption that linear superpositions of different positions spontaneously decay, either by jumps [@GRW] or by continuous transitions [@pearle]. The form of this decay is chosen to satisfy the condition that it takes very long time for microsystems, so that the standard quantum mechanics is a good approximation and it needs very short time for macrosystems, so that the transition from $|\Psi\rangle$ to $|r\rangle$ results. In this way, a simple explanation of the definite positions of macroscopic systems and of the pointers of meters is achieved.
The DRP makes the state reduction well-defined by choosing a particular frame for it: the $Q$-representation. This leads to a breaking of the symmetry with respect to all unitary transformations that is not only a beautiful but also a practical feature of the standard quantum mechanics. Moreover, the choice of a definite space may lead to a conflict with special relativity.
Another example of collapse scenario is the approach started in Refs.[@hajicek2; @hajicek4; @models]. The state reduction (collapse of wave function) is considered as a physical phenomenon and postulated similarly as done by von Neumann [@JvN] or by the DRP. It is however assumed to occur only if a meter interacts with a microscopic system. The structure of such meter is analysed and classified. The form of the state reduction is then uniquely determined by this structure. The result is more specific that that by von Neumann but less radical than that by DRP. It just corrects the Schrödinger equation in well-defined cases in such a way that the general unitary symmetry (including the transformation between position and momentum representations) is preserved. An important role is played by the assumption that state-reduction events are correlated with changes of so-called [*separation status*]{}.
A new definition of separation status is given in [@haj1]. There, quantum theory of indistinguishable particles is shown to imply that a complete meter registering quantum system ${\mathcal S}$ must be drown out by the noise due to particles in the environment that are of the same type as ${\mathcal S}$. A [*complete meter*]{} is defined as such that reacts to all states of the registered system. Thus, only incomplete meters can measure because they need not react to states of the particles in the environment. The states to which the meter does react, have been called [*domain*]{} of the meter. Moreover, a measurement can only be successful if ${\mathcal S}$ is prepared in a state that is sufficiently different from states of particles of the same type in the environment, that is, the state has a separation status. It is shown in [@haj1] that there are two equivalent ways of describing states of all particles indistinguishable from ${\mathcal S}$ if ${\mathcal S}$ has a separation status: First, states that are tensor products of the prepared state of ${\mathcal S}$ and of the state of other particles of the same type and second, states that are totally (anti-)symmetrized over all existing particles of the same type as ${\mathcal S}$.
The notion of separation status used in [@hajicek2; @hajicek4] was defined by the position of the registered particle. In this way, the position representation was given a special importance. The new definition concentrates on the process of energy dissipation rather than on the particle position. This makes three improvements possible: First, it allows a representation-independent formulation of the theory that preserves the invariance of quantum mechanics with respect to general unitary transformations. Second, the theory works not only for macroscopic detectors (containing about $10^{23}$ particles) but also for much smaller ones such as emulsion grains etc. Finally, the new definition provides a clear distinction between a scattering and an absorption of a particle by a macroscopic body. Thus, a theory that could be originally applied only to registration processes can be extended to a general theory of state reduction.
Screening
=========
Screens are no detectors and usually do not send any signals. Thus, they do not come under the theory of state reduction as given in [@hajicek2; @hajicek4]. Moreover, a study of screening can provide a new motivation for our theory of meters because screens are more simple than detectors.
Screens are used in most preparation procedures. For example, in optical experiments, such as [@RSH], polarisers, such as Glan-Thompson ones, are employed. A polariser is a macroscopic body that decomposes the coming light into two orthogonal-polarisation parts. One part disappears inside an absorber and the other is left through. Similarly, the Stern-Gerlach experiment (see, e.g., [@peres], p. 14) can be modified so that the beam corresponding to spin down is blocked out by an absorber and the other beam is left through. Finally, there are several screens in the modern two-slit experiments (see, e.g., [@tonomura]), which are just walls with holes. Generally, a screen is a macroscopic body that decomposes the incoming, already prepared, beam into one part that disappears inside the body and the other part consists of particles in the state prepared by the screening.
Let ${\mathcal S}$ be a particle with Hilbert space ${\mathbf H}$ and $\psi$ the prepared wave function of its initial state. We assume that a registration will be performed after the particle passes the screen, that the meter defines a separation status and that state $\psi$ has this separation status (see [@haj1]). Let screen ${\mathcal M}$ be a macroscopic quantum system with Hilbert space ${\mathbf H}_{\mathcal M}$. We assume that the initial state of ${\mathcal S}$ can be decomposed as follows: $$\label{decomp1} \psi = c_{\text{thr}} \psi_{\text{thr}} +
c_{\text{sw}} \psi_{\text{sw}}\ ,$$ where $\psi_{\text{thr}}$ is a normalised wave function of the part that will be left through and $\psi_{\text{sw}}$ that that will be absorbed by ${\mathcal M}$. That is, if $\psi_{\text{thr}}$ were prepared initially instead of $\psi$ then ${\mathcal S}$ will pass the screen without interaction with it (for two possible descriptions of such a process, see [@haj1]) and if $\psi_{\text{sw}}$ were, then ${\mathcal S}$ would be absorbed.
Decomposition (\[decomp1\]) is determined by the nature of ${\mathcal M}$: for a polariser, $\psi_{\text{thr}}$ and $\psi_{\text{sw}}$ are the two orthogonal polarisation states, and for a simple screen consisting of a wall and a hole, these can be calculated from the geometry of ${\mathcal M}$ and the incoming beam, but the two wave functions are then determined only approximately and the decomposition (\[decomp1\]) is only approximately unique. That is, the norm of the difference of any two possible “sw” or “thr” wave functions is small.
Let us study the behaviour of $\psi_{\text{sw}}$. The process of disappearance of a quantum system ${\mathcal S}$ in a macroscopic body ${\mathcal M}$ can be decomposed into three steps. First, ${\mathcal S}$ is prepared in a state that has a separation status so that a further preparation or registration (in which the screen participate) can be made. Second, such ${\mathcal S}$ enters ${\mathcal M}$ and ditch most of its kinetic energy somewhere inside ${\mathcal M}$. Third, the energy passed to ${\mathcal M}$ is dissipated and distributed homogeneously through ${\mathcal M}$, e.g., in a process aiming at thermodynamic equilibrium. Then, particle ${\mathcal S}$ does not possess any state of its own after being absorbed if there are any particles of the same type within ${\mathcal M}$ as has been explained in [@haj1]. It loses its separation status. Even if, originally, no particle of the same type as ${\mathcal S}$ is within ${\mathcal M}$, in the course of the experiment, ${\mathcal M}$ will be polluted by many of them. The body is assumed to be a perfect absorber so that ${\mathcal S}$ does not leave it. The absorption process is (or can be in principle) observable. For instance, the increase of the temperature of ${\mathcal M}$ due to the absorbed particles can be measured. That is, either a single particle ${\mathcal S}$ has enough kinetic energy to cause an observable temperature change, or there is a cumulative effect of many absorbed particles. In any case, the initial and final states of ${\mathcal M}$ cannot be described by wave functions, nor the registered values of the temperature are eigenvalues of any operator on the Hilbert space of ${\mathcal M}$.
The initial state of ${\mathcal M}$ is a state described by a state operator ${\mathsf T}$ (additional motivation for using states that are not pure is given in [@hajicek; @haj3]). Then the initial state for the evolution of the composite is $$\bar{\mathsf T}_{\text{swi}} = \nu {\mathsf \Pi}_{\mathcal
S}(|\psi_{\text{sw}}\rangle \langle\psi_{\text{sw}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S}\ ,$$ where $\nu^{-1} = tr[{\mathsf \Pi}_{\mathcal S}(|\psi_{\text{sw}}\rangle
\langle\psi_{\text{sw}}| \otimes {\mathsf T}){\mathsf \Pi}_{\mathcal S}]$ and ${\mathsf \Pi}_{\mathcal S}$ is the symmetrization over all particles indistinguishable from ${\mathcal S}$ within the composite system ${\mathcal
S} + {\mathcal M}$. It is an orthogonal projection operator on the Hilbert space ${\mathbf H} \otimes {\mathbf H}_{\mathcal M}$.
Let the evolution of the composite ${\mathcal S} + {\mathcal M}$ be described by operator ${\mathsf U}$. It leads to the end state of the system including the absorption and dissipation process. ${\mathsf U}$ is a unitary operator on the Hilbert space ${\mathbf H} \otimes {\mathbf H}_{\mathcal M}$ of the composite. It is independent of the choice of the initial state. After the process is finished, we obtain $$\bar{\mathsf T}' = \nu{\mathsf U}{\mathsf \Pi}_{\mathcal S}
(|\psi_{\text{sw}}\rangle \langle\psi_{\text{sw}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S}{\mathsf U}^\dagger\ .$$
Next, let us study the behaviour of $\psi_{\text{thr}}$. The initial state now is $$\bar{\mathsf T}_{\text{thri}} = \nu {\mathsf \Pi}_{\mathcal
S}(|\psi_{\text{thr}}\rangle \langle\psi_{\text{thr}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S}\ ,$$ and its evolution by ${\mathsf U}$ is similar to what has been described in [@haj1]. Thus, the final state of this evolution has two equivalent descriptions: The second one is $$\bar{\mathsf T}_{\text{thrf}2} = {\mathsf U}\nu {\mathsf \Pi}_{\mathcal
S}(|\psi_{\text{thr}}\rangle \langle\psi_{\text{thr}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S}{\mathsf U}^\dagger$$ while the first one is $$|\psi'_{\text{thr}}\rangle \langle\psi'_{\text{thr}}| \otimes {\mathsf
T}_{\text{thr}}\ ,$$ where $\psi'_{\text{thr}}$ and ${\mathsf T}_{\text{thr}}$ are the states of ${\mathcal S}$ and ${\mathcal M}$ after having been evolved by ${\mathsf U}$ independently from each other. ${\mathsf T}_{\text{thr}}$ is a state of the screen that is approximately equal to its initial state ${\mathsf T}$.
As yet, we have described the two partial “channels” of evolution in which the problem has been decomposed. Now, we have to deal with the whole process. The initial state of the composite is then $$\label{screeni} \bar{\mathsf T}_i = |\psi\rangle \langle\psi|
\otimes {\mathsf T}\ .$$ Using decomposition (\[decomp1\]), we can write $$\begin{gathered}
\label{formscreeni} \bar{\mathsf T}_{\text{fei}} = \nu
\Big(c^*_{\text{thr}}c_{\text{thr}}{\mathsf \Pi}_{\mathcal
S}(|\psi_{\text{thr}}\rangle \langle\psi_{\text{thr}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S} + c_{\text{thr}}c^*_{\text{sw}}{\mathsf
\Pi}_{\mathcal S} (|\psi_{\text{thr}}\rangle \langle\psi_{\text{sw}}| \otimes
{\mathsf T}){\mathsf \Pi}_{\mathcal S} \\ +
c_{\text{sw}}c^*_{\text{thr}}{\mathsf \Pi}_{\mathcal S}
(|\psi_{\text{sw}}\rangle \langle\psi_{\text{thr}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S} + c^*_{\text{sw}}c_{\text{sw}} {\mathsf
\Pi}_{\mathcal S} (|\psi_{\text{sw}}\rangle \langle\psi_{\text{sw}}| \otimes
{\mathsf T}){\mathsf \Pi}_{\mathcal S}\Big)\ .\end{gathered}$$ After the process of evolution by ${\mathsf U}$ is finished, the end state is $$\begin{gathered}
\label{formscreenf} \bar{\mathsf T}_{\text{fef}} = \nu
c^*_{\text{thr}}c_{\text{thr}}{\mathsf \Pi}_{\mathcal
S}(|\psi'_{\text{thr}}\rangle \langle\psi'_{\text{thr}}| \otimes {\mathsf
T}_{\text{thr}}){\mathsf \Pi}_{\mathcal S} +
c^*_{\text{sw}}c_{\text{sw}}\bar{\mathsf T}' \\ +
\nu\Big(c_{\text{thr}}c^*_{\text{sw}} {\mathsf U}{\mathsf \Pi}_{\mathcal S}
(|\psi_{\text{thr}}\rangle \langle\psi_{\text{sw}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S}{\mathsf U}^\dagger +
c_{\text{sw}}c^*_{\text{thr}} {\mathsf U}{\mathsf \Pi}_{\mathcal S}
(|\psi_{\text{sw}}\rangle \langle \psi_{\text{thr}}| \otimes {\mathsf
T}){\mathsf \Pi}_{\mathcal S}{\mathsf U}^\dagger \Big)\ .\end{gathered}$$ The first two terms form a convex composition of the two end states of the two channels. The rest is an operator with trace zero that describes some correlations between systems ${\mathcal S}$ and ${\mathcal
M}$. In any case, state (\[formscreenf\]) describes the screen as being in two macroscopically different state simultaneously. This is never observed and hence the Schrödinger equation must be corrected. Not only the last term must vanish but the convex combination must be replaced by a proper mixture [@hajicek2]. The proper mixture (also called “Gemenge” [@BLM] or “direct mixture” [@ludwig1]) of states ${\mathsf T}_1$, …${\mathsf T}_N$, $${\mathsf T} = \left(\sum_k\right)_p c_k {\mathsf T}_k\ ,$$ where $\sum_k c_k = 1$, is such that each individual system in state ${\mathsf
T}$ is simultaneously in one of the states ${\mathsf T}_1$, …${\mathsf
T}_N$, and the probability of being in ${\mathsf T}_k$ is $c_k$[^1]. We distinguish proper mixtures by the sign $(+)_p$ or $(\sum)_p$ (on more details on the properties of the operation $(+)_p$, see [@hajicek5]).
Hence our crucial model assumption is:
\[asstredscreen\] The final state of the true evolution is $$\label{screenf} \bar{\mathsf T}_f = {\mathrm P}_{\text{thr}}
|\psi'_{\text{thr}}\rangle \langle\psi'_{\text{thr}}| \otimes {\mathsf
T}_{\text{thr}}\ (+)_p\ {\mathrm P}_{\text{sw}} \bar{\mathsf T}'\ ,$$ where $${\mathrm P}_{\text{thr}} = c^*_{\text{thr}}c_{\text{thr}}\ ,\quad {\mathrm
P}_{\text{sw}} = c^*_{\text{sw}}c_{\text{sw}}\ .$$
The state reduction is not a unitary transformation: First, the non-diagonal terms in (\[formscreenf\]) have been erased. Second, we have also assumed that state $\psi'_{\text{thr}}$ is the state of ${\mathcal S}$ that has been [*prepared*]{} by the screening. This means for us that it is a real state with a separation status. Hence, operator ${\mathsf \Pi}_{\mathcal
S}$ has been left out in Formula (\[screenf\]) (see [@haj1]). This is, of course, another violation of unitarity.
However, all operations that constitute the transformation from the initial state (\[screeni\]) to the final one (\[screenf\]) are independent of representation: they can be written in $Q$- as well as in $P$-representation. More generally, they are covariant with respect to any unitary transformation. Our theory of state reduction preserves the unitary-transformation symmetry of quantum mechanics.
The disappearance of ${\mathcal S}$ in ${\mathcal M}$ is a physical process that have a definite time and place. One could therefore assume presumably without any bad consequences that the state reduction occurs at the time and the place of the possible absorption of the particle in ${\mathcal M}$. The possible absorption had to be viewed as a part of the whole process even in the case that the individual particle is not absorbed but goes through. Indeed, that an individual particle goes through is only a result of the state reduction, which is a change from the linear superposition of the left-through and the absorption states.
This idea is utilised for a definite choice of the state reduction as postulated in Assumption \[asstredscreen\]. The choice is determined by the experimental arrangement and the resulting two alternatives, one of which leads to the absorption of the object system and to the loss of separation status.
Models of meter reading
=======================
Let us now explain how some of the ideas on screening can also be applicable to registrations. We analyse the meters and find some structures that can play a similar role as screens do.
The structure of meters
-----------------------
The theoretical description of meters that can be found in the literature is relatively simple and “clean” (see, e.g., [@BLM]): the meter is a quantum system with a “pointer” observable. Our general hypotheses will, however, connect the occurrence and form of state reduction with some construction details of meters. Thus, we must refine the language: the words that name parts or structural elements of meters will be [*field, screen, ancilla*]{} and [*detector*]{} [@hajicek4].
Screens have been dealt with in the foregoing section. It is also more or less clear what are the fields: for example, in the Stern-Gerlach experiment, the beam is split by an inhomogeneous magnetic field. Or, in some optical experiments, various crystals are used that enable to split different polarisations from each other or to split the beam into two mutually entangled beams such as it is done by the down-conversion process in a crystal of KNbO$_3$ [@MW]. The corresponding crystals can also be considered as fields. In any case, the crystals and fields are macroscopic systems in which the energy of the incident particles does not dissipate.
In many modern experiments, in particular in non-demolition [@braginsky] and weak measurements [@aharonov], but not only in these, the following idea is employed. The object system $\mathcal S$ interacts first with a microscopic quantum system $\mathcal A$ that is prepared in a suitable state. After $\mathcal S$ and $\mathcal A$ become entangled, $\mathcal A$ is subject to further registration and, in this way, some information on $\mathcal S$ is obtained. Subsequently, or simultaneously, another measurements on $\mathcal S$ can but need not be made. In any case, the state of $\mathcal S$ is influenced by the registration just because of its entanglement with $\mathcal A$. The auxiliary system $\mathcal A$ is usually called [*ancilla*]{}.
Finally, very important parts of meters are [*detectors*]{}. Indeed, even a registration of an ancilla needs a detector. It seems that any registration on microscopic systems has to use [*detectors*]{} in order to make features of microscopic systems visible to humans. Detector is a many-particle system (the particle number being not necessarily of the order of $10^{23}$) in a metastable state that appreciably changes during the interaction with the registered system[^2]. It contains [*active volume*]{} $\mathcal D$ and [*signal collector*]{} $\mathcal C$ in a state of metastable equilibrium. Notice that the active volume is a physical system, not just a volume of space. Interaction of the detected systems with $\mathcal D$ triggers a relaxation process leading to observable changes in the detector that are called [*detector signals*]{}. For some theory of detectors, see, e.g., [@leo; @stefan].
In the so-called cryogenic detectors [@stefan], ${\mathcal S}$ interacts, e.g., with superheated superconducting granules. It is scattered by a nucleus in a granule and the resulting phonons induce the phase transition from the superconducting into the normally conducting phase. An active volume can contain very many granules (typically $10^9$) in order to enhance the probability of such scattering if the interaction between ${\mathcal S}$ and the nuclei is very weak (${\mathcal S}$ may be a weakly interacting massive particle, neutrino). Then, there is signal collector: a solenoid around the active volume and an independent strong magnetic field. The phase transition of only one granule leads to a change in magnetic current through the solenoid giving a perceptible electronic signal.
Modern detectors are constructed so that their signals are electronic. For example, to a scintillation film, a photomultiplier is attached (as, e.g., in Tonomura experiment [@tonomura]). We assume that there is a signal collected immediately after the sensitive matter falls down from its metastable state, which we call [*primary*]{} signal. Primary signal may still be amplified and filtered by other electronic apparatuses to transform it into the final signal of the detector. For example, the light signal of a scintillation film in Tonomura experiment is a primary signal. It is then transformed into an electronic signal by a photocathode and the resulting electronic signal is further amplified by a photomultiplier.
One of new but rather obvious ideas of [@hajicek2] was that ancillas and detectors within meters ought to be distinguished from each other. To make this aim easier, we have slightly modified the current notion of detector: the detectors as defined above are more specific than what may be sometimes understood as detectors.
Three assumptions about registration
------------------------------------
The foregoing analysis motivated the introduction of some general features of meter models that are summarised in the following assumptions.
\[aspointerh\] Any meter for microsystems must contain at least one detector and every reading of the meter can be identified with a primary signal from a detector.
A similar assumption has been first formulated in [@hajicek2]. Assumption \[aspointerh\] makes the reading of meters less mysterious. Observational facts together with the ideas of Section 2 suggest that the state reductions might take place in detectors:
\[assh\] Let ${\mathcal M}$ be a meter registering a quantum system ${\mathcal S}$. Let the Schrödinger equation for the composite ${\mathcal S} + {\mathcal M}$ leads to a linear superposition of alternative evolutions such that each alternative is associated with a definite signal state of a detector. Then, it must be corrected so that the linear superposition is replaced by the proper mixture of the alternatives.
For similar assumptions, see [@hajicek2; @hajicek4]. Both detectors and screens, where the state reductions occur, are many-particle systems, but there are many-particle systems in which no state reduction occurs if a microsystem interacts with them. For example, during the scattering of neutrons with ferromagnetic crystals, no state reduction seems to happen (see [@hajicek4]). It may be a loss of separation status that makes the difference:
\[asavh\] During the interaction of the registered system ${\mathcal S}$ with the active volume ${\mathcal D}$ of a detector, the energy of ${\mathcal S}$ dissipates and becomes distributed among many particles of the composite ${\mathcal S} + {\mathcal D}$. This leads to a loss of separation status of ${\mathcal S}$.
The dissipation is necessary to accomplish the loss. The dissipation process does not have anything mysterious about it. It can be a usual thermodynamic relaxation process in a macroscopic system or a similar process of the statistical thermodynamics generalised to nano-systems (see, e.g., [@horodecki]). ${\mathcal S}$ might be the objects system or an ancilla of the original experiment. The loss of separation status is an objective process and the significance of Assumption \[asavh\] is that it formulates an objective condition for the applicability of an alternative kind of dynamics.
Assumption \[assh\] defines a rule that determines the correction to unitary evolution [*uniquely*]{} in a large class of scattering and registration processes. This has been shown in [@hajicek4].
The three assumptions form a basis of our theory of registration. This theory just refines and completes the standard quantum mechanics that includes state reduction as formulated by von Neumann [@JvN]. It generalises some empirical experience, is rather specific and, therefore, testable. That is, it cannot be disproved by purely logical argument but rather by an experimental example. For the same reason, it also shows a specific direction in which experiments are to be proposed and analysed: if there is a state reduction, does then a loss of separation status take part in the process? What system loses its status?
In fact, our theory remains rather phenomenological. It does not suggests any causal chain leading from a separation status loss to a state reduction. A model of such a chain might require some new physics and we believe that hints of what this new physics could be would come from attempts to answer the above two questions for some suitable experiments.
Stern-Gerlach story retold
==========================
In this section, we shall modify the textbook description (e.g., [@peres], pp. 14 and 375) of the Stern-Gerlach experiment. In this way, the above ideas can be explained and illustrated.
The experiment measures the spin of silver atoms. A silver atom consists of 47 protons and 61 neutrons in the nucleus and of 47 electrons around it, but we consider only its mass-centre and spin degrees of freedom. We denote the system with these degrees of freedom by ${\mathcal S}$ and its Hilbert space by ${\mathbf H}$. Let $\vec{\mathsf x}$ be its position, $\vec{\mathsf p}$ its momentum and ${\mathsf S}_z$ the $z$-component of its spin with eigenvectors $|j\rangle$ and eigenvalues $j \hbar/2$, $j = \pm$.
Let ${\mathcal M}$ be a Stern-Gerlach apparatus with an inhomogeneous magnetic field in a region $D$ that separates different $z$-components of spin of a silver atom arriving in $D$. Let the detector of the apparatus be a photo-emulsion film with energy threshold $E_0$ placed roughly orthogonally to the split beam. The emulsion is the active volume ${\mathcal D}$ of ${\mathcal
M}$. The emulsion grains are sufficiently large so that the hit ones can be made directly visible and can therefore also be considered as a signal collectors. ${\mathcal D}$ is a macroscopic quantum system with Hilbert space ${\mathbf H}_{\mathcal D}$.
First, let ${\mathcal S}$ be prepared at time $t_1$ in a definite spin state $$\label{sin} |\vec{p},\Delta \vec{p}\rangle \otimes |j \rangle\
,$$ where $|\vec{p},\Delta \vec{p}\rangle$ is a Gaussian wave packet corresponding to fixed average $\vec{p}$ and variance $\Delta \vec{p}$ of momentum $\vec{\mathsf p}$ such that ${\mathcal S}$ can be registered by ${\mathcal M}$ within some time interval $(t_1,t_2)$. That is, the direction of $\vec{p}$ is suitably restricted and its magnitude must respect the energy threshold $E_0$. Such states lie in the domain of the apparatus ${\mathcal
M}$. Hence, state (\[sin\]) has a separation status at $t_1$ (see [@haj1]). Initially, let ${\mathcal D}$ be in metastable state ${\mathsf
T}_{\mathcal D}(t_1)$ at $t_1$.
Let the measurement coupling be denoted by ${\mathsf U}$. It contains the interaction of ${\mathcal S}$ with the magnetic field and with the active volume ${\mathcal D}$. The time evolution within $(t_1,t_2)$ is: $${\mathsf U}\nu{\mathsf \Pi}_{\mathcal S}\Bigl(|\vec{p},\Delta
\vec{p}\rangle\langle \vec{p},\Delta \vec{p}| \otimes |j \rangle \langle j
|\otimes {\mathsf T}_{\mathcal D}(t_1)\Bigr){\mathsf \Pi}_{\mathcal S}{\mathsf
U}^\dagger = \bar{\mathsf T}_{j}(t_2)\ ,$$ where ${\mathsf \Pi}_{\mathcal S}$ is antisymmetrisation on the Hilbert space of silver atom part of ${\mathcal S} + {\mathcal D}$ and $\nu$ is a normalisation factor because ${\mathsf \Pi}_{\mathcal S}$ does not preserve normalisation. Hence, state $\bar{\mathsf T}_{j}(t_2)$ is uniquely determined for each $j$. States $\bar{\mathsf T}_{j}(t_2)$ are operators on ${\mathsf
\Pi}_{\mathcal S}({\mathbf H} \times {\mathbf H}_{\mathcal D})$.
Evolution ${\mathsf U}$ includes a thermodynamic relaxation of ${\mathcal D}$ with ${\mathcal S}$ inside ${\mathcal D}$ and detector signals that are concentrated within one of two strips on the film, each strip corresponding to one value of $j$. ${\mathsf U}$ also describes the loss of separation status of ${\mathcal S}$ in ${\mathcal D}$. This is clear because the silver atoms are both the registered systems and components of the detector. Then, individual states of ${\mathcal S}$ do not make sense and the physical object ${\mathcal S}$ has been lost, that is, it has ceased to exist (see [@haj1]).
Suppose next that the initial state of ${\mathcal S}$ at $t_1$ is a linear superposition of the spin states: $$|\vec{p},\Delta \vec{p}\rangle \otimes \left(\sum_j c_j |j\rangle\right)$$ with $$\sum_j|c_j|^2 = 1\ .$$
Evolution ${\mathsf U}$ of this state is: $$\begin{gathered}
\nu{\mathsf U}{\mathsf \Pi}_{\mathcal
S}\left[|\vec{p},\Delta \vec{p}\rangle \langle\vec{p},\Delta \vec{p}| \otimes
\left(\sum_jc_j|j \rangle\right) \left(\sum_{j'}c^*_{j'}\langle
j'|\right)\otimes {\mathsf T}_{\mathcal D}(t_1)\right]{\mathsf \Pi}_{\mathcal
S}{\mathsf U}^\dagger \\ = \sum_{jj'}c_jc^*_{j'} \bar{\mathsf T}_{jj'}(t_2)\ .\end{gathered}$$ The right-hand side is a quadratic form in $\{c_j\} \in
{\mathbb C}^2$. Coefficients $\bar{\mathsf T}_{jj'}(t_2)$ of the form are operators on the Hilbert space ${\mathsf \Pi}_{\mathcal S}({\mathbf H} \times
{\mathbf H}_{\mathcal D})$.
The operator coefficients are state operators only for $j' = j$. From the linearity of ${\mathsf U}$, it further follows that $$\bar{\mathsf T}_{jj}(t_2) = \bar{\mathsf T}_{j}(t_2)$$ for all $j$.
Now we postulate a correction to the Schrödinger equation:
1. The loss of separation status of ${\mathcal S}$ disturbs the standard quantum evolution so that, instead of $$\sum_{j,j'} c_jc^*_{j'} \bar{\mathsf T}_{jj'}(t_2)\ ,$$ state $$\left(\sum_j\right)_p\ |c_j|^2 \bar{\mathsf T}_{j}(t_2)$$ results.
2. States $\bar{\mathsf T}_{j}(t_2)$ are uniquely determined by the experimental arrangement: the measurement coupling and the losses of separation status in the meter.
3. The sum is not only a convex combination but also a proper mixture of the signal states $\bar{\mathsf T}_{j}(t_2)$. That is, the system ${\mathcal
S} + {\mathcal M}$ is always in one particular state $\bar{\mathsf
T}_{j}(t_2)$ after each individual registration and the probability for that is $|c_j|^2$.
4. The correction is manifestly independent of any representation that may be used to calculate it. Thus, the theory is covariant with respect to the general unitary transformation group.
The present account differs from that of Ref. [@peres] in the description of the meter and in the role of the detector. In Peres’ words (p. 17):
> The microscopic object under investigation is the magnetic moment $\mathbf \mu$ of an atom.... The macroscopic degree of freedom to which it is coupled in this model is the centre of mass position $\mathbf r$... I call this degree of freedom [*macroscopic*]{} because different final values of $\mathbf r$ can be directly distinguished by macroscopic means, such as the detector... From here on, the situation is simple and unambiguous, because we have entered the macroscopic world: The type of detectors and the detail of their functioning are deemed irrelevant.
The root of such notion of meter may be found among some ideas of the grounding fathers of quantum mechanics. For example, Ref. [@pauli], p. 64, describes a measurement of energy eigenvalues with the help of scattering similar to Stern-Gerlach experiment, and Pauli explicitly states:
> We can consider the centre of mass as a ’special’ measuring apparatus...
The point of view of the present paper is that the spin degree of freedom can be considered as the registered system, the centre-of-mass one as the ancilla and there must be a detector to register the ancilla.
Experiments with superconductor currents
========================================
Finally, let us briefly describe a possible instance of experiments for which the analysis proposed at the end of Section 3.2 could either disprove our theory or give some new insight on the nature of state reduction.
Consider a single Josephson junction SQUID ring [@leggett], that is a superconducting ring interrupted by transversal layer of oxide which allows the electrons to pass through by tunnelling. A quantum model of the Bose-Einstein condensate of the Cooper pairs in the superconductor, based on a number of simplifications, has been constructed in [@leggett]. We call it “Quantum model of superconductor currents” (QMSC). We shall not go into details of the construction but refer the interested reader to [@leggett].
QMSC has one degree of freedom which is chosen to be the magnetic flux $\Phi$ through the ring created by the current. Thus the states of the model can be described by wave functions of the form $\psi(\Phi)$. The Schrödinger equation of the model is: $$\label{leggett1} i\hbar\frac{\partial \psi}{\partial t} =
-\frac{\hbar^2}{2C}\ \frac{\partial^2\psi}{\partial \Phi^2} + V(\Phi)\psi\ ,$$ where $$\label{leggett2} V = \frac{(\Phi - \Phi_{\text{ext}})^2}{2L} -
\frac{I_c\varphi_0}{2\pi}\ \cos\frac{2\pi\Phi}{\varphi_0}\ .$$ The numerical parameters have the following physical meaning: $C$ is the capacitance and $L$ is the self-inductance of the ring, $I_c$ is the critical current, i.e., the maximum current that can be transmitted through the junction without dissipation, $\varphi_0 = \pi\hbar/e$ is the so-called superconducting flux quantum and $\Phi_{\text{ext}}$ is the magnetic current through the ring that is applied externally (for more details on the physics of the potential function and the parameters, see [@leggett]).
For some suitable values of the parameters, the potential function $V(\Phi)$ has two local minima at $\Phi_1$ and $\Phi_2$ of the equal depths divided by a barrier of the height $\bar{V}$. The two metastable quantum states concentrated around the minima define two values $E_1$ and $E_2$ of the total energy (described by the operator on the right-hand side of equation (\[leggett1\])). According to [@LS; @laloe], the quantum fluxes $\Phi_1$ and $\Phi_2$ can be sufficiently strong to be considered macroscopically distinct so that they could be distinguished by “naked eye”, or by a “tiny magnetic needle”.
The original aim was to prove or disprove that linear superposition of macroscopically distinct states are possible. The hope was that, by a suitable arrangement of the SQUID experiment, one can prepare a state of the condensate that is a linear superposition of the two metastable states. Some measurements were proposed that would prove the existence of the linear superposition.
A possible existence of linear superposition of quantum states of macroscopic systems does not lead to any difficulties for our theory of classical properties ([@hajicek; @haj3]). However, a side aspect of such measurements that is important for us is that the measurements are supposed to be usual quantum measurements inducing a collapse of the wave function. In this case, it will be the collapse to one of the linear superposition components. Now, a difficulty with such collapse emphasized by [@LS; @laloe] is to imagine the tiny needle influencing the macroscopic magnetic field so that it could change appreciably. We add a further question: Where is the separation status change that would be necessary for the collapse if our theory were true?
However, the notion of a tiny needle occurs only in thought experiments described in [@LS; @laloe]. If the experiments that have been really carried out are carefully studied, one finds that they are organised along completely different lines [@squid1; @squid2]. They still confirm the existence of linear superposition but they have less clear results about the wave function collapse. Let us look at some details.
The measurements [@squid1; @squid2] are made spectroscopically. A microwave radiation is applied to the superconductors and one looks for a resonance at the frequency of the energy difference $E_2 - E_1$ between the metastable states. The energy that the radiation imparts to the superconducting device is smaller than the height $\bar{V}$ of the potential barrier between the metastable states, hence it requires the tunnelling between them and this in turn requires a linear superposition of the states.
Thus, a different measurement structure emerges: the registered system is a photon rather than the BEC and the meter seems to use (a part of) the superconductor device rather than any tiny needle. The most important result for us is that the observed resonance lines show an appreciable width. This is interpreted as a dissipation effect that occurs somewhere within the superconducting device. However, the dissipation is not well understood, in particular, it does not occur within the framework of QMSC [@leggett].
Clearly, Assumption \[asavh\] suggests a new direction of investigation that concentrates on the dissipation effect. Indeed, some dissipation and some metastable states are postulated by our theory of measurement. The dissipation may occur within a definite subsystem of the meter and one can then ask whether a loss of separation status occurs there, what system loses its status and whether the status loss is associated with any state reduction. One could also try to suggest another experimental set-up that were better adapted to the task of answering these questions.
Conclusion
==========
The basic idea on the structure of meters and the role of detectors as explained in [@hajicek2; @hajicek4] has been adapted to the new definition [@haj1] of separation status. Three main improvements resulted.
First, the restriction of state reduction to registration processes has been removed and a general theory of state reduction has been introduced and explained by the example of screening. For such generalisation, a clear distinction between scattering and partial and complete absorption of a particle is necessary and it is provided by the presence or absence of dissipation.
Second, the restriction of [@hajicek2; @hajicek4] to macroscopic meters can be abandoned because dissipation processes are possible also in much smaller detectors. Thus, our theory becomes applicable to many modern experiments.
Third, papers [@hajicek2; @hajicek4] used the notion of separation status in an incorrect way because the their misleading limitation to the geometrical aspects of the experimental arrangements. The present paper has shown that the local aspects of separation status are not always important. Hence, our theory of state reduction is independent of representation used to formulate it so that it is covariant with respect to unitary transformations.
Finally, the example of superconducting rings have also suggested that an experimental check of the theory is possible. In summary, a better understanding of the notion of state reduction has resulted.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author is indebted to Nicolas Gisin, Stefan Jánoš, Petr Jizba and Jiří Tolar for useful discussions.
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[^1]: Of course, such a definition can be conceptually clean only within a realist interpretation of quantum mechanics, such as [@hajicek5], that considers states as objective (and deterministic) properties of quantum systems.
[^2]: We consider even relatively small systems as detectors, for example, the nanowire single photon detector [@natarayan].
|
-1.0cm
**Constraints on scalar and vector leptoquarks\
from the LHC Higgs data**
[**Jian Zhang $^{a}$**]{}[^1] [**Chong-Xing Yue $^{a}$**]{}[^2] [**Chun-Hua Li $^{a}$**]{} and [**Shuo Yang $^{b}$**]{}\
[$^a$Department of Physics, Liaoning Normal University, Dalian 116029, China\
$^b$Department of Physics, Dalian University, Dalian 116622, China ]{}
**Abstract**
Introduction {#intro}
============
In the last few years accumulated experimental results of semileptonic $B$-meson decays point to lepton flavour universality violation (LFUV). In the case of flavor changing neutral current (FCNC) transition $b \to s \mu^+ \mu^-$, ratios $R_{K^{(*)}} = \frac{\mathcal{B}(\bar{B} \to K^{(*)} \mu^+ \mu^-)}{\mathcal{B}(\bar{B} \to K^{(*)} e^+ e^-)}$ measured by the LHCb collaboration are lower than the SM expectations by $\sim 2.6 \sigma$ [@Aaij:2014ora; @Aaij:2017vbb; @Bordone:2016gaq; @Capdevila:2017ert]. For $b \to c \ell \nu$ ($\ell = e, \mu$) charged current case, measurements of $R_{D} = \frac{\mathcal{B}(\bar{B} \to D \tau^- \bar{\nu})}{\mathcal{B}(\bar{B} \to D \ell^- \bar{\nu})}$ and $R_{D^*} = \frac{\mathcal{B}(\bar{B} \to D^* \tau^- \bar{\nu})}{\mathcal{B}(\bar{B} \to D^* \ell^- \bar{\nu})}$ from experiments are higher than the SM expectations by $\sim 2.3 \sigma$ and $\sim 3.4 \sigma$, respectively [@Aaij:2015yra; @Huschle:2015rga; @Sato:2016svk; @Hirose:2016wfn; @Lees:2012xj; @Lees:2013uzd; @HFAG2017; @Fajfer:2012vx; @Aoki:2016frl]. As popular candidates for explaining $B$-anomalies, leptoquarks (LQs) are extensively discussed in specific ultraviolet (UV) theories or model-independently (see, e.g, [@Gripaios:2014tna; @Georgi:2016xhm; @Becirevic:2016yqi; @Becirevic:2017jtw; @Diaz:2017lit; @Buttazzo:2017ixm; @Guo:2017gxp; @DiLuzio:2017vat; @Calibbi:2017qbu; @Blanke:2018sro; @Fajfer:2018bfj; @Matsuzaki:2018jui; @Hati:2018fzc; @Becirevic:2018afm; @Crivellin:2018yvo; @deMedeirosVarzielas:2018bcy; @Azatov:2018kzb; @DiLuzio:2018zxy; @Faber:2018qon; @Heeck:2018ntp; @Angelescu:2018tyl; @Balaji:2018zna; @Watanabe:2018jhh; @Schmaltz:2018nls; @Bansal:2018nwp; @Iguro:2018vqb; @Fajfer:2018hbq; @Fornal:2018dqn; @DaRold:2018moy; @deMedeirosVarzielas:2019lgb; @Zhang:2019hth; @Aydemir:2019ynb; @Cata:2019wbu; @Bhattacharya:2019olg; @Adam:2019oes; @Aebischer:2019mlg; @Cornella:2019hct; @Barbieri:2019zdz]).
LQs are hypothetical color-triplet bosons that carry both baryon and lepton numbers [@Buchmuller:1986zs; @Dorsner:2016wpm; @Tanabashi:2018oca]. They naturally appear in many extensions of the Standard Model (SM) such as Pati-Salam model [@Pati-Salam], grand unification theories based on $SU(5)$ [@Georgi:1974sy] and $SO(10)$ [@Georgi:1974my], extended technicolor [@technicolor], and compositeness [@composite]. According to their properties under the Lorentz transformations, LQs can be either scalar (spin 0) or vector (spin 1). Several models suggest LQs mass of TeV-scale.
LQs can also couple to Higgs boson and considerably modify loop-induced Higgs processes, gluon fusion production ($ggF$) and $h \to \gamma \gamma$ decay, without appreciably changing kinematics of theses process. Scalar LQs interact with the Higgs boson at tree level via Higgs portal interactions. Their contributions to loop-induced Higgs processes can be studied model-independently [@Dorsner:2016wpm; @Dorsner:2015mja; @Chang:2012ta]. Vector LQs, as gauge fields in full fledged models, make contributions to the loop processes that are sensitive to the gauge sector of the ultraviolet (UV) theories which they belong to. $ggF$ predominates the Higgs production processes at the LHC. And the LHC is sensitive to $h \to \gamma \gamma$ decay process. After the discovery of the 125 GeV Higgs boson by the ATLAS [@atlas-higgs] and CMS [@cms-higgs] experiments in 2012, precisely measuring properties of the Higss boson are then performed by the ATLAS and CMS experiments with LHC Run I and II data sets [@Khachatryan:2016vau; @ATLAS:2019slw; @Sirunyan:2018koj]. Globally analyzing these measurements, in some sense, can guide us for LQs study.
Constraints on scalar LQs are obtained by Ref. [@Dorsner:2016wpm] via analyzing Higgs data from the LHC Run I reported by the ATLAS and CMS collaborations [@TheATLASandCMSCollaborations:2015bln; @CMS:2015kwa]. We update these results via comprehensively analyzing Higgs data from the LHC Run I and II [@Khachatryan:2016vau; @ATLAS:2019slw; @Sirunyan:2018koj].
Since interactions between vector LQ and the Higgs boson as well as other gauge fields are sensitive to the UV theories which the LQ belongs to, the contributions of vector LQ to the loop-induced Higgs processes should be studed in a specific model. Of particular note is that $U_1(\mathbf{3},\mathbf{1},2/3)$ with mass of several TeV performs quite well in explaining both anomalies of $R_{D^{(*)}}$ and $R_{K^{(*)}}$ [@Buttazzo:2017ixm]. In this article, we study $U_1(\mathbf{3},\mathbf{1},2/3)$ originating from a particular theory, namely the ’4321’ model. One of purposes of the model is to explain $B$-anomalies [@DiLuzio:2017vat; @DiLuzio:2018zxy]. Besides obtaining the constraints on the size of vector LQ interactions to the Higgs boson from current LHC Higgs data, we also provide a method to determine vacuum expectation values (VEVs) $\upsilon_3$ and $\upsilon_1$ of the new scalar fields $\Omega_3$ and $\Omega_1$ in the ’4321’ model via the combination of Higgs data and measurements of $R_{D^{(*)}}$ and $R_{K^{(*)}}$.
The article is organized as follows: we first review current Higgs data from the LHC Run I and II in Section \[sec:2\]. In Section \[sec:3\], we model-independently study the contributions of single scalar LQ to loop-induced Higgs processes, $ggF$ production and $h \to \gamma \gamma$ decay. Contributions of the vector LQ $U_1$ to these loop processes are discussed in framework of the ’4321’ model in Section \[sec:4\]. In the same section, we also discuss the determination of VEVs $\upsilon_3$ and $\upsilon_1$ of this model. Finally, conclusions for this work are given in Section \[sec:5\].
The LHC Higgs data {#sec:2}
==================
The discovery of the 125 GeV Higgs boson by the ATLAS [@atlas-higgs] and CMS [@cms-higgs] experiments in 2012 is one of the greatest achievements in the history of particle physics. Precise measurements of the Higss boson properties are then performed by these experiments. At the LHC, only products of cross sections and branching fractions are measured. In the narrow-width approximation, the signal cross section of an individual channel, e.g. $\sigma (gg \to H \to \gamma\gamma)$, can be factorized as [@Heinemeyer:2013tqa] $$\begin{aligned}
\sigma (gg \to H \to \gamma\gamma) &=& \frac{\sigma_{ggF} \cdot \Gamma^{\gamma\gamma}}{\Gamma_h} \nonumber \\
&=& (\sigma_{ggF} \cdot B^{\gamma\gamma})_{SM}\dfrac{\kappa_g^2 \cdot \kappa_{\gamma}^2}{\kappa_h^2},\end{aligned}$$ where $\sigma_i$ and $\Gamma^j$ represent measured values of $i \to h$ production and $h \to j$ decay, respectively, and $\sigma^{SM}_i$ and $\Gamma^j_{SM}$ are their SM expectations, $\kappa_i$ are the so called ’coupling modifiers’ defined as $\kappa^2_i = \sigma_i/\sigma^{SM}_{i}$ or $\kappa^2_i = \Gamma^i/\Gamma^i_{SM}$ ( all $\kappa_i$ values equal unity in the SM ), and $\Gamma_h$ denotes the total width of the Higgs boson.
In 2016, the ATLAS and CMS collaborations reported measurements of the Higgs boson production and decay rates as well as constraints on its couplings to vector bosons and fermions by using the LHC Run I data recorded in 2011 and 2012 [@Khachatryan:2016vau]. The integrated luminosities in each experiment are about 5 fb$^{-1}$ at $\sqrt{s} = 7$ TeV and 20 fb$^{-1}$ at $\sqrt{s} = 8$ TeV. The measurements are based on five main Higgs boson production processes (gluon fusion, vector boson fusion, and associated production with a $W$ or a $Z$ boson or pair of top quarks) and six decay modes ( $h \to ZZ, WW, \gamma \gamma, \tau \tau, bb \ {\rm and} \ \mu \mu$ ).
In 2019, the similar measurements are reported by the ATLAS and CMS collaborations via using the Run II data set recorded by the ATLAS detector during 2015, 2016 and 2017 with the integrated luminosity of 79.8 fb$^{-1}$ at $\sqrt{s} = 13$ TeV [@ATLAS:2019slw] and the CMS detector in 2016 at $\sqrt{s} = 13$ TeV with the integrated luminosity of 35.9 fb$^{-1}$ [@Sirunyan:2018koj], respectively.
The Higgs boson with mass of $m_h = 125.09$ GeV is assumed in all the above experimental analyses. These measurements normalized to the SM predictions are listed in Table \[tab:measurements\]. From Table \[tab:measurements\] we can see that measurements obtained by each experiment from the LHC Run I or Run II are precisely consistent within error with their SM predictions. This implies that NP properly lies in a scale much higher than the mass of Higgs boson, and new heavy particles carrying electric and colour charge may still be present in the loop-induced Higgs processes, $ggF$ production and $h \to \gamma \gamma$ decay, without appreciably changing kinematics of theses process [@Chang:2012ta; @Enkhbat:2013oba; @Djouadi:2005gj; @Carena:2012xa; @Dorsner:2012pp; @Agrawal:1999bk; @Gori:2013mia].
[lr@[0.4ex]{}lr@[0.4ex]{}lr@[0.4ex]{}l]{}\
&\
&\
& & &\
& & &\
$\sigma_{ggF}\cdot B_{ZZ} $ & $1.16$&$ {}_{- \ 0.24}^{+ \
0.26}$ & $1.13$&${}_{- \ 0.13}^{+ \ 0.13}$ & $1.07$&${}_{- \ 0.18}^{+ \ 0.20}$\
$\sigma_{VBF}/\sigma_{ggF}$ & $1.33$&${}_{-\ 0.36}^{+\ 0.44}$ & $1.23$&${}_{-\ 0.27}^{+\ 0.32}$ & $0.6$&${}_{-\ 0.24}^{+\ 0.30}$\
$\sigma_{WH}/\sigma_{ggF}$ & $0.84$&${}_{-\ 0.71}^{+\ 0.76}$ & $1.26$&${}_{-\ 0.45}^{+\ 0.59}$ & $2.19$&${}_{-\ 0.69}^{+\ 0.86}$\
$\sigma_{ZH}/\sigma_{ggF}$ & $3.06$&${}_{-\ 1.48}^{+\ 1.84}$ & $1.01$&${}_{-\ 0.35}^{+\ 0.47}$ & $0.88$&${}_{-\ 0.27}^{+\ 0.34}$\
$\sigma_{ttH+tH}/\sigma_{ggF}$ & $3.28$&${}_{-\ 1.02}^{+\ 1.15}$ & $1.20$&${}_{-\ 0.27}^{+\ 0.31}$ & $1.06$&${}_{-\ 0.27}^{+\ 0.34}$\
$\mathcal{B}_{\gamma\gamma}/\mathcal{B}_{ZZ}$ & $0.81$&${}_{-\ 0.16}^{+\ 0.21}$ & $0.87$&${}_{-\ 0.12}^{+\ 0.14}$ & $1.14$&${}_{-\ 0.20}^{+\ 0.28}$\
$\mathcal{B}_{WW}/\mathcal{B}_{ZZ}$ & $0.83$&${}_{-\ 0.16}^{+\ 0.20}$ & $0.85$&${}_{-\ 0.15}^{+\ 0.18}$ & $1.23$&${}_{-\ 0.22}^{+\ 0.27}$\
$\mathcal{B}_{\tau\tau}/\mathcal{B}_{ZZ}$ & $0.76$&${}_{-\ 0.21}^{+\ 0.26}$ & $0.86$&${}_{-\ 0.22}^{+\ 0.26}$ & $1.07$&${}_{-\ 0.30}^{+\ 0.37}$\
$\mathcal{B}_{bb}/\mathcal{B}_{ZZ}$ & $0.20$&${}_{-\ 0.12}^{+\ 0.21}$ & $0.93$&${}_{-\ 0.28}^{+\ 0.38}$ & $0.84$ &${}_{-\ 0.27}^{+\ 0.37}$\
$\mathcal{B}_{\mu\mu}/\mathcal{B}_{ZZ}$ & $-$ & ${}$ &$-$& & $0.63$&${}_{-\ 1.21}^{+\ 1.24} $\
To test our point of view, we perform a fit to these measurements by minimizing a $\chi^2$ function, which is defined as $$\begin{aligned}
\label{eq:xihiggs}
\chi^2_{\rm Higgs} = \sum_{i=1}^{28} \sum_{j=1}^{28}[E_i -T_i]C_{ij}^{-1}[E_j -T_j],\end{aligned}$$ where $E_i$ denotes experimentally measured $\sigma_{ggF}\cdot B_{ZZ} $, $\sigma_i / \sigma_{ggF}$ or $\mathcal{B}_i / \mathcal{B}_{ZZ}$ and $T_i$ is its theoretical expectation. $C$ is a $28\times28$ covariance matrix, which can be constructed by using the standard errors and corresponding correlations between these measurements obtained from the original articles published.
Assuming that BSM contributes to the loop processes only, we have $$\begin{aligned}
\label{eq:gamh}
\kappa_h^2 = \frac{\Gamma_h}{\Gamma_h^{SM}}.\end{aligned}$$ In this case, the coupling modifiers $\kappa_{\gamma}$ and $\kappa_g$ are free and other $\kappa_i$ are fixed to unity. The best fit to the measurements yields $$\begin{aligned}
\kappa_{\gamma} = 1.008 \pm 0.042, \ \kappa_g = 1.025 \pm 0.040,\end{aligned}$$ with the correlation between the two quantities $\rho=-0.34$. Two dimensional likelihood contours at 68% and 95% C.L. in ($\kappa_g , \ \kappa_{\gamma}$) plane are shown in Fig. \[fig:kg-kr\]. The fitting results are in good agreement within error with the SM predictions ( errors of $\kappa_{\gamma}$ and $\kappa_g$ are both reduced to about 4% ), which further support our argument of NP only modifying loop-induced Higgs processes.
The appropriate cumulative distribution functions are used to obtain the upper bounds for this and following analysis, namely, 68% (95%) best-fit region satisfies $\chi^2 -\chi_{min}^2 \leq 0.99 \ (3.84)$ for one parameter, and $\chi^2 -\chi_{min}^2 \leq 2.28 \ (5.99)$ for two parameters.
Scalar LQs {#sec:3}
==========
By using transformations under the SM gauge group $\mathcal{G}_{SM} = SU(3)_c\times SU(2)_L\times U(1)_Y$ as the classification criterion, there are six possible scalar LQ multiplets [@Dorsner:2016wpm]: $S_{3}(\overline{\mathbf{3}},\mathbf{3},1/3), R_{2}(\mathbf{3},\mathbf{2},7/6), \widetilde{R}_{2}(\mathbf{3},\mathbf{2},1/6),\\\widetilde{S}_{1}(\overline{\mathbf{3}},\mathbf{1},4/3),S_{1}(\overline{\mathbf{3}},\mathbf{1},1/3), \bar{S}_{1}(\overline{\mathbf{3}}, \mathbf{1},-2/3)$. The first number, the second one and the last one within each brackets indicates the QCD representation, the weak isospin representation and the weak hypercharge, respectively.
The colorless vacuum requires that these colored scalars cannot acquire their masses via spontaneous symmetry breaking [@Chang:2012ta]. Assuming weak components of single scalar LQ multiplet ($S$) to be degenerate at the electroweak scale, namely the mass of scalar LQ, $m_S$, is a free parameter, the Higgs portal interaction reads [@Dorsner:2016wpm] $$\begin{aligned}
\mathcal{L} \ni -\lambda_{S}(S_{ia}^{\dagger} S_{ia})(H_j^{\dagger}H_j)=-\lambda_{S} \upsilon (S_{ia}^{\dagger} S_{ia})h,
\label{eq:higgsportal}\end{aligned}$$ where $i, j$ are weak indices, $a$ represents color index, $\lambda_{S}$ is the coupling constant for the LQ-Higgs-LQ vertex, $\upsilon$ is vacuum expectation of the Higgs boson with $\upsilon=246.22$ GeV.
Contributions of $S$ to loop-induced Higgs processes arise from Eq. (\[eq:higgsportal\]), and are described by only two independent parameters, $\lambda_S$ and $m_S$. For convenience, a new parameter $\xi_{S}(\lambda_{S}, m_{S}^2) \equiv \lambda_{S}(\upsilon/m_{S})^2$ is introduced.
In the SM, $W$ boson and top quark loops dominate the partial decay width of $ h \to \gamma \gamma$ decay. The partial decay width in presence of single scalar LQ $S$ is given by $$\begin{aligned}
\label{eq:hrr}
\Gamma(h \to \gamma \gamma) = \frac{G_F \alpha_{em}^2 m_h^3}{128 \sqrt{2} \pi^3} \left| F_1(x_W) + \frac{4}{3} F_{1/2}(x_t) + \sum_{i} \frac{\xi_{S}}{2} d(r_{S})Q_{S_i}^2 F_0(x_{S}) \right|^2,\end{aligned}$$ where $G_{F}$ and $\alpha_{em}$ are the Fermi and fine-structure constants, respectively, $Q_{S_i}$ is electric charge of the weak component $S_i$ of single representation $S$, the sum of $i$ is taken over the weak components, $d(r_{S})$ represents the dimension of the color representation, and $x_i\equiv m_h^2/(4m_i^2)$ (i= W, t, $S$). The one-loop functions $F(x)$ read $$\begin{aligned}
F_1(x) &=& \left[ x(2x+3) + 3(2x-1)f(x) \right]x^{-2}, \nonumber \\
F_{1/2}(x) &=& -2\left[ x + (x-1)f(x) \right]x^{-2}, \\
F_0(x) &=& \left[ x - f(x) \right]x^{-2}, \nonumber\end{aligned}$$ with the function $$\begin{aligned}
f({x})&=&\left\{ \begin{array}{cc}
\arcsin^{2}\sqrt{{x}} & {x} \leqslant 1\\
-\frac{1}{4}\left(\log\frac{1+\sqrt{1-{x^{-1}}}}{1-\sqrt{1-{x^{-1}}}}-i\pi\right)^{2} & {x}<1
\end{array}\right..\end{aligned}$$ Then, one can obtain normalized modification of partial decay width of $h \to \gamma \gamma$ decay induced by single scalar LQ, which is expressed as [@Dorsner:2016wpm] $$\begin{aligned}
\label{kappa-hrr-S}
\frac{\Gamma_{h \to \gamma \gamma}}{\Gamma^{SM}_{h \to \gamma \gamma}} = |\kappa_{\gamma}|^2, \ {\rm where} \ \kappa_{\gamma} = 1-0.026 \xi_{S} d(r_S)\sum_{i}Q_{S_i}^2.\end{aligned}$$
In the SM, top quark loop dominates the $ggF$ Higgs production cross section. In presence of single scalar LQ $S$, the leading order parton cross section of $ g g \to h$ at the partonic center mass of energy $\sqrt{\hat{s}}$ can be expressed as $$\begin{aligned}
\hat{\sigma}_{LO}(g g \to h)= \frac{\sigma_0}{m_h^2}\delta(\hat{s}-m_h^2),\end{aligned}$$ where $\sigma_0$ is proportional to the partial decay width of $h \to g g$ decay, which is given by $$\begin{aligned}
\sigma_0 &=& \frac{8\pi^2}{m_h^3}\Gamma_{LO}(h \to g g) \nonumber \\
&=& \frac{G_F \alpha_s^2(\mu^2)}{512\sqrt{2}\pi} \left|F_{1/2}(x_t) + \sum_{i}^{N_{S_i}} \xi_{S} C(r_{S}) F_0(x_{S}) \right|^2,
\label{eq:ggh}\end{aligned}$$ where $\alpha_s^2(\mu^2)$ represents the strong coupling constant, $F_0$ term induced by single scalar LQ $S$. $C(r_S)$ is the index of color representation of $S$ ( $C(r_S) = 1/2$ for color triplet ) and $N_{S_i}$ is the number of weak components of $S$. Effects of higher order QCD are neglected, since the ratio, $\sigma/(\sigma)_{SM}$, is found to be less sensitive to that [@Gori:2013mia]. The normalized modification of $ggF$ Higgs production cross section induced by single scalar LQ is given by [@Dorsner:2016wpm] $$\begin{aligned}
\label{kappa-hgg-S}
\frac{\sigma_{gg \to h}}{\sigma^{SM}_{gg \to h}} = |\kappa_{g}|^2, \ {\rm where} \ \kappa_{g} = 1 + 0.24 \xi_{S} N_{S_i} C(r_S).\end{aligned}$$
Thus for the case of single scalar LQ representation $S$ modifies the loop-induced Higgs processes, there is only one free parameter $\xi_S$ left.
To obtain $\xi_S$, we re-perform the Higgs fit by using $\xi_S$ to replace $\kappa_{\gamma}$ and $\kappa_g$ via Eqs. (\[kappa-hrr-S\]) and (\[kappa-hgg-S\]). Best values of $\xi_S$ with standard errors and 95% C.L. intervals for all six scalar LQ representations are shown in Table \[tab:fitreslut-slq\]. Errors of $\xi_S$ for all scalar LQs obtained in this analysis are reduced more than half compared with previous analysis in Ref. [@Dorsner:2016wpm]. But constraints on $\xi_S$ for all scalar LQs are still too loose to acquire exact information for scalar LQs with TeV-scale masses. Table \[SLQmasslimit\] shows best values of scalar LQs masses and their lower limits at 95% C.L. in the assumption of the portal coupling $\lambda_S = 1.0$. If LQs are insensitive to generation as well as their decay modes, the most stringent limits on the mass of scalar LQs reads $m_S > 1560$ GeV reported by the ATLAS collaboration [@Aaboud:2019jcc]. Assuming $m_{S} = 1560$ GeV, best values of portal couplings $\lambda_S$ and their upper limits at 95% C.L. obtained from Higgs fit are shown in Table \[SLQcouplinglimit\].
The results are expected to be significantly improved at High Luminosity (HL)-LHC. Ref. [@Cepeda:2019klc] reported the projections for Higgs couplings determinations at HL-LHC with an integrated luminosity of 3000 fb$^{-1}$. The precision on $\kappa_{\gamma}$ and $\kappa_g$ is expected to be 2.4% and 3.1% at the ATLAS experiment while that is 2.0% and 2.5% at the CMS experiment. Thus the precision on $\kappa_{\gamma}$ and $\kappa_g$ is expected to be 1.5% and 1.9% at HL-LHC by combining the ATLAS and CMS measurements of $\kappa_{\gamma}$ and $\kappa_g$. Then we can obtain the precision on $\xi_S$ expected at the HL-LHC via Eqs. (\[kappa-hrr-S\]) and (\[kappa-hgg-S\]). The approximate relation between errors of $\xi_S$ and $\kappa_{\gamma}$ and $\kappa_g$ read $$\begin{aligned}
\delta_{\xi_S} \approx \left[ 0.24 N_{S_i} C(r_S) - 0.026 d(r_S)\sum_{i}Q_{S_i}^2 \right] \sqrt{\delta^2_{\kappa_g}+\delta^2_{\kappa_{\gamma}}}.\end{aligned}$$ Compared to the present precision on $\xi_S$, the situation is expected to improve by a factor of 2.4 at the HL-LHC.
------------------------------------------------------- ------------------- -------------------
best fit 95% C.L.
$S_3(\bar{\mathbf{3}}, \mathbf{3}, 1/3)$ 0.060 $\pm$ 0.108 \[-0.173, 0.294\]
$R_2(\mathbf{3}, \mathbf{2}, 7/6)$ 0.032 $\pm$ 0.134 \[-0.241, 0.326\]
$\tilde{R}_{2}(\mathbf{3}, \mathbf{2},1/6)$ 0.115 $\pm$ 0.162 \[-0.237, 0.456\]
$\tilde{S}_{1}(\bar{\mathbf{3}}, \mathbf{1},4/3)$ 0.048 $\pm$ 0.245 \[-0.465, 0.604\]
$S_{1}(\bar{\mathbf{3}},\mathbf{1},1/3)$ 0.234 $\pm$ 0.316 \[-0.452, 0.895\]
$\bar{S}_{1}(\overline{\mathbf{3}}, \mathbf{1},-2/3)$ 0.220 $\pm$ 0.329 \[-0.494, 0.917\]
------------------------------------------------------- ------------------- -------------------
: Constraints on LQs from the LHC Run I and II Higgs data for all scalar LQ representations, where $\xi_{S} = \lambda_S \upsilon^2 / m^2_S$.
\[tab:fitreslut-slq\]
------------------------------------------------------- ---------- --------------
best fit 95% C.L.
$S_3(\bar{\mathbf{3}}, \mathbf{3}, 1/3)$ 1005 GeV $>$ 454 GeV
$R_2(\mathbf{3}, \mathbf{2}, 7/6)$ 1376 GeV $>$ 431 GeV
$\tilde{R}_{2}(\mathbf{3}, \mathbf{2},1/6)$ 726 GeV $>$ 364 GeV
$\tilde{S}_{1}(\bar{\mathbf{3}}, \mathbf{1},4/3)$ 1124 GeV $>$ 317 GeV
$S_{1}(\bar{\mathbf{3}},\mathbf{1},1/3)$ 509 GeV $>$ 260 GeV
$\bar{S}_{1}(\overline{\mathbf{3}}, \mathbf{1},-2/3)$ 525 GeV $>$ 257 GeV
------------------------------------------------------- ---------- --------------
: For the LQ-Higgs coupling $\lambda_S = 1.0$, best values and lower limits at 95% C.L. of scalar LQs masses obtained from Higgs fit.
\[SLQmasslimit\]
------------------------------------------------------- ------------ ----------
best fit 95% C.L.
$S_3(\bar{\mathbf{3}}, \mathbf{3}, 1/3)$ 2.4 $<$ 11.8
$R_2(\mathbf{3}, \mathbf{2}, 7/6)$ 1.3 $<$ 13.1
$\tilde{R}_{2}(\mathbf{3}, \mathbf{2},1/6)$ 4.6 $<$ 18.3
$\tilde{S}_{1}(\bar{\mathbf{3}}, \mathbf{1},4/3)$ 1.9 $<$ 24.2
$S_{1}(\bar{\mathbf{3}},\mathbf{1},1/3)$ 9.4 $<$ 35.9
$\bar{S}_{1}(\overline{\mathbf{3}}, \mathbf{1},-2/3)$ 8.8 $<$ 36.8
------------------------------------------------------- ------------ ----------
: For $m_{S} = 1000$ GeV, best values and upper limits at 95% C.L. of the size of LQ-Higgs coupling $|\lambda_S|$ for scalar LQs obtained from Higgs fit.
\[SLQcouplinglimit\]
Vector LQ $U_1(\mathbf{3},\mathbf{1},2/3)$ in the ’4321’ model {#sec:4}
==============================================================
Now we consider contributions of vector LQ $U_1(\mathbf{3},\mathbf{1},2/3)$ to the loop-induced Higgs processes $gg \to h$ and $h \to \gamma \gamma$, which LQ performs quite well in explaining both anomalies of $R_{D^{(*)}}$ and $R_{K^{(*)}}$. Our study in framework of the ’4321’ model [@DiLuzio:2017vat; @DiLuzio:2018zxy]. We first briefly review the ’4321’ model, then we study contributions of $U_1$ to the loop-induced Higgs processes. Constraints on the interactions of $U_1$ with the Higgs boson from LHC Higgs data is obtained. Further more, we obtain constraints on the VEVs $\upsilon_3$ and $\upsilon_1$ of new scalar fields $\Omega_3$ and $\Omega_1$ in the model.
The ’4321’ model
----------------
The model gauge group is expressed as $\mathcal{G}_{4321} = SU(4)\times SU(3)'\times SU(2)_L\times U(1)'$, for which $H^{\alpha}_{\mu}, G'^a_{\mu}, W^i_{\mu}, \\ B'_{\mu}$ denote corresponding gauge fields, $g_4, g_3, g_2, g_1$ the gauge couplings and $T^{\alpha}, T^{a}, T^i, Y'$ the generators, where the indices $\alpha = 1,...,15,\ a= 1,...,8,\ i=1,...,3$. The generators are normalized in such a way that $\text{Tr} T^A T^B = \frac{1}{2}\delta^{AB}$. The SM gauge symmetry $SU(3)_c\times U(1)_Y$ is embedded in $SU(4)\times SU(3)'\times U(1)'$.
The model comprises four scalar representations: $\Omega_3(\bar{\mathbf{4}},\mathbf{3},\mathbf{1},1/6)$, $\Omega_1(\bar{\mathbf{4}},\mathbf{1},\mathbf{1},-1/2)$, $\Omega_{15}(\overline{\mathbf{15}},\mathbf{1},\mathbf{1},0)$ and $\Phi(\mathbf{1},\mathbf{1},\mathbf{2},1/2)$, where $\Omega_3$ and $\Omega_1$ are respectively a $4\times3$ matrix and a $4$-vector transforming as $\Omega_3 \to U^*_4 \Omega_3 U^T_{3'}$ and $\Omega_1 \to U^*_4 \Omega_1$ under $SU(4)\times SU(3)'$ and $H$ is the Higgs doublet ( in this analysis we neglect the effect of $\Omega_{15}$ ). Phenomenological considerations suggest : $ \langle \Omega_3 \rangle > \langle \Omega_1 \rangle > \langle \Phi \rangle $. According to Ref. [@DiLuzio:2018zxy], the most general scalar potential involving $\Omega_{3,1}$ and $H$ can be written as $$\begin{aligned}
\label{scalar-potential}
V = &+& \mu_3^2{\rm Tr}(\Omega_{3}^{\dagger}\Omega_{3}) +
\lambda_1 \left({\rm Tr}(\Omega_{3}^{\dagger}\Omega_{3}) - \frac{3}{2}\upsilon_3^2 \right)^2 + \lambda_2
{\rm Tr} \left(\Omega_{3}^{\dagger}\Omega_{3} - \frac{1}{2}\upsilon_3^2 \right)^2 \nonumber \\&+&\mu_1^2|\Omega_{1}|^2 + \lambda_3 \left( |\Omega_{1}|^2 - \frac{1}{2}\upsilon_1^2 \right)^2 + \lambda_4 \left({\rm Tr}(\Omega_{3}^{\dagger}\Omega_{3}) - \frac{3}{2}\upsilon_3^2 \right) \left( |\Omega_{1}|^2 - \frac{1}{2}\upsilon_1^2 \right) \nonumber \\
&+& \lambda_5 \Omega_{1}^{\dagger}\Omega_{3}\Omega_{3}^{\dagger}\Omega_{1} + \lambda_6 \left( [\Omega_{3} \Omega_{3} \Omega_{3} \Omega_{1}]_1 + {\rm h.c.} \right) + \mu^2_{\Phi}\Phi^{\dagger}\Phi + \lambda_7 \left(\Phi^{\dagger}\Phi - \frac{\upsilon^2}{2} \right)^2 \nonumber \\
&+& \lambda_8 \left({\rm Tr}(\Omega_{3}^{\dagger}\Omega_{3}) - \frac{3}{2}\upsilon_3^2 \right) \left(\Phi^{\dagger}\Phi - \frac{\upsilon^2}{2} \right) + \lambda_9 \left( |\Omega_{1}|^2 - \frac{1}{2}\upsilon_1^2 \right) \left(\Phi^{\dagger}\Phi - \frac{\upsilon^2}{2} \right) .\end{aligned}$$ where $\left[ \Omega_3 \Omega_3 \Omega_3 \Omega_1 \right]_1 \equiv \epsilon_{\alpha \beta \gamma \delta} \epsilon^{a b c} \left( \Omega_3 \right)^{\alpha}_a \left( \Omega_3 \right)^{\beta}_b \left( \Omega_3 \right)^{\gamma}_c \left( \Omega_1 \right)^{\delta}$. VEV configurations [@DiLuzio:2018zxy] $$\begin{aligned}
\langle \Omega_3 \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix}
\upsilon_3 & 0 & 0 \\
0 & \upsilon_3 & 0 \\
0 & 0 & \upsilon_3 \\
0 & 0 & 0
\end{pmatrix} \ , \ \langle \Omega_1 \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix}
0\\
0\\
0\\
\upsilon_1
\end{pmatrix},\end{aligned}$$ together with $\mu^2_3 = -3\lambda_6 \upsilon_3 \upsilon_1$, $\mu^2_1 = - 3\lambda_6 \upsilon^2_3 / \upsilon_1$ and $\mu^2_h = 0$ in Eq. (\[scalar-potential\]) ensure the proper $\mathcal{G}_{4321} \to \mathcal{G}_{SM}$ breaking. Under $\mathcal{G}_{SM}$, $\Omega_3$ and $\Omega_1$ decomposed as: $\Omega_3 \to \mathbb{S}_3(\mathbf{1},\mathbf{1},0)\oplus \mathbb{T}_3(\mathbf{3},\mathbf{1},2/3)\oplus \mathbb{O}_3(\mathbf{8},\mathbf{1},0)$ and $\Omega_1 \to \mathbb{S}_1(\mathbf{1},\mathbf{1},0)\oplus \mathbb{T}_1^*(\mathbf{3},\mathbf{1},2/3)$. The final breaking of $\mathcal{G}_{SM}$ proceeds via the Higgs doublet field acquiring a VEV $\langle \Phi \rangle = (0 \ \upsilon)^T/\sqrt{2}$, with $\upsilon = 246.22$ GeV.
The covariant derivatives of $\Omega_3$, $\Omega_1$ and $\Phi$ are given by $$\begin{aligned}
D_{\mu} \Omega_3 &=& \partial_{\mu} \Omega_3 + {\rm i}g_4 H^{\alpha}_{\mu} T^{*\alpha} \Omega_3 - {\rm i}g_3 G^{'a}_{\mu} T^{a} \Omega_3 - \frac{1}{6}{\rm i} g_1 B'_{\mu} \Omega_3, \nonumber \\
D_{\mu} \Omega_1 &=& \partial_{\mu} \Omega_1 + {\rm i}g_4 H^{\alpha}_{\mu} T^{*\alpha} \Omega_1 + \frac{1}{2} {\rm i} g_1 B'_{\mu} \Omega_1, \nonumber \\
D_{\mu} \Phi_{~} &=& \partial_{\mu} \Phi_{~} - {\rm i}g_2 W^{i}_{\mu} T^{i} \Phi - \frac{1}{2} {\rm i} g_1 B'_{\mu} \Phi \label{eq:CDS}\end{aligned}$$
In the model, the mass of $U_1$ and corresponding mass eigenstate expressed in terms of the original gauge fields are given by [@DiLuzio:2018zxy] $$\begin{aligned}
m_U = \dfrac{1}{2} g_4 \sqrt{\upsilon_3^2 + \upsilon_1^2}, \label{eq:mU}\end{aligned}$$ and $$\begin{aligned}
U_{1\mu}^{1,2,3} = \frac{1}{2} \left( H_{\mu}^{9,11,13} - {\rm i}H_{\mu}^{10,12,14} \right).\end{aligned}$$ Then we obtain Feynman rules of $U_1$ interactions to scalars $$\begin{aligned}
\begin{pmatrix}U^{~}_{1 \mu} \\ U^{*}_{1 \nu}\\ \mathbb{S}^{(*)}_{3}\end{pmatrix} &:& ~\dfrac{\rm i}{2} g^2_4 \frac{\upsilon_{3}}{4\sqrt{3}} g_{\mu \nu} ,\qquad \begin{pmatrix}U^{~}_{1 \mu} \\ U^{*}_{1 \nu}\\ \mathbb{S}^{(*)}_{1}\end{pmatrix} : ~\dfrac{\rm i}{2} g^2_4 \frac{\upsilon_{1}}{2\sqrt{2}} g_{\mu \nu} , \qquad \qquad \quad \label{V-uus}\end{aligned}$$ From Eq. (\[eq:CDS\]) we can see that $U_1$ can not couple to the Higgs doublet $\Phi$ directly. $U_1$ interacts with the Higgs boson $h$ via the mixing of $\phi^{(*)}$ and representations $\mathbb{S}^{(*)}_{3,1}(\mathbf{1},\mathbf{1},0)$ after the final SM breaking, where $\phi$ represents the neutral component of the Higgs doublet and $\mathbb{S}_{3,1}(\mathbf{1},\mathbf{1},0)$ are decompositions of $\Omega_{3,1}$ under the SM symmetry. In the basis ($\mathbb{S}_3, \mathbb{S}^*_3, \mathbb{S}_1, \mathbb{S}^*_1, \phi,\phi^*$), singlet spectrum are expressed as
$$\begin{gathered}
\mathcal{M}^2_{S} = \\
\left(
\begin{array}{cccccc}
\mathcal{M}^2_1 &
\mathcal{M}^2_2 &
\mathcal{M}^2_3 &
\frac{1}{2} \sqrt{\frac{3}{2}} \lambda_4 v_1 v_3 & \frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 & \frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 \\
\mathcal{M}^2_2 &
\mathcal{M}^2_1 &
\frac{1}{2} \sqrt{\frac{3}{2}} \lambda_4 v_1 v_3 &
\mathcal{M}^2_3 & \frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 & \frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 \\
\mathcal{M}^2_3 &
\frac{1}{2} \sqrt{\frac{3}{2}} \lambda_4 v_1 v_3 &
\lambda_3 v_1^2 &
\lambda_3 v_1^2-3\lambda_6\frac{v_3^3}{v_1} & \frac{1}{2} \lambda_9 v v_1 & \frac{1}{2} \lambda_9 v v_1 \\
\frac{1}{2} \sqrt{\frac{3}{2}} \lambda_4 v_1 v_3 &
\mathcal{M}^2_3 &
\lambda_3 v_1^2-3\lambda_6\frac{v_3^3}{v_1} &
\lambda_3 v_1^2 & \frac{1}{2} \lambda_9 v v_1 & \frac{1}{2} \lambda_9 v v_1 \\
\frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 & \frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 & \frac{1}{2} \lambda_9 v v_1 & \frac{1}{2} \lambda_9 v v_1 & \lambda_7 \upsilon^2 & \lambda_7 \upsilon^2 \\
\frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 & \frac{1}{2} \sqrt{\frac{3}{2}} \lambda_8 v v_3 & \frac{1}{2} \lambda_9 v v_1 & \frac{1}{2} \lambda_9 v v_1 & \lambda_7 \upsilon^2 & \lambda_7 \upsilon^2
\end{array}
\right),
\end{gathered}$$
where $$\begin{aligned}
\mathcal{M}^2_1 = \frac{1}{2} \left( 3 \lambda_1 + \lambda_2 \right) v_3^2+3 \lambda_6 v_1 v_3, \ \mathcal{M}^2_2 = \frac{1}{2} \left( 3 \lambda_1 + \lambda_2 \right) v_3^2-\frac{3}{2} \lambda_6 v_1 v_3, \ \mathcal{M}^2_3 = \sqrt{\frac{3}{2}} \left( 3 \lambda_6 v_3^2 + \frac{1}{2} \lambda_4 v_1 v_3 \right). \nonumber\end{aligned}$$ It turns out $\mathcal{M}^2_{S} = 4$. Two massless modes correspond to eigenvectors $$\begin{aligned}
S^Z_{GB} = \dfrac{1}{\sqrt{2}} \left( 0, 0, 0, 0, -1, 1 \right),\end{aligned}$$ and $$\begin{aligned}
S^{Z'}_{GB} = \dfrac{1}{\sqrt{\frac{2}{3}\upsilon^2_3 + \upsilon^2_1}} \left( \dfrac{\upsilon_3}{\sqrt{3}}, \ -\dfrac{\upsilon_3}{\sqrt{3}}, \ -\dfrac{\upsilon_1}{\sqrt{2}}, \ \dfrac{\upsilon_1}{\sqrt{2}}, \ 0, \ 0 \right),\end{aligned}$$ which are associated to the longitudinal degrees of freedom of the $Z$ and $Z'$, respectively. One of four non-zero eigenvalues as well as corresponding eigenvector can be also easily obtained as $$\begin{aligned}
\label{eq:ms0}
\mathcal{M}^2_{S_0} = \dfrac{3\lambda_6 \upsilon_3 \left( 2\upsilon^2_3 + 3\upsilon^2_1 \right)}{2 \upsilon_1},\end{aligned}$$ and $$\begin{aligned}
S_0 =\dfrac{1}{\sqrt{\frac{2}{3}\upsilon^2_3 + \upsilon^2_1}} \left( -\dfrac{\upsilon_1}{\sqrt{2}}, \ \dfrac{\upsilon_1}{\sqrt{2}}, \ -\dfrac{\upsilon_3}{\sqrt{3}}, \ \dfrac{\upsilon_3}{\sqrt{3}}, \ 0, \ 0 \right).\end{aligned}$$
Precisely acquiring mass eigenvalue of the would-be Higgs boson and corresponding eigenvector is difficult, unless conditions such as precise value of the Higgs mass obtained from experiment as well as other constraints are applied. We assume the Higgs boson with mass value of $125.09$ GeV corresponds to normalized eigenvector $$\begin{aligned}
\label{eq:heigenvector}
h = \left( \lambda_{\mathbb{S}_3}, \ \lambda_{\mathbb{S}^*_3}, \ \lambda_{\mathbb{S}_1}, \ \lambda_{\mathbb{S}^*_1}, \ \lambda_{\phi}, \ \lambda_{\phi^*} \right),\end{aligned}$$ where $\lambda_{\mathbb{S}^{(*)}_{3}}$, $\lambda_{\mathbb{S}^{(*)}_{1}}$ and $\lambda_{\phi^{(*)}}$ represent mixing constants with $\lambda_{\mathbb{S}_{3}} = \lambda_{\mathbb{S}^{*}_{3}}$, $\lambda_{\mathbb{S}_{1}} = \lambda_{\mathbb{S}^{*}_{1}}$ and $\lambda_{\phi} = \lambda_{\phi^{*}}$, since $h$ is a real field. According to Eq. (\[V-uus\]), Feynman rule of $U_{1 \mu} U^{*}_{1 \nu}h$ should be expressed as $$\begin{aligned}
\label{eq:uuh1}
U_{1 \mu} U^{*}_{1 \nu}h \ : \quad \frac{\rm i}{2} g^2_4 \left( \frac{\upsilon_{3}}{2\sqrt{3}} \lambda_{\mathbb{S}_3} + \frac{\upsilon_{1}}{\sqrt{2}} \lambda_{\mathbb{S}_1} \right) g_{\mu \nu}.\end{aligned}$$ We do not intend to further solve these mixing parameters $\lambda_{\mathbb{S}^{(*)}_{3,1}}$. For convenience, we re-express the Feynman rule as $$\begin{aligned}
U_{1 \mu} U^{*}_{1 \nu}h \ : \quad \frac{\rm i}{2} g^2_4 \upsilon g_{\mu \nu} \dfrac{\upsilon_3}{\upsilon} \lambda_V, \label{V-uuh}\end{aligned}$$ where the $U_1$-Higgs coupling $\lambda_V = \frac{\lambda_{\mathbb{S}_3}}{2\sqrt{3}} + \frac{\upsilon_1}{\upsilon_3}\frac{\lambda_{\mathbb{S}_1}}{\sqrt{2}}$, which is expected to be small according to current Higgs measurements analyses in Section \[sec:2\].
We now consider interactions among gauge bosons. The interactions are obtained from the gauge kinetic term [@DiLuzio:2018zxy] $$\begin{aligned}
\mathcal{L}_{gauge} = -\dfrac{1}{4} H^{\alpha}_{\mu \nu} H^{\alpha,\mu \nu} -\dfrac{1}{4} G'^{a}_{\mu \nu} G'^{a,\mu \nu} -\dfrac{1}{4} W^{i}_{\mu \nu} W^{i,\mu \nu} -\dfrac{1}{4} B'_{\mu \nu} B'^{\mu \nu},\end{aligned}$$ where definitions of field strengths $H^{\alpha}_{\mu \nu}$, $G'^{a}_{\mu \nu}$, $W^{i}_{\mu \nu}$ and $B'_{\mu \nu}$ see [@DiLuzio:2018zxy].
Prior to electroweak symmetry breaking, the massless $SU(3)_c\times U(1)_Y$ degrees of freedom of $\mathcal{G}_{SM}$ expressed in terms of the original gauge fields are given by [@DiLuzio:2018zxy] $$\begin{aligned}
&g_{\mu}^a = \dfrac{g_3 H_{\mu}^a + g_4 G'^a_{\mu}}{\sqrt{g_4^2 + g_3^2}} , \\ &B_{\mu} = \dfrac{\sqrt{\frac{2}{3}}g_1 H_{\mu}^{15} + g_4 B'_{\mu}}{\sqrt{g_4^2 + \frac{2}{3}g_1^2}}.\end{aligned}$$ The SM gauge couplings are matched as [@DiLuzio:2018zxy] $$\begin{aligned}
&g_s = \dfrac{g_4 g_3}{\sqrt{g_4^2 + g_3^2}} , \label{eq:gsg4g3} \\
&g_Y = \dfrac{g_4 g_1}{\sqrt{g_4^2 + \frac{2}{3}g_1^2}}. \label{eq:gsg4g1}\end{aligned}$$
Then, one can obtain Feynman rules related to $U_1$ interactions to the SM gauge boson $\gamma$ and $g$, $$\begin{aligned}
\begin{pmatrix}
U^{~}_{1 \mu}(k_1) \\ U^{*}_{1 \nu}(k_2) \\ A_{\rho}(k_3)
\end{pmatrix} &:& -{\rm i} \dfrac{\frac{2}{3}g_4g_1 {\rm cos}(\theta_W)}{\sqrt{g_4^2 + \frac{2}{3}g_1^2}} V_{\mu \nu \rho} \left(k_1, k_2, k_3\right) = - {\rm i} e Q_{U}V_{\mu \nu \rho} \left(k_1, k_2, k_3\right), \label{V-uua}\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}U^{~}_{1 \mu}(k_1) \\ U^{*}_{1 \nu}(k_2) \\ A_{\rho}(k_3) \\ A_{\sigma}(k_4) \end{pmatrix} &:& ~
{\rm i} (e Q_{U})^2 \left( g_{\mu \rho}g_{\nu \sigma} + g_{\mu \sigma} g_{\nu \rho} - 2g_{\mu \nu}g_{\rho \sigma}\right), \qquad \qquad \qquad \qquad \label{V-uuaa}\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}U^i_{1 \mu}(k_1) \\ U^{*j}_{1 \nu}(k_2) \\ g^a_{\rho}(k_3) \end{pmatrix} &:&
- {\rm i} \dfrac{g_4g_3 }{\sqrt{g_4^2 + g_3^2}}T^a_{ij} V_{\mu \nu \rho} \left(k_1, k_2, k_3\right) = - {\rm i} g_s T^a_{ij} V_{\mu \nu \rho} \left(k_1, k_2, k_3\right), \label{V-uug}\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}U^i_{1 \mu}(k_1) \\ U^{*j}_{1 \nu}(k_2) \\ g^a_{\rho}(k_3) \\ g^b_{\sigma}(k_4)\end{pmatrix} &:& ~{\rm i} g^2_s \delta_{ij} \dfrac{\delta_{ab}}{4} \left(g_{\mu \rho}g_{\nu \sigma} + g_{\mu \sigma} g_{\nu \rho} - 2g_{\mu \nu}g_{\rho \sigma} \right), \qquad \qquad \qquad \qquad \label{V-uugg}\end{aligned}$$ where $\theta_W$ is the Weinberg angle, the function $V_{\mu \nu \rho} \left(k_1, k_2, k_3\right)$ is defined as $$\begin{aligned}
V_{\mu \nu \rho} \left(k_1, k_2, k_3\right)
= g_{\mu \nu} (k_2 - k_1)_{\rho} + g_{\nu \rho}(k_3 - k_2)_{\mu} - g_{\rho \mu}(k_1 - k_3)_{\nu}, \nonumber\end{aligned}$$ with $k_i$ being four-momentum of the $i$-th particle ( direction towards the vertex is specified to be positive ).
Constraints on $U_1$ from Higgs data
------------------------------------
By using Eqs. (27,33-36), we obtain the partial decay width of $ h \to \gamma \gamma$ and cross section of $ g g \to h$ in presence of $U_1$ $$\begin{aligned}
\label{eq:hrr-U}
\Gamma(h \to \gamma \gamma) = \frac{G_F \alpha_{em}^2 m_h^3}{128 \sqrt{2} \pi^3} \left| F_1(x_W) + \frac{4}{3} F_{1/2}(x_t) + \xi_V d(r_{U})Q_{U}^2 F_1(x_{U}) \right|^2 ,\end{aligned}$$ and $$\begin{aligned}
\sigma_0 = \frac{G_F \alpha_s^2(\mu^2)}{512\sqrt{2}\pi} \left|F_{1/2}(x_t) + \xi_V F_1(x_{U}) \right|^2,
\label{eq:ggh-U}\end{aligned}$$ where $$\begin{aligned}
\xi_V = \dfrac{g^2_4 \lambda_V \upsilon \upsilon_3}{4 m^2_U} = \dfrac{\lambda_V \upsilon \upsilon_3}{\upsilon^2_3 + \upsilon^2_1} .
\label{eq:Xi_V}\end{aligned}$$ In obtaining Eq. (\[eq:Xi\_V\]), we have used mass expression Eq. (\[eq:mU\]). Eq. (\[eq:Xi\_V\]) shows that the ’4321’ model’s $U_1$ modifications to the loop-induced Higgs processes depend on $U_1$-Higgs coupling $\lambda_V$ and new VEVs $\upsilon_3$ and $\upsilon_1$ in the model rather than the mass of $U_1$ and gauge coupling $g_4$. This means that once $\xi_V$ is determined from the Higgs fit one can determine $\upsilon_3$ and $\upsilon_1$ by using $\xi_V$ together with other condition such as the mass of $U_1$ determined from colliders.
For single vector LQ $U_1$ modifying partial decay width of $ h \to \gamma \gamma$ and cross section of $ g g \to h$, coupling modifiers $\kappa_{\gamma}$ and $\kappa_g$ are expressed with $\xi_V$, which read $$\begin{aligned}
\label{kappa-rr-gg-v}
\kappa_{\gamma} = 1+1.44 \xi_V \ {\rm and} \ \kappa_g = 1 - 5.09 \xi_V.\end{aligned}$$
To obtain the size of $U_1$ interaction with the Higgs boson, we re-analyze the Higgs data by using Eq. (\[kappa-rr-gg-v\]). The best value with standard error and 95% C.L. intervals of $\xi_V$ obtained from the Higgs fit are $$\begin{aligned}
\label{limit-V}
\xi_V = -0.005 \pm 0.008 ,\end{aligned}$$ and $$\begin{aligned}
\label{limit-V95}
\xi_V \in \left[ -0.021, \ 0.011 \right].\end{aligned}$$
For $U_1$-Higgs coupling with value of one-third (-tenth) of the electromagnetic coupling strength, $|\lambda_V| = 0.1 (0.03)$, $\xi_V$ varying as a function of $\upsilon_3$ for a fixed value of $\upsilon_1$ and combined limits on $\upsilon_3$ and $\upsilon_1$ from the condition, $\upsilon_3 > \upsilon_1 > \upsilon$, as well as current Higgs data, are shown in Fig. \[fig:v3-XiV\]. From Fig. \[fig:v3-XiV\] one can see that we still need more precise Higgs measurements, since at least the sign of $\lambda_V$ has not been determined yet from current Higgs data. It should be noted that the result of precision on $\xi_S$ is also applicable to $\xi_V$, which means the precision on $\xi_V$ is expected to improve by a factor of 2.4 compared with present situation at HL-LHC.
Combined limits on the relation of $\lambda_V$ and $\upsilon_{3,1}$
-------------------------------------------------------------------
We can further constrain the relation of $\lambda_V$ and $\upsilon_{3,1}$ by combined limits on $\xi_V$ obtained in this analysis and $m_U$ obtained from direct searches at colliders as well as measurements of $R_{D^{(*)}}$ and $R_{K^{(*)}}$. Details of obtaining the constraints on $U_1$ from $B$-anomalies are shown in appendix \[sec:A\]
Current lower limits on masses of vector LQs with decay mode $LQ \to t \nu / b \tau$ is $m_{LQ} > 1530$ GeV reported by the CMS collaboration [@Sirunyan:2018kzh]. For $g_4 = 3.5$, $$\begin{aligned}
\upsilon^2_3 +\upsilon^2_1 = \frac{4}{g^2_4} m^2_U > (874 \ {\rm GeV})^2,\end{aligned}$$ which is looser than the constraints from $B$-anomalies. Thus we consider combined constraints from the LHC Higgs data and $B$-anomalies measurements, which is performed via minimizing $$\begin{aligned}
\chi^2 = \chi^2_{\rm Higgs} + \chi^2_B,\end{aligned}$$ where $\chi^2_{\rm Higgs}$ has been shown in Eq. (\[eq:xihiggs\]) and $\chi^2_B$ is explained in Eq. (\[eq:xiB\]). Assuming tree level contributions induced by $U_1$ dominant the NP contributions to $B$-anomalies. Fig. [\[fig:vvlv\]]{} shows two dimensional likelihood contours at 68% and 95% C.L. in ($\upsilon^2_3 + \upsilon^2_1 , \ \lambda_V \upsilon_3$) plane obtained from combination of the LHC Higgs data together with measurements of $B$-anomalies. Best values of $\upsilon^2_3 + \upsilon^2_1$ and $\lambda_V \upsilon_3$ read $$\begin{aligned}
\upsilon^2_3 + \upsilon^2_1 = 1.496 \pm 0.250 \ {\rm TeV^2} \ {\rm and} \ \lambda_V \upsilon_3 = -0.0315 \pm 0.0473 \ {\rm TeV}.\end{aligned}$$
Assuming $\lambda_{V} = -0.1 (-0.03)$, we show the constraints on $\upsilon_{3,1}$ in Fig. \[fig:vvv\], which are obtained from combined limits of LHC Higgs data and $B$-anomalies as well as the condition $\upsilon_3 > \upsilon_1 > \upsilon$. The best value of $\upsilon_3$ obtained under the assumption of $\lambda_{V} = -0.1$ does not in the allowed region as shown in Fig. \[fig:vvv\] (a), while that does for $\lambda_{V} = -0.03$ (see Fig. \[fig:vvv\] (b)).
If the Higgs coupling and $R_{D^{(*)}}$ and $R_{K^{(*)}}$ precisely measured in the future, we can determine VEVs $\upsilon_3$ and $\upsilon_1$. For $\lambda_{V} = -0.03$, at the best fit value point, we obtain $$\begin{aligned}
\upsilon_3 = 1.051 \ {\rm TeV} \ {\rm and} \ \upsilon_1 = 0.625 \ {\rm TeV}.\end{aligned}$$ Equivalently, we obtain the mass of $U_1$ $$\begin{aligned}
m_U = \dfrac{1}{2} g_4 \sqrt{\upsilon_3^2 + \upsilon_1^2} = 2.140 {\rm TeV}\end{aligned}$$
Then, one can determine or constrain other parameters in the model directly by using $\upsilon_3 = 1.051 \ {\rm GeV} \ {\rm and} \ \upsilon_1 = 0.625 \ {\rm GeV}$, or together with other constraints. For example, we can directly determine the masses of the other two new gauge particles $g'$ and $Z'$ in the model [@DiLuzio:2018zxy]. Assuming $g_4 = 3.5$ and $g_3 = 1.07$ as well as $g_1 = 0.364$, we obtain $$\begin{aligned}
m_{g'} &=& \sqrt{\dfrac{1}{2} (g^2_4 + g^2_3)\upsilon^2_3} = 2.72 \ \text{TeV} , \\
m_{Z'} &=& \sqrt{\dfrac{1}{4} (\dfrac{3}{2}g^2_4 + g^2_1)(\dfrac{1}{3}\upsilon^2_3 + v^2_1)} = 1.88 \ \text{TeV} .\end{aligned}$$
Alternatively, once two of the three massive particles $U_1$, $g'$ and $Z'$ are found at the LHC or future colliders, one can use these masses together with $\xi_V$ to determine the $U_1$-Higgs coupling $\lambda_{V}$.
Conclusions {#sec:5}
===========
$B$-anomalies may be a long-awaited new physical signal, and is discussed extensively as a hot topic. The good performance in explaining $B$-anomalies indicates that LQ may be discovered in the near future.
LQs with mass value of TeV-scale can considerably modify loop-induced Higgs processes, $ggF$ production and $h \to \gamma \gamma$ decay, which depending on the coupling size of LQ interactions with the Higgs boson. We study contributions of single scalar or vector LQ to loop-induced Higgs processes by analyzing current LHC Higgs data. Scalar LQs are studied model-independently while vector LQ, $U_1(\mathbf{3},\mathbf{1}, 2/3)$, is discussed in so called the ’4321’model.
Constraints on sizes of portal interactions, $\lambda_S = \lambda_S (\upsilon / m_{S})^2$, of all possible scalar LQs are obtained. Currently, the constraints for all scalar LQs are still too loose to acquire exact information for scalar LQ with mass of TeV scale, although accuracy of the result in this analysis is more than doubled compared with previous one by analyzing Higgs data from LHC Run I [@Dorsner:2016wpm].
For vector LQ, $U_1(\mathbf{3},\mathbf{1}, 2/3)$, the size of interaction between $U_1$ and Higgs boson is parameterized as $$\begin{aligned}
\xi_V = \dfrac{\lambda_V \upsilon \upsilon_3}{\upsilon^2_3 + \upsilon^2_1}, \nonumber\end{aligned}$$ where $\lambda_V$ is the $U_1$-Higgs coupling constant. The best value with standard error and 95% C.L. intervals for $\xi_V$ obtained from the Higgs fit read $$\begin{aligned}
\xi_V = -0.005 \pm 0.008 , \ \xi_V \in \left[ -0.021, \ 0.011 \right]. \nonumber\end{aligned}$$ The LQ coupling $\lambda_V$ is constrained to be small ($< 0.3$) for TeV-scale mass $U_1$, which is in accordance with the prediction of the ’4321’ model.
Compared to the present precision on $\xi_{S(V)}$, the situation is expected to improve by a factor of 2.4 at the HL-LHC.
We provide a method to determine VEVs, $\upsilon_3$ and $\upsilon_1$, of new scalar fields, $\Omega_3$ and $\Omega_1$ in the ’4321’model, via the combination of the relation $\xi_V = \lambda_V \upsilon \upsilon_3/(\upsilon^2_3 + \upsilon^2_1)$ together with direct searches of $U_1$ at colliders as well as other constraints such as measurements of $R_{D^{(*)}}$ and $R_{K^{(*)}}$.
For conclusion, loop-induced Higgs processes $ggF$ production and $h \to \gamma \gamma$ decay are important processes that contributions of new heavy particles such as LQs may hidden in. We expect more precise measurements of Higgs properties in the future to guide us in the direction for LQ study.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Ilja Doršner for discussions. This work is supported in part by the National Natural Science Foundation of China under Grants No.11875157 and 11847303 (C. X. Yue and J. Zhang), and Liaoning Revitalization Talents Program No.60618009 (C. H. Li).
{#sec:A}
Contributions of $U_1$ to $R_{D^{(*)}}$ and $R_{K^{(*)}}$ {#contributions-of-u_1-to-r_d-and-r_k .unnumbered}
---------------------------------------------------------
For the charged current $b \to c \ell \nu$, $U_1$ modifies $R_{D^{(*)}} = \frac{\mathcal{B}(\bar{B} \to D^{(*)} \tau^- \bar{\nu})}{\mathcal{B}(\bar{B} \to D^{(*) \ell^- \bar{\nu})}}$ ($\ell = e, \mu $) by [@DiLuzio:2018zxy] $$\begin{aligned}
\Delta R_{D^{(*)}} &=& \frac{R^{\rm exp}_{D^{(*)}}}{R^{\rm SM}_{D^{(*)}}} -1 \nonumber \\
&\approx& 0.2 \left( \dfrac{2 \ {\rm TeV}}{m_U} \right)^2 \left( \frac{g_4}{3.5} \right)^2 {\rm sin} (2 \theta_{LQ}) \left( \frac{s_{\ell_3}}{0.8} \right)^2 \left( \frac{s_{q_3}}{0.8} \right) \left( \frac{s_{q_2}}{0.35} \right).\end{aligned}$$ Setting $\theta_{LQ} = \pi/4$, $s_{\ell_3} = s_{q_3} = 0.8$, $s_{q_2} = 0.35$ and $g_4 = 3.5$, we obtain $$\begin{aligned}
\label{eq:delRD}
\Delta R_{D^{(*)}} \approx 0.2 \left( \frac{2000 \ {\rm GeV}}{m_U} \right)^2 \approx 0.2 \dfrac{ \left( 1143 \ {\rm GeV} \right)^2}{ \upsilon^2_3 +\upsilon^2_1 }.\end{aligned}$$
For neutral currents $b \to s \ell \ell$ case, $U_1$’s tree level contributions to Wilson coefficients $C^{\mu \mu}_9$ and $C^{\mu \mu}_{10}$ ($C_i = C^{SM}_i + \Delta C_i$) in the ’4321’ model are given by [@DiLuzio:2018zxy] $$\begin{aligned}
\left. \Delta C^{\mu \mu}_9 \right|_{\rm tree} = - \left. \Delta C^{\mu \mu}_{10} \right|_{\rm tree} = \frac{2 \pi}{\alpha_{em} V_{tb}V^*_{ts}} C_U \beta_{s \mu} \beta^*_{b \mu} ,\end{aligned}$$ where $C_U = g^2_4 \upsilon^2 / (4 m^2_U)$, $\beta_{s \mu} = c_{\theta_{LQ}} s_{q_2} s_{\ell_2}$, $\beta_{b \mu} = -s_{\theta_{LQ}} s_{q_3} s_{\ell_2}$. For $\theta_{LQ} = \pi/4$, $s_{\ell_2} = 0.06$, $s_{q_3} = 0.8$, $s_{q_2} = 0.35$ and $g_4 = 3.5$, we obtain $$\begin{aligned}
\label{eq:c9tree}
\left. \Delta C^{\mu \mu}_9 \right|_{\rm tree} = - \left. \Delta C^{\mu \mu}_{10} \right|_{\rm tree} = -0.46 \frac{\left( 1143 \ {\rm GeV} \right)^2}{\upsilon^2_3 + \upsilon^2_1}\end{aligned}$$
One-loop log-enhanced processes at the scale of the bottom mass may also contribute to the neutral currents sizeable. The contribution of the loops only to $C^{\ell \ell}_9$, which, in the $\beta_{b \tau}|V_{ts}|\ll \beta_{s \tau}$ limit, is given by [@DiLuzio:2018zxy] $$\begin{aligned}
\label{eq:c9loop}
\left. \Delta C^{\ell \ell}_9 \right|_{\rm loop} \left( m^2_b \right) \approx \frac{1}{3} \Delta R_{D^{(*)}} \left( {\rm log}x_b - \frac{1}{s^2_{\tau}} {\rm log}x_{E_2} \right),\end{aligned}$$ where $x_{\alpha} = m^2_{\alpha}/m^2_{U}$, $E_2$ is a vector-like lepton introduced in the model. The contribution is universal for all leptons. Taking Eq. (\[eq:delRD\]) in to the above equation and setting $s_{\tau} = 0.8$, $m_{E_2} = 850$ GeV, we have $$\begin{aligned}
\label{eq:delCll}
\left. \Delta C^{\ell \ell}_9 \right|_{\rm loop} \left( m^2_b \right) \approx \frac{0.2}{3} \dfrac{ \left(1143 \ {\rm GeV}\right)^2}{\upsilon^2_3 +\upsilon^2_1 } \left( {\rm log} \frac{ \left(4.8 \times 10^{-3} \ {\rm TeV}\right)^2 }{ \upsilon^2_3 +\upsilon^2_1 } - \frac{1}{0.8^2} {\rm log}\frac{\left(0.97 \ {\rm TeV}\right)^2}{ \upsilon^2_3 +\upsilon^2_1 } \right).\end{aligned}$$
Thus $U_1$ modifies the $b \to s \ell \ell$ processes via $$\begin{aligned}
\delta C^{\mu \mu}_9 &=& \left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm tree} + \left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm loop}, \\
\delta C^{\mu \mu}_{10} &=& - \left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm tree},\\
\delta C^{e e}_9 &=& \left. \Delta C^{\mu \mu}_9 \right|_{\rm loop}, \\
\delta C^{e e}_{10} &=& 0.\end{aligned}$$
In this analysis, we consider the $U_1$ contributions to $b \to s \ell \ell$ processes in the case of
$\mathbf{scenario \ A}$. only via tree level contributions (Eq. (\[eq:c9tree\])), i.e. $$\begin{aligned}
\delta C^{\mu \mu}_9 &=& - \delta C^{\mu \mu}_{10} = \left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm tree}, \\ \nonumber
\delta C^{e e}_9 &=& \ ~\delta C^{e e}_{10} = 0.\nonumber\end{aligned}$$
$\mathbf{scenario \ B}$. only via loop contributions (Eq. (\[eq:c9loop\])), i.e. $$\begin{aligned}
\delta C^{\mu \mu}_9 &=& \delta C^{e e}_9 = \left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm loop}, \\ \nonumber
\delta C^{\mu \mu}_{10} &=& \delta C^{e e}_{10} = 0.\nonumber\end{aligned}$$
Fit to $R_{D^{(*)}}$ and $R_{K^{(*)}}$ measurements {#fit-to-r_d-and-r_k-measurements .unnumbered}
---------------------------------------------------
The newest average values of $R_{D}$ and $R_{D^{*}}$ including preliminary results at Belle II experiment [@Abdesselam:2019dgh] are given by [@Murgui:2019czp] $$\begin{aligned}
R_{D} = 0.337 \pm 0.030 \quad {\rm and} \quad R_{D^{*}} = 0.299 \pm 0.013,\end{aligned}$$ with a correlation of -0.36. The SM predictions of these two measurements read $$\begin{aligned}
R^{SM}_{D} = 0.300^{+0.005}_{-0.004} \quad {\rm and} \quad R^{SM}_{D^{*}} = 0.251^{+0.004}_{-0.003}.\end{aligned}$$ Then we obtain $$\begin{aligned}
\label{eq:drde}
\Delta R_{D} = 0.123 \pm 0.101 \quad {\rm and} \quad \Delta R_{D^{*}} = 0.191\pm 0.054,\end{aligned}$$ the correlation between the two quantities reads -0.34.
Ref. [@Arbey:2019duh] has updated the $b \to s$ anomalies by including newest measurements of $R_{K}$ measured by the LHCb collaboration [@Aaij:2019wad], $R_{K^{*}}$ measured by the Belle collaboration [@Abdesselam:2019wac] as well as $B_{s,d} \to \mu^+ \mu^-$ measured by the ATLAS collaboration [@Aaboud:2018mst]. The best fit values of $\left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm tree}$ and $\left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm loop}$ read respectively $$\begin{aligned}
\label{eq:dc9}
\left. \Delta C^{\mu \mu}_{9, U_1} \right|_{\rm tree} = -0.41 \pm 0.10 \quad {\rm and} \quad \left. \Delta C_{9, U_1} \right|_{\rm loop} = -1.01 \pm 0.20.\end{aligned}$$
To obtain $\upsilon^2_3 + \upsilon^2_1$, we perform our fit to measurements in Eqs. (\[eq:drde\]) and (\[eq:dc9\]) by minimizing $$\begin{aligned}
\label{eq:xiB}
\chi^2_B = \left( \Delta R^{exp} - \Delta R^{the} \right) C^{-1}_{\Delta R} \left( \Delta R^{exp} - \Delta R^{the} \right) + \frac{\left(\Delta C^{exp}_{9} - \Delta C^{the}_{9} \right)^2}{ \left(\delta C^{\mu \mu}_{9}\right)^2},\end{aligned}$$ where $\Delta R^{exp}$ denotes the measurement of $\Delta R_{D^{(*)}}$ and $\Delta R^{the}$ represents its theoretical prediction as shown in Eq. [\[eq:delRD\]]{}. Similarly, $\Delta C^{\mu \mu, exp}_{9}$ denotes $\Delta C^{\mu \mu}_{9}$ measured at experiments and $\Delta C^{\mu \mu, the}_{9}$ is its theoretical prediction as shown in Eq. [\[eq:c9tree\]]{} or Eq. [\[eq:c9loop\]]{}.
Then, we obtain the best fit value and preferred 95% C.L. intervals of $\upsilon^2_3 + \upsilon^2_1$, for the case of scenario $\mathbf{A}$ $$\begin{aligned}
\upsilon^2_3 + \upsilon^2_1 &=& 1.496 \pm 0.250 \ {\rm TeV}^2, \\
\upsilon^2_3 + \upsilon^2_1 &\in& \left[ 1.127, 2.226 \right] \ {\rm TeV}^2 \ {\rm at \ 95\% C.L.},\end{aligned}$$ for the case of scenario $\mathbf{B}$ $$\begin{aligned}
\upsilon^2_3 + \upsilon^2_1 &=& 1.220 \pm 0.187 \ {\rm TeV}^2, \\
\upsilon^2_3 + \upsilon^2_1 &\in& \left[ 0.939, 1.748 \right] \ {\rm TeV}^2 \ {\rm at \ 95\% C.L.}.\end{aligned}$$
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[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
|
---
address: |
Instituto de Física Teórica CSIC-UAM and Departamento de Física Teórica,\
Universidad Autónoma de Madrid, Cantoblanco 28049 Madrid, Spain\
Département de Physique Théorique, Université de Genève,\
24 quai Ernest Ansermet, CH–1211 Genève 4, Switzerland
author:
- 'JUAN GARCIA-BELLIDO'
title: |
PRIMORDIAL GRAVITATIONAL WAVES AND THE LOCAL\
B-MODE POLARIZATION OF THE CMB
---
Cosmological Inflation [@LindeBook; @MukhanovBook] naturally generates a spectrum of density fluctuations responsible for large scale structure formation which is consistent with the observed CMB anisotropies.[@Komatsu2010] It also generates a spectrum of gravitational waves, whose amplitude is directly related to the energy scale during inflation and which induces a distinct B-mode polarization pattern in the CMB.[@DurrerBook] Moreover, Inflation typically ends in a violent process at preheating,[@preheating] where large density waves collide at relativistic speeds generating a stochastic background of GW [@GWpreh] with a non-thermal spectrum characterized by a prominent peak at GHz frequencies for GUT-scale models of inflation (or at mHz-kHz for low scale models of inflation), and an amplitude proportional to the square of the mass scale driving/ending inflation. Such a background could be detected with future GW observatories like Adv-LIGO [@ligo], LISA [@lisa], BBO [@bbo], etc. Furthermore, if inflation ended with a global phase transition, like in certain scenarios of hybrid inflation, then there is also a GWB due to the continuous self-ordering of the Goldstone modes at the scale of the horizon,[@Krauss] which is also scale-invariant on subhorizon scales,[@FFDGB] with an amplitude proportional to the quartic power of the symmetry breaking scale, that could be detectable with laser interferometers as well as indirectly with the B-mode polarization of the CMB.[@GBDFFK]
Gravitational waves produced during inflation arise exclusively due to the quasi-exponential expansion of the Universe [@MukhanovBook], and are not sourced by the inflaton fluctuations, to first order in perturbation theory. They have an approximately scale invariant and Gaussian spectrum whose amplitude is proportional to the energy density during inflation. GUT scale inflation has good chances to be discovered (or ruled out) by the next generation of CMB anisotropies probes, Planck [@Planck] and CMBpol [@CMBpol], see Fig. 1 for present bounds. At the end of inflation, reheating typically takes place in several stages. There is first a rapid (explosive) conversion of energy from the inflaton condensate to the fields that couple to it. This epoch is known as preheating [@preheating] and occurs in most models of inflation. It can be particularly violent in the context of hybrid inflation, where the end of inflation is associated with a symmetry breaking scenario, with a huge range of possibilities, from GUT scale physics down to the Electroweak scale. Gravitational waves are copiously produced at preheating from the violent collisions of high density waves moving and colliding at relativistic speeds [@GWpreh], see Fig. 1. The GW spectrum is highly peaked at the mass scale corresponding to the symmetry breaking field, which could be very different from the Hubble scale.
In low-scale models of hybrid inflation it is possible to attain a significant GWB at the range of frequencies and sensitivities of LIGO or BBO. The origin of these GW is very different from that of inflation. Here the space-time is essentially static, but there are very large inhomogeneities in the symmetry breaking (Higgs) field due to the random spinodal growth during preheating. Although the transition is not first order, “bubbles" form due to the oscillations of the Higgs field around its minimum. The subsequent collisions of the quasi-bubble walls produce a rapid growth of the GW amplitude due to large field gradients, which source the anisotropic stress-tensor [@GWpreh]. The relevant degrees of freedom are those of the Higgs field, for which there are exact analytical solutions in the spinodal growth stage, which later can be input into lattice numerical simulations in order to follow the highly non-linear and out-of-equilibirum stage of bubble collisions and turbulence. However, the process of GW production at preheating lasts only a short period of time around symmetry breaking. Soon the amplitude of GW saturates during the turbulent stage and then can be directly extrapolated to the present with the usual cosmic redshift scaling. Such a GWB spectrum from preheating would have a characteristic bump, worth searching for with GW observatories based on laser interferometry, although the scales would be too small for leaving any indirect signature in the CMB polarization anisotropies. Moreover, the mechanism generating GW at preheating is also active in models where the SB scenario is a local one, with gauge fields present in the plasma, and possibly related to the production of magnetic field flux tubes [@PMFpreh]. In such a case, one could try to correlate the GWB amplitude and the magnitude and correlation length of the primordial magnetic field seed.
In the case that inflation ends with a global or local symmetry breaking mechanism, then there generically appear cosmic defects associated with the topology of the vacuum manifold. For instance, global cosmic strings are copiously produced during preheating if the Higgs field is a complex scalar with a U(1) global symmetry [@preheating]. In principle, all kinds of topological and non-topological defects could form at the end of inflation and during preheating. Such defects will have contributions to all three different metric perturbations: scalar, vector and tensor, with similar amplitudes [@defrev]. In a recent work [@FFDGB], we analyzed the production of GW at preheating for a model with O(N) symmetry. The dynamics at subhorizon scales was identical to that of the usual tachyonic preheating. However, in this model even though the Higgs potential fixed the radial component to its vev, there remained the free (massless) Goldstone modes to orient themselves in an uncorrelated way on scales larger than the causal horizon. In the subsequent evolution of the Universe, as the horizon grows, spatial gradients at the horizon will tend to reorder these Goldstone modes in the field direction of the subhorizon domain. This self-ordering of the fields induces an anisotropic stress-tensor which sources GW production. In the limit of large N components, it is possible to compute exactly the scaling solutions, and thus the amplitude and shape of the GWB spectrum. It turns out that the GWB has a scale-invariant spectrum on subhorizon scales [@Krauss] and a $k^3$ infrared tail on large scales [@FFDGB], which can be used to distinguish between inflation and these non-topological defects.
Apart from the IR tail, the main difference between inflationary and global defect contributions to the CMB anisotropies arises from the fact that defects generically contribute with all modes: scalar, vector and tensor modes, with similar amplitudes, while inflationary tensor modes could be negligible if the scale of inflation is well below the GUT scale. Since (curl) B-modes of the polarization anisotropies only get contributions from the vector and tensor modes, the detection of the B-mode from inflation may be challenging, and dedicated experiments like Planck and CMBpol have been designed to look for them. On the other hand, defects’ contribution to the temperature anisotropies have a characteristic smooth hump in the angular power spectrum, which allows one to bound their amplitude (and thus the scale of symmetry breaking) below $10^{16}$ GeV. [@defCMB] However, the contribution to the B-mode coming from defects have both tensor and vector components, and the latter can be up to ten times larger than the former, and actually peaks at a scale somewhat below the horizon at last scattering (in harmonic space the corresponding multipole is $\ell\sim1000$).
In a recent paper [@GBDFFK] we analyzed the possibility of disentangling the different contributions to the B-mode polarization coming from defects versus that from inflation. The main difficulty, for both defects and tensor modes from inflation, is that the B-mode power spectrum is “contaminated" by the effect of lensing from the intervening matter distribution on the dominant E-mode contribution on similar angular scales. Using the temperature power spectrum to determine the underlying matter perturbation from evolved large scale structures responsible for CMB lensing, it is possible to engineer an iterative scheme to clean the primordial B-modes from lensed E-modes [@SeljakHirata]. This procedure leaves a significantly smaller polarization noise background $\Delta_{P,{\rm eff}}$ which allows one to detect the GW background at high confidence level (3-$\sigma$) if the scale of inflation or that of symmetry breaking is high enough. What we realized is that the usual E- and B-modes used for computing the angular power spectra are complicated non-local functions of the Stokes parameters, involving both partial differentiation and inverse laplacian integration. Such a non-local function requires knowledge of the global polarization on scales as large as the horizon, where the B-mode angular correlation function is negligible and thus prone to large systematic errors. In contrast, the so-called “local" E- and B-modes [@DurrerBook; @BZ] can be constructed directly from the Stokes parameters and do not involve any non-local inversion. A direct consequence in this change of variables is that the angular power spectrum of local B-modes has a extra factor $n_\ell = (\ell+2)!/(\ell-1)! \sim \ell^4$, which boosts the high-$\ell$ peak in the defects’ power spectra. When compared with the angular correlation function of inflation, it gives a significant advantage to the defects’ prospects for detection in future CMB experiments, see Fig. 3 and Table 1.
$S/N=3$ Inflation Strings Semilocal Textures Large-N
---------------- ----------- -------------------- -------------------- -------------------- --------------------
Planck $0.03$ $1.2\cdot10^{-7}$ $1.1\cdot10^{-7}$ $1.0\cdot10^{-7}$ $1.6\cdot10^{-7}$
CMBpol $10^{-4}$ $7.7\cdot10^{-9}$ $6.9\cdot10^{-9}$ $6.3\cdot10^{-9}$ $1.0\cdot10^{-8}$
$\tilde B$ exp $10^{-7}$ $1.1\cdot10^{-10}$ $1.0\cdot10^{-10}$ $0.9\cdot10^{-10}$ $1.4\cdot10^{-10}$
: The limiting amplitude for inflation (r=T/S) and various defects (${\epsilon}=Gv^2$), at 3-$\sigma$ in the range $\theta\in[0,1^o]$, for Planck ($\Delta_{P,{\rm eff}}=
11.2\,\mu$K$\cdot$arcmin), CMBpol-like exp. ($\Delta_{P,{\rm eff}}=0.7\,\mu$K$\cdot$arcmin) and a dedicated CMB experiment with $\Delta_{P,{\rm eff}}=0.01\,\mu$K$\cdot$arcmin. We have assumed $f_{\rm sky}=0.7$ for all CMB experiments. \[tab:3sigma\]
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Ruth Durrer, Martin Kunz, Elisa Fenu and Daniel Figueroa for a very enjoyable collaboration. This work is supported by the Spanish MICINN under project AYA2009-13936-C06-06.
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abstract: '$F-$Lie algebras are natural generalisations of Lie algebras ($F=1$) and Lie superalgebras ($F=2$). When $F>2$ not many finite-dimensional examples are known. In this paper we construct finite-dimensional $F-$Lie algebras $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of $F-$Lie algebras constructed in this way from $\mathfrak{su}(n), \mathfrak{sp}(2n)$ $\mathfrak{so}(n)$ and $\mathfrak{sl}(n|m)$, $\mathfrak{osp}(2|m)$ are given. We obtain non-trivial extensions of the Poincaré algebra by Inönü-Wigner contraction of certain $F-$Lie algebras with $F>2$.'
author:
- |
M. Rausch de Traubenberg [^1]$\,\,$${}^{a,b}$ and M.J. Slupinski [^2]$\,\,$${}^{c}$\
\
[${}^{a}$[*Laboratoire de Physique Théorique, Université Louis Pasteur*]{}]{}\
\
[${}^{b}$[*Laboratoire de Physique Mathématique et Théorique, Université Montpellier II,*]{}]{}\
\
[${}^c$[*Institut de Recherches en Mathématique Avancée*]{}]{}\
\
title: 'Finite-dimensional Lie algebras of order $F$'
---
6000 6000
Introduction
============
The classification of algebraic objects satisfying certain axioms may be considered a fundamental objective on purely mathematical grounds. If in addition, these objects turn out to to be relevant for the description of the possible symmetries of a physical system, such a classification takes on a whole new meaning. The main question is, of course, what are the mathematical structures which are useful in describing the laws of physics. Simple complex finite-dimensional Lie algebras were classified at the end of the 19th century by W. Killing and E. Cartan well before any physical applications were known. Since then, Lie algebras have become essential for the description of space-time symmetries and fundamental interactions. On the other hand, it was the discovery of supersymmetry in relativistic quantum field theory or as a possible extension of Poincaré invariance [@susy] which gave rise to the concept of Lie superalgebras and their subsequent classification [@super; @fk].
It is generally accepted that because of the theorems of Coleman $\&$ Mandula [@cm] and Haag, Lopuszanski $\&$ Sohnius [@hls], one cannot go beyond supersymmetry. However, if one weakens the hypotheses of these two theorems, one can imagine symmetries which go beyond supersymmetry [@ker; @luis; @para; @fsusy; @fsusy1d; @am; @fr; @qfsusy; @hek; @asm; @prs; @fsusy2d; @ad; @kk; @fvir; @fvir2; @fsusy3d; @fsusyh; @flie; @brs; @poly; @infty; @flie2], the idea being that then the generators of the Poincaré algebra can be obtained as an appropriate product of more than two fundamental additional symmetries. These new generators are in a representation of the Lorentz algebra which is neither bosonic nor fermionic. Two kinds of representations are generally taken: parafermionic representations [@paraferm], or infinite-dimensional representations (Verma module) [@verma].
Fractional supersymmetry (FSUSY) [@fsusy; @fsusy1d; @am; @fr; @qfsusy; @hek; @asm; @prs; @fsusy2d; @ad; @kk; @fvir; @fvir2; @fsusy3d; @fsusyh; @flie; @brs; @poly; @infty; @flie2] is among the possible extensions of supersymmetry which have been studied in the literature. Basically, in such extensions, the generators of the Poincaré algebra are obtained as $F-$fold ($F \in \bb N^\star$) symmetric products of more fundamental generators. A natural generalisation of Lie (super)algebras which is relevant for the algebraic description of FSUSY was defined in [@flie; @flie2] and called an $F-$Lie algebra. An $F-$Lie algebra admits a $\bb Z_F-$gradation, the zero-graded part being a Lie algebra. An $F-$fold symmetric product (playing the role of the anticommutator in the case $F=2$) expresses the zero graded part in terms of the non-zero graded part.
The purpose of this paper is to show how one can construct many examples of finite-dimensional $F-$Lie algebras by an inductive process starting from Lie algebras and Lie superalgebras. Some preliminary results is this direction were given in [@flie2]. Two types of finite-dimensional $F-$Lie algebras will be constructed. The first family of examples, which we call trivial, are obtained by taking the direct sum of a Lie (super)algebra with the trivial representation. The second family is more interesting: by an inductive procedure we show how one can give the underlying vector space of any Lie algebra or any classical Lie superalgebra the structure of an $F-$Lie algebra. This procedure involves Casimir operators in the case of Lie algebras and invariant symmetric forms on the odd part of the algebra in the case of Lie superalgebras.
The paper is organized as follows. In section 2 we recall the definition of an $F-$Lie algebra and show how one can construct an $F-$Lie algebra of order $F_1+F_2$ from an $F-$Lie algebra of order $F_1 \ge 2$ and an invariant symmetric form of order $F_2$ on its non-zero graded part ([*c.f.*]{} Theorem \[tensor\]). In section 3 we introduce the notion of a graded $1-$Lie algebra in order to prove a version of theorem \[tensor\] when $F_1=1$ (theorem \[tensor-bis\]), and give some explicit examples of $F-$Lie algebras associated to Lie algebras. In section 4 we give explicit examples of $F-$Lie algebras associated to Lie superalgebras. In section 5 we obtain FSUSY extensions of the Poincaré algebra by Inönü-Wigner contraction of certain $F-$Lie algebras constructed in the two previous sections. In section 6 we define a notion of simplicity for $F-$Lie algebras and give examples of simple and non-simple $F-$Lie algebras. Finally, in section 7 we give finite-dimensional matrix realisations of the $F-$Lie algebras of section 4 induced from $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(2|2n)$ and a quadratic form. Using finite-dimensional matrices, we also show that the underlying vector spaces of the graded $1-$Lie algebras $\mathfrak{su}(n) \oplus \mathfrak{su}(n),
\mathfrak{so}(n) \oplus \mathfrak{so}(n)$ and $\mathfrak{sp}(2n) \oplus\mathfrak{sp}(2n)$ can be given $F-$Lie algebra structures which cannot be obtained by our inductive process.
$F-$Lie algebras
================
Definition of $F-$Lie algebras
------------------------------
In this section, we recall briefly the definition of $F-$Lie algebras given in [@flie; @flie2]. Let $F$ be a positive integer and let $q=e^{\frac{ 2 \pi i}{F}}$. We consider $S$ a complex vector space and $\varepsilon$ an automorphism of $S$ satisfying $\varepsilon^F=1$. We set ${\cal A}_k=S_{q^k}, 1 \le k \le F-1$ and ${\cal B}=S_1$ ($S_{q^k}$ is the eigenspace corresponding to the eigenvalue $q^k$ of $\varepsilon$). Then we have $S= {\cal B }\oplus_{k=1}^{F-1} {\cal A}_k$.
\[f-lie\] $S= {\cal B} \oplus_{k=1}^{F-1} {\cal A}_k$ is called a (complex) $F-$Lie algebra if it is endowed with the following structure:
1. ${\cal B}$ is a (complex) Lie algebra and ${\cal A}_k, 1 \le k \le F-1$ are representations of ${\cal B}$. If $[ , ]$ denotes the bracket on ${\cal B}$ and the action of ${\cal B}$ on $S$ it is clear that $\forall b \in {\cal B}, \forall s \in S, [\varepsilon(b),\varepsilon(s)]=
\varepsilon\left([b,s]\right)$.
2. There exist multilinear ${\cal B}-$equivariant maps $\left\{~~, \cdots,~~ \right\}: {\cal S}^F\left({\cal A}_k\right)
\rightarrow {\cal B}$, where ${ \cal S}^F(D)$ denotes the $F-$fold symmetric product of $D$. It is easy to see that $\left\{\varepsilon(a_1), \cdots, \varepsilon(a_F)\right\}=
\varepsilon\left(\left\{a_1, \cdots, a_F\right\}\right),
\forall a_1, \cdots, a_F \in {\cal A}_k.$
3. For $b_i \in {\cal B}$ and $a_j \in {\cal A}_k$ the following “Jacobi identities” hold:
\[eq:jac\] &&+ + =0\
&&+ + =0\
&&= {,…,a\_F} + …+ {a\_1,…,}\
&&\_[i=1]{}\^[F+1]{} =0. (J\_4)
Note that the three first identities are automatic but the fourth, which we will refer to as $J_4$, is an extra constraint.
An $F-$Lie algebra is more than a Lie algebra $\mathfrak{g}_0$, a representation $\mathfrak{g}_1$ of $\mathfrak{g}_0$ and a $\mathfrak{g}_0-$valued $\mathfrak{g}_0-$equivariant symmetric form on $\mathfrak{g}_1$. Indeed, although the three first Jacobi identities are manifest in this situation, the fourth is not necessarily true. As an example, consider $\mathfrak{g}_0 = \mathfrak{sl}(2, \mathbb C)$ and $\mathfrak{g}_1
={\cal S}_{2k+1}, \ \ k \in \mathbb N$ (the irreducible representation of dimension $2k+2$). From the decomposition ${\cal S}^2 \left({\cal S}_{2k+1} \right) =
{\cal S}_{4k+2} \oplus {\cal S}_{4k-2} \oplus \cdots \oplus {\cal S}_2$ one has an $\mathfrak{sl}(2, \mathbb C)-$equivariant mapping from ${\cal S}^2 \left({\cal S}_{2k+1} \right) \longrightarrow {\cal S}_2
\longrightarrow \mathfrak{sl}(2, \mathbb C)$. But $\mathfrak{g} = \mathfrak{sl}(2, \mathbb C) \oplus {\cal S}_{2k+1}$ is not a Lie superalgebra (the fourth Jacobi identity is not satisfied) except when $k =0$ where it reduces to $\mathfrak{osp}(1|2)$.
\[1-2-Lie\] A $1-$Lie algebra is a Lie algebra, and a $2-$Lie algebra is a Lie superalgebra. We will also refer to these objects as $F-$Lie algebras of order one and two respectively.
\[irred-flie\] Notice that $\{a_1, \cdots, a_F \}$ is only defined if the $a_i$ are in the [*same*]{} ${\cal A}_k$ and that $\forall k=1,\cdots, F-1$, the spaces $S_k={\cal B}
\oplus {\cal A}_k$ are $F-$Lie algebras.\
[**From now on, we consider only $F-$Lie algebras $S={\cal B} \oplus {\cal
A}$ such that ${\cal A}$ is an eigenspace of $\varepsilon$.**]{}
\[cartan-weil\] If ${\mathfrak h} \subset {\cal B}$ is a Cartan subalgebra and $F_{\lambda_1}, \cdots, F_{\lambda_F} \in {\cal F}$ are respectively of weight $\lambda_1, \cdots, \lambda_F$, then $\left\{F_{\lambda_1}, \cdots, F_{\lambda_F}\right\} \in {\cal B}$ is of weight $\lambda_1 + \cdots + \lambda_F$. In particular, if $\lambda_1 + \cdots + \lambda_F \ne 0$ is not a root of ${\cal B}$ this bracket is zero.
This structure can be seen as a possible generalisation of Lie algebras ($F=1$) or Lie superalgebras ($F=2$) and can be compared, in some sense, to the ternary algebras ($F=3$) considered in [@tern], and to the $n-$ary algebras ($F=n$) introduced in [@vk] but in a different context. We have shown [@flie; @flie2] that all examples of FSUSY considered in the literature can be described within the framework of $F-$Lie algebras.
An inductive construction of $F-$Lie algebras
---------------------------------------------
Let $\mathfrak{g}$ be a complex Lie algebra and let ${\mathfrak{ r}},
{\mathfrak{ r^\prime}}$ be representations of $\mathfrak{g}$ such that there is a $\mathfrak{g}-$equivariant map $\mu_F: S^F(\mathfrak{{r}}) \rightarrow \mathfrak{{r^\prime}}$. We set:
$$S={\cal B}\oplus {\cal A}_1 = (\mathfrak{g} \oplus {\mathfrak{ r^\prime}})
\oplus {\mathfrak{ r}}.$$
Then, ${\cal B}=\mathfrak{g} \oplus {\mathfrak{ r^\prime}}$ is a Lie algebra as the semi-direct product of $\mathfrak{g}$ and $\mathfrak{{r^\prime}}$ (the latter with the trivial bracket). We can extend the action of $\mathfrak{g}$ on ${\mathfrak{ r}}$ to an action of ${\cal B}$ on ${\mathfrak{ r}}$ by letting ${\mathfrak{ r^\prime}}$ act trivially on ${\mathfrak{ r}}$. This defines the bracket $[~,~]$ on $S$. For the map $\left\{\cdots \right\}$ we take $\mu_F$. The first three Jacobi identities are clearly satisfied, and the fourth is also satisfied as each term in the expression on the L.H.S. of $J_4$ vanishes. Hence $S$ is an $F-$Lie algebra. There are two essentially opposite ways of giving explicit examples of $F-$Lie algebras of this type. One can either start from ${\mathfrak g}$ and $\mathfrak{{r^\prime}}$ and extract an “$F-$root” of $\mathfrak{{r^\prime}}$, or one can decompose ${\cal S}^F({\mathfrak{ r}})$ into irreducible summands and project onto one of them [@flie]. The first approach is the more difficult since, in general, it involves infinite-dimensional representation theory. For example if one starts with $\mathfrak{{r^\prime}}=\mathfrak{D}_{\mu_1}$, the vector representation of $\mathfrak{so}(1,d-1)$ of highest weight $\mu_1$, the representation $\mathfrak{{r}}=\mathfrak{D}_{\frac{\mu_1}{F}}$ of highest weight $\frac{\mu_1}{F}$, is not exponentialisable (see [*e.g.*]{} [@kpr]) and does not define a representation of the Lie group $\overline{SO(1,d-1)}$, except when $d=3$ where such representations describe relativistic anyons [@anyon]. The second approach on the other hand will always give finite-dimensional $F-$Lie algebras if one starts from finite-dimensional representations.
The following theorem gives an inductive procedure for constructing finite-dimensional $F-$Lie algebras.
\[tensor\] Let $\mathfrak{g_0}$ be a Lie algebra and $\mathfrak{g}_1$ a representation of $\mathfrak{g_0}$ such that
\(i) $S_1=\mathfrak{g_0} \oplus \mathfrak{g_1}$ is an $F-$Lie algebra of order $F_1>1$;
\(ii) $\mathfrak{g_1}$ admits a $\mathfrak{g_0}-$equivariant symmetric form $\mu_2$ of order $F_2 \ge 1$.
Then $S=\mathfrak{g_0} \oplus \mathfrak{g_1}$ admits an $F-$Lie algebra structure of order $F_1+F_2$, which we call the $F-$Lie algebra induced from $S_1$ and $\mu_2$.
[**Proof:**]{} By hypothesis, there exist $\mathfrak{g_0}-$equivariant maps $\mu_1:{\cal S}^{F_1}\left(\mathfrak{g_1}\right) \longrightarrow
\mathfrak{g}_0$ and $\mu_2:{\cal S}^{F_2}\left(\mathfrak{g_1}\right) \longrightarrow
\mathbb C$. Now, consider $\mu:{\cal S}^{F_1+ F_2}\left(\mathfrak{g_1}\right)
\longrightarrow \mathfrak{g}_0 \otimes\mathbb C \cong \mathfrak{g}_0$ defined by
$$\begin{aligned}
\label{eq:tensor}
&& \hskip 4truecm
\mu(f_1,\cdots,f_{F_1+F_2})= \\
&&\frac{1}{F_1 !}\frac{1}{F_2 !} \sum \limits_{\sigma \in S_{F_1 + F_2}}
\mu_1(f_{\sigma(1)},\cdots,f_{\sigma({f_{F_1}})}) \otimes
\mu_2(f_{\sigma(f_{F_1+1})},\cdots,f_{\sigma(f_{F_1+F_2})}),
\nonumber\end{aligned}$$
where $f_1,\cdots,f_{F_1+F_2} \in \mathfrak{g_1}$ and $S_{F_1 + F_2}$ is the group of permutations on $F_1 + F_2$ elements. By construction, this is a $\mathfrak{g_0}-$equivariant map from ${\cal S}^{F_1+ F_2}\left(\mathfrak{g_1}\right) \longrightarrow
\mathfrak{g_0}$, thus the three first Jacobi identities are satisfied. The last Jacobi identity $J_4$, is more difficult to check and is a consequence of $J_4$ for the $F-$Lie algebra $S_1$ and a factorisation property. Indeed, setting $F=F_1+F_2$, the identity $J_4$ for the terms in
$$\sum\limits_{i=0}^{F} \left[ f_i,\mu\left(f_1,\dots,
f_{i-1},
f_{i+1},\dots,f_{F}\right) \right],$$
of the form $\mu_1(f_{\sigma(1)},\cdots,f_{\sigma({f_{F_1}})}) \otimes
\mu_2(f_{\sigma(f_{F_1+1})},\cdots,f_{\sigma(f_{F_1+F_2})}$ with $\sigma \in S_{F_1+F_2 +1}$, reduces to
$$\begin{aligned}
\sum\limits_{i=0}^{F_1}
\left[ f_{\sigma(i)},\mu_1\left(f_{\sigma(1)},\cdots,
f_{\sigma(i-1)},f_{\sigma(i+1)},\cdots,f_{\sigma({f_{F_1}})}
\right) \right]
\otimes
\mu_2(f_{\sigma(f_{F_1+1})},\cdots,f_{\sigma(f_{F_1+F_2})}=0,
\nonumber\end{aligned}$$
using $\mu_2(f_{\sigma(f_{F_1+1})},\cdots,f_{\sigma(f_{F_1+F_2})}
\in \mathbb C$. But the L.H.S. vanishes by $J_4$ for the $F-$Lie algebra $S_1$. A similar argument works for the other terms and hence $J_4$ is satisfied and $S$ is an $F-$Lie algebra of order $F_1 + F_2$. [**QED**]{}
Theorem \[tensor\] is equivalent to the fact that the product of two $\mathfrak{g_0}-$equivariant symmetric forms satisfying $J_4$ also satisfies $J_4$ if one of them is scalar-valued.
Finite-dimensional $F-$Lie algebras associated to Lie algebras
==============================================================
In this section we first introduce the notion of a graded $1-$ Lie algebra in order to have a version of \[tensor\] when $F_1=1$.
Graded $1-$Lie algebras
-----------------------
\[1-lie\] A graded $1-$Lie algebra is a $\mathbb Z_2-$graded vector space $S={\cal B} \oplus {\cal F}$ such that:
1. ${\cal B}$ is a Lie algebra;
2. ${\cal F}$ is a representation of ${\cal B}$;
3. there is a ${\cal B}-$equivariant map $\mu : {\cal F} \to {\cal B}$;
4. $[\mu(f_1),f_2]+ [\mu(f_2),f_1]=0$, $\forall f_1, f_2 \in {\cal F}$.
\[graded\] Let $\mathfrak{g}$ be a Lie algebra. Set ${\cal B} = \mathfrak{g}$, ${\cal F} = \mathrm {ad } \ \mathfrak{g}$ and $S= {\cal B} \oplus {\cal F}$. If $\mu : {\cal F} \to {\cal B}$ is the identity then $(S,\mu)$ is a graded $1-$Lie algebra.
\[natural-lie\] A graded $1-$Lie algebra is not [*a priori*]{} a Lie algebra but it easy to see that, in fact, it has a natural graded Lie algebra structure.
${\mathrm Ker} \mu$ is a ${\cal B}-$invariant subspace of ${\cal F}$ and ${\mathrm Im} \mu$ is a ${\cal B}-$invariant subspace of ${\cal B}$. In particular, if ${\cal B}$ is simple, ${\cal F}$ irreducible and $\mu$ non-trivial, then $\mu$ defines a ${\cal B}-$equivariant isomorphism between ${\cal F}$ and ${\cal B}$.
A graded $1-$Lie algebra is a graded Lie algebra in the usual sense. In general, however, a graded Lie algebra is not a graded $1-$Lie algebra since there is no preferred map from the odd to the even part.
Let $\mathfrak{g}=\mathfrak{g}_+ \oplus \mathfrak{g}_- $ be a graded Lie algebra, and let $\mu : \mathfrak{g}
\to \mathfrak{g} $ be an odd $\mathfrak{g}_+-$equivariant map of $\mathfrak{g}$ such that $\mu $ is injective on $[\mathfrak{g}_+, \mathfrak{g}_-]$. Then $(\mathfrak{g}, \mu)$ is a graded $1-$Lie algebra.
[**Proof:**]{} One only has to check \[1-lie\](4). One has $\forall f_1,f_2 \in\mathfrak{g}_-$, $\mu\left([\mu(f_1),f_2] + [\mu(f_2),f_1]\right)=
[\mu(f_1),\mu(f_2)] + [\mu(f_2),\mu(f_1)]=0$. Since $\mu$ is injective on $[\mathfrak{g}_+, \mathfrak{g}_-]$ this implies \[1-lie\](4). [**QED**]{}
\[tensor-bis\] Let $\mathfrak{g_0}$ be a Lie algebra and $\mathfrak{g}_1$ a representation of $\mathfrak{g_0}$ such that
\(i) $S_1=\mathfrak{g_0} \oplus \mathfrak{g_1}$ is an graded $1-$Lie algebra;
\(ii) $\mathfrak{g_1}$ admits a $\mathfrak{g_0}-$equivariant symmetric $\mu_2$ form of order $F_2 \ge 1$.
Then $S=\mathfrak{g_0} \oplus \mathfrak{g_1}$ admits an $F-$Lie algebra structure of order $1+F_2$ which we call the $F-$Lie algebra induced from $S_1$ and $\mu_2$.
[**Proof:**]{} Analogous to \[tensor\]. [**QED**]{}
Trivial and induced $F-$Lie algebras
------------------------------------
Consider the graded $1-$Lie algebra $S=\mathfrak{g}_0 \oplus \mathfrak{g}_1$ where $\mathfrak{g}_0$ is a Lie algebra, $\mathfrak{g}_1$ is the adjoint representation of $\mathfrak{g}_0$ and $\mu : \mathfrak{g}_1 \to \mathfrak{g}_0$ is the identity. Let $J_1,\cdots, J_{\mathrm{dim}\mathfrak{g}_0}$ be a basis of $\mathfrak{g}_0$, and $ A_1,\cdots,
A_{\mathrm{dim}\mathfrak{g}_0}$ the corresponding basis of $\mathfrak{g}_1$. The graded $1-$Lie algebra structure on $S$ is then:
$$\begin{aligned}
\label{eq:1-lie}
\left[J_a, J_b \right] = f_{ab}^{\ \ \ c} J_c, \qquad
\left[J_a, A_b \right] = f_{ab}^{\ \ \ c} A_c, \qquad
\mu(A_a)= J_a,\end{aligned}$$
where $f_{ab}^{\ \ \ c} $ are the structure constant of $\mathfrak{g}_0$, Two types of $F-$Lie algebras associated to $S$ will be defined.
The first type of $F-$Lie algebras associated to $S$, will be called trivial and are constructed as follows:
\[lie-trivial\] Let $\mathfrak{g}_0$ be a Lie algebra and let $F \ge 1$ be an integer. Then $S=\mathfrak{g}_0 \oplus \left( \mathfrak{g}_1 \oplus
\mathbb C \right)$ can be given the structure of an $F-$Lie algebra (graded $1-$Lie algebra if $F=1$) where $\mathfrak{g}_1 $ is the adjoint representation of $\mathfrak{g}_0$ and $ \mathbb C$ is the trivial representation.
[**Proof:**]{}. The map $\mu: {\cal S}^F(\mathfrak{g}_1 \oplus {\mathbb C})
\longrightarrow \mathfrak{g}_0$ is given by projection on $\mathfrak{g}_1$ in the decomposition ${\cal S}^F(\mathfrak{g}_1 \oplus {\mathbb C })=
{\cal S}^F(\mathfrak{g}_1) \oplus
{\cal S}^{F-1} (\mathfrak{g}_1) \oplus \cdots \oplus
{\cal S}^{2} (\mathfrak{g}_1)
\oplus \mathfrak{g}_1
\oplus \mathbb C$, followed by the identification of $\mathfrak{g}_1$ with $\mathfrak{g}_0$.
With the notations of (\[eq:1-lie\]) the brackets are:
\[eq:t-lie\] {, , }&=&0\
{,,, A\_a, }&=& J\_a\
&&\
{,,, A\_[a\_1]{}, , A\_[a\_k]{} }&=&0, 1 < k F\
&&\
{A\_[a\_1]{},, A\_[a\_F]{} }&=&0.
with $A_a, \in \mathfrak{g}_1, \lambda \in \mathbb C$, $J_a \in \mathfrak{g}_0$.
It is easy to check that the four Jacobi identities are satisfied. [**QED**]{}\
The second type of $F-$Lie algebras associated to $S$ are those induced from $S$ and Casimir operators of $\mathfrak{g}_0$ (see \[tensor-bis\]). It is well known that the invariant tensors on $\mathfrak{g}_0^\star$ are generated by primitive invariant tensors which are either fully symmetric or fully antisymmetric [@ec]. By duality, symmetric invariant tensors are related to the Casimir operators of $\mathfrak{g}_0$, and it is well known that for a rank $r$ Lie algebra one can find $r$ independent primitive Casimir operators.
\[casimir\] Let $\mathfrak{g}_0$ be a simple (complex) Lie algebra and $\mathfrak{g}_1$ be the adjoint representation of $\mathfrak{g}_0$. Then a Casimir operator of $\mathfrak{g}_0$ of order $m$ induces the structure of an $F-$Lie algebra of order $m+1$ on $S_{m+1}= \mathfrak{g}_0 \oplus \mathfrak{g}_1$.
[**Proof:**]{} By example \[graded\] $\mathfrak{g}_0 \oplus \mathfrak{g}_1$ is a graded $1-$Lie algebra and the result follows from \[tensor-bis\]. [**QED**]{}
\[induced-graded\] One can give explicit formulae for the bracket of these $F-$Lie algebras as follows. Let $J_a, a=1,\cdots, \mathrm{dim}(\mathfrak{g}_0)$ and let $A_a, a=1,\cdots, \mathrm{dim}(\mathfrak{g}_0) $ be bases as at the beginning of this section. Let $h_{a_1 \cdots a_{m}}$ be a Casimir operator of order $m$ (for $m=2$, the Killing form $g_{ab}=\mathrm{Tr}(A_a A_b)$ is a primitive Casimir of order two). Then, the $F-$bracket of the $F-$Lie algebra is
$$\begin{aligned}
\label{eq:mi-lie}
\left\{A_{a_1}, A_{a_2}, \cdots, A_{a_{m+1}} \right\} =
\sum \limits_{\ell =1}^{m+1}
h_{a_1 \cdots a_{\ell-1} a_{\ell +1} \cdots a_{m+1}} J_{a_\ell}\end{aligned}$$
For the Killing form this gives
$$\begin{aligned}
\label{eq:3-lie}
\left\{A_a, A_b, A_c \right\} =
g_{ab} J_c + g_{ac} J_b + g_{bc} J_a.\end{aligned}$$
If $\mathfrak{g}_0= \mathfrak{sl}(2)$, the $F-$Lie algebra of order three induced from the Killing form is the $F-$Lie algebra of [@ayu].
Finite-dimensional $F-$Lie algebras associated to Lie superalgebras
===================================================================
In this section we will consider some $F-$Lie algebras which can be associated to Lie superalgebras using Theorem \[tensor\].
Lie superalgebras
-----------------
We first recall some basic results on simple complex Lie superalgebras (for more details see [@r; @fss]). Simple Lie superalgebras can be divided into two types: classical and the Cartan-type. Classical Lie superalgebras can be further divided into two families: basic and strange. A basic Lie superalgebra $\mathfrak{g}= \mathfrak{g}_0 \oplus \mathfrak{g}_1$ is said to be respectively of type I or type II depending on whether the $\mathfrak{g}_0-$module $\mathfrak{g}_1$ is respectively reducible or irreducible. Here is the complete list of simple classical Lie superalgebras [@super; @fk]. In the statement of 1(i) the symbol $(\overline{\mathbf { m+1}}, {\mathbf {n+1}})^+
\oplus ({\mathbf { m+1}}, \overline{\mathbf {n+1}})^-$ denotes $\left({\mathbb C}^{m+1\ \star} \otimes {\mathbb C}^{n+1}
\otimes {\mathbb C} \right)
\oplus \left({\mathbb C}^{m+1\ } \otimes {\mathbb C}^{n+1 \star}
\otimes {\mathbb C}^\star \right)$, where ${\mathbb C}^{m+1 }$ is the fundamental representation of $\mathfrak{sl}(m+1)$, ${\mathbb C}^{m+1\star}$ its dual representation and ${\mathbb C}$ is the standard one dimensional representation of $\mathfrak{gl}(1)$. In the rest of the theorem we use analogous notation.
\[super\] Let $\mathfrak{g}= \mathfrak{g}_0 \oplus \mathfrak{g}_1$ be a classical simple complex Lie superalgebra. Then $\mathfrak{g}$ is isomorphic to one of the following:
1. (Basic of type I)
\(i) $A(m,n)$: $m > n \ge 0,
\mathfrak{g}_0= \mathfrak{sl}(m+1) \oplus \mathfrak{sl}(n+1)
\oplus \mathfrak{gl}(1),
\mathfrak{g}_1=(\overline{\mathbf { m+1}}, {\mathbf {n+1}})^+
\oplus (\mathbf{m+1}, \overline {{\mathbf {n+1}}})^-$
\(ii) $A(n,n): n \ge 1, \mathfrak{g}_0= \mathfrak{sl}(n+1)
\oplus \mathfrak{sl}(n+1),
\mathfrak{g}_1=(\overline{{\mathbf {n+1}}}, \mathbf{n+1}) \oplus
({\mathbf {n+1}}, \overline {{\mathbf {n+1}}})$;
\(iii) $C(n+1): n \ge 1, \mathfrak{g}_0=\mathfrak{sp}(2n)
\oplus \mathfrak{gl}(1),
\mathfrak{g}_1={\mathbf {2n}}^+ \oplus {\mathbf {2n}^-}$.
2. ( Basic of type II)
\(i) $B(m,n): m \ge 0, n \ge 1, \mathfrak{g}_0=
\mathfrak{so}(2m+1) \oplus
\mathfrak{sp}(2n), \mathfrak{g}_1=(\mathbf{2m+1}, \mathbf{2n})$;
\(ii) $D(m,n): m \ge 2, n \ge 1, m \ne n+1, \mathfrak{g}_0=
\mathfrak{so}(2m) \oplus
\mathfrak{sp}(2n), \mathfrak{g}_1= (\mathbf{2m}, \mathbf{2n})$;
\(iii) $D(n+1,n): \mathfrak{g}_0= \mathfrak{so}(2(n+1))
\oplus
\mathfrak{sp}(2n), \mathfrak{g}_1 =(\mathbf{2(n+1)}, \mathbf{2n})$;
\(iv) $D(2,1; \alpha): \alpha \in \bb C-\left\{0,-1\right\},
\mathfrak{g}_0= \mathfrak{sl}(2) \oplus \mathfrak{sl}(2) \oplus
\mathfrak{sl}(2),
\mathfrak{g}_1=(\mathbf{2},\mathbf{2},\mathbf{2})$;
\(v) for $F(4): \mathfrak{g}_0=\mathfrak{sl}(2) \oplus
\mathfrak{so}(7),
\mathfrak{g}_1=(\mathbf{2}, \mathbf{8})$;
\(vi) for $G(3): \mathfrak{g}_0=\mathfrak{sl}(2) \oplus G_2,
\mathfrak{g}_1 =(\mathbf{2}, \mathbf{7})$.
3. (Strange)
\(i) $Q(n): n>1 \mathfrak{g}_0=\mathfrak{sl}(n),
\mathfrak{g}_1 =\mathrm{ad}(\mathfrak{sl}(n))$, with ad the adjoint representation;
\(ii) $P(n): n>1 \mathfrak{g}_0=\mathfrak{sl}(n),
\mathfrak{g}_1 = [2] \oplus [1^{n-2}]$, where $[2]$ denotes ${\cal S}^2\left({\mathbb C}^n\right)$ the two-fold symmetric representation and $[1^{n-2}]$ denotes $\Lambda^{n-2}\left(\mathbb C^n\right)$ the $(n-2)-$fold antisymmetric representation.
(The superscript in 1(i) and 1(iii) indicates the $\mathfrak{gl}(1)$ charge).
Symmetric invariant forms {#sym-inv}
-------------------------
By Theorem \[tensor\] one can construct an $F-$Lie algebra from a Lie superalgebra $\mathfrak{g}=\mathfrak{g}_0 \oplus \mathfrak{g}_1$ and a $\mathfrak{g}_0-$invariant symmetric form on $\mathfrak{g}_1$. In general determining [*all*]{} invariant symmetric forms on a given representation of a given Lie algebra is very difficult. However, for the Lie superalgebras given in the above list we will show how one can construct many invariant symmetric forms. The key observation is that for each basic Lie superalgebra in the list, the odd part $\mathfrak{g}_1$ is either a tensor product (type II) or a sum of two dual tensor products (type I) as a representation of $\mathfrak{g}_0$. Thus, to find $\mathfrak{g}_0-$invariant symmetric forms on $\mathfrak{g}_1$ one can use the following well known isomorphisms of representations of $GL(A) \times GL(B)$ [@fulton-harris]:
\[sum\] [S]{}\^p (A B )&=& \_[k=0]{}\^p [S]{}\^k (A )\^[p-k]{} (B)\
\[Young\] [S]{}\^p (A B)&=& \_ [\$]{}\^ (A) \^ (B),
where the second sum is taken over all Young diagrams $\Gamma$ of length $p$ and ${\$}^{\Gamma}\left(A\right)$ denotes the irreducible representation of $GL(A)$ corresponding to the Young symmetriser of $\Gamma$.
### Type I {#type-i .unnumbered}
We consider the Lie superalgebra $A(m,n)$. The case of the other basic type I Lie superalgebras is similar. Then $\mathfrak{g}_0= \mathfrak{sl}(m+1) \oplus \mathfrak{sl}(n+1)
\oplus \mathfrak{gl}(1)$ and $\mathfrak{g}_1= \left({\mathbb C}^{m+1\ \star} \otimes {\mathbb C}^{n+1}
\otimes {\mathbb C} \right) \oplus
\left({\mathbb C}^{m+1 \star } \otimes {\mathbb C}^{n+1}
\otimes {\mathbb C} \right)^\star$. Using the formulae (\[sum\]) and (\[Young\]), one sees that ${\cal S}^p({\mathfrak{g}}_1)$ is a direct sum of terms of the form:
\^([C]{}\^[m+1 ]{}) \^[\^]{} ([C]{}\^[m+1 ]{})\^([C]{}\^[n+1 ]{})\^[\^]{} ([C]{}\^[n+1 ]{})\^[||-|\^|]{},
where $|\Gamma|$ is the length of $\Gamma$ and $|\Gamma | + |\Gamma^\prime| =p$. If this term contains the trivial representation then $n$ must be even and $|\Gamma | =|\Gamma^\prime|$. Furthermore the dimension of the vector space of $\mathfrak{g}_0$ invariants is then
I\_[, \^]{}= \_[(m+1)]{} ( [\$]{}\^[\^]{} ([C]{}\^[m+1 ]{}), [\$]{}\^ ([C]{}\^[m+1 ]{})) \_[(n+1)]{} ( [\$]{}\^[\^]{} ([C]{}\^[n+1 ]{}), [\$]{}\^ ([C]{}\^[n+1 ]{})),
where $\mathrm {Hom}_{\mathfrak{sl}(m+1)}$ denotes homomorphisms which are $\mathfrak{sl}(n+1)$ equivariant. One can calculate the dimensions of these spaces using well known results [@fulton-harris]. If $\Gamma= \Gamma^\prime$ then $I_{\Gamma, \Gamma^\prime} \ge 1$; if $\Gamma=\Gamma^\prime$ and $|\Gamma|=|\Gamma^\prime|=1$ then $I_{\Gamma, \Gamma^\prime} = 1$ and the invariant quadratic form corresponds to the tautological metric on $\mathfrak{g}_1$. In [@flie2] $F-$Lie algebras were constructed using this symmetric form.
### Type II {#type-ii .unnumbered}
All basic type II Lie superalgebras except (iv) have $\mathfrak{g}_0=
\mathfrak{g}_0^\prime \oplus \mathfrak{g}_0^{\prime \prime}$ and $\mathfrak{g}_1= {\cal D}^\prime \otimes {\cal D}^{\prime \prime}$, where ${\cal D}^\prime$ and ${\cal D}^{\prime \prime}$ are irreducible self-dual representations of respectively $\mathfrak{g}_0^\prime$ and $\mathfrak{g}_0^{\prime \prime}$. Therefore ${\cal S}^p\left(\mathfrak{g}_1 \right)$ is the direct sum of terms of the form:
\[II\] [\$]{}\^([D]{}\^) \^([D]{}\^)
where $|\Gamma|=p$. The dimension of the vector space of $\mathfrak{g}_0$ invariants is I\_= \^([D]{}\^)\^[\_0\^]{} [\$]{}\^([D]{}\^ )\^[\_0\^]{},
where ${\$}^\Gamma\left({\cal D}^\prime\right)^{\mathfrak{g}_0^\prime}$ denotes the space of $\mathfrak{g}_0^\prime$ invariant vectors in ${\$}^\Gamma\left({\cal D}^\prime\right)$. Although the respective factors in the product (\[II\]) are irreducible for $GL({\cal D}^\prime)$ and $GL({\cal D}^{\prime \prime})$, they may become reducible for $\mathfrak{g}_0^\prime$, $\mathfrak{g}_0^{\prime \prime}$. For example the representations associated to the Young diagram $\begin{tabular}{|c|c|}\hline
& \\ \hline
& \\
\cline{1-2}
\end{tabular}$ are reducible for both $\mathfrak{g}_0^\prime = \mathfrak{so}(m)$ and $\mathfrak{g}_0^{\prime \prime} = \mathfrak{sp}(2n)$.
### The strange superalgebra $Q(n)$ {#the-strange-superalgebra-qn .unnumbered}
Up to duality ${\cal S}^\star (\mathfrak{g}_1)$ (the symmetric algebra on $\mathfrak{g}_1$) is generated by the Casimir operators of $\mathfrak{sl}(n)$ (see section 3.).
### The strange superalgebra $P(n)$ {#the-strange-superalgebra-pn .unnumbered}
In this case ${\cal S}^\star \left(\mathfrak{g}_1\right)$ is a direct sum of terms of the form
\[Pn\] [S]{}\^k( [S]{}\^2 ([C]{}\^n ) ) \^[p-k]{}( \^[n-2]{} ([C]{}\^n) )
This representation is in general reducible but we do not know of a simple general formula for the dimension of $\mathfrak{sl}(n)$ invariants.
Trivial and induced $F-$Lie algebras
------------------------------------
In this section $F-$Lie algebras associated to Lie superalgebras will be constructed explicitly. To fix our notations, consider $\mathfrak{g}= \mathfrak{g}_0 \oplus \mathfrak{g}_1$ a classical Lie superalgebra. Let $J_a, 1 \le a \le \mathrm {dim~ }\mathfrak{g}_0$ be a basis of $\mathfrak{g}_0$ and $F_\alpha,
1 \le \alpha \le \mathrm {dim~ }\mathfrak{g}_1$ be a basis of $\mathfrak{g}_1$. The structure constants of $\mathfrak{g}$ are given by
\[eq:lie\] &=& f\_[ab]{}\^[ c]{} J\_c\
&=& (R\_[a]{})\_\^[ ]{} F\_,\
{F\_, F\_}& =& E\_ = S\_\^[a]{} J\_a
The structure constants are given [*e. g.*]{} in [@fss] for particular choices of bases.
The first type of $F-$Lie algebras associated to $\mathfrak{g}$ will be called trivial and are constructed as follows:
\[trivial\] Let $\mathfrak{g}=\mathfrak{g}_0 \oplus \mathfrak{g}_1$ be a Lie superalgebra and let $F \ge 1$ be an integer. Then $S=\mathfrak{g}_0 \oplus \left( \mathfrak{g}_1 \oplus
\mathbb C \right)$ (with $ \mathbb C$ the trivial representation of $\mathfrak{g}_0$) can be given the structure of an $F-$Lie algebra.
[**Proof:**]{}. The proof is analogous to the proof of \[lie-trivial\]. [**QED**]{}
The second type of $F-$Lie algebras associated to $\mathfrak{g}$ are those induced from $\mathfrak{g}$ and symmetric forms on $\mathfrak{g}_1$. Let $\mathfrak{g}= \mathfrak{g}_0 \oplus \mathfrak{g}_1$ be one of the classical Lie superalgebras in the statement of \[super\] and let $g$ be a $\mathfrak{g}_0$ invariant symmetric form of order $m$ on $\mathfrak{g}_1$. The bracket of the associated $F-$Lie algebra of order $m+2$ in the above basis is given by (\[eq:tensor\])
\[super-F\] {F\_[\_1]{}, , F\_[\_[m+2]{}]{} }= \_[ i < j]{} g\_[\_1 \_[i-1]{} \_[i+1]{} \_[j-1]{} \_[j+1]{} \_[m+2]{}]{} E\_[\_i \_j]{}
\[sl\] We denote by $S$ the $F-$Lie algebra of order $4$ induced from the Lie superalgebra
$$A(m-1,n-1) =
\Big(
\mathfrak{sl}(m) \oplus \mathfrak{sl}(n) \oplus \mathfrak{gl}(1)
\Big) \oplus \left(
{\mathbb C}^{m } \otimes {\mathbb C}^{n \star } \otimes {\mathbb C}
\right) \oplus \left(
{\mathbb C}^{m } \otimes {\mathbb C}^{n \star } \otimes {\mathbb C}
\right)^\star,$$
and the tautological quadratic form on $\left(
{\mathbb C}^{m } \otimes {\mathbb C}^{n \star } \otimes {\mathbb C}
\right) \oplus \left(
{\mathbb C}^{m } \otimes {\mathbb C}^{n \star } \otimes {\mathbb C}
\right)^\star$. Let $\left\{E_{IJ}\right\}_{\begin{tiny}\begin{array}{l}
1 \le I \le m \\ 1 \le J \le m
\end{array}\end{tiny}}$ and $\left\{E_{IJ}\right\}_{\begin{tiny}\begin{array}{l}
m+1 \le I \le m+ n \\ m+ 1 \le J \le m+ n
\end{array}\end{tiny}}$ be the standard bases of $\mathfrak{gl}(m)$ and $\mathfrak{gl}(n)$ respectively. Let $\left\{F_{IJ}\right\}_{\begin{tiny}\begin{array}{l}
1 \le I \le m \\ m+1 \le J \le m+n
\end{array}\end{tiny}}$ and $\left\{F_{IJ}\right\}_{\begin{tiny}\begin{array}{l}
m+1 \le I \le m+ n \\ 1 \le J \le m
\end{array}\end{tiny}}$ be bases of $(\overline{\mathbf{m}}, \mathbf{n})^+,$ and $(\mathbf{m}, \overline{\mathbf{n}})^-$ respectively.
Then the four brackets of $S$ have the following simple form:
\[eq:unitary\] {F\_[I\_1 J\_1]{},F\_[I\_2 J\_2]{},F\_[I\_3 J\_3]{},F\_[I\_4 J\_4]{} }& =& \_[I\_1 I\_2]{} \_[J\_1 J\_2]{} (\_[I\_3 J\_4]{} E\_[J\_3 I\_4]{} + \_[J\_3 I\_4]{} E\_[I\_3 J\_4]{} )\
&+& \_[I\_1 I\_3]{} \_[J\_1 J\_3]{} (\_[I\_2 J\_4]{} E\_[J\_2 I\_4]{} + \_[J\_2 I\_4]{} E\_[I\_2 J\_4]{} )\
&+& \_[I\_1 I\_4]{} \_[J\_1 J\_4]{} (\_[I\_2 J\_3]{} E\_[J\_2 I\_3]{} + \_[J\_2 I\_3]{} E\_[I\_2 J\_3]{} )\
&+& \_[I\_2 I\_3]{} \_[J\_2 J\_3]{} (\_[I\_1 J\_4]{} E\_[J\_1 I\_4]{} + \_[J\_1 I\_4]{} E\_[I\_1 J\_4]{} )\
&+& \_[I\_2 I\_4]{} \_[J\_2 J\_4]{} (\_[I\_1 J\_3]{} E\_[J\_1 I\_3]{} + \_[J\_1 I\_3]{} E\_[I\_1 J\_3]{} )\
&+& \_[I\_3 I\_4]{} \_[J\_3 J\_4]{} (\_[I\_1 J\_2]{} E\_[J\_1 I\_2]{} + \_[J\_1 I\_2]{} E\_[I\_1 J\_2]{} ).
The fact that the R.H.S is in $\mathfrak{sl}(m) \oplus
\mathfrak{sl}(n)\oplus \mathfrak{gl}(1)$ is a consequence of theorem \[tensor\].
\[osp\] We denote by $S$ the $F-$Lie algebra of order $4$ induced from the Lie superalgebra
$$\mathfrak{osp}(2|2m)= \left(\mathfrak{so}(2) \oplus \mathfrak{sp}(2m)\right)
\oplus {\mathbb C}^2 \otimes {\mathbb C}^{2m},$$
and the quadratic form $g=\varepsilon \otimes \Omega$, where $\varepsilon$ is the invariant symplectic form on ${\mathbb C}^2$ and $\Omega$ the invariant symplectic form on ${\mathbb C}^{2m}$. Let $\left\{S_{\alpha \beta} = S_{\beta \alpha }
\right\}_{\begin{tiny}\begin{array}{l}
1 \le \alpha \le 2m \\ 1 \le \beta \le 2 m
\end{array}\end{tiny}}$ be a basis of $\mathfrak{sp}(2m)$ and $\left\{h \right\}$ be a basis of $\mathfrak{so}(2)$. Let $\left\{F_{q \alpha}\right\}_{\begin{tiny}\begin{array}{l}
q=\pm 1\\ 1 \le \alpha \le 2m
\end{array}\end{tiny}}$ be a basis of ${\mathbb C}^2 \otimes {\mathbb C}^{2m}$. Then the four brackets of $S$ take the following form
\[eq:flie-orth\] &&3.truecm {F\_[q\_1 \_1]{}, F\_[q\_2 \_2]{}, F\_[q\_3 \_3]{}, F\_[q\_4 \_4]{} }=\
&&\_[q\_1 q\_2]{} \_[\_1 \_2]{} (\_[q\_3 + q\_4]{} S\_[\_3 \_4]{} + \_[q\_3 + q\_4]{} \_[\_3 \_4]{} h ) + \_[q\_1 q\_3]{} \_[\_1 \_3]{} (\_[q\_2 + q\_4]{} S\_[\_2 \_4]{} + \_[q\_2 +q\_4]{} \_[\_2 \_4]{} h )\
&+ & \_[q\_1 q\_4]{} \_[\_1 \_4]{} (\_[q\_2 + q\_3]{} S\_[\_2 \_3]{} + \_[q\_2 + q\_3]{} \_[\_2 \_3]{} h ) + \_[q\_2 q\_3]{} \_[\_2 \_3]{} ( \_[q\_1 + q\_4]{} S\_[\_1 \_4]{} + \_[q\_1 + q\_4]{} \_[\_1 \_4]{} h )\
&+& \_[q\_2 q\_4]{} \_[\_2 \_4]{} (\_[q\_1 + q\_3]{} S\_[\_1 \_3]{} + \_[q\_1 + q\_3]{} \_[\_1 \_3]{} h ) + \_[q\_3 q\_4]{} \_[\_3 \_4]{} (\_[q\_1 + q\_2]{} S\_[\_1 \_2]{} + \_[q\_1 + q\_2]{} \_[\_1 \_2]{} h ).
Other extensions of Lie superalgebras have been considered in the literature. For instance extensions of the orthosymplectic superalgebra $\mathfrak{osp}(1|4)$ or the unitary $\mathfrak{sl}(4|1)$ were constructed by means of parafermions and parabosons [@lr]. The first example of an $F-$Lie algebra was considered in [@fvir; @fvir2] as a possible extension of the Virasoro algebra. In [@ayu] an example of a “trivial” $F-$Lie algebra, related to the superalgebra $\mathfrak{osp}(1|2)$ was constructed.
By repeated application of theorem \[tensor\] one construct $F-$Lie algebras of higher and higher order.
Finite-dimensional FSUSY extensions of the Poincaré algebra
===========================================================
It is well known that supersymmetric extensions of the Poincaré algebra can be obtained by Inönü-Wigner contraction of certain Lie superalgebras. In fact, one can also obtain FSUSY extensions of the Poincaré algebra by Inönü-Wigner contraction of certain $F-$Lie algebras as we now show with two examples.
For the first example, we let $S_3= \mathfrak{sp}(4) \oplus \mathrm{ad} \ \mathfrak{sp}(4)$ be the real $F-$lie algebra of order three (see Remark \[induced-graded\]) induced from the real graded $1-$Lie algebra $S_1= \mathfrak{sp}(4) \oplus \mathrm{ad} \ \mathfrak{sp}(4)$ (see Example \[graded\]) and the Killing form on $\mathrm {ad} \ {\mathfrak sp}(4)$. Using vector indices of $\mathfrak{so}(1,3)$ coming from the inclusion $\mathfrak{so}(1,3) \subset \mathfrak{so}(2,3)
\cong \mathfrak{sp}(4)$, the bosonic part of $S_3$ is generated by $M_{\mu \nu}, M_{\mu 4}$, with $\mu, \nu =0,1,2,3$ and the graded part by $J_{\mu \nu}, J_{4 \mu}$. Letting $\lambda \to 0$ after the Inönü-Wigner contraction,
[ll]{} M\_ L\_,& M\_[4]{} P\_\
J\_ Q\_,& J\_[4 ]{} Q\_,
one sees that $L_{\mu \nu }$ and $P_\mu$ generate the $(1+3)D$ Poincaré algebra and that $Q_{\mu \nu}, Q_\mu$ are the fractional supercharges in respectively the adjoint and vector representations of $\mathfrak{so}(1,3)$. This $F-$Lie algebra of order three is therefore a non-trivial extension of the Poincaré algebra where translations are cubes of more fundamental generators. The subspace generated by $L_{\mu \nu}, P_\mu, Q_\mu$ is also an $F-$Lie algebra of order three extending the Poincaré algebra in which the trilinear symmetric brackets have the simple form:
{Q\_, Q\_, Q\_}= \_ P\_+ \_ P\_+ \_ P\_,
where $\eta_{\mu \nu}$ is the Minkowski metric. This algebra should be compared to the algebra recently obtained in a different context, where a “trilinear” extension of the Poincaré algebra involving “supercharges” in the vector representation was constructed [@wt].
For the second example, we let $S_4=\left( \mathfrak{so}(2)\oplus \mathfrak{sp}(4) \right)
\oplus \underline{{\mathbf 2}} \otimes \underline{{\mathbf 4}}$ be the real $F-$Lie algebra of order four induced from $\mathfrak{osp}(2|4)$ and the symmetric form $\varepsilon \otimes \Omega$ , where $\Omega$ is the symplectic form on $\underline{{\mathbf 4}}$ and $\varepsilon$ the antisymmetric two-form on $\underline{{\mathbf 2}}$. Using spinor indices coming from $\mathfrak{sl}(2,\bb C)
\cong \mathfrak{so}(1,3) \subset \mathfrak{so}(2,3)$ the bosonic part is generated by $E_{\alpha \beta},
E_{\dot \alpha \dot \beta},
E_{\dot \alpha \beta}$ and the fermionic part by $F_\alpha^\pm,
\bar F_{\dot \alpha}^\pm, \alpha, \beta =1,2 $ and $\dot \alpha,
\dot \beta= \dot 1,
\dot 2$. Letting $\lambda \to 0$ after the Inönü-Wigner contraction \[eq:iw\]
[llll]{} E\_ L\_& E\_ L\_ & E\_ P\_ & h Z &F\_\^ Q\_\^ &|F\_\^ \_\^,
one sees that $L_{\alpha \beta},L_{\dot \alpha \dot \beta}$ and $ P_{\alpha \dot \alpha}$ generate the $(1+3)D$ Poincaré algebra, that $Z$ is central and that $Q_\alpha^\pm,
\overline{Q}_{\dot \alpha}^\pm$ are the fractional-supercharges in the spinor representations of $\mathfrak{so}(1,3)$. This $F-$Lie algebra of order four is therefore a non-trivial extension of the Poincaré algebra where translations are fourth powers of more fundamental generators. The four bracket can be expressed simply if we introduce the following notation: $\sigma_{\alpha \dot \alpha}^\mu, \overline{\sigma}^{\mu \dot \alpha \alpha}$ are the Dirac matrices, $\sigma^{\mu \nu}_{\alpha \beta}$, $\bar \sigma^{\mu \nu}_{\dot \alpha \dot \beta}$ and $P_\mu$ are the Poincaré generators (for details [*e.g.*]{} [@wb]). One then has:
\[eq:poincare\] && -1.truecm{Q\_[\_1]{}\^[q\_1]{},Q\_[\_2]{}\^[q\_2]{}, Q\_[\_3]{}\^[q\_3]{}, Q\_[\_4]{}\^[q\_4]{} }=\
&& 2 \^[q\_1 q\_2]{} \^[q\_3 q\_4]{} \_[\_1 \_2]{} \_[\_3 \_4]{} Z + 2 \^[q\_1 q\_4]{} \^[q\_2 q\_3]{} \_[\_1 \_4]{} \_[\_2 \_3]{} Z + 2 \^[q\_1 q\_3]{} \^[q\_2 q\_4]{} \_[\_1 \_3]{}\_[\_2 \_4]{} Z\
\
&&-.85truecm{Q\_[\_1]{}\^[q\_1]{},Q\_[\_2]{}\^[q\_2]{}, Q\_[\_3]{}\^[q\_3]{}, \_[\_4]{}\^[q\_4]{} }= \^[q\_1 +q\_4]{} \^[q\_2 q\_3]{} \_[\_2 \_3]{} \^\_[\_1 \_4]{} P\_\
&&2.95truecm+ \^[q\_2 +q\_4]{} \^[q\_1 q\_3]{} \_[\_1 \_3]{} \^\_[\_2 \_4]{} P\_\
&&2.95truecm+ \^[q\_3 +q\_4]{} \^[q\_1 q\_2]{} \_[\_1 \_2]{} \^\_[\_3 \_4]{} P\_\
\
&&-.75truecm {Q\_[\_1]{}\^[q\_1]{},Q\_[\_2]{}\^[q\_2]{}, \_[\_3]{}\^[q\_3]{}, \_[\_4]{}\^[q\_4]{} }= 0,
together with similar relations involving $\left\{Q_{\alpha_1}^{q_1},\overline{Q}_{\dot \alpha_2}^{q_2},
\overline{Q}_{\dot \alpha_3}^{q_3}, \overline{Q}_{\dot \alpha_4}^{q_4}
\right\}$ and $\left\{\overline{Q}_{\dot \alpha_1}^{q_1},\overline{Q}_{\dot \alpha_2}^{q_2},
\overline{Q}_{\dot \alpha_3}^{q_3}, \overline{Q}_{\dot \alpha_4}^{q_4}
\right\}$.\
Analogous constructions lead to FSUSY extensions of the Poincaré algebra in any space-time dimensions.
Simple $F-$Lie algebras
=======================
By analogy with the case of Lie (super)algebras we define ideals and the notion of simplicity for $F-$Lie algebras.
\[ideal\] Let $S={\cal B} \oplus {\cal F}$ be an $F-$Lie algebra, or a graded $1-$Lie algebra. Then $\mathfrak{I}={\cal B}^\prime \oplus {\cal F}^\prime$ is an ideal of $S$ if and only if
\(i) $\forall f^\prime_1 \in {\cal F}^\prime,\forall f_2,\cdots, f_F \in
{\cal F}:$ $\left\{f_1^\prime, f_2,\cdots, f_F \right\} \in {\cal B}^\prime;$
\(ii) ${\cal B}^\prime$ is an ideal of ${\cal B}$ ( $\forall b^\prime\in {\cal B}^\prime,\forall b \in {\cal B},
[b, b^\prime] \in {\cal B}^\prime$);
\(iii) $\forall b \in {\cal B},\forall f^\prime \in {\cal F}^\prime$ $ [b,f^\prime] \in {\cal F}^\prime;$
\(iv) $\forall b^\prime \in {\cal B}^\prime,\forall f \in {\cal F}$ $ [b^\prime,f] \in {\cal F}^\prime.$
For a graded $1-$Lie algebra $S={\cal B} \oplus {\cal F}$, denoting $\mu$ the map from ${\cal F}$ to ${\cal B}$, the property (i) of \[ideal\] becomes $\mathrm {Im} \mu \subset {\cal B}^\prime$.
\[surjective\] By \[ideal\], $\mathrm{ Im} \mu \oplus {\cal F}$ is an ideal of $S$ ($\mu$ denotes the ${\cal B}-$equivariant map from ${\cal S}^F\left({\cal F}\right)$ $ \longrightarrow {\cal B}$).
In the case of Lie algebras and Lie superalgebras, this is the usual definition. In the case of a graded $1-$Lie algebra $S={\cal B} \oplus {\cal F}$, $S^\prime={\cal B^\prime} \oplus {\cal F^\prime}$ is an ideal if and only if it is a $\mathbb Z_2-$graded ideal for the natural Lie bracket on $S$ ([*c.f.*]{} \[natural-lie\]).
An $F-$Lie algebra $S$ is said to be simple if and only if its only ideals are $S$ and $\left\{0\right\}$, and $\mu :{\cal S}^F\left({\cal F}\right) \to {\cal B}$ is non-zero.
Let $S= {\cal B} \oplus {\cal F}$ be a graded $1-$Lie algebra such that $\mu :{\cal F} \to {\cal B}$ is non-zero. Then, $S$ is simple if and only if ${\cal B}$ is a simple Lie algebra and ${\cal F}$ is an irreducible representation of ${\cal B}$.
If $\mathfrak{g}$ is a simple Lie algebra, and $S= \mathfrak{g} \oplus
\mathrm{ad~} \mathfrak{g}$ is the graded $1-$Lie algebra of Example \[graded\], then $S$ is simple as a graded $1-$Lie algebra but is not simple as a Lie algebra, with respect to the natural Lie bracket \[natural-lie\].
\[simple-bis\] Let $S= {\cal B} \oplus {\cal F}$ be an $F-$Lie algebra such that (i) ${\cal B}$ is semi-simple, (ii) the map $\mu :$ ${\cal S}^F \left({\cal F}\right) \longrightarrow {\cal B}$ is a surjection and (iii) no non-zero ideal of ${\cal B}$ has non-zero fixed points in ${\cal F}$. Then:
1. $S$ is simple.
2. The $F-$Lie algebra of order $(F+2)$ induced from a ${\cal B}-$equivariant non-degenerate quadratic form on ${\cal F}$ (see \[tensor\]-\[tensor-bis\]) also satisfies (i) and (ii).
[**Proof :**]{} Let $\mathfrak{I}= {\cal B}^\prime \oplus {\cal F}^\prime$ be a non-trivial ideal of $S$. Then ${\cal B}^\prime$ is an ideal of ${\cal B}$ and $[{\cal B}^\prime, {\cal F}] \subset
{\cal F}^\prime$. But if ${\cal F} ={\cal F}^\prime \oplus
{\cal F}^{\prime \prime}$ as ${\cal B}^\prime-$modules then $[{\cal B}^\prime, {\cal F}^{\prime \prime}]=0$ and therefore, ${\cal F}^{\prime \prime}= \{0\}$ since by hypothesis ${\cal B}^\prime$ does not admit non-zero fixed points. This proves (a).\
To prove (b). it is enough to prove that the induced $(F+2)-$bracket is surjective. Since the $F-$bracket $\mu : {\cal S}^{F}\left({\cal F}\right)
\to {\cal B}$ is surjective, by diagonalising the quadratic form, it is easy to see that the $(F+2)-$bracket (\[eq:tensor\]) is also surjective. [**QED**]{}
If $\mathfrak{g}$ is a simple Lie algebra, the graded $1-$Lie algebras $\mathfrak{g} \oplus {\mathrm {ad}
} \mathfrak{g}$ satisfies (i), (ii) and (iii) above. As one can check, the Lie superalgebras in the list \[super\] also satisfy (i), (ii) and (iii). Thus the induced $F-$Lie algebras associated to non-degenerate quadratic forms and these graded $1-$Lie algebras or Lie superalgebras are always simple.
The trivial $F-$Lie algebras associated to graded $1-$Lie algebras or Lie superalgebras \[lie-trivial\]-\[trivial\] are not simple since in both cases ${\mathfrak{g}_0} \oplus {\mathfrak{g}_1 }$ is an ideal of $S$. In particular, when $F=2$, the trivial Lie superalgebras associated to graded $1-$Lie algebras are not simple. The direct sum of two simple $F-$Lie algebras of the same order is clearly not simple. These two kinds of examples of non-simple $F-$Lie algebras indicate that probably, as for Lie superalgebras, there are different inequivalent ways to define semi-simple Lie $F-$Lie algebras.
Representations
===============
A representation of an $F-$Lie algebra $S$ is a linear map $\rho : ~ S \to \mathrm{End}(H)$, and a automorphism $\hat \varepsilon$ such that $ \hat \varepsilon^F=1$ which satisfy
\[eq:rep\]
[ll]{} & ()= (x) (y)- (y)(x) & {a\_1.,a\_F}= \_[S\_F]{} (a\_[(1)]{}) (a\_[(F)]{}) & (s) \^[-1]{} = ((s))
($S_F$ being the group of permutations of $F$ elements).
As a consequence of these properties, since the eigenvalues of $\hat \varepsilon$ are $\mathrm{F}^{\mathrm{th}}-$ roots of unity, we have the following decomposition
$$H= \bigoplus \limits_{k=0}^{F-1} H_k,$$
where $H_k=\left\{\left|h\right> \in H ~:~
\hat \varepsilon\left|h\right>=q^k \left|h\right> \right\}$. The operator $N \in \mathrm{End}(H)$ defined by $N\left|h\right>=k
\left| h \right>$ if $\left|h\right> \in H_k$ is the “number operator” (obviously $q^N=\hat \varepsilon$). Since $\hat \varepsilon \rho(b)= \rho(b) \hat \varepsilon, \forall b \in {\cal B}$ each $H_k$ provides a representation of the Lie algebra ${\cal B}$. Furthermore, for $a \in {\cal A}_\ell$, $\hat \varepsilon \rho(a)=q^\ell \rho(a) \hat \varepsilon$ and so we have $\rho(a) .H_k\ \subseteq
H_{k+\ell ({\mathrm{mod~} F)}}$\
\[so-sp\] Let $X,Y,Z$ be $n\times n$ (resp. $2n \times 2n$) matrices in $\mathfrak{so}(n)$ (resp. $\mathfrak{sp}(2n)$). Then, it is easy to see that $\{X,Y,Z\}$ is also in $\mathfrak{so}(n)$ (resp. $\mathfrak{sp}(2n)$). Consequently, $S=\mathfrak{so}(n)\oplus \mathfrak{so}(n)$ (resp. $S=\mathfrak{sp}(2n) \oplus \mathfrak{sp}(2n)$), is an $F-$Lie algebra of order $3$ (the only non-trivial point to be checked is the Jacobi identity (J4) in Definition \[f-lie\]). A similar property is true for any odd number of matrices. We will calculate the structure constants in the case of $\mathfrak{so}(n)$, the calculation for $\mathfrak{sp}(2n)$ being analogous. If $X_a, 1 \le a \le {\mathrm {dim}} \ \mathfrak{so}(n)$ is a basis of $\mathfrak{so}(n)$, then the $3-$bracket of $S$ is given by
\[eq:so-sp\] {X\_a,X\_b,X\_c }= k\_[abc]{}\^[ d]{} X\_d.
Writing $\left\{X_a,X_b,X_c,X_d\right\}= \Big(
\left\{X_a,X_b,X_c\right\}X_d +
\left\{X_a,X_b,X_d\right\}X_c +
\left\{X_a,X_c,X_d\right\}X_b +
\left\{X_b,X_c,X_d\right\}X_a \Big)$ and taking the trace using (\[eq:so-sp\]), we get $4 k_{abc}^{\ \ \ \ d} tr(X_d X_e) = \mathrm{Tr}\left(\left\{X_a,X_b,X_c,X_e
\right\} \right)$. Since the trace defines a metric on $\mathfrak{so}(n)$ this gives $k_{abc}^{\ \ \ \ d}= \frac{1}{4} {\mathrm Tr }
\left\{X_a,X_b,X_c,X_d\right\} g^{de}$.
This $F-$Lie algebra of order three is [*not* ]{} induced from the graded $1-$Lie algebra $\mathfrak{so}(n) \oplus \mathfrak{so}(n)$ and the Killing form: if this where the case we would have $\left\{X_a,X_b,X_c\right\}= \mathrm{Tr}\left(X_a X_b \right) X_c +
\mathrm{Tr}\left(X_a X_c \right) X_b + \mathrm{Tr}\left(X_b X_c \right) X_a$ which is clearly false if $a=b=c$. However, by proposition \[simple-bis\] $S$ is simple.
We can construct a representation of $S$ in ${\mathbb C}^n \otimes
{\mathbb C}^3$ as follows: define $\rho: S \to {\mathrm End}
\left({\mathbb C}^n \otimes {\mathbb C}^3\right)$ by
\(X) = {
[ll]{} X & [ if ]{} X [ is in the first ]{} (n) X Q& [ if ]{} X [ is in the second ]{} (n),
.
where $Q: {\mathbb C}^3 \to {\mathbb C}^3$ is any linear map whose minimal polynomial is $\lambda^3-1$ ([*i.e.*]{}, $Q^3=\mathbf {Id}$ and $Q$ has three distinct eigenvalues).
Related results were obtained for $\mathfrak{so}(n)$ and $\mathfrak{sp}(2n)$ in [@multi-lie].
\[sl-n\] Let $X,Y,Z$ be three $n\times n$ matrices in $\mathfrak{u}(n)$. Then, it is easy to see that $\{X,Y,Z\}$ is also in $\mathfrak{u}(n)$. As in the previous example, this simple observation enables us to give $\mathfrak{u}(n) \oplus \mathfrak{u}(n)$ or $\mathfrak{u}(n) \oplus \mathfrak{su}(n)$ the structure of an $F-$Lie algebra of order $3$.
\[mat-sl\] Let $A(m-1,n-1)$, $n \ne m$ be the Lie superalgebra of $(n+m)\times(n+m)$ matrices [@fk; @fss],
$$M=\begin{pmatrix}
E_{mm}& F_{mn} \cr
F_{nm} &E_{nn}
\end{pmatrix},$$
of supertrace zero ([*i.e.*]{}, $\mathrm{sTr} M=\mathrm {tr}E_{mm}-
\mathrm {tr}E_{nn}=0$).
If $J_{i_1}, \cdots, J_{i_{2F}}$ are arbitrary matrices then
\[eq:trace\] {J\_[i\_1]{}, ,J\_[i\_[2F]{}]{}}= \^F\_
[l]{}a < b =1 a b
{{J\_[i\_a]{},J\_[i\_b]{} }, {J\_[i\_a]{}, J\_[i\_b]{} ,J\_[i\_1]{}, ,J\_[i\_[2F]{}]{}} }.
Applying this formula to $2F$ odd matrices in $A(m-1,n-1)$ one sees by an induction that the supertrace of the $2F-$bracket (\[eq:trace\]) vanishes. Using the $\mathbb Z_2$ graduation of $A(m-1,n-1)$ one sees that this bracket belongs to the even part of the algebra and hence defines the structure of an $F-$Lie algebra of order $2F$ on the underlying vector space of $A(m-1,n-1)$. For $F=4$ this is just the the $F-$Lie algebra of order 4 induced by the tautological quadratic form of Example \[sl\]. Indeed, let $V={\mathbb C}^{\star n} \otimes {\mathbb C}^{ m} \otimes {\mathbb C}$ and let $\mathfrak{g}_0= \mathfrak{sl}(n) \oplus \mathfrak{sl}(m) \oplus
\mathfrak{gl}(1)$. Then, comparing $\mathfrak{gl}(1)$ charges, we have $\mathrm{Hom}_{\mathfrak{g}_0}
\left({\cal S}^4 \Big(V \oplus V^\star\Big), \mathfrak{g}_0\right)=
\mathrm{Hom}_{\mathfrak{g}_0}
\left({\cal S}^2 (V) \otimes {\cal S}^2 ( V^\star), \mathfrak{g}_0\right)$. Since,
$${\cal S}^2 \Big(V \Big)
\otimes {\cal S}^2 \Big( V^\star\Big) \cong
\Big({\cal S}^2({\mathbb C}^{\star n}) \otimes {\cal S}^2({\mathbb C}^{m})
\oplus \Lambda^2({\mathbb C}^{\star n}) \otimes \Lambda^2({\mathbb C}^{m})
\Big) \otimes
\Big({\cal S}^2({\mathbb C}^{ n}) \otimes {\cal S}^2({\mathbb C}^{\star m})
\oplus \Lambda^2({\mathbb C}^{n}) \otimes \Lambda^2({\mathbb C}^{\star m})
\Big)$$ and since the representations ${\mathbf 1}$ and $\mathfrak{sl}(n)$ occur exactly once in ${\cal S}^2({\mathbb C}^{ n})\otimes
{\cal S}^2({\mathbb C}^{\star n})$ and not at all in ${\Lambda}^2({\mathbb C}^{ n})\otimes {\Lambda}^2({\mathbb C}^{\star n})$, we deduce that $\mathrm{Hom}_{\mathfrak{g}_0}
\left({\cal S}^4 \Big(V \oplus V^\star\Big), \mathfrak{g}_0\right)$ is of dimension one.
By definition, the fundamental $(n+m)\times (n+m)$ matrix representation of the Lie superalgebra $A(m-1,n-1)$ is also a representation of the $F-$Lie algebra of order $2F$ constructed above. In general, this is not true: for instance if $m=2, n=1$, one can check that the $6-$dimensional representation of $A(2,1)$ is not a representation of the associated $F-$Lie algebra of order $4$.
\[mat-osp\]
Let $S$ be the set of all matrices of the form \[eq:osp-flie\] M=
q&0&F\_+ 0&-q&F\_- -F\_-\^t&-i F\_+\^t&S
,
where $q$ is a complex number, $F_\pm$ are two $1 \times 2n$ matrices, $\Omega$ is the standard $2n \times 2n$ symplectic form on ${\mathbb C}^{2n}$ and $S$ is a $2n \times2n$ matrix in $\mathfrak{sp}(2n)$, [*i.e*]{}, $S^t=\Omega S \Omega$. Let ${\cal B}=\left\{
\begin{pmatrix}
q&0&0 \cr
0&-q&0 \cr
0&0&S
\end{pmatrix}, q\in {\mathbb C}, S \in \mathfrak{sp}(2n) \right\}
\cong \mathfrak{so}(2) \oplus \mathfrak{sp}(2n)$ and let ${\cal F}= \left\{
\begin{pmatrix}
0&0&F_+ \cr
0&0&F_- \cr
-\Omega F_-^t&-i \Omega F_+^t&0
\end{pmatrix}, F_\pm \in {\cal M}_{1,2n}\left({\mathbb C}\right) \right\}$. If one now takes
$${\cal F}_{a +}=\begin{pmatrix}
0&0&F_{a+} \cr
0&0&0 \cr
0&-i \Omega F_{a+}^t&0
\end{pmatrix}, \ \
{\cal F}_{a -}=\begin{pmatrix}
0&0&0 \cr
0&0&F_{a-} \cr
-\Omega F_{a-}^t&0&0
\end{pmatrix}$$ and ${\cal F}_{a}= {\cal F}_{a +}+ {\cal F}_{a -}$ we get $\left\{{\cal F}_{a}, {\cal F}_{b} \right\}=
\begin{pmatrix} \alpha_{ab} & 0 &0 \cr
0&-i \alpha_{ab} &0 \cr
0&0&A_{ab}
\end{pmatrix}$, where $A_{ab}=-\Omega F_{a-}^t F_{b+} - i \Omega F_{a+}^t F_{b-}
-\Omega F_{b-}^t F_{a+} - i \Omega F_{b+}^t F_{a-}$ and where $\alpha_{ab}= -F_{a+} \Omega F_{b-}^t - F_{b+} \Omega F_{a-}^t.$ This shows that ${\cal B} \oplus {\cal F}$ is not closed under the superbracket. From the formula $\left\{{\cal F}_{a_1}, {\cal F}_{a_2},{\cal F}_{a_3},
{\cal F}_{a_4} \right\}
=\left\{ \left\{{\cal F}_{ a_1}, {\cal F}_{ a_2} \right\},
\left\{{\cal F}_{a_3}, {\cal F}_{a_4} \right\} \right\} +
\left\{ \left\{{\cal F}_{a_1}, {\cal F}_{a_3} \right\},
\left\{{\cal F}_{a_2}, {\cal F}_{a_4} \right\} \right\} +
\left\{ \left\{{\cal F}_{a_1}, {\cal F}_{a_4} \right\},
\left\{{\cal F}_{a_2}, {\cal F}_{a_3} \right\} \right\}$, observing that $\left\{{\cal F}_{a +},{\cal F}_{b +}\right\}=
\left\{{\cal F}_{a-},{\cal F}_{b-}\right\}=0$ the four bracket $\left\{{\cal F}_{ a_1 q_1},{\cal F}_{ a_2 q_2},
{\cal F}_{ a_3 q_3}, {\cal F}_{ a_4 q_4} \right\}=0$ if $q_1+q_2+q_3+q_4 \ne 0$. We then calculate $4$-brackets for $q_1=q_2=-q_3=-q_4=1$ and obtain
\[mat-osp2\] && {[F]{}\_[a +]{},[F]{}\_[b +]{}, [F]{}\_[c -]{}, [F]{}\_[d -]{} }=
q&0&0 0&-q&0 0&0&S
,\
q&=&2 (F\_[a +]{} F\_[c-]{}\^t ) ( F\_[b +]{} F\_[d-]{}\^t ) + 2 (F\_[a +]{} F\_[d-]{}\^t ) ( F\_[b +]{} F\_[c-]{}\^t )\
S&=& F\_[a +]{} F\_[d-]{}\^t (F\_[c-]{}\^t F\_[b+]{} + F\_[b +]{}\^t F\_[c -]{} ) + F\_[a +]{} F\_[c-]{}\^t (F\_[d-]{}\^t F\_[b+]{} + F\_[b +]{}\^t F\_[d -]{} )\
&+& F\_[b +]{} F\_[d-]{}\^t (F\_[c-]{}\^t F\_[a+]{} + F\_[a +]{}\^t F\_[c -]{} ) + F\_[b +]{} F\_[c-]{}\^t (F\_[d-]{}\^t F\_[a+]{} + F\_[a +]{}\^t F\_[d -]{} ).
This shows that ${\cal B} \oplus {\cal F}$ is an $F-$Lie algebra of order $4$ since $S^t= \Omega S \Omega$.
In fact, the matrices of ${\cal B}\oplus {\cal F}$ define a representation of the $F-$Lie algebra of order $4$ induced from $\mathfrak{osp}(2|2m)$ and $\varepsilon \otimes \Omega$ (see \[osp\]). Indeed setting $$\overline{{\cal F}}_{a +}=\begin{pmatrix}
0&0&F_{a+} \cr
0&0&0 \cr
0&- \Omega F_{a+}^t&0
\end{pmatrix}, \ \
\overline{{\cal F}}_{a -}=\begin{pmatrix}
0&0&0 \cr
0&0&F_{a-} \cr
-\Omega F_{a-}^t&0&0
\end{pmatrix},$$ we see that ${\cal B} \oplus \overline{{\cal F}} \cong \mathfrak{osp}(2|2m)$ and that $\left\{{\cal F}_{a + }, {\cal F}_{b + },{\cal F}_{c -},
{\cal F}_{d -} \right\}=
<\overline{{\cal F}}_{a+},\overline{{\cal F}}_{c-}>
\left\{\overline{{\cal F}}_{b+},\overline{{\cal F}}_{d-}\right\}+
<\overline{{\cal F}}_{b+},\overline{{\cal F}}_{d-}>
\left\{\overline{{\cal F}}_{a+},\overline{{\cal F}}_{c-}\right\}+
<\overline{{\cal F}}_{a+},\overline{{\cal F}}_{d-}>
\left\{\overline{{\cal F}}_{b+},\overline{{\cal F}}_{c-}\right\}+
<\overline{{\cal F}}_{b+},\overline{{\cal F}}_{c-}>
\left\{\overline{{\cal F}}_{a+},\overline{{\cal F}}_{d -}\right\}$
where $<\overline{{\cal F}}_{a+},\overline{{\cal F}}_{c-}>$ denotes the $\varepsilon \otimes \Omega$ invariant form.
Given an $F-$Lie algebra $S={\cal B} \oplus {\cal F}$ one can define the universal enveloping algebra ${\cal U}(S)$ by taking the quotient of the tensor algebra ${\cal T}(S)$ by the two-sided ideal generated by (see definition \[f-lie\])
$$\begin{aligned}
\label{eq:universal}
\left\{
\begin{array}{l}
\sum\limits_{\sigma \in \Sigma_F}
a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(F)}
-\left\{a_1,\cdots, a_F \right\}, \\
b_1 \otimes b_2 -b_2 \otimes b_1- \left[b_1,b_2\right], \\
b_1 \otimes a_2 -a_2 \otimes b_1-\left[b_1,a_2\right], \\
\end{array}
\right.\end{aligned}$$
with $a_1, \cdots, a_F \in {\cal A}_1, b_1,b_2 \in {\cal B}$. It is not necessary to impose the Jacobi identity (J4) since it is true in ${\cal T}(S)$.
The natural filtration of ${\cal T}(S)$ factors to a filtration of ${\cal U}(S)$ and, denoting the associated graded algebra by $\mathrm{gr}({\cal U}(S))$, we conjecture the following:
1. $\mathrm{gr}({\cal U}(S))$ is isomorphic to $ {\cal T}(S)/\bar I$, where $\bar I$ is the two-sided ideal generated by
$$\begin{aligned}
\label{eq:universal2}
\left\{
\begin{array}{l}
\sum\limits_{\sigma \in \Sigma_F}
a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(F)},\\
b_1 \otimes b_2 -b_2 \otimes b_1, \\
b_1 \otimes a_2 -a_2 \otimes b_1.
\end{array}
\right. \nonumber \end{aligned}$$
(This would then imply that $\mathrm{gr}({\cal U}(S)) \cong S({\cal B}) \otimes \Lambda_F({\cal F})$, where $S({\cal B})$ is the symmetric algebra on ${\cal B}$ and $\Lambda_F({\cal F})$ is the $F-$exterior algebra on ${\cal F}$ [@cliff]).
2. The natural map $\pi: {\cal U}(S) \to \mathrm{gr}({\cal U}(S))$ is a linear isomorphism. (This would be an analogue of the Poincaré-Birkhoff-Witt theorem).
In the usual way, the representations of $S$ are in bijective correspondence with the representations of the associative algebra ${\cal U}(S)$. Consequently, if ${\cal I} \subset {\cal U}(S)$ is a two-sided ideal, then the quotient $ {\cal U}(S)/{\cal I}$ gives a representation of $S$. It would be very convenient to have a theory of “Cartan sub-algebras”, “roots” and “weights” for $S$. However, even for simple Lie superalgebras this kind of theory only works well for basic Lie superalgebras [@fss]. One might expect $F-$Lie algebras induced from basic Lie superalgebras to be amenable to this approach. This seems not to be the case. Indeed, recall that if $S$ is a basic Lie superalgebra with Borel decomposition $S={\mathfrak h} \oplus
{\mathfrak n}_+ \oplus {\mathfrak n}_-$ and $\lambda \in {\mathfrak h}^\star$ is a dominant weight, then ${\cal V}_\lambda={\cal U}/{\cal I}_\mu$ (where ${\cal I}_\lambda$ is the ideal corresponding to $\lambda$) is (i) generated by the action of ${\mathfrak n}_+$ on the vacuum and (ii) has a unique quotient ${\cal D_\lambda}$ on which the action of ${\mathfrak n}_+$ is nilpotent and which is therefore finite-dimensional. However, if $S_g$ is the $F-$Lie algebra induced from $S$ and a symmetric form $g$, the quotient ${\cal V}_\lambda^\prime = {\cal U}(S_g)/{\cal I}_\lambda^\prime$ is (i) not generated by the action of ${\mathfrak n}_+$ on the vacuum and (ii) the nilpotence of the action of ${\mathfrak n}_+$ in a quotient does not guarantee finite-dimensionality. This means that in finite-dimensional representations of $S$, as in the examples of section 7, the elements of ${\mathfrak n}_+$ are not only nilpotent but also satisfy additional relations.\
Conclusion
==========
The mathematical structure underlying supersymmetry is that of a Lie superalgebra. Given the classification of Lie superalgebras, one can list the possible supersymmetric extensions of the Poincaré algebra. These extensions have had a wide range of applications in physics.
Fractional supersymmetries were first studied in the early 1990’s in relation with low dimensional physics $(D \le 3)$ where fields which are neither bosonic nor fermionic [@anyon] do exist. It was understood a few years later that FSUSY can be considered in arbitrary dimensions and the definition of an $F-$Lie algebra, the underlying mathematical structure, was given [@flie]. However when $F>2$, most of the examples of $F-$Lie algebras which have been found since then are of infinite-dimensions. In this paper, we show how one can construct many finite-dimensional $F-$Lie algebras starting from Lie algebras or Lie superalgebras equipped with appropriate symmetric forms. We define a notion of simplicity in this context and show that some of our examples are simple. Furthermore, we construct the first finite-dimensional FSUSY extensions of the Poincaré algebra by Inönü-Wigner contraction of certain $F-$Lie algebras.
These results can be seen as a first step in classifying $F-$Lie algebras.
#### Acknowledgements {#acknowledgements .unnumbered}
J. Lukierski and Ph. Revoy are gratefully acknowledged for useful discussions and remarks.
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[^1]: [email protected]
[^2]: [email protected]
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abstract: |
The magnitude of the angular momentum ($J^2$) in quantum mechanics is larger than expected from a classical model. We explain this deviation in terms of quantum fluctuations. A standard quantum mechanical calculation gives the correct interpretation of the components of the angular momentum in the vector model in terms of projections and fluctuations. We show that the addition of angular momentum in quantum mechanics gives results consistent with the classical intuition in this vector model.
La magnitud del momento angular ($J^2$) en mecánica cuántica es mas grande que lo esperado en un modelo clásico. Explicamos esta diferencia en términos de las fluctuaciones cuánticas. Un cálculo estándar de mecánica cuántica da la interpretación correcta a las componentes del momento angular en el modelo vectorial en términos de proyecciones y fluctuaciones. Mostramos que la suma de momento angular en mecánica cuántica da resultados consistentes con la intuición clásica en este modelo vectorial.
author:
- 'E. Gomez'
title: |
The meaning of 1 in [*j(j+1)*]{}
El significado del 1 en [*j(j+1)*]{}
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Introduction
============
The operator of angular momentum in quantum mechanics is always a confusing topic for new students. The quantum description of angular momentum involves differential operators or new algebra rules that seem to be disconnected from the classical intuition. For small values of angular momentum one needs a quantum description because the quantum fluctuations are as big as the angular momentum itself. In this regime, the simple classical models generally do not give the right result. In this paper I describe the use of fluctuations in the angular momentum components to produce a vector model compatible with the quantum mechanical result. I show that the addition of angular momenta from a standard quantum mechanical calculation is consistent with the classical intuition using the vector model. The paper is organized as follows: Section \[vector\] shows the problems encountered with the vector model, section \[spin\] works out the details for a spin 1/2 particle, section \[addition\] explains the addition of angular momenta for two spin 1/2 particles, section \[general\] describes the general case of addition of angular momenta and I give some conclusions at the end.
\[vector\]Vector model of angular momentum
==========================================
The presentation of angular momentum in quantum mechanics textbooks demonstrates the following relations [@sakurai]
$$\begin{aligned}
\langle J^2 \rangle &= j(j+1)\hbar^2
\label{magnitudej} \\
\langle J_z \rangle &= m\hbar,
\label{projectionj}\end{aligned}$$
with $-j \leq m \leq j$. It is usually said that the angular momentum comes in units of $\hbar$. This is consistent with Eq. \[projectionj\] since $m$ is an integer, but not with Eq. \[magnitudej\]. For example, if we have one unit of angular momentum ($j=1$), then $\langle J^2 \rangle = 2\hbar^2$, that is, the magnitude of the angular momentum is $\sqrt{2}$ rather than 1 (from now on we express angular momentum in units of $\hbar$). Only the $z$ projection of the angular momentum comes in units of $\hbar$ and not the magnitude. How can we reconcile both expressions? There is a nice derivation that explains the expression for $J^2$ by averaging the value of $J_z^2$.[@mcgervey91; @milonni89; @feynman] There are also ways to give an heuristic derivation of the properties of angular momentum.[@leblond76] We would like to gain some intuition as to where the extra 1 in Eq. \[magnitudej\] comes from.
The vector model is often introduced to give a classical analogy to the quantum angular momentum.[@cohentannoudji] To describe the angular momentum classically by a vector, we must specify its three components ${\cal J}_x$, ${\cal J}_y$ and ${\cal J}_z$. The magnitude of the vector is obtained from the components. The problem with that scheme in quantum mechanics is that it is impossible to measure with absolute precision the three components of the angular momentum. If one measures $J_z$ and $J_y$ exactly then the uncertainty in $J_x$ grows, that is, there is an uncertainty relation for the components of the angular momentum analogous to the uncertainty relation between position and momentum.
The natural choice for the components of angular momentum in the vector model would be ${\cal J}=(\langle J_x \rangle,\langle J_y \rangle,\langle J_z \rangle)$. We will show that this choice (choice A) gives the incorrect value for ${\cal J}^2$. A better choice (choice B) for the angular momentum vector is ${\cal J}=(\langle J_x^2 \rangle^{\frac{1}{2}},\langle J_y^2 \rangle^{\frac{1}{2}},\langle J_z^2 \rangle^{\frac{1}{2}})$. With this choice the magnitude square of the angular momentum vector gives the correct value ${\cal J}^2=\langle J_x^2 \rangle + \langle J_y^2 \rangle + \langle J_z^2 \rangle$. In the next section we give a classical interpretation of the components of the angular momentum vector in terms of fluctiations and we use this interpretation to explain the origin of the extra 1 in Eq. \[magnitudej\].
\[spin\]Spin 1/2 case
=====================
The key point to explain Eq. \[magnitudej\] lies in the fluctuations. Take the case of a state with spin 1/2 ($j=s=1/2$) and $m_s=1/2$. The values of $\langle S_x \rangle$, $\langle S_y \rangle$ and $\langle S_z \rangle$ are 0, 0 and 1/2 respectively. Choice A for the vector model gives ${\cal S}=(0,0,1/2)$ and the magnitude square of this vector is ${\cal S}^2=1/4$, which differs from the result $\langle S^2 \rangle=3/4$ obtained from Eq. \[magnitudej\]. Choice B gives right value for ${\cal S}^2$ since it was constructed that way. What is the meaning of each component? ${\cal S}_z=\langle S_z^2 \rangle^{\frac{1}{2}}=(\langle S_z \rangle^2)^{\frac{1}{2}}=\langle S_z \rangle$ and this component reduces to the $z$ projection of the operator S. For ${\cal S}_x$ we cannot use the same trick since we are not using an eigenstate of $S_x$. Still we can relate that component to the fluctuations. The fluctuations of an operator $A$ in quantum mechanics are given by [@sakurai] $$\Delta A^2 = \langle A^2 \rangle - \langle A \rangle^2.
\label{variance}$$ For the present state and the operator $S_x$ the result is $$\Delta S_x^2 = \langle S_x^2 \rangle.
\label{uncertaintys}$$ Then ${\cal S}_x=\langle S_x^2 \rangle^{\frac{1}{2}}=(\Delta S_x^2)^{\frac{1}{2}}=\Delta S_x$ and this component is equal to the fluctuations in the $x$ axis of the operator $S$. The $y$ component gives the same result. The meaning of the vector components in choice B is that ${\cal S}_x$ and ${\cal S}_y$ are fluctuations and ${\cal S}_z$ is the projection in the corresponding axis. The quantum mechanical calculation of the fluctuations gives $$\Delta S_x^2=\langle \frac{1}{2} | ~S_x^2~ | \frac{1}{2} \rangle = \frac{1}{4}, \label{fluctuations12}$$ then $\Delta S_x=1/2$, and similarly $\Delta S_y=1/2$. The vector is ${\cal S}=(1/2,1/2,1/2)$ and the magnitude square of the vector is ${\cal S}^2=3/4$ which is the correct value. The value of ${\cal S}^2$ in choice A is $s^2=1/4$. Instead in choice B, ${\cal S}_x$ and ${\cal S}_y$ contribute to ${\cal S}^2$ through the fluctuations giving the value of $s(s+1)=3/4$.
\[addition\]Addition of angular momenta
=======================================
We construct any value of angular momentum by adding several spin 1/2 particles. We show how the vector model works for two spin 1/2 particles. The sum of two spin 1/2 particles gives a total angular momentum of $j=1$ or $j=0$. Take first the case of the state with $j=1$ and $m=1$. The state is represented in quantum mechanics by $|\frac{1}{2},\frac{1}{2}\rangle$ where the numbers represent the $z$ projection of the spin of particles 1 and 2 respectively. The objective is to calculate the value of $\langle J^2 \rangle$, with $J=S_1+S_2$, the sum of the spin contributions. The quantum mechanical result from Eq. \[magnitudej\] is $\langle J^2 \rangle=2$, and we want to explain this in terms of the vector model.
The expression for $J^2$ is $$J^2 = J_x^2 + J_y^2 + J_z^2 = (S_{x1}+S_{x2})^2 + (S_{y1}+S_{y2})^2 + (S_{z1}+S_{z2})^2,
\label{jdecomposed}$$ where the index 1 and 2 refer to particles 1 and 2 respectively. There is no question as to how to calculate the expectation values in quantum mechanics, but if we think in terms of the vector model we are in trouble since we have to add two vectors that are a mixture of projections and fluctuations. We show the correct recipe for adding this vectors from a quantum mechanical calculation and show that it is consistent with the classical intuition.
Take ${\cal J}_z$ first. The sum is again simplified since we use an eigenstate of the operator. We have ${\cal J}_z=(\langle (S_{z1}+S_{z2})^2 \rangle)^{\frac{1}{2}} = \langle S_{z1} \rangle + \langle S_{z2} \rangle$, that is, ${\cal J}_z$ is just the direct sum of the individual projections. The $x$ component gives $${\cal J}_x=(\langle (S_{x1}+S_{x2})^2 \rangle)^{\frac{1}{2}} = (\langle S_{x1}^2 \rangle + 2 \langle S_{x1}S_{x2} \rangle + \langle S_{x2}^2 \rangle)^{\frac{1}{2}} = (\Delta S_{x1}^2 + \Delta S_{x2}^2)^{\frac{1}{2}}.
\label{xsumspin}$$ The two contributions add up in quadrature. This is expected since the $x$ component for each spin in the vector model corresponds to fluctuations (or noise), and the proper way to add uncorrelated noise is in quadrature. For a classical variable $w=u+v$, where $u$ and $v$ are fluctuating variables, the noise in $w$ is given by [@bevington] $$\sigma_w^2 = \sigma_u^2 + \sigma_v^2 + 2 \sigma_{uv}^2.
\label{noisefullformula}$$ The quantum mechanical expression for the fluctuations of $J_x=S_{x1}+S_{x2}$ for the present state is $$\Delta J_x^2 = \langle S_{x1}^2 \rangle + \langle S_{x2}^2 \rangle + 2\langle S_{x1}S_{x2} \rangle,
\label{noisesxsum}$$ where the similarity between the last two expressions is evident. The state we are considering has the two spins aligned. Since the two spins are independent, we expect their noise to be uncorrelated. The calculation of the correlation term (last term in Eq. \[noisesxsum\]) gives $$\langle \frac{1}{2},\frac{1}{2}| ~S_{x1}S_{x2}~ |\frac{1}{2},\frac{1}{2}\rangle = 0, \label{correlationterm}$$ and the sum for ${\cal J}_x$ reduces to Eq. \[xsumspin\].
We can understand the addition of angular momentum in the vector model: the components that are projections add up directly whereas the components that are fluctuations add up as noise. The noise can have different degrees of correlation as calculated by the last term in Eq. \[noisesxsum\]. The noise for the present state happens to be uncorrelated (Eq. \[correlationterm\]). The vectors for the individual spins are ${\cal S}_1={\cal S}_2=(1/2,1/2,1/2)$ and their sum gives ${\cal J}=(1/\sqrt{2},1/\sqrt{2},1)$ where we have added the $x$ and $y$ components in quadrature and the $z$ components directly. The magnitude square of the vector gives ${\cal J}^2=2$ in accordance with Eq. \[magnitudej\]. The result should be contrasted with a naive addition of the vectors ${\cal S}_1+{\cal S}_2=(1,1,1)$, that gives a magnitude square of 3. The case for the state with $j=1$ and $m=-1$ works the same way.
The state with $j=1$ and $m=0$ is more interesting. The state is the symmetric combination of the spins, $(|\frac{1}{2},-\frac{1}{2} \rangle + |-\frac{1}{2},\frac{1}{2} \rangle)/\sqrt{2}$. The vectors for the individual spins are ${\cal S}_1=(1/2,1/2,1/2)$ and ${\cal S}_2=(1/2,1/2,-1/2)$. We take the negative value of the square root in ${\cal S}_{2z}$ since the $z$ component of the two spins point in opposite directions. We choose ${\cal S}_{1z}$ (${\cal S}_{2z}$) positive (negative), but the opposite is equally correct. In the direct sum of the $z$ components ${\cal S}_{1z}$ and ${\cal S}_{2z}$ cancel each other giving 0. The correlation term in the $x$ component for this state gives $$\frac{1}{\sqrt{2}} \left( \langle \frac{1}{2},-\frac{1}{2}|+ \langle -\frac{1}{2},\frac{1}{2}| \right) ~S_{x1}S_{x2}~ \left( |\frac{1}{2},-\frac{1}{2}\rangle + |-\frac{1}{2},\frac{1}{2}\rangle \right) \frac{1}{\sqrt{2}} = \frac{1}{4}, \label{correlationterm2}$$ and the calculation for the fluctuations from Eq. \[noisesxsum\] gives $\Delta J_x^2=\Delta J_y^2=1$. The sum vector is ${\cal J}=(1,1,0)$ that gives the correct result for the magnitude square ${\cal J}^2=2$. The noise calculation for ${\cal J}_x$ and ${\cal J}_y$ tells us that we have perfectly correlated noise, so instead of adding the two contributions in quadrature, we add them directly (each one equal to 1/2 giving a total of 1). It is not unexpected that the noise behaves in a correlated manner since we use the symmetric combination of the spins.
Finally the case with $j=0$ and $m=0$. The state is the anti-symmetric combination of the spins and we expect the noise to be anti-correlated. The vectors for the indivicual spins are still ${\cal S}_1=(1/2,1/2,1/2)$ and ${\cal S}_2=(1/2,1/2,-1/2)$. The correlation term in the $x$ component for this state is now $$\frac{1}{\sqrt{2}} \left( \langle \frac{1}{2},-\frac{1}{2}|- \langle -\frac{1}{2},\frac{1}{2}| \right) ~S_{x1}S_{x2}~ \left( |\frac{1}{2},-\frac{1}{2}\rangle - |-\frac{1}{2},\frac{1}{2}\rangle \right) \frac{1}{\sqrt{2}} = -\frac{1}{4}, \label{correlationterm3}$$ and the calculation for the fluctuations from Eq. \[noisesxsum\] gives $\Delta J_x^2=\Delta J_y^2=0$. The sum vector is ${\cal J}=(0,0,0)$ with a magnitude square of ${\cal J}^2=0$ as expected. The anti-symmetric combination of the spins results in noise that is perfectly anti-correlated (due to the minus sign in the wave function). The noise subtracts directly ($\frac{1}{2}-\frac{1}{2}=0$) and not in quadrature for the $x$ and $y$ components. It seems that the noise in $J_x$, $J_y$ and $J_z$ is zero for the state. From the point of view of the sum, the individual perpendicular fluctuations are actually not zero, it is because of the correlations that the fluctuations of $J$ become zero.
\[general\]General case
=======================
Any other value of angular momentum can be constructed using the same scheme. For example, to obtain $j=3/2$ we add three spin 1/2 particles. Each particle contributes some amount to the value of ${\cal J}_z$ and also to the fluctuations in the perpendicular components. There is some degree of correlation between the spins depending on the $m$ value chosen. The correlation between spins can be calculated from the crossed term in Eq. \[noisesxsum\]. The correlation term between spins $i$ and $k$ in the x component for the state with angular momentum $j$ and projection $m$ is $$\langle j,m| ~S_{xi}S_{xk}~ |j,m \rangle. \label{correlationtermij}$$ It is not trivial to predict the result of this calculation except for the maximum and minimum projections. All the spins are uncorrelated if $m=j$ or $m=-j$. For any other projections there will be some intermediate degree of correlation between spins that can be calculated from Eq. \[correlationtermij\]. For the maximum projection, the $x$ (and $y$) components of all the individual spins add up in quadrature to give $${\cal J}_x=\sqrt{\Delta S_{x1}^2+\Delta S_{x2}^2+...+\Delta S_{x(2j)}^2}=\sqrt{2j(1/4)}=\sqrt{j/2}. \label{generalsum}$$ The vector sum is ${\cal J}=(\sqrt{j/2},\sqrt{j/2},j)$ with a magnitude square ${\cal J}^2=j/2+j/2+j^2=j(j+1)$, where the 1 comes from the perpendicular components.
\[conclusions\]Conclusions
==========================
We explain the 1 in the expression $j(j+1)$ in terms of the quantum fluctuations of the $x$ and $y$ components of the angular momentum. We include the fluctuations to describe the addition of angular momenta in the vector model. The vector components can be projections or fluctuations and they have different formulas for addition. The correlations in the fluctuations cannot be ignored. Formula \[magnitudej\] tells us that angular momentum does not come in units of $\hbar$, but instead it comes in units of $\sqrt{1+(1/j)}\hbar$. This is not even a uniform unit, but depends on the value of $j$ in a complicated way. This happens because some of the components of $J$ add up directly and others in quadrature. Only in the limit of big $j$ we recover the well known $\hbar$ unit of angular momentum. At small $j$ the quantum noise cannot be ignored.
I would like to thank A. P[é]{}rez, E. Ugalde and J. Ur[í]{}as for helpful discussions.
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abstract: 'We study deposition dynamics of Na and Na$_2$ on an Ar substrate, both species neutral as well as charged. The system is modeled by a hierarchical approach describing the Na valence electrons by time-dependent density-functional theory while Na core, Ar atoms and their dynamical polarizability are treated by molecular dynamics. We explore effects of Na charge and initial kinetic energy of the impinging Na system. We find that neutral Na is captured into a loosely bound adsorbate state for sufficiently low impact energy. The charged monomers are more efficiently captured and the cation Na$^+$ even penetrates the surface layer. For charged dimers, we come to different final configurations depending on the process, direct deposit of Na$_2^+$ as a whole, or sequential deposit. In any case, charge dramatically amplifies the excitation of the matrix, in particular at the side of the Ar dipoles. The presence of a charge also enhances the binding to the surface and favours accumulation of larger compounds.'
address:
- '$^a$Laboratoire de Physique Théorique, Université Paul Sabatier, CNRS, F-31062 Toulouse Cédex, France'
- '$^b$Institut f[ü]{}r Theoretische Physik, Universit[ä]{}t Erlangen, D-91058 Erlangen, Germany'
author:
- 'P. M. Dinh$^a$'
- ', F. Fehrer$^b$, P.-G. Reinhard$^b$, E. Suraud$^a$'
title: 'Deposition dynamics of Na monomers and dimers on an Ar(001) substrate'
---
TDDFT ,hierarchical approach ,deposition dynamics ,rare gas surface 31.15.ee ,31.70.Hq ,34.35.+a ,36.40.Wa ,61.46.Bc
Introduction
============
Clusters on surfaces are a much studied subject due to its interesting perspectives for basic research and for applications to nano-structured materials [@Hab94b; @Bin01]. One important aspect is here the synthesis of deposited clusters. Two different techniques have been developed, namely controlled growth of elementary units on a surface by molecular beam epitaxy (for a brief review, see e.g. [@Bru00]) or direct deposition of size-selected clusters on a substrate (see e.g. [@Har00]). An interesting aspect also concerns a non-destructive deposition technique of metal clusters on metal surfaces that can be achieved by means of a thin rare gas film above the metal surface (see e.g. [@Lau05] and refs. therein). We take up this scenario and aim here at a theoretical study of deposition of Na on a rare gas surface. Thereby we concentrate on the first stages of growth, the capture of atoms and molecules with particular emphasis on charged projectiles.
The theoretical description of deposition dynamics employs predominantly classical molecular dynamics with effective atom-atom forces, see [@Xir02]. This was done, e.g., for the deposition dynamics of Cu clusters on metal [@Che94] or Ar [@Rat99] surfaces, and of Al or Au clusters on SiO$_2$ [@Tak01a]. That, however, does ignore possible effects from electronic degrees of freedom, as it can become crucial in metal clusters, and the more so if a finite net charge is involved. One then better uses models which take care of the electronic degrees of freedom. Fully detailed calculations have been undertaken, e.g., for the structure of small Na clusters on NaCl [@Hak96b] or the deposit dynamics of Pd clusters on a MgO substrate [@Mos02a]. But the expense for such fully fledged quantum simulations grows huge. These subtle models are hardly extendable to truly dynamical situations, to larger clusters or substrates, and to systematic explorations for broad variations of conditions. There thus exists a great manifold of approximations which aim at an affordable compromise between reliability and expense, often called quantum-mechanical-molecular-mechanical (QM/MM) models. They have been applied for instance to chromophores in bio-molecules [@Gre96a; @Tap07], surface physics [@Nas01a; @Inn06], materials physics [@Rub93; @Kur96; @Ler98; @Ler00], embedded molecules [@Sul05a] and ion channels of cell membranes [@Buc06]. We take up here a QM/MM modeling which was developed particularly for the combination of Na clusters with Ar substrate [@Dup96; @Ger04b; @Feh05a]. This method has already been successfully applied to deposition dynamics on finite Ar clusters [@Din07a] or on Ar surfaces [@Din07b]. The originality of this approach lies in the fact that the substrate polarizability is treated dynamically, a key aspect as soon as charged species are considered. In this paper, as stated above, we focus on the basic initial stages, that is, deposition of a single Na atom or a Na dimer. We study the effect of the initial kinetic energy given to the deposited system and of its charge. We aim at the observation of different possible energy thresholds between regimes of dynamical bouncing, binding or inclusion of the Na in the Ar matrix. We also compare the direct deposition of Na$_2$ with the sequential process where the system is deposited atom by atom, in the spirit of the technique of atomic layer epitaxy or deposition [@Sun90].
Model {#sec:model}
=====
We start with a very brief summary of the hierachical description of the combined NaAr system. We treat the metal atoms in full microscopic detail at the level of Time Dependent Local Density Appromixation (TDLDA) for the valence electrons, coupled to Molecular Dynamics (MD) for the ions. Details on the successful TDLDA-MD approach for free clusters can be found in [@Cal00; @Rei03a]. The substrate consists out of Ar atoms to which we associate classical degrees of freedom for position and dipole moment. The latter serves to take into account the dynamical polarizability of the substrate. The Ar atoms are coupled to the Na by long range polarization and some short range repulsion to account for the Pauli blocking of cluster electrons in the vicinity of the Ar cores. The model is calibrated to measured properties of typical Na-Ar systems. We refer the reader to [@Dup96; @Ger04b; @Feh05b] for a detailed description of the model.
The Ar(001) surface is modeled through six layers of 8$\times$8 Ar atoms. The atoms in the two lowest layers are frozen at bulk crystal positions. The layers are periodically repeated in both lateral directions, thus simulating bulk material in these two dimensions. The six-layer sample in vertical direction is finite but is sufficiently large. We have counterchecked that by repeating some calculations for eight layers (making 512 atoms). That did not make much a difference and the now dynamically free fifth and sixth layers did not acquire any sizeable amount of kinetic energy. Thus freezing them in the 384 atom sample is a good approximation, at least for qualitative purposes. The dynamics is initialized by placing the projectile (Na atom, ion or dimer) at a distance of 20 a$_0$ from the surface and boosting it with a given initial kinetic energy $E_0$, towards the substrate and along the direction (denoted by $z$ in the following) normal to it. We analyze the subsequent dynamics in terms of detailed ionic and atomic coordinates as well as of the various parts of the kinetic energy.
Dynamical deposition of neutral Na on Ar surface
================================================
As a first test case, we study the deposition of a neutral Na atom. Figure \[fig:NaArsurf\] shows results for three different impact energies. The atom is captured by the surface for all initial kinetic energies $E_0\leq 0.14$ eV. Above this value, after impact, the Na acquires a positive escaped velocity which does not change of sign with time later on. The projectile is thus reflected and the process turns into an inelastic collision. The threshold value of 0.14 eV looks low at first glance. It is, however, already larger than the energies of Ar binding (typically 0.05 eV) and of the NaAr dimer (0.005 eV). And even in the regime of capture, the atom can still have huge amplitudes in its first bouncing oscillations reaching far away from the surface which leaves these initial stages somewhat vulnerable against perturbations. Safe deposit with immediate binding would require very low impact energies. The Ar material is, in fact, rather repelling to one single neutral Na atom. The latter is just loosely tied to the surface and insertion inside is energetically much unfavourable. That changes for Na clusters. Already small clusters as Na$_6$ or Na$_8$ are tightly captured in a wide range of impact energies [@Din07a; @Din07b] and they are also favourably embedded deep inside Ar material [@Feh07c]. The difference stands in a larger polarizability of the cluster, due to the cooperative response of the valence electron cloud, while one single and tightly bound electron in the Na atom is too weak to develop a strong polarization. This difference shows the enormous importance of the polarization interaction in material combinations where metals and polarizable media are involved. We will see that again when considering charged projectiles farther below.
The reaction of the matrix seems weak when looking at the local positions in the left panels of figure \[fig:NaArsurf\]. Nonetheless, one can spot a faint sound wave propagating through the layers and some oscillations. A more telling view of energy transport is provided by the right panels of figure \[fig:NaArsurf\] which show the evolution of kinetic energies for various impact energies. The pattern are to some extent all similar. About half of the initial kinetic energy of the Na atom is very quickly transferred to Ar at first impact followed by a phase where kinetic energy is flowing away from the Na at a slower time scale. This second energy loss is due to the Na atom trying to escape against the attractive dipole force of the surface. A small fraction of the potential energy thus worked up is further transmitted to the kinetic energies of the Ar atom. After 2-3 ps, we have the typical result that almost all Na energy is transferred to the Ar substrate which seems to share it half and half into kinetic and potential energy. A quick note on the lowest right panel. It looks as if the Na atom had lost all its energy although the spatial picture (lowest left panel) shows final reflection. At second glance, we see that a small amount of kinetic energy remains steadily in the Na atom. It is obvious that just above threshold, we encounter a very inelastic collision.
Dynamical deposition of charged Na on Ar surface
================================================
The cation case
---------------
As clusters are usually manipulated as cations, it is especially interesting to consider the deposition problem with such charged species. We shall now consider this case in detail.
It should be noted that treating charged species requires a proper handling of the surface degrees of freedom. Accounting for the polarizability of Ar atoms dynamically, as we do in our model (see section \[sec:model\]), is here a crucial ingredient. It will thus also be interesting to look at how Ar dipoles respond to a deposition. A simple measure for this effect is the excitation energy of the dipoles which, in our model, scales with the square of the Ar dipole amplitudes. [[ When depositing]{}]{} a neutral Na (cf. figure \[fig:NaArsurf\]), the dipole excitation energy turns out to be vanishingly small, whatever the deposition energy. Thus we did not show it on the figure. Still, the effect of dipoles is known to be decisive, even at low energies, see for example in the analysis of optical properties of embedded metal clusters [@Feh05a].
### Time evolution of positions
[[ We first view the deposition process in real space and plot again the $z$ coordinates as a function of time for the deposition of Na$^+$ on Ar(100). [[ Figure \[fig:napatoms\] shows results for]{}]{} two typical deposition energies, one below and one above deposition threshold. ]{}]{} The lowest energy [[ would correspond for the neutral Na atom to a situation]{}]{} just below threshold for capture (compare with middle panel in figure \[fig:NaArsurf\]). The charge of the Na$^+$ enhances the attraction to the surface due to the Ar polarizability. This leads in the incoming stage to a much larger acceleration of the Na$^+$ ion towards the surface as compared with the neutral Na atom. But the now much larger kinetic energy at contact time (about 1 ps) does not cause immediate reflection. The large attraction enhances, in fact, capture. The ion uses its high kinetic energy at impact to overcome the short range repulsion of the Ar atoms and penetrates the first layer. It is then caught, after some oscillations forth and back, between first and second layers. In the first bounce back at around 3 ps, it makes space by kicking one Ar atom out of its position. With the creation of this vacancy, some rearrangements occur in the highest layers at a very slow time scale such that finally the Na$^+$ resides between first and second layers while one Ar atom is shifted out of the surface to what could be called the next upper Ar layer. There is however some uncertainty about the final fate of this atom. Indeed, although the way to equilibration is visible in the time evolution of the matrix kinetic energy (see top panel of figure \[fig:napekin\], full thick line), the matrix is surely not thermalized yet. At 30 ps, the adatom still looks like an atom loosely adsorbed on the surface and will very probably remain so. But there is no guarantee that it does not finally escape by thermal agitation. Any weak external perturbation may destabilize that adsorbate.
[[ The higher energy case, presented in the bottom panel of figure \[fig:napatoms\], displays reflection. In that case, the Na$^+$ is quickly ejected together with a few nearby Ar atoms. The remaining surface atoms accomodate the perturbation in a way similar to the lower energy case,]{}]{} [[ however with somewhat larger oscillations. ]{}]{}
### Time evolution of energies
The evolution of atomic and ionic kinetic energies is plotted in the top panel of figure \[fig:napekin\]. It again shows first a large initial acceleration of the charged projectile caused by the attractive polarization interaction with the substrate. This effect is much larger than in the previous example with a neutral projectile. Much similar to the previous case is the immediate and large energy transfer to the substrate at the time of closest impact. The figure does now also show the kinetic energy of the Ar dipoles (see bottom panel).
[[ Let us first discuss the low energy case.]{}]{} For the charged projectile, the energies of atoms and dipoles are of the same order of magnitude : For Ar atoms, about two thirds of the ion kinetic energy at impact ($\simeq 0.9$ eV), while the Ar dipoles get an excitation energy up to about 1 eV. The energy gathered by the substrate comes from three sources : [*i)*]{} From direct conversion of the ion kinetic energy, [*ii)*]{} from release of potential energy due to deformation and rearrangments of the matrix, and [*iii)*]{} from the electrostatic influence of the inclusion of a positive charge into the matrix. Later, the remaining Na ion kinetic energy is quickly and almost completely taken up by the matrix in the tight bounces between the layers. It is remarkable to observe that the excitation of the matrix is as high for the atoms as for the dipoles. This demonstrates that the dynamic of dipole polarization plays a crucial role in the process and that a theoretical description has properly to take that into account. Omitting these degrees of freedom changes the deposition process completely, as has been also shown in the case of Na clusters [@Din07b].
[[ The higher energy case displays another interesting feature. At first glance, the dipole energy vanishes asymptotically, [[ different from]{}]{} the low energy case. In fact, a closer look at the curve shows that this energy reaches temporarily very high values, before the few Ar atoms are emitted. It is thus likely that the few emitted atoms were precisely the ones which had the largest dipoles. A more detailed inspection of the]{}]{} [[ spatial distribution of dipole energies confirms that. ]{}]{}
### Systematics in deposition energy
[[ It is finally instructive to [[ analyze the trends of]{}]{} deposition dynamics by considering the energetics as function of the initial kinetic energy of the projectile $E_0$. [[ Figure \[fig:napsystemat\] shows]{}]{} the asymptotic kinetic energies of Na$^+$, Ar atoms and excitation energy of Ar dipoles as a function of $E_0$. ]{}]{} [[ The pattern indicate a dramatic change around $E_0\approx 2.7$ eV, which is the transition point from capture to reflection of the impinging Na$^+$. Above that critical energy, the outgoing kinetic energy of the Na$^+$ increases linearly with $E_0$. The same holds for the energy of the Ar atoms because it is then dominated by the few atoms which are accompagning the departing ion.]{}]{} Below the threshold, these quantities also show a monotonous increase with impact energy. [[ Most remarkable is the behaviour of the Ar dipoles which differs essentially from the two other quantities. [[ Below threshold, when]{}]{} Na$^+$ ion is [[ captured by]{}]{} the substrate, the Ar dipoles acquire an energy which seems to depend only on the net charge and not on the [[ initial kinetic]{}]{} energy. [[ Above threshold, the Ar dipole energy vanishes because]{}]{} the Na$^+$ finally escapes from the surface. Comparing with the bottom panel of figure \[fig:napekin\], sharp peaks in the Ar dipole energy appear precisely when the Na$^+$ is in the vicinity of the surface and later on, the excitation energy rapidly decreases towards zero as the Na$^+$ moves away. ]{}]{}
A final word has to be added about the reflection threshold we found around 2.7 eV. In the bottom panel of figure \[fig:napatoms\], we clearly notice the bouncing of the propagating wave in the Ar substrate at the level of the fifth layer. We recall that this layer, as well as the sixth layer, are fixed, for reasons of computational [[ expense. The value for the reflection threshold, found here at $E_0 \approx 2.7$ eV, has thus to be taken with care. We checked the case of deposition of Na$^+$ with precisely this initial kinetic enery on Ar$_{512}$ (six active layers plus two fixed ones) rather than Ar$_{384}$. Here the projectile is still captured and reflection emerges at higher initial energies. However, the qualitative features of deposition dynamics and trends remain the same although the threshold value is somewhat shifted. ]{}]{}
The anion case
--------------
The other choice for a charged monomer is a Na$^-$ anion. Figure \[fig:NamAr384\] shows the result, this time again at the threshold energy for the neutral case, $E_0 = 0.136$ eV. The attraction for polarization potentials is as large as it was for the Na$^+$ cation. But the doubly charged electron cloud experiences a full load of the Pauli repulsion from the Ar cores which is built into the short-range part of the effective electron-Ar interaction. As a consequence, the Na$^-$ is blocked by the surface and will not penetrate into the material. On the other hand, the long-range attraction persists. Thus the anion is very efficiently captured at a safe distance from the surface. In comparison with the neutral case, there are no noteworthy amplitudes in the bouncing oscillations. Thus the anion comes quickly to a rest. In comparison to the cation case, the impact phase is much earlier stopped such that the anion could not acquire so much kinetic energy. This, in turn, leaves less energy to be absorbed by the substrate. The pattern of energy transport in the Ar atoms are similar to the neutral case : Half of the energy is immediately transferred at first impact and the other bits with each bounce. However, because of the negative charge, the Ar dipoles experience a much larger excitation and acquire a kinetic energy five or six orders of magnitude higher than in the case of the neutral Na deposition. Note finally that the time scale is also different because the bounces recur much more frequently. For both charged cases, we mention that the threshold for capture is much higher than for the neutral cluster (not shown). And there is actually no regime of inelastic reflections. Enhancing the impact energy further to force reflection leads into a regime where a whole surface area is destroyed. This is similar to our findings for deposition of Na clusters [@Din07b].
A simple analysis of the results
================================
[[ It is, finally, instructive to interpret the above observed findings in terms of energy surfaces. To that end, we have computed the energy of Na, Na$^+$ or Na$^-$ for systematically varied distances to the surface, keeping the distance and the atomic positions frozen while allowing the polarizabilities to adjust to the given configurations. This will provide an estimate of how far or close we are to (a)diabaticity in the deposition scenarios explored above]{}]{}. [[ The energy surfaces for Na, Na$^+$ and Na$^-$ on Ar(001) are plotted in figure \[fig:bo\]. The “static” results are qualitatively compatible with]{}]{} [[ our fully dynamical calculations.]{}]{} Let us, for example, take the case of neutral Na. A faint minimum is found at a distance of about 7.5 $a_0$ with a binding energy of $-52$ meV. This has to be compared with the threshold for reflection at 136 meV that we found in our dynamical calculations. This is 2–3 times larger than the “static” value but the orders of magnitude are similar. The same qualitative conclusion holds true for Na$^+$ and Na$^-$.
[[ At a quantitative level, one has to note that there remains sizable differences between the static and dynamical results. This is an expected and welcome feature showing that, at least in the deposition energy range we consider, there remain genuine dynamical effects which cannot be simply evaluated without a proper account of dynamics. The point was rather obvious from the behavior of Ar atoms and dipoles as is particularly clear from figures \[fig:napatoms\] and \[fig:napekin\]. ]{}]{}
The site dependence on the deposition has also been explored in this “static” way. The dynamical results presented in the previous sections all start from a projectile initially positioned above a hollow site in the first layer, but above on Ar atom in the second layer. We checked that changing the deposition site of course modifies the threshold in initial kinetic energy $E_0$ for the observation of the reflection. This, however, does not change qualitatively our findings. This is also compatible with the results on deposition of Na$_6$ on Ar clusters [@Din07a] and Ar surface [@Din07b], which show only weak dependence on implantation site.
Dynamical deposition of Na dimer
================================
Not surprisingly, the effect of the charge is thus determinant for the fate of the deposited atom. In the same spirit, we have compared the cases for the dimers, Na$_2$ and Na$_2^+$. The results are shown in the top panels of figure \[fig:Na2\]. As for a single atom, the charged dimer sticks more closely to the Ar surface than the neutral Na$_2$. Note that in both cases the dimers are not strongly perturbed. More precisely, the dimer bond lengths exhibit some oscillations but remain almost unchanged as compared to the free value : the Na$_2^+$ is slightly longer ($+2$ %), while the Na$_2$ bond length decreases by 4.5 %. The matrix shows more perturbation than in the previous cases with a single Na projectile. Two atoms have simply more impact which is fully downloaded into the matrix. Comparing the two dimers, we see that Na$_2^+$ attaches more tightly to the Ar surface which is, again, due to the larger attraction. However, the positively charged Na$_2^+$ stays above the surface and does not manage to dive below as the Na$^+$ did.
There is an alternative option to bring a dimer onto the surface, that is, sequential deposit of monomers. Particularly interesting is here the case where a Na$^+$ cation was deposited first and where, in a second round, a neutral Na atom is attached. The left lower panel of figure \[fig:Na2\] shows that process. To produce the corresponding initial state, we take the final state of Na$^+$ deposition (see top panel of figure \[fig:napatoms\]), remove the Ar adatom, relax the remaining configuration by cooling, and inject a neutral Na atom a distance of 15 a$_0$ from the surface with our meanwhile standard impact energy of 0.14 eV. The new Na atom is captured with small remaining oscillations. Compared with the direct deposition of Na$_2^+$ (top left panel of same figure), the charged dimer is more deeply bound with one leg residing below the surface. We also observe that its bond length is almost unchanged ($-0.23$ %) with respect to the value of the free charged dimer. The example thus demonstrates that the final state can depend sensitively on the production process.
One could now hope that neutralization of that immersed Na$_2^+$ leads to an equally deep bound Na$_2$ dimer. To that end, we take the final state from the previous sequential deposit (at the end time in the lower left panel), cool the obtained configuration, and add an electron in the electronic ground state of the tied dimer with yet fixed ionic configuration. Then we release the system to fully free electronic, ionic, and atomic dynamics. The result is shown in the right lower panel of figure \[fig:Na2\]. The now neutral dimer pops up out of the surface and performs bouncing oscillations with large amplitude about the final stage of Na$_2$Ar$_{383}$ which was also obtained by direct deposit of Na$_2$, see upper right panel. The minor difference with the now missing adatom plays little role for the comparison. There are fast oscillations of the dimer bond length. These emerge because the neutral dimer has a smaller bond length than the charged one from which it was started. Note that these bond-length oscillations persist for long. This indicates that energy transfer from intrinsic ionic motion to the substrate is very slow, a feature which was also observed for larger clusters in Ar substrate [@Feh05b].
In order to quantify the dynamics of dimer deposition in simple terms, we have plotted in figure \[fig:Na2\_analysis\] their center-of-mass (c.o.m.) $z$ coordinate as a function of time. This complements the previous, more detailed, figures and allows a more direct comparison between the various cases. The lower panel shows direct deposit of neutral dimer as well as charged dimer and sequential deposit leading to a charged dimer. The trend is obvious : Binding stays 28 % closer to the surface for the charged dimer compared with the neutral one, and sequential deposit brings it even closer (of about 60 %) so that the dimers center lies almost at the surface. For completeness, we also show in the upper panel the evolution of the re-neutralized charged dimer. The return to the equilibrium position of the directly deposited neutral dimer is visible as well as the still large oscillations about that point. Significant energy transfer happens only at the bouncing points and the long time span per bounce lets us predict a very slow relaxation needing about hundreds of ps.
Conclusion
==========
To conclude this paper, we have studied deposition of Na atoms, Na ions and Na dimers on Ar(001) substrate, using a hierarchical approach with time-dependent density-functional theory for the Na electrons coupled to molecular dynamics for the Na ions and Ar atoms as well as dipole moments. We have paid particular attention to the effect of charged projectiles. We have found that the neutral Na is not likely to penetrate into the Ar matrix and sticks loosely to the Ar surface for initial kinetic energy lower than 0.14 eV while it is inelastically reflected for larger energies. A Na$^+$ cation behaves much differently. It is tightly captured and even penetrates the surface to reside finally between surface and next layer. The Ar surface undergoes strong pertubations and displaces one Ar atom to an adatom site. [[ At a given size of Ar substrate, we found a reflection threshold twenty times larger for Na$^+$ than for neutral Na.]{}]{} Different is the behavior for the negatively charged Na$^-$ anion. It is also tightly bound. But the strong electron-Ar repulsion keeps it safely above the surface. The deposition of Na dimers shows similar trends as for the atom. The charged dimer is closer bound than the neutral one. It stays, however, fully outside the substrate. The alternative process of sequential deposit produces a different final state for the charged Na$_2^+$ dimer. The lower ion of the dimer is placed now below the surface while the upper one stays just above. The final state obviously depends on the pathway of the process. It was, however, not possible to keep a neutral dimer in that close contact with the surface. Re-neutralizing the close Na$_2^+$ configuration leads back to the Na$_2$ outside the surface at a distance which was obtained also by direct deposit of Na$_2$. After all, the results show that charge makes a huge difference in connection with polarizable media as, e.g., Ar substrate. It acts to some extent as a catalyst for capture. The studies will be continued with larger samples to explore different scenarios for producing deposited clusters.
Acknowledgments: This work was supported by the DFG, project nr. RE 322/10-1, the French-German exchange program PROCOPE nr. 07523TE, the CNRS Programme “Matériaux” (CPR-ISMIR), Institut Universitaire de France, the Humboldt foundation, a Gay-Lussac price, and the French computational facilities CalMip (Calcul en Midi-Pyrénées), IDRIS and CINES.
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abstract: 'We point out the high physical correctness of the use and the concept of the crystal-field approach, even if is used to metallic magnetic materials of transition-metal 3d/4f/5f compounds. We discuss the place of the crystal-field theory in modern solid-state physics and we point out the necessity to consider the crystal-field approach with the spin-orbit coupling and strong electron correlations, as a contrast to the single-electron version of the crystal field customarily used for 3d electrons. We have extended the strongly-correlated crystal-field theory to a Quantum Atomistic Solid-State Theory (QUASST) to account for the translational symmetry and inter-site spin-dependent interactions indispensable for formation of magnetically-ordered state. We have correlated macroscopic magnetic and electronic properties with the atomic-scale electronic structure for ErNi$_5$, UPd$_2$Al$_3$, FeBr$_2$, LaCoO$_3$ and LaMnO$_3$. In QUASST we have made unification of 3d and rare-earth compounds in description of the low-energy electronic structures and magnetism of open 3d-/4f-/5f-shell electrons. QUASST offers consistent description of zero-temperature properties and thermodynamic properties of 4f-/5f-/3d-atom containing compounds. Our studies indicate that it is the highest time to unquench the orbital magnetism in 3d oxides.'
author:
- 'R. J. Radwanski'
- 'Z. Ropka'
title: 'On the crystal field in the modern solid-state theory$^\spadesuit$'
---
Introduction
============
The use and the concept of the crystal-field approach, if used to metallic magnetic materials, has been recognized as erroneous by the Highest Scientific Council of the Polish Government (CK ds SiTN, later in short CK) in its decision BCK-V-O-819/03 on 31.05.2004. The full decision, in Polish and in part in English, is available on www.css-physics.edu.pl.. This reproach we denote as No 1. Such a formulation of the reproach should not be read that this Council agrees that the use and the concept of the crystal-field approach, if used to nonmetallic magnetic materials, is correct. The above decision is somehow in a common line of depreciation of the crystal field (CEF) theory in the modern solid-state theories. Let mention, that a recently edited (2003) book of Mohn “ Magnetism in the solid state” \[1\], being an overview of presently-in-fashion magnetic theories, mentions only one position on the localized magnetism: a book of Van Vleck from 1932 \[2\] (apart of this shortage the book is very nice and we highly recommend it). Diagrams of Tanabe-Sugano \[3\], known already 50 years, are not exploited in the modern solid-state physics theories for description of 3d-ion compounds and the orbital magnetic moment only recently starts to draw the proper attention. Theoretical approaches yielding continuous wide 1-10 eV energy bands for 3d/4f/5f states overwhelm the present solid-state theory. On other side the CEF approach, yielding the discrete electronic structure for 3d/4f/5f electrons with details at least 1000 times smaller, below 1 meV, is often recalled by experimentalists in order to analyze obtained experimental results. Thus, one can say that there is at present a large gap between theory and experiment in description of 3d/4f/5f states.
In this paper we would like to discuss the place of the crystal-field theory in the modern solid-state physics, to clarify our understanding of the crystal field approach and to inform about the administrative interference to Physics in judging the physical correctness, rather incorrectness, of the crystal-field theory to metallic magnetic materials. We claim that the simplest and most natural theoretical approach, as the CEF theory is, has not been exploited enough for showing its physical adequacy and its total theoretical rejection is premature.
We by years openly formulate the need of taking into account the crystal field in description of 3d/4f/5f compounds, even these exhibiting the heavy-fermion phenomena (mainly 4f or 5f compounds) and insulating 3d oxides. In order to avoid undeserved critics we do not claim that CEF explains everything but we claim that CEF effects should be clarified at first (properly!!!) in any analysis of physical properties of any transition-metal compound. Also we do not claim to invent the crystal-field theory - we point out its importance in the specific social conditions of the end of the XX and the beginning of the XXI century, when the crystal-field theory is somehow prohibited and rejected from magnetic theories.
The biggest problem in this discussion is related to a fact that the crystal-field theory has within the magnetic community in last 30 years a special place - being continuously rejected from the scientific life permanently appears as an unavoided approach for explanation of properties of real compounds. The crystal-field theory is in the modern solid-state theory like an unwilling child, 75 years old already.
Further administrative details
==============================
The above mentioned objection was the only one scientific reproach in the administrative decision CK-04 towards the disqualification of the scientific activity of R. J. Radwanski. These scientific achievements account for a date of 7 May 2001 126 publications in international journals from the SCI philadelphia list and two promoted doctors. The full list of these publications is available on www.css-physics.edu.pl.. There are also further 19 publications to 31.12.2004. See also 51 internet papers in ArXiv/cond-mat. In an earlier decision BCK-V-P-1262/02 of 24.03.2003, denoted latter as CK-03, CK has formulated another reproach, No 2, that “the used by Radwanski crystal-field approach is oversimplified, and the agreement of calculations with experiments is accidental.”
These two decisions have been undertaken by means of opinions with the negative conclusion of Prof. Prof. H. Szymczak (November 2001), J. Sznajd (February 2003), Prof. A. M. Oles (December 2003) and of J. Klamut (April 2004). For the final decision two last opinions have been crucial. All of the referees belong to the best polish solid-state physicists and magneticians, so such opinions deserve on the serious attention by the magnetic community, not only polish but the international one. Opinions with the positive conclusion of Prof. Prof. K. Krop (October 2001), R. Micnas (April 2002) and K. Wysokinski (December 2002) have been found insufficient, in light of four negative opinions, to provide evidence for substantial scientific achievements required by the Polish law.
Crystal field in modern solid-state physics and its extension to Quantum Atomistic Solid State Theory
=====================================================================================================
These decisions and objections become a part of the long-lasting world discussion going on about the use and the applicability of the CEF approach. This discussion lasts already 75 years as the CEF theory has been started in 1929 by Bethe and followed by Kramers, Van Vleck and many, many others. Despite of 75 years and in meantime (1936-1938) formulation of the competitive band theory there is no consensus within the modern solid-state physics on the description of compounds containing transition-metal atoms with open 3d/4f/5f shells. These compounds exhibit so exciting phenomena like heavy-fermion behaviour at low temperatures and unexpected, in frame of band models, insulating ground state of 3d monoxides. In such scientific circumstances the decision of CK disqualifying the crystal-field theory is, according to us, premature and simply harmful to Physics. The most important is a fact, that important polish physicists by means of CK like to solve a serious scientific problem by means of the administration decision. From such a point of view this decision is a curious one as the European civilization already 370 years ago has learnt that no administrative inquisition-like decision, even of the highest level, can solve any scientific problem. We add that nobody has questioned in a scientific way anyone of our published papers!!! We admit that we suffer often an unscientific treatment of our submissions by Editors, who often find them simply not suitable without a clear scientific formulation of objections. We are sure, however, that a good science will always win, i.e. will show its physical adequacy and the conceptual fertility, and we also know from the history of science that good theories suffer often seriously for a pretty long time. Thus we continue our studies by more than 20 years and we are optimists. As violation of scientific rules we presume the rejection to publish a Comment, that corrects a recently published paper. In a consequence, for instance, oversimplified electronic structures of 3d ions, without strong correlations and without the spin-orbit coupling, still appear in Phys. Rev. Lett. and Phys. Rev. B despite of our (not suitable) submission “Relativistic effects in the electronic structure for 3d paramagnetic ions” PRL-LS 6925 from 1997 (available at ArXiv cond-mat/9907140).
Theoretical hypothesis of our 20 years research can be formulated as: macroscopic properties of compounds containing open-shell 3d/4f/5f atoms are predominantly determined by the low-energy discrete electronic structure, with separations below 1 meV. These states originate from atomic-like energy states of 3d/4f/5f ions. For description of these atomic-like states the local surroundings, crystal-field, spin-orbit and strong intra-atomic correlations have to be taken into account. As for description of a crystalline solid it is necessary to consider at least the translational symmetry and inter-site interactions, in particular spin-dependent interactions indispensable for formation of magnetically-ordered state, we have extended the CEF theory to an Quantum Atomistic Solid State Theory (QUASST) [@4; @5]. The CEF theory is, however, the basic ingredient of QUASST, particularly important for the physical understanding and the overall scientific paradigm. Coming out with QUASST we would like to skip somehow the crystal-field theory that has got a negative meaning in the solid-state physics, becoming a synonym of the oversimplified point-charge model. We hope that magnetic theoreticians give some credit for QUASST to allow showing its applicability and usefulness for understanding of transition-metal compounds.
The crystal field gives explanation for the physical origin of the observed low-energy electronic structure, yielding their nature and a well-defined number. Strong intra-atomic correlations assure that these states are describable for an atom being the full part of a solid like in the free ion. It means that in QUASST we assume that the atomic-like integrity is preserved even when the given atom becomes the full part of a solid. It is a very strong assumption but taking into account that it is based on the generally-accepted concept of the atomistic construction of matter surely is worth to be thoroughly studied. We are consequently doing it in the Center of Solid State Physics in Krakow. The valency of the atom in a solid depends on the partner(s) and the stoichiometry. It can be 3+ in case of Pr$_2$O$_3$, but 4+ for PrO$_2$. In metallic PrNi$_5$, without judging the formal stoichiometry of Pr and Ni atoms, the observed discrete electronic structure turns out to be related to the 4f$^2$ configuration occurring formally in the Pr$^{3+}$ ion.
The CEF approach and QUASST in conventional 4$\textrm{f}$
=========================================================
Among others Radwanski and Franse in years 1984-1995 has put a substantial contribution to show the physical adequacy of the CEF approach to conventional 4f compounds, like Ho$_2$Co$_{17}$, Dy$_2$Co$_{17}$, Nd$_2$Fe$_{14}$B, ErNi$_5$, DyNi$_5$, NdNi$_5$, PrNi$_5$, .. All of them are metallic. All are magnetic, apart of PrNi$_5$ down to 1 K. By physical adequacy we understand a highly consistent description of physical properties. Let focus on (anisotropic) magnetic properties of all above mentioned compounds. For it we correlated macroscopic properties, like value of the magnetic moment and its direction in the crystal, with atomic-scale properties like localized states with (low-)energies and eigenfunctions. By it we could prove that the observed huge anisotropy is predominantly of the single-ion origin. The derived CEF-like electronic structure from high-field magnetization measurements have been later positively verified by specific heat measurements [@6]. A conical structure of Nd$_2$Fe$_{14}$B below 140 K has been nicely described within the CEF theory revealing the importance of higher-order CEF interactions [@7]. The importance of higher-order CEF interactions is manifest again in the first-order metamagnetic transition at 17 T.
The model analysis of the overall temperature dependence of the specific heat is shown in Fig. 1. Fig. 2 shows the fine electronic structure of the 4f$^{11}$ configuration (the Er$^{3+}$ ion) associated with the $^4$I$_{15/2}$ ground multiplet. The perfect description both in magnetic and paramagnetic state with the $\lambda$ peak at T$_c$ should be noted. Concluding ErNi$_5$ we say that in metallic magnetic compound coexist localized electrons having discrete states with conduction electrons originating from outer shells of Er and Ni. Magnetic and electronic properties are predominantly governed by localized electrons with states determined by CEF interactions.
The CEF theory at the start points two things. One, that a solid is not a homogeneous jellium (magma) but there exists varied in space the electrostatic potential obviously due to charge polarized atoms and electrons (the simplest version is a charge point ionic model). Secondly, a 3d/4f/5f paramagnetic atom serves as an atomic-scale agent to probe this potential. The CEF theory points out the multipolar character of this electrostatic potential. It is reflected in subsequent orders of CEF parameters (quadrupolar - B$_2$$^0$, B$_2$$^2$ parameters; octupolar - B$_4$$^0$, B$_4$$^4$, …, dodehexapolar B$_6$$^0$, B$_6$$^6$, ..). For a paramagnetic ion this multipolar potential causes the splitting of its ionic electronic structure. This splitting is a hallmark of the CEF theory. This splitting in case of 4f compounds is surprisingly well describable making use of the total angular momentum quantum number J as the good quantum number. Actually, we should work with the all-term electronic structure instead of the one, Hund’s rule, ground multiplet only. The successful approximation with only one multiplet is due to the strong spin-orbit coupling that causes the excited multiplet to lie at least 0.3 eV above the ground multiplet preventing its substantial thermal population at, say, room temperature. Energies of this electronic structure can be later verified by, for instance, specific heat measurements and by spectroscopic measurements using inelastic neutron scatterings. The eigenfunction of the ground state bears information about the magnetic moment, its value and the direction, a fact that we strongly employ in our studies.
As a strong confirmation of the CEF approach we take the possibility of prediction of magnetic properties of the isostructural compound with another rare-earth atom. Using the single-ion scaling we had predicted in 1986, for instance, a value of the transition field of 26 T for Dy$_2$Co$_{17}$ basing on 19 T for Ho$_2$Co$_{17}$. In years 1991-1995 a remarkably consistent description within the CEF approach has been obtained for the RNi$_5$ series, both zero-temperature properties and thermodynamics. All of the above mentioned compounds, except PrNi$_5$, are magnetically ordered. Thus, the calling the applied approach as the CEF approach is only a nick-name pointing out the fundamental role of CEF states for the magnetic and electronic properties. Of course, a magnetic order cannot be obtained within the purely CEF approach. However, we know what happens to CEF states when the magnetic order is formed. The magnetic state develops on the CEF states. In fact, all of the analysis of RNi$_5$ compounds illustrate the action of the QUASST theory.
Sub-Conclusion: there is wide experimental evidence for the existence of CEF states in rare-earth (4f) compounds, both metallic and ionic. From our studies of ionic compounds we can mention Nd$_2$CuO$_4$ and ErBa$_2$CuO$_7$.
Extension of the CEF approach to actinides (5$\textrm{f}$ compounds)
====================================================================
Just after the first experimental results on newly discovered in group of Prof. Frank Steglich heavy-fermion metal UPd$_2$Al$_3$ Radwanski and Franse in 1992 have described the specific heat, from 4 to 300 K, as related to the 5f$^3$ (U$^{3+}$) configuration. We have managed to describe the overall temperature dependence with a Schoottky-like peak at 50 K and a $\lambda$-type peak related to the antiferromagnetic state formed at T$_N$ of 14 K. This energy level scheme has been confirmed by INS experiment of Krimmel/Steglich in 1996 [@9] as we pointed out in year of 2000 [@10]. The observation of well-defined localized CEF excitations in heavy-fermion metal UPd$_2$Al$_3$ we take as great confirmation of our atomistic approach. With great pleasure we have noted in year of 2001 a change of mind of Fulde and Zwicknagl from the itinerant picture for all f electrons to a dual model with two fully localized f electrons [@11]. Two or three localized electrons we treat as a minor problem, because the main theoretical difference is related to the itinerant or localized point of view. For the scientific honesty we have to mention that the problem of localized states in UPd$_2$Al$_3$ is not yet over - another German group of Lander with coworkers quite recently claim that there is no evidence for the localized states in UPd$_2$Al$_3$ [@12]. Just after appearance of these doubts we again clearly defined our point of view and our interpretation with the 5f$^3$ configuration [@13].
We prefer 3 f electrons owing to the intrinsic dynamics of the Kramers system, states of which are established by the atomic physics (in particular the number of states and their many-electron atomic-like nature). We can add that no one succeeded in description of the observed transitions and other properties to the 5f$^2$ configuration with the similar consistency to ours.
The derived electronic structure accounts, apart of the INS excitations, also surprisingly well for the overall temperature dependence of the heat capacity, the substantial uranium magnetic moment and its direction. We make use of a single-ion like Hamiltonian, the same as has been used for ErNi$_5$, for the ground multiplet J=9/2 [@6; @16]:
$H=H_{CF}+H_{f-f}=\sum \sum B_n^mO_n^m+n_{RR}g^2\mu _B^2\left(
-J\left\langle J\right\rangle +\frac 12\left\langle J\right\rangle
^2\right) $
The first term is the crystal-field Hamiltonian. The second term takes into account intersite spin-dependent interactions (n$_{RR}$ - molecular field coefficient, $g$=11/8 - Lande factor) that produce the magnetic order below T$_N$ what is seen in Fig. 3 as the appearance of the splitting of the Kramers doublets and in experiment as the $\lambda$-peak in the heat capacity at T$_{N}$.
The splitting energy between two conjugate Kramers ground state agrees surprisingly well, both the value of the energy and its temperature dependence, to a low-energy excitation of 1.7 meV at T=0 K observed by Sato [*et al.*]{} [@14; @15], which has been attributed by them to a magnetic exciton. Thus, we are convinced that the 5f$^{3} $(U$^{3+}$) scheme provides a clear physical explanation for the 1.7 meV excitation (magnetic exciton) - this excitation is associated to the removal of the Kramers-doublet ground state degeneracy in the antiferromagnetic state.
For actinides we also should mention the consistent description (5f$^3$, U$^{3+}$) of a ferromagnetic metal UGa$_2$, T$_c$ of 125 K, both zero-temperature properties (magnetic moment of 2.7 $\mu_B$ lying in the hexagonal plane, T$_c$=125 K) and thermodynamics (temperature dependence of the specific heat and of anisotropic paramagnetic susceptibility) [@16]. Later this description has been extended to NpGa$_2$ (5f$^4$, Np$^{3+}$), isostructural easy axis ferromagnet properties of which has been described using the single-ion correlation (Stevens factors) [@17].
Recently in autumn of 2003 a well-defined localized excitation has been observed in heavy-fermion metal YbRh$_2$Si$_2$. Sichelschmidt from F. Steglich group has managed to observe in this heavy-fermion metal at temperature T=1.5 K an ESR signal typical for the localized Yb$^{3+}$ ion [@18]. The observation of the ESR signal is a large surprise as YbRh$_2$Si$_2$ was regarded as a prominent heavy-fermion metal with the Kondo temperature T$_K$, of 25-30 K. The Kondo model does not expect localized states to exist at temperatures lower than T$_K$, whereas temperature of 1.5 K is more than 10 times smaller than T$_K$. Surely, such the observation calls for the rejection, or at least a substantial revision of the Kondo lattice theory. We are convinced that this revision will go to our CEF based understanding of the heavy-fermion phenomena with the importance of the local Kramers doublet ground state. Just after the Sichelschmidt/Steglich discovery we have described the g tensor and derived two sets of CEF parameters for $\Gamma$$_6$ and $\Gamma$$_7$ CEF ground states [@19].
Sub-conclusion: We take these examples as further evidence for the applicability of the CEF-based approach to actinides and anomalous 4f/5f compounds. Our basic idea for the localized CEF origin of heavy-fermion phenomena has been formulated already in 1992 [@20]. A report of CSSP-4/95 “Physics of heavy-fermion phenomena” [@21] has been widely distributed to the leading scientists over 400 copies, including the International Board of SCES-94 and SCES-95. Our CEF-based interpretation of the heavy-fermion phenomena has been put in 1995 to the scientific protection of the Prezes of the Polish Academy of Science.
Anomalous properties and heavy-fermion behavior
===============================================
In our understanding of anomalous 4f/5f compounds the localized Kramers doublet ground state plays the essential role [@20; @21]. A lattice of Kramers ions with the local Kramers doublet ground state is the physical realization of the anisotropic spin liquid postulated [*ad hoc*]{} in heavy-fermion theories. According to us, the heavy-fermion behavior is related to difficulties in the removal of the Kramers doublet degeneracy. The local Kramers doublet is always formed for a strongly-correlated odd-number electron system. The removal of the Kramers degeneracy is equivalent to the formation of the magnetic state, characterized by breaking of the time-reversal symmetry. There can be different reason for this difficulty in the removal of the Kramers degeneracy (this difficulty can be called as a quantum entanglement of two Kramers conjugate states) causing its removal at low temperatures only. The Kramers-doublet degeneracy has to be removed before the system approaches zero temperature. In this view heavy-fermion state is a magnetic state. In contrast to well-defined magnetic/paramagnetic transition characterized by the lambda-type peak in the specific heat the magnetic state in heavy-fermion compounds is not uniformed, being of the spin-fluctuation type. There is a site-to-site change of value of the Kramers doublet splitting. Associated with it is a site-to-site change of the value of the local magnetic moment and its direction. Thus one can model such magnetic state by a statistical distribution of the 0-0.3 meV splittings and of Kondo temperatures. In QUASST heavy-fermion excitations are neutral spin-like excitations between conjugate local Kramers states. These thermal excitations are associated with the reversal of spin. In our picture f electrons (exactly f electron states) are localized , whereas f excitations looks like itinerant (no one can say which exactly atom becomes excited). Our explanation with localized f electrons is unpopular within the magnetic community which prefers itinerant f electrons. If f electrons would be really itinerant than the conductivity of a heavy-fermion compound would be larger than the reference La/Y/Lu compound. But in experiment is always opposite - the resistivity of a heavy-fermion compound is always larger than the reference system. Finally we can add that in QUASST CEF-like f states do not lie at the Fermi energy. The Fermi surface is established for itinerant conduction electrons only. Moreover, in QUASST the heavy-fermion like phenomena at low temperatures can occur also in ionic compounds. This analysis of anomalous and heavy-fermion behavior in transition-metal compounds was not a subject of evaluation by CK - here it was added for the completeness reasons.
3$\textrm{d}$ ionic compounds
=============================
In 1996 we have realized that a standard approach to electronic structures and magnetism of 3d-ion compounds substantially differ from that used in rare-earth compounds. Namely, owing to the weakness of the spin-orbit coupling, in 3d compounds the spin-orbit coupling has been customarily ignored. As a consequence the magnetic moment was essentially of the spin-type only whereas the electronic structure was built from the orbital-only states. Moreover, the concept of electronic structures and magnetism was built on single-electron states, t$_{2g}$ and eg orbitals known from the octahedral crystal field, with neglecting intra-atomic electron correlations among d shell. In 1997 we have performed calculations for the spin-orbit effect on the electronic states of 3d paramagnetic ions in the octahedral crystal field revealing a variety of low-energy states, Fig. 4 [@22; @23]. For these calculations we have taken into account strong correlations among electrons in the 3d shell by considering many-electron states and two Hund’s rules. These structures have been put in 2000 to the scientific protection of the President of the American Physical Society.
According to the Quantum Atomistic Solid-State theory the atomic-like electronic structures, shown in (c), are preserved also in a solid. The shown states are many electron states of the whole d$^n$ configuration. At zero temperature only the lowest state is occupied. The higher states become populated with the increasing temperature. In Fig. 4 on the lowest levels the magnetic moment (in $\mu _{B}$ ) are written. Their are not integer. It means that a general conviction that the localized model gives the magnetic moment of the unpaired n localized d electrons as 2n$\mu _{B}$ (or (10-2n)$\mu _{B}$) is not true.
This approach called a strongly-correlated crystal field approach \[23\] is in contrast to the single-electron crystal-field approach customarily presently used. By doing it we have made unification of 3d and rare-earth compounds in description of the low-energy electronic structures and magnetism, of course keeping the relevant strength of the spin-orbit coupling. We have calculated the low-energy electronic structure and correlate it with magnetic and electronic properties, e.g. $^{3}T$$_{1g}$ of the V$^{3+}$ ion in LaVO$_3$ and $^{5}$E$_g$ of the Mn$^{3+}$ ion in LaMnO$_3$. In SCES-02 there was a reproach to us that these ground subterms are incorrect owing to literature t$^2$$_{2g}$ and t$^3$$_{2g}$e$_{g}$ configuration, with the e$_{g}$ orbital higher whereas derived by us $^{5}$E$_g$ subterm is lower. Despite of our long explanations, explaining lower and capital symbols, the papers have been rejected - the International Advisory Board have been informed about this controversy by the Chairman of SCES-02. Just after, our solution with the ground subterm $^{3}T$$_{1g}$ for the V$^{3+}$ ion (3d$^2$ configuration) in LaVO$_3$ or V$_2$O$_3$ and $^{5}E$$_{g}$ for the Mn$^{3+}$ ion (3d$^4$ configuration) in LaMnO$_3$ has been put to the scientific protection of the Rector of the Jagiellonian University in Krakow and of the AGH University of Mining and Metallurgy. Recently A. M. Oles, the vice chairman of SCES-02, has admitted the correctness of our ground states in LaVO$_3$ (V$_2$O$_3$) and in LaMnO$_3$. We await for further scientific steps.
We have clarified the electronic structure and magnetism of LaCoO$_3$ [@24]. It turns out that relatively strong octahedral crystal field yields the breaking of Hund’s rules establishing the ground subterm $^{1}$A$_1$ originating from $^1$I term that in the free Co$^{3+}$ ion lies 4.45 eV above the ground term. However, the octahedral crystal field interactions are too weak to break intra-atomic correlations and to create conditions for the applicability of the single-electron approach. It means that the experimentally observed states are still described within the atomistic QUASST approach. By perfect reproduction of the ESR results of Noguchi from 2002 [@25] on the excited triplet we have proved that this triplet is a part of the $^5$T$_{2g}$ sub-term, originating from the high-spin $^5$D term. It means that there is no intermediate spin-state, with S=1, despite of theoretical LDA-U calculations of Korotin et al. from 1996 [@26] and a numerous literature on this subject. Thus, we have confirmed the substantial physical applicability of the atomistic CEF-based Tanabe-Sugano diagrams, existing already 50 years, applicability of which have been questioned by band-structure calculations. The breaking of the Hund’s rules in LaCoO$_3$ results from the extraordinary small Co-O distance, of 192 pm. In CoO, for instance, the Co-O distance amounts to 217 pm. This smaller distance by 13$\%$ causes increase of B$_4$$^0$ parameter by 85$\%$ owing to the R$^{-5}$ dependence of the octupolar CEF interactions.
We have calculated the orbital moment in NiO [@27], CoO [@28], FeBr$_2$ and LaMnO$_3$[@29]. We have derived highly anisotropic properties of these compounds in full agreement with the experimental evidence.
In Fig. 5 we present the calculated temperature dependence of the specific heat of FeBr$_2$ both in the paramagnetic and antiferromagnetic state with the $\lambda$- peak at T$_N$ of 14 K [@30]. For description we use the similar Hamiltonian as shown above. The formation of the magnetic state is related with the splitting of the lowest quasi triplet. It is worth to add that this quasi-triplet is the excited quasi-triplet discussed above for LaCoO$_3$. The Fe$^{2+}$ and Co$^{3+}$ ions are isoelectronic 3d$^6$ systems. Despite of the hexagonal lattice symmetry of FeBr$_2$ and the slightly distorted cubic structure of LaCoO$_3$ in both these compounds the local symmetry is octahedral. We would like to turn attention that the good reproduction of the overall specific heat means, in fact, the counting of atoms. The reproduction of the absolute value indicate that all atoms equally contribute to the observed property. It means, though it could sound unbelievedly, that all Fe atoms have the same electronic structure. We think that it is an effect of the blind action of the simple physical laws.
Recently we describe consistently NiO within the strongly-correlated CEF approach reconciling its insulating ground state, the value and the direction of the magnetic moment in the antiferromagnetic state below 525 K and thermodynamic properties [@31]. In particular we have calculated the overall electronic specific heat with the lambda-type peak at T$_N$ and a substantial heat with the overall entropy of Rln3 fully released at T$_N$. We have quantified crystal-field (the leading parameter B$_{4}$ = +21 K), spin-orbit (-480 K, i.e. like in the free ion [@32]) and magnetic interactions (B$_{mol}$ of 503 T and n= -200 T/$\mu_{B}$). In our approach E$_{dd}$ $\gg$ E$_{CF}$(=2.0 eV)$\gg$E$_{s-o}$(=0.29 eV)$\gg$E$_{mag}$(=0.07 eV). The orbital and spin moment of the Ni$^{2+}$ ion in NiO has been calculated within the quasi-atomic approach. The orbital moment of 0.54 $\mu _{B}$ amounts at 0 K in the magnetically-ordered state, to about 20% of the total moment (2.53 $\mu _{B}$). Despite of using the full atomic orbital quantum number $L$=3 and $S$=1, the calculated effective moment from the temperature dependence of the susceptibility amounts to 3.5-3.8 $\mu _{B}$, i.e. only 20 $\%$ larger value than a spin-only value of 2.83 $\mu _{B}$.
We take as great confirmation of the strongly-correlated crystal field approach that the electronic structure through the series of compounds results from the symmetry of the transition-metal surroundings. For instance, in all of above mentioned compounds the closest surroundings has predominantly the octahedral symmetry. As a consequence the ground state subterm alternates as the octupolar moment of the 3d$^n$ configuration.
Development of physics
======================
Everybody can see that our understanding of the electronic structure and magnetism of transition-metal compounds is along the well-established in solid-state physics paradigm. By pointing out the importance of the crystal field we call for a larger attention to local atomic-scale effects. In the present situation of the administrative reproaching of the use and the concept of the crystal-field approach if applied to magnetic metallic compounds we call to members of the international physical community. We have to turn to the international community, because by years Polish physicists can not settle up a problem and by years after the first negative Szymczak’s opinion of 2001 subsequent opinions are written in very loosely language. For instance in the opinion of A.M. Oles, crucial for the final negative decision, there is only one (!!) verifable statement, that fitting of the only one physical property is not reliable. We fully agree with this general Oles statement, but it does not touch the scientific approach of Radwanski, where always we analyze many properties for one compound and look for the consistency with other isostructural compounds. Other reproaching statements of Oles are not scientifically conclusive in the sense that these statements contain always “seems to”, “probably” or “likely”. Even a reproach of H. Szymczak, obviously erroneous, about description of the trigonal off-octahedral distortion used by Radwanski cannot be clarified. In our paper by Ropka and Radwanski in Phys. Rev. B (63 (2001) 172404), Ref. 1 on the list of our publication from 2001, we have described the trigonal off-octahedral distortion, at the end of the first page, by the B$_2$$^0$O$_2$$^0$ term added to the octahedral Hamiltonian written for the z axis along the cube diagonal (reproach No 3). This procedure has been simply erroneously questioned by Szymczak in 2001, but there was no one to clarify it. According to normal scientific rules Szymczak has been obliged to write a Comment to the Editor of Phys. Rev. B and to inform publicly the scientific community about incorrectness before writing an unsubstantiated reproach to the Governmental Scientific Committee. (In fact, the simplest was to send e-mail to Radwanski, and trying to explain the controversy.) An unscientifically made reproach cannot be clarified now. Later, four referees did not clarify this erroneous reproach - we take it as an evidence for their scientific dishonesty. At present instead of the simple clarifying the erroneous decision of CK, the easiest would be the correction by Szymczak, the Polish magnetic community keeps a long-lasting splendid quiet. This unscientific behavior is partly related to the dominant administrative position of H. Szymczak in the Polish physics and a lack of respect for basic scientific rules in Polish physics. We hope that these problems will be discussed in the coming magnetic conferences in Poland, in Wroclaw, 19-21 May 2005, on “Anomalous properties of strongly correlated systems” chaired by D. Kaczorowski and in Poznan, 25-29 June 2005 on “Physics of Magnetism” chaired by Krompiewski and R. Micnas.
We submit this problem also to Prof. E. Bauer, the Chairman of the incoming SCES-05 Conference to be held in 26-30 July 2005 in Vienna, and to all members of the International Advisory Board. In particular, we turn to well-experienced physicists: M. Abd-Elmeguid, P. Alekseev, J. Allen, M.C. Aronson, P. Coleman, M. Continentino, B. Coqblin, A. de Visser, C. Di Castro, Z. Fisk, J. Flouquet, A. Fujimori, P. Fulde, J. Gomez-Sal, H. Harima, H. Johannesson, B. Johansson, C. Lacroix, A. Loidl, G. Lonzarich, M.B. Maple, F. Marabelli, K.A. McEven, A.J. Millis, J.A. Mydosh, Y. Onuki, G. Oomi, R. Osborn, M. Reiffers, T.M. Rice, T.F. Rosenbaum, E.V. Sampathkumaran, H. Sato, G.A. Sawatzky, V. Sechovsky, J. Sereni, M. Sigrist, J. Spalek, F. Steglich, T. Takabatake, J.D. Thompson, K. Ueda, D. Vollhardt, H. v. Lohneysen, V. Zlatic. All of us knows that the magnetism of transition-metal compounds is still under scientific discussion and any administration decision about the incorrectness of the crystal-field-based approach to magnetic metallic materials is premature and harmful to physics. A reproach that “the used (crystal-field) approach is too simplified” with a simultaneous statement that “the obtained agreements with experimental data are accidental” is illogical. We are ashamed that such strong administrative interference to Physics happens in Poland, a country of the long tradition of freedom. We are lucky that apart of the administration we have another great authority - the Pope John Paul II. By last 20 years he teaches about the truth, the values, the dignity and the freedom in everyday life and in Science.
We turn to the international physical community as we believe that all members of this community share our view that Science can develop only in the truth and in freedom. We call to physicists, our colleagues in searching for the scientific truth: We can differ in approaching to Physics and Science - but all of us agree that Science and Physics can develop only without administration interference for judging correctness of any scientific theory. We believe that the future proves the incorrectness of the administrative interference to physics and for the restoration of normal scientific conditions in physics. Independently on it, we will continue with the highest integrity our research on the magnetism and electronic structures of transition-metal compounds being open for scientific discussions and critics.
We are grateful to all opponents - thanks them our studies turn out to be scientifically important despite of using at the start well-known atomistic approach and the 75 years old crystal-field theory. Being grateful to our opponents we cannot, however, accept discrimination and inquisition methods used in doing science. It is obvious that Science without ethic values becomes empty. If somebody is able to prove errors in the crystal-field approach is welcome to publish it openly. If somebody has something against me and my scientific activity is welcome to say it publicly, not to work with the help of administration methods like rejection of papers from the publication or the presentation on conferences. A great ethic problem appears if later he publishes quite similar things. We declare our deep will to cooperate with everybody in serving to built the scientific magnetic community and to search for the scientific truth.
Remarks on the crystal-field theory
===================================
A strange scientific climate about the crystal-field theory in the modern solid-state paradigm comes from a widely-spread view within the magnetic community that it does not have the proper theoretical justification. An oversimplified point charge model is treated as an essence of the crystal-field theory. An indication in some cases that the point charge model is not sufficient to account for the crystal-field splittings was taken as the conclusive proof for the incorrectness of the crystal-field concept at all. According to us the theoretical background for the crystal-field theory is the atomic construction of matter. Simply, atoms constituting a solid preserve much of their atomic properties. One can say that the atomic-like integrity is preserved, after giving up partly or fully some electrons, and then the atomic identity serves as the good quantum number of the electron system. We are quite satisfied that the point-charge model provides the proper variation the ground states going on from one to another 3d/4f/5f atom. Different ionic states we consider as different states of the atom, though it is better instead of the ionic state to say about the electron configurations and their different contributions to magnetic, electronic, spectroscopic and optical properties. For instance, in metallic ErNi$_{5}$ there exists 4f$^{11}$ electron configuration, often written as the Er$^{3+}$ ion, that is found to be predominantly responsible for the magnetism and the electronic structure of the whole compound [@6]; the other electrons of Er and Ni are responsible for the metallic behavior. We point out the multipolar character of the electric potential in a solid. It is very fortunate situation when a solid, with milliard of milliard of atoms (in America billion of billions), can be described with the single electronic structure. It is true, that the crystal-field theory being itself a single-ion theory cannot describe a solid with collective interactions. For this reason we came out with the Quantum Atomistic Solid State Theory and completed the crystal-field theory with strong intra-atomic correlations and intersite spin-dependent interactions. By pointing out the importance of the CEF theory we would like to put attention to the fundamental importance of the atomic physics (Hund’s rules, spin-orbit coupling, ....) and local single-ion effects. It is worth remind that the source of a collective phenomenon, the magnetism of a solid, are atoms constituting this solid. Properties of these potentially-active atoms (open-shell atoms) are determined by local surroundings and local symmetry. Subsequently, these atomic moments, with spin and orbital parts, enter to the collective game in a solid. If somebody thinks that CEF and QUASST is too simple (in fact, it is not simple!!) should not blame authors for it, but Nature. Nature turns out to be simpler than could be!
Another wrong conviction about the crystal-field theory is that it was exploited already completely. In order to shown that this thinking is wrong we turn the reader’s attention that the crystal-field approach used within the rare-earth and actinide community (4f and 5f systems) fundamentally differs from that used within the 3d community. The 4f/5f community works with $J$ as the good quantum number whereas the 3d community “quenches” the orbital moment and works with only the spin $S$. Our description of a 3d-atom compound like FeBr$_{2}$ and LaCoO$_{3}$ one can find in Refs [@30] and [@24]. In case of the strongly-correlated crystal-field approach we work with many-electron states of the whole 4f$^{n}$, 5f$^{n}$ 3d$^{n}$ configuration in contrary to single-electron states used in 3d magnetism and LDA, LSDA, and many other so-called [*ab initio*]{} approaches. Technically, strong correlations are put within the CEF theory, and in QUASST, by application of two Hund’s rules. The [*ab initio*]{} calculations will meet the CEF (QUASST) theory in the evaluation of the detailed charge distribution within the unit cell and after taking into account strong intra-atomic correlations among electrons of incomplete shells and the spin-orbit coupling in order to reproduce the CEF conditions (two Hund’s rules, also the third one for rare-earths and actinides).
We would like to mention that we are fully aware that used by us the Russell-Saunders LS coupling can show some shortages in case of actinides related to the growing importance of the j-j coupling. We are aware of many other physical problems which we could not mention here due to the length problem - finally we mention only that we can reverse scientific problem in the solid state physics and use 4f/5f/3d compounds as a laboratory for the atomic physics for study 3d/4f/5f atoms in extremal electric and magnetic fields. In the solid-state physics we study the lowest part of the atomic structure, but extremely exactly.
The detailed electronic structure is predominantly determined by conventional interactions in a solid: the Stark-like effect by the crystalline electric field potential due to 3-dimensional array of charges in a crystal acting on the aspherical incomplete shell, and the Zeeman-like effect due to spin-dependent interactions of the incomplete-shell spin (atomic-like moment) with self-consistently induced spin surroundings. These states can become broaden in energy by different interactions (lowering of the local symmetry, thermal expansion, appearance of a few inequivalent sites, lattice imperfection, surface effects and other solid-state effects). Obviously, we should not think that discrete crystal field states mean that they are extremely thin lines. 3 or even 10 meV broad lines are still of the crystal-field origin. Underlying by us by many, many years the importance of the crystal field we have treated as an opposite view to the overwhelmed band structure view yielding the spreading of the f-electron (and mostly 3d-electron) spectrum by 2-5 eV.
Conclusions
===========
We advocate for the high adequacy of use and the concept of the crystal field approach, even if applied to metallic magnetic materials of transition-metal compounds. By extension of the crystal-field theory to a Quantum Atomistic Solid-State theory (QUASST) we have made the unification of 3d and rare-earth compounds in description of the low-energy electronic structures and magnetism of open 3d/4f/5f shell electrons taking into account the local crystal field, the intra-atomic spin-orbit coupling and strong intra-atomic correlations. QUASST offers consistent description of zero-temperature properties and thermodynamic properties of 3d-ion containing compounds. We have calculated the orbital moment in 3d oxides (NiO, CoO, LaCoO$_3$, LaMnO$_3$, FeBr$_2$). Our studies indicate that it is the highest time to unquench the orbital magnetism in 3d -ion compounds. We claim that the first-principles and ab initio studies will be as long not successful as strong correlations assumed in the CEF approach will be not incorporated in the calculations. We do not claim to (re-)invent crystal-field theory, but we do not agree for the depreciation of the crystal field theory. We are convinced that the crystal-field theory with strong correlations is a fundamental ingredient of the modern solid-state paradigm.
$^\spadesuit$ dedicated to the Pope John Paul II, a man of freedom in life and in Science.
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abstract: 'A coupled map lattice of generalized Lotka-Volterra equations in the presence of colored multiplicative noise is used to analyze the spatiotemporal evolution of three interacting species: one predator and two preys symmetrically competing each other. The correlation of the species concentration over the grid as a function of time and of the noise intensity is investigated. The presence of noise induces pattern formation, whose dimensions show a nonmonotonic behavior as a function of the noise intensity. The colored noise induces a greater dimension of the patterns with respect to the white noise case and a shift of the maximum of its area towards higher values of the noise intensity.'
title: 'NONMONOTONIC PATTERN FORMATION IN THREE SPECIES LOTKA-VOLTERRA SYSTEM WITH COLORED NOISE'
---
Introduction
============
The addition of noise in mathematical models of population dynamics can be useful to describe the observed phenomenology in a realistic and relatively simple form. This noise contribution can give rise to non trivial effects, modifying sometimes in an unexpected way the deterministic dynamics. Examples of noise induced phenomena are stochastic resonance, noise delayed extinction, temporal oscillations and noise-induced pattern formation [@Val; @Lutz; @Sancho; @Garcia-Sancho; @Katja]. Biological complex systems can be modelled as open systems in which interactions between the components are nonlinear and a noisy interaction with the environment is present [@Ciuchi]. Recently it has been found that nonlinear interaction and the presence of multiplicative noise can give rise to pattern formation in population dynamics of spatially extended systems [@Spa; @Spa1; @Ale1]. The real noise sources are correlated and their effects on spatially extended systems have been investigated in Refs. [@Sancho] (see cited Refs. there) and [@Garcia-Sancho]. In this paper we study the spatio-temporal evolution of an ecosystem of three interacting species: two competing preys and one predator, in the presence of a colored multiplicative noise. We find a nonmonotonic behavior of the average size of the patterns as a function of the noise intensity. The effects induced by the colored noise, in comparison with the white noise case [@Ale1], are: (i) pattern formation with a greater dimension of the average area, (ii) a shift of the maximum of the area of the patterns towards higher values of the multiplicative noise intensity.
The model
=========
To describe the dynamics of our spatially distributed system, we use a coupled map lattice (CML) [@Ale1; @cml] with a multiplicative noise
$$\begin{aligned}
x_{i,j}^{n+1} & = & \mu x_{i,j}^n (1 - x_{i,j}^n-\beta^n
y_{i,j}^n-\alpha z_{i,j}^n)+ x_{i,j}^n X_{i,j}^n + D\sum_p
(x_{p}^n-x_{i,j}^n),
\nonumber \\
y_{i,j}^{n+1} & = & \mu y_{i,j}^n (1 - y_{i,j}^n-\beta^n
x_{i,j}^n-\alpha z_{i,j}^n)+ y_{i,j}^n Y_{i,j}^n +D\sum_p
(y_{p}^n-y_{i,j}^n),
\nonumber \\
z_{i,j}^{n+1} & = & \mu_z z_{i,j}^n
[-1+\gamma(x_{i,j}^n+y_{i,j}^n)] + z_{i,j}^n Z_{i,j}^n + D\sum_p
(z_{p}^n-z_{i,j}^n),
\label{eqset}\end{aligned}$$
where $x_{i,j}^n$, $y_{i,j}^n$ and $z_{i,j}^n$ are respectively the densities of preys $x$, $y$ and the predator $z$ in the site $(i,j)$ at the time step $n$. Here $\alpha$ and $\gamma$ are the interaction parameters between preys and predator, $D$ is the diffusion coefficient, $\mu$ and $\mu_z$ are scale factors. $\sum_{p}$ indicates the sum over the four nearest neighbors in the map lattice. $X(t), Y(t), Z(t)$ are Ornstein-Uhlenbeck processes with the statistical properties
$$\langle \chi(t) \rangle = 0, \:\:\:\:\: \langle \langle \chi(t)
\chi(t+\tau) \rangle = \frac{q}{2 \tau_c}
e^{-\tau / \tau_c}, \label{meanou}$$
and $$\langle X^n_{i,j} Y^m_{i,j} \rangle = \langle X^n_{i,j} Z^m_{i,j}\rangle =
\langle Y^n_{i,j} Z^m_{i,j} \rangle = 0 \;\;\; \forall \; n,m,i,j$$ where $\tau_c$ is the correlation time of the process, $q$ is the noise intensity, and $\chi(t)$ represents the three continuous stochastic variables ($X(t), Y(t), Z(t)$), taken at time step $n$. The boundary conditions are such that no interaction is present out of lattice. Because of the environment temperature, the interaction parameter $\beta(t)$ between the two preys can be modelled as a periodical function of time
$$\beta(t)=1 + \epsilon + \eta cos(\omega t).
\label{betat}$$
Here $\eta = 0.2$, $\omega =
\pi 10^{-3}$ and $\epsilon=-0.1$. The interaction parameter $\beta(t)$ oscillates around the critical value $\beta_c=1$ in such a way that the dynamical regime of Lotka-Volterra model for two competing species changes from coexistence of the two preys ($\beta<1$) to exclusion of one of them ($\beta>1$). The parameters used in our simulations are the same of [@Ale1], in order to compare the results with the white noise case. Specifically they are: $\mu = 2$; $\alpha = 0.03$; $\mu_z = 0.02$, $\gamma = 205$ and $D = 0.1$. The noise intensity $q$ varies between $10^{-11}$ and $10^{-2}$. With this choice of parameters the intraspecies competition among the two prey populations is stronger compared to the interspecies interaction preys-predator ($\beta \gg \alpha$), and both prey populations can therefore stably coexist in the presence of the predator [@Baz]. To evaluate the species correlation over the grid we consider the correlation coefficient $r^n$ between a couple of them at the step $n$ as
$$r^n = \frac{\sum_{i,j}^N (w_{i,j}^n - \bar{w}^n) (k_{i,j}^n -
\bar{k}^n)}{\left[\sum_{i,j}^N (w_{i,j}^n - \bar{w}^n)^2
\sum_{i,j}^N (k_{i,j}^n - \bar{k}^n)^2 \right]^{1/2}},
\label{r}$$
where $N$ is the number of sites in the grid ($100\mathrm{x}100$), the symbols $w^n, k^n$ represent one of the three species concentration $x, y, z$, and $\bar{w}^n,\bar{k}^n$ are the mean values of the same quantities in all the lattice at the time step $n$. From the definition (\[r\]) it follows that $ -1 \leq r^n
\leq 1$.
Colored Noise effects
=====================
We quantify our analysis by considering the maximum patterns, defined as the ensemble of adjoining sites in the lattice for which the density of the species belongs to the interval $[3/4 \;
max, max]$, where $max$ is the absolute maximum of density in the specific grid. The various quantities, such as pattern area and correlation parameter, have been averaged over 50 realizations, obtaining the mean values below reported. We evaluated for each spatial distribution, in a temporal step and for a given noise intensity value, the following quantities referring to the maximum pattern (MP): mean area of the various MPs found in the lattice and correlation $r$ between two preys, and between preys and predator.
From the deterministic analysis we observe: (i) for $\epsilon < 0$ ($\beta < 1$) a coexistence regime of the two preys, characterized in the lattice by a strong correlation between them and the predator lightly anti-correlated with the two preys; (ii) for $\epsilon > 0$ ($\beta > 1$) wide exclusion zones in the lattice, characterized by a strong anti-correlation between preys. Because of the periodic variation of the interaction parameter $\beta(t)$, an interesting activation phenomenon for $\epsilon < 0$ takes place: the two preys, after an initial transient, remain strongly correlated for all the time, in spite of the fact that the parameter $\beta(t)$ takes values greater than $1$ during the periodical evolution. We focus on this dynamical regime to analyze the effect of the noise. We found that the noise acts as a trigger of the oscillating behavior of the species correlation $r$ giving rise to periodical alternation of coexistence and exclusion regime. Even a very small amount of noise is able to destroy the coexistence regime periodically in time. This gives rise to a periodical time behavior of the correlation parameter $r$, with the same periodicity of the interaction parameter $\beta(t)$ (see Eq.(\[betat\])), which turns out almost independent of the noise intensity and of the correlation time $\tau_c$ (see Fig.\[cor\](a)). This periodicity reflects the periodical time behavior of the mean area of the patterns. A nonmonotonic behavior of the pattern area as a function of time is observed for all values of noise intensity investigated. This behavior becomes periodically in time for lower values of noise intensity, when higher values of correlation time $\tau_c$ are considered. In Figs.\[cor\](b-d) we show the time evolution of the mean area of the maximum patterns, for $q = 10^{-4}$ and for three values of correlation time, namely $\tau_c = 1, 10, 100$. The periodicity of the nonmonotonic behavior of the area of MPs is clearly observed.
-0.5cm {height="8cm"} -0.3cm
-0.5cm \[cor\]
To analyze the noise induced pattern formation we focus on the correlation regime between preys $r_{12} = 1$, where pattern formation appears. In fact when the preys are highly anticorrelated with species correlation parameter $r_{12} = -1$, a big clusterization of preys is observed, with large patches of preys enlarging to all the available space of the lattice. This scenario, observed also in the white noise case [@Ale1], is confirmed by the analysis of the time series of the species. These large patches appear, in the anticorrelation regime corresponding to the exclusion regime of the two preys, with smooth contours and low intensity of species density for lower noise intensities and higher correlation time values.
The study of the area of the pattern formation as a function of noise intensity with colored noise shows two main effects: 1) the increase of the pattern dimension and 2) a shift of the maximum toward higher values of the noise intensity. As expected, for low values of the correlation time we observe the same results than in the white noise case. These effects are well visible in Fig. \[aree\] where the three curves show the nonmonotonic behavior of the area of the maximum pattern as a function of noise intensity. The interaction step here considered is 1400, which correspond to the biggest pattern area found in our calculations. The first curve ($\tau_c=1$) is quite the same found in the white noise case. The value of maximum in the third curve ($\tau_c=100$) is not so different from the previous one ($\tau_c=10$), because its value is approaching the maximum possible value of 10.000 into the used grid.
{height="6cm"} -0.3cm
-0.5cm \[aree\]
{height="9cm"} -0.3cm
-0.8cm \[pat\]
The pattern formation is visible in Fig. \[pat\], where we report three patterns of the two preys and the predator for the following values of noise intensity: $q=10^{-11}, 10^{-9},
10^{-7}$ and $\tau_c=1$. The initial spatial distribution is homogeneous and equal for all species, that is $x_{ij}^{init}=y_{ij}^{init}=z_{ij}^{init}= 0.25$ for all sites ($i,j$). We see that a spatial structure emerges with increasing noise intensity. At very low noise intensity ($q = 10^{-11}$), the spatial distribution appears almost homogeneous without strong pattern formation (see Fig. \[pat\]a). We considered here only structured pattern, avoiding big clusterization of density visible in the case of anticorrelated preys. At intermediate noise intensity ($q = 10^{-9}$) spatial patterns appear. As we can see the structure disappears by increasing the noise intensity (see Fig. \[pat\]c). Consistently with Fig. \[aree\], we find that for higher correlation time $\tau_c$ the qualitative shape of the patterns shown in Fig. \[pat\] are repeated, but with a shift of the maximum area (darkest patterns) toward higher values of the noise intensity.
Conclusions
===========
The noise-induced pattern formation in a coupled map lattice of three interacting species, described by generalized Lotka-Volterra equations in the presence of multiplicative colored noise, has been investigated. We find nonmonotonic behavior of the mean area of the maximum patterns as a function of noise intensity for all the correlation time investigated. For increasing values of the correlation time $\tau_c$ we observe an increase of the area of the pattern and a shift of the maximum value towards higher values of the noise intensity. The nonmonotonic behavior is also found for the area of the patterns as a function of the evolution time. 0.2cm This work was supported by , by INFM and MIUR.
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abstract: |
In this paper, we address the theoretical limitations in reconstructing sparse signals (in a known complete basis) using compressed sensing framework. We also divide the CS to non-blind and blind cases. Then, we compute the Bayesian Cramer-Rao bound for estimating the sparse coefficients while the measurement matrix elements are independent zero mean random variables. Simulation results show a large gap between the lower bound and the performance of the practical algorithms when the number of measurements are low.
*Index Terms*-Compressed sensing, Sparse component analysis, Blind source separation, Cramer-Rao bound.
author:
- |
H. Zayyani [${^{1}}$]{}, M. Babaie-Zadeh[${^{1}}$]{} , and C. Jutten [${^{2}}$]{} \
EDICS:SAS-ICAB or SAS-STAT [^1] [^2] [^3] [^4] [^5] [^6]
title: 'Bayesian Cramér-Rao Bound for Noisy Non-Blind and Blind Compressed Sensing '
---
Introduction {#sec:intro}
============
Compressed Sensing or Compressive Sampling (CS) [@CandT06], [@Dono06] is an emerging field in signal processing. The theory of CS suggests to use only a few random linear measurement of a sparse signal (in a basis) for reconstructing the original signal. The mathematical model of noise free CS is: $${{\textbf{y}}}=\boldsymbol{\Phi}{{\textbf{x}}}$$ where ${{\textbf{x}}}=\boldsymbol{\Psi}{{\textbf{w}}}$ is the original signal with length $m$ and is sparse in the basis $\boldsymbol{\Psi}$ (${i.e.,\xspace}||{{\textbf{w}}}||_0<K$ and $K$ is defined as sparsity level) and $\boldsymbol{\Phi}$ is an $n\times m$ random measurement matrix where $n<m$. For near perfect recovery, in addition to the signal sparsity, the incoherence of the random measurement matrix $\boldsymbol{\Phi}$ with the basis $\boldsymbol{\Psi}$ is needed. The incoherence is satisfied with high probability for some types of random matrices such as i.i.d Gaussian elements or i.i.d Bernoulli $\pm1$ elements. Recent theoretical results show that under these two conditions (sparsity and incoherence), the original signal can be recovered from only a few linear measurements of the signal within a controllable error, even in the case of noisy measurements [@CandT06], [@Dono06], [@HaupN06], [@AkcaT07].
In [@HaupN06], some error bounds are introduced for reconstructing the original sparse (or compressible) signal in the noisy CS framework. In [@AkcaT07], the performance limits of noisy CS is investigated by definition of some performance metrics which are of Shannon Theoretic spirit. [@CandT06] considers the no noise CS and finds an upper bound on reconstruction error in terms of Mean Square Error (MSE) only for [$\ell^{1}$]{}-minimization recovery algorithm. But, [@HaupN06] finds some upper bounds in the noisy CS and for general recovery algorithms. [@AkcaT07] is also investigated its own decoder which is derived based on joint typicality. Moreover, some information theoretic bounds are derived in [@AeroZS07].
In this paper, we derive a Bayesian Cramer-Rao Bound (BCRB) ([@Vant68], [@TichMN98]), which is a lower bound, for noisy CS by a statistical view to the CS problem. This BCRB bounds the performance of any parametric estimator (whether biased or unbiased) of the sparse coefficient vector in terms of mean square estimation error [@Vant68], [@TichMN98]. We also introduce the notion of blind CS in contrast to the traditional CS to whom we refer on the non-blind CS. We compute BCRB for both non-blind and blind CS, where in the latter, we do not know the measurement matrix in advance. In a related direction of research, a CRB is obtained for mixing matrix estimation in Sparse Component Analysis (SCA) [@ZayyBHJ08].
Non-blind and blind noisy CS
============================
Consider the noisy CS problem: $$\label{eq: ncs}
{{\textbf{y}}}=\boldsymbol{\Phi}\boldsymbol{\Psi}{{\textbf{w}}}+{{\textbf{e}}}={{\textbf{D}}}{{\textbf{w}}}+{{\textbf{e}}}$$ where ${{\textbf{D}}}=\boldsymbol{\Phi}\boldsymbol{\Psi}$, ${{\textbf{w}}}$ is a sparse vector and ${{\textbf{e}}}$ is a Gaussian zero-mean noise vector with the covariance $\sigma^2_e{{\textbf{I}}}$. In CS framework, we want to estimate ${{\textbf{w}}}$, from which, ${{\textbf{x}}}=\boldsymbol{\Psi}{{\textbf{w}}}$ can be reconstructed from the measurement vector ${{\textbf{y}}}$.
We nominate the traditional CS problem as non-blind CS since we know the basis $\boldsymbol{\Psi}$ and the measurement matrix $\boldsymbol{\Phi}$ and hence ${{\textbf{D}}}$ in advance. In some cases, we have no prior information about the signals in addition to their sparsity. As such, we do not know the basis $\boldsymbol{\Psi}$, in which the signals are sparse. One application is a blind interceptor who intercepts the signals. The only information is that the signals have been received are sparse in some unknown domain. In these cases, we nominate the problem as blind CS which is inspired from the well known problem of Blind Source Separation (BSS). As such, each measurement will be: $$\label{eq: blind1}
y=\boldsymbol{\phi}^T\boldsymbol{\Psi}{{\textbf{w}}}+{{\textbf{e}}}={{\textbf{d}}}^T{{\textbf{w}}}+{{\textbf{e}}}$$ where $\boldsymbol{\phi}^T$ is the random measurement vector and a row of $\boldsymbol{\Phi}$) and ${{\textbf{d}}}^T=\boldsymbol{\phi}^T\boldsymbol{\Psi}$ is the corresponding row in ${{\textbf{D}}}$ and an unknown random vector.
Bayesian Cramer-Rao Bound {#sec: CRLB}
=========================
The Posterior Cramer-Rao Bound (PCRB) or Bayesian Cramer-Rao Bound (BCRB) of a vector of parameters $\boldsymbol{\theta}$ estimated from data vector ${{\textbf{y}}}$ is the inverse of the Fisher information matrix, and bounds the estimation error in the following form [@TichMN98]: $$\label{eq: BCRB}
E\left[(\boldsymbol{\theta}-\hat{\boldsymbol{\theta}})(\boldsymbol{\theta}-\hat{\boldsymbol{\theta}})^T\right]\ge{{\textbf{J}}}^{-1}$$ where $\boldsymbol{\hat{\theta}}$ is the estimate of $\boldsymbol{\theta}$ and ${{\textbf{J}}}$ is the Fisher information matrix with the elements [@TichMN98]: $$J_{ij}=E_{{{\textbf{y}}},\boldsymbol{\theta}}\left[-\frac{\partial^2\log
p({{\textbf{y}}},\boldsymbol{\theta})}{\partial\theta_i\partial\theta_j}
\right],$$ where $p({{\textbf{y}}},\boldsymbol{\theta})$ is the joint probability between the observations and the parameters. Unlike CRB, the BCRB (\[eq: BCRB\]) is satisfied for any estimator (even for biased estimators) under some mild conditions [@Vant68], [@TichMN98] which we assume that are fulfilled in our problem. Using Bayes rule, the Fisher information matrix can be decomposed into two matrices [@TichMN98]: $${{\textbf{J}}}={{\textbf{J}}}_{D}+{{\textbf{J}}}_P,$$ where ${{\textbf{J}}}_{D}$ represents data information matrix and ${{\textbf{J}}}_P$ represents prior information matrix which their elements are [@TichMN98]: $$J_{D_{ij}}\triangleq
E_{{{\textbf{y}}},\boldsymbol{\theta}}\left[-\frac{\partial^2\log
p({{\textbf{y}}}|\boldsymbol{\theta})}{\partial\theta_i\partial\theta_j}\right]=E_{\boldsymbol{\theta}}(J_{s_{ij}})$$ $$\label{eq: pfim} J _{P_{ij}}\triangleq
E_{\boldsymbol{\theta}}\left[-\frac{\partial^2\log
p(\boldsymbol{\theta})}{\partial\theta_i\partial\theta_j}\right]$$ where ${{\textbf{J}}}_{s}\triangleq
E_{{{\textbf{y}}}|\boldsymbol{\theta}}[-\frac{\partial^2\log
p({{\textbf{y}}}|\boldsymbol{\theta})}{\partial\theta_i\partial\theta_j}]$ is the standard Fisher information matrix [@Kay93] and $p(\boldsymbol{\theta})$ is the prior distribution of the parameter vector.
In this paper, we use this BCRB for our problem because we have a sparse prior information about the parameter which is estimated. We compute BCRB for two blind and non-blind cases.
Computing BCRB in non-blind CS
------------------------------
In the non-blind CS case, the matrices $\boldsymbol{\Phi}$ and $\boldsymbol{\Psi}$ are assumed to be known and $\boldsymbol{\Phi}$ is a random matrix while $\boldsymbol{\Psi}$ is a fixed basis matrix. Similar to [@WiesEY08], since $\boldsymbol{\Phi}$ is assumed to be known and random, $\boldsymbol{\Phi}$ can be added as an additional observation. Hence, the data information matrix elements ${{\textbf{J}}}_{D_{ij}}$ from model (\[eq: ncs\]) are of the form: $$\label{eq: expec}
J_{D_{ij}}=E_{{{\textbf{y}}},{{\textbf{w}}},\boldsymbol{\Phi}}\left[-\frac{\partial^2\log
p({{\textbf{y}}},\boldsymbol{\Phi}|{{\textbf{w}}})}{\partial\w_i\partial\w_j}\right].$$ since $p({{\textbf{y}}},\boldsymbol{\Phi}|{{\textbf{w}}})=p(\boldsymbol{\Phi})p({{\textbf{y}}}|\boldsymbol{\Phi},{{\textbf{w}}})$, $p(\boldsymbol{\Phi})$ is independent of ${{\textbf{w}}}$ and $p({{\textbf{y}}}|\boldsymbol{\Phi},{{\textbf{w}}})=(2\pi\sigma^2_e)^{\frac{-n}{2}}\exp(\frac{-1}{2\sigma^2_e}||{{\textbf{y}}}-{{\textbf{D}}}{{\textbf{w}}}||^2_2)$, we can write $\frac{\partial \log
p({{\textbf{y}}},\boldsymbol{\Phi}|{{\textbf{w}}})}{\partial{{\textbf{w}}}}=\frac{-1}{2\sigma^2_e}(-2{{\textbf{y}}}^T{{\textbf{D}}}+2{{\textbf{D}}}^T{{\textbf{D}}}{{\textbf{w}}})$. So, we have $\frac{\partial \log
p({{\textbf{y}}},\boldsymbol{\Phi}|{{\textbf{w}}})}{\partial
w_i}=\frac{1}{\sigma^2_e}({{\textbf{y}}}^T{{\textbf{D}}})_i-\frac{1}{\sigma^2_e}\sum_{r=1}^m
g_{ir}w_r$ where $g_{ij}$ denotes the elements of the matrix ${{\textbf{G}}}={{\textbf{D}}}^T{{\textbf{D}}}$. Hence, we have $\frac{\partial^2\log
p({{\textbf{y}}},\boldsymbol{\Phi}|{{\textbf{w}}})}{\partial w_i\partial
w_j}=\frac{-1}{\sigma^2_e}g_{ij}$. So, the expectation (\[eq: expec\]) will be $J_{D_{ij}}=E_{{{\textbf{y}}},{{\textbf{w}}},\boldsymbol{\Phi}}\left[
\frac{1}{\sigma^2_e}g_{ij}\right]=\frac{1}{\sigma^2_e}E_{\boldsymbol{\Phi}}\{g_{ij}\}=J_{D_{ij}}=\frac{1}{\sigma^2_e}\sum_{r=1}^n
E_{\boldsymbol{\Phi}}\{d_{ri}d_{rj}\} $. Some simple manipulations show that under assumption that the elements of $\boldsymbol{\Phi}$ are zero mean and independent random variables, the data information matrix will be: $${{\textbf{J}}}_{D}=n\frac{\sigma^2_r}{\sigma^2_e}\boldsymbol{\Psi}^T\boldsymbol{\Psi}$$ where $\sigma^2_r=E(\phi^2_{ij})$ is the variance of the random measurement matrix elements. If $\Psi$ is an orthonormal basis then $\boldsymbol{\Psi}^T\boldsymbol{\Psi}={{\textbf{I}}}$ and hence ${{\textbf{J}}}_{D}=n\frac{\sigma^2_r}{\sigma^2_e}{{\textbf{I}}}$.
To compute the prior information matrix ${{\textbf{J}}}_{P}$ from (\[eq: pfim\]), we should assume a sparse prior distribution for our parameter vector elements $w_i$. Similarly to [@WipfR04], we assume $w_i$’s are independent and have a parameterized Gaussian distribution: $$\label{eq: pr}
p(w_i)=\frac{1}{\sigma_i\sqrt{2\pi}}\exp(-\frac{w^2_i}{2\sigma^2_i}),$$ In (\[eq: pr\]), the variance $\sigma^2_i$ enforce the sparsity of the corresponding coefficient: a small variance means that the coefficient is inactive and a large value means the activity of the coefficient. It can be easily seen that in this case, the prior information matrix is ${{\textbf{J}}}_P=\diag(\frac{1}{\sigma^2_i})$. Finally, for orthonormal bases for $\boldsymbol{\Psi}$ and for prior distribution (\[eq: pr\]), the BCRB results in: $$\label{eq: NBCRB} E\left[(w_i-\hat{w_i})^2\right]\ge
\left(n\frac{\sigma^2_r}{\sigma^2_e}+\frac{1}{\sigma^2_i}\right)^{-1}.$$
Computing BCRB in blind CS
--------------------------
In the blind CS case, the matrix $\boldsymbol{\Psi}$ is not known in advance and hence the elements of matrix ${{\textbf{D}}}$ are random and unknown with zero mean. If we restrict ourselves to Gaussian measurements matrix elements ($\phi_{ij}$ is a zero-mean Gaussian) then different measurement samples of $y$ are also Gaussian and independent of each other. Hence, we can compute the data information matrix from only one measurement (\[eq: blind1\]). Then, the information matrix elements $J_{D_{ij}}=E_{y,{{\textbf{w}}}}\left[-\frac{\partial^2\log
p(y|{{\textbf{w}}})}{\partial\w_i\partial\w_j}\right]$ will be equal to (refer to [@Kay93]): $$\label{eq: bbcrb} J_{D_{ij}}=E_{y,{{\textbf{w}}}}\left[\frac{\partial\log
p(y|{{\textbf{w}}})}{\partial\w_i}\frac{\partial\log p(y|{{\textbf{w}}})}{\partial\w_j}
\right].$$ If the elements of $\boldsymbol{\phi}$ are assumed to be random with a Gaussian distribution of zero mean and variance $\sigma^2_r$ and the columns of the basis matrix $\boldsymbol{\Psi}$ have unit norms, then: $$\label{eq: blind}
p(y|{{\textbf{w}}})=\frac{1}{\sqrt{2\pi\sigma^2({{\textbf{w}}})}}\exp(-\frac{y^2}{2\sigma^2({{\textbf{w}}})})$$ where $\sigma^2({{\textbf{w}}})\triangleq\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2$. Simple manipulations show: $$\frac{\partial\log p(y|{{\textbf{w}}})}{\partial
w_i}=-\frac{w_i\sigma^2_r}{\sigma^4({{\textbf{w}}})}\left(\sigma^2({{\textbf{w}}})-y^2\right)$$ and from (\[eq: bbcrb\]) we should compute: $$J_{D_{ij}}=\sigma^4_r\int_{{{\textbf{w}}}}\frac{w_iw_j}{\sigma^8({{\textbf{w}}})}
\left[\int_{y}(\sigma^2({{\textbf{w}}})-y^2)^2p(y|{{\textbf{w}}})dy\right]p({{\textbf{w}}})d{{\textbf{w}}}$$ where the internal integral is $\int_{y}(\sigma^2({{\textbf{w}}})-y^2)^2p(y|{{\textbf{w}}})dy=m_4-2\sigma^2({{\textbf{w}}})m_2+\sigma^4({{\textbf{w}}})$ in which $m_2$ and $m_4$ are the second and fourth order moments equal to $m_2=\sigma^2({{\textbf{w}}})$ and $m_4=3\sigma^4({{\textbf{w}}})$. So, we have $\int_{y}(\sigma^2({{\textbf{w}}})-y^2)^2p(y|{{\textbf{w}}})dy=2\sigma^4({{\textbf{w}}})$ and then: $$J_{D_{ij}}=2\sigma^4_r\int_{{{\textbf{w}}}}\frac{w_iw_j}{\sigma^4({{\textbf{w}}})}p({{\textbf{w}}})d{{\textbf{w}}}$$ where the off diagonal terms are zeros $J_{D_{ij}}=0,j\neq i$ because the integrand is an odd function. The diagonal terms are: $$J_{D_{ii}}=2\sigma^4_r\int_{{{\textbf{w}}}}\frac{w^2_i}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)^2}p({{\textbf{w}}})d{{\textbf{w}}}$$
Following Appendix \[app1\], the diagonal elements are simplified as: $$\label{eq: final}
J_{D_{ii}}=\frac{2\sigma^2_r}{m}\left(A_1-\sigma^2_eA_2\right)$$ where $A_1$ and $A_2$ are defined and calculated in Appendix \[app1\].
The prior information matrix for BG distribution $p(w_i)=p\delta(w_i)+(1-p)\frac{1}{\sigma\sqrt{2\pi}}\exp(-\frac{w^2_i}{2\sigma^2})$ is calculated in Appendix \[app2\]: $${{\textbf{J}}}_P=\frac{1-p}{\sigma^2}{{\textbf{I}}}$$ Finally, the Blind BCRB is calculated as: $$\label{eq: BBCRB} E\left[(w_i-\hat{w_i})^2\right]\ge
\left(2\frac{\sigma^2_r}{m}\left(A_1-\sigma^2_eA_2\right)+\frac{1-p}{\sigma^2}\right)^{-1}$$
Simulation results {#sec: simresult}
==================
In this section, we compare the CRB’s with the results of some of the state-of-the-art algorithms for signal reconstruction in CS. In our simulations, we used sparse signals with the length $m=512$ in the time domain where $\boldsymbol{\Psi}={{\textbf{I}}}$. We used a BG distribution with the probability of being nonzero equal to $1-p=0.1$ and the variance for nonzero coefficients is equal to $\sigma^2=(0.5)^2$. So, in average there were 51 active coefficients. We used a Gaussian random measurement matrix with elements drawn from zero mean Gaussian distribution with variance equal to $\sigma^2_r=1$. The number of measurements are varied between 60 to 200. We computed the Mean Square Error (MSE) for sparse coefficient vector over 100 different runs of the experiment: $$\mbox{MSE}\triangleq10\log_{10}\left(\frac{1}{100}\sum_{r=1}^{100}||{{\textbf{w}}}_r-\hat{{{\textbf{w}}}}_r||^2_2\right)$$ where $r$ is the experiment index. We compared this measure for various algorithms with the average value of BCRB for non-blind case which is equal to $\frac{1}{m}\mbox{trace}({{\textbf{J}}}^{-1})$. The algorithms used for our simulation are Orthogonal Matching Pursuit (OMP) [@PatiRK93], Basis Pursuit (BP) [@ChenDS98], Bayesian Compressive Sampling (BCS) [@JiXC08] and Smoothed-L0 (SL0) [^7] [@MohiBJ08]. We also computed the BCRB for blind case (\[eq: BBCRB\]) to compare the BCRB’s in both blind and non-blind case. Figure \[fig1\] shows the results of the simulation. It can be seen that in the low number of measurements, there is a gap between the BCRB and the performance of algorithms while one of the algorithms approximately reaches the BCRB for large number of measurements. Moreover, the difference between the BCRB’s for the non-blind and blind cases are very large. It shows that the blind case needs much more linear measurements than the non-blind case.
To verify the approximation $D_1\approx 0$ and $D_2\approx 0$ (refer to Appendix \[app2\]), we calculated the integrals numerically with parameters $p=0.9$ and $\sigma=1$. When $\sigma_0=10^{-5}$ then $D_1=4.7990\times 10^{-25}$ and $D_2=2.7673\times 10^{-19}$. It shows that our approximations are true for sufficiently small value of $\sigma_0$.
![ MSE versus number of measurements for a sparse signal in the time domain ($\boldsymbol{\Psi}={{\textbf{I}}}$) with length $m=512$ and with the BG distribution with parameters $p=0.9$, $\sigma_1=0.5$ and $\sigma_2=0$. Measurement matrix elements are unit variance Gaussian random variables.[]{data-label="fig1"}](exp1.eps){width="7cm"}
Conclusions
===========
In this paper, the CS problem is divided into non-blind and blind cases and the Bayesian Cramer-Rao bound for estimating the sparse vector of the signal was calculated in the two cases. The simulation results show a large gap between the lower bound and the performance of the practical algorithms when the number of measurements are low. There was also a large gap between the BCRB in both non-blind and blind cases. It also shows that in the blind CS framework, much more blind linear measurements of the sparse signal are needed for perfect recovery of the signal.
Computing the integral {#app1}
======================
Let define $I_i=\int_{{{\textbf{w}}}}\frac{w^2_i}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)^2}p({{\textbf{w}}})d{{\textbf{w}}}$ and assume an equal prior distribution for all coefficients $w_i$, then all $I_i$’s are the same because of the symmetry of the integral. So, we can add all the integrals and write: $$\begin{split}
m\sigma^2_rI_i=\int_{{{\textbf{w}}}}\frac{\sigma^2_r||{{\textbf{w}}}||^2_2}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)^2}p({{\textbf{w}}})d{{\textbf{w}}}=\\
\int_{{{\textbf{w}}}}\frac{p({{\textbf{w}}})}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)}d{{\textbf{w}}}-\sigma^2_e\int_{{{\textbf{w}}}}\frac{p({{\textbf{w}}})}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)^2}d{{\textbf{w}}}\end{split}$$ Then, if we nominate the two above integrals as $A_1=\int_{{{\textbf{w}}}}\frac{p({{\textbf{w}}})}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)}d{{\textbf{w}}}$ and $A_2=\int_{{{\textbf{w}}}}\frac{p({{\textbf{w}}})}{(\sigma^2_e+\sigma^2_r||{{\textbf{w}}}||^2_2)^2}d{{\textbf{w}}}$, the integral $I_i$ is computed as $I_i=\frac{1}{m\sigma^2_r}\left(A_1-\sigma^2_eA_2\right)$. To compute $A_1$ and $A_2$, we approximate the joint probability distribution of coefficients as: $$\begin{split}
p({{\textbf{w}}})=\prod_{i=1}^mp(w_i)\approx p^m\prod_{i=1}^m\delta(w_i)+\\
p^{m-1}(1-p)\sum_{r=1}^m\frac{\prod_{i=1,i\neq
r}^m\delta(w_i)}{\sigma\sqrt{2\pi}}\exp\Big(-\frac{w^2_i}{2\sigma^2_2}\Big)
\end{split}$$ This approximation is based on the assumption that the value of $(1-p)$ which is the activity probability is very small and so we can neglect the higher order powers of $(1-p)$. By this approximation, the two integrals will be approximately: $$\label{eq: A1}
A_1=\frac{p^m}{\sigma^2_e}+\frac{mp^{m-1}(1-p)}{\sigma\sqrt{2\pi}}B_1$$ $$\label{eq: A2}
A_2=\frac{p^m}{\sigma^2_4}+\frac{mp^{m-1}(1-p)}{\sigma\sqrt{2\pi}}B_2$$ where the two integrals are $B_1=\int_{w}\frac{\exp(-\frac{w^2}{2\sigma^2})}{(\sigma^2_e+\sigma^2_rw)}dw$ and $B_2=\int_{w}\frac{\exp(-\frac{w^2}{2\sigma^2})}{(\sigma^2_e+\sigma^2_rw)^2}dw$. By change of variable $x=\frac{w}{\sigma\sqrt{2}}$, the two integrals are equal to: $$\label{eq: B1}
B_1=\frac{1}{\sqrt{2}\sigma\sigma^2_r}\int_{-\infty}^{+\infty}\frac{\exp(-x^2)}{a^2+x^2}dx=\frac{1}{\sqrt{2}\sigma\sigma^2_r}C_1$$ $$\label{eq: B2}
B_2=\frac{1}{2\sqrt{2}\sigma^3\sigma^4_r}\int_{-\infty}^{+\infty}\frac{\exp(-x^2)}{(a^2+x^2)^2}dx=\frac{1}{2\sqrt{2}\sigma^3\sigma^4_r}C_2$$ where $a^2=\frac{\sigma^2_e}{2\sigma^2_r\sigma^2}$. The above integrals are equal to[^8]: $$C_1=\frac{\pi}{a}\exp(a^2)\left[1-\mbox{erf}(a)\right]$$ $$C_2=\frac{\pi\exp(a^2)}{2a^3}\left[1-2a^2+\frac{2\sqrt{\pi}}{\pi\exp(a^2)}-\mbox{erf}(a)+2a^2\mbox{erf}(a)\right]$$ where $\mbox{erf}(x)$ is the error function, defined as $\mbox{erf}(x)\triangleq\frac{2}{\sqrt{\pi}}\int_0^x
\exp(-t^2)dt$.
Prior information matrix for BG distribution {#app2}
============================================
Since the coefficients $w_i$’s are independent, the off diagonal terms $J_{P_{ij}},i\neq j$ are zero. Because of the independence of $w_i$’s, we can write $J_{P_{ii}}=E_{w_i}\{-\frac{\partial^2\log p(w_i)}{\partial^2 w_i}
\}$. To calculate this term, we use a Gaussian distribution with small variance $\sigma^2_0$ instead of delta function $\delta(w_i)$. So, the prior is: $$p(w_i)=A\exp\Big(-\frac{w^2_i}{2\sigma^2_0}\Big)+B\exp\Big(-\frac{w^2_i}{2\sigma^2}\Big)$$ where $A=\frac{p}{\sigma_0\sqrt{2\pi}}$, $B=\frac{1-p}{\sigma\sqrt{2\pi}}$ and $\sigma_0\rightarrow 0$. The partial derivative can be calculated as: $$\frac{\partial^2\log p(w_i)}{\partial
w^2_i}=\frac{1}{p(w_i)}\frac{\partial^2p(w_i)}{\partial
w^2_i}-\frac{1}{p^2(w_i)}\Big(\frac{\partial p(w_i)}{\partial
w_i}\Big)^2$$ Hence, we have: $$\label{eq: jbp}
J_{P_{ii}}=-\int_{-\infty}^{+\infty}\frac{\partial^2p(w_i)}{\partial
w^2_i}dw_i+\int_{-\infty}^{+\infty}\frac{1}{p(w_i)}\Big(\frac{\partial
p(w_i)}{\partial w_i}\Big)^2dw_i$$ To compute the above integrals, the partial derivatives are $\frac{\partial p(w_i)}{\partial
w_i}=-\frac{Aw_i}{\sigma^2_0}\exp(-\frac{w^2_i}{2\sigma^2_0})-\frac{Bw_i}{\sigma^2}\exp(-\frac{w^2_i}{2\sigma^2})$ and $\frac{\partial^2 p(w_i)}{\partial
w^2_i}=-\frac{A}{\sigma^2_0}\exp(-\frac{w^2_i}{2\sigma^2_0})+\frac{Aw^2_i}{\sigma^4_0}\exp(-\frac{w^2_i}{2\sigma^2_0})
-\frac{B}{\sigma^2}\exp(-\frac{w^2_i}{2\sigma^2})+\frac{Bw^2_i}{\sigma^4}\exp(-\frac{w^2_i}{2\sigma^2})$. Simple calculations show that $\int\frac{\partial^2p(w_i)}{\partial w^2_i}dw_i=0$ and hence: $$J_{P_{ii}}=\int_{-\infty}^{+\infty}\frac{[-\frac{Aw_i}{\sigma^2_0}\exp(-\frac{w^2_i}{2\sigma^2_0})-\frac{Bw_i}{\sigma^2}\exp(-\frac{w^2_i}{2\sigma^2})]^2}{A\exp(-\frac{w^2_i}{2\sigma^2_0})+B\exp(-\frac{w^2_i}{2\sigma^2})}dw_i$$ where the above integral can be decomposed to three integrals which are $D_1=\int_{-\infty}^{+\infty}\frac{\frac{-A^2w^2_i}{\sigma^4_0}\exp(-\frac{w^2_i}{2\sigma^2_0})}{A\exp(-\frac{w^2_i}{2\sigma^2_0})+B\exp(-\frac{w^2_i}{2\sigma^2})}dw_i$, $D_2=\int_{-\infty}^{+\infty}\frac{\frac{-ABw^2_i}{\sigma^2_0\sigma^2}\exp(-\frac{w^2_i}{2\sigma^2_0}-\frac{w^2_i}{2\sigma^2})}{A\exp(-\frac{w^2_i}{2\sigma^2_0})+B\exp(-\frac{w^2_i}{2\sigma^2})}dw_i$ and $D_3=\int_{-\infty}^{+\infty}\frac{\frac{-B^2w^2_i}{\sigma^4}\exp(-\frac{w^2_i}{2\sigma^2})}{A\exp(-\frac{w^2_i}{2\sigma^2_0})+B\exp(-\frac{w^2_i}{2\sigma^2})}dw_i$. Since we have a term $w^2_i$ in the numerator of the above integrals and the Gaussian term with small variance is large near zero, we can neglect the Gaussian term with small variance (delta function) in the denominator. So, the integrals $D_1$ and $D_2$ with neglecting this term will be approximately zero. We verify this approximation in the simulation results by computing these integrals numerically. Finally, the third integral will be approximately $D_3\approx\int_{-\infty}^{+\infty}\frac{\frac{-B^2w^2_i}{\sigma^4}\exp(-\frac{w^2_i}{2\sigma^2})}{B\exp(-\frac{w^2_i}{2\sigma^2})}dw_i$. Calculating this integral results is $J_{P_{ii}}\approx
D_3\approx\frac{1-p}{\sigma^2}$.
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[^1]: $^1$Electrical engineering department, Sharif university of technology, Tehran, Iran.
[^2]: $^2$GIPSA-LAB, Grenoble, and Institut Universitaire de France, France.
[^3]: This work has been partially funded by Iran NSF (INSF) under contract number 86/994, by Iran Telecom Research Center (ITRC), and also by center for International Research and Collaboration (ISMO) and French embassy in Tehran in the framework of a GundiShapour collaboration program.
[^4]: First author: Hadi Zayyani, email: [[email protected]]{}, Tel: +98 21 66164125, Fax: +98 21 66023261.
[^5]: Second author: Masoud Babaie-zadeh, email: [[email protected]]{}, Tel: +98 21 66165925, Fax: +98 21 66023261.
[^6]: Fourth author: Christian Jutten, email: [[email protected]]{}, Tel: +33 (0)4 76574351, Fax: +33 (0)4 76574790.
[^7]: We used the OMP code from http://sparselab.stanford.edu with 50 iterations, the BP code from http://www.acm.caltech.edu/l1magic/l1eq-pd.m with pdtol=1e-6 and its default parameters, the BCS code from http://people.ee.duke.edu/\~lihan/cs with its default parameters and the SL0 code from http://ee.sharif.edu/\~SLzero with parameters =0.001 and =0.9.
[^8]: We used Maple software to compute the integrals analytically.
|
---
abstract: |
A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical postprocessing, and thermodynamic projector.
Thermodynamic projector is the tool to transform almost arbitrary anzatz to a thermodynamically consistent model, the postprocessing is the cheapest way to improve the solution, obtained by the Legendre integrators. Legendre Integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (MaxEnt) approximations.
The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to FENE-P model of polymer kinetic theory. The multy-peak model of polymer dynamics is developed. The simplest example, discussed in details, is the two peaks model for Gaussian manifold instability in polymer dynamics. This type of models opens a way to create the computational models for the “molecular individualism".
author:
- 'Alexander N. Gorban$^{1,2}$[^1],'
- 'Pavel A. Gorban$^{1,3**}\!\!$,'
- |
and Iliya V. Karlin$^{1,2***}\!\!$,\
$^{1}$ ETH-Zentrum, Department of Materials, Institute of Polymers,\
Sonneggstr. 3, ML J19, CH-8092 Z[ü]{}rich, Switzerland;\
$^{2}$ Institute of Computational Modeling SB RAS,\
Akademgorodok, Krasnoyarsk 660036, Russia;\
$^{3}$ Omsk State University, Omsk, Russia
title: '**Legendre Integrators, Post-Processing and Quasiequilibrium**'
---
**Introduction**
================
There are many attempts to fill the gap between the microscopic and the macroscopic models (the famous micro-macro gap), and to construct closed macroscopic equations. Most of the closure assumptions have a relatively narrow domain of applicability, and their usage has the following problems:
1\) Violation of the basic physics (thermodynamics) laws;
2\) Absence of the accuracy control procedures;
3\) Absence of the successive step-by-step procedure of the refinement of a model.
The main object of investigation is the evolution equation $$\begin{aligned}
\label{eqn1}
\dot{\Psi}=J(\Psi);\end{aligned}$$ where $J$ is some operator, and $\Psi$ is the distribution function over the phase space.
The constructed methods are aimed at extracting the dynamics of the macroscopic variables from the microscopic equations (\[eqn1\]). The prototypes of these methods are the quasiequilibrium approximation, dual integrators and the thermodynamic projector.
The quasiequilibrium closure for the set of the macroscopic variables $M(\Psi)$ is built with the help of the solution to the variation problem (MaxEnt approximation)[^2]: $$\begin{aligned}
\label{eqn2}
S(\Psi)\rightarrow max\nonumber\\
\\
M(\Psi)=M,\nonumber\end{aligned}$$ where $S(\Psi)$ is the entropy.
The quasiequilibrium closure is always thermodynamically consistent, but the problem 2 (the absence of the accuracy control) remains unsolved, and the problem 3 (the absence of the refinement procedures) can be solved by adding new macroscopic variables to the problem (\[eqn2\]). But uncontrolled enlargement of the macroscopic variables set give us no guarantee of the accuracy improvement. There exists one more specific problem for the quasiequilibrium approximation (\[eqn2\]). Usually while solving the variation problem (\[eqn2\]) we can find explicit dependencies $\Psi(\Lambda)$ and $M(\Lambda)$, where $\Lambda$ are the corresponding Lagrange multipliers (dual variables), more or less easily. Much more difficult is to find the dependencies $\Lambda(M)$ and $\Psi(M)$ which we need for the closure of the macroscopic equations.
The method of the Legendre integrators consists of building and solving the equations of motion for the dual variables. The methods of the first order, based on this idea were suggested and tested in the papers [@IKOePhA02; @IKOePhA03; @GKIOeNONNEWT2001]
The method of the thermodynamic projector let us to represent every ansatz-manifold as the solution to the variation problem (\[eqn2\]) with the specially chosen constraints.
The thermodynamic projector is the unique operator which transforms the arbitrary vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function (and also this happens on any manifold which is not tangent to the level of the Lyapunov function).
Equations which are derived by the method of the thermodynamic projector are always [**thermodynamically consistent**]{}. Although this idea was published in the year 1992 [@GK1], the full construction is published only recently in application to the chemical kinetics [@InChLANL].
One of the problems, discussed in this paper, is to construct the method of the thermodynamic projector for the derivation of the physically consistent macroscopic equations for the polymer dynamics. In the process of building the thermodynamic projector and the quasiequilibrium approximation is involved the Lyapunov function for the equations (\[eqn1\]) which is the entropy $S$. The equations for the polymer dynamics (Fokker-Planck equation) allows us to use the huge amount of different Lyapunov functions and each of them can be formally chosen to describe the macroscopic processes. We need to analyze the different Lyapunov functions for the Fokker-Planck equation.
The problem of the accuracy estimation of the resulting approximations and their further improvement is suggested to solve with the procedures of the post-processing.
Suppose that for the dynamical system (\[eqn1\]) the approximate invariant manifold has been constructed and the approximate slow motion equations $\Psi_{M}(t)$ have been derived:
$$\label{slag}
\frac{d\Psi_{M}}{dt} = P_{\Psi_{M}}(J(\Psi_{M})),$$
where $P_{\Psi_{M}}$ is the corresponding projector onto the tangent space $T_{\Psi_{M}}$ of $\Psi_{M}$. Suppose that we have solved the system (\[slag\]) and have obtained $\Psi_{M}(t)$. Let’s consider the following two questions:
- [How well this solution approximates the true solution $\Psi(t)$ given the same initial conditions?]{}
- [How is it possible to use the solution $\Psi_{M}(t)$ for it’s refinement without solving the system (\[slag\]) again?]{}
These two questions are interconnected. The first question states the problem of the accuracy estimation. The second one states the problem of post-processing.
The corresponding methods to answer these questions are developed and described in this work.
**Elimination of fast variables with the help of the Lyapunov function**
========================================================================
The most popular way to investigate the dynamics of complicated systems is to split the motion into the slow and the fast components, and then to exclude the fast component. As a result, one gets a system of equations that describes the evolution of the slow variables. The necessary conditions of usefulness of this method are usually formulated as a set of restrictions for the possible dynamics of the “fast subsystem". Here the “fast subsystem" is the subsystem which describes the evolution of the fast variables with an assumption that slow variables are constant.
Unfortunately, often appear situations where we cannot avoid using this method, and there is no proof that it is valid. These situations appear almost everywhere in physical kinetics. Here one follows with the same scheme: the relaxation processes are splitted into slow and fast. In spite of the fact that in most cases the proofs of validity of this scheme are absent, the experience helps to avoid fatal errors.
In this section the method to obtain the equations of the macrokinetics from the microdescription is demonstrated. The basis of the analysis is the assumption that if the macroscopic variables are chosen in the proper way, then all other variables relax fast: the probability distribution of the microscopic variables after a small period of time is determined with good accuracy by the macroscopic variables. Let us call this assumption the “quasiequilibrium hypothesis".
The notion “macroscopic variables" is a somewhat relative and is introduced to stress the difference of these variables from “everything else". For example, one-particle distribution function can be “macroscopic" for the full description of the system.
The goal of this section is to describe the most primitive procedure of derivation of the equations for the slow variables and to discuss the form of these equations.
In this paper the reduction of description goes on with the help of the Lyapunov functions. This formalism is the case of the known principle of the conditional maximum of entropy with given values of the macroscopic variables.
Let us review the basic notions of the convex analysis which are used here.
The subset $U$ of the vector space $E$ is convex, if for every two points $x_{1},x_{2}\in U$ it contains the segment between $x_{1}$ and $x_{2}$: for every $\lambda\in[0,1]$ $$\label{oprvypmn}
\lambda x_{1}+(1-\lambda)x_{2}\in U.$$ The intersection of any number of the convex sets is convex.
The convex envelope of the subset $M$ of a vector space $E$ is the smallest convex set $co M\subset
E$, that includes $M$. It is the intersection of all the convex sets, that include $M$.
If the set $U\subset E$ is convex , $x_{1},...,x_{k}\in U$, $\lambda_{1},...,\lambda_{k}\geq0$, $\sum_{i}\lambda_{i}=1$, then $\sum_{i}\lambda_{i}x_{i}\in U$. It leads to another definition of the convex envelope: $$\label{oprconenv}
coM=\{\sum_{i=1}^{k}\lambda_{i}x_{i}|x_{1},...,x_{k}\in M,\:\:
\lambda_{1},...,\lambda_{k}\geq0,\:\:\sum_{i}\lambda_{i}=1, k<\infty\}.$$ If $dimE=n$, then in the equation (\[oprconenv\]) it is sufficient to take $k\leq n+1$ (Carthedory theorem).
The function $f$, defined on the convex set $U\subset E$, is convex, if its epigraph, i.e. the set of pairs $$\label{oprnadgr}
Epif=\{(x,g)|x\in U, g\geq f(x)\},$$ is the convex set in $E\times R$. Sometimes it is convenient to consider functions which can reach the value $f=\infty$. If there occurs a necessity to study the functions $f$ which are defined on the non-convex set $V\subset E$, then it is supposed that $f$ is convex, if the restriction of $f$ onto every convex subset of $V$ is convex. If the restriction of $f$ onto every line segment from the region of definition is convex, then $f$ is convex. The differentiable function $f$ of the class $C^{2}$ is convex if and only if the matrix of the second derivatives $\partial^{2}f/\partial
x_{i}\partial x_{j}$ is nonnegative defined (i.e. all its eigenvalues are nonnegative). The smooth convex function $f$ on the convex set $U\subset R^{n}$ satisfies the inequality $$\label{neqconv}
f(x^{1})-f(x^{2})\geq(\nabla f|_{x^{2}},x^{1}-x^{2})=\sum_{i}(\partial f/\partial
x_{i})_{x=x^{2}}(x^{1}_{i}-x^{2}_{i}), (x^{1},x^{2}\in U).$$ Geometrically it means, that the graph of $f$ is located above the hyperplane, tangent at the point $x=x^{2}$.
The function $f$ is called strictly convex if in the domain of the definition there is no line segment, on which it is constant and finite ($f(x)=const\neq\infty$). The sufficient condition for the differentiable function $f$ of the $C^{2}$ class to be strictly convex is that the matrix of the second derivatives $\partial^{2}f/\partial x_{i}\partial x_{j}$ is positive defined.
In the set of the maximum points of the convex function $f$ on the compact set $U$ ($U$ may be not convex) there are some boundary points of $U$, and if $U$ is convex, then there are some extreme points of $U$. The set of the minimum points of $f$ on the convex set $U$ is convex (but may be empty). The strictly convex continuous function has its maximum only in the boundary points of $U$, and if $U$ is convex, then in the extreme points. The strictly convex function may have the finite minimum only in one point.
The function $f$ called concave if the function $-f$ is convex.
Every bounded convex function on the open subset of $R^{n}$ is continuous.
Let the $C^{2}$-smooth function $H$ be defined in the domain $U\subset R^{n}$. Let us correspond the vector $\mu=\nabla_{x}H:\mu_{i}=\partial H/\partial x_{i}$ to every point $x\in U$. If the matrix $\partial\mu_{i}/\partial x_{j}=\partial^{2}H/\partial x_{i}\partial x_{j}$ is non-degenerated, then for the transform $x\rightarrow\mu$ there locally (in the neighborhood of every point) exist the differentiable inverse transform. The variables $\mu$ are often called conjugated variables, and the transform $x\rightarrow\mu$ is called “transition to the conjugated coordinates". Let the transform $x\rightarrow\mu$ be invertible on the open set $V\subset U$. This means that the function $x(\mu)$ is defined on $V$. Assuming the smoothness of this function, we describe the inverse transform $\mu\rightarrow x$ in the same way as the direct. For this purpose we introduce a function $$\begin{aligned}
\label{legtrans}
G(\mu)=(\mu,x(\mu))-H(x(\mu))=\sum_{i}\mu_{i}x_{i}(\mu)-H(x(\mu)), \nonumber \\ \frac{\partial
G}{\partial\mu_{i}}=x_{i}+\sum_{j}\mu_{j}\frac{\partial
x_{j}}{\partial\mu_{i}}-\sum_{j}\frac{\partial H}{\partial x_{j}}\frac{\partial
x_{j}}{\partial\mu_{i}}=x_{i}.\end{aligned}$$ The function $G$ called the Legendre transform of $H$.
With the help of the conjugated coordinates it is possible to write down the necessary conditions of the extremum for the problems with the linear constraints on the open set in a very simple way: $$\label{mtsk}
\begin{array}{l}
H(x)\rightarrow min,\\ \sum_{j}m_{ij}x_{j}=M_{i},\: (i=1,...,k),\: x\in U.
\end{array}$$ With the method of Lagrange multipliers we get the system of the equations which is giving us the necessary conditions for the solution to the problem (\[mtsk\]): $$\label{nmcond}
\begin{array}{l}
\mu_{j}=\sum_{i}\lambda_{i}m_{ij}, \: j=1,...,n,\\ \sum_{j}m_{ij}x_{j}=M_{i}, \: (i=1,...,k),
\end{array}$$ where the $\lambda_{i}$ are the Lagrange multipliers. The necessary conditions for the extremum are given by the system of the equations (\[nmcond\]). One part of the system is linear in the $x$ coordinates, and the other part is linear in the conjugated coordinates $\mu$.
Let us have the Legendre transform $G(\mu)$ for the function $H(x)$, let the transform $x\rightarrow\mu$ have the smooth reverse, and let the solution to the problem (\[mtsk\]) be unique for some open set of the values of the vector $(M_{1},...,M_{k})\in R^{n}$. Also let the point of the minimum $x_{min}$, and, consequently, the minimal value of $H$ be smooth dependent on $M$, $H_{min}=H(M)$. Let us denote $\mu_{M_{i}}=\partial H(M)/\partial M_{i}$, $\mu_{M}=(\mu_{M_{1}},...,\mu_{M_{k}})$. Let us get some information about the function $H(M)$ from the functions $H(x)$ and $G(\mu)$ without solving any equations. With the known value of the vector $\mu_{M}$ we can immediately find the vector $\mu$ in the corresponding point of the conditional minimum, $\mu_{j}=\sum_{i}\mu_{M{i}}m_{ij}$. From this equality we get $$\label{xotmum}
x(\mu_{M})=(\nabla_{\mu}G(\mu))|_{\mu_{j}=\sum_{i}\mu_{M_{i}}m_{ij}}.$$ From $x_{\mu_{M}}$ we obtain $M(\mu_{M})$ and $H(M(\mu_{M}))$: $$\label{himotxmum}
M_{i}(\mu_{M})=\sum_{j}m_{ij}x_{j}(\mu_{M}), H(M(\mu_{M}))=H(x(\mu_{M})).$$ Finally, the Legendre transform $G(\mu_{M})$ for the function $H(M)$ is: $$\label{ltrotxmu}
G(\mu_{M})=(\mu_{M},M(\mu_{M}))-H(M(\mu_{M}))=G(\mu(\mu_{M})).$$ So, we can find dependencies $\mu(\mu_{M})$, $x(\mu_{M})$, $M(\mu_{M})$, $H(\mu_{M})$ and $G(\mu_{M})$ from the functions $H(x)$ and $G(x)$ without solving any equations. We hope, that the similar notations for $H(x)$ and corresponding conditional minimum function $H(M)$, and for their Legendre transforms $G(\mu)$ and $G(\mu_{M})$ will not cause a confusion. Let us note, that with our assumptions the reversibility of the transform $M\rightarrow\mu_{M}$ follows from the reversibility of the transform $x\rightarrow\mu$, and moreover, the function $M(\mu_{M})$ can be found explicitly.
The convexity of the function $H(x)$ usually makes our assumptions (existence and uniqueness of the conditional minimum, global reversibility of the transform $x\rightarrow\mu$, smoothness of the function $H(M)$) easier to check. Note, that the convexity of the function $H(M)$ is neither necessary nor sufficient condition for our assumptions. If $H(x)$ is convex, then the function of the conditional minimum $H(M)$ is convex too.
Now we proceed to the problem of elimination of the fast variables. Let us have the system of differential equations with smooth right-hand sides $$\label{sisdif}
\dot{x}=F(x),$$ in the convex domain $U\subset R^{n}$, and moreover let the linear transform $x\rightarrow M$, $M_{i}=\sum_{j}m_{ij}x_{j}$ from the phase space to the space of the slow variables $M$ be defined. We can assume, that we have no linearly dependent rows in the matrix $m_{ij}$, because it is always possible to eliminate the linear dependent functions $M_{i}(x)$, if they are present.
Let us assume that in the interesting for us domain of the initial conditions $x_{0}$ the solutions $x(t)$ of the equations (\[sisdif\]) are developing in the following way: the vector $x(t)$ is going fast to the value which is defined by the slow variables $M$; after that $x$ can be represented as the function of $M$ with a good accuracy, and this function is unique for every initial conditions. So,
A)For each value of the slow variables $M\in M(U)$ there exist such $x=x^{*}(M)$, that if $M(x^{0})=M^{0}$, then $x(t)$ is going very fast to some small neighborhood of the $x^{*}(M^{0})$, and during that $M(x(t))$ is almost constant.
B)In the process of the further evolution, $x(t)$ stays in the small neighborhood of the value of $x$ which corresponds to $M(x(t))$, so $x$ is close to $x^{*}(M(x(t)))$.
It is usually impossible to give a strong proof for A and B for the situations of real complexity in the nonequilibrium thermodynamics, so this assumptions are, probably, the weakest point of all the construction. We are accepting them because we are sure that the evolution of the macroscopic variables is possible to describe by the autonomous system of differential equations of the first order (if it is impossible, then, probably, one should extend the list of the macroscopic variables with respect to the physical properties of the investigated process). There is another way to deal with this problem: to equip the obtained approximations by the [*postprocessing*]{}. The postprocessing helps us to correct the errors, if they are not too big, and gives us a signal if they are too big.
If we know the function $x^{*}(M)$, then we can write $$\label{equaproc}
\dot{M}=mF(x^{*}(M)),\:\: \dot{M}_{i}=\sum_{j}m_{ij}F_{j}(x^{*}(M)).$$ In general, this equation can be used only for short periods of time which do not exceed some limit. The right-hand side $mF(x^{*}(M))$ of the equations (\[equaproc\]) is not exactly the $mF(x(t))$, and it may cause the error increment, and as a result the solution of the equations (\[equaproc\]) will divert from the true solution strongly. The exclusion is the case when in accordance to the equations (\[equaproc\]) $M(t)$ tends to the only stable fixed point when $t\rightarrow\infty$. If the solution of the equations (\[equaproc\]) and the real values of $M(x(t))$ are not succeed in getting far one from another during the time in which the solution of the equations (\[equaproc\]) is coming in the small neighborhood of the fixed point, then the equations (\[equaproc\]) can be used also for $t\rightarrow\infty$.
The function $x^{*}(M)$ for the particular system is not unique, but the range of choice is small in that sense, in which the neighborhood of $x^*(M(x(t)))$ (in which the evolution goes after the short period of time) is small.
Let us have the Lyapunov function $H(x)$ for the system (\[sisdif\]) which is decreasing along the trajectories. We can try to find the dependence $x^{*}(M)$ as the solution to the problem $H(x)\rightarrow min$, $mx=M$. This way seems to be natural, but it does not follow directly from the assumptions A and B. For example, there could be a situation in which $H$ is very sensitive to small changes of the slow variables, and not sensitive to the changes of the fast variables. In this situation the assumption, that $x^{*}(M)$ is the point of conditional minimum of the function $H$, may not give the desired result. The following idea does not solve the problem, but it can be useful: In the applications, the system (\[sisdif\]) usually dependens on some parameters. It seems to be more reasonable to use the Lyapunov function which does not depend on these parameters, if there exists such a function. It is most important in the case when among the parameters we have such, that their values are determining, whether is it possible to split the variables to fast and slow, or not.
So, the fast variables will be eliminated with the help of the Lyapunov function. Let us have the Lyapunov function $H$ for the initial system, let the transform $x\rightarrow\mu=\nabla_{x}H$ have the smooth inverse, and let us know the Legendre transform $G(\mu)$ for the function $H(x)$. Here it is also assumed that for every $M\in M(U)$ the problem (\[mtsk\]) has the unique solution, and the minimum point $x^{*}(M)$, and the function of the conditional minimum $H(M)$ smoothly depend on $M$. With the value $\mu_{M}=\nabla_{M}H(M)$ it is possible to find $\mu(\mu_{M})$, $x(\mu(\mu_{M}))$ (look at the (\[xotmum\]-\[ltrotxmu\])). The result is $$\label{mtchkn}
\dot{M}=mF(\nabla_{\mu}G(\mu))|_{\mu=\mu_{M}m},$$ where $\mu_{M}m$ is the product of the row vector $\mu_{M}$ and the matrix $m$: $$\begin{aligned}
(\mu_{M}m)_{j}=\sum_{i}\mu_{M_{i}}m_{ij},\end{aligned}$$ $\nabla_{\mu}G$ is the vector with the components $\partial G/\partial\mu_{i}$, and all derivatives are taken in the point $\mu=\mu_{M}m$. The right-hand sides of (\[mtchkn\]) are defined as the functions of $\mu_{M}$. In order to define them as functions of $M$, one needs to make the Legendre transform, find the function $H(M)$ and, respectively, $\mu_{M}=\nabla_{M}H(M)$ from the function $G(\mu_{M})$ (\[ltrotxmu\]). It is impossible to make these calculations explicitly in such a general case. It seems to be a very natural and convenient to define the right-hand sides of the kinetic equation as the functions of the conjugated variables. If in the beginning the right-hand sides of the equation (\[sisdif\]) are defined as the functions of $\mu$ (i.e. $\dot{x}=J(\mu)$), then the equations (\[mtchkn\]) have a very simple form: $$\label{mtchkn1}
\dot{M}=mJ(\mu_{M}m).$$ $H(M)$ is the Lyapunov function for (\[mtchkn\]), its time derivative due to the system (\[mtchkn\]) is not positive: $$\label{teors}
\dot{H}(M)=(\mu_{M}, mJ(\mu_{M}m))=(\mu_{M}m, J(\mu_{M}m))\leq0,$$ because $(\mu,J(\mu))=\dot{H}(x)\leq0$.
Let us call the systems dissipative, if $\dot{H}\leq0$ and conservative, if $\dot{H}=0$. For the dissipative system we have $\dot{H}(M)\leq0$ (\[teors\]), and if the system is conservative, then for all values of $\mu$ we have $(\mu,J(\mu))=\dot{H}(x)=0$, and then from the equation (\[teors\]) we get $\dot{H}(M)=(\mu_{M}, mJ(\mu_{M}m))=(\mu_{M}m, J(\mu_{M}m))=0$. So, we proved the following
[**Theorem**]{}[^3]. [*The Lyapunov function for the microscopic system remains the Lyapunov function for the macroscopic system, and if the microscopic system is conservative, then its projection to the space of the macroscopic variables remains conservative.*]{}
If necessary, it is easy to perform further exclusion of the variables in the equations (\[mtchkn\]) with the help of the function $H(M)$. The right-hand sides of the resulting equations will be defined again as the functions on the conjugated variables, and the function of the conditional minimum will be the Lyapunov function again. Let us note that in (\[mtchkn1\]) we have neither $H$ nor $G$ in the explicit form (they occur only when we need to find the connections between $M$ and $\mu_{M}$ or $x$ and $\mu$).
Convexity of $H$ was never used above, but the natural domain of applicability of the described formalism are systems with convex Lyapunov functions $H(x)$, or at least with such $H$, that the sets $\{x|H(x)<h\}$ are convex. Otherwise there exist such linear manifolds, that the local minimum of $H$ is not unique on them, and further considerations are required to select the relevant minima. The finite dimensionality of the phase space is not so important, because everything said above can be applied to the infinite-dimension case with proper restrictions. Let $E$ be the Banach space, $U\subset E$ be the convex open set, $H:U\rightarrow R$ be $C^{2}$-smooth function. With every point $x\in U$ we associate the linear functional $\mu_{x}\in E^{*}$: $\mu_{x}=\nabla_{x}H$ which is the differential of $H$ in the point $x$. Let $V$ be the set of the values of $\mu_{x}$ for $x\in U$ and let us have the smooth mapping $J$ from $E^{*}$ to $E$ in the neighborhood of $V$. The system $(U,H,J)$ determines the system of equations: $$\label{sdinf}
\dot{x}=J(\mu_{x}).$$ Let $L$ be the closed subset of $E$ and for every $M\in U/L$ let the problem $H(x)\rightarrow min,\:
x/L=M,\: x\in U$ have the unique solution $x_{min}$ which is $C^{2}$-smooth dependent on $M$, $H(M)=H(x_{min})$. Denoting $\mu_{M}=\nabla_{M}H(M)\in(E/L)^{*}\in E^{*}$ we can define the factor-system which is the exact analogue of (\[mtchkn\]): $$\label{faksis}
\dot{M}=J(\mu_{M})/L.$$ Here the argument $J$ is the linear functional on $\mu_{M}$: $\mu_{M}x=\mu_{M}(x/L)$
The described procedure of the elimination of variables has one very important commutativity property: If one makes a further simplification and transact to the variables $N=N(M)$, then after the application of the described formalism to the system (\[faksis\]) with the function $H(M)$, one get the same result as after the application of this formalism directly to the reduction from $x$ to $N(x)=N(M(x))$. So, the chain of exclusions $x\rightarrow M\rightarrow N$ gives us the same result as the direct exclusion $x\rightarrow N$.
**The main problems in usage of the quasiequilibrium approximations**
=====================================================================
Our problem is to build the closed system $$\begin{aligned}
\dot{M}=J(M),\end{aligned}$$ from the initial system (\[eqn1\]) and its Lyapunov function.
If we know the function $x^{*}(M)$ then it is sufficient to calculate $m(F(x^{*}(M)))$. This problem is the problem of the calculation of the projection of the microscopic vector field $F$ on the macroscopic variables $M$ in known point $x^{*}(M)$. Let us call this problem the problem about the macroscopic projection. If the right-hand parts are expressed through $\mu$ then we have the problem about the macroscopic projection too.
Another problem is to find $\mu_{M}$. Usually it is necessary to solve the system of non-linear equations (if the function $H$ is not quadratic) to solve this problem. Indeed, let us consider the conditions for the conditional extremum of $H$ with given values of the moments $M$. From the functions $H(x)$, $G(\mu)$ we get $\mu(\mu_{M})$, $x(\mu_{M})$, $M(\mu_{M})$, $H(M(\mu_{M}))$. But in this list we have no function $\mu_{M}(M)$. We can find this function as the solution of the equation $$\label{eq16}
M(\mu_{M}m)=M.$$
Let us give a few examples.
[*One-particle approximation*]{}. Let $x$ be the $N$-particle distribution function, $f_{N}(\xi_{1},...,\xi_{N})$, where $\xi_{i}$ is vector of coordinates and momenta of the $i$-th particle, and let the evolution of this function be described by the linear equation $$\label{eq17}
\frac{\partial f_{N}}{\partial t}=Lf.$$ Furthermore, let $M$ be one-particle distribution function $$f_{1}(\xi)=N\int f_{N}(\xi,\xi_{2},...,\xi_{N})d\xi_{2}...d\xi_{N},$$ and $H$ be the entropy (we use the $H$-function which is equal to the minus entropy) $$\label{eq18}
H(f_{N})=\int f_{N}(\ln f_{N}-1)d^{N}\xi,$$ For given $f_{N}$, $H$, $f$ we get $\mu=\ln f_{N}$, $f_{N}=\exp{\mu}$, $$\label{eq19}
G(\mu)=\int\exp{\mu(\xi_{1},...,\xi_{N})}d^{N}\xi,$$ $m(f_{N})=\int\sum_{i=1}^{N}\delta(\xi-\xi_{i})f_{N}(\xi_{1},...,\xi_{N})d^{N}\xi$; the extremum conditions (\[nmcond\]) are of the form $$\begin{aligned}
\label{eq20}
\mu(\xi_{1},...,\xi_{N})=\int d\xi
\mu_{1}(\xi)\sum_{i=1}^{N}\delta(\xi-\xi_{i})=\sum_{i}\mu_{1}(\xi_{i}), \nonumber \\
f_{N}=\exp{\sum_{i}\mu_{1}(\xi_{i})}.\end{aligned}$$
The normalization condition here is $\int f_{N}d^{N}\xi=1$, that is $$\int\exp{\mu_{1}(\xi)}d\xi=1.$$ Connection between the macroscopic variables $f_{1}$ (that is $M$) and the quasiequilibrium values of the microscopic variables $f^{*}_{N}$ (that is, $x^{*}_{M}$) is given by well known formula: $$\label{eq21}
f_{N}(\xi_{1},...,\xi_{N})=\frac{1}{N^{N}}f_{1}(\xi_{1})...f_{1}(\xi_{N}).$$ Projection of the microscopic vector field (\[eq17\]) can be found by direct integration.
[*Two-particle distribution function as the macroscopic variable*]{}. One-particle distribution function $f_{1}(\xi)$ is often not sufficient because, for example, the energy of the interaction of pairs of particles cannot be found from this function. Much more detailed description is given by the two-particle distribution function $$\label{eq22}
f_{2}(\xi_{1},\xi_{2})=N(N-1)\int f_{N}(\xi_{1},...,\xi_{N})d\xi_{3}...d\xi{N}.$$ We can easily find the expression $$\begin{aligned}
\mu(\xi_{1},...,\xi_{N})=\sum_{i,j,i\neq j}\mu_{2}(\xi_{i},\xi_{j});\\
f_{N}(\xi_{1},...,\xi_{N})=\exp{\mu}=\exp{\sum_{i,j,i\neq j}\mu_{2}(\xi_{i},\xi_{j})}.\end{aligned}$$ But it is difficult to find the connection between $\mu_{2}$ and $f_{2}$ explicitly. Only a series expansion for it in the neighborhood of the uncorrelated state is known [@BGKTMF]. The problem about the macroscopic projection becomes hard too: the necessary integrals in general case are impossible to find analytically. For two-particle distribution functions as well as for majority of the most interesting variables the transform $M\leftrightarrow\mu_{M}$ is very complicated in the direct direction and not very simple (as simple as the derivation of $f_{2}$ from $f_{N}$) in the opposite direction.
So, we need to avoid the necessity to calculate $\mu_{M}(M)$ (and, if possible, to make less calculations to find $M(\mu_{M})$).
The first of these two problems (avoiding calculation of $\mu_{M}(M)$) is solved by the method of the Legendre integrators which is developed by us [@IKOePhA02].
**Legendre integrators**
========================
The main idea of the Legendre integrators is to find some alternate way to solve the macroscopic equations $\dot{M}=J(x)$: a way to find their solution in the absence of the explicit form of these equations.
First of all, note, that we have a linear connection between $\dot{M}$ and $\dot{\mu}_{M}$: $$\label{conn}
\frac{dM}{dt}=(m(D^{2}_{x}S(x))^{-1}m^{T})\frac{d\mu_{M}}{dt};$$ $$\begin{aligned}
\label{mjumu}
\dot{M}=\frac{d}{dt}(mx(\mu_{M}m))=m(D_{\mu}x)m^{T}\mu;\nonumber\\\\
D_{\mu}x=(D_{x}\mu)^{-1}=(D^{2}_{x}S(x))^{-1}\nonumber.\end{aligned}$$
Calculation of the functions $m(F(x))$ is the standard problem of the macroscopic projection. Dependencies $x(\mu_{M})$ are usually quite simple. We suggest the following advancing in time to solve (unknown) equations $\dot{M}=\Phi(M)$: $$\label{dual}
\mu_{M}(t)\rightarrow x=x(\mu_{M})\rightarrow
\dot{M}\rightarrow\dot{\mu}_{M}\rightarrow\mu_{M}(t+\Delta t)\rightarrow M(t+\Delta t).$$ In the sequence (\[dual\]) there is one operation of macroscopic projection and one operation of solving the system of linear equations (\[conn\]).
Formally, it is possible to write down the equations for $\mu_{M}$: $$\label{mudot}
\frac{d\mu_{M}}{dt}=(m(D^{2}_{x}S(x))^{-1}m^{T})^{-1}mF(x),$$ where $x=x^{*}_{M}$.
Nevertheless, explicit inversion of the operator in the right-hand part of the equation (\[mudot\]) is usually difficult and one should use the chain of computations (\[dual\]). In our first calculations using of the Legendre integrators [@IKOePhA02; @IKOePhA03] the methods of the first order of accuracy were used. This is not the principal restriction: the scheme (\[dual\]) gives us a possibility to calculate $\dot{\mu}_{M}$ for any given $\mu_{M}$, so all known methods of the higher order can be used (for example, the Runge-Kutta method with the different procedures of the automatic step selection [@Gustafsson; @Hairer; @Hairer2]).
**Lyapunov functions for the Fokker-Planck equation**
=====================================================
The Fokker-Planck equation (FPE) in the absence of the drive forces has the form $$\label{FPE}
\frac{\partial\Psi(q,t)}{\partial t}=\nabla_{q}\{D(\Psi(q,t)\nabla_{q}U(q)+\nabla_{q}\Psi(q,t))\},$$ where $\Psi$ is the probability density over the configuration space, $q$ is the point of this space, $\Psi(q)$ is the function of the time $t$, $U(q)$ is the normalized potential energy ($U=U_{potential}/kT$), $D(q)$ is positively semidefinite diffusion operator ($(y_{i},D_{y})\geq0$).
The FPE has two important properties:
1\) Conservation of the total probability: $$\label{procons}
\frac{d}{dt}\int\Psi(q,t)dq\equiv0.$$
2\) Dissipation: for every convex function of one variable $h(a)$ ($h''(a)>0, a\geq0$) the following functional $S[\Psi]$ is monotonically non-increasing in time: $$\label{Lyapfunc}
S[\Psi]=-\int\Psi^{*}(q)h\left(\frac{\Psi(q)}{\Psi^{*}(q)}\right)dq,$$ where $$\label{equilFPE}
\Psi^{*}(q)=const\cdot\exp(-U(q)),$$ is the Boltzmann-Gibbs distribution.
For $h(a)=a\ln a$, the functional $S[\Psi]$ is the usual Boltzmann-Gibbs-Shannon entropy: $$\label{SBGS}
S[\Psi]=-\int\Psi^{*}(q)\ln\left(\frac{\Psi(q)}{\Psi^{*}(q)}\right)dq,$$ Let us calculate the time derivative of $S[\Psi]$ due to FPE (\[FPE\]). Note, that $$\begin{aligned}
\nabla_{q}\left(\frac{\Psi(q)}{\Psi^{*}(q)}\right)=\frac{\nabla_{q}\Psi(q)+\Psi(q)\nabla_{q}U}{\Psi^{*}(q)},\end{aligned}$$ so we can rewrite FPE as follows: $$\begin{aligned}
\frac{\partial\Psi(q,t)}{\partial
t}=\nabla_{q}D\left(\Psi^{*}(q)\nabla_{q}\left(\frac{\Psi(q)}{\Psi^{*}(q)}\right)\right).\end{aligned}$$ Let us consider FPE in the domain $\Omega$. Function $dS/dt$ consists of two summands: The first is the integral of the local “production of $S$", $\int\sigma(q)dq$, and the second is the flow through the boundary of the domain $\Omega$: $$\begin{aligned}
\frac{dS(\Psi)}{dt}=-\int_{\Omega}h'\left(\frac{\Psi}{\Psi^{*}}\right)\nabla_{q}\left(D\Psi^{*}\left(\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right)\right)\right)dq=\\
-\int_{\Omega}div
\left[h'\left(\frac{\Psi}{\Psi^{*}}\right)D\Psi^{*}\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right)\right]dq+\int_{\Omega}\Psi^{*}h''\left(\frac{\Psi}{\Psi^{*}}\right)\left(\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right),D\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right)\right)dq=\\
\int_{\partial\Omega}\Psi^{*}h'\left(\frac{\Psi}{\Psi^{*}}\right)\left(\nu_{q},D\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right)\right)dw+\int_{\Omega}\sigma(q)dq,\end{aligned}$$ where $dw$ is the differential of the area, $\nu_{q}$ is a vector of the unitary normal to $\partial\Omega$ in the point $q$, $\sigma(q)$ is the entropy $S$ production: $$\label{sigma}
\sigma(q)=\Psi^{*}h''\left(\frac{\Psi}{\Psi^{*}}\right)\left(\frac{\Psi}{\Psi^{*}},D\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right)\right)\geq0.$$ Let the flow of $\Psi$ through the boundary $\partial\Omega$ be equal to zero: $$\begin{aligned}
\left(\nu_{q},D\nabla_{q}\left(\frac{\Psi}{\Psi^{*}}\right)\right)=0,\end{aligned}$$ at all points of $\partial\Omega$. Then $$\begin{aligned}
\frac{dS}{dt}=\int_{\Omega}\sigma(q)dq\geq0.\end{aligned}$$ The most important cases of $S$ selection are:
$h(q)=a\ln a$, $S$ is the Boltzmann-Shannon-Gibbs entropy;
$h(a)=a\ln ax-\alpha\ln ax$ is the maximal family of [*additive trace-form*]{} entropies [@ENTR1; @ENTR2; @ENTR3] (these entropies are additive for composition of independent subsystems);
$h(a)=\frac{1-a^{\alpha}}{1-\alpha}, \alpha\neq1$ is the Tsallis entropy [@Abe]. These entropies are not additive, but become additive after nonlinear monotonous transformation. This property can serve as definition of the Tsallis entropies in the class of generalized entropies (\[Lyapfunc\]) [@ENTR3].
**Macroscopic variables and quasiequilibrium distribution functions for FPE**
=============================================================================
The set of the macroscopic variables can be continuous or discrete. Let $\alpha$ be the discrete or continuous parameter, that enumerates the macroscopic variables, and $M_{\alpha}$ be the corresponding variables. Every macroscopic value $M_{\alpha}$ is defined by its microscopic density $m_{\alpha}(q):$ $$\label{macro}
M_{\alpha}=\int_{\Omega}m_{\alpha}(q)\Psi(q)dq$$ The choice of the domain $\Omega$, in which we are solving the FPE, needs to be discussed separately. We can suppose formally, that $\Omega=R^{n}$, but for the calculations it is better to make it as small as possible with the preservation of the accuracy. Usually, when $\|q\|\rightarrow\infty$ the function $\Psi(q)$ tends to zero faster, then exponential, and we can a priori select the bounded domain $\Omega$, out of which $\Psi$ is negligibly small.
We shall do the calculations for the general form of $S$ (see equation (\[Lyapfunc\])) and give the examples for the most popular choice (\[SBGS\]) of $S$.
Quasiequilibrium function $M_{\alpha}$ for the given Lyapunov function $S$ (\[Lyapfunc\]) is defined as the solution to the problem $$\label{maxent}
\left\{\begin{array}{lc}S(\Psi)\rightarrow\max&\\ \int
m_{\alpha}(q)\Psi(q)dq=M_{\alpha}&\end{array}\right.,$$
Due to the convexity of $h$ (and, consequently, concavity of $S$), it is sufficient to investigate the conditions of the local extremum: $$\label{extnes}
D_{\Psi}S=\sum_{\alpha}m_{\alpha}(q)\mu_{\alpha},$$ where $\mu_{\alpha}$ are variables, dual to $M_{\alpha}$ ($\mu_{M}$). For continuous parameter the sum in the equation (\[extnes\]) is replaced by integration on $\alpha$.
Next, we use the standard Riesz representation of functionals (through the $L^{2}$ scalar product). Let us write $$\begin{aligned}
D_{\Psi}S(\Psi)=-h'\left(\frac{\Psi}{\Psi^{*}}\right);\\
h'\left(\frac{\Psi}{\Psi^{*}}\right)=-\sum_{\alpha}m_{\alpha}(q)\mu_{\alpha}.\end{aligned}$$ For the quasiequilibrium distribution we have $$\label{QEFPE}
\Psi=\Psi^{*}g\left(-\sum_{\alpha}m_{\alpha}(q)\mu_{\alpha}\right),$$ where $g(a)$ is a function of one variable, inverse to $h'(b)$. Note, that $h'(b)$ is a monotonous increasing function (because $h$ is convex), so $g(a)$ is a monotonous increasing function too, and $g'(a)=(h''(g(a)))^{-1}$.
Let us denote the quasiequilibrium distribution function (\[QEFPE\]) as $\Psi^{qe}(\{\mu_{\alpha}\},q)$.
For the BGS entropy $h(b)=b(\ln b-1)$, $h'(b)=\ln b$, $g(a)=\exp a$, and the equations (\[QEFPE\]) transforms into the following equation: $$\label{QEFPEB}
\Psi^{qe}(\{\mu_{\alpha}\},q)=\Psi^{*}\exp\left(-\sum_{\alpha}m_{\alpha}(q)\mu_{\alpha}\right).$$
For the next steps it is convenient to consider the temperature dependence explicitly (i.e. write $\beta U$ instead of $U$ in FPE, $\beta=1/kT$), then we have $\Psi^{*}=const\cdot\exp(-\beta U)$.
For the classical BGS entropy (\[SBGS\]) the quasiequilibrium distribution will take the simplest form: $$\label{psiqe}
\Psi^{qe}(\{\mu_{\alpha}\},q)=\exp\left(-\mu_{0}-\mu_{U}U-\sum_{\alpha}m_{\alpha}(q)\mu_{\alpha}\right),$$ where $\mu_{U}=\beta=1/kT$, $\mu_{0}$ is a variable, conjugated to $M_{0}=\int_{\Omega}\Psi
dq\equiv1$. The function (\[psiqe\]) is a solution to the problem: $$\label{Clas}
\left\{\begin{array}{l}-\int_{\Omega}\Psi\ln\Psi dq\rightarrow\max\\
M_{0}(\Psi)=\int_{\Omega}\Psi(q)dq\nonumber=1\\
M_{U}(\Psi)=\int_{\Omega}U(q)\Psi(q)dq\nonumber=M_{U}\\
M_{\alpha}(\Psi)=\int_{\Omega}m_{\alpha}(q)\Psi(q)dq\nonumber=M_{\alpha}\end{array}\right.$$ In the problem (\[Clas\]) we move from the relative (so-called Kullback) entropy to the absolute entropy.
Selection of the macroscopic variables is the most critical point in construction of the quasiequilibrium approximations. It is always necessary to select them, basing on the specific of the problem. Nevertheless, there are some simple general recommendations about construction of the set of variables for the Legendre integrators.
1\) It is necessary to include $M_{0}$ in the list of variables, because $\mu_{0}$ is not constant in time;
2)It is useful to include $M_{U}$ in the list of variables. With this variable in the process of the relaxation all other $\mu_{\alpha}\rightarrow0$, and $\mu_{U}\rightarrow1/kT$.
3\) It is better for the set of the functions $m_{\alpha}(q)$ to be linearly independent.
For the classical entropy we have $$\label{QEPsi}
\Psi^{qe}(\{\mu\},q,t)=\exp\left(-\mu_{0}(t)-\mu_{U}(t)U(q)-\sum_{\alpha}m_{\alpha}(q)\mu_{\alpha}(t)\right).$$ Due to the equation (\[QEPsi\]) we have $$\label{Psidot1}
\frac{\partial\Psi}{\partial
t}=-\Psi\left[\frac{d\mu_{0}}{dt}+U(q)\frac{dM_{U}}{dt}+\sum_{\alpha}m_{\alpha}(q)\frac{\mu_{\alpha}}{dt}\right];$$ The FPE gives us $$\begin{aligned}
\label{Psidot2}
&&\frac{\partial\Psi}{\partial t}=\nabla D\left(\Psi^{*}\nabla\frac{\Psi}{\Psi^{*}}\right)=
-\Psi\left.\mbox{\Huge [}(\mu_{U}-\beta)(\nabla,D\nabla)U(q)+\right. \nonumber\\
&&\sum_{\alpha}\mu_{\alpha}(\nabla,D\nabla)m_{\alpha}(q)-
\sum_{\alpha}(2\mu_{U}\mu_{\alpha}-\beta\mu_{\alpha})(\nabla U(q),D\nabla m_{\alpha}(q))-\\
&&\left.\sum_{\alpha,\alpha'}\mu_{\alpha}\mu_{\alpha'}(\nabla m_{\alpha}(q),D\nabla
m_{\alpha'}(q))\right.\mbox{\Huge ]}\nonumber.\end{aligned}$$ To calculate $\frac{dM}{dt}(\{\mu\})$ means to calculate the following integrals: $$\begin{aligned}
\frac{dM_{U}}{dt}=\int_{\Omega}U(q)\frac{\partial\Psi(q)}{\partial t}dq;\\
\frac{dM_{\alpha}}{dt}\int_{\Omega}m_{\alpha}(q)\frac{\partial\Psi(q)}{\partial t}dq,\end{aligned}$$ where $\frac{\partial\Psi}{\partial t}$ is calculated due to equation (\[Psidot2\]), $dM_{0}/dt=0$.
From the equation (\[Psidot1\]) we get the conditions for derivation of $\dot{\mu}$ $$\begin{aligned}
\label{mudoteq}
-\frac{d \mu_{0}}{d
t}-M_{U}\frac{d\mu_{U}}{dt}-\sum_{\alpha}M_{\alpha}\frac{d\mu_{\alpha}}{dt}=\dot{M}_{0}=0;\nonumber\\
-M_{U}\frac{d\mu_{0}}{dt}-\langle U^{2}\rangle_{\Psi}\frac{d\mu_{U}}{dt}-\sum_{\alpha}\langle
Um_{\alpha}\rangle_{\Psi}=\dot{M}_{U};\\ -M_{\alpha}\frac{d\mu_{0}}{dt}-\langle
Um_{\alpha}\rangle_{\Psi}\frac{d\mu_{U}}{dt}-\sum_{\gamma}\langle
m_{\gamma}m_{\alpha}\rangle_{\Psi}\frac{d\mu_{\gamma}}{dt}=\dot{M}_{\alpha},\nonumber\end{aligned}$$ where by $\langle f(q)g(q)\rangle_{\Psi}$ we denote the averaging $\langle
fg\rangle_{\Psi}=\int_{\Omega}f(q)g(q)\Psi(q)dq$.
We get the closed system for derivation of the dynamics of $\mu$. But the question about the choice of the macroscopic variables still remains open.
In the problem of the quasiequilibrium we find the projections of $\Psi$ to the given set of the functions (linear space), afterwards we calculate $\Psi$ due to the maximum entropy condition.
It seems to be physically sensible to choose the additional variables to $M_{0}, M_{U}$ as [*the projections of $\Psi$ onto some equilibrium states*]{}: $$\label{malpha}
M_{\alpha}(\Psi)=\int_{\Omega}e^{-\alpha U(q)}\Psi(q)dq.$$ There are two classical choices of macroscopic variables:
1\) $\alpha=R_{+}$ (Laplace transform of the energy distribution density)
2\) $\alpha=ik, k\in R$ (Fourier transform of the energy distribution density).
The variable $M_{U}$ is the average energy in the potential well $U(q)$. In analogue to this, the variable $M_{\alpha}(\Psi)$ (\[malpha\]) for the real $\alpha>0$ can be considered as the energy in the potential well $e^{-\alpha U(q)}$. This potential is gained by the monotonous nonlinear deformation of the energy scale $U\rightarrow e^{-\alpha U(q)}$. For imaginary $\alpha$ this nonlinear deformation is given by the periodical functions $U\rightarrow \cos(kU)+i\sin(kU)$
A benefit of usage of (\[malpha\]) is also in that $\langle m_{ \alpha}m_{ \alpha'} \rangle =M_{
\alpha+ \alpha'}$, and we have to perform less calculations in (\[mudoteq\]). This set of the deformed energies can be used for both the initial potential $U$ and the set of additional potentials.
Is this set of macroscopic variables sufficient for description of nonequilibrium kinetics of polymers in presence of flow? Probability densities for all the quasiequilibrium distributions which can be constructed with this macroscopic variables have the form $\Psi(q)=\varphi(U(q))$, where $\varphi(U)$ is a function of one variable. Is this class of distributions sufficient for the specific problem? This question can be answered only after specification the problem. But what is possible to do, if the closure with these variables gives too big error (the estimation of accuracy is discussed below)? There are at least two ways: to extend the list of variables or to improve the quasiequilibrium manifold [@GKIOeNONNEWT2001; @GKTTSP94] (application of the methods of invariant manifolds to improving the quasiequlibrium closure for dynamics of dilute polymeric solution is presented in [@ZKD2000]). The extension of the list of variables is the central method of the extended irreversible thermodynamics [@EIT]. It is possible to combine the potential energy $U(q)$, the vector of the configuration space $q$, and the gradient of $U(q)$, $\nabla U(q)=- F(q)$, ($F(q)$ is the force) and to obtain a huge amount of densities $m(q)$ which can be scalars, vectors, or tensors. The corresponding “macroscopic variables" are $\int_{\Omega}m(q)\Psi(q) dq$.
The best hint for a choice of new macroscopic variables is the analysis of the right hand side of dynamic equations [@GKPRE96]. The well known distinguished macroscopic variable associated with the polymeric kinetic equations is the polymeric stress tensor [@Bird; @Martin]. This variable is not the conserved quantity but nevertheless it should be treated as a relevant slow variable because it actually contributes to the macroscopic (hydrodynamic) equations. Equations for the stress tensor are known as the constitutive equations, and the problem of reduced description for the polymeric models consists in deriving such equations from the kinetic equation.
The tensor $$\label{tau_p} {{\boldmath \mbox{$\tau$}}}_{{\mbox{\scriptsize p}}\, ij}=k_{{\mbox{\scriptsize B}}}T \left(\delta_{ij} - \int_{\Omega}F_iq_j\Psi(q) dq\right)$$ gives a contribution to stresses caused by the presence of polymer molecules for unit density. Here $F(q)=-\nabla U(q)$ is the force vector, $\delta_{ij}$ is the Kronecker symbol. For spherically symmetric potentials ($U(q)=u(q^2)$) this tensor is symmetric. The tensor of dencities $m_{ij}(q)=F_i(q)q_j$ is the first addition to the dencities which depend only of $U(q)$.
**Macroscopic variables and boundary conditions**
=================================================
There is a standard technique to solve the boundary value and initial-boundary value problems of mathematical physics: first to build the space of the functions which satisfy the boundary conditions, and then to find the solution in this space.
When one uses the Legendre integrators, a special technique is needed to satisfy the boundary conditions.
FPE describes the evolution of the probability distribution. It conserves the total probability. The natural boundary conditions for the FPE is the absence of the flow through the boundary of $\Omega$: $$\label{BC}
\Psi^{qe}\left(\nu_q,D\nabla_{q}\left(\frac{\Psi}{\Psi_{q}}\right)\right)=0,$$ on $\partial\Omega$, where $\nu_q$ is a vector of outlet normal to $\partial\Omega$ in the point $q$.
Quasiequilibrium distribution functions (\[QEPsi\]) satisfy the condition (\[BC\]), if $$\label{BCQE}
\left\{\begin{array}{l} (\nu_q,D\nabla_{q}U(q))=0\\
(\nu_q,D\nabla_{q}m_{\alpha}(q))=0\end{array}\right.,$$ for all $\alpha$.
There is also a different way to satisfy conditions (\[BC\]): to make $\Psi^{*}|_{\partial\Omega}=0$. It is possible to do by making $U(q)\rightarrow\infty$ while $q\rightarrow q_{0}\in\partial\Omega$. But this choice leads to the singularities and is very inconvenient from the numerical point of view.
Conditions (\[BCQE\]) look somewhat surprisingly, if considered without the context of the quasiequilibrium approximations: for the quasiequilibrium solutions the absence of the flow through the barrier follows not from the infinite heights of the barrier, but from the fact, that the normal derivatives of $U$ and $m_{\alpha}$ are zeros.
To satisfy the condition (\[BCQE\]) it may be necessary to deform the initial potential $U$ and densities $m(q)$. This deformation will be the smoothing of $U$ near $\partial\Omega$. The error, introduced by this deformation is usually not very big (because of the smallness of $\Psi^{*}$ near $\partial\Omega$) and can be estimated easily.
So, the quasiequilibrium approximation and the Legendre Integrators of any order of accuracy are built, and the way to satisfy the boundary conditions is suggested. First numerical experiments [@IKOePhA02; @IKOePhA03] proved the effectiveness of this idea.
The main computational challenge in this method is to calculate the integrals of the form $$\label{Int}
\int_{\Omega}\left(\sum\lambda_{k}\varphi_{k}(q)\right)\exp\left(\sum\gamma_{i}\psi_{i}(q)\right)dq$$ where $\varphi_{k}(q), \psi_{i}(q)$ are known functions (usually they are given analytically). For the problems of the polymer physics the complexity of the problem (\[Int\]) is dependent on few characteristics:
1\) The quantity of the different functions $\varphi_{k}(q), \psi_{i}(q)$ is usually 5-10;
2\) The dimension of the space in which the integration goes is usually 10-100.
**Thermodynamic projector and Galerkin approximations**
=======================================================
Almost every manifold of the functions can be represented as the solution to the quasiequilibrium problem (\[maxent\]), if this manifold is not tangent to the level surface of the entropy $S=const$ [@GK1]. For this representation only the right system of restrictions is needed. By the simple parameterization with the moments $M(\Psi)$ it is possible to get only the classical quasiequilibrium manifolds (\[maxent\]). The restrictions which are necessary to represent manifold $\Omega$ as the quasiequilibrium manifold are built as follows. Let $f\in\Omega$, and $T_{f}$ be the tangent space to $\Omega$ in the point $f$. On the space of the distribution functions $E$ we define the projector $P_{f}:E\rightarrow T_{f}$. Operator $P_{f}$ depends smoothly on the point $f$ and on $T_{f}$. The problem of the quasiequilibrium is posed as follows: $$\label{QEPre}
\left\{\begin{array}{l} S(\Psi)\rightarrow\max\\ P_{f}(\Psi-f)=0\end{array}\right.$$ The necessary and sufficient condition for $f$ to be the unique solution to the problem (\[QEPre\]) is [@GK1]: $$\label{CondQE}
\ker P_{f} \subseteq \ker DS|_{f},$$ that is, if $P_{f}(\varphi)=0$, then $DS|_{f}(\varphi)=0$. For the classical entropy $DS|_{f}(\varphi)=-\int\varphi(q)\ln f(q)dq$ and the condition (\[CondQE\]) takes the form: $$\label{CondQE1}
\mbox{If }P_{f}(\varphi)=0,\mbox{ then }\int\varphi\ln fdq=0.$$
Among all projectors which satisfy the condition (\[CondQE\]) there is unique projector which has the following property: let us have the appropriate equation $$\begin{aligned}
\dot{\Psi}=J(\Psi),\end{aligned}$$ for which $dS[\Psi]/dt\geq0$. Then for the projected equation on $\Omega$ $$\label{TDPro}
\dot{f}=P_{f}(J(f)),$$ we also have $dS[f]/dt\geq0$.
This projector was introduced in the paper [@InChLANL], and there it is also proved its uniqueness. It is built as follows.
Let us require that the field of projectors, $P(\Psi,T)$, is defined for any $\Psi$ and $T$, if
$$\label{transversality}
T\not{\!\subset} \ker D_{\Psi}S.$$
From these conditions it follows immediately that in the equilibrium, $P(\Psi^*,T)$ is the orthogonal projector onto $T$ (orthogonality with respect to entropic scalar product $\langle |
\rangle_{\Psi^*}$).
The field of projectors is constructed in the neighborhood of the equilibrium based on the requirement of maximal smoothness of $P$ as a function of $g_{\Psi}=D_{\Psi}S$ and $\Psi$. It turns out that to the first order in the deviations $\Psi-\Psi^*$ and $g_{\Psi}-g_{\Psi^*}$, the projector is defined uniquely. Let us first describe the construction of the projector, and next discuss its uniqueness.
Let the subspace $T\subset E$, the point $\Psi$, and the differential of the entropy in this point, $g_{\Psi}=D_{\Psi}S$, be defined such that the transversality condition (\[transversality\]) is satisfied. Let us define $T_0=T\bigcap\ker g_{\Psi}$. By the condition (\[transversality\]), $T_0\neq T$. Let us denote, $e_g=e_g(T)\in T$ the vector in $T$, such that $e_g$ is orthogonal to $T_0$, and is normalized by the condition $g(e_g)=1$. Vector $e_g$ is defined unambiguously. Projector $P_{S,\Psi}=P(\Psi,T)$ is defined as follows: For any $z\in E$,
$$\label{projgen}
P_{S,\Psi}(f)=P_0(z)+e_gg_{\Psi}(f),$$
where $P_0$ is the orthogonal projector on $T_0$ (orthogonality with respect to the entropic scalar product $\langle |\rangle_{\Psi}$). Entropic projector (\[projgen\]) depends on the point $q$ through the $\Psi$-dependence of the scalar product $\langle |\rangle_{\Psi}$, and also through the differential of $S$ in $\Psi$, the functional $g_{\Psi}$.
Obviously, $P(f)=0$ implies $g(f)=0$, that is, the thermodynamicity requirement is satisfied. Uniqueness of the thermodynamic projector (\[projgen\]) is supported by the requirement of the *maximal smoothness* (analyticity) [@InChLANL] of the projector as a function of $g_{\Psi}$ and $\langle |\rangle_{\Psi}$, and is done in two steps which we sketch here:
1. Considering the expansion of the entropy in the equilibrium up to the quadratic terms, one shows that in the equilibrium the thermodynamic projector is the orthogonal projector with respect to the scalar product $\langle|\rangle_{\Psi^*}$.
2. For a given $g$, one considers auxiliary dissipative dynamic systems which satisfy the condition: For every $\Psi'\in U$, it holds, $g_{\Psi}(J(\Psi'))=0$, that is, $g_{\Psi}$ defines an additional linear conservation law for the auxiliary systems. For the auxiliary systems, the point $\Psi$ is the equilibrium. Eliminating the linear conservation law $g_{\Psi}$, and using the result of the previous point, we end up with the formula (\[projgen\]).
Thermodynamic projector allows to use almost arbitrary manifolds as quasiequilibrium closure assumption. If the projection of FPE (\[TDPro\]) is built with the thermodynamic projector, then $dS/dt$ conserves (not only the sign, but also the value). The only restriction is that the manifold must not be tangent to the level surfaces of $S$ (and must contain the equilibrium point).
Let us write down the explicit formulas for the closure assumption of the form $$\label{Ga}
f(q)=\Psi^{*}(q)+\sum_{\alpha}f_{\alpha}(q)\mu_{\alpha}.$$ Due to probability conservation for all $\alpha$ we have $\int f_{\alpha}(q)dq=0$.
Tangent spaces to the manifold (\[Ga\]) in all points coincide and have the form $T=\{\sum_{\alpha}\mu_{\alpha}f_{\alpha}(q)\}$. The natural coordinates in $T$ are $\mu_{\alpha}$. For every $f(q)$ of the form (\[Ga\]) there is the entropic scalar product, defined in $T$: $$\begin{aligned}
\langle\varphi|\psi\rangle_{f}=-\langle\varphi|(D^{2}S|_{f})\psi\rangle=\int\frac{\varphi(q)\psi(q)}{f(q)}dq\end{aligned}$$ In the coordinates $\mu_{\alpha}$ this scalar product has the form $$\begin{aligned}
\langle\sum_{\alpha}f_{\alpha}(q)\mu_{\alpha}|\sum_{\beta}f_{\beta}(q)\mu'_{\beta}\rangle_{f}=\sum_{\alpha,\beta}g_{\alpha,\beta}\mu_{\alpha}\mu'_{\beta},\end{aligned}$$ where $$\begin{aligned}
g_{\alpha,\beta}=\int\frac{f_{\alpha}(q)f_{\beta}(q)}{f(q)}dq.\end{aligned}$$
We will need the orthonormalized basis of the subspace $T\bigcap\ker(DS|_{f})$. This subspace is defined by the equation $$\begin{aligned}
\int\sum_{\alpha}f_{\alpha}(q)\mu_{\alpha}\ln\frac{f(q)}{\Psi^{*}(q)}dq=0.\end{aligned}$$ Let be $\int f_{1}(q)\ln\frac{f(q)}{\Psi^{*}(q)}dq\neq0$ bor the definiteness. Suppose for $\alpha>1$ $$\begin{aligned}
\label{proor}
q_{\alpha}=f_{\alpha}-\nu_{\alpha}f_{1},\\\mbox{where } \nu_{\alpha}=\frac{\int
f_{\alpha}(q)\ln\frac{f(q)}{\Psi^{*}(q)}dq}{\int f_{1}(q)\ln\frac{f(q)}{\Psi^{*}(q)}dq}\nonumber\end{aligned}$$ Let us orthogonalize the family of the vectors $q_{\alpha}$ ($\alpha>1$) with respect to the scalar product $\langle\cdot|\cdot\rangle_{f}$. We will get the orthogonal basis in $T\bigcap\ker(DS|_{f})$: $\{e_{\alpha}\} (\alpha>1)$.
Let $e_{1}\in T$ be the vector, orthogonal to all $e_{\alpha}$ (for example, $e_{1}=a(f_{1}-\sum_{\alpha>1}e_{\alpha}\langle f_{1}|e_{\alpha}\rangle_{f})$) and let $e_{1}$ be normalized in the following way: $\int e_{1}(q)\ln\frac{f_{1}(q)}{\Psi^{*}(q)}dq=1$. The projection of the vector $J$ on $T$ is defined in this way: $$\label{PrTD}
P^{th}_{f}J=e_{1}\int
J(q)\ln\frac{f_{1}(q)}{\Psi^{*}(q)}dq+\sum_{\alpha>1}e_{\alpha}\int\frac{J(q)e_{\alpha}(q)}{f(q)}dq.$$
Projector (\[PrTD\]) allows to consider every manifold of the form (\[Ga\]) which is not tangent to the level surface of the entropy $S$, as the quasiequilibrium manifold. If the vector field is projected with the operator (\[PrTD\]), then the dissipation is conserved.
As we can see, there is a “law of the difficulty conservation": for the quasiequilibrium with the moment parameterization the manifold is not explicit, and it can be difficult to calculate it. Thermodynamic projector completely eliminates this difficulty. From the other side, on the quasiequilibrium manifold with the moment parameterization (if it is found) it is easy to find the dynamics: simply write $\dot{M}_{\alpha}=\int\mu_{\alpha}Jdq$. The building of the thermodynamic projector may require some efforts.
Finally, for each of the distributions $\Psi$ it is easy to find its projection on the classical quasiequilibrium manifold $\Psi\rightarrow\Psi^{qe}_{M(\Psi)}$: it requires just calculation of the moments $M(\Psi)$. The analogue projection for the general thermodynamic projector is rather difficult: $\Psi\rightarrow f$ with the condition $P^{th}_{f}(\Psi-f)=0$. This equation defines the projection of some neighborhood of the manifold $\Omega$ on $\Omega$, but the solution of this equation is rather difficult. Fortunately, we need to build such operators only to analyze the fast processes of the initial relaxation layer, and it is not necessary to investigate the slow dynamics.
**A few words about the specifics of the computational difficulties**
=====================================================================
From the computational point of view, the main difficulties in realization of the described methods are in the calculation of the integrals of the form $$\begin{aligned}
\int_{\Omega}\sum a_{i}f_{i}(q)F(\sum b_{j}f_{j}(q))dq\end{aligned}$$ where $f_{i}$ are given functions of the vector $q$, $a_{i}, b_{i}$ are the numbers, $F$ is a function of one variable. The usual $F$ are $F(z)=e^{z}; F(z)=1/z,...$. The usual dimension of $\Omega$ in polymer physics is a few hundreds, number of different $f_{i}$ is a few dozens.
In any case, the transition from the integration of the whole FPE to solution of the moment equations gives a considerable decrease of the computation time.
In the methods of Legendre integrators and thermodynamic projector the computational problems of the linear algebra are present: the solution of the system of linear equations $C\dot{\mu}=\dot{M}$ (\[mjumu\]), the problem of the orthogonalisation of vectors in $T_{f}$ (\[proor\]) and so on. All these problems have the data which depends smooth on the current state of $\Psi$, and, consequently, on the time $t$. So, it is possible to solve these problems with the help of the perturbation theory and the methods of parametric continuation. These methods of the computational linear algebra are widely used and their details are well-known, so we are not discussing it here ([@Cont; @Lin].
**Accuracy estimation and postprocessing in invariant manifolds constructing**
==============================================================================
Suppose that for the dynamical system (\[eqn1\]) the approximate invariant manifold has been constructed and the slow motion equations have been derived:
$$\label{slag1}
\frac{dx_{sl}}{dt} = P_{x_{sl}}(J(x_{sl})), x_{sl}\in M,$$
where $P_{x_{sl}}$ is the corresponding projector onto the tangent space $T_{x_{sl}}$ of $M$. Suppose that we have solved the system (\[slag1\]) and have obtained $x_{sl}(t)$. Let’s consider the following two questions:
- [How well this solution approximates the real solution $x(t)$ given the same initial conditions?]{}
- [How is it possible to use the solution $x_{sl}(t)$ for it’s refinement without solving the system (\[slag\]) again?]{}
These two questions are interconnected. The first question states the problem of the [*accuracy estimation*]{}. The second one states the problem of [*postprocessing*]{}.
The simplest (“naive") estimation is given by the “invariance defect":
$$\label{defag}
\Delta_{x_{sl}} = (1-P_{x_{sl}})J(x_{sl})$$
compared with $J(x_{sl})$. For example, this estimation is given by $\epsilon =
\|\Delta_{x_{sl}}\|/\|J(x_{sl})\|$ using some appropriate norm.
Probably, the most comprehensive answer to this question can be given by solving the following equation:
$$\label{varia}
\frac{d(\delta x)}{dt}=\Delta_{x_{sl}(t)}+D_xJ(x)|_{x_{sl}(t)}\delta x.$$
This linear equation describes the dynamics of the deviation $\delta x(t) = x(t) - x_{sl}(t)$ using the linear approximation. The solution with zero initial conditions $\delta x(0) = 0$ allows estimating $x_{sl}$ robustness as well as the error value. Substituting $x_{sl}(t)$ for $x_{sl}(t)+\delta x(t)$ gives the required solution refinement. This [*dynamical postprocessing*]{} [@MaTiWyPP] allows to refine the solution substantially and to estimate it’s accuracy and robustness. However, the price for this is solving the equation (\[varia\]) with variable coefficients. Thus, this dynamical postprocessing can be followed by a whole hierarchy of simplifications, both dynamical and static. Let’s mention some of them, starting from the dynamical ones.
1\) [**Freezing the coefficients**]{}. In the equation (\[varia\]) the linear operator $D_xJ(x)|_{x_{sl}(t)}$ is replaced by it’s value in some distinguished point $x^*$ (for example, in the equilibrium) or it is frozen somehow else. As a result, one gets the equation with constant coefficients and the explicit integration formula:
$$\label{duam}
\delta x(t) = \int_0^t{exp(D^*(t-\tau))\Delta_{x_{sl}(\tau)}d\tau},$$
where $D^*$ is the “frozen" operator and $\delta x(0)=0$.
Another important way of freezing is substituting (\[varia\]) for some [*model equation*]{}, i.e. substituting $D_xJ(x)$ for $-\frac{1}{\tau^*}$, where $\tau^*$ is the relaxation time. In this case the formula for $\delta x(t)$ has a very simple form:
$$\label{duam1}
\delta x(t) = \int_0^t{e^{\frac{\tau-t}{\tau^*}}\Delta_{x_{sl}(\tau)}d\tau}.$$
2\) [**One-dimensional Galerkin-type approximation.**]{} Another “scalar" approximation is given by projecting (\[varia\]) on $\Delta(t)= \Delta_{x_{sl}(t)}$:
$$\label{1var}
\delta x(t) = \delta(t)\cdot \Delta(t), \; \frac{d\delta(t)}{dt} =
1+\delta\frac{\langle\Delta|D\Delta\rangle -
\langle\Delta|\dot{\Delta}\rangle}{\langle\Delta|\Delta\rangle},$$
where $\langle\hspace{1pt}|\rangle$ is an appropriate scalar product which can depend on the point $x_{sl}$ (for example, the entropic scalar product), $D=D_xJ(x)|_{x_{sl}(t)}$ or the self-adjoint linearization of this operator, or some approximation of it.
The “hybrid" between equations (\[1var\]) and (\[varia\]) has the simplest form (but is more difficult for computation than eq. (\[1var\])):
$$\label{hybrid}
\frac{d(\delta
x)}{dt}=\Delta(t)+\frac{\langle\Delta|D\Delta\rangle}{\langle\Delta|\Delta\rangle}\delta x.$$
Here one uses the normalized matrix element $\frac{\langle\Delta|D\Delta\rangle}{\langle\Delta|\Delta\rangle}$ instead of the linear operator $D=D_xJ(x)|_{x_{sl}(t)}$.
Both equations (\[1var\]) and (\[hybrid\]) can be solved explicitly:
$$\begin{aligned}
\delta(t)&=&\int_0^t d \tau \exp\left(\int_{\tau}^t k(\theta)d\theta \right), \\ \delta x(t)&=&
\int_0^t \Delta(\tau)d \tau \exp\left(\int_{\tau}^t k_1(\theta)d\theta \right),\end{aligned}$$
where $k(t)=\frac{\langle\Delta|D\Delta\rangle -
\langle\Delta|\dot{\Delta}\rangle}{\langle\Delta|\Delta\rangle},$ $k_1(t)=\frac{\langle\Delta|D\Delta\rangle}{\langle\Delta|\Delta\rangle}.$
The projection of $\Delta_{x_{sl}}(t)$ on the slow motion is zero, hence, for post-processing analysis of the slow motion, the one-dimensional model (\[1var\]) should be supplemented by one more iteration: $$\begin{aligned}
{d(\delta x_{sl}(t)) \over dt} = \delta(t) P_{x_{sl}(t)}(D_xJ(x_{sl}(t)))(\Delta(t)); \nonumber \\
\delta x_{sl}(t)= \int_0^t \delta(\tau) P_{x_{sl}(\tau)}(D_xJ(x_{sl}(\tau)))(\Delta(\tau))d\tau.\end{aligned}$$ where $\delta(t)$ is the solution of (\[1var\]).
3\) For a [**static postprocessing**]{} one uses stationary points of dynamical equations (\[varia\]) or their simplified versions (\[duam\]),(\[1var\]). Instead of (\[varia\]) one gets:
$$\label{stvar}
D_xJ(x)|_{x_{sl}(t)}\delta x = -\Delta_{x_{sl}(t)}$$
with one additional condition, $P_{x_{sl}}\delta x=0$. This is exactly the iteration equation of the Newton’s method in solving the invariance equation.
The corresponding stationary problems for the model equations and for the projections of (\[varia\]) on $\Delta$ are evident. We only mention that in the projection on $\Delta$ one gets a step of the relaxation method for the invariant manifold construction.
For the static postprocessing with frozen parameters the “naive" estimation given by the “invariance defect" (\[defag\]) makes sense.
**Example: Dumbell model, explosion of the Gaussian anzatz and polymer stretching in flow**
===========================================================================================
Here is an example of application of the thermodynamic projector method. In this example we consider the following simplest one-dimensional kinetic equation for the configuration distribution function $\Psi(q,t)$, where $q$ is the reduced vector connecting the beads of the dumbell. This equation is slightly different from the FPE considered above. It is nonlinear, because of the dependence of $U$ on the moment $M_{2}[\Psi]=\int q^{2}\Psi(q) dq$. This dependence allows us to get the exact quasiequilibrium equations on $M_{2}$, but this equations are not solving the problem: this quasiequilibrium manifold may become unstable when the flow is present [@IK00]. Here is this model: $$\label{530}
\partial_{t}\Psi=-\partial_{q}\{\alpha(t)q\Psi\}+\frac{1}{2}\partial^{2}_{q}\Psi.$$ Here $$\label{531}
\alpha(t)=\kappa(t)-\frac{1}{2}f(M_{2}(t)),$$ $\kappa(t)$ is the given time-independent velocity gradient, $t$ is the reduced time, and the function $-fq$ is the reduced spring force. Function $f$ may depend on the second moment of the distribution function $M_{2}=\int q^{2}\Psi(q,t)dq$. In particular, the case $f\equiv1$ corresponds to the linear Hookean spring, while $f=[1-M_{2}(t)/b]^{-1}$ corresponds to the self-consistent finite extension nonlinear elastic spring (the FENE-P model, first introduced in [@FENEP]). The second moment $M_{2}$ occurs in the FENE-P force $f$ as the result of the pre-averaging approximation to the original FENE model (with nonlinear spring force $f=[1-q^{2}/b]^{-1}$). Leading to closed constitutive equations, the FENE-P model is frequently used in simulations of complex rheological flows as the reference for more sophisticated closures to the FENE model [@HCO; @Keu98; @Martin]. The parameter $b$ changes the characteristics of the force law from Hookean at small extensions to a confining force for $q^{2}\rightarrow b$. Parameter $b$ is roughly equal to the number of monomer units represented by the dumbell and should therefore be a large number. In the limit $b\rightarrow\infty$, the Hookean spring is recovered. Recently, it has been demonstrated that FENE-P model appears as first approximation within a systematic self-confident expansion of nonlinear forces [@IKOe99; @GKIOeNONNEWT2001].
Equation (\[530\]) describes an ensemble of non-interacting dumbells subject to a pseudo-elongational flow with fixed kinematics. As is well known, the Gaussian distribution function, $$\label{532}
\Psi^{G}(M_{2})=\frac{1}{\sqrt{2\pi M_{2}}}\exp\left[-\frac{q^{2}}{2M_{2}}\right],$$ solves equation (\[530\]) provided the second moment $M_{2}$ satisfies $$\label{533}
\frac{dM_{2}}{dt}=1+2\alpha(t)M_{2}.$$ Solution (\[532\]) and (\[533\]) is the valid macroscopic description if all other solutions of the equation (\[530\]) are rapidly attracted to the family of Gaussian distributions (\[532\]). In other words [@GKTTSP94], the special solution (\[532\]) and (\[533\]) is the macroscopic description if equation (\[532\]) is the stable invariant manifold of the kinetic equation (\[530\]). If not, then the Gaussian solution is just a member of the family of solutions, and equation (\[533\]) has no meaning of the macroscopic equation. Thus, the complete answer to the question of validity of the equation (\[533\]) as the macroscopic equation requires a study of dynamics in the neighborhood of the manifold (\[532\]). Because of the simplicity of the model (\[530\]), this is possible to a satisfactory level even for $M_{2}$-dependent spring forces.
In the paper [@IK00] it was shown, that there is a possibility of “explosion" of the Gaussian manifold: with the small initial deviation from it, the solutions of the equation (\[530\]) are very fast going far from, and then slowly come back to the stationary point which is located on the Gaussian manifold. The distribution function $\Psi$ is stretched fast, but looses the Gaussian form, and after that the Gaussian form recovers slowly with the new value of $M_{2}$. Let us describe briefly the results of [@IK00].
Let $M_{2n}=\int q^{2n}\Psi dq$ denote the even moments (odd moments vanish by symmetry). We consider deviations $\mu_{2n}=M_{2n}-M_{2n}^{\rm G}$, where $M_{2n}^{\rm G}=\int q^{2n} \Psi^{\rm G}dq$ are moments of the Gaussian distribution function (\[532\]). Let $\Psi(q,t_0)$ be the initial condition to the Eq. (\[530\]) at time $t=t_0$. Introducing functions, $$\label{result0} p_{2n}(t,t_0)=\exp\left[4n\int_{t_0}^{t}\alpha(t')dt'\right],$$ where $t\ge t_0$, and $2n \ge 2$, the [*exact*]{} time evolution of the deviations $\mu_{2n}$ for $2n\ge 2$ reads $$\label{result1}
\mu_4(t)=p_4(t,t_0)\mu_4(t_0),$$ and $$\label{result2}
\mu_{2n}(t)=\left[ \mu_{2n}(t_0) + 2n(4n-1)\int_{t_0}^t
\mu_{2n-2}(t')p_{2n}^{-1}(t',t_0)dt' \right] p_{2n}(t,t_0),$$ for $2n\ge 3$. Equations (\[result0\]), (\[result1\]) and (\[result2\]) describe evolution near the Gaussian solution for arbitrary initial condition $\Psi(q,t_0)$. Notice that explicit evaluation of the integral in the Eq. (\[result0\]) requires solution to the moment equation (\[533\]) which is not available in the analytical form for the FENE-P model.
It is straightforward to conclude that any solution with a non-Gaussian initial condition converges to the Gaussian solution asymptotically as $t\to\infty$ if
$$\label{result3} \lim_{t\to\infty}\int_{t_0}^t\alpha(t')dt'<0.$$
However, even if this asymptotic condition is met, deviations from the Gaussian solution may survive for considerable [*finite*]{} times. For example, if for some finite time $T$, the integral in the Eq. (\[result0\]) is estimated as $\int_{t_0}^t\alpha(t')dt'>\alpha (t-t_0)$, $\alpha>0$, $t\le
T$, then the Gaussian solution becomes exponentially unstable during this time interval. If this is the case, the moment equation (\[533\]) cannot be regarded as the macroscopic equation. Let us consider specific examples.
For the Hookean spring ($f\equiv 1$) under a constant elongation ($\kappa={\rm const}$), the Gaussian solution is exponentially stable for $\kappa<0.5$, and it becomes exponentially unstable for $\kappa>0.5$. The exponential instability in this case is accompanied by the well known breakdown of the solution to the Eq. (\[533\]) due to infinite stretching of the dumbbell. Similar instability has been found numerically in three-dimensional flows for high Weissenberg numbers [@PGA851; @PGA852].
Eqs. (\[533\]) and (\[result1\]) were integrated by the 5-th order Runge-Kutta method with adaptive time step. The FENE-P parameter $b$ was set equal to 50. The initial condition was $\Psi(q,0)=C(1-q^2/b)^{b/2}$, where $C$ is the normalization (the equilibrium of the FENE model, notoriously close to the FENE-P equilibrium [@Herrchen]). For this initial condition, in particular, $\mu_4(0)=-6b^2/[(b+3)^2(b+5)]$ which is about 4$\%$ of the value of $M_4$ in the Gaussian equilibrium for $b=50$. In Fig. \[EPJ713\_fig\] we demonstrate deviation $\mu_4(t)$ as a function of time for several values of the flow. Function $M_2(t)$ is also given for comparison. For small enough $\kappa$ we find an adiabatic regime, that is $\mu_4$ relaxes exponentially to zero. For stronger flows, we observe an initial [*fast runaway*]{} from the invariant manifold with $|\mu_4|$ growing over three orders of magnitude compared to its initial value. After the maximum deviation has been reached, $\mu_4$ relaxes to zero. This relaxation is exponential as soon as the solution to Eq.(\[533\]) approaches the steady state. However, the time constant for this exponential relaxation $|\alpha_{\infty}|$ is very small. Specifically, for large $\kappa$, $$\label{alpha_lim} \alpha_{\infty}=\lim_{t\to\infty}\alpha(t)=-\frac{1}{2b}+O(\kappa^{-1}).$$ Thus, the steady state solution is unique and Gaussian but the stronger is the flow, the larger is the initial runaway from the Gaussian solution, while the return to it thereafter becomes flow-independent. Our observation demonstrates that, though the stability condition (\[result3\]) is met, [*significant deviations from the Gaussian solution persist over the times when the solution of Eq.*]{} (\[533\]) [*is already reasonably close to the stationary state.*]{} If we accept the usually quoted physically reasonable minimal value of parameter $b$ of the order $20$ then the minimal relaxation time is of order $40$ in the reduced time units of Fig. \[EPJ713\_fig\]. We should also stress that the two limits, $\kappa\to\infty$ and $b\to\infty$, are not commutative, thus it is not surprising that the estimation (\[alpha\_lim\]) does not reduce to the above mentioned Hookean result as $b\to\infty$. Finally, peculiarities of convergence to the Gaussian solution are even furthered if we consider more complicated (in particular, oscillating) flows $\kappa(t)$.
In accordance with [@IK00] the anzatz for $\Psi$ can be suggested in the following form: $$\label{Anz}
\Psi^{An}(\{\sigma,\varsigma\},q)=\frac{1}{2\sigma\sqrt{2\pi}}\left(e^{-\frac{(q+\varsigma)^{2}}{2\sigma^{2}}}+e^{-\frac{(q-\varsigma)^{2}}{2\sigma^{2}}}\right).$$ Natural inner coordinates on this manifold are $\sigma$ and $\varsigma$. Note, that now $\sigma^{2}\neq M_{2}$. The value $\sigma^{2}$ is a dispersion of one of the Gaussian summands in (\[Anz\]), $$\begin{aligned}
M_{2}(\Psi^{An}(\{\sigma,\varsigma\},q))=\sigma^{2}+\varsigma^{2}.\end{aligned}$$ To build the thermodynamic projector on the manifold (\[Anz\]), the thermodynamic Lyapunov function is necessary. It is necessary to emphasize, that equations (\[530\]) are nonlinear. For such equations, the arbitrarity in the choice of the thermodynamic Lyapunov function is much smaller. Nevertheless, such function exists. It is the free energy $$\label{Free}
F=U(M_{2}[\Psi])-TS[\Psi],$$ where $$\begin{aligned}
S[\Psi]=-\int\Psi(\ln\Psi-1)dq,\end{aligned}$$ $U(M_{2}[\Psi])$ is the potential energy in the mean field approximation, $T$ is the temperature (further we assume that $T=1$). The thermodynamic properties of the mean field models in polymer physics are studied in the recent paper [@MaKaHa2003]
Note, that Kullback-form entropy $S_{k}=-\int\Psi\ln\left(\frac{\Psi}{\Psi^{*}}\right)$ also has the form $S_{k}=-F/T$: $$\begin{aligned}
\Psi^{*}=\exp(-U),\\ S_{k}[\Psi]=-\langle U\rangle-\int\Psi\ln\Psi dq.\end{aligned}$$ If $U(M_{2}[\Psi])$ in the mean field approximation is the convex function of $M_{2}$, then the free energy (\[Free\]) is the convex functional too.
For the FENE-P model $U=-\ln[1-M_{2}/b]$.
In accordance to the thermodynamics the vector of flow of $\Psi$ must be proportional to the gradient of the corresponding chemical potential $\mu$: $$\label{Flux}
J=-B(\Psi)\nabla_{q}\mu,$$ where $\mu=\frac{\delta F}{\delta\Psi}$, $B\geq0$. From the equation (\[Free\]) it follows, that $$\begin{aligned}
\label{muflux}
\mu=\frac{d U(M_{2})}{d M_{2}}\cdot q^{2}+\ln\Psi\nonumber\\ J=-B(\Psi)\left[2\frac{dU}{dM_{2}}\cdot
q+\Psi^{-1}\nabla_{q}\Psi\right].\end{aligned}$$ If we suppose here $B=\frac{D}{2}\Psi$, then we get $$\begin{aligned}
\label{TDeq}
J=-D\left[\frac{dU}{dM_{2}}\cdot q\Psi+\frac{1}{2}\nabla_{q}\Psi\right]\nonumber\\
\frac{\partial\Psi}{\partial t}=div_{q}J=D\frac{d U(M_{2})}{d
M_{2}}\partial_{q}(q\Psi)+\frac{D}{2}\partial^{2}q\Psi,\end{aligned}$$ When $D=1$ this equations coincide with (\[530\]) in the absence of the flow: due to equation (\[TDeq\]) $dF/dt\leq0$.
Let us construct the thermodynamic projector with the help of the thermodynamic Lyapunov function $F$ (\[Free\]). Corresponding entropic scalar product in the point $\Psi$ has the form $$\label{Scal}
\left.\langle f|g\rangle=\frac{d^{2}U}{dM_{2}^{2}}\right|_{M_{2}=M_{2}[\Psi]}\cdot\int
q^{2}f(q)dq\cdot\int q^{2}g(q)dq+\int\frac{f(q)g(q)}{\Psi(q)}dq$$ During the investigation of the anzatz (\[Anz\]) the scalar product (\[Scal\]), constructed for the corresponding point of the Gaussian manifold with $M_{2}=\sigma^{2}$, will be used. It will let us to investigate the neighborhood of the Gaussian manifold (and to get all the results in the analytical form): $$\label{ScalG}
\left.\langle f|g\rangle_{\sigma^{2}}=\frac{d^{2}U}{dM_{2}^{2}}\right|_{M_{2}=\sigma^{2}}\cdot\int
q^{2}f(q)dq\cdot\int q^{2}g(q)dq+\sigma\sqrt{2\pi}\int e^{\frac{q^{2}}{2\sigma^{2}}}f(q)g(q)dq$$ Also we will need to know the functional $DF$ in the point of Gaussian manifold: $$\label{Prod}
\left.DF_{\sigma^{2}}(f)=\left(\frac{d U(M_{2})}{dM_{2}}\right|_{M_{2}=\sigma^{2}}
-\frac{1}{2\sigma^{2}}\right)\int q^{2}f(q)dq,$$ (with the condition $\int f(q)dq=0$). The point $$\begin{aligned}
\left.\frac{d U(M_{2})}{dM_{2}}\right|_{M_{2}=\sigma^{2}}=\frac{1}{2\sigma^{2}},\end{aligned}$$ corresponds to the equilibrium.
The tangent space to the manifold (\[Anz\]) is spanned by the vectors $$\begin{aligned}
\label{basis}
&&f_{\sigma}=\frac{\partial\Psi^{An}}{\partial(\sigma^{2})}; \:
f_{\varsigma}=\frac{\partial\Psi^{An}}{\partial(\varsigma^{2})};\nonumber\\
f_{\sigma}&=&\frac{1}{4\sigma^{3}\sqrt{2\pi}}\left[e^{-\frac{(q+\varsigma)^{2}}{2\sigma^{2}}}
\frac{(q+\varsigma)^{2}-\sigma^{2}}{\sigma^{2}}+e^{-\frac{(q-\varsigma)^{2}}{2\sigma^{2}}}
\frac{(q-\varsigma)^{2}-\sigma^{2}}{\sigma^{2}} \right];\\
f_{\varsigma}&=&\frac{1}{4\sigma^{2}\varsigma\sqrt{2\pi}}\left[-e^{-\frac{(q+\varsigma)^{2}}{2\sigma^{2}}}
\frac{q+\varsigma}{\sigma}+e^{-\frac{(q-\varsigma)^{2}}{2\sigma^{2}}} \frac{(q-\varsigma)}{\sigma}
\right];\nonumber\end{aligned}$$ The Gaussian entropy (free energy) production in the directions $f_{\sigma}$ and $f_{\varsigma}$ (\[Prod\]) has a very simple form: $$\begin{aligned}
\label{Fpro}
\left.DF_{\sigma^{2}}(f_{\varsigma})=DF_{\sigma^{2}}(f_{\sigma})=\frac{d
U(M_{2})}{dM_{2}}\right|_{M_{2}=\sigma^{2}}-\frac{1}{2\sigma^{2}}.\end{aligned}$$ The linear subspace $\ker DF_{\sigma^{2}}$ in $lin\{f_{\sigma},f_{\varsigma}\}$ is spanned by the vector $f_{\varsigma}-f_{\sigma}$.
Let us have the given vector field $d\Psi/dt=\Phi(\Psi)$ in the point $\Psi(\{\sigma,\varsigma\})$. We need to build the projection of $\Phi$ onto the tangent space $T_{\sigma,\varsigma}$ in the point $\Psi(\{\sigma,\varsigma\})$: $$\label{Prosigma}
P^{th}_{\sigma,\varsigma}(\Phi)=\varphi_{\sigma}f_{\sigma}+\varphi_{\varsigma}f_{\varsigma}.$$ This equation means, that the equations for $\sigma^{2}$ and $\varsigma^{2}$ will have the form $$\label{eqsigma}
\frac{d\sigma^{2}}{dt}=\varphi_{\sigma};\:\: \frac{d\varsigma^{2}}{dt}=\varphi_{\varsigma}$$ Projection $(\varphi_{\sigma},\varphi_{\varsigma})$ can be found from the following two equations: $$\begin{aligned}
\label{psieq}
\varphi_{\sigma}+\varphi_{\varsigma}=\int q^{2}\Phi(\Psi)(q)dq\nonumber;\\
\langle\varphi_{\sigma}f_{\sigma}+\varphi_{\varsigma}f_{\varsigma}|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}}
=\langle\Phi(\Psi)|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}},\end{aligned}$$ where $\langle
f|g\rangle_{\sigma^{2}}=\langle\Phi(\Psi)|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}}$, (\[Scal\]). First equation of (\[psieq\]) means, that the time derivative $dM_{2}/dt$ is the same for the initial and the reduced equations. Due to the formula for the dissipation of the free energy (\[Prod\]), this equality is equivalent to the persistence of the dissipation in the neighborhood of the Gaussian manifold.
The second equation in (\[psieq\]) means, that $\Phi$ is projected orthogonally on $\ker DS\bigcap
T_{\sigma,\varsigma}$. Let us use the orthogonality with respect to the entropic scalar product (\[ScalG\]). The solution of equations (\[psieq\]) has the form $$\begin{aligned}
\label{projphi}
\frac{d\sigma^{2}}{dt}=\varphi_{\sigma}=\frac{\langle\Phi|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}}+M_{2}(\Phi)(\langle
f_{\varsigma}|f_{\varsigma}\rangle_{\sigma^{2}}-\langle
f_{\sigma}|f_{\varsigma}\rangle_{\sigma^{2}})}{\langle
f_{\sigma}-f_{\varsigma}|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}}}\nonumber,\\\\
\frac{d\varsigma^{2}}{dt}=\varphi_{\varsigma}=\frac{-\langle\Phi|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}}+M_{2}(\Phi)(\langle
f_{\sigma}|f_{\sigma}\rangle_{\sigma^{2}}-\langle
f_{\sigma}|f_{\varsigma}\rangle_{\sigma^{2}})}{\langle
f_{\sigma}-f_{\varsigma}|f_{\sigma}-f_{\varsigma}\rangle_{\sigma^{2}}}\nonumber,\end{aligned}$$ where $\Phi=\Phi(\Psi)$, $M_{2}(\Phi)=\int q^{2}\Phi(\Psi)dq$.
It is easy to check, that the formulas (\[projphi\]) are indeed defining the projector: if $f_{\sigma}$ (or $f_{\varsigma}$) is substituted there instead of the function $\Phi$, then we will get $\varphi_{\sigma}=1, \varphi_{\varsigma}=0$ (or $\varphi_{\sigma}=0, \varphi_{\varsigma}=1$, respectively). Let us substitute the right part of the initial kinetic equations (\[530\]), calculated in the point $\Psi(q)=\Psi(\{\sigma,\varsigma\},q)$ (see the equation (\[Anz\])) in the equation (\[projphi\]) instead of $\Phi$. We will get the closed system of equations on $\sigma^{2}, \varsigma^{2}$ in the neighborhood of the Gaussian manifold.
This system describes the dynamics of the distribution function $\Psi$. The distribution function is represented as the half-sum of two Gaussian distributions with the averages of distribution $\pm\varsigma$ and mean-square deviations $\sigma$. All integrals in the right-hand part of (\[projphi\]) are possible to calculate analytically.
Basis $(f_{\sigma},f_{\varsigma})$ is convenient to use everywhere, except the points in the Gaussian manifold, $\varsigma=0$, because if $\varsigma\rightarrow0$, then $$\begin{aligned}
f_{\sigma}-f_{\varsigma}=O\left(\frac{\sigma^{2}}{\varsigma^{2}}\right)\rightarrow0.\end{aligned}$$ To analyze the relaxation in the small neighborhood of the Gaussian manifold it is more convenient to use another basis: $$\begin{aligned}
F^{+}=f_{\sigma}+f_{\varsigma}\\ F^{-}=\frac{\sigma^{2}}{\varsigma^{2}}(f_{\sigma}-f_{\varsigma}).\end{aligned}$$ It corresponds to a reparametrization of the initial manifold (\[Anz\]): $$\label{Anz1}
\Psi(\{\xi,\varsigma\},q)=\frac{1}{2\sqrt{2\pi}\sqrt{\xi^{2}-\varsigma^{2}}}\left(e^{-\frac{(q+\varsigma)^{2}}{2(\xi^{2}-\varsigma^{2})}}+e^{-\frac{(q-\varsigma)^{2}}{2(\xi^{2}-\varsigma^{2})}}\right).$$ Let us analyze the stability of the Gaussian manifold to the “dissociation" of the Gaussian peak in two peaks (\[Anz\]). To do this, it is necessary to find first nonzero term in the Taylor expansion in $\varsigma^{2}$ of the right-hand side of the second equation in the system (\[projphi\]). The denominator has the order of $\varsigma^{4}$, the numerator has, as it is easy to see, the order not less, than $\varsigma^{6}$ (because the Gaussian manifold is invariant with respect to the initial system).
Let us denote $G_{\sigma}=\frac{1}{\sqrt{2\pi}}e^{-\frac{q^{2}}{\sigma^{2}}}$. Then we get $$\begin{aligned}
\Psi(\{\sigma,\varsigma\},q)=G_{\sigma}(q)\left[1+\frac{1}{2}\frac{\varsigma^{2}}{\sigma^{2}}\left(\frac{q^{2}}{\sigma^{2}}-1\right)+\frac{1}{4}\frac{\varsigma^{4}}{\sigma^{4}}\left(\frac{1}{2}-\frac{q^{2}}{\sigma^{2}}+\frac{1}{6}\frac{q^{4}}{\sigma^{4}}\right)\right]+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right);\\
f_{\sigma}=\frac{G_{\sigma}(q)}{2\sigma^{2}}\left[\frac{q^{2}}{\sigma^{2}}-1+\frac{\varsigma^{2}}{\sigma^{2}}\left(\frac{1}{2}\frac{q^{4}}{\sigma^{4}}-3\frac{q^{2}}{\sigma^{2}}+\frac{3}{2}\right)+\frac{\varsigma^{4}}{\sigma^{4}}\left(\frac{1}{24}\frac{q^{6}}{\sigma^{6}}-\frac{15}{24}\frac{q^{4}}{\sigma^{4}}+\frac{15}{8}\frac{q^{2}}{\sigma^{2}}-\frac{5}{8}\right)\right]+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right);\\
f_{\varsigma}=\frac{G_{\sigma}(q)}{2\sigma^{2}}\left[\frac{q^{2}}{\sigma^{2}}-1+\frac{\varsigma^{2}}{\sigma^{2}}\left(\frac{1}{6}\frac{q^{4}}{\sigma^{4}}-\frac{q^{2}}{\sigma^{2}}+\frac{1}{2}\right)+\frac{\varsigma^{4}}{\sigma^{4}}\left(\frac{1}{120}\frac{q^{6}}{\sigma^{6}}-\frac{1}{8}\frac{q^{4}}{\sigma^{4}}+\frac{3}{8}\frac{q^{2}}{\sigma^{2}}-\frac{1}{8}\right)\right]+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right);\\
f_{\sigma}-f_{\varsigma}=\frac{\varsigma^{2}}{\sigma^{2}}\frac{1}{2\sigma^{2}}G_{\sigma}(q)\left[\frac{1}{3}\frac{q^{4}}{\sigma^{4}}-2\frac{q^{2}}{\sigma^{2}}+1+\frac{\varsigma^{2}}{\sigma^{2}}\left(\frac{1}{30}\frac{q^{6}}{\sigma^{6}}-\frac{1}{2}\frac{q^{4}}{\sigma^{4}}+\frac{3}{2}\frac{q^{2}}{\sigma^{2}}-\frac{1}{2}\right)\right]+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right).\end{aligned}$$ Let us calculate $\partial_{t}\Psi=\Phi(\Psi(\{\sigma,\varsigma\}))$ with the accuracy up to $\varsigma^{4}$: $$\begin{aligned}
\frac{1}{2}\partial^{2}_{q}\Psi(\{\sigma,\varsigma\})=f_{\sigma};\\
M_{2}(\frac{1}{2}\partial^{2}_{q}\Psi(\{\sigma,\varsigma\}))=1;\\
M_{2}(\Psi(\{\sigma,\varsigma\}))=\sigma^{2}+\varsigma^{2};\\
-\alpha\partial_{q}(q\Psi(\{\sigma,\varsigma\}))=\alpha
G_{\sigma}(q)\left[\frac{q^{2}}{\sigma^{2}}-1+\frac{\varsigma^{2}}{\sigma^{2}}\left(\frac{1}{2}\frac{q^{4}}{\sigma^{4}}-2\frac{q^{2}}{\sigma^{2}}+\frac{1}{2}\right)+\right.\\
\left.\frac{\varsigma^{4}}{\sigma^{4}}\left(\frac{1}{24}\frac{q^{6}}{\sigma^{6}}-\frac{11}{24}\frac{q^{4}}{\sigma^{4}}+\frac{7}{8}\frac{q^{2}}{\sigma^{2}}-\frac{1}{8}\right)\right]+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right)\\
M_{2}(-\alpha\partial_{q}(q\Psi(\{\sigma,\varsigma\})))=2\alpha(\sigma^{2}+\varsigma^{2})+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right).\end{aligned}$$ The diffusion part gives the zero contribution to the numerator of the equation (\[projphi\]): $$\begin{aligned}
-\langle f_{\sigma}|f_{\sigma}-f_{\varsigma}\rangle+\langle
f_{\sigma}|f_{\sigma}-f_{\varsigma}\rangle=0,\end{aligned}$$ therefore to find $d\varsigma/dt$ it is sufficient to use $\Phi_{1}=-\alpha\partial_{q}(q\Psi)$, so we get $$\begin{aligned}
M_{2}(\Phi_{1}(\Psi(\{\sigma,\varsigma\})))f_{\sigma}-\Phi_{1}(\Psi(\{\sigma,\varsigma\}))=\alpha
G_{\sigma}(q)\frac{\varsigma^{4}}{\sigma^{4}}\left(\frac{1}{3}\frac{q^{4}}{\sigma^{4}}-2\frac{q^{2}}{\sigma^{2}}+1\right)+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right)\\
=2\alpha\sigma^{2}\frac{\varsigma^{2}}{\sigma^{2}}(f_{\sigma}-f_{\varsigma})+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right).\end{aligned}$$ Thus $$\label{itog}
\frac{1}{\sigma^{2}}\frac{d\varsigma^{2}}{dt}=2\alpha\frac{\varsigma^{2}}{\sigma^{2}}+o\left(\frac{\varsigma^{4}}{\sigma^{4}}\right).$$ So, if $\alpha>0$, then $\varsigma^{2}$ grows exponentially ($\varsigma\sim e^{\alpha t}$) and the Gaussian manifold is unstable; if $\alpha<0$, then $\varsigma^{2}$ decreases exponentially and the Gaussian manifold is stable.
The form of the phase trajectories is shown qualitative on the figure \[figFENEP\].
![Phase trajectories for two-peak approximation, FENE-P model. The vertical axis ($\varsigma=0$) corresponds to the Gaussian manifold. The triangle with $\alpha(M_2)>0$ is the domain of exponential instability. []{data-label="figFENEP"}](F1.eps){width="160mm" height="127mm"}
![Phase trajectories for two-peak approximation, FENE model: [**a)**]{} A stable equilibrium on the vertical axis, one stable peak; [**b)**]{} A stable equilibrium with $\varsigma>0$, stable two-peak configuration.[]{data-label="figFENE"}](F2.eps){width="160mm" height="87mm"}
For the real equation FPE (for example, with the FENE potential) the motion in presence of the flow can be represented as the motion in the effective potential well $\tilde{U}(q)=U(q)-\kappa q^{2}$. Different variants of the phase portrait for the FENE potential are present on the figure \[figFENE\]. Instability and dissociation of the unimodal distribution functions (“peaks") for the FPE is the general effect when the flow is present.
The instability occurs when the matrix $\partial^{2}\tilde{U}/\partial q_{i}\partial q_{j}$ starts to have negative eigenvalues ($\tilde{U}$ is the effective potential energy, $\tilde{U}(q)=U(q)-\sum_{i,j}\kappa_{i,j}q_{i}q_{j}$).
The stationary polymodal distribution corresponds to the persistence of several local minima of the function $\tilde{U}(q)$. The multidimensional case is different from one-dimensional because it has the huge amount of possible configurations. All normal forms of the catastrophe of “birth of the critical point" are well investigated and known [@ArVarGZ1995-1998]. Every dissociation of the peak is connected with such a catastrophe. The number of the new peaks is equal to the number of the new local minima of $U$.
It is not very difficult to perform the analysis of the equations (\[projphi\]) for every quantity of peaks and every potential. Moreover, for the polynomial potentials all the necessary integrals are possible to calculate analytically (if the coefficients of the scalar product and entropy production are taken in the Gaussian point). The same situation is also for the general Gaussian distributions: $$\begin{aligned}
G_{\Sigma}=const\cdot\exp\left(-\frac{1}{2}\sum_{i,j}(\Sigma^{-1})_{ij}q_{i}q_{j}\right),\end{aligned}$$ where $\Sigma$ is the covariance matrix. Here in the equation for the effective energy we have the symmetric part of the tensor $\kappa_{ij}={\partial^2 U}/{\partial q_{i}\partial q_{j}}$. The presence of the unsimmetric part may lead to the relaxation oscillations (both for FPE and for the peak dynamics).
For the modeling of dynamics of the multimodal distributions for FPE with the presence of the flow (the flow may be nonstationary) it seems to be useful to use the physically clear modeling of the distribution function as a sum of the finite number of the Gaussian peaks. Thermodynamic projector gives us an opportunity to make this models thermodynamically consistent.
**Conclusion**
==============
In this work we presented a toolbox for the development and reduction of the dynamical models of nonequilibrium systems with the persistence of the correct dissipation.
The basic notions of this toolbox are: entropy, quasiequilibrium (MaxEnt) distribution, dual variables, thermodynamic projector.
The main technical ideas are: Legendre Integrators, dynamical postprocessing, transformation of almost arbitrary anzatz to a thermodynamically consistent model via thermodynamic projector.
The Legendre Integrators are based on a simple, but very useful idea: to write and solve dynamic equations for dual variables. This idea is efficient, because to obtain the dynamic equations for dual variables it is necessary to solve linear equations. To get the usual quasiequilibrium dynamical equations for the moments, we should solve nonlinear (transcendent) equations. Sometimes it happens that these equations can be written down in the explicit form (Vlasov equation, Euler equation, ten moments Gaussian approximation in gas kinetics [@Ko; @LPi]), but usually these equations remain in implicit form with right-hand sides derived by a system of transcendent equations.
The post-processing is necessary for accuracy estimation. It gives us the cheapest way to improve the solution obtained by the Legendre Integrators.
Termodynamic projector allows to transform almost arbitrary anzatz into a physically consistent dynamic model with persistence of dissipation. The simplest example, discussed in details, is the two peaks model for Gaussian manifold instability in polymer dynamics. This type of models opens a way to create the computational models for the “molecular individualism" [@DeGenne; @Chu; @LeHa1999].
The simplest model of the molecular individualism is the “Gaussian parallelepiped". The distribution function is represented as a sum of $2^m$ Gaussian peaks located in the vertixes of centrally symmetrical parallelepiped:
$$\begin{aligned}
\Psi(q)={1 \over 2^m(2\pi)^{n/2}\sqrt{\det \Sigma}} \sum_{\varepsilon_i=\pm 1, \, (i=1, \ldots, m)}
\exp\left(-\frac{1}{2}\left(\Sigma^{-1}\left(q+\sum_{i=1}^m \varepsilon_i \varsigma_i \right), \:
q+\sum_{i=1}^m \varepsilon_i \varsigma_i\right)\right),\end{aligned}$$
where $n$ is dimension of configuration space, $2\varsigma_i$ is the vector of the $i$th edge of the parallelepiped, $\Sigma$ is the one peak covariance matrix (in this model $\Sigma$ is the same for all peaks). The macroscopic variables for this model are:
1. The covariance matrix $\Sigma$;
2. The set of vectors $\varsigma_i$ (or the parallelepiped edges).
The dimension is $n(n+1)/2+mn$.
The number $m \: (m\leq n) $ is the estimated number of nonstable directions of motion (dimension of instability). To include the nongaussian equilibrium the “Gaussian parallelepiped" should be deformed to nongaussian “peaks parallelepiped". Technical details will be discussed in the separate paper. The structure of “peaks parallelepiped" leads to the molecular individualism in such a way: each individual molecule belongs to a domain of a peak in configuration space. The number of these peaks grows significantly with the dimension of instability, as $2^m$, and even if $m=3$, than the number of peaks is 8, and one should discover 8 distinguished sorts of molecular configurations. On the other hand, in projection on a line this amount of peaks can form a distribution without a sign about peak structure, hence, the study of properties of ensembles (viscosity, stress coefficient, etc.) can be without any hint to a cluster structure in configuration space.
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[^1]: agorban$@$mat.ethz.ch, $^{**}[email protected], $^{***}$ikarlin$@$mat.ethz.ch
[^2]: From time to time it is discussed in the literature, who was the first to introduce the quasiequilibrium approximations, and how to interpret them. At least a part of the discussion is due to a different rôle the quasiequilibrium plays in the entropy–conserving and the dissipative dynamics. The very first use of the entropy maximization dates back to the classical work of G. W. Gibbs [@Gibb], but it was first claimed for a principle by E. T. Jaynes [@Janes1]. Probably the first explicit and systematic use of quasiequilibria to derive dissipation from entropy–conserving systems is due to the works of D. N. Zubarev. Recent detailed exposition is given in [@Zubarev]. For dissipative systems, the use of the quasiequilibrium to reduce description can be traced to the works of H. Grad on the Boltzmann equation [@Grad]. The viewpoint of two of the present authors (ANG and IVK) was influenced by the papers by L. I. Rozonoer and co-workers, in particular, [@KoRoz; @Ko; @Roz]. A detailed exposition of the quasiequilibrium approximation for Markov chains is given in the book [@G1] (Chapter 3, [*Quasiequilibrium and entropy maximum*]{}, pp. 92-122), and for the BBGKY hierarchy in the paper [@Kark]. We have applied maximum entropy principle to the description the universal dependence the 3-particle distribution function $F_3$ on the 2-particle distribution function $F_2$ in classical systems with pair interactions [@BGKTMF]. A very general discussion of the maximum entropy principle with applications to dissipative kinetics is given in the review [@Bal]. The methods for corrections the quasiequilibrium approximations are developed in [@GK1; @GKTTSP94; @KTGOePhA2003; @Plenka].
[^3]: This is a rather old theorem, one of us had published this theorem in 1984 already as textbook material ([@G1], chapter 3 “Quasiequilibrium and entropy maximum", p. 37, see also the paper [@GKIOeNONNEWT2001]), but from time to time different particular cases of this theorem are continued to be published as new results.
|
---
abstract: 'Long-established results for the low-energy photon-photon scattering, $\gamma\gamma\to\gamma\gamma$, have recently been questioned. We analyze that claim and demonstrate that it is inconsistent with experience. We demonstrate that the mistake originates from an erroneous manipulation of divergent integrals and discuss the connection with another recent claim about the Higgs decay into two photons. We show a simple way of correctly computing the low-energy $\gamma\gamma$ scattering.'
author:
- Yi Liang and Andrzej Czarnecki
title: |
Photon-photon scattering: a tutorial
Alberta Thy 12-11
---
*Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1*
PACS Numbers: 13.60.Fz, 13.40.-f, 12.15.Lk
Introduction
============
After Dirac proposed the theory of negative energy solutions of his equation [@Dirac:1930ek], it was realized that photons can interact with other photons by polarizing the vacuum. Photon-photon scattering was qualitatively considered in this context by Halpern [@Halpern33], and its cross section, for the case of photon energies low compared to the electron mass, was determined by Euler and Kockel in 1935 [@EulerKockel35; @Euler36]. If the energy of each of the colliding photons is $\omega$ in the frame in which their total momentum vanishes, the low-energy differential cross section is $$\frac{d\sigma}{d\Omega}=\frac{139\alpha^{4}}{\left(180\pi\right)^{2}}\frac{\omega^{6}}{m^8}\left(3+\cos^{2}\theta\right)^{2},
\label{eq:correctCross}$$ where $\alpha\simeq1/137$ is the fine structure constant and $m$ is the electron mass.
High energy scattering was considered soon afterward [@Akhi36; @Akhi37]. A thorough analysis of the scattering at all energies, including partial cross sections for various polarization states, was carried out in [@KarplusNeuman1; @KarplusNeuman2], using the then new diagrammatic technique of Feynman. Since then, the photon-photon scattering cross section has been confirmed with other methods, and even higher-order QED corrections have been computed [@Dittrich:2000zu]. Results obtained up to 1971 are reviewed in [@Costantini:1971cj] and more recent developments are summarized in [@Martin:2003gb].
Very recently, the classic result for the low-energy cross section (\[eq:correctCross\]) has been questioned [@Kanda:2011vu; @Fujita:2011rd]. In those papers, the cross section is found to be many orders of magnitude larger, since it is not suppressed by powers of $\left(\omega/m\right)$, but is proportional to $1/\omega^{2}$,$$\frac{d\sigma_{\mbox{{\scriptsize FK}}}}{d\Omega}=\frac{\alpha^{4}}{\left(12\pi\right)^{2}\omega^{2}}\left(3+2\cos^{2}\theta+\cos^{4}\theta\right).\label{eq:CrossFK}$$ As we will demonstrate in this paper, this claim is incorrect. It has already been pointed out [@Bernard:2011vz] that it contradicts existing laboratory bounds on the the photon-photon cross section, obtained by colliding laser beams. We show in addition that a cross section increasing with the inverse squared energy of the colliding photons limits the mean free path of visible light due to collisions with the cosmic microwave background radiation (CMBR) to less than the distance between Earth and Jupiter. Thus the fact that we can sharply see much more distant astronomical objects proves that the low-energy photon-photon scattering must be significantly suppressed, as predicted by eq. (\[eq:correctCross\]).
The matrix element for the photon-photon scattering is absent at the tree level since photons are neutral. It arises only at the loop level. The sum of all contributing loop diagrams must be finite since there is no parameter in the QED Lagrangian whose renormalization could absorb a divergence. Refs. [@Kanda:2011vu; @Fujita:2011rd] found an incorrect result because of the assumption that if the sum of those diagrams is finite, they can be calculated without any regularization. In fact, even though the sum of the diagrams is finite, each of them separately is divergent. Calculating the sum is somewhat delicate and is easiest done with regularized loop integrals (see, however, an alternative calculation in [@Schwinger:II] and another point of view on avoiding regularization in [@Jackiw:1999qq]).
Interestingly, a similar error [@Gastmans:2011ks; @Gastmans:2011wh] has recently cast doubt over the rate of the Higgs boson decay into two photons. That process, too, is loop induced, and the sum of contributing loops is finite. But individual loop integrals are divergent and must be regularized, as has already been thoroughly discussed in this context [@Shifman:2011ri; @Huang:2011yf; @Marciano:2011gm; @Jegerlehner:2011jm; @Shao:2011wx].
Mean free path of photons in a microwave background
===================================================
The CMBR is a gas of photons with the spectrum of a black body at a temperature of $1/\beta=2.725$ K. Here we want to compute how far a visible-light photon with energy $E_{\gamma}\simeq2.5$ eV can travel in such a gas before scattering, from the point of view of an observer in whose frame the CMBR is isotropic. (We will call it the LAB frame. For the purposes of this discussion an Earth-based observer is a good approximation.) Consider one mode of the CMBR radiation, characterized by its energy $E$ and inclination angle $\theta$ with respect to the direction from which the visible photon is incident. The relative velocity of the two photons (as seen in the LAB frame) is $\vec{v}_{1}+\vec{v}_{2}=\left(1+\cos\theta,-\sin\theta\right)$, $\left|\vec{v}_{1}+\vec{v}_{2}\right|=2\cos\frac{\theta}{2}$ (we use the units $c=\hbar=k_{B}=1$). In the frame where the total momentum of the photons vanishes, each has the energy $\omega$ given by$$\omega=\frac{1}{2}\sqrt{2EE_{\gamma}\left(1+\cos\theta\right)}=\cos\frac{\theta}{2}\sqrt{EE_{\gamma}}.\label{eq:omega}$$ That energy determines the scattering cross section. Collisions with photons in this particular mode will occur at the rate$$\mbox{d}\Gamma_{E\theta}=\left|\vec{v}_{1}+\vec{v}_{2}\right|\sigma\mbox{d}\rho\left(E\right)\label{eq:partialRate}$$ where $$\mbox{d}\rho\left(E\right)=\frac{E^{2}}{2\pi^{2}}\frac{\mbox{d}E\mbox{d}\cos\theta}{\exp\left(\beta E\right)-1}\label{eq:density}$$ is the density of CMBR photons with energy $E$, and $\sigma$ is the scattering cross section. Integrating over the energies and directions of the CMBR photons we find the mean free path. Between collisions, the visible-light photon will travel on average the distance $$\lambda=\pi^{2}\left[\int_{0}^{\infty}\mbox{d}E\int_{-1}^{1}\mbox{d}\cos\theta\cos\frac{\theta}{2}\frac{E^{2}\sigma}{\exp\left(\beta E\right)-1}\right]^{-1}.\label{eq:meanFreePath}$$ We now consider the two formulas for the low-energy cross section. If we use the classical result (\[eq:correctCross\]), we find the total cross section$$\sigma\left(\gamma\gamma\to\gamma\gamma\right)=\frac{973\alpha^{4}\omega^{6}}{10125\pi m^{8}},\label{eq:totalCrossCorrect}$$ and the mean free path$$\begin{aligned}
\lambda & = & \pi^{2}\left[\frac{973\alpha^{4}E_{\gamma}^{3}}{10125\pi m^{8}}\int_{-1}^{1}\mbox{d}\cos\theta\cos^{7}\frac{\theta}{2}\int_{0}^{\infty}\mbox{d}E\frac{E^{5}}{\exp\left(\beta E\right)-1}\right]^{-1}\nonumber \\
& = & \pi^{2}\left[\frac{973\alpha^{4}E_{\gamma}^{3}}{10125\pi m^{8}}\cdot\frac{4}{9}\cdot\frac{8\pi^{6}}{63\beta^{6}}\right]^{-1}=\frac{820125m^{8}\beta^{6}}{4448\pi^{3}\alpha^{4}E_{\gamma}^{3}}.\label{eq:freePathClassic}\end{aligned}$$ Using $m=0.511$ MeV we find $\lambda\simeq7\cdot10^{68}$ meters, a distance that would take light about $10^{43}$ times more time to travel than the age of the Universe. In other words, the CMBR is a rather transparent medium at visible frequencies.
However, if we take instead the cross section suggested in [@Kanda:2011vu; @Fujita:2011rd], we find from eq. $$\sigma_{\mbox{\scriptsize FK}}=\frac{29\alpha^{4}}{540\pi\omega^{2}},\label{eq:totalFK}$$ which gives a much shorter mean free path,
$$\begin{aligned}
\lambda_{\mbox{\scriptsize FK}} & = & \pi^{2}\left[\frac{29\alpha^{4}}{540\pi E_{\gamma}}\int_{-1}^{1}\frac{\mbox{d}\cos\theta}{\cos\frac{\theta}{2}}\int_{0}^{\infty}\mbox{d}E\frac{E}{\exp\left(\beta E\right)-1}\right]^{-1}\nonumber \\
& = & \pi^{2}\left[\frac{29\alpha^{4}}{540\pi E_{\gamma}}\cdot4\cdot\frac{\pi^{2}}{6\beta^{2}}\right]^{-1}=\frac{810\pi\beta^{2}E_{\gamma}}{29\alpha^{4}},\label{eq:freePathFK}\end{aligned}$$
or $\lambda_{\mbox{\scriptsize FK}}=3\cdot10^{11}$ meters, equivalent to about 15 light minutes. For comparison, the orbital radius of Jupiter is about $8\cdot10^{11}$ meters. If the mean free path of the visible light were so much shorter than even the radius of Jupiter’s orbit, no stars would be visible on the night sky. Clearly, the result eq. is at odds with experience.
The situation with eq. is actually even worse. Since the cross section is not suppressed by the mass of the electron, there would be additional positive contributions from other charged fermions that would differ only by the coupling constant, and would further decrease the mean free path. This lack of suppression by the inverse mass of the loop particle contradicts the Appelquist-Carazzone decoupling theorem [@Appelquist:1974tg].
Determination of the photon-photon scattering
=============================================
In this section we present a derivation of the photon-photon scattering matrix element in two regularization schemes: dimensional and Pauli-Villars. We consider the box diagram shown in Fig. \[fig:Virtual-electron-loop\]. External photons carry momenta $k_{1},\ldots,k_{4}$ which we will consider as incoming, $k_{1}+k_{2}+k_{3}+k_{4}=0$.
![Virtual electron loop inducing the four-photon coupling. \[fig:Virtual-electron-loop\]](boxLBL.ps)
There are six ways in which the four momenta can be arranged around the oriented electron loop. However, diagrams that differ only by the direction of the electron line give identical results so it is enough to compute three of them, corresponding to three cyclic permutations of $k_{1},k_{2},$ and $k_{3}$. (If there was an odd number of photons coupling to the electron loop, the diagrams differing by the direction of the electron would cancel one other, resulting in a vanishing amplitude. This is the theorem due to Furry [@FurryTheorem].)
At low energies of external photons it is especially easy to compute the diagrams in Fig. \[fig:Virtual-electron-loop\]. We simply Taylor-expand the electron propagators in the external momenta, so that each propagator’s denominator becomes simply $\left({q\hspace{-0.45em}/}-m\right)^{-1}=\left({q\hspace{-0.45em}/}+m\right)/\left(q^{2}-m^{2}\right)$ where $q$ is the loop momentum. Such expansion does not lead to any spurious divergences, and commutes with the integration over $q$. We now explain how this integration is performed in two regularization schemes.
Dimensional regularization
--------------------------
Now that $q$ is present in the denominators only through $q^{2}$, also in the numerator we can replace all scalar products of $q$ with other vectors by powers of $q^{2}$ times products not involving $q$,$$q^{\mu_{1}}\ldots q^{\mu_{2n}}\to\frac{\Gamma\left(\frac{D}{2}\right)}{2^{n}\Gamma\left(\frac{D}{2}+n\right)}\left(q^{2}\right)^{n}S\left(g^{\mu_{1}\mu_{2}}\ldots g^{\mu_{2n-1}\mu_{2n}}\right).\label{eq:average}$$ Here $D$ is the space-time dimension and $S\left(g^{\mu_{1}\mu_{2}}\ldots g^{\mu_{2n-1}\mu_{2n}}\right)$ is a sum of products of $n$ metric tensors $g$, totally symmetric in all indices $\mu_{i}$; it has $\left(2n-1\right)!!$ terms. Terms odd in $q$ vanish upon integration.
The powers of $q^{2}$ resulting from can be canceled against the denominators and the loop integration can be completed using$$\int\frac{\dd^{D}q}{\left(2\pi\right)^{D}}\frac{1}{\left(q^{2}-m^{2}+i0\right)^{a}}=\frac{\left(-1\right)^{a}i}{\left(4\pi\right)^{D/2}}m^{D-2a}\frac{\Gamma\left(a-\frac{D}{2}\right)}{\Gamma\left(a\right)}.\label{eq:loopInt}$$ Each of the three diagrams contains terms with the exponent $a=2$, leading to a divergence $\Gamma\left(2-\frac{D}{2}\right)\sim1/\left(D-4\right)$. The divergences cancel when we add all three contributions. But individual diagrams containing singularities $1/\left(D-4\right)$ have also $D$-dependent factors, arising from the averaging in eq. . The resulting finite contributions *do not* cancel among themselves.
How do these remaining terms depend on $m$? We remember that they arise from the $a=2$ sector, therefore they scale like $m^{0}$ (the overall dimension of the $\gamma\gamma\to\gamma\gamma$ amplitude). There are other terms that scale with this power, arising from convergent integrals like $m^{2}\int\dd^{4}q/\left(q^{2}-m^{2}\right)^{3}$. The essential point is that the sum of all $m^{0}$ terms, including the remnants of singularities, adds up to zero. The total result for the amplitude turns out to be suppressed by four powers of $1/m$.
Pauli-Villars regularization
----------------------------
Another way of carrying out this calculation is to stay in four dimensions but add another amplitude, with the electron replaced by a very heavy particle of mass $M$, and with an opposite sign than the electron loop. The calculation proceeds very similarly to the case of dimensional regularization, with two changes. In averaging over the loop momentum directions we replace $\frac{\Gamma\left(D+2n-1\right)}{\Gamma\left(D\right)}$ by its value at $D=4$, $\left(2n+2\right)!/6$. The formula for the loop integration is also replaced by its $D=4$ value, except in the divergent case $a=2$. In the dimensional regularization, this divergent integral gives $m^{D-4}\Gamma\left(2-\frac{D}{2}\right)\to\frac{2}{4-D}-\ln m^{2}$. In the Pauli-Villars approach one finds a convergent combination$$\int\frac{\dd^{4}q}{\left(2\pi\right)^{4}}\left[\frac{1}{\left(q^{2}-m^{2}+i0\right)^{2}}-\frac{1}{\left(q^{2}-M^{2}+i0\right)^{2}}\right]=\frac{i}{16\pi^{2}}\ln\frac{M^{2}}{m^{2}}.\label{eq:log}$$ When the three diagrams are added, this logarithm cancels, but now there are no finite remnants of the singularities. Instead, the $m^{0}$ terms from the convergent diagrams are canceled by the $M^{0}$ terms from the Pauli-Villars subtraction. Since they are independent of the electron mass, they are the same in the amplitudes with the electron and with the very heavy particle, and cancel in the difference.
In both regularization schemes, the only remaining result is suppressed by the electron mass.
Potential error from neglecting regularization
----------------------------------------------
We have just seen that the regularization is crucial in computing the photon-photon scattering amplitude, even though the final result does not contain divergences. We now want to inspect more closely the part of the amplitude that does not contain external photon momenta, and thus scales like the zeroth power of the electron mass,
$$\begin{aligned}
{\cal M}_{m^{0}} & \sim & \int\frac{\dd^{D}q}{\left(q^{2}-m^{2}+i0\right)^{4}}\Big[m^{4}S_{1}^{\mu\nu\rho\sigma}+2m^{2}\left(2S_{2}^{\mu\nu\rho\sigma}-q^{2}S_{1}^{\mu\nu\rho\sigma}\right)\nonumber \\
& & +24q^{\mu}q^{\nu}q^{\rho}q^{\sigma}+\left(q^{2}\right)^{2}S_{1}^{\mu\nu\rho\sigma}-4q^{2}S_{2}^{\mu\nu\rho\sigma}\Big]\epsilon_{1\mu}\epsilon_{2\nu}\epsilon_{3\rho}\epsilon_{4\sigma},\label{eq:noPowerOfM}\end{aligned}$$
with
$$\begin{aligned}
S_{1}^{\mu\nu\rho\sigma} & = & g^{\mu\nu}g^{\rho\sigma}+g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\rho\nu},\\
S_{2}^{\mu\nu\rho\sigma} & = & g^{\mu\nu}q^{\rho}q^{\sigma}+\mbox{five other terms, }\end{aligned}$$
where the terms not shown in $S_{2}$ have the other five distributions of indices so that both $S_{1}$ and $S_{2}$ are totally symmetric in $\mu,\nu,\rho,\sigma$. The second line in contains four powers of the loop momentum $q$ and thus represents divergent integrals. Without regularization, these divergent integralssimply do not have a meaning. If we apply the averaging procedure to these terms, we find $$\begin{aligned}
\left\langle S_{2}^{\mu\nu\rho\sigma}\right\rangle & = & \frac{2q^{2}}{D}S_{1}^{\mu\nu\rho\sigma},\nonumber \\
\left\langle q^{\mu}q^{\nu}q^{\rho}q^{\sigma}\right\rangle & = & \frac{\left(q^{2}\right)^{2}}{D\left(D+2\right)}S_{1}^{\mu\nu\rho\sigma},\label{eq:averagesS}\end{aligned}$$ so that if $D=4$, the second line of vanishes, as does the term $\sim m^{2}$ in its first line. Thus, if the regularization is neglected, one is left with only the first term $m^{4}S_{1}$ which, after the $q$ integration, gives a result independent of the electron mass, scaling like $m^{0}$,$$\begin{aligned}
i\mathcal{M}_{m^{0}} & = & -\frac{4}{3}\alpha^{2}S_{1}^{\mu\nu\rho\sigma}\epsilon_{1\mu}\epsilon_{2\nu}\epsilon_{3\rho}\epsilon_{4\sigma}\label{eq:amplWrongA}\\
& = & -\frac{4}{3}\alpha^{2}\left(\epsilon_{1}\cdot\epsilon_{2}\,\epsilon_{3}\cdot\epsilon_{4}+\epsilon_{1}\cdot\epsilon_{3}\,\epsilon_{2}\cdot\epsilon_{4}+\epsilon_{1}\cdot\epsilon_{4}\,\epsilon_{2}\cdot\epsilon_{3}\right),\label{eq:amplWrong}\end{aligned}$$ where $\epsilon_{i}$ are the polarization vectors of the four photons. This dependence of the amplitude only on the polarization vectors (and not on the photon momenta) means that the induced coupling of the photons involves only their vector potential (the induced effective operator is proportional to $(A^2)^2$), and not its derivatives. It is not possible to construct such a coupling in a gauge invariant way.
This violation of gauge invariance may also generate photon’s mass. For example, if two of the external photon lines in Fig. \[fig:Virtual-electron-loop\] are contracted, the resulting two-loop diagram generates an operator $\sim A^2$, thus giving the photon a mass.
In order to see how the cross section in eq. follows from the amplitude , we define two transverse polarization vectors for each photon, $\vec{\epsilon}_{i}^{1,2}$, with $\vec{\epsilon}^{1}$ perpendicular to the scattering plane and $\vec{\epsilon}^{2}$ lying in that plane. We do not include here the longitudinal photon polarizations, present if the photon becomes massive, even though they may dominate the cross section; however, our goal here is merely to explain how the result is related to the gauge-invariance violating amplitude . For the eq. to give a non-zero result, each polarization must be represented an even number of times (otherwise there will always be a factor 0 in every term). There are eight possible such polarization assignments, giving the following values of the three terms in ,$$\begin{aligned}
{\cal M}_{1111} & \sim & 1+1+1,\\
{\cal M}_{2222} & \sim & 1+\cos^{2}\theta+\cos^{2}\theta,\\
{\cal M}_{1122} & \sim & 1+0+0,\\
{\cal M}_{1212} & \sim & 0+\cos\theta+0,\\
{\cal M}_{1221} & \sim & 0+0+\cos\theta,\end{aligned}$$ and the last three amplitudes enter with a weight factor of 2, due to the symmetry $1\leftrightarrow2$. The various amplitudes differ by the polarization of some photons, so they do not interfere. The sum of their squares gives $3^{2}+\left(1+2\cos^{2}\theta\right)^{2}+2+4\cos^{2}\theta=4\left(3+2\cos^{2}\theta+\cos^{4}\theta\right)$, the angular structure of the (incorrect) result quoted in . In fact, a sum over the polarizations of the final state photons and an average over the polarizations of the initial state photons, leads to the cross section given in eq. .
What went wrong in the above procedure? The formulas cannot be applied to the divergent integrals in the second line of in $D=4$, without regularization. If we stay in $D$ dimensions, the terms we found to be zero in the $D\to4$ limit given finite contributions that cancel against the first term of the integrand, $\sim m^{4}S_{1}$. In this correct treatment the resulting amplitude is suppressed by four powers of $1/m$.
The recent incorrect claim about the decay $H\to\gamma\gamma$ [@Gastmans:2011ks; @Gastmans:2011wh] originated with a similar, but somewhat simpler integral. An example of a contribution to that process is shown in Fig. . There are only three propagators, and the divergent integrals are present in the combination [@Gastmans:2011wh]$$I_{\mu\nu}\left(D\right)=\int\dd^{D}q\frac{q^{2}g_{\mu\nu}-4q_{\mu}q_{\nu}}{\left(q^{2}-m^{2}+i0\right)^{3}}.$$ Without dimensional regularization, if we take $D=4$, it seems that this integral vanishes after averaging over $q$ with help of . As we have seen with the example of $\gamma\gamma$ scattering, and as has already been discussed in the literature [@Marciano:2011gm; @Shifman:2011ri; @Jackiw:1999qq], such manipulations with unregulated, divergent integrals are unjustified. In the case of the Higgs decay, they lead to the incorrect conclusion that $I_{\mu\nu}\left(D\to4\right)$ vanishes. In fact, in the limit of a very heavy Higgs boson, the correct finite result of $I_{\mu\nu}$ gives the most important contribution.
![An example of a $W$ boson loop mediating the Higgs boson decay into two photons. \[fig:Virtual-electron-loop-1\]](higgs2photons.ps)
Results for polarized photons
-----------------------------
The correct result of the loop integration in the $\gamma\gamma\to\gamma\gamma$ amplitude contains scalar products among photon momenta $k_{i}$, in addition to the polarization vectors $\epsilon_{i}^{\lambda}$. The effective photon-photon coupling induced in this way is described by operators involving the electromagnetic field tensor and is gauge invariant. We know calculate the scattering cross sections for various polarization situations. Instead of the linear polarizations we have just considered, the scattering amplitudes will be presented in terms of circular polarization states. Thus we introduce$$\vec{\epsilon}^{\pm}=\frac{1}{\sqrt{2}}\left(\vec{\epsilon}^{1}\pm i\vec{\epsilon}^{2}\right),$$ describing right- and left-handed polarization states, respectively. There are four possible initial polarization states, but it is sufficient to consider just two of them, $++$ and $+-$. We get three independent scattering amplitudes, ${\cal M}_{++++}$, ${\cal M}_{+++-}$, and ${\cal M}_{++--}$. We describe kinematics in terms of Mandelstam variables $s=\left(k_{1}+k_{2}\right)^{2}=4E^{2}$ and $t=\left(k_{1}+k_{3}\right)^{2}=-2E^{2}\left(1-\cos\theta\right)$ and find
$$\begin{aligned}
i{\cal M}_{++++} & = & \frac{2\alpha^{2}\left(s^{2}+st+t^{2}\right)}{15m^{4}}=\frac{8\alpha^{2}\omega^{4}\left(3+\cos^{2}\theta\right)}{15m^{4}},\nonumber \\
i{\cal M}_{+++-} & = & -\frac{\alpha^{2}st\left(s+t\right)}{315m^{6}}=-\frac{16\alpha^{2}\sin^{2}\theta\omega^{6}}{315m^{6}},\nonumber \\
i{\cal M}_{++--} & = & -\frac{11\alpha^{2}s^{2}}{45m^{4}}=-\frac{176\alpha^{2}\omega^{4}}{45m^{4}}.\label{eq:amplits}\end{aligned}$$
We note that the amplitude ${\cal M}_{+++-}$ vanishes at the leading order in $E/m$ expansion at which the other amplitudes are finite. In order to compute it, we have to evaluate two more terms in the Taylor expansion.
All other amplitudes can be obtained from eq. using space and/or time reversal and the crossing symmetry. For example, $i{\cal M}_{+-+-}=-\frac{11\alpha^{2}t^{2}}{45m^{4}}$ and $i{\cal M}_{+--+}=-\frac{11\alpha^{2}\left(s+t\right)^{2}}{45m^{4}}$
Total cross section
-------------------
Once the polarized amplitudes have been evaluated, the unpolarized cross section can be easily found. We quote here only the leading low-energy result (thus we neglect ${\cal M}_{+++-}$ and seven amplitudes related to it) for the cross section averaged over initial and summed over final polarizations,$$\begin{aligned}
\frac{\dd\sigma\left(\gamma\gamma\to\gamma\gamma\right)}{\dd\Omega} & = & \frac{1}{256\pi^{2}\omega^{2}}\cdot\frac{\left|{\cal M}_{++++}\right|^{2}+\left|{\cal M}_{++--}\right|^{2}+\left|{\cal M}_{+-+-}\right|^{2}+\left|{\cal M}_{+--+}\right|^{2}}{2}\\
& = & \frac{139\alpha^{4}\omega^{6}}{\left(180\pi\right)^{2}m^8}\left(3+\cos^{2}\theta\right)^{2},\end{aligned}$$ in agreement with the classic result . When integrated over both $\theta$ and $\phi$ from 0 to $\pi$ (we integrate only over one hemisphere since the two final-state photons are identical), this gives the total photon-photon scattering cross section, $$\sigma\left(\gamma\gamma\to\gamma\gamma\right)=\frac{973\alpha^{4}\omega^{6}}{10125\pi
m^8},$$ in agreement with [@Landau4]. Other texts seem to have misprints in these results [@IZ; @ab65].
Conclusions
===========
We have demonstrated that the recently claimed result for the $\gamma\gamma\to\gamma\gamma$ scattering cross-section must be wrong. The photon-photon coupling is induced by virtual loops with charged particles and is suppressed at low photon energy by the inverse power of the electron mass. The result lacks this suppression and yields a very large cross-section, therefore a short mean free path of visible photons even in the rare cosmic microwave radiation background. Such short path would obscure all astronomical objects as close as Jupiter.
We have showed that the error resulted from manipulating unregulated divergent integrals. A similar error misled the authors of [@Gastmans:2011ks; @Gastmans:2011wh] in the context of the Higgs decay to two photons. Both processes arise only at the loop level and their amplitudes must be finite, since there are no parameters in the Lagrangian that could absorb a divergence. However, both processes are usually computed from a sum of several diagrams, among which some are divergent. For this reason, a regularization of individual contributions is necessary.
Interestingly, the mistakes in these recent studies of $\gamma\gamma\to\gamma\gamma$ and $H\to\gamma\gamma$ led to confusions about various types of decoupling. The correct result for the former process does respect Appelquist-Carazzone decoupling theorem in the limit of low photon energies or large electron mass, whereas the incorrect result of [@Kanda:2011vu; @Fujita:2011rd] does not. On the other hand, the correct result for the Higgs decay does not vanish, as one could naively expect, in the limit of large Higgs mass [@Shifman:2011ri] (or, equivalently, low $W$ boson mass; this type of decoupling affects for example quarks but not the longitudinal $W$ components), while part of the reason why [@Gastmans:2011ks; @Gastmans:2011wh] believed their result was that it did vanish in that limit.
We have also showed how the $\gamma\gamma\to\gamma\gamma$ amplitude can be calculated in the low-energy regime, and how an expansion in powers of the photon energy to the electron mass ratio can be organized. This tutorial illustrates useful techniques of loop calculations: averaging over loop momentum direction, loop momentum integration, and various regularizations. We hope it will be helpful for other similar loop calculations.
Acknowledgment {#acknowledgment .unnumbered}
--------------
We gratefully acknowledge helpful discussions with Alexander Penin and Arkady Vainshtein, and support of this research by Science and Engineering Research Canada (NSERC). We thank Ben O’Leary for pointing out typographical mistakes in the first version of this paper.
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abstract: 'A discussion on the interaction between skyrmions in a bi-layer system connected by a non-magnetic metal is presented. From considering a free charge carrier model, we have shown that the Ruderman-Kittel-Kasuya-Yosida ($RKKY$) interaction can induce attractive or repulsive interaction between the skyrmions depending on the spacer thickness. We have also shown that due to an increasing in RKKY energy when the skyrmions are far from each other, their widths are diminished. Finally, we have obtained analytical solutions to the skyrmion position when the in-plane distance between the skyrmions is small and it is shown that an attractive RKKY interaction yields a skyrmion precessory motion. This RKKY-induced coupling could be used as a skyrmion drag mechanism to displace skyrmions in multilayers.'
author:
- 'R. Cacilhas'
- 'S. Vojkovic'
- 'V.L. Carvalho-Santos'
- 'E. B. Carvalho'
- 'A.R. Pereira'
- 'D. Altbir'
- 'Á. S. Núñez'
title: Coupling of skyrmions mediated by RKKY interaction
---
The possibility of using magnetic patterns such as vortices [@Vortex], domain walls [@Catalan] and skyrmions [@Tomasello-SciRep-2014] in spintronic devices has resulted in an increasing interest in studying statical and dynamical properties of these magnetization collective modes. In particular, skyrmions are topological spin textures that may appear as groundstate in non-centrosymmetric crystals in the presence of the bulk Dzyaloshinskii-Moriya interaction ($DMI$) [@Jiang; @Nagaosa-NatNano-2013; @Muhlbauer-Science-2009; @Yu-NatMat-2011; @Jonietz-Science-2010]. Due to their topological stability, small size and low driving magnetic field/current density [@Fert-NatNano-2013; @Kang-SciRep-2016], skyrmions are also promising candidates to compose spintronic devices based on the interesting concept of racetrack memory [@Tomasello-SciRep-2014; @Parkin-Science-2008; @Allwood-Science-2002], as well as in logic devices [@Zhang-1; @Zhang-2] and spin transfer nano-oscillators [@Garcia-NJP-2016; @Chui-AIP-2015].
Nevertheless, as a consequence of the skyrmion Hall effect, the use of these objects in racetrack devices is strongly hampered because a skyrmion cannot move in a straight line along the driving current or external magnetic field direction. Therefore, magnetic skyrmions can be destroyed at the edges of nanostripes[@Purkana-SciRep-2015; @Zhang-3]. A possible way to avoid the skyrmion Hall effect is the coupling between two skyrmions lying in different layers. In fact, when two skyrmions are on separate planes, changes in the interlayer exchange interaction and the signs of the $DMI$ can induce different statical and dynamical properties of the magnetization [@Koshibae-SciRep-2017]. In this context, it has been shown that the Skyrmion Hall effect can be suppressed by considering two perpendicularly magnetized ferromagnetic sublayers strongly coupled via an antiferromagnetic exchange interaction [@Zhang-4]. Furthermore, skyrmions belonging to different layers can move simultaneously. That is, the skyrmion in the layer without current follows the motion of the skyrmion belonging to the layer in which an electrical current is injected [@Zhang-4]. From experimental point of view, a superimposition of skyrmions can be obtained from the strong dipolar stray field of two skyrmions, which behave as a single particle [@Sampaio-NatCom].
In this paper, we study the statical and dynamical properties of two skyrmions in superimposed layers connected by a non-magnetic conductor material. The presence of the conductor causes the magnetic layers interact through the Ruderman-Kittel-Kasuya-Yosida($RKKY$) interaction[@Kitel; @Kasuya; @Yosida; @Bruno-PRB-46; @Aristov-PRB]. The $RKKY$ interaction is one of the most important and frequently discussed couplings between the localized magnetic moments in solids and adatoms interactions[@Stepanyuk; @Stepanyuk-2; @Brovko; @Ako-Nature]. Particularly, concerning topological objects, this interaction has been proposed as a mechanism to stabilize an isolated magnetic skyrmion in a magnetic monolayer on a nonmagnetic conducting substrate [@Bez-PRB]. Here, we show that due to the oscillatory signal of $RKKY$ interaction as a function of the spacer thickness, the interaction between skyrmions placed in different layers can be attractive or repulsive. Additionally, we show that the skyrmion radius diminishes when the skyrmions are far from each other, recovering their widths of isolated skyrmions when they are superimposed. Finally, we obtain an interaction potential for the case of ferromagnetic $RKKY$ coupling and solve the Thiele’s equation aiming to describe the skyrmion dynamics when the in-plane distance between the two skyrmions is small.
The considered magnetic system consists of two rectangular monolayer with dimensions $2\ell_1$ and $2\ell_2$ separated by a non-magnetic metal with thickness $d$ (See Fig. \[Coordinates\]). The layer spacer consists of a conductor material and the electrons are described as free charge carrier, whose formula to $RKKY$ interaction is well established [@Aristov-PRB; @Karol-2008]. Without lost of generality, it is assumed that the skyrmion placed in the lower layer is located at the origin of the adopted coordinate system in such way that the magnetization pattern of the layers comprises a skyrmion positioned in the coordinate $\mathbf{r}_1=(\zeta,-\xi,d)$ and a skyrmion at $\mathbf{r}_2=(0,0,0)$, where the subindices $1$ and $2$ refer respectively to the upper and lower layer.
![Adopted coordinate system to describe the skyrmions position. Vertical bar shows the $z$-component of the magnetization profile of the skyrmions. Gray region represents the non-magnetic metal separating the layers.[]{data-label="Coordinates"}](CoordenadasMat "fig:") ![Adopted coordinate system to describe the skyrmions position. Vertical bar shows the $z$-component of the magnetization profile of the skyrmions. Gray region represents the non-magnetic metal separating the layers.[]{data-label="Coordinates"}](Legend "fig:")
The magnetic energy of an arbitrary magnetization profile will be given by $E=E_{x}+E_{dmi}+E_{rkky}$, where $E_x$, $E_{dmi}$ e $E_{rkky}$ are respectively the energies coming from exchange, Dzyaloshinskii-Moriya and $RKKY$ interactions. It is important to note that if the magnetic properties of the system is described only in terms of exchange and DMI interactions, helical states are predicted to appear [@Nagaosa-PRB]. Nevertheless, despite anisotropy and Zeeman interactions are important to ensure skyrmion stability, for our purposes, the knowledge on exchange and DMI energies is enough to describe our results. For more details and analysis of another terms to the magnetic energy, see Ref. [@Aranda-JMMM].
In the continuous limit, the exchange and $DMI$ energies are respectively given by \[ExModelDisc\] E\_x=J()\^[2]{} dS, and \[DMIModelDisk\] E\_[dmi]{}=DdS,\
where $\mathbf{m}=\mathbf{M}/M_S$ is the unitary vector describing the magnetization direction, $M_S$ is the saturation magnetization of the material, $J$ is the exchange constant and $D$ is the $DMI$ constant. The $RKKY$ interaction will be determined from a continuous approximation considering that the properties of the non-magnetic material connecting the layers can be well described by a free electron gas. In this context, the energy coming from the $RKKY$ interaction between two magnetic moments is given by $H_{rkky}=F(k_F\rho)\,\mathbf{m}_1\cdot\mathbf{m}_2$ [@Kitel; @Bruno-PRB-46; @Aristov-PRB], where $\mathbf{m}_1$ is a magnetic moment in layer $1$, $\mathbf{m}_2$ is a magnetic moment in layer $2$, $F(k_F\rho)$ is the function that determines the coupling between two magnetic moments belonging to different layers, $k_F$ is the Fermi vector of the conductor material and $\rho=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+d^2}$ is the distance between two magnetic moments in different layers. For a free electron gas, the function $F(k_F\rho)$ is given by [@Karol-2008] F(k\_[F]{}R) = \_3(), where $\mathcal{C}_3={A_3^2\mathcal{M}k_F^4}/{8\pi h^2}$, $A_3$ is the exchange interaction between electrons in the magnetic layer and conduction electrons, $\mathcal{M}$ is the effective mass of the conduction electrons and $h$ is the Planck constant. It can be observed that, due to the periodicity of trigonometric functions, the coupling between two magnetic moments can be ferromagnetic or antiferromagnetic, depending on the distance between them. The total $RKKY$ interaction energy is given by the sum of all pairs of magnetic moments of the bi-layer. Thus, in a continuous approximation, the $RKKY$ energy can be written as $$\label{RKKYModel}
E_{rkky}= \frac{1}{\mathcal{S}^2}\iint \mathbf{m}_{1}.\mathbf{m}_{2}'\,F(k_fR)\, dS_{1}dS_{2}'\,,$$ where $\mathcal{S}$ is the surface area of a unitary cell of the magnetic material and the integrals are performed along the surfaces of the two magnetic layers. It is important to note that the interaction is inversely proportional to $\rho^4$ and for $\rho\gg \sqrt{a}$, $F(k_F\rho)$ oscillates with a period $T = \Lambda_f/2$ and decays as $\rho^3\,\,$ [@Bruno-PRB-46].
In the adopted model, the magnetization vector field $\mathbf{M}(\mathbf{r})$ is considered as a continuous function depending on the position inside the magnetic layer. There are several ansatz describing the magnetization profile of a skyrmion [@Beg-Arxiv; @Mourafis-PRB-2006; @Finazzi-PRL-2013; @Vagson-JMMM-2015; @Vagson-JAP-2015]. In this work, we use the ansatz of Refs. [@Guslienko-IEEEMagLett-2015; @Aranda-JMMM], in which the magnetization is parametrized as $\mathbf{m}_j=\sin\Theta_j\cos\Phi_j\,\mathbf{x}+\sin\Theta_j\sin\Phi_j\,\mathbf{y}+\cos\Theta_j\,\mathbf{z}$, with \[ansatz\] \_j=(),=(), where $\lambda$ is the skyrmion characteristic length and $\mathcal{R}_j=\sqrt{(x-x_j)^2+(y-y_j)^2}$. The subindex $j=(1,2)$ describes the layer in which the skyrmion $j$ lies, that is, $(x_j,y_j)=(\zeta,-\xi)$ for layer $1$ and $(x_j,y_j)=(0,0)$ for layer $2$. Therefore, Eq.(\[ExModelDisc\]) can be rewritten as \[ExModel\] E\_[x]{}= J\_[i=1,2]{}dS\_j. Subindex $i$ in the sum refers to $x$ (1) and $y$ (2) coordinates. In addition, Eq. (\[DMIModelDisk\]) is also rewritten as \[DMIModel\] E\_[dmi]{}= D dS\_j.
From the described model we are in position to calculate the total magnetic energy of the bi-layer system. Therefore, from considering that the skyrmions are far from the borders of the stripe, the exchange and $DMI$ energies of the described skyrmion profile are given respectively by $ E_{x_j}=4\pi\lambda^2
J\mathcal{G}_j$ and $E_{dmi_j} =
2\pi\lambda^3D\mathcal{G}_j$, where $$\mathcal{G}_j=\frac{1}{\mathcal{R}_j^2+\lambda^2}-\frac{1}{R_j^2+\lambda^2},$$$$R_j=\text{min}[(L_1-x_j)^2+(L_1-y_j)^2,(L_2-x_j)^2+(L_2-y_j)^2].$$ It can be noted that, in the limit $R_j\gg\lambda$, the exchange energy of the skyrmions is $E_x\approx8\pi J$ and $E_{dmi}\approx4\pi\lambda D$, which is in accord to the energy of solitonic solutions of the non-linear $\sigma$-model [@Rajaraman]. Figure \[SkWidth\] shows the skyrmion energy as a function of its width for different $J/D$. It can be noted that, despite the energy decreases when $D$ increases, there is no qualitative changes of the $\lambda$ value that minimizes the magnetic energy. This is associated with the fact that we are not considering anisotropy or Zeeman interactions in this work. For more details and analysis on the skyrmion size, see Ref. [@Wang-Comm-Phys].
![Qualitative description of $E_x+E_{dmi}$ energy given as a function of $\lambda$ for different relations $J/D$. In this case, we have used $J=1$ and $D=J/4$ (blue), $D=J/3$ (black) and $D=J/2$ (red).[]{data-label="SkWidth"}](EneLamb)
![$RKKY$ energy between skyrmions as a function of the interlayer distance. Figure (a) shows the $RKKY$ energy as a function of the distance between the skyrmions for an interlayer distance of $4$ nm. Inset is a view of the region in which the in-plene distance between skyrmions is less than $10$ nm. Fig. (b) presents $RKKY$ interaction when the interlayer distance is $2.5$ nm. []{data-label="RKKYEnergy"}](D200 "fig:")\
![$RKKY$ energy between skyrmions as a function of the interlayer distance. Figure (a) shows the $RKKY$ energy as a function of the distance between the skyrmions for an interlayer distance of $4$ nm. Inset is a view of the region in which the in-plene distance between skyrmions is less than $10$ nm. Fig. (b) presents $RKKY$ interaction when the interlayer distance is $2.5$ nm. []{data-label="RKKYEnergy"}](D085 "fig:")
![$RKKY$ energy for different values of skyrmion widths for $d=4$ nm. It can be noted that the skyrmions energy increases with $\lambda$ for large distances. However, when the distance between the skyrmions is $\approx d$, the $RKKY$ energy is independent on the skyrmions width.[]{data-label="SkLenght"}](SirmionRadiusL)
Our main goal in this work is to determine the magnetic energy when the skyrmions are separated by a conductor layer. Nevertheless, an expression for $RKKY$ energy is hard to be obtained analytically and so, it is numerically solved by the integration of Eq.(\[RKKYModel\]). We consider that the conductor material is a copper layer for which $k_f=1.36\times10^{10}m^{-1}$. The integration along all the region of the stripe demands a very long computational time. Therefore, aiming to obtain numerical results faster, we have performed the numerical integration by cutting the integration region. That is, we have calculated the $RKKY$ energy of one magnetic moment located in layer $1$ with the magnetic moments located inside a square of side $\rho$ in the layer $2$. This procedure was performed for all magnetic moments in layer $1$. It is noted that differences in the obtained $RKKY$ energy values starts to be negligible when the square side is on the order of $1.2$ nm. Thus, by using this value to the square side, we have calculated the $RKKY$ energy as a function of the spacer thickness and the skyrmion position along $y$-direction. The main results are summarized in Fig.\[RKKYEnergy\], in which it can be noted that due to the periodicity of the $RKKY$ interaction, the skyrmion-skyrmion interaction can be attractive or repulsive, depending on the interlayer thickness. We have also performed the numerical integration of Eq.(\[RKKYModel\]) for different values of $\lambda$ aiming to understand how $RKKY$ interaction influences the skyrmion width; the results are shown in Fig.\[SkLenght\]. We have obtained that the $RKKY$ energy depends on the skyrmon width only when they are far away one to another. That is, for large distances, the skyrmion radius diminishes aiming to reduce $RKKY$ energy. However, when the distance between them decreases, $RKKY$ energy is independent of the skyrmion width and then, when the skyrmions are superimposed, their widths must be determined by the interplay among uniaxial anisotropy, exchange and $DMI$ interactions, being equal to the isolated skyrmion width [@Aranda-JMMM; @Wang-Comm-Phys].
We will now describe the skyrmion dynamics from considering two layers in the absence of currents and external magnetic fields. Additionaly, we will assume that the in-plane skymion distance is on the order of $\lambda$ because, for larger distances, the potential coming from $RKKY$ interaction is practically constant. In our analytical model, we neglect the dynamical deformation of the skyrmions in such way that, the Landau-Lifshitz-Gilbert equation can be reduced to the Thiele equation[@Thiele-Work], written as \[ThieleEq\] -g\_j(t)+\_j(t)=-U(\_1-\_2), where subscripts $j$ label the layers ($1$ and $2$). The first term in the above equation describes the Magnus force exerted by the magnetic texture in the magnetic skyrmion located in the $j$-th layer, which displaces with velocity $\mathbf{v}_j$. Once we are considering the dynamics of a highly symmetrical skyrmion, we have that $g=-4\pi M_s/\gamma$, where $\gamma$ is the gyromagnetic ratio. The second term represents the dissipative force action in each magnetic skyrmion. The right side of Eq. represents the force that determines the skyrmions dynamics, which contains a contribution from the potential $U(\mathbf{r}_1-\mathbf{r}_2)$, originated from the skyrmion-skyrmion interaction. From the results obtained to the $RKKY$ energy, we can state that when $r=|\mathbf{r}_1-\mathbf{r}_2|$ is very small, the interaction energy between skyrmions is given by a harmonic potential $U_{r\rightarrow0}\approx\kappa_1 r^2$, where $\kappa_1$ depends on the layer distance and the conductor parameters and $r$ is the distance between the skyrmions along the $xy$-plane. From Fig. \[RKKYEnergy\] one can note that this approximation is very good when the skyrmion centers are separated by a distance $r\lesssim20$ nm. On the other hand, in the asymptotic limit $r\gtrsim\lambda$, the interaction energy is almost constant and comes from the interaction of the skyrmion in layer $1$ with the ferromagnetic state in layer $2$ and vice-versa, with $U_{r\gtrsim\lambda}=\kappa_3$. In these asymptotic limits, we can obtain the analytical solution to Eq.(\[ThieleEq\]) and it is presented into the Appendix \[Appendix\]. By using the result given in Fig.\[RKKYEnergy\], we can also assume that the potential in the intermediary zone (that is, $20\,\text{nm}<r<70\,\text{nm}$) can be very well represented by $U=\kappa_1
r^2/(\kappa_2+r^2)+\kappa_3$ (See Fig. \[RKKYEnergy\]a). From estimating $\mathcal{C}_3\mathcal{S}^2\sim10^{15}$ Jm, we have obtained the position of the skyrmions for $d=4$ nm as a function of time and the results can be viewed in Fig. \[SkPosition\]. It can be noted that when the in-plane distance between skyrmions is on the order of $\lambda$, a drag effect is observed and an attractive interaction is observed in such way that the two skyrmions become coupled, precessing one around other with frequency $\omega={4\kappa_1 tg}/{(g^2+\alpha^2)}$. This precessory motion is resulted from the interplay between skyrmion-skyrmion attraction and the Magnus force exerted by the magnetic texture in the magnetic skyrmions. In this way, when the skyrmions move one towards the other, the skyrmion Hall effect produces a displacement of skyrmions in opposite directions and the resulting motion is that one shown in Fig. \[SkPosition\]b. The skyrmions can be decoupled by changing the interlayer distance when an antiferromagnetic $RKKY$ interaction appears.
![Fig.a shows the positions along y-axis of skyrmions as a function of time. Fig.b shows the precessory motion of both skyrmions around each other in the in-plane perspective. []{data-label="SkPosition"}](SkPosition "fig:")\
![Fig.a shows the positions along y-axis of skyrmions as a function of time. Fig.b shows the precessory motion of both skyrmions around each other in the in-plane perspective. []{data-label="SkPosition"}](SkPositionPre "fig:")
The observed skyrmion coupling is a very interesting result because it opens the possibility to realize devices based on the concept of skyrmion drag effects. That is, if it is possible to apply different in-plane magnetic fields or different anisotropies in layers $1$ and $2$, the skyrmions can displace with different velocities; however, when the distance between them is small, skyrmions couple and can displace together, which can diminishes the skyrmion Hall effect due to the increasing of the effective skyrmion mass. In addition, a magnetoresistive device can be proposed because there is a decreasing in the resistence when the skyrmion are superimposed. In fact, the value of an electric current passing through the multilayer system in the direction perpendicular to the planes depends on the skyrmion positions because magnetoresistive effects take place and the resistence diminishes when the skyrmions are superimposed.
In conclusion, we have described the coupling between skyrmions in different layers separated by a conductor non-magnetic material. It has been shown that the $RKKY$ interaction can lead to a skyrmion coupling/decoupling, depending on the spacer thickness. The skyrmion radius is also affected by $RKKY$ interaction in such way that it diminishes when the skyrmions are far from each other and it increases when they are in an small in-plane distance. The dynamics of this system show that when skyrmions in different layers are put near one to another, they execute a precession motion around each other. The drag effect mediated by $RKKY$ can be used to move skyrmions in different layers with the same velocity.
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Solution of Thiele Equation for interacting skyrmions {#Appendix}
=====================================================
The dynamics of the skyrmion pair is determined from the solution of Thiele’s equation (\[ThieleEq\]). In the limit of $r\rightarrow0$ the potential can be approximated by a harmonic oscillator of the kind $U=\kappa_1 r^2$. The analytical solution to the Thiele equation is given by
x\_[1\_[r0]{}]{}={+\^}
y\_[1\_[r0]{}]{}={--\^}
x\_[2\_[r0]{}]{}={+\^} y\_[2\_[r0]{}]{}={-+\^}
|
---
abstract: |
We study Dirac-Born-Infeld type effective field theory of a complex tachyon and U(1)$\times$U(1) gauge fields describing a D3${\bar {\rm D}}$3 system. Classical solutions of straight global and local DF-strings with quantized vorticity are found and are classified into two types by the asymptotic behavior of the tachyon amplitude. For sufficiently large radial distances, one has linearly-growing tachyon amplitude and the other logarithmically-growing tachyon amplitude. A constant radial electric flux density denoting the fundamental-string background makes the obtained DF-strings thick. The other electric flux density parallel to the strings is localized, which represents localization of fundamental strings in the D1-F1 bound states. Since these DF-strings are formed in the coincidence limit of the D3${\bar {\rm D}}$3, these cosmic DF-strings are safe from inflation induced by the approach of the separated D3 and ${\bar
{\rm D}}3$.
---
=.22in
[hep-th/0510218\
KIAS-P05058\
SNUTP 05-015]{}
[[**DF-strings from D3${\bar {\bf D}}$3 as Cosmic Strings**]{}\
Inyong Cho\
[*Center for Theoretical Physics, School of Physics,\
Seoul National University, Seoul 151-747, Korea*]{}\
[[email protected]]{}\
Yoonbai Kim\
[*BK21 Physics Research Division and Institute of Basic Science,*]{}\
[*Sungkyunkwan University, Suwon 440-746, Korea*]{}\
[[email protected]]{}\
Bumseok Kyae\
[*School of Physics, Korea Institute for Advanced Study,\
207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-012, Korea*]{}\
[[email protected]]{} ]{}
Introduction
============
When we have a system of a D3-brane and an anti-D3-brane, its dynamics is well described by the effective field theory of a complex tachyon field and U(1)$\times$U(1) gauge fields [@Kraus:2000nj; @Jones:2002si; @Sen:2003tm]. While the D3 and ${\bar {\rm D}}3$ approach each other from apart, the Universe undergoes an inflationary era due to the gravitational effect [@Dvali:1998pa]. When the D-brane coincides with the ${\bar {\rm D}}3$-brane, the system reaches the top of the tachyon potential and the main inflation ends. Then, this unphysical symmetric vacuum state at the zero tachyon amplitude restarts to decay to the true U(1) degenerate vacua at an infinite tachyon amplitude.
When the D3-brane and ${\bar {\rm D}}3$-brane are annihilated in their coincidence limit, perturbative open string degrees living on the branes disappear, but nonperturbative open string degrees can survive in a form of fundamental strings, or of lower-dimensional D-branes of codimension-two with closed string degrees. In terms of effective field theory, one species among those generated through a cosmological phase transition are nothing but vortex-strings [@VS; @Kibble:2004hq] carrying D1- (vorticity or quantized magnetic flux) and fundamental string charge (electric flux along the string). Since inflation already ended, the produced D1 and D1-F1 bound states [@Dvali:2003zj; @Copeland:2003bj; @Leblond:2004uc; @Blanco-Pillado:2005xx; @Kim:2005tw; @Polchinski:2004ia] can remain as relics of the cosmic superstrings [@Witten:1985fp] in the present Universe.
In this paper we consider the DBI type effective action of a complex tachyon and U(1)$\times$U(1) gauge fields, and find straight global and local vortex-string solutions with an electric flux. As shown in [@Sen:2003tm; @Kim:2005tw], there exist static global and local D-vortex solutions in the coincidence limit of D2$\rm \bar{D}2$. While only singular D-vortex solutions are possible without DBI electromagnetic fields [@Sen:2003tm], the regular solutions are allowed when an electric flux is turned on in the radial direction [@Kim:2005tw]. The point-like D-vortices could be readily extended to become D-strings of the D3$\rm \bar{D}3$ system. In this paper we will also turn on a constant electric flux along the string direction, and find that its conjugate momentum density is well localized along the string. The obtained static soliton configurations turn out to be identified with DF-strings from a system of D3${\bar{\rm D}}3$ with fundamental string fluid. In addition to the known D-string solutions with linearly-growing tachyon amplitude, we find new D- and DF-string solutions with logarithmically-growing tachyon amplitude.
Specific contents in each section are as follows. In section 2, we introduce the effective action with a tachyon potential and briefly discuss perturbative degrees and their fate on the D$p{\bar {\rm D}}p$ system. In section 3, we first discuss global DF-strings and then local DF-strings in details. We conclude in section 4 with summary of the obtained results and discussions on a few topics for future studies.
Setup and Perturbative Physics
==============================
We consider a D$p{\bar {\rm D}}p$ system in the coincidence limit of two branes, where the individual branes have the same transverse coordinates. The brane-antibrane system possesses a complex tachyon field $(T,\bar{T})$ describing instability of this system and two massless gauge fields of U(1)$\times$U(1) symmetry $A^{a}_{\mu},\;a=1,2$ living on each brane. Two representative nonlocal effective actions have been used as tachyon actions, i.e., one is derived from boundary string field theory (BSFT) [@Kraus:2000nj; @Jones:2003ae] and the other is Dirac-Born-Infeld (DBI) type proposed in Ref. [@Sen:2003tm]. In this paper we shall employ the latter, $$\begin{aligned}
\label{actl}
S=-{\cal T}_{p}\int d^{p+1}x\, V(\tau)\left[\,\sqrt{-\det(X^{+}_{\mu\nu})}
+\sqrt{-\det(X^{-}_{\mu\nu})}\,\,\right],\end{aligned}$$ where ${\cal T}_{p}$ is tension of the D$p$-brane, $T=\tau e^{i\chi}$, and $$\label{Xpm}
X^{\pm}_{\mu\nu}=g_{\mu\nu}+F_{\mu\nu}\pm C_{\mu\nu}
+({\overline {D_{\mu}T}}D_{\nu}T +{\overline {D_{\nu}T}}D_{\mu}T)/2 .$$ Our notations are $A_{\mu}=(A^{1}_{\mu}+A^{2}_{\mu})/2$ with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, $C_{\mu}=(A^{1}_{\mu}-A^{2}_{\mu})/2$ with $C_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}$, and $D_{\mu}T=(\partial_{\mu} -2iC_{\mu})T$.
Since DF-strings as codimension-two objects are of interest, we consider D3${\bar {\rm D}}$3 system. $-\det
(X^{\pm}_{\mu\nu})$ in the action with $p=3$ takes the following form; $$\begin{aligned}
-\det (X^{\pm}_{\mu\nu})&=&-\det
(g_{\mu\nu})\left[(1+S^\mu_\mu)\left(1+\frac{1}{2}{\cal
F}^{\pm}_{\rho\sigma}{\cal
F}^{\pm\rho\sigma}\right)+\frac{1}{2}A_{\mu\nu}A^{\mu\nu}+S^\mu_\nu
{\cal F}^{\pm}_{\mu\rho}{\cal F}^{\pm\rho\nu}\right]
\nonumber \\
&&-\frac{1}{64}\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma\delta}
{\cal F}^{\pm}_{\mu\nu}{\cal F}^{\pm}_{\rho\sigma}{\cal
F}^{\pm}_{\alpha\beta}{\cal F}^{\pm}_{\gamma\delta}-\frac{1}{16}
\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma\delta}
{\cal F}^{\pm}_{\mu\nu}A_{\rho\sigma}{\cal
F}^{\pm}_{\alpha\beta}A_{\gamma\delta}\, ,
\label{xde}\end{aligned}$$ where ${\cal F}^{\pm}_{\mu\nu}\equiv F_{\mu\nu}\pm C_{\mu\nu}$, $S_{\mu\nu}\equiv (\overline{D_\mu T}D_\nu T+\overline{D_\nu
T}D_\mu T)/2$, and $A_{\mu\nu}\equiv (\overline{D_\mu T}D_\nu
T-\overline{D_\nu T}D_\mu T)/2i$, respectively. Up to the quadratic terms in the gauge fields and derivative of the tachyon amplitude, the Lagrange density in (1+3) dimensions becomes $$\begin{aligned}
\label{la}
{\cal L}\approx -2{\cal T}_{3}V(\tau)\left[\left(\frac{1}{2}\partial_{\mu}\tau
\partial^{\mu}\tau+1\right)+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
+\left(\frac{1}{4}C_{\mu\nu}C^{\mu\nu}+2\tau^{2}\tilde{C}_{\mu}\tilde{C}^{\mu}
\right) \right] ,\end{aligned}$$ where the unitary gauge, $\tilde{C}_{\mu}=C_{\mu}-\partial_{\mu}\Omega/2$, is chosen for topologically trivial sector with $C_{\mu\nu}=\partial_{\mu}\tilde{C}_{\nu}-\partial_{\nu}\tilde{C}_{\mu}$. Note that a cross term $F_{\mu\nu}C^{\mu\nu}$ does not appear in the approximated Lagrange density (\[la\]).
From Ref. [@Sen:1999mg; @Sen:1999xm], universally allowed conditions of the tachyon potential $V$ for the ${\rm D}p\bar{{\rm
D}}p$ system are monotonically decaying property connecting smoothly the maximum of $V(\tau=0)=1$ and minimum of $V(\tau=\infty)=0$. To support perturbative spectrum in superstring theory, we choose $d^{2}V/d\tau^{2}|_{\tau=0}=-1/R^{2}=-1/2$. In the DBI type effective action, exponentially decaying property for large $\tau$, $V(\tau)\sim e^{-\tau/R}$ is usually assumed [@Sen:2002an]. For the analysis of DF-string solutions with the cylindrical symmetry, the above properties are enough at both string core and asymptotic region. For numerical analysis, however, we will use a specific potential satisfying all the above conditions for convenience [@Kutasov:2003er; @Buchel:2002tj] $$\label{V3}
V(\tau)=\frac{1}{{\rm cosh}\left(\frac{\tau}{R}\right)}.$$ Let us read possible perturbative spectra from the Lagrange density (\[la\]). Before the ${\rm D}3\bar{{\rm D}}3$ decays, the complex scalar fields, $T$ and $\bar{T}$, are tachyonic, and both gauge fields, $A_{\mu}$ and $C_{\mu}$, are massless. When it decays completely, a ring of degenerate minima at infinite tachyon amplitude is formed. Naively $A_{\mu}$ seems to remain massless and $\tilde{C}_{\mu}$, absorbing the Goldstone degree $\Omega$, becomes massive due to nonzero vacuum expectation value of $\tau$. Different from usual field theory results, all the tachyon and the gauge fields cannot survive due to vanishing tachyon potential which is an overall factor in the Lagrange density (\[la\]). This phenomenon is easily expected because all the perturbative open string degrees should disappear after the decay of ${\rm
D}3\bar{{\rm D}}3$. On the other hand, nonperturbative degrees including codimension-two branes and fundamental strings can be formed, so that the runaway nature of above tachyon potential should play an indispensable role for determining characters of the generated topological solitons.
DF-strings
==========
In this section we study static DF-string solutions of the classical equations, which are identified as the codimension-two DF-composites from D3${\bar {\rm D}}$3. The obtained nonsingular configurations are classified into the following four types by two crossed borderlines, i.e., (i) global U(1) DF-vortices with critical boundary value of electric field at infinity $F_{tr}(r=\infty)$, (ii) global U(1) DF-vortices with noncritical boundary values of $F_{tr}(r=\infty)$, (iii) local U(1) DF-vortices with critical boundary value of $F_{tr}(r=\infty)$, and (iv) local U(1) DF-vortices with noncritical boundary values of $F_{tr}(r=\infty)$. Since there is one-to-one correspondence between the obtained DF-vortex solution and the D-vortex solution in Ref. [@Kim:2005tw], the newly-obtained DF-vortices with critical boundary value imply the existence of additional D-vortices with the same critical electric field.
Straight strings along the $z$-axis is conveniently described in the cylindrical coordinates $(t,r,\theta,z)$. The ansatz for the $n$ strings superimposed at the origin $r=0$ is $$\label{ant}
T=\tau(r) e^{in\theta}.$$ In order to obtain regular DF-strings, we assume the minimal configuration of the DBI electromagnetic fields $F_{\mu\nu}$ as $$\label{ane}
F_{tr}(r)=E_{r}(r),\quad F_{tz}(r)=E_{z}(r),\quad \mbox{others}=0.$$ Introduction of the gauge field $C_{\theta}$ replaces global strings by local strings $$\begin{aligned}
\label{anc}
C_{\mu}=\delta_{\mu \theta}C_{\theta}(r), \quad (C_{r\theta}=C_{\theta}').\end{aligned}$$
Insertion of the ansätze (\[ant\])–(\[anc\]) into the determinant (\[xde\]) gives $$\begin{aligned}
\label{det}
&&-\det (X^{\pm}_{\mu\nu})
\equiv r^2X \\
&&=-\det
(g_{\mu\nu})\left\{\left[1+\frac{\tau^2}{r^2}(n-2C_{\theta})^2\right]\left[
\left(1-E_{z}^2\right) \left(1+{\tau'}^2\right) -E_{r}^2 \right]
+\left(1-E_{z}^2\right)\frac{{C_{\theta}'}^2}{r^2}\right\} ,
\label{det1}\end{aligned}$$ which simplifies the action (\[actl\]) as $$S=-2{\cal T}_{3}\int dt dr d\theta dz r\, V(\tau) \sqrt{X}\, .$$
Bianchi identity, $\partial_\mu F_{\nu\rho}+\partial_\nu
F_{\rho\mu}+\partial_\rho F_{\mu\nu}=0$, requires $E_z$ to be a constant. When $E_{z}^{2}>1$, $X$ becomes negative and the action (\[actl\]) becomes imaginary, which is physically unacceptable. When $E_{z}^{2}=1$, derivative of the tachyon amplitude disappears in (\[det1\]) and then no nontrivial solution is supported. When $E_{z}^{2}<1$, introduction of new variables, $$\begin{aligned}
\label{corr}
{\tilde E}_{r}(r)=\frac{E_{r}}{\sqrt{1-E_{z}^2}},\qquad
{\tilde {\cal T}}_{3}=\sqrt{1-E_{z}^2}\, {\cal T}_{3},\qquad
\tilde{X}=\frac{X}{1-E_{z}^2},\end{aligned}$$ show that we have $\tilde{X}=X|_{E_z=0}$ and thereby the system with nonvanishing constant $E_{z}$ is formally equivalent to that with vanishing $E_{z}$ under the correspondence (\[corr\]).
For the gauge field $A_{i}$ and conjugate momentum $\Pi^{i}$, the only nontrivial equation is $(r\Pi^r)'=0$ which is rewritten by introducing constant charge density $Q_{{\rm F}1}$ per unit length along $z$-axis as $$\label{Pir}
\Pi^{r}\equiv \frac{1}{\sqrt{-g}}\frac{\delta S}{\delta
(\partial_{t}A_{r})} =\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta
E_{r}}= \frac{1}{\sqrt{1-E_{z}^2}}\frac{2{\tilde {\cal
T}}_3V}{\sqrt{\tilde{X}}}\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]
{\tilde E}_r =\frac{Q_{{\rm F}1}}{r}.$$ Equation of motion for the tachyon amplitude $\tau$ is $$\begin{aligned}
\label{Teq}
\frac{1}{r}\frac{d}{dr}\left\{
\frac{rV}{\sqrt{\tilde{X}}}\left[1+
\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]\tau' \right\}
-\frac{V}{\sqrt{\tilde{X}}}\left(
1+\tau^{'2} -\tilde{E}_{r}^2
\right)\frac{(n-2C_\theta)^2}{r^{2}}\tau
=\sqrt{\tilde{X}}\frac{dV}{d\tau} ,\end{aligned}$$ and that for the gauge field $C_{\theta}$ is $$\begin{aligned}
\label{Ceq}
\frac{1}{r}\frac{d}{dr}\left( \frac{rV}{\sqrt{\tilde{X}}}
\frac{{C_{\theta}'}}{r^2}\right)
+2\frac{V}{\sqrt{\tilde{X}}}\left( 1+\tau^{'2}-\tilde{E}_{r}^2
\right)\frac{\tau^2(n-2C_\theta)}{r^{2}}=0 .\end{aligned}$$ From (\[Pir\]) we obtain an algebraic expression for $E_r$ (or equivalently ${\tilde E}_{r}$) $$\begin{aligned}
\label{Er}
{\tilde E}_{r}(r)^{2}=\frac{E_r(r)^{2}}{1-E_z^2} =\frac{(1+\tau^{'2})
\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]
+\frac{C_\theta^{'2}}{r^2}}
{\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]
\left\{1+\left(\frac{2{\cal T}_3 rV}{Q_{{\rm F}1}}\right)^2
\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]\right\}} .\end{aligned}$$
The $\tau$- and $C_\theta$-equations (\[Teq\])–(\[Ceq\]) with $E_z\ne 0$ is exactly the same as the equations with $E_z=0$. The solutions have been discussed in Ref. [@Kim:2005tw] $$\tau (r) =\tau (r) |_{E_z=0} ,\qquad
C_\theta(r) = C_\theta (r)|_{E_z=0} .$$ The functional form of $E_{r}(r)|_{E_z=0}$ has been also discussed in [@Kim:2005tw], which is the same as $E_r(r)$ in (\[Er\]) except for an overall factor $(1-E_z^2)$. According to Ref. [@Kim:2005tw], the obtained D-vortex solutions are classified as follows. With nonvanishing $E_{r}$ regular vortex solutions are obtained, but with vanishing $E_{r}$ only singular configurations are constructed [@Sen:2003tm]. Characters of the obtained vortices are divided by the gauge field $C_{\mu}$, i.e., global vortices for $C_{\mu}=0$ and local vortices for $C_{\mu}\ne 0$. Since the extension from the point-like D-vortices to the straight D-strings along $z$-direction is straightforward, the aforementioned properties of D-vortex solutions hold also for the DF-strings of our interest.
In the above we have explained similarity between the vortex solutions without $E_{z}$ and those with $E_{z}$. Let us discuss the quantities how to distinguish DF-vortices with $E_{z}$ from D-vortices without $E_{z}$ in what follows. Once we obtain profiles of the tachyon amplitude $\tau$ and the gauge field $C_{\theta}$ for given constant $Q_{{\rm F}1}$ and $E_{z}$, the fundamental string charge density per unit length distributed along the straight DF-string is given by conjugate momentum $\Pi^z$ of the gauge field $A_{z}$ $$\begin{aligned}
\Pi^z(r)^2&\equiv& \left[\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta
(\partial_{t}A_{z})}\right]^2=\left(\frac{1}{r}\frac{\delta
S}{\delta E_{z}}\right)^2
\nonumber\\
&&\hspace{-22mm}=\frac{Q_{{\rm F}1}^{2}E_z^2}{1-E_z^2} \,
\frac{(1+\tau^{'2})
\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]
+\frac{C_\theta^{'2}}{r^2}}{r^2\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2
\right]} \,
\left\{1+\left(\frac{2{\cal T }_3rV}{Q_{{\rm F}1}}\right)^2
\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]\right\} .
\label{Piz}\end{aligned}$$ To be identified as a DF-string, $\Pi^z$ should be localized on the D-string stretched along $z$-direction. Although the shape of the fundamental string charge density per unit length keeps the same form (\[Pir\]) irrespective of its charge $Q_{{\rm F}1}$, that of the DF-strings changes its shape by $Q_{{\rm F}1}$.
To understand detailed property of the DF-strings we also need to investigate U(1) current $j^{\theta}$ $$\begin{aligned}
\label{jth}
j^\theta = \frac{2{\tilde {\cal T}}_3V}{\sqrt{{\tilde X}}}\left(
1+\tau^{'2}-{\tilde E}_r^2\right) \frac{\tau^2}{r^2}(n-2C_\theta),\end{aligned}$$ and nonvanishing components of the energy-momentum tensor $$\begin{aligned}
T^t_{\; t} &=& -\frac{2{\tilde {\cal T}}_3V}{(1-E_z^2)\sqrt{{\tilde
X}}} \left\{\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]
(1+\tau^{'2})+\frac{C_\theta^{'2}}{r^2}\right\} ,
\label{Tt}\\
T^r_{\; r} &=& -\frac{2{\tilde {\cal T}}_3V}{
\sqrt{{\tilde X}}}\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right],
\label{Tr}\\
T^\theta_{\;\theta} &=& -\frac{2{\tilde {\cal T}}_3V}{
\sqrt{{\tilde X}}}\left(1+\tau^{'2}-{\tilde E}_r^2\right) ,
\label{Tth}\\
T^z_{\; z} &=& -\frac{2\tilde {\cal T}_3V}{(1-E_z^2)\sqrt{ \tilde
X}}\left\{\left[1+\frac{\tau^2}{r^2}(n-2C_\theta)^2\right]
(1+\tau^{'2}-E_r^2)+\frac{C_\theta^{'2}}{r^2}\right\} .
\label{Tz}\end{aligned}$$
Global DF-strings
-----------------
Global DF-vortex solutions are attained by choosing constant gauge field $C_{\theta}=0$ in the previous part of the section 2 with neglecting the gauge field equation (\[Ceq\]). Then the only nontrivial differential equation is that of the tachyon amplitude (\[Teq\]). For every regular vortex solution of $n\ne 0$, boundary conditions for the tachyon amplitude are $$\label{bd}
\tau(r=0)=0,\qquad \tau(r\rightarrow \infty)\rightarrow \infty.$$ The runaway nature of the tachyon potential dictates that $\tau(r)$ of a DF-string solution should be a monotonically-increasing function which connects smoothly the boundaries with the conditions (\[bd\]).
Near the origin, a consistent power-series expansion leads to increasing $\tau$, $$\label{tr00}
\tau(r)\approx
\left\{
\begin{array}{ll}
\tau_0 r \left[1 -\frac{{\cal T}_3^2(1+\tau_0^2)^2}{5 Q_{\rm F1}^2R^2}r^4
+ \cdots\right], & (n=1), \\
\tau_{0}r\left[1+\frac{2{\cal T}_3^{2}}{3Q_{{\rm
F}1}^{2}}(1+\tau_0^2)(n^2-1)r^2 - {\cal O}(r^4)\right], & (n \ge 2).
\end{array}
\right.$$ Inserting (\[tr00\]) into (\[Er\]) and (\[Piz\]), we have decreasing $E_{r}^{2}$ from a constant value and decreasing $\Pi^{z2}$ from the infinity, $$\begin{aligned}
E_{r}^{2} &\approx & (1-E_{z}^{2}) (1+\tau_0^2)\left[
1-\frac{4{\cal T}_p^{2}}{Q_{{\rm F}1}^{2}}(1+\tau_0^2) r^2 +\cdots
\right]
, \label{ee0}\\
\Pi^{z2}&\approx & \frac{E_z^2(1+\tau_0^2)}{1-E_z^2}
\left(\frac{Q_{{\rm F}1}}{r}\right)^{2}\left\{1 +\frac{4{\cal T}_3^2}{Q_{\rm
F1}^2}\left[1+\tau_0^2(2n^2-1)\right]r^2 +\cdots\right\} .\end{aligned}$$ In addition, we obtain the current density (\[jth\]), $$\begin{aligned}
\label{jth0}
j^{\theta}&\approx & 4n\tilde{{\cal
T}}_3^2\tau_0^2 \sqrt{\frac{1+\tau_0^2}{1-E_z^2}}~
\frac{r}{|Q_{\rm F1}|} + \cdots ,\end{aligned}$$ and the energy-momentum tensor (\[Tt\])–(\[Tz\]), $$\begin{aligned}
T^{t}_{\; t}&\approx &
-\sqrt{\frac{1+\tau_0^2}{1-E_z^2}}~\frac{|Q_{\rm F1}|}{r}
\left\{1+\frac{2{\cal T}_3^2}{Q_{\rm F1}^2}\left[1+\tau_0^2(2n^2-1)\right]r^2
+\cdots\right\} ,
\label{tt0}\\
T^{r}_{\; r}&\approx &
-\sqrt{\frac{1-E_z^2}{1+\tau_0^2}}~\frac{|Q_{\rm F1}|}{r}
\left[1+\frac{2{\cal T}_3^2}{Q_{\rm F1}^2}(1+\tau_0^2)r^2 +\cdots\right] ,\\
T^{\theta}_{\; \theta}&\approx & -4{\tilde{\cal
T}}_3^2\sqrt{\frac{1+\tau_0^2}{1-E_z^2}}
~\frac{r}{|Q_{\rm F1}|} + \cdots ,
\label{tth0} \\
T^{z}_{\; z}&\approx &
-\sqrt{\frac{1+\tau_0^2}{1-E_z^2}}~\frac{|Q_{\rm
F1}|}{r}\left\{E_z^2 + \frac{2{\cal T}_3^2}{Q_{\rm F1
}^2}[2(1+\tau_0^2n^2)-E_z^2(1+\tau_0^2)]r^2 +\cdots \right\} .
\label{tz0}\end{aligned}$$ As it was expected, the angular component of U(1) current $j^{\theta}$ and the pressure $T^{\theta}_{\theta}$ vanish at the origin. In $T^{t}_{t}$, $T^{r}_{r}$, and $T^{z}_{z}$, the leading term shows singular behavior due to the background fundamental string charge $Q_{{\rm F1}}$. As this fundamental-string charge decreases to zero, the leading term goes to zero, but the slope of the second term proportional to $1/Q_{{\rm F}1}$ becomes steep. It is consistent with the observation that only the singular global vortex solution exists in the absence of the background fundamental-string charge [@Kim:2005tw].
At sufficiently large $r$, we solve the tachyon equation (\[Teq\]). We try to get the tachyon solutions with (i) power-law behavior $\tau\sim \tau_\infty r^k$ ($k> 0$) and (ii) logarithmic behavior $\tau\sim \ln r$.
[**(i)** ]{}: If we substitute the power-law behavior into the tachyon equation (\[Teq\]), only the linearly increasing $\tau$ solution is allowed at leading order. The power-series expansion gives $$\begin{aligned}
\label{solT}
\tau &\approx& \tau_\infty r + \delta - \frac{4{\cal T
}_3^2R}{\tau_\infty^2Q_{\rm
F1}^2}(1+\tau_\infty^2)(1+\tau_\infty^2n^2)
r^2e^{-2\frac{\tau_\infty r+\delta}{R}} + \cdots ,\end{aligned}$$ where $\tau_\infty$ ($>0$) and $\delta$ are undetermined, but $\tau_\infty$ is related with $\tau_{0}$ near the origin. Numerical works show that $\tau_{\infty}$ is almost proportional to $\tau_{0}$ for large $\tau_{0}$’s as in Fig. \[fig1\]. Specifically, for $n=1$, $Q_{{\rm F1}}/{\cal T}_{3}=2$, and $R=\sqrt{2}$, $\displaystyle{\lim_{\tau_{0}\rightarrow \infty}(\tau_{\infty}/
\tau_{0})\rightarrow 0.4566}$.
Inserting (\[solT\]) into the various physical quantities (\[Er\]), (\[Piz\]), (\[jth\])–(\[Tz\]), we read the followings. First, the radial electric field approaches rapidly a constant boundary value $|E_{r}(\infty)|=\sqrt{(1-E_{z}^{2})(1+\tau_{\infty}^{2})}\,$ , $$\begin{aligned}
\label{solE}
E_r^2&\approx& (1-E_{z}^{2})
(1+\tau_\infty^2)\left\{1-\frac{16{\cal T}_3^2R}{\tau_\infty
Q_{\rm
F1}^2}\left[1+\tau_\infty^2n^2\left(1+\frac{2\delta}{R}\right)
\right]re^{-2\frac{\tau_\infty r+\delta}{R}} + \cdots\right\} .\end{aligned}$$ Second, the leading terms of $\Pi^{z}$, $T^{t}_{t}$, $T^{r}_{r}$, $T^{z}_{z}$ are all proportional to $\Pi^{r}$ ($=Q_{{\rm F}1}/r$) and the subleading terms exponentially suppressed, $$\begin{aligned}
\Pi^{z2}&\approx & \frac{E_z^2(1+\tau_\infty^2)}{1-E_z^2}
\left(\frac{Q_{\rm F1}}{r}\right)^{2} \left[1 + \frac{32{\cal
T}_3^2}{Q_{\rm F1}^2}(1+\tau_\infty^2n^2)r^2
e^{-2\frac{\tau_\infty r+\delta}{R}}
+\cdots \right] ,
\label{pzz}\\
T^{t}_{\; t}&\approx & -\sqrt{\frac{1+\tau_{\infty}^{2}}{1-E_z^2}}
\,~\frac{|Q_{{\rm F1}}|}{r}\left[1+\frac{16{\cal T }_3^2}{Q_{\rm
F1}^2}(1+\tau_\infty^2n^2)r^2e^{-2\frac{\tau_\infty
r+\delta}{R}}+ \cdots \right],
\label{ett2}\\
T^{r}_{\; r}&\approx & -\sqrt{\frac{1-E_z^2}{1+\tau_{\infty}^{2} }}
\,~\frac{|Q_{{\rm F1}}|}{r}\left\{1+\frac{8{\cal T
}_3^2R}{\tau_\infty Q_{\rm
F1}^2}\left[1+\tau_\infty^2n^2\left(1+\frac{2\delta}{R}\right)\right]
re^{-2\frac{\tau_\infty r+\delta}{R}} +\cdots\right\},\\
T^{z}_{\; z}&\approx &
-\sqrt{\frac{1+\tau_\infty^2}{1-E_z^2}}~\frac{|Q_{\rm
F1}|}{r}\left[E_z^2 + \frac{16{\cal T}_3^2}{Q_{\rm
F1}^2}(1+\tau_\infty^2n^2)r^2e^{-2\frac{\tau_\infty
r+\delta}{R}}+\cdots \right] .
\label{tzu}\end{aligned}$$ Third, the angular components $j^{\theta}$ and $T^{\theta}_{\theta}$ exponentially decreasing so that, with (\[jth0\]) and (\[tth0\]), their shapes in $(r,\theta)$-plane look like a ring, $$\begin{aligned}
j^{\theta}&\approx & 16n\tilde{{\cal
T}}_{3}^2\tau_{\infty}^2\sqrt{\frac{1+\tau_{\infty}^{2}}{1-E_z^2}
}~\frac{|Q_{\rm F1}|}{r} e^{-2\frac{\tau_\infty r+\delta}{R}}+\cdots ,
\label{jtt}\\
T^{\theta}_{\;\theta}&\approx & -16\tilde{{\cal
T}}_{3}^2\sqrt{\frac{1+\tau_{\infty}^{2}}{1-E_z^2}}~\frac{r}{|Q_{\rm
F1}|}e^{-2\frac{\tau_\infty r+\delta}{R}}+\cdots .
\label{tthu}\end{aligned}$$ Here, we do not present the results of numerical analysis since the obtained configurations are exactly the same as those of D-strings [@Kim:2005tw] except for the fundamental-string charge density $\Pi^{z}$ in Fig. \[fig2\].
[**(ii)** ]{}: If $\tau_{0}$ in the tachyon field near the origin (\[tr00\]) is sufficiently small, then this solution cannot reach $\tau(r=\infty)=\infty$. It means that there exists a critical value of $\tau_{0}$ which corresponds to $\tau'(\infty
)\rightarrow 0$ in (\[solT\]), and it also requires a critical-charge density $(Q_{{\rm F}1}/{\cal T}_{3})^{2}=
8/R^{2}$. In this limit, a natural asymptotic behavior of the tachyon amplitude is logarithmic, $\tau(r)\sim \tau_{\infty}{\rm
ln}r$.[^1] If we try this configuration, the field equation (\[Teq\]) with (\[Er\]) fixes the value of $\tau_{\infty}$ to $\tau_{\infty}=2R$, which leads the tachyon potential to a power-law decay, $V\approx 2/r^{2}$; $$\begin{aligned}
\label{logsolT}
\tau(r)\approx 2R\, {\rm ln}r\left(1 -2n^2R^2\frac{{\rm
ln}r}{r^2} + \cdots\right) .\end{aligned}$$ Note that $\ln r$ is not well-defined at the entire region $(0\le
r\le \infty)$, regularity of the obtained solution needs further mathematically-rigorous study. It turns out, in this case, that the radial component of the electric field $E_{r}$ (\[Er\]) approaches a critical value at infinity with a power-law ${\cal
O}(1/r^{2})$, $E_{r}^2(r=\infty)=1-E_{z}^{2}$, $$\begin{aligned}
\label{loer}
E_{r}^{2}(r)\approx (1-E_z^2)\left(1+ \frac{2R^{2}}{r^{2}}
+ \cdots \right) .\end{aligned}$$ This looks similar to the case of the thick single topological BPS tachyon kink [@Kim:2003in].
Inserting (\[logsolT\]) into (\[Piz\]), (\[Tt\]), (\[Tr\]), and (\[Tz\]), we have again ${\cal O}(1/r)$ leading term in $\Pi^{z}$, $T^{t}_{\; t}$, $T^{r}_{\; r}$, and $T^{z}_{z}$, $$\begin{aligned}
\Pi^{z2}&\approx & \frac{E_z^2}{1-E_z^2}\left(\frac{Q_{\rm
F1}}{r}\right)^2\left(1+\frac{6R^2}{r^2}
+\cdots \right) ,
\label{logsolE}\\
T^{t}_{\; t}&\approx & -\frac{1}{\sqrt{1-E_z^2}}
\,\frac{|Q_{\rm F1}|}{r}\left(1+\frac{3R^2}{r^2}+\cdots\right),
\label{trin}\\
T^{r}_{\; r} &\approx & -\sqrt{1-E_z^2} \,\frac{|Q_{\rm
F1}|}{r}\left(1-\frac{R^2}{r^2}+\cdots\right),
\\
T^{z}_{z}&\approx & -\frac{1}{\sqrt{1-E_z^2}}\frac{|Q_{\rm
F1}|}{r}\left[E_z^2 +(2+E_z^2)\frac{R^2}{r^2}+\cdots \right].
\label{ltz}\end{aligned}$$ The coefficients of the leading terms can be understood as the $\tau_{\infty}
\rightarrow 0$ limit of (\[pzz\])–(\[tzu\]) for the power-law solution. However, the subleading terms exhibit also a power-law behavior in (\[logsolE\])–(\[ltz\]), instead of the exponential decay in (\[pzz\])–(\[tzu\]). This ${\cal O}(1/r)$ term makes its energy diverge linearly. On the other hand, the angular components of the current $j^{\theta}$ (\[jth\]) and the pressure $T^{\theta}_{\;\theta}$ (\[Tth\]) have different behaviors for the leading terms. They have a power-law decay in this case while those for the linearly-growing tachyon have an exponential decay (\[logsolT\]), $$\begin{aligned}
j^{\theta}&\approx & \frac{64 n\,\tilde{{\cal
T}_{3}^2}R^{2}}{\sqrt{1-E_z^2}~|Q_{\rm F1}|} \, \frac{(\ln
r)^{2}}{r^{5}} + \cdots ,
\\
T^{\theta}_{\; \theta}&\approx & -\frac{16\tilde{{\cal
T}}_3^2}{\sqrt{1-E_z^2}~|Q_{\rm F1}|} \,\frac{1}{r^{3}} +\cdots .\end{aligned}$$
The numerical solution for the logarithmic tachyon amplitude connecting the origin and large $r$ is shown in Fig. \[fig3\]-(a). We read $\tau_{0}$ as $\tau_{0}
=2.46327$. The profile of the radial electric field $E_{r}$ decreases monotonically from a nonzero value larger than unity at the origin to unity at infinity as shown in Fig. \[fig3\]-(b). The angular components of the current $j^{\theta}$ and the pressure $T^{\theta}_{\theta}$ have a ring shape connecting zeros at both boundaries, and $j^{\theta}$ is plotted in Fig. \[fig3\]-(c). Since $\Pi^{z}$, $T^{t}_{t}$, $T^{r}_{r}$, and $T^{z}_{z}$ behave in a similar way, we only draw the figure of $\Pi^{z}$ which has ${\cal O}(1/r)$ singularity at the origin and decreases monotonically to zero as shown in Fig. \[fig2\]. Both the power-law solution (\[solT\]) and the logarithmic solution (\[logsolT\]) share similar shapes for $\Pi^{z}$ as are given by the solid and dashed lines in Fig. \[fig2\].
The leading linear divergence in the energy of the obtained DF-string configurations per unit length along the $z$-direction can be read off from (\[ett2\]) and (\[trin\]), $$\label{Eg}
\frac{E}{\int dz} =\int_0^{R_{{\rm IR}}}dr r d\theta \, T_{tt}=
2\pi\sqrt{\frac{1+\tau_{\infty}^{2}}{1-E_{z}^{2}}}\, |Q_{{\rm
F1}}|R_{{\rm IR}}+({\rm finite}) .$$ Since the divergent part is linearly proportional to the fundamental-string charge density at the origin, a possible source of this infra-red divergence is different from the familiar nature of logarithmically divergent energy of the global vortex. For a given fundamental-string charge density $Q_{{\rm F1}}$, the energy spectrum of each solution is classified by $\tau_{\infty}$. In that sense, the logarithmic solution with $\tau_{\infty}=0$ in (\[Eg\]) is the minimum energy solution of the D- or DF-strings.
If we take the limit of vanishing fundamental-string charge density $Q_{{\rm F}1}\rightarrow 0$, the first terms proportional to $Q_{{\rm F}1}^{2}$ in $T^{t}_{\;t}$, (\[tt0\]) and (\[ett2\]) approach zero for the linearly-growing solution (\[solT\]). From behavior of the second $Q_{{\rm
F}1}$-independent terms in (\[tt0\]) and (\[ett2\]), we may read the limit of $\delta$-function like configuration in the limit of $\tau_{\infty}\rightarrow \infty$. This phenomenon is consistent with the existence of singular vortex solution in the absence of $Q_{{\rm F}1}^{2}$ [@Sen:2003tm]. When the electric field $E_{z}$ approaches critical value, various densities including $\Pi^{z}$ (\[Piz\]), $T^{t}_{\;t}$ (\[Tt\]), $T^{z}_{\; z}$ (\[Tz\]), $j^{\theta}$ (\[jth\]), and $T^{\theta}_{\;\theta}$ diverge with the finite fundamental-string charge density $Q_{{\rm F}1}$, but $E_{r}$ (\[Er\]) and $T^{r}_{\; r}$ (\[Tr\]) go to zero. These can easily be checked also by the expanded expressions given in this subsection. This singularity was expected from the beginning if we see the expression of determinant (\[det1\]) in the action (\[actl\]).
There is another coupling to the bulk RR fields given by the Wess-Zumino term, and, for the global DF-strings from D$3{\bar {\rm D}}3$ [@Kraus:2000nj; @Jones:2002si; @Sen:2003tm; @Kennedy:1999nn], it is $$\begin{aligned}
S_{\rm WZ}&=& \mu ~{\rm Str}\int_{\Sigma_4}C_{\rm RR}\wedge{\rm
exp}\left(\begin{array}{cc} F^{1}-T\bar{T} & i^{3/2}~\partial T \\
-i^{3/2}~\partial\overline{T} & F^{2}-\bar{T}T
\end{array}\right)
\nonumber \\
&=&-n\mu\int_{\Sigma_4} \frac{de^{-\tau^{2}}}{dr}
\left(C_{\rm RR}^{(1)}\wedge dr\wedge d\theta +
\frac{E_{z}}{3}C_{\rm RR}^{(-1)}\wedge dt\wedge dr\wedge d\theta\wedge dz
\right),
\label{rtwz}\\
&\propto & n,\end{aligned}$$ where $\mu$ is a real constant and the supertrace ${\rm Str}$ is defined to be a trace with $\sigma_3$ inserted. The first term in (\[rtwz\]) means the charge of a D1-brane stretched along the $z$-axis, which is proportional to the vorticity $n$. Thus the charge density of the D1-brane per unit length is identified as the topological charge of which current density is defined by $$\label{d1c}
j_{{\rm D}1}^{\mu}=\frac{{\bar T}\partial^{\mu}T-T\partial^{\mu}{\bar T}}{
4\pi i{\bar T}T}.$$ Although the second term proportional to both $n$ and $E_{z}$ in (\[rtwz\]) implies an (Minkowski) instanton charge, its possible physical meaning will be discussed in the next subsection. In addition to the D1 charge (\[d1c\]), the stringy object of interest carries the charge of fundamental strings along the $z$-axis, which is denoted by the localized electric flux $\Pi^{z}$ (\[Piz\]). Since the point charge $Q_{{\rm F}1}$ (\[Pir\]) at $r=0$ is nothing but the background charge distribution of fundamental strings coming from a transverse direction, and is ending on a point along the $z$-axis, the stringy object carrying the vortex charge $n$ and the localized electric flux $\Pi^{z}$ is identified as a DF-string or a $(p,q)$-string (composite of D1F1) from D3${\bar {\rm D}}3$ system with fundamental strings. What we obtained is summarized schematically in Fig. \[fig4\]. If the early Universe involved a D3${\bar {\rm D}}3$, the obtained DF-string can remain as a cosmic fossil named as the cosmic global DF-string.
Local DF-strings
----------------
When the gauge field $C_{\mu}$ (\[anc\]) is turned on, the character of DF-strings becomes local. In usual local field theories, e.g., the Abelian-Higgs model, a role of the gauge field is to make energy of the local vortex (energy density of the vortex-string per unit length along the string) finite by trimming the logarithmically-divergent energy of the global vortex [@VS]. This phenomenon was not observed in D-vortices from D2${\bar {\rm D}}$2 system with fundamental strings; the energy of the D-vortex is linearly-divergent, but its source is fundamental string charges [@Kim:2005tw]. Although this sort of energy-trimming seems unlikely also for local DF-strings of our interest, we investigate the existence and the property of local DF-strings in this subsection.
Since the inclusion of the gauge field $C_{\theta}$ (\[anc\]) requires the analysis of the coupled equations (\[Teq\])–(\[Ceq\]), we need boundary conditions for the gauge field in addition to those for the tachyon (\[bd\]), $$\label{cbd}
C_{\theta}(0)=0,\qquad C_{\theta}(\infty)=\frac{n}{2}.$$ From now on, we examine the differential equations (\[Teq\])–(\[Ceq\]) and the expressions (\[Er\]) and (\[Piz\]) for $E_{r}$ and $\Pi^{z}$, and find local DF-string solutions satisfying the boundary conditions (\[bd\]) and (\[cbd\]).
Near the origin, the power-series expansion of $\tau(r)$ and $C_{\theta}(r)$ for DF-string solutions gives $$\label{tao}
\tau(r)\approx
\left\{
\begin{array}{ll}
\tau_0 r \left[1 -\frac{{\cal T}_3^2(1+\tau_0^2)^2}{5 Q_{\rm F1}^2R^2}r^4
+ \cdots\right], & n=1 \\
\tau_{0}r\left[1+\frac{(n^{2}-1)\alpha}{6} r^2+ \cdots\right], & n \ge 2
\end{array}
\right.$$ $$\label{cto}
C_{\theta}(r)\approx
C_{0}r^{3}\left[1-\frac{3+\tau_0^2 n^2(5-2n^2)}{10(1+\tau_0^2n^2)}\alpha r^2+
\cdots \right], \qquad n\ge 1$$ where $\alpha$ is $$\begin{aligned}
\label{alp}
\alpha=\frac{1}{(1+\tau_0^2n^2)^2}\left[\frac{4{\cal
T }_3^2}{Q_{\rm F1}^2}(1+\tau_0^2)(1+\tau_0^2n^2)^2-9C_0^2\right] .\end{aligned}$$ For the local DF-strings with unit vorticity, the increasing rate of the tachyon amplitude (\[tao\]) is not affected by $C_{0}$ up to the second order. On the other hand, the signature in front of $9C_{0}^{2}$ in (\[alp\]) is opposite to that of the first term which is proportional to ${\cal T}_{3}^{2}/Q_{{\rm F}1}^{2}$, so the increasing rate of the tachyon amplitude (\[tao\]) becomes smaller for local DF-strings.
Inserting the expansions (\[tao\])–(\[cto\]) into the radial electric field $E_{r}$ (\[Er\]) and the fundamental-string charge density $\Pi^{z}$ (\[Piz\]), we have a nonzero value, $(1-E_z^2)(1+\tau_{0}^2)$, for $E_{r}$ and singular $|\Pi^{z}|$ at the origin $$\begin{aligned}
E_{r}^{2}(r)&\approx & (1-E_z^2)(1+\tau_{0}^2)(1-\alpha r^2+
\cdots), \label{ero}\\
\Pi^{z2} &\approx &
\frac{E_z^2(1+\tau_0^2)}{(1-E_z^2)}\left(\frac{Q_{\rm
F1}}{r}\right)^2\left( 1 - \beta r^{2}+ \cdots \right),
\label{piz0}\end{aligned}$$ where $\beta$ is $$\begin{aligned}
\beta =- \frac{1}{(1+\tau_0^2n^2)^2}\left[
\frac{4{\cal T
}_3^2}{Q_{\rm F1}^2}(1+\tau_0^2(2n^2-1))(1+\tau_0^2n^2 )^2 + 9C_0^2
\right].\end{aligned}$$ The current density (\[jth\]) again increases from zero $$\begin{aligned}
j^{\theta}\approx
\sqrt{\frac{1-E_z^2}{1+\tau_0^2}}~\tau_0^2n|Q_{\rm
F1}|\alpha r + \cdots ,\end{aligned}$$ and nonvanishing components of the energy-momentum density (\[Tt\])–(\[Tz\]) show the following behavior which is similar to the case of global DF-strings (\[tt0\])–(\[tz0\]) $$\begin{aligned}
T^{t}_{\;t}&\approx &
-\sqrt{\frac{1+\tau_{0}^2}{1-E_z^2}}~\frac{|Q_{\rm
F1}|}{r}\left(1-\frac{\beta}{2}r^{2}+ \cdots \right),
\\
T^{r}_{\;r}&\approx &
-\sqrt{\frac{1-E_z^2}{1+\tau_{0}^2}}~\frac{|Q_{\rm F1}|}{r}
\left( 1 +\frac{\alpha}{2}r^{2}+ \cdots \right),
\\
T^{\theta}_{\;\theta}&\approx &
-\sqrt{\frac{1-E_z^2}{1+\tau_0^2}}\, |Q_{\rm F1}|\alpha r + \cdots
,
\\
T^z_{\;z}&\approx & -E_z^2
\sqrt{\frac{1+\tau_0^2}{1-E_z^2}}~\frac{|Q_{\rm
F1}|}{r}\left\{1-\left[\frac{\alpha}{2}-\frac{4(1+\tau_{0}^{2}n^{2})}{E_{z}^{2}
}\frac{{\cal T}_{3}^{2}}{Q_{{\rm F}1}^{2}}
\right]r^{2}+\cdots\right\}.\end{aligned}$$
The near-origin behavior of the local DF-string solutions is parameterized smoothly by $\tau_{0}$ in (\[tao\]) and $C_{0}$ in (\[cto\]). At asymptotic region, the tachyon amplitude of every local DF-string will be proven to approach universally the vacuum, but the approaching behavior is sorted into two; (i) linear divergence $\tau \sim \tau_\infty r$ and (ii) logarithmic divergence $\tau \sim \tau_\infty \ln r$, as were for the global DF-strings in the previous subsection. We analyze the DF-string solutions with the linearly-divergent $\tau$ and the logarithmically-divergent $\tau$ separately in what follows.
[**(i)** ]{}: If we examine carefully the coupled differential equations (\[Teq\])–(\[Ceq\]), the leading asymptotic behavior of the tachyon amplitude is either linearly-growing or logarithmically-growing. First, we consider the linearly-growing case. The subleading term of the tachyon amplitude is decreasing exponentially $$\begin{aligned}
\label{tan}
\tau(r)&\approx& \tau_{\infty}r +\delta -\frac{4{\cal
T}_3^2R(1+\tau_\infty^2)}{\tau_\infty^2Q_{\rm
F1}^2}~r^2e^{-2\frac{\tau_\infty r+\delta }{R}}+\cdots ,\end{aligned}$$ where $\tau_{\infty}$ and $\delta$ are undetermined constants which are governed by the behavior near the origin. For the gauge field $C_{\theta}$, we consider small $\delta C_\theta$ at the asymptotic region, $$\begin{aligned}
\label{ctn}
C_{\theta}(r)&\approx& \frac{n}{2} + \delta C_\theta .\end{aligned}$$ Substituting this into the equation for the gauge field, we obtain a linear equation, $$\begin{aligned}
\label{Ne}
M(t)\frac{d^2\delta C_\theta}{dt^2}
=-\frac{d}{d\delta C_\theta}U(\delta C_\theta),\qquad
t=\kappa r^3 , \quad \left(
\kappa=\frac{4{\cal T}_p\tau_\infty}{3|Q_{\rm F1}|}
\sqrt{1+\tau_\infty^2}e^{-\delta/R}
\right),\end{aligned}$$ where $M(t)=e^{2\tau_\infty t^{1/3}/(R\kappa^{1/3})}$ and $U=-(\delta C_\theta)^2/2$. If we identify $\delta C_\theta$ as a one-dimensional position of a hypothetical particle, Eq. (\[Ne\]) is interpreted as a Newtonian equation with a variable mass $M(t)$ and a conservative potential $U(\delta C_\theta)$. The nontrivial analytic solution satisfying $\delta C_\theta (r=\infty)=0$ is not known yet, but the existence of such solution can easily be read from the properties of $U(\delta C_\theta)$; max\[$U(\delta C_\theta)$\]=0 at $\delta C_\theta=0$ and min\[$U(\delta C_\theta)]=-\infty$ at $\delta C_\theta=\pm\infty$.
With the asymptotic behavior of the tachyon amplitude (\[tan\]) and the gauge field (\[ctn\]), the radial electric field approaches its boundary value exponentially, $$\begin{aligned}
\label{ern}
E_{r}^{2}(r)&\approx& (1-E_z^2)(1+\tau_{\infty}^2)
\left(1-\frac{16{\cal T}_3^2R}{\tau_\infty Q_{\rm
F1}^2}re^{-2\frac{\tau_\infty r+\delta }{R}}+\cdots \right),\end{aligned}$$ and the U(1) current and the angular component of the energy-momentum tensor decay exponentially to zero, $$\begin{aligned}
j^{\theta}&\approx & -32{\tilde{\cal
T}}_3^2\tau_\infty^2\sqrt{\frac{1+\tau_\infty^2}{1-E_z^2}}~\frac{r}{|Q_{\rm
F1}|}e^{-2\frac{\tau_\infty r +\delta}{R}}\delta C_{\theta} ,
\label{jtil} \\
T^{\theta}_{\;\theta}&\approx & -16\tilde{{\cal
T}}_3^2\sqrt{\frac{1+\tau_\infty^2}{1-E_z^2}}~\frac{r}{|Q_{\rm
F1}|}e^{-2\frac{\tau_\infty r+\delta }{R}}
+ \cdots .
\label{thhil}\end{aligned}$$ As it was the case of global DF-strings in the previous subsection, the leading terms of $\Pi^{z}$, $T^{t}_{\;t}$, $T^{r}_{\;r}$, and $T^z_{\;z}$ are commonly proportional to $Q_{\rm F1}/r$ but the subleading terms decrease exponentially, $$\begin{aligned}
\Pi^{z2} &\approx&
\frac{E_z^2(1+\tau_\infty^2)}{1-E_z^2}\left(\frac{Q_{\rm
F1}}{r}\right)^2\left(1+\frac{32{\cal T }_3^2}{Q_{\rm F1
}^2}r^2e^{-2\frac{\tau_\infty r+\delta }{R}}+\cdots \right) ,
\label{pizi}\\
T^{t}_{\;t}&\approx &
-\sqrt{\frac{1+\tau_{\infty}^2}{1-E_z^2}}~\frac{|Q_{\rm F1}|}{r}
\left(1+\frac{16{\cal T}_3^2}{Q_{\rm
F1}^2}~r^2e^{-2\frac{\tau_\infty r+\delta }{R}}+\cdots \right),
\label{etil}\\
T^{r}_{\;r}&\approx &
-\sqrt{\frac{1-E_z^2}{1+\tau_{\infty}^2}}~\frac{|Q_{\rm F1}|}{r}
\left(1+\frac{8{\cal T}_3^2R}{\tau_\infty Q_{\rm F1}^2}
re^{-2\frac{\tau_\infty r+\delta }{R}} + \cdots \right),
\label{eril}\\
T^z_{\;z}&\approx&
-\sqrt{\frac{1+\tau_\infty^2}{1-E_z^2}}~\frac{|Q_{\rm
F1}|}{r}\left(E_z^2+\frac{16{\cal T}_3^2}{Q_{\rm
F1}^2}r^2e^{-2\frac{\tau_\infty r+\delta }{R}} +\cdots \right) .\end{aligned}$$ Note that the leading long-range terms of local DF-strings are the same as those of global DF-strings, and that the limiting behaviors for the large, or small $Q_{{\rm F}1}^{2}$, and for the critical electric field $|E_{z}|\rightarrow 1$, are also the same.
[**(ii)** ]{}: Lastly let us discuss logarithmic $\tau$-solution of the local DF-string. For sufficiently large $r$, the leading logarithmic term has the same coefficient with the global DF-string (\[logsolT\]), but the subleading term does not contain logarithmic term; $$\begin{aligned}
\tau(r)&\approx& 2R ~{\rm ln}r -\frac{4R^3}{r^2} +\cdots .\end{aligned}$$ As we observed in the global string case, the first subleading term of $E_{r}$ in (\[loer\]) is governed only by the leading term of the tachyon amplitude in (\[logsolT\]). Therefore, the expansion for the logarithmic $\tau$-solution is the same for the local string. If we try to get the asymptotic solution of the gauge field $C_{\theta}$ similarly to the case of linearly-growing solution (\[ctn\]), the linearized equation for $\delta C_\theta$ is $$\frac{d}{dr}\left(\frac{\delta C_\theta'}{r^2}\right)
\approx 32R^4\frac{({\rm ln}r)^2}{r^4}\delta C_\theta .$$ We find a solution which decreases rapidly to zero, $$\delta C_\theta (r)\approx C_{1}\frac{r^{\frac{3}{2}}}{\sqrt{\ln r}}\,
{\rm WhittakerW}\left(-\frac{9}{128}\frac{\sqrt{2}}{R},\, \frac{1}{4},\,
4\sqrt{2}R(\ln r)^{2}\right),$$ where $C_{1}$ is an integration constant which is to be set by the boundary conditions at the asymptotic. As shown in Fig. \[fig5\], the gauge field $C_{\theta}$ (the dashed line) reaches its boundary value $n/2$ rapidly and the tachyon amplitude $\tau$ (the solid line) grows logarithmically. Due to the mixture of slowly-growing and rapidly-growing functional behaviors for single numerical analysis, the results of numerical work need further improvement for this logarithmic case.
This rapidly-decreasing behavior of the gauge field $C_{\theta}$ affects that of the U(1) current (\[jth\]), but does not appear in the radial pressure (\[Tr\]) up to the leading order, $$\begin{aligned}
j^{\theta}&\approx & -\frac{128\,\tilde{{\cal
T}_{3}^2}R^{2}}{\sqrt{1-E_z^2}~|Q_{\rm F1}|} \,
\frac{(\ln r)^{2}}{r^{5}} \delta C_\theta +\cdots , \\
T^{\theta}_{\; \theta}&\approx & -\frac{16\tilde{{\cal
T}}_3^2}{\sqrt{1-E_z^2}~|Q_{\rm F1}|}
\,\frac{1}{r^{3}}+\cdots .\end{aligned}$$ Similar to $E_{r}$, all the components of the energy-momentum tensor and the $z$-component of the electric flux density for the local DF-string have the same functional forms with those of the corresponding components of the global DF-string up to the second order terms. Therefore, the leading energy density of the local DF-string for the logarithmically-growing tachyon amplitude shares that of the local DF-string for the linearly-growing tachyon amplitude and that of the global strings. It means that the discussion on the energy of the global DF-strings can also be applied to that of the local DF-strings; the role of the gauge field $C_{\theta}$ in localizing the energy of the string is negligible, which is very different from that in the case of Nielsen-Olesen vortex-string in the Abelian-Higgs model.
Due to the gauge field $C_{\theta}$, the Wess-Zumino term of D$3{\bar {\rm D}}3$ describing coupling to the bulk RR fields becomes slightly different from that of the global DF-strings [@Kraus:2000nj; @Jones:2002si; @Sen:2003tm; @Kennedy:1999nn], $$\begin{aligned}
S_{\rm WZ}&=& \mu ~{\rm Str}\int_{\Sigma_4}C_{\rm RR}\wedge{\rm
exp}\left(\begin{array}{cc} F^{1}-T\bar{T} & i^{3/2}~D T \\
-i^{3/2}~\overline{DT} & F^{2}-\bar{T}T
\end{array}\right)
\nonumber \\
&=&2\mu\int_{\Sigma_4} \left\{\frac{d}{dr}\left[e^{-\tau^{2}}\left(C_{\theta}-
\frac{n}{2}\right)\right]
\left(C_{\rm RR}^{(1)}+\frac{E_{z}}{3}C_{\rm RR}^{(-1)}\wedge dt\wedge dz
\right)\wedge dr\wedge d\theta
\right. \nonumber\\
&&\hspace{15mm} \left. -\frac{1}{3}E_{z}C_{r\theta}
C_{\rm RR}^{(-1)}\wedge dt\wedge dr\wedge d\theta\wedge dz
\right\}.
\label{rtz}\end{aligned}$$ The term of $C_{\rm RR}^{(1)}$ coupling is proportional to the vorticity $n$ so that the local DF-string carries the quantized magnetic flux as a D1 charge density per unit length along the $z$-axis, $$\begin{aligned}
\label{flu}
\Phi=\int_{0}^{\infty} dr\int_{0}^{2\pi}d\theta \,
C_{r\theta} =\pi n.\end{aligned}$$ Note that the second term in (\[rtz\]) tells us that $C_{\rm RR}^{(-1)}$ coupling is nothing but an axion coupling. Since we have additionally the fundamental-string charge density $\Pi^{z}$ localized along the string direction (the $z$-axis in our case), the obtained stringy objects are local DF-strings, or local $(p,q)$-strings from D3${\bar {\rm D}}$3 system with fundamental strings.
Conclusions
===========
The system of D3${\bar {\rm D}}3$ with fundamental strings has been considered in the coincidence limit of a brane and an anti-brane. In the scheme of effective field theory, it is described by the DBI type effective action of a complex tachyon field and U(1)$\times$U(1) gauge fields. The runaway tachyon potential has U(1) vacua at an infinite tachyon amplitude, which supports topological vortex-strings of codimension-two. Specifically, we study straight string solutions by examining field equations.
The topological charge of the string represented by vorticity is interpreted as the RR-charge density of D-string (D1-brane). (See the circles in Fig. \[fig4\].) Introduction of the radial DBI electric field coupled nonminimally to the tachyon is indispensable to obtain a thick D-string, which implies that background fundamental strings live in an extra-dimension with a fluid form and end at each vortex-string origin. (See the radial arrows in Fig. \[fig4\].)
According to asymptotic behavior of the tachyon amplitude at infinity, we obtained either linearly-growing tachyon configurations, or a newly-found lograrithmically-growing tachyon configuration which represents the minimum energy configuration. We additionally turn on the constant DBI electric field parallel to the string. Then its conjugate momentum density is localized along the string. (See the straight arrows along the $z$-direction in Fig. \[fig4\].) This confined electric flux density tells us that the stringy object of interest carries a fundamental string charge density, so we find it a DF-string (or a $(p,q)$-string). Lastly the nonvanishing gauge field coupled minimally to the tachyon replaces the global DF-string by a local DF-string carrying a quantized magnetic flux density as a D1 charge density.
Now that we have global and local, D- and DF-strings as soliton solutions in the context of effective field theory [@Kim:2005tw], the future tasks to construct a viable cosmic superstring become more tractable. Dynamical issues [@Dvali:2003zj; @Copeland:2003bj; @Leblond:2004uc] involve (i) head-on collision of two D-vortices for checking the intercommuting (reconnecting) property of two D-strings, (ii) collision of two DF-strings leading to a tree structure composed of a pair of trilinear vertices, which is to form a cosmic DF-string network [@Hashimoto:2002xe], and (iii) the stability of long macroscopic D- and DF-strings.
On cosmological aspects, we would gravitate the obtained static stringy defects and see the resultant spacetime structure. This may provide a basis to tackle the possibility of observing its effect astrophysically including viable density fluctuations and quintessence [@Cho:1998jk]. Inclusion of time-dependence is also important to understand how the D- and DF-strings are generated, and whether or not they survive during the inflationary era induced by the separation of D3 and ${\bar {\rm D}}3$. To lower the scale from the fundamental string scale to a scale to pass observational tests such as the cosmic microwave background, the pulsar timing, and the gravitational radiation, it is also intriguing to take into account the D- and DF-strings obtained in the background of various string (inspired) models [@Sarangi:2002yt].
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Jungjai Lee, Sangmin Lee, and Jin Hur for helpful discussions. This work is the result of research activities (Astrophysical Research Center for the Structure and Evolution of the Cosmos (ARCSEC)) supported by Korea Science $\&$ Engineering Foundation (Y.K.), and the BK 21 project of the Ministry of Education and Human Resources Development, Korea (I.C.).
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[^1]: In the context of 4 dimensional supergravity, a logarithm-type behavior at asymptotic region was found in the dilaton field ($\sim \ln r$) and the Higgs field ($\sim {\rm VEV}-{\cal O}(1/{\rm ln} r)$), and the obtained vortices carry finite tension [@Blanco-Pillado:2005xx].
|
---
abstract: 'In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem \[12theorem\]). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. \[weakInvariant\]). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary \[trivialCondition\]).'
author:
- Noboru Ito Yusuke Takimura
title: '(1, 2) and weak (1, 3) homotopies on knot projections'
---
Addendum: added 2014 {#addendum-added-2014 .unnumbered}
====================
After this article was published, the following information about doodles was pointed out by Roger Fenn. A doodle was introduced by Fenn and Taylor \[2\], which is a finite collection of closed curves without triple intersections on a closed oriented surface considered up to the second flat Reidemeister moves with the condition ($\ast$) that each component has no self-intersections. Khovanov \[4\] introduced doodle groups, and for his process, he considered doodles under a more generalized setting (i.e., removing the condition ($\ast$) and permitting the first flat Reidemeister moves). He showed \[4, Theorem 2.2\], a result similar to our \[3, Theorem 2.2 (c)\]. He also pointed out that \[1, Corollary 2.8.9\] gives a result similar to \[4, Theorem 2.2\]. The authors first noticed the above results by Fenn and Khovanov via personal communication with Fenn, and therefore, the authors would like to thank Roger Fenn for these references.
[9]{}
R. Fenn, Techniques of geometric topology, London Mathematical Society Lecture Note Series, 57. *Cambridge University Press, Cambridge*, 1983.
R. Fenn and P. Taylor, Introducing doodles, *Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1997)*, pp. 37–43, Lecture Notes in Math., 722, *Springer, Berlin*, 1979.
N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, *J. Knot Theory Ramifications* [**22**]{} (2013), 1350085, 14 pp.
M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. [**349**]{} (1997), 2297–2315.
Introduction {#intro}
============
Throughout this paper, we consider objects in the smooth category. A [*[knot]{}*]{} is defined as a circle smoothly embedding into ${\mathbb{R}}^{3}$ and its [*[knot projection]{}*]{} is a [*[regular projection]{}*]{} of the knot to a sphere. Here, the term [*[regular projection]{}*]{} means a projection to a sphere in which the image has only transversal double points of self-intersection. If each double point of a knot projection is specified by over-crossing and under-crossing branches, we call the knot projection a [*[knot diagram]{}*]{}. Therefore, for a given knot projection that has $n$ double points, it is possible to consider $2^{n}$ knot diagrams. Indeed, knot projections have been studied by this approach [@taniyama1; @taniyama2].
Knot isotopy classes are often interpreted as equivalence classes of knot diagrams under first, second, and third Reidemeister moves defined by Fig. \[reidemeister\]. The diagrams in Fig. \[reidemeister\] show that the local replacements on the neighborhoods and the exterior of the neighborhoods are the same for both diagrams of each move.
(0,0) (23,50)[$\Omega_1$]{} (121,50)[$\Omega_2$]{} (245,50)[$\Omega_3$]{}
![Reidemeister moves $\Omega_1$, $\Omega_2$, and $\Omega_3$. []{data-label="reidemeister"}](reidemeister.eps){width="11cm"}
As shown in Fig. \[ProjReidemeister\], we can define local moves of knot projections, called homotopy moves in this paper, by seeing projection images of Reidemeister moves of knot diagrams. We call $H_1$, $H_2$, and $H_3$ of Fig. \[ProjReidemeister\] the first, second, and third homotopy moves or simply $H_1$, $H_2$, and $H_3$ moves.
(0,0) (23,52)[$H_1$]{} (130,52)[$H_2$]{} (253,52)[$H_3$]{}
Arnold introduced invariants of knot projections under the second or third homotopy moves, called [*[perestroikas]{}*]{} and found low-ordered invariants [@arnold1; @arnold2] by using concepts similar to Vassiliev’s ordered invariants of knots [@vassiliev].
This paper was motivated by attempts to solve the problem that determines which knot projections can be trivialized by the first and third moves. Starting with Arnold’s work, we can choose any two kinds of homotopy moves from the $H_1$, $H_2$, and $H_3$ moves. First, we take $H_1$ and $H_2$ moves and consider the equivalence relation between knot projections by $H_1$ and $H_2$ moves. We obtain a necessary and sufficient condition that two knot projections are equivalent under $H_1$ and $H_2$ moves (Theorem \[12theorem\]). Second, we take $H_2$ and $H_3$, which is nothing but regular homotopy, and knot projections under regular homotopy are classified by rotation numbers [@whitney]. The last possibility is the case of $H_1$ and $H_3$ moves, which includes open problems: which two knot projections are equivalent under relations generated by $H_1$ and $H_3$ moves is unknown, and even which knot projection can be trivialized by $H_1$ and $H_3$ moves is unknown. In this paper, as a first step, we give a necessary and sufficient condition that a knot projection can be trivialized under relations by $H_1$ and restricted $H_3$ moves.
The restricted $H_3$ moves are [*[weak third homotopy moves]{}*]{}, or simply weak $H_3$ moves, defined by Fig. \[weakPerestroika\] for knot projections.
(0,0) (44,45)[weak $H_3$ move]{} (210,45)[strong $H_3$ move]{}
The weak third homotopy move is a positive weak perestroika, and its inverse move is based on the work of Viro [@viro] who defined high-ordered invariants by generalizing Arnold invariants. Viro introduced [*[weak]{}*]{} and [*[strong]{}*]{} triple point perestroikas (Fig. \[weakPerestroika\]). Arnold’s (and subsequently Viro’s) triple point perestroikas have positive directions that depend on the connecting branches of triple points (for details, see [@arnold2 p. 6]).
For all knot projections, we consider equivalence classes under the first homotopy move and the weak third homotopy move, called [*[weak (1, 3) homotopy]{}*]{}. In particular, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary \[trivialCondition\]).
At the end of this section, we discuss the remarkable work of Hagge and Yazinski [@HY] with regard to this paper. We investigate the equivalence relation of knot projections by the first and third homotopy moves; i.e. [*[(1, 3) homotopy]{}*]{}. Hagge and Yazinski [@HY] studied the non-triviality of (1, 3) homotopy classes of knot projections. However, we still do not have any numerical invariants for all knot projections that exhibit the non-triviality of knot projections under (1, 3) homotopy.
(1, 2) homotopy classes of knot projections
===========================================
We define [*[(1, 2) homotopy]{}*]{} as the equivalence relation generated by $H_1$ and $H_2$ moves for all knot projections. In this section, we determine how to detect two knot projections under (1, 2) homotopy (Theorem \[12theorem\]). The $H_1$ (resp. $H_2$) move consists of generating and removing a $1$-gon (resp. $2$-gon) as shown in Fig. \[taki\_reide1\]. Every generation (resp. removal) of a $1$-gon is called a $1a$ move or denoted by $1a$ (resp. $1b$ move or $1b$), and every generation (resp. removal) of a $2$-gon is called a $2a$ move or denoted by $2a$ (resp. $2b$ move or $2b$), as in Fig. \[taki\_reide1\].
(0,0) (30,35)[$1b$]{} (15,123)[$1a$]{} (125,123)[$2a$]{} (125,35)[$2b$]{}
![Our conventions $1a$, $1b$, $2a$, and $2b$. []{data-label="taki_reide1"}](taki_reide1.eps){width="6cm"}
\[1reduced\] For an arbitrary knot projection $P$ having $n$ individual $1$-gons and $2$-gons, the [*[reduced projection]{}*]{} $P^{r}$ is the knot projection obtained through any sequences of $1b$ and $2b$ deleting $n$ individual $1$-gons and $2$-gons arbitrarily.
We consider two special cases of this definition to define [*[1-homotopy]{}*]{} (resp. [*[2-homotopy]{}*]{}) as the equivalence relation generated by $H_1$ (resp. $H_2$) moves for all knot projections. For an arbitrary knot projection $P$ having $n$ individual $1$-gons (resp. $2$-gons), the [*[reduced projection]{}*]{} $P^{1r}$ (resp. $P^{2r}$) is the knot projection obtained through any sequences of $1b$ (resp. $2b$) deleting $n$ individual of $1$-gons (resp. $2$-gons).
Definition \[1reduced\] looks like that the definition depends on the way of deleting $1$- and $2$-gons (Fig. \[takimuex\]). However, the reduced knot projection does not depend on the way of deleting $1$- and $2$-gons. In other words, the uniqueness of the reduced knot projection is well defined by Theorem \[12theorem\] (Corollary \[12uniqueness\]).
(0,0) (28,-10)[$P_2$]{} (28,70)[$P_1$]{} (113,70)[${P_1}^{r}$]{} (113,-10)[${P_2}^{r}$]{}
![Examples of knot projections and their reduced knot projections (by using Theorem \[12theorem\], ${P_1}^{r} \neq {P_2}^{r}$ under (1, 2) homotopy).[]{data-label="takimuex"}](takimuex.eps){width="5cm"}
\[12theorem\]
1. Two knot projections $P$ and $P'$ are equivalent under 1-homotopy if and only if $P^{1r}$ and ${P'}^{1r}$ are equivalent under isotopy on $S^{2}$. \[1stStatement\]
2. Two knot projections $P$ and $P'$ are equivalent under 2-homotopy if and only if $P^{2r}$ and ${P'}^{2r}$ are equivalent under isotopy on $S^{2}$. \[2ndStatement\]
3. Two arbitrary knot projections $P$ and $P'$ are equivalent under (1, 2) homotopy if and only if ${P}^{r}$ and ${P'}^{r}$ are equivalent under isotopy on $S^{2}$. \[3rdStatement\]
We will obtain the proof of (\[3rdStatement\]) which includes the proofs of (\[1stStatement\]) and (\[2ndStatement\]). As shown below, Theorem \[12theorem\] is derived from Lemma \[12lemma\].
\[12lemma\] Any finite sequence generated by $H_1$ and $H_2$ moves between an arbitrary knot projection and an arbitrary reduced knot projection can be replaced with a sequence of only $1a$ and $2a$ moves or only $1b$ and $2b$ moves.
Let $n$ be an arbitrary integer, with $n \ge 2$, and let $x$ be a sequence of $n-2$ moves consisting of $1a$ and $2a$. We use the convention that the sequence $x$ followed by a $1a$ move is denoted by $x(1a)$. For the other moves (e.g. $1b$, $2a$, or $(2a)(1b)$), the same convention is applied. Let $P_i$ be the $i$th knot projection appearing in a sequence of $H_1$ and $H_2$ moves of length $n$. In the discussion below, we often use the symbol $Q$, which stands for a knot projection. We also use the convention that if the sequence $x(1a)(1b)$ can be replaced with $x$, we denote this by $x(1a)(1b)$ $=$ $x$. We apply the same convention to all similar cases that appear below.
Below we make claims about four cases of the first appearance of $1b$ or $2b$ in the sequence $P_1$ $\to$ $P_2$ $\to \dots \to$ $P_n$ $\to$ $P_{n+1}$ of $H_1$ and $H_2$ moves.
- Case 1: $x(1a)(1b)$ $=$ $x$ or $x(1b)(1a)$.
- Case 2: $x(2a)(1b)$ $=$ $x(1a)$ or $x(1b)(2a)$.
- Case 3: $x(1a)(2b)$ $=$ $x(1b)$ or $x(2b)(1a)$.
- Case 4: $x(2a)(2b)$ $=$ $x$ or $x(2b)(2a)$.
Case 1: The last two moves $(1a)(1b)$ can be presented as in Fig. \[taki1-1\]. The symbols $\delta_x$ and $\delta_y$ denote $1$-gons with boundaries, as shown in Fig. \[taki1-1\].
(0,0) (45,16)[$\delta_x$]{} (45,73)[$\delta_y$]{} (20,25)[$1a$]{} (65,80)[$1b$]{}
![The last two moves of $x(1a)(1b)$ of Case 1.[]{data-label="taki1-1"}](taki1-1.eps){width="3cm"}
1. If $\delta_x$ $\cap$ $\delta_y$ $\neq$ $\emptyset$, there are two cases of the pair $\delta_x$ and $\delta_y$ as shown in Fig. \[taki1-2\].
![Case 1–(i). The case $\delta_x$ $=$ $\delta_y$ (left) and the case $\delta_x$ $\cap$ $\delta_y$ $=$ $\{{\text{one vertex}}\}$ (right).[]{data-label="taki1-2"}](taki1-2.eps){width="3cm"}
In both of these cases, we have $x(1a)(1b)$ $=$ $x$ by Fig. \[taki1-3\].
![Case 1–(i). The sequence $x(1a)(1b)$ $=$ $x$.[]{data-label="taki1-3"}](taki1-3.eps){width="3cm"}
2. If $\delta_x$ $\cap$ $\delta_y$ $=$ $\emptyset$, then Fig. \[taki1-4\] implies $x(1a)(1b)(1b)$ $=$ $x(1b)$. Here, we can find the special move $1b$, corresponding to the inverse move of given $1a$. The move $1a$ follows this sequence, and we have $x(1a)(1b)(1b)(1a)$ $=$ $x(1b)(1a)$ as in Fig. \[taki1-4\]. Therefore, $x(1a)(1b)$ $=$ $x(1b)(1a)$.
(0,0) (0,79)[$P_{n-1}$]{} (17,87)[${1a}$]{} (30,79)[$P_n$]{} (50,87)[${1b}$]{} (60,79)[$P_{n+1}$]{} (87,87)[$1b$]{} (105,79)[$Q$]{} (120,87)[$1a$]{} (130,79)[$P_{n+1}$]{} (50,110)[$1b$]{}
![Case 1–(ii). The sequence $P_{n-1}$ $\stackrel{1a}{\to}$ $P_{n}$ $\stackrel{1b}{\to}$ $P_{n+1}$ $\stackrel{1b}{\to}$ $Q$ shows that $x(1a)(1b)(1b)$ $=$ $x(1b)$.[]{data-label="taki1-4"}](takimu1-4.eps){width="5cm"}
Case 2: The last two moves $(2a)(1b)$ can be presented as in Fig. \[taki2-1\]. The symbol $\delta_x$ (resp. $\delta_y$) denotes a $2$-gon (resp. $1$-gon) with a boundary, as shown in Fig. \[taki2-1\].
(0,0) (46,16)[$\delta_x$]{} (49,57)[$\delta_y$]{} (22,22)[$2a$]{} (65,65)[$1b$]{}
![The last two moves of $x(2a)(1b)$ of Case 2.[]{data-label="taki2-1"}](taki2-1.eps){width="3cm"}
1. If $\delta_x$ $\cap$ $\delta_y$ $\neq$ $\emptyset$, the pair $\delta_x$ and $\delta_y$ appears as in Fig. \[taki2-2\].
(0,0) (9,18)[$\delta_x$]{} (23,19)[$\delta_y$]{}
![Case 2-(i). []{data-label="taki2-2"}](taki2-2.eps){width="1cm"}
In this case, we have $x(2a)(1b)$ $=$ $x(1a)$ by Fig. \[taki2-3\].
![Case 2-(i). The sequence $x(2a)(1b)$ $=$ $x(1a)$.[]{data-label="taki2-3"}](taki2-3.eps){width="3cm"}
2. If $\delta_x$ $\cap$ $\delta_y$ $=$ $\emptyset$, then Fig. \[taki2-4\] implies that $x(2a)(1b)(2b)$ $=$ $x(1b)$. Here, we can find the special move $2b$, corresponding to the inverse move of given $2a$. The move $2a$ follows this sequence, and we have $x(2a)(1b)(2b)(2a)$ $=$ $x(1b)(2a)$ as in Fig. \[taki2-4\]. Therefore, $x(2a)(1b)$ $=$ $x(1b)(2a)$.
(0,0) (0,59)[$P_{n-1}$]{} (18,67)[$2a$]{} (30,59)[$P_n$]{} (47,67)[$1b$]{} (60,59)[$P_{n+1}$]{} (80,67)[$2b$]{} (93,59)[$Q$]{} (107,67)[$2a$]{} (120,59)[$P_{n+1}$]{} (50,90)[$1b$]{}
![Case 2-(ii). The sequence $P_{n-1}$ $\stackrel{2a}{\to}$ $P_{n}$ $\stackrel{1b}{\to}$ $P_{n+1}$ $\stackrel{2b}{\to}$ $Q$ shows that $x(2a)(1b)(2b)$ $=$ $x(1b)$.[]{data-label="taki2-4"}](takimu2-4.eps){width="4.5cm"}
Case 3: The last two moves $(1a)(2b)$ can be presented as in Fig. \[taki3-1\]. The $\delta_x$ (resp. $\delta_y$) denotes a $1$-gon ($2$-gon) with a boundary, as shown in Fig. \[taki3-1\].
(0,0) (41,15)[$\delta_x$]{} (37,57)[$\delta_y$]{} (16,23)[$1a$]{} (57,63)[$2b$]{}
![The last two moves of $x(1a)(2b)$ of Case 3.[]{data-label="taki3-1"}](taki3-1.eps){width="3cm"}
1. If $\delta_x$ $\cap$ $\delta_y$ $\neq$ $\emptyset$, the pair $\delta_x$ and $\delta_y$ appears as in Fig. \[taki3-2\].
(0,0) (9,19)[$\delta_y$]{} (23,18)[$\delta_x$]{}
![Case 3-(i).[]{data-label="taki3-2"}](taki3-2.eps){width="1cm"}
In this case, $x(1a)(2b)$ $=$ $x(1b)$ by Fig. \[taki3-3\].
![Case 3-(i). The sequence $x(1a)(2b)$ $=$ $x(1b)$. []{data-label="taki3-3"}](taki3-3.eps){width="3cm"}
2. If $\delta_x$ $\cap$ $\delta_y$ $=$ $\emptyset$, then Fig. \[taki3-4\] implies that $x(1a)(2b)(1b)$ $=$ $x(2b)$. Here, we can find the special move $1b$, corresponding to the inverse move of given $1a$. The move $1a$ follows this sequence, and we have $x(1a)(2b)(1b)(1a)$ $=$ $x(2b)(1a)$ as in Fig. \[taki3-4\]. Therefore, $x(1a)(2b)$ $=$ $x(2b)(1a)$.
(0,0) (-4,63)[$P_{n-1}$]{} (17,69)[$1a$]{} (30,63)[$P_n$]{} (48,69)[$2b$]{} (58,63)[$P_{n+1}$]{} (79,69)[$1b$]{} (92,63)[$Q$]{} (107,69)[$1a$]{} (122,63)[$P_{n+1}$]{} (50,88)[$2b$]{}
![Case 3-(ii). The sequence $P_{n-1}$ $\stackrel{1a}{\to}$ $P_{n}$ $\stackrel{2b}{\to}$ $P_{n+1}$ $\stackrel{1b}{\to}$ $Q$ shows that $x(1a)(2b)(1b)$ $=$ $x(2b)$.[]{data-label="taki3-4"}](takimu3-4.eps){width="4.5cm"}
Case 4: The last two moves $(2a)(2b)$ can be presented as in Fig. \[taki4-1\]. The symbols $\delta_x$ and $\delta_y$ denote $2$-gons with boundaries, as shown in Fig. \[taki4-1\].
(0,0) (43,16)[$\delta_x$]{} (43,60)[$\delta_y$]{} (23,20)[$2a$]{} (58,65)[$2b$]{}
![The last two moves of $x(2a)(2b)$ of Case 4.[]{data-label="taki4-1"}](taki4-1.eps){width="3cm"}
1. If $\delta_x$ $\cap$ $\delta_y$ $\neq$ $\emptyset$, there are two cases of the pair $\delta_x$ and $\delta_y$, as shown in Fig. \[taki4-2\].
(0,0) (58,16)[$\delta_x$]{} (73,16)[$\delta_y$]{}
![Case 4-(i). The case $\delta_x$ $=$ $\delta_y$ (left) and the case $\delta_x$ $\cap$ $\delta_y$ $=$ $\{{\text{one vertex}}\}$.[]{data-label="taki4-2"}](taki4-2.eps){width="3cm"}
In both of these cases, we have $x(2a)(2b)$ $=$ $x$ as in Fig. \[taki4-3\].
(0,0) (64,8)[$\delta_x$]{} (81,8)[$\delta_y$]{}
![Case 4-(i). The sequence $x(2a)(2b)$ $=$ $x$. []{data-label="taki4-3"}](taki4-3.eps){width="5cm"}
2. If $\delta_x$ $\cap$ $\delta_y$ $=$ $\emptyset$, then Fig. \[taki4-4\] implies that $x(2a)(2b)(2b)$ $=$ $x(2b)$. Here, we can find the special move $2b$, corresponding to the inverse move of given $2a$. The move $2a$ follows this sequence, and we have $x(2a)(2b)(2b)(2a)$ $=$ $x(2b)(2a)$ as in Fig. \[taki4-4\]. Therefore, $x(2a)(2b)$ $=$ $x(2b)(2a)$.
(0,0) (-4,61)[$P_{n-1}$]{} (16,66)[$2a$]{} (30,61)[$P_n$]{} (48,66)[$2b$]{} (58,61)[$P_{n+1}$]{} (79,66)[$2b$]{} (93,61)[$Q$]{} (107,66)[$2a$]{} (118,61)[$P_{n+1}$]{} (50,87)[$2b$]{}
![Case 4-(ii). The sequence $P_{n-1}$ $\stackrel{2a}{\to}$ $P_n$ $\stackrel{2b}{\to}$ $P_{n+1}$ $\stackrel{2b}{\to}$ $Q$ shows that $x(2a)(2b)(2b)$ $=$ $x(2b)$.[]{data-label="taki4-4"}](takimu4-4.eps){width="4.5cm"}
Thus, we have shown that the above claims about the four cases are true.
Next, we show that the statement of Lemma \[12lemma\] is true. Let us consider a given sequence $s$ of $1a$, $1b$, $2a$, and $2b$ from the left with the given reduced knot projection. Here, $1b$ and $2b$ are called $b$ moves. We focus on the first appearance of any $b$ move, which is called the first $b$ move. The first $b$ move cannot be the first move of the sequence $s$ since we start from the left with the reduced knot projection that does not have any $1$-gons and $2$-gons. If the first $b$ move is $1b$, we use the discussions of Cases 1-(ii) and 2-(ii) and move to the left (if necessary) until the $b$ move encounters Case 1-(i) or 2-(i), either of which eliminates the $b$ moves. Here, note that the $b$ move must encounter Case 1-(i) or 2-(i) because the $b$ move must not the first move of $s$. If the first $b$ move is $2b$, we use the discussions of Cases 3-(ii) and 4-(ii) and move to the left (if necessary) until the $b$ move encounters Case 3-(i) or 4-(i), either of which eliminates the $2b$ move. Cases 3-(i) or 4-(i) eliminate $2b$ and permit the replacement $y(2b)$ with $z(1b)$, where $y$ and $z$ are sequences entirely consisting of $1a$ and $2a$ moves. However, $z(1b)$ belongs to Case 1 or 2, and so the $b$ move $1b$ is eliminated. This completes the proof.
Now we will prove Theorem \[12theorem\].
For two arbitrary knot projections $P$ and $P'$, we take the projections $P^{r}$ and ${P'}^{r}$ arbitrarily. For $P^{r}$ and ${P'}^{r}$, we apply Lemma \[12lemma\]. If $P^{r}$ $\neq$ ${P'}^{r}$ under isotopy on $S^{2}$, there exists a non-empty sequence of only $1a$ and $2a$ moves from $P^{r}$ to ${P'}^{r}$ or from ${P'}^{r}$ to $P^{r}$. However, neither $P^{r}$ nor ${P'}^{r}$ has any $1$-gon or $2$-gon. This contradicts that the non-empty sequence consists of only $1a$ and $2a$ moves. Then, the assumption that $P^{r}$ $\neq$ ${P'}^{r}$ under isotopy on $S^{2}$ is false. Therefore, our claim is true. The proof of the statement (\[1stStatement\]) (resp. (\[2ndStatement\])) of Theorem \[12theorem\] is obtained by considering Case 1 (resp. Case 4) of Lemma \[12lemma\].
Theorem \[12theorem\] implies Corollary \[12uniqueness\].
\[12uniqueness\] For an arbitrary knot projection $P$, the reduced knot projection $P^{r}$ is uniquely determined; that is, $P^{r}$ does not depend on the way in which $1$-gons and $2$-gons of $P$ are deleted.
Reduced projections having no $1$-gons and $2$-gons, produced by Definition \[1reduced\], are called [*[lune-free graphs]{}*]{} [@EHK; @AST]. The knot projection ${P_{2}}^{r}$ of Fig. \[takimuex\] appears in [@EHK; @AST].
Positive resolutions and weak (1, 3) homotopy invariants. {#weakInvariant}
=========================================================
In the rest of this paper, unless otherwise specified, we adopt unoriented knot projections, and so the sphere containing knot projections does not have its orientation. Moreover, by invoking isotopy on $S^{2}$, if necessary, we can assume without loss of generality that every double point of the knot projection $P$ consists of two orthogonal branches.
In this section, we define a map from weak (1, 3) homotopy classes to knot isotopy classes. Take an arbitrary knot projection $P$ and give $P$ any orientation. Let us define [*[crossings]{}*]{} as double points of knot diagrams. We replace the neighborhood of every double point by that of the crossing of knot diagrams as shown in Fig. \[positiveResolution\].
![Positive resolution.[]{data-label="positiveResolution"}](positiveResolution.eps){width="8cm"}
(0,0) (-127,35)[$\mapsto$]{}
This replacement does not depend on the orientation of $P$. Then, the replacements define the map from knot projections to knot diagrams.
Polyak [@polyak] introduced this map and called each replacement a [*[positive resolution]{}*]{} when every double point is regarded as a singular point. That is, this map gives resolutions of singularities of double points. Using this map, Polyak defined finite type invariants of plane curves. We consider further applications.
\[theorem1\] For an arbitrary knot projection $P$, the positive resolution of all double points of $P$ defines the map from weak (1, 3) homotopy classes of knot projections to knot diagrams.
Let us denote by $p$ the map defined by the positive resolutions of all double points of $P$ and sending knot projections to knot diagrams. We will check the behavior of $p$ of the first and third homotopy moves.
- $H_1$ moves.
We denote by $D_1$ (resp. $D_2$) the local diagram defined by the left (resp. right) side of the $H_1$ move in Fig. \[ProjReidemeister\]. All possibilities for $D_2$ are shown in Fig. \[firstMove\]. In every case, $p(D_2)$ is transferred to $p(D_1)$ by the first Reidemeister move $\Omega_1$ of Fig. \[reidemeister\].
(0,0) (43,23)[$\mapsto$]{} (137,23)[$\mapsto$]{}
![The first homotopy move $H_1$ and positive resolutions.[]{data-label="firstMove"}](firstMove.eps){width="6cm"}
- Weak $H_3$ moves.
We denote by $D_3$ (resp. $D_4$) the local diagram defined by the left (resp. right) side of the weak $H_3$ move in Fig. \[weakPerestroika\]. As Fig. \[thirdMove\] shows, $p(D_3)$ is transferred to $p(D_4)$ by the third Reidemeister move $\Omega_3$ of Fig. \[reidemeister\].
(0,0) (60,35)[$\mapsto$]{} (220,35)[$\mapsto$]{}
![The third homotopy move $H_3$ and positive resolutions.[]{data-label="thirdMove"}](thirdMove.eps){width="10cm"}
Then, an arbitrary representative of a weak (1, 3) homotopy class is sent by the map $p$ to an isotopy class of a knot. This completes the proof.
In what follows, we permit the same symbol $p$ to denote the map defined by Theorem \[theorem1\] from weak (1,3) homotopy classes of knot projections to isotopy classes of knots.
Let $I$ be an arbitrary knot invariant and let $p$ be the map defined by Theorem \[theorem1\] from weak (1,3) homotopy classes of knot projections to knot isotopy classes. Then, $I \circ p$ is an invariant under weak (1, 3) homotopy.
As is well known, tricolorability is a knot invariant. Let $I_1$ be the map $I_1(D)$ $=$ $1$ if a knot diagram $D$ is tricolorable and the map $I_1(D)$ $=$ $0$ otherwise. Let $P_5$, $P_6$, and $P_7$ be the knot projections shown from the left to right as in Fig. \[knotProjections\]. Thus, $I_1 \circ p (P_5)$ $=$ $0$, $I_1 \circ p (P_6)$ $=$ $1$, and $I_1 \circ p (P_7)$ $=$ $1$. The diagram $P_7$ appears in [@HY].
(0,0) (28,-10)[$P_5$]{} (134,-10)[$P_6$]{} (270,-10)[$P_7$]{}
![Examples of knot projections.[]{data-label="knotProjections"}](knotProjections.eps){width="12cm"}
Triviality of knot projections under weak (1, 3) homotopy
=========================================================
Let us call the knot projection of $P_5$ in Fig. \[knotProjections\] [*[the trivial diagram]{}*]{}. If a knot projection $P$ and the trivial diagram can be related by a finite sequences of $H_1$ and $H_3$ moves, the knot projection $P$ is described as [*[trivial under weak (1, 3) homotopy]{}*]{}. If a diagram the same as $P_5$ is a knot diagram, we also call the knot diagram [*[the trivial diagram]{}*]{}.
\[trivialCondition\] Let $P$ be an arbitrary knot projection. The knot projection $P$ is trivial under weak (1, 3) homotopy if and only if $P$ and the trivial diagram can be related by a finite sequence consisting of the first homotopy moves.
By Theorem \[theorem1\], there exists the map from weak (1,3) homotopy classes of knot projections to knot isotopy classes, which is denoted by $p$. Then if $T$ is a weak (1, 3) homotopy classes of a knot projection containing the trivial diagram, then $p(T)$ is the unknot that is defined as the knot isotopy classes containing the trivial diagram.
We will prove that the converse of the claim. If $P$ is an arbitrary weak homotopy classes of an arbitrary knot projection, then $p(P)$ is a positive knot by the definition of $p$. As is well known, if a positive knot diagram belongs to the isotopy class of the unknot, there exists a finite sequence of $\Omega_1$ moves (Fig. \[reidemeister\]) between the positive knot diagram and the trivial diagram (e.g. [@PT]). Neglecting the information on overpasses and underpasses at double points, we can regard the sequence as one that consists of $H_1$ moves between a knot projection and the trivial diagram. This completes the proof.
Theorem \[theorem1\] and Corollary \[trivialCondition\] imply the following.
Let $p$ the map defined by Theorem \[theorem1\] from a weak (1,3) homotopy class of knot projections to a knot isotopy class. If the weak (1, 3) homotopy class containing the trivial diagram is denoted by $T$ and the unknot is denoted by $U$, then $$p(T) = U~{\text{and}}~p^{-1}(U) = T.$$
The first equation is assured by Theorem \[theorem1\] and the second equation is obtained from Corollary \[trivialCondition\].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Professor Kouki Taniyama for his fruitful comments. The work on this paper by N. Ito was partially supported by JSPS KAKENHI Grant Number 23740062.
[99]{} C. Adams, R. Shinjo, Tanaka, Complementary regions of knot and link diagram. [*[Ann. Comb.]{}*]{} [**[15]{}**]{} (2011), 549–563. V. I. Arnold, Plane curves, their invariants, perestroikas and classifications. With an appendix by F. Aicardi. Adv. Soviet Math., 21, [*[Singularities and bifurcations,]{}*]{} 33–91, [*[Amer. Math. Soc. Providence, RI,]{}*]{} 1994. V. I. Arnold, Topological invariants of plane curves and caustics. University Lecture Series, 5. [*[American Mathematical Society, Providence, RI,]{}*]{} 1994. S. Eliahou, F. Harary, and L. H. Kauffman, Lune-free knot graphs, [*[J. Knot Theory Ramifications]{}*]{} [**[17]{}**]{} (2008), 55–74. T. J. Hagge and J. Yazinski, On the necessity of Reidemeister move 2 for simplifying immersed planar curves, arXiv: 0812.1241. M. Polyak, Invariants of curves and fronts via Gauss diagrams. [*[Topology]{}*]{} [**[37]{}**]{} (1998), 989–1009. J. H. Przytycki and K. Taniyama, Almost positive links have negative signature, [*[J. Knot Theory Ramifications]{}*]{} [**[19]{}**]{} (2010), 187–289. K. Taniyama, A partial order of knots. [*[Tokyo J. Math.]{}*]{} [**[12]{}**]{} (1989), 205–229. K. Taniyama, A partial order of links. [*[Tokyo J. Math.]{}*]{} [**[12]{}**]{} (1989), 475–484. V. A. Vassiliev, Cohomology of knot spaces. [*[Theory of singularities and its applications,]{}*]{} 23–69, Adv. Soviet Math., 1, [*[Amer. Math. Soc., Providence, RI,]{}*]{} 1990. O. Viro, Generic immersions of the circle to surfaces and the complex topology of real algebraic curves. [*[Topology of real algebraic varieties and related topics,]{}*]{} 231–252, Amer. Math. Soc. Transl. Ser. 2, 173, [*[Amer. Math. Soc., Providence, RI,]{}*]{} 1996. H. Whitney, On regular closed curves in the plane. [*[Compositio Math.]{}*]{} 4 (1937), 276–284.
Waseda Institute for Advanced Study, 1-6-1, Nishi Waseda Shinjuku-ku Tokyo 169-8050, Japan\
[*[E-mail address]{}*]{}: [`[email protected]`]{}
Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishi Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan\
[*[E-mail address]{}*]{}: [`[email protected]`]{}
|
---
abstract: 'We present a study of thermal conductivity, $\kappa$, in undoped and doped strontium titanate in a wide temperature range (2-400 K) and detecting different regimes of heat flow. In undoped SrTiO$_{3}$, $\kappa$ evolves faster than cubic with temperature below its peak and in a narrow temperature window. Such a behavior, previously observed in a handful of solids, has been attributed to a Poiseuille flow of phonons, expected to arise when momentum-conserving scattering events outweigh momentum-degrading ones. The effect disappears in presence of dopants. In SrTi$_{1-x}$Nb$_{x}$O$_{3}$, a significant reduction in lattice thermal conductivity starts below the temperature at which the average interdopant distance and the thermal wavelength of acoustic phonons become comparable. In the high-temperature regime, thermal diffusivity becomes proportional to the inverse of temperature, with a prefactor set by sound velocity and Planckian time ($\tau_{p}=\frac{\hbar}{k_{B}T}$).'
author:
- 'Valentina Martelli$^{1}$, Julio Larrea Jiménez$^{2}$, Mucio Continentino$^{1}$, Elisa Baggio-Saitovitch$^{1}$ and Kamran Behnia$^{3,4}$'
date: 'January 26, 2018'
title: Thermal transport and phonon hydrodynamics in strontium titanate
---
Heat travels in insulators thanks to phonons. This has been described by the Peierls-Boltzmann equation, which quantifies the spatial variation in phonon population caused by the temperature gradient. In recent years, thanks to improved computing performance and new theoretical techniques, a quantitative account of intrinsic thermal conductivity of semiconductors is accessible to first-principle theory[@Lindsay:2013]. When most scattering events conserve momentum and do not decay heat flux, collective phonon excitations, dubbed relaxons, become fundamental heat carriers [@Cepellotti:2016]. This hydrodynamic regime of phonon flow, identified decades ago[@Sussmann:1963; @Guyer:1966; @Gurzhi:1968; @Beck:1974], is gaining renewed attention in the context of graphene-like two-dimensional systems[@Cepellotti:2015; @Lee:2015].
![ a) Crystal structure of strontium titanate; b) The cubic Brillouin zone and its high-symmetry points. c) The temperature dependence of the two soft modes according to the neutron scattering studies[@Shirane:1969], hyper-Raman [@Vogt:1979] and Brillouin scattering spectroscopy[@Hehlen:1999]. d) Thermal conductivity of a SrTiO$_{3}$ crystal(closed red squares) in a log-log plot (For a linear plot see Fig. 3a). Different regimes of thermal transport are identified. Solid lines represent the expected behaviors in these regimes. An additional window due to enhanced Umklapp scattering opens up in the vicinity of the Antiferrodistortive (AFD) transition.[]{data-label="Fig1"}](Fig1.pdf){width="42.00000%"}
The perovskyte SrTiO$_{3}$ is a quantum paraelectric[@Muller:1979], which owes its very existence to zero-point quantum fluctuations. First-principle calculations find imaginary phonon modes[@Aschauer:2014], which hinder a quantitative understanding of the lattice thermal transport [@Steigmeier:1968]. This insulator turns to a metal upon the introduction of a tiny concentration of dopants. The metal has a dilute superconducting ground state[@Lin:2013] and an intriguing room-temperature charge transport[@Lin:2017]. Its thermal conductivity has remained largely unexplored, in contrast to electric[@Lin:2015] and thermoelectric[@Cain:2013] transport.
In this Letter, we present an extensive study of thermal conductivity, $\kappa$, of undoped and doped SrTiO$_{3}$ crystals and report on three new findings. First of all, in a narrow temperature range, thermal conductivity evolves faster than cubic. This behavior had only been reported in a handful of solids[@Beck:1974] and attributed to a Poiseuille flow of phonons. We argue that the emergence of phonon hydrodynamics results from the multiplication of momentum-conserving scattering events due to the presence of a ferroelectric soft mode, as suggested decades ago [@Gurevich:1988]. It lends support to previous reports on the observation of the second sound in this system[@Hehlen:1995; @Koreeda:2007], which has been controversial[@Scott:2000]. Second, our study finds that a random distribution of dopants drastically reduces thermal conductivity below a temperature which tunes the heat-carrying phonon wavelength to the average interdopant distance. Finally, we put under scrutiny the thermal diffusivity of the system near room temperature and link its magnitude and temperature dependence to the so-called Planckian scattering time[@Bruin:2013], in the context of the ongoing debate on a possible boundary to diffusivity[@Hartnoll:2015; @Zhang:2017].
The cubic elementary cell of strontium titanate encloses a TiO$_{6}$ octahedra and has strontium atoms at its vertices (Fig. 1a). Neutron and Raman scattering studies have identified two distinct soft modes. The first is associated with the antiferrodistortive (AFD) transition, which leads to the loss of cubic symmetry at 105 K[@Shirane:1969] by tilting two adjacent TiO$_{6}$ octahedra in opposite orientations. It is centered at the R-point of the Brillouin zone (Fig. 1b). The second soft mode [@Yamada:1969], located at the zone center, is associated with the aborted ferroelectricity. Fig. 1c presents the temperature dependence of the two modes established by converging spectroscopic tools [@Shirane:1969; @Vogt:1979; @Hehlen:1999]. In common solids, only acoustic branches can host thermally-excited phonons at low temperatures. Here, phonons associated with these soft modes remain relevant down to fairly low temperatures.
We used a standard one-heater-two-thermometers technique to measure the thermal conductivity of commercial single crystal of Sr$_{1-x}$Nb$_{x}$TiO$_{3}$[@supplement]. The results, presented in Fig.1d, reveal different regimes of heat transport classified by previous authors[@Guyer:1966; @Beck:1974; @Lee:2015]. Simply put, thermal conductivity is the product of specific heat, mean-free-path, and velocity[@Berman:1976]. At one extreme, i.e. at low temperature, the phonon mean-free-path saturates, the system enters the ballistic regime and $\kappa$ becomes cubic in temperature. In the other extreme, at high temperature, the specific heat saturates and thermal conductivity, reflecting the temperature dependence of the mean-free-path, follows $T^{-1}$. In this kinetic regime, the wave-vector of thermally-excited phonons is large enough to allow Umklapp scattering events. Well below the Debye temperature, such events become rare and $\kappa$ increases exponentially. This is this Ziman regime.
The AFD transition has visible consequences for heat transport. First of all, it attenuates $\kappa$ near T$_{AFD}$, impeding a smooth evolution between T$^{-1}$ and exponential regimes. The R-point soft mode associated with the AFD transition provides additional Umklapp scattering at low energy cost. Interestingly, fitting $\kappa \propto exp(\frac{E_{D}}{T})$ in the Ziman regime, one finds $E_{D}\simeq$ 20 K, an energy scale comparable to the AFD soft mode. The second consequence of the AFD transition is to generate multiple tetragonal domains in an unstrained crystal[@Tao:2016]. Given that the typical size of tetragonal domains is a few microns[@Buckley:1999], the upper boundary to the ballistic mean-free-path of phonons can be much lower than the sample dimensions.
![Thermal conductivity, $\kappa$, as a function of $T^{3}$ in silicon (a) (after ref.[@Glass:1964]) and in KTaO$_{3}$(b). In both, $\kappa$ deviates downward from the $T^{3}$ line. (c) In bismuth (after ref. [@Kopylov:1974]) it deviates upward. In three different crystals of SrTiO$_{3}$ (d,e,f) the deviation is upward. g) Thermal conductivity and specific heat of SrTiO$_{3}$ evolve faster than cubic in this temperature range. But in a narrow window, thermal conductivity increases more rapidly. h) The apparent mean-free-path in both Bi and SrTiO$_{3}$ present a local peak, the hallmark of Poiseuille flow.[]{data-label="Fig2"}](Fig2.pdf){width="45.00000%"}
We found a $\kappa$ varying faster than $T^{3}$ in a narrow (6 K $< T <$13 K) temperature window just below the peak. Usually, the ballistic regime ends with a downward deviation of $\kappa$ from its cubic temperature dependence. This happens in silicon[@Glass:1964] (Fig. 2a) or in KTaO$_{3}$ (Fig. 2b). This is not the case of bismuth where it shows an upward deviation between the ballistic regime and the peak (Fig. 2c). This has been identified as a signature of Poiseuille flow of phonons[@Kopylov:1974].
The Poiseuille regime emerges when energy exchange between phonons is frequent enough to keep the local temperature well-defined and Umklapp collisions are so rare that the flow is mainly impeded by boundary scattering. Without viscosity, no external temperature gradient would be then required to sustain the phonon drift[@Gurzhi:1968]. This picture, developed decades ago[@Sussmann:1963; @Guyer:1966; @Gurzhi:1968], requires a hierarchy of time scales. The time separating two normal scattering events, $\tau_{N}$, should become much shorter than the time between boundary scattering events, $\tau_{B}$, and the latter much shorter than the time between resistive scattering events, $\tau_{R}$, which are due to either Umklapp or impurity scattering. The same hierarchy ($\tau_{N}\ll\tau_{B}\ll\tau_{R}$) is required for second sound, a wave-like propagation of temperature and entropy, which has been observed in bismuth as well as in other solids displaying Poiseuille flow[@Beck:1974].
We confirmed a faster than cubic $\kappa$ in three different SrTiO$_{3}$ crystals (Fig.2 d-f). Here, the identification of this behavior with Poiseuille flow is less straightforward since the specific heat of SrTiO$_{3}$ also evolves faster than cubic between 4K and 20K[@Ahrens:2007]. This is because the Debye approximation is inadequate in the presence of soft modes and one needs to consider Einstein terms of the soft optical modes. In order to address this concern, we measured the specific heat of our cleanest crystal and found that the thermal conductivity increases faster than specific heat (Fig. 2g). The effective mean-free-path, $\ell_{Ph}=\frac{3\kappa C_{p}}{v_{s}}$, extracted from the specific heat, $C_{p}$, and the sound velocity, v$_s$, was found to show a peak comparable to what was found in bismuth[@Kopylov:1974] (Fig. 2h). In both cases, $\ell_{Ph}$ presents a local maximum 1.3 times the Knudsen minimum. The magnitude of the latter is slighly smaller than the crystal dimensions in bismuth, and to the typical size of tetragonal domains in strontium titanate, which have been found to be of the order of a micrometer[@Buckley:1999]. As far as we know, the only available explanation for a local peak in $\ell_{Ph}$ is Poiseuille flow.
Neither in bismuth nor in strontium titanate, the chemical purity is exceptionally high. The same is true of black phosphorus, where a faster-than-cubic $\kappa$ was recently observed[@bp]. Therefore, in these cases, in contrast to He crystals, the Poiseuille flow is presumably caused by a large three-phonon phase space[@Lindsay:2008] for momentum-conserving (compared to momentum-degrading) scattering events. We note that the low-temperature validity of the $\tau_{N}\ll\tau_{R}$ inequality in strontium titanate was previously confirmed by low-frequency light-scattering experiments[@Koreeda:2007]. Anomalies detected by Brillouin scattering experiments[@Hehlen:1999] are believed to be caused by strong anharmonic coupling between acoustic and optical modes at low temperatures. A strong hybridization between acoustic and transverse optical phonons was theoretically confirmed[@Bussmann:1997] and is expected to flatten the phonon dispersion. This would pave the way for frequent normal momentum exchange. It would also pull down the phonon velocity, providing an alternative explanation for an unusually short apparent mean-free-path.
Let us turn our attention to the effect of atomic substitution. Fig. 3a shows thermal conductivity of SrTi$_{1-x}$Nb$_{x}$O$_{3}$. The magnitude of $\kappa$ smoothly decreases with increasing dopant concentration. Only at lower temperatures, additional contribution by electrons outweighs the reduction in lattice thermal conductivity. In this range, we resolve a finite $T$-linear component in thermal conductivity of metallic samples due to the electronic component of thermal conductivity, $\kappa_{e}$. This is in agreement with a previous study focused on temperatures below 0.5 K[@Lin:2014], which verified the validity of the Wiedemann-Franz(WF) law in the zero-temperature limit, namely: $\kappa_{e}\rho/T=L_{0}$, where $\rho$ is the electric resistivity and L$_{0}=2.45\times 10^{-8} V^{2}/K^{2}$ is the Lorenz number. Assuming the validity of the WF law at finite temperatures, one can separate the electronic, $\kappa_{e}$, and the phononic, $\kappa_{ph}$, components of the total thermal conductivity. At finite temperature, because of inelastic scattering, one expects $\kappa_{e}\rho/TL_{0}\leq 1$ and electric resistivity provides only a rough measure of $\kappa_{e}$, which, as seen in Fig. 3b, becomes rapidly much smaller than $\kappa_{ph}$ with rising temperature.
![ a) $\kappa$ as a function of temperature in SrTi$_{1-x}$Nb$_{x}$O$_3$. b) Electronic, $\kappa_{e}$ and phononic, $\kappa_{ph}$, components of the thermal conductivity in three doped samples compared to undoped strontium titanate. Note the persistence of a $T^{3}$ behavior over a wide temperature window with a drastically reduced magnitude. c) Relative attenuation in phonon thermal conductivity, $\Delta\kappa_{ph}=1-\frac{\kappa_{ph} (x\neq0) }{\kappa_{ph} (x=0)}$ in SrTi$_{1-x}$Nb$_{x}$O$_3$ (top) and in Sr$_{1-x}$Ca$_{x}$ TiO$_3$ (x=0.0045) and in SrTiO$_{3-\delta}$ (n=7$\times$ 10$^{17}cm^{-3}$). Small arrows represent T$_{qn}$ (See text).[]{data-label="fig3"}](Fig3.pdf){width="45.00000%"}
The first consequence of the disorder, introduced by this tiny substitution for $\kappa_{ph}$, is the loss of the faster than cubic regime associated with Poiseuille flow. As seen in Fig. 3b, reminiscent of what was observed in doped silicon and germanium[@Carruthers:; @1957], doping drastically damps $\kappa_{ph}$ at low temperature. The temperature-dependence of attenuation of phonon thermal conductivity caused by substitution: $\Delta\kappa_{ph}= 1- \kappa_{ph}(x\neq0)/ \kappa_{ph}(x=0)$, presented in Fig. 3c displays a regular pattern. For small substitution (x= 0.0004), the lattice thermal conductivity is reduced by 8 percent at room temperature, by as much as 70 percent at 20 K and by 20 percent at 3 K. In other words, the maximum attenuation occurs in an intermediate temperature window. With increasing Nb concentration, the pattern is similar, but it shifts to higher temperatures. As seen in the lower panel of Fig. 3c, our measurements on an oxygen-reduced and a calcium-substituted sample produce similar patterns. Since Ca substitution[@Rischau:2017] keeps the system an insulator, one can conclude that the drastic reduction in lattice conductivity is mainly due to the random distribution of substituting atoms and *not* to the scattering by mobile electrons.
A rigorous account of the temperature dependence of $\Delta\kappa_{ph}$ is missing. We note, however, that $\Delta\kappa_{ph}$ drastically enhances at a temperature, which shifts upward as the the concentration increases (see upward arrows in Fig. 3c). Consider that with decreasing temperature, the typical wave-vector of thermally-excited phonons shrinks, following: $q_{ph}= \frac{k_{B}T}{\hbar v_{s}}$. Therefore, at high-temperature, the phonon wave-length is shorter the average distance between dopants and the effect of disorder is limited. The random distribution of dopants begins to matter when the phonon wavelength becomes comparable to the average interdopant distance. In contrast to electrons, Anderson localization of phonons[@Luckyanova:2016] is not expected to impede diffusive transport[@Sheng:1994]. Theoretically, tiny level of disorder is sufficient to transform some phonon modes from propagating waves (propagons) to diffusons, which travel diffusively, or to fully localized locons[@Seyf:2016]. One expects phonons with a wavelength much shorter or much longer than randomness length to be less affected. As a consequence, attenuation is to be more pronounced in the temperature window where the most-concerned phonons happen to be dominant thermally-excited carriers of heat. For each concentration, $n$, a temperature, $T_{qn} = hv_{s}/\ell_{dd}k_{B}$, can be defined, which corresponds to equality between the typical acoustic phonon wavelength, $\lambda_{ph}=2\pi/q_{ph}$ and interdopant distance, $\ell_{dd}=n^{-1/3}$. As one can see in Fig. 3c, $T_{qn}$ is close to where $\Delta\kappa_{ph}$ becomes large. Such a crude picture based on the Debye approximation, should not be taken too literally in presence of soft modes.
In principle, *Ab Initio* calculations[@Lindsay:2013] can give an account of heat transport near room temperature. Recently, two groups [@Feng:2015; @Tadano:2015] succeeded in determining the phonon spectrum of strontium titanate free of the commonly-found imaginary frequencies[@Aschauer:2014] and computing the intrinsic lattice conductivity of the cubic phase. Fig. 4a compares our high-temperature data with these calculations [@Feng:2015; @Tadano:2015] as well as previous experimental reports[@Steigmeier:1968; @Muta:2005; @Yu:2008]. As one can see in the figure, there is a broad agreement between experimental results. Theoretical calculation using the Generalized Gradient Approximation (GGA)[@Feng:2015] are very close to the experimental data above 250 K. On the other hand, the experimental slope matches more the theory based on microscopic anharmonic force constants[@Tadano:2015].
![ a) Thermal conductivity at high-temperature compared to previous experimental reports [@Steigmeier:1968; @Muta:2005; @Yu:2008] and theoretical calculations[@Feng:2015; @Tadano:2015]; b) Thermal diffusivity, $D$, extracted from thermal conductivity and specific heat data as a function of temperature in SrTiO$_{3}$ (solid blue circles) together with data from ref.[@Hofmeister:2010] (open circles), compared to silicon and PbTe. Solid lines represent $D= s v_{s}^{2}\tau_{P}$ (See text).[]{data-label="Fig4"}](Fig4.pdf){width="40.00000%"}
Let us conclude by a short discussion of thermal diffusivity, $D =\frac{\kappa}{C_{p}}$ in this regime. We can extract $D$ by combining our thermal conductivity data and the specific heat. Fig. 4b presents the temperature dependence of thermal diffusivity. One can see that, at room temperature and above, thermal diffusivity tends to be proportional to $T^{-1}$. Our data is in good agreement with reported values of thermal diffusivity at high temperature[@Hofmeister:2010]. In the vicinity of room temperature and above, thermal diffusivity becomes proportional to the inverse of temperature. The thermal diffusivity of a good conductor of heat, silicon and a very bad one, PbTe, are also shown. Remarkably, in the two bad conductors, the magnitude and the temperature dependence of $D$ in the high-temperature regime can be expressed in a very simple way:
$$D= s v^2_{s} \tau_{p}$$
Here, $\tau_{p}=\frac{\hbar}{k_{B}T}$ is the Planckian scattering time[@Bruin:2013], and $s$ a dimensionless parameter (See table I). In PbTe and SrTiO$_{3}$, $s$ is close to unity and the temperature dependence is set by $\tau_{p}$. It emerges as a useful parameter for comparing the thermal conductivity of different cubic insulators. In many perovskytes, recently studied by Hofmeister[@Hofmeister:2010], $D$ has a comparable magnitude and temperature dependence. On the other hand, in a highly conducting cubic insulator such as silicon, $D$ is much larger and drops faster, presumably because the phase space for three-phonon scattering Umklapp events[@Lindsay:2008] is smaller.
Eq. 1 is strikingly similar to the suggested universal boundary on diffusivity suggested by Hartnoll[@Hartnoll:2015], with sound velocity replacing the Fermi velocity. The experimental motivation for Hartnoll’s proposal[@Hartnoll:2015] was the fact that $\tau_{p}$ is the average scattering rate of electrons in numerous metals with linear resistivity[@Bruin:2013]. Is there a boundary to thermal transport by phonons in insulators? In other, words, is there a fundamental reason for $s$ to remain larger than unity? These are the questions raised by our observation.
JLJ acknowledges the Science Without Borders program of CNPq/MCTI-Brazil and VM and KB acknowledge FAPERJ fellowships (Nota 10 and Visitante). KB is also supported by Fonds ESPCI and by a QuantEmX grant (GBMF5305) from ICAM and the Gordon and Betty Moore Foundation. We thank B. Fauqué, Y. Fuseya, S. A. Hartnoll and A. Kapitulnik for stimulating discussions.
\[tab:2\]
system D$_{300K}$ (mm$^{2}$/s) v$_{sl}$(100)(km/s) s
------------- ------------------------- --------------------- -----
SrTiO$_{3}$ 4.0 7.87 2.6
PbTe 1.9 3.59 5.9
Si 91 8.43 51
: Room-temperature thermal diffusivity ($D_{300K}$), longitudinal sound velocity ($v_{sl}$) and the parameter $s$ quantifying the slope of high-temperature thermal diffusivity, in three cubic solids.
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Methods
=======
Thermal conductivity and specific heat measurements were performed into the commercial system Dynacool PPMS (Physical Property Measurement System) that allows to perform experiments between 1.7 and 400K.
Thermal conductivity was measured with a standard two-thermometers-one heater configuration. The power supplied through the heater established a temperature difference ($\Delta T=T_1-T_2$) along the sample that was kept below 1$\%$ of the average temperature ($T_{av}=(T_1+T_2)/2$). This condition guarantees to have a negligible thermal flow along the electrical wires of the sensors and the heater, and to obtain an accurate determination of the thermal conductivity. The major source of experimental error comes from the determination of the geometrical factor that can be up to 6$\%$.
Specific heat was measured with the standard commercial platform compatible with Dynacool. Temperature rises were limited to 1-2% of base temperature. Due to the low heat capacity of SrTiO$_{3}$ at the lowest temperature, a minimum sample mass of 45mg was necessary to obtain a sizeable contribution of the sample heat capacity respect to the addenda (N-apiezon).
Samples
=======
The samples investigated (SrTiO$_{3}$, SrTi$_{1-x}$Nb$_{x}$O$_{3}$, Sr$_{1-x}$Ca$_{x}$TiO$_{3}$ and KTaO$_{3}$) in this work are all commercial single crystal specimens. For SrTiO$_{3}$, two of the three samples of SrTiO$_{3}$ came from two different batches of the same supplier (sample-1 and sample-3), whereas the other one came from a second supplier (sample-2).
The carrier density was determined by measuring the Hall resistivity and was found to be in good agreement with the nominal Nb content. Table I summarizes relevant values of both electrical and thermal transport measurements. $\rho_0$ and $A$ are obtained fitting the resistivity curve at the lowest measured temperature with the function $\rho=\rho_0$+AT$^2$ (see solid lines in Fig. \[Fig:R\]b).
\[tab:1\]
-------- ------------------------ ------------------- ----------------- ------ --------------------- --------------- ------------------------- --------------------- -------------------
**x** **$n$** **$\rho_{300K}$** **$\rho_{2K}$** RRR **$R_H$** **$\rho_0$** **$A$** **$\kappa_{300K}$** **$\kappa_{2K}$**
**** (cm$^{-3}$) (m$\Omega$cm) (m$\Omega$cm) **** (cm$^{-3}$/C) (m$\Omega$cm) ($\mu$$\Omega$cm/K$^2$) (W/cmK) (W/cmK)
0.0004 0.53(4)$\cdot 10^{19}$ 277 0.088 2579 1.18 0.078(4) 0.85(1) 0.1 0.0013
0.001 1.4(1)$\cdot 10^{19}$ 59.1 0.08 739 $4.33\cdot 10^{-1}$ 0.077(4) 0.26(1) 0.099 0.0011
0.01 9.4(7)$\cdot 10^{19}$ 9.46 0.064 148 $6.59\cdot 10^{-2}$ 0.060(2) 0.062(1) 0.093 7.5$\cdot10^{-4}$
0.014 n.a. 4.22 0.087 49 n.a. 0.075(5) 0.030(8) 0.098 0.0014
0.02 2.5(9)$\cdot 10^{20}$ 3.53 0.077 46 $2.45\cdot 10^{-2}$ 0.072(4) 0.029(8) 0.088 0.0011
-------- ------------------------ ------------------- ----------------- ------ --------------------- --------------- ------------------------- --------------------- -------------------
Thickness dependence of thermal conductivity
============================================
Sample-3, initially 500$\mu$m-thick, was thinned down to 150$\mu$m in order to perform the thickness-dependence measurement of thermal conductivity (Fig. \[Figs5:size\]). We observed that the thermal conductivity decreases in the ballistic regimes and in the temperature range that we identified as Poiseuille, but not at the peak temperature and above.
The lower thermal conductivity at low temperatures implies a lower mean free path, when phonon thermal conductivity is expressed as $k=\frac{1}{3}c_{ph}v_{ls}$. As discussed in the main text, the mean free path of SrTiO$_3$ in the ballistic regime is much lower than the sample size and of the order of magnitude of the domain size\[S1\]. Therefore, the modest reduction of the mean free path by reducing thickness suggests a either a correlation between domain and sample dimensions or the existence of a small subset of phonons which can travel across domain boundaries.
{width="80.00000%"}
Thermal conductivity and its sensitivity to disorder
====================================================
{width="80.00000%"}
Fig. \[Figs1:k\] presents the thermal conductivity of the three samples compared with an early study \[S2\]. One can see an overall agreement between the data sets. The peak thermal conductivity, $\kappa_{peak}$ (showed by downward arrows), appears to be sample-dependent. Since controlled substitution drastically affects $\kappa_{peak}$ (see below), it is reasonable to assume that $\kappa_{peak}$ is very sensitive to disorder and is largest in the cleanest samples.
Fig. \[Fig:k\_s\] compares the thermal conductivity of one of our cleanest SrTiO$_3$ samples with a reduced (SrTiO$_{3-\delta}$ ), two niobium-doped and one calcium substituted sample. One can see that the effect of oxygen reduction and Ca substitution is similar to the effect of Nb substitution, which was studied *in extenso* (See Fig. 3 of the main text). An extremely low level of atomic substitution drastically reduces the $\kappa_{peak}$. Ca substitution, oxygen reduction and Nb substitution have a very similar conseqiences. Now, the latter two turn the system to a metal, but the former does not introduce mobile electrons). Therefore, as argued in the main text, the main reason for the observed reduction in lattice thermal conductivity is not electron scattering but disorder.
{width="80.00000%"}
{width="60.00000%"}
Electric resistivity
====================
Electrical resistivity and Hall-resistivity were also carried out in the PPMS system. The data are similar to what was reported previously \[S3, S4\]. Figure \[Fig:R\] shows the resistivity of SrTi$_{1-x}$Nb$_{x}$O$_{3}$ as a function of temperature (panel (a)) and as a function of $T^2$ (panel (b)). As found previously\[S4\], resistivity follows a $T^2$ behavior at low temperature and then a faster than $T^2$ (close to cubic) at higher temperatures.
{width="80.00000%"}
Thermal diffusivity in three cubic solids
=========================================
Figure \[Fig:D\] shows the thermal conductivity and specific heat of SrTiO$_{3}$, Si and PbTe. This data was used to extract the thermal diffusivity of these three cubic semiconductors shown in Fig. 4 of the main text. To quantify the specific heat per volume, the molar volume was used as specified in Table II. The magnitude of $s$ is the main text was extracted using equation 1 and taking v$_s$ to be the longitudinal sound velocity along 100 as specified in the table.
\[tab:2\]
system V$_m$(cm$^{3}$/mol v$_{sl}$(100) (km/s) v$_{st}$(100) (km/s) $\kappa_{300K}$(W/m.K) C$_{300K}$ (J/cm$^{3}$K) D$_{300K}$ (cm$^{2}$/s)
------------- -------------------- ---------------------- ---------------------- ------------------------ -------------------------- -------------------------
SrTiO$_{3}$ 35.7 7.87 4.9 11.0 2.75 0.04
Si 12.1 8.43 5.84 150 1.65 0.91
PbTe 40.9 3.59 1.26 2.37 1.25 0.019
References
==========
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|
---
abstract: 'This letter is concerned with asymptotic analysis of a PDE model for motility of a eukaryotic cell on a substrate. This model was introduced in [@ZieSwaAra11], where it was shown numerically that it successfully reproduces experimentally observed phenomena of cell-motility such as a discontinuous onset of motion and shape oscillations. The model consists of a parabolic PDE for a scalar phase-field function coupled with a vectorial parabolic PDE for the actin filament network (cytoskeleton). We formally derive the sharp interface limit (SIL), which describes the motion of the cell membrane and show that it is a volume preserving curvature driven motion with an additional nonlinear term due to adhesion to the substrate and protrusion by the cytoskeleton. In a 1D model problem we rigorously justify the SIL, and, using numerical simulations, observe some surprising features such as discontinuity of interface velocities and hysteresis. We show that nontrivial traveling wave solutions appear when the key physical parameter exceeds a certain critical value and the potential in the equation for phase field function possesses certain asymmetry.'
address:
- 'Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA'
- 'Mathematical Division, B. Verkin Institute for Low Temperature, Physics and Engineering of National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkiv, Ukraine'
author:
- Leonid Berlyand
- Mykhailo Potomkin
- Volodymyr Rybalko
bibliography:
- 'cell.bib'
title: 'Phase-Field Model of Cell Motility: Traveling Waves and Sharp Interface Limit'
---
phase field system with gradient coupling ,curvature driven motion ,traveling waves ,cell motility
Introduction {#section:intro}
============
An initially symmetric cell on a substrate may exhibit spontaneous breaking of symmetry or self-propagation along the straight line maintaining the same shape over many times of its length [@KerPinAllBarMarMogThe08; @BarLeeKerMogThe11]. Understanding the initiation of steady motion of a biological cell as well as the mechanism of symmetry breaking is a fundamental issue in cell biology.
In [@ZieSwaAra11; @ZieAra13] a phase-field model was proposed to describe motility of a eukaryotic cell on a substrate. We consider a simplified version of that model without myosin contraction ($\gamma=0$ in [@ZieSwaAra11]), which consists of two coupled PDEs $$\begin{aligned}
&&\frac{\partial \rho_\ve}{\partial t}=\Delta \rho_\ve
-\frac{1}{\ve^2}W^{\prime}(\rho_\ve)%\rho(\rho-1)(\rho-1/2)
-P_\ve\cdot \nabla \rho_\ve +\lambda_\ve(t),\quad x\in \Omega, \;t>0,
\label{eq1}\\
&&\frac{\partial P_\ve}{\partial t}=\ve\Delta P_\ve -\frac{1}{\ve}P_\ve
-\beta \nabla \rho_\ve
%-\textcolor{red}{\gamma_\ve} (\nabla \rho\cdot P)\, P
\label{eq2}\end{aligned}$$ in a bounded domain $\Omega\subset \mathbb{R}^2$, where the unknowns are the phase-field function $\rho_\ve$ and the vector field $P_\ve$ modeling average orientation of the actin network. System - is obtained by diffusive scaling of equations from [@ZieSwaAra11] to study a sharp interface limit (SIL) of that model under special scaling assumptions on the parameters. We introduce the volume preservation constraint via the Lagrange multiplier $$\label{lagrange}
\lambda_\ve(t)=\frac{1}{|\Omega|}\int_\Omega\left(\frac{1}{\ve^2}W^\prime(\rho_\ve)
+ P_\ve\cdot \nabla \rho_\ve \right)\, dx$$ in place of the volume constraint originally introduced in the potential [@ZieSwaAra11]. The function $W^{\prime}(\rho)$ in is the derivative of a double equal well potential (e.g., $W(\rho)=\frac{1}{4}\rho^2(1-\rho)^2$).
The phase-field function $\rho_\ve$ takes values close to the wells of the potential $1$ and $0$ for sufficiently small $\ve>0$ everywhere in $\Omega$ except for a thin transition layer. The corresponding subdomains are interpreted as the inside cell and the outside cell regions, while the transition layer models the cell membrane. In , $\beta>0$ is a fixed parameter responsible for the creation of the field $P_\ve$ near the interface. The boundary conditions $\partial_\nu \rho_\ve=0$ and $P_\ve=0$ are imposed on the boundary $\partial \Omega$. We study system - in the sharp interface limit $\ve\to 0$. Well known approaches in the study of sharp interface limits of phase field models such as viscosity solutions techniques and the $\Gamma$- convergence method, see, e.g., [@LioKimSle04; @Gol97; @Ser10], are not readily applied to - because of the coupling through the terms $P_\ve\cdot \nabla \rho_\ve$ and $\nabla \rho_\ve$. The comparison principle, necessary for the viscosity solutions technique, does not apply for -. Also this system is not a gradient flow for an energy functional which makes the $\Gamma$-convergence techniques inapplicable. Another analytical approach, based on formal asymptotic expansions was developed for different phase field models in [@Che94; @MotSha95; @CheHilLog10]. Some ingredients of this approach are also used in the present study. We also mention here an alternative approach to cell motility based on numerical study of free boundary value problems developed in [@KerPinAllBarMarMogThe08; @RubJacMog05; @BarLeeAllTheMog15; @RecTru13; @RecPutTru15], and numerical studies of different phase field models of cell motility [@CamZhaLiLevRap13].
In this work we first show that solutions of - do not blow up on finite time intervals for sufficiently small $\ve$ by establishing energy type and pointwise bounds, next we formally derive a law of motion of the interface postulating a two-scale ansatz in the spirit of [@MotSha95]. Then we prove the existence of nontrivial traveling waves in a one-dimensional version of - in the case when the potential $W$ has certain assymmetry. This is done by an asymptotic reduction to a finite dimensional system for $V$ and $\lambda$, and applying the Schauder fixed point theorem. Finally in a one-dimensional dynamical system we rigorously prove that the interface velocity satisfies a simple nonlinear equation and demonstrate existence of a hysteresis loop in the system by numerical simulations.
Existence of Solutions and Sharp Interface Limit in 2D Model {#sec_sil_2D}
============================================================
The first result of this work demonstrates that for sufficiently small $\ve>0$ a unique solution $\rho_\ve$, $P_\ve$ of - exists and $\rho_\ve$ maintains the structure of a sharp interface between two phases $0$ and $1$, provided that initial data are well prepared. To formulate this result we introduce the following auxiliary (energy-type) functionals: $$\label{energy}
\begin{array}{l}
E_\ve(t):=\frac{\ve}{2} \int_\Omega |\nabla \rho_\ve(x,t)|^2dx+\frac{1}{\ve}\int_\Omega W( \rho_\ve(x,t) )dx,\\ \\
F_{\ve}(t):= \int_\Omega \Bigl(| P_\ve(x,t)|^2+|P_\ve(x,t)|^4\Bigr)dx.
\end{array}$$
\[wp\_theorem\] Assume that the system - is supplied with initial data that satisfy $-\ve^{1/4}< \rho_\ve (x,0)<1+\ve^{1/4}$, and $$E_\ve(0)+F_\ve(0)\leq C_1.
\label{IniEnBound}$$ Then for any $T>0$ there exists a solution $\rho_\ve$, $P_\ve$ of - on the time interval $(0,T)$ when $\ve>0$ is sufficiently small, $\ve<\ve_0(T)$. Moreover, $-\ve^{1/4}\leq \rho_\ve (x,t)\leq 1+\ve^{1/4}$ and $$\label{noblowup1}
\ve \int_0^T\int_\Omega \Bigl(\frac{\partial\rho_\ve}{\partial t} \Bigr)^2dxdt\leq C_2,
\quad
E_\ve(t)+F_\ve(t)\leq C_2\quad \forall t\in(0,T),$$ where $C_2$ is independent of $t$ and $\ve$.
This theorem shows that there is no blow up of the solution on the given time interval $(0,T)$, also it proves that if the initial data have sharp interface structure, this sharp interface structure is preserved by the solution on the whole time interval $(0,T)$. The claim of Theorem \[wp\_theorem\] is nontrivial due to the presence of the quadratic term $P_{\ve}\cdot \nabla\rho_{\ve}$ in which, in general, could lead to a finite time blow up. The main idea behind the existence proof is to find and utilize a bound for $\rho_\ve$ in $L^\infty((0,T)\times \Omega)$, which is obtained by combining the maximum principle and energy estimates.
Next we study the SIL $\ve\to 0$ for the system (\[eq1\])-(\[eq2\]). We seek solutions in the form of ansatz (locally in a neighborhood of the interface) $$\label{ANSATZ}
\rho_\ve=\theta_0(d/\ve)+\ve
\theta_1(d/\ve,S) +\dots,\quad
P_\ve=\nu\Psi_0(d/\ve,
S)+\dots,$$ where $d=d(x,t)$ is the (signed) distance to a unknown evolving interface curve $\Gamma(t)$, $S=s(p(x,t),t)$ with $p(x,t)$ being the projection of $x$ on $\Gamma(t)$ and $s(\xi,t)$ being a parametrization of $\Gamma(t)$, $\nu=\nu(p(x,t),t)$ is the inward pointing normal to $\Gamma(t)$ at $p(x,t)\in\Gamma(t)$. The key choice here is the interface curve $\Gamma(t)$ that allows for appropriate estimates. We substitute this ansatz in (\[eq1\]) to find, after collecting terms (formally) of the order $\ve^{-2}$, that $\theta_0$ satisfies $\theta_0^{\prime\prime}=W^\prime(\theta_0)$. It is known that there exists a unique (up to a translation) solution (standing wave) $\theta_0(z)$ which tends to $0$ or $1$ when $z\to-\infty$ or $z\to+\infty$. For the potential $W(\rho)=\frac14\rho^2(\rho-1)^2$ the function $\theta_0$ is explicitly given by $\theta_0(z)=\frac{1}{2}\left(1+\tanh \frac{z}{2\sqrt{2}}\right)$. Then substitute in (\[eq2\]) and consider the leading (of the order $\ve^{-1}$) term. Denoting by $V(x,t)$ the (inward) normal velocity of the curve $\Gamma(t)$ at $x\in \Gamma(t)$ we obtain that the scalar function $\Psi_0(z)$ solves $$\label{eq_for_Psi}
-\frac{\partial^2\Psi_0}{\partial z^2}-V\frac{\partial \Psi_0}{\partial z}+\Psi_0+
\beta\theta_0^\prime(z)=0.$$ Finally, assuming that the leading term of the expansion of $\lambda_\ve$ is of the order ${\ve}^{-1}$, $\lambda_\ve =\lambda(t)/\ve+\dots$, and collecting terms of the order $\ve^{-1}$ in we are led to the following equation $$-\frac{\partial^2 \theta_1}{\partial
z^2}+W^{\prime\prime}(\theta_0)\theta_1=(V-\kappa)\frac{\partial
\theta_0}{\partial z}-\Psi_0\frac{\partial \theta_0}{\partial
z}+\lambda(t),$$ where $\kappa$ denotes the curvature of $\Gamma(t)$. The solvability condition for this equation (orthogonality to the eigenfunction $\theta_0^\prime$ of the linearized Allen-Cahn equation) yields the desired sharp interface equation $$\label{SharpInterEq}
V(x,t)= \kappa(x,t) +
%\frac{\beta}{c_0}
\frac{1}{c_0}\Phi_\beta(V(x,t))-\lambda(t),\quad x\in\Gamma(t),$$ where $c_0=\displaystyle\int\left(\theta_0^\prime\right)^2dz$, and $\Phi_\beta(V)$ is given by $$\label{def_of_Phi}
\Phi_\beta(V)=\int\limits_{\mathbb R} \Psi_0\left(\theta_0^\prime(z)\right)^2dz.$$ From the volume preservation condition $\int_{\Gamma(t)}Vds=0$ it follows that $\lambda(t)=\frac{1}{c_0}\fint_{\Gamma(t)}(c_0\kappa+\Phi_\beta(V))ds$.
The above formal derivation of the sharp interface limit is rigorously justified in 1D (see Theorem \[1Dinterface\] below) because of significant technical difficulties due to the curvature in 2D. Solvability of was shown in [@MizBerRybZha15] for $\beta$ less than some critical value, moreover was proved to enjoy a parabolic regularization feature. However for large $\beta$, the equation might have multiple solutions. To obtain a selection criterion and elucidate the role of the parameter $\beta$ in the cell interface motion we consider a 1D model of the cell-motility in the next sections.
Traveling wave solutions in 1D {#sec_tw_1D}
==============================
In this section we show that solutions of system - exhibit significant qualitative changes when the parameter $\beta$ increases and the potential $W(\rho)$ has certain asymmetry, e.g. $W(\rho)=\frac14\rho^2(\rho-1)^2(1+\rho^2)$. Here we look for traveling wave solutions in 1D model, considering (\[eq1\])-(\[eq2\]) with $\Omega=\mathbb{R}^1$. In other words we are interested in nontrivial spatially localized solutions of - of the form $\rho_{\ve} =\rho_{\ve}(x-Vt)$, $P_{\ve}=P_{\ve}(x-Vt)$. This leads to the stationary equations with unknown (constant) velocity $V$ and constant $\lambda$: $$\begin{aligned}
0&=&\partial^2_x \rho_{\ve} +V\partial_x\rho_{\ve}-\frac{W'(\rho_{\ve})}{\ve^2}-P_{\ve}\partial_x\rho_{\ve} +\dfrac{\lambda}{\ve}\label{tw_rho},\\
0&=&\ve \partial_x^2P_{\ve}+V\partial_xP_{\ve} -\frac{1}{\ve} P_{\ve} -\beta\partial_x\rho_{\ve}. \label{tw_P}\end{aligned}$$
We are interested in solutions of - that are essentially localized on the interval $(-a,a)$, for a given $a>0$. We look for such solutions for sufficiently small $\ve>0$ with the phase field function $\rho_\ve$ of the form $$\label{repr}
\rho_{\ve} =\theta_0((x+a)/\ve)\theta_0((a-x)/\ve)+\ve\psi_\ve+\ve u_{\ve},$$ where constant $\psi_\ve$ is the smallest solution of $W^\prime(\ve\psi)=\ve\lambda$ and $u_{\ve}$ is the new unknown function vanishing at $\pm\infty$. Observe that the first term $\theta_0((x+a)/\ve)\theta_0((a-x)/\ve)$ has “$\Pi$” shape and becomes the characteristic function of the interval $(-a,a)$ in the limit $\ve\to 0$.
\[prop\_1\] For any real $\beta\geq 0$ and sufficiently small $\ve$ there exists a localized standing wave solution (with $V=0$) of - . It is localized in the sense that the representation holds with $u_{\ve}\in L^2(\mathbb{R})\cap L^\infty(\mathbb{R})$ and $\|u_{\ve}\|_{L^\infty}\leq C$.
Proposition \[prop\_1\] justifies expected existence of standing wave solutions (immobilized cells) in the class of functions with the symmetry $\rho(-x)=\rho(x)$ and $P(-x)=-P(x)$, so that the polarization field on the front and back has the same magnitude but is oriented in opposite directions. This field, loosely speaking, is trying to push front and back in opposite directions with the same velocities, thus, cell does not move. Indeed, the relation between $P_\ve$ and $V$ can be obtained from the second equation in , and .
We show, however, that not all localized solutions of - are necessarily standing waves. Assuming that there exists a traveling wave solution with a nonzero velocity, e.g. $V>0$, and passing to the sharp interface limit $\ve\to 0$ in - at the back and front transition layers ($x=\pm a$ in ) we formally obtain two relations for the velocity $V$ and the constant $\lambda$ $$\label{tw_front_and_back}
c_0V=\Phi_\beta(V)-\lambda, \text{ and } -c_0V=\Phi_\beta(-V)-\lambda.$$ Then eliminating $\lambda$ we obtain the equation for the velocity $V$: $$\label{eq_for_nonzero_tw}
2c_0V =\Phi_\beta(V)-\Phi_\beta(-V).$$ This equation always has one root $V=0$ which corresponds to the standing wave solution whose existence for system - is established in Proposition \[prop\_1\]. Two more roots, say $V_0$, and $-V_0$ appear for sufficiently large $\beta>0$ in the case when $\Phi_\beta(V)>\Phi_\beta(-V)$ for $V>0$, thanks to the fact that $\Phi_{\beta}$ is proportional to $\beta$ (note that if $W(\rho)=\frac{1}{4}\rho^2(\rho-1)^2$ then $\Phi_\beta$ is an even function, so the RHS of vanishes for arbitrary $\beta$ and thus $V$ is necessarily $0$). This heuristic argument can be made rigorous by proving the following:
\[theorem\_2\] Let $W(\rho)$ and $\beta$ be such that has a root $V=V_0>0$ and $\Phi_\beta^\prime(V_0)+\Phi_\beta^\prime(-V_0)\not=2c_0$ (nondegenerate root). Then for sufficiently small $\ve>0$ there exists a localized solution of - with $V=V_\ve\neq 0$, moreover $V_{\ve}\to V_0\neq 0$ as $\ve \to 0$ (as above localized solution means that representation holds with $u_{\ve}\in L^2(\mathbb{R})\cap L^\infty(\mathbb{R})$ and $\|u_{\ve}\|_{L^\infty}\leq C$).
[**Remark.**]{} In Theorem 2, it is crucial that has a non-zero solution $V_0$ which is impossible for the symmetric potential $W(\rho)=\frac{1}{4}\rho^2(\rho-1)^2$, but does hold for an asymmetric potential, e.g., $W(\rho)=\frac{1}{4}\rho^2(\rho-1)^2(1+\rho^2)$. In the case of smaller diffusion in equation one can prove that $\int_0^1 W''(\rho)dW^{3/2}(\rho)>0$ is a sufficient condition for existence of $V_0\neq 0$. We conjecture that this remains true for -.
Theorem \[theorem\_2\] guarantees existence of non-trivial traveling waves that describe steady motion without external stimuli. Thus our analysis of - is consistent with experimental observations of motility on keratocyte cells [@KerPinAllBarMarMogThe08].
The proof of Theorem \[theorem\_2\] is carried out in two steps. In the first step we use to rewrite - as a single equation of the form $\mathcal{A}_{\ve}u_{\ve}+\ve B_\ve(V,\lambda)+\ve^2 C_\ve(u_{\ve},V,\lambda)=0$, where $\mathcal{A}_\ve u:=\ve^2\partial_x^2u-
W^{\prime\prime}(\theta_0((x+a)/\ve)\theta_0((a-x)/\ve))u$ is the Allen-Cahn operator linearized around the first term in . We rewrite this equation as a fixed point problem $u_{\ve}=-\ve\mathcal{A}_{\ve}^{-1}( B_\ve(V,\lambda)+\ve C_\ve(u_{\ve},V,\lambda))$. The operator $\mathcal{A}_{\ve}$ has zero eigenvalue of multiplicity two (up to a proper $o(\ve^2)$ perturbation). This leads to solvability conditions which to the leading term coincide with . In the second step we apply the Schauder fixed point theorem to establish existence of solutions of -.
Sharp interface limit in a 1D model problem and hysteresis {#sec_sil_1D}
==========================================================
This section is devoted to the asymptotic analysis as $\ve \to 0$ of the following 1D problem $$\begin{aligned}
&&
\frac{\partial \rho_{\varepsilon}}{\partial t}=\partial^2_{x}\rho_{\varepsilon}-\frac{W'(\rho_{\varepsilon})}{\varepsilon^2}-P_{\varepsilon}\partial_x\rho_{\varepsilon}+\frac{F(t)}{\varepsilon}, \label{1D_rho}
\\
&&\frac{\partial P_{\varepsilon}}{\partial t}=\varepsilon \partial_{x}^2P_{\varepsilon}-\frac{1}{\varepsilon}P_{\varepsilon}-\beta \partial_{x}\rho_{\varepsilon},
\label{1D_P}\end{aligned}$$ $x\in \mathbb{R}^1$, $t>0$, for a given function $F:(0,+\infty)\to \mathbb{R}^1$. This is a model problem to develop rigorous mathematical tools for -, and it describes a normal cross-section of the transition layer (interface) between $0$ and $1$ phases. The variable $x\in \mathbb{R}$ corresponds to the re-scaled signed distance $d$ (see Section \[sec\_sil\_2D\]). The function $F(t)$ models forces due to the curvature of the interface and the mass preservation constraint $\lambda_\ve$, and for technical simplicity $F(t)$ is chosen to be independent of $x$.
Similar to Section \[sec\_tw\_1D\], we seek the solution of - in the form $$\label{eq_form}
\rho_{\ve}(x,t)=\theta_0(y)+\ve\psi_{\ve}(y,t)+\ve u_{\ve}(y,t), \;\;y=\frac{x-x_{\ve}(t)}{\ve},$$
where $\theta_0$ and $\psi_{\ve}$ are known functions, and $u_{\ve}$ is a new unknown function. Function $\psi_{\ve}(y,t)$ is defined by $$\nonumber
\psi_{\ve}(y,t)=\psi^-_{\ve}(t)+\theta_0(y)(\psi^+_{\ve}(t)-\psi^-_{\ve}(t)), \quad\text{ where}\quad \partial_t (\ve \psi^{\pm}_{\ve})=-\frac{W'((1\pm 1)/2+\ve\psi_{\ve}^{\pm})}{\ve^2}+\frac{F(t)}{\ve},\; \psi^{\pm}_{\ve}(0)=0.$$ Existence of the $x_{\ve}(t)$ (describing the location of the interface) together with estimates on $u_{\ve}$ uniform in $\ve$ and $t$ are established in the following
\[ansatz\_existence\] Let $\rho_{\ve},P_{\ve}$ be a solution of Problem - with initial data for $\rho_\ve$ and $P_{\ve}$ satisfying “well-prepared” initial conditions: $$\label{ic}
\rho_{\ve}(x,0)=\theta_0\left(x/{\ve}\right)+\ve v_{\ve}\left({x}/{\ve}\right),$$ where $\|v_{\ve}\|^2_{L^2}=\int_{\mathbb R}|v_{\ve}(y)|^2 dy<C$, $\|v_\ve\|_{L^{\infty}(\mathbb R)}\leq C/\ve$, and $P_{\ve}(x,0)=p_\ve (\frac{x}{\ve})$ such that $$%\|v_\ve\|_{L^2(\mathbb R)}<C,\;\; \|v_\ve\|_{L^{\infty}(\mathbb R)}\leq C/\ve, \;\;
%\text{ and }
\|p_\ve\|_{L^2(\mathbb R)}+ \| p_\ve\|_{L^{\infty}(\mathbb R)}+\| \partial_y p_\ve\|_{L^{\infty}(\mathbb R)}<C.$$ Then there exists $x_{\ve}(t)$ such that expansion holds with $\|u_{\ve}(\cdot,t)\|_{L^2(\mathbb R)}~<~C$ for $t\in [0,T]$ and $\int_{\mathbb R} u_{\ve}\theta_0' dy =0$. Moreover, assuming that $\int_{\mathbb R} v_{\ve}\theta_0' dy =0$ , the interface velocity $V_\ve=\dot x_\ve(t)$ is determined by the following system:
[align]{} (c\_0+\_(t))V\_(t)&=(\_0’)\^2 \_dy-F(t)+\_(t),\[eq\_for\_reduced\_V\]\
&= +V\_(t)-\_-(y),\[eq\_for\_reduced\_A\]
where ${\tilde{\mathcal{O}}}_{\ve}(t)$ and $\mathcal{O}_{\ve}(t)$ are bounded in $L^{\infty}(0,T)$.
The reduced system - can be further simplified by taking the limit $\ve\to 0$. Formal passing to the limit in leads to equation whose unique solution depends on the parameter $V$. Substituting this solution into in place of $\Psi_\ve$ we obtain the equation $$\label{1D_sil}
c_0V_0(t)=\Phi_\beta(V_0(t))-F(t)$$ for the limiting velocity $V_0=\lim_{\ve\to 0} V_\ve$. However, in general, equation is not uniquely solvable. The plot of the function $c_0V-\Phi_\beta(V)$ for sufficiently large $\beta$ is depicted on the Figure 1, where one sees that has two or three solutions when $F\in [F_{\rm min},F_{\rm max}]$. In order to justify and select a correct solution we reduce system - to a single nonlinear equation substituting expression for $V_\ve$ from into . Then rescaling time and neglecting terms of the order $\ve$ we arrive at the equation $\partial_t U=\partial_y^2 U+\frac{1}{c_0}(
\int (\theta^\prime_0)^2Udy-F)\partial_yU-
%(1-\mu\theta_0^2)
U-\beta \theta_0^\prime$ whose long time behavior has to be analyzed in order to obtain the limit of - as $\ve\to 0$. This is done by spectral analysis of the linearized operator $\mathcal{A}_V U=\partial_y^2 U+V\partial_y U-
%(1-\mu\theta_0^2)
U-\frac{1}{c_0}\partial_y \Psi_0
\int (\theta^\prime_0(z))^2U(z)dz$ about steady states $\Psi_0$ of the above nonlinear equation, where $\Psi_0$ are obtained by finding roots $V$ of the ordinary equation $c_0V=\Phi_\beta(V)-F$ and then solving the PDE .
\[def\_stable\] Define the set of stable velocities $\mathcal{S}$ by $\mathcal{S}=\{V \in \mathbb{R};\ \sigma(\mathcal{A}_V)\subset \{ \lambda\in \mathbb{C};{\rm Re}\lambda <0\}\}$, where $\sigma(\mathcal{A}_V)$ denotes the spectrum of the operator $\mathcal{A}_V$ (note that $\mathcal{S}$ is an open set).
\[1Dinterface\] Let $F(t)$ be a continuous function and assume that $V_0\in \mathcal{S}$ solves $c_0V_0=\Phi_\beta(V_0)-F(0)$. Assume also that $\|p_\ve-\Psi_0\|_{L^2} \leq\delta$, where $\Psi_0$ is the solution of with $V=V_0$ and $\delta>0$ is some small number depending on $V_0$ but independent of $\ve$. Then $V_\ve(t)=\dot x_\ve(t)$ defined in Theorem \[ansatz\_existence\] converges to the continuous solution of the equation $c_0V(t)=\Phi_\beta(V(t))-F(t)$ with $V(0)=V_0$ on every finite time interval $[0,T]$ where such a solution exists and $V(t)\in\mathcal{S}$ $\forall t\in [0,T]$.
We conjecture that stability of velocities is related to monotonicity intervals of the function $c_0V-\Phi_\beta(V)$. This conjecture is supported by the following result.
If $c_0\leq\Phi^\prime_\beta(V)$, then $V$ is not a stable velocity.
In general $\Phi^\prime_\beta(0)$ is nonzero if the potential $W(\rho)$ is asymmetric. In particular, for $W(\rho)=\frac{1}{4}\rho^2(1-\rho)^2(1+\rho^2)$ we have $c_0<\Phi^\prime_\beta(0)$ when $\beta>\beta_{critical}>0$, therefore zero velocity is not stable in this case. For 2D problem this would imply instability of initial circular shape leading to a spontaneous breaking of symmetry observed in experiments.
In the particular case $W(\rho)=\frac{1}{4}\rho^2(\rho-1)^2$ we prove that $(-\infty,\sqrt{2})\cap \left\{V;\; c_0>\Phi_\beta^\prime(V) \right\}\subset \mathcal{S}$. We also establish $\mathcal{S}=\left\{V;\; c_0>\Phi_\beta^\prime(V) \right\}$ via verifying numerically a technical inequality.
While Theorem \[1Dinterface\] describes local in time continuous evolution of the interface velocity according to the law $c_0V=\Phi_\beta(V)-F(t)$ until $V$ leaves the set of stable velocities $\mathcal{S}$, we conjecture that this law remains valid even after the time when the solution $V$ reaches an endpoint of a connected component of $\mathcal{S}$. Consider a particular example of $\beta=150$, the corresponding plot of the function $c_0V-\Phi_\beta(V)$ is depicted on Fig. 1. Thus we conjecture that system has a hysteresis loop, this conjecture is verified by numerical simulations for the sharp interface limit as well as the original system - for small $\ve$. The results of the latter simulations with $\ve=0.01$ are depicted on Fig. 1, right.
![Hysteresis loop in the problem of cell motility. (Left) The sketch of the plot for $c_0V- \Phi_\beta(V)$; (Center,Right) Simulations of $V=V(F)$, (Center): solution of (Right): solution of PDE system -. On both figures (Center) and (Right) arrows show in what direction the system $(V(t),F(t))$ evolves as time $t$ grows; blue curve is for $F_{\downarrow}(t)$, red curve is for $F_{\uparrow}(t)$.[]{data-label="fig:hysteresis"}](scheme_phi_2.pdf "fig:"){width="30.00000%"} ![Hysteresis loop in the problem of cell motility. (Left) The sketch of the plot for $c_0V- \Phi_\beta(V)$; (Center,Right) Simulations of $V=V(F)$, (Center): solution of (Right): solution of PDE system -. On both figures (Center) and (Right) arrows show in what direction the system $(V(t),F(t))$ evolves as time $t$ grows; blue curve is for $F_{\downarrow}(t)$, red curve is for $F_{\uparrow}(t)$.[]{data-label="fig:hysteresis"}](hysteresis_sil_beta_150_arrows.pdf "fig:"){width="34.00000%"} ![Hysteresis loop in the problem of cell motility. (Left) The sketch of the plot for $c_0V- \Phi_\beta(V)$; (Center,Right) Simulations of $V=V(F)$, (Center): solution of (Right): solution of PDE system -. On both figures (Center) and (Right) arrows show in what direction the system $(V(t),F(t))$ evolves as time $t$ grows; blue curve is for $F_{\downarrow}(t)$, red curve is for $F_{\uparrow}(t)$.[]{data-label="fig:hysteresis"}](hysteresis_pde_eps_0_01_beta_150_arrows.pdf "fig:"){width="34.00000%"}
Acknowledgments {#acknowledgments .unnumbered}
===============
This work of LB and VR was partially supported by NSF grants DMS-1106666 and DMS-1405769. The work of MP was partially supported by the NSF grant DMS-1106666.
References {#references .unnumbered}
==========
|
---
abstract: 'We investigate experimentally and theoretically the nonlinear propagation of $^{87}$Rb Bose Einstein condensates in a trap with cylindrical symmetry. An additional weak periodic potential which encloses an angle with the symmetry axis of the waveguide is applied. The observed complex wave packet dynamics results from the coupling of transverse and longitudinal motion. We show that the experimental observations can be understood applying the concept of effective mass, which also allows to model numerically the three dimensional problem with a one dimensional equation. Within this framework the observed slowly spreading wave packets are a consequence of the continuous change of dispersion. The observed splitting of wave packets is very well described by the developed model and results from the nonlinear effect of transient solitonic propagation.'
address: 'Kirchhoff Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg'
author:
- 'Th. Anker, M. Albiez, B. Eiermann, M. Taglieber and M. K. Oberthaler'
title: Linear and nonlinear dynamics of matter wave packets in periodic potentials
---
“Bose-Einstein condensation in atomic gases”, ed. by M. Inguscio, S. Stringari, and C. Wieman, (IOS Press, Amsterdam 1999)
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P. Meystre, “Atom Optics” (Springer Verlag, New York, 2001) p 205, and references therein.
The experimental realization in our group will be published elsewhere.
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A.A. Sukhorukov, D. Neshev, W. Krolikowski, and Y.S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically-induced lattices”, nlin.PS/0309075.
B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion Management for Atomic Matter Waves”, Phys.Rev.Lett. [**91**]{} 060402 (2003).
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O. Morsch, J. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices”, Phys. Rev. Lett. [**87**]{} 140402 (2001).
C.F. Bharucha, K.W. Madison, P.R. Morrow, S.R. Wilkinson, Bala Sundaram, and M.G. Raizen, “Observation of atomic tunneling from an accelerating optical potential”, Phys.Rev. A [**55**]{} R857 (1997)
L. Salasnich, A. Parola, and L. Reatto, “Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates”, Phys.Rev. A [**65**]{} 043614 (2002).
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Introduction
============
The experimental investigation of nonlinear matter wave dynamics is feasible since the realization of Bose-Einstein-condensation of dilute gases [@BEC_general]. The combination of this new matter wave source with periodic potentials allows for the realization of many nonlinear propagation phenomena. The dynamics depends critically on the modulation depth of the potential. For deep periodic potentials the physics is described locally taking into account mean field effects and tunneling between adjacent potential wells. In this context wave packet dynamics in Josephson junction arrays have been demonstrated experimentally [@Inguscio] and nonlinear self trapping has been predicted theoretically [@Trombettoni01]. In the limit of weak periodic potentials and moderate nonlinearity rich wave packet dynamics result due to the modification of dispersion which can be described applying band structure theory [@Steel98]. Especially matter wave packets subjected to anomalous dispersion (negative effective mass) or vanishing dispersion (diverging mass) are of great interest. In the negative mass regime gap solitons have been predicted theoretically [@gapsoliton] and have been observed recently [@obergapoliton]. Also modulation instabilities can occur [@modulationinstability].
The experiments described in this work reveal wave dynamics in the linear and nonlinear regime for weak periodic potentials. The observed behavior is a consequence of the special preparation of the wave packet leading to a continuous change of the effective mass and thus the dispersion during the propagation. The initial propagation is dominated by the atom-atom interaction leading to complex wave dynamics. After a certain time of propagation slowly spreading atomic wave packets are formed which are well described by linear theory. In this work we focus on the mechanisms governing the initial stage of propagation.
The paper is organized as follows: in section [\[Effectivemassconcept\]]{} we describe the effective mass and dispersion concept. In section [\[ExperimentalSetup\]]{} we present our experimental setup and in section [\[preparation\]]{} the employed wave packet preparation schemes are discussed in detail. In section [\[ExperimentalResults\]]{} the experimental results are compared with numerical simulations. We show that some features of the complex dynamics can be identified with well known nonlinear mechanisms. We conclude in section [\[Conclusion\]]{}.
Effective mass and dispersion concept {#Effectivemassconcept}
=====================================
In our experiments we employ a weak periodic potential which leads to a dispersion relation $E_n(q)$ shown in Fig. \[fig:1\](a). This relation is well known in the context of electrons in crystals [@Ashcroftenglish76] and exhibits a band structure. It shows the eigenenergies of the Bloch states as a function of the quasi-momentum $q$. The modified dispersion relation leads to a change of wavepacket dynamics due to the change in group velocity $v_g(q)=1/\hbar \;\partial E/\partial q$ (see Fig. \[fig:1\](b)), and the group velocity dispersion described by the effective mass $m_{eff}=\hbar^2(\partial^2 E/\partial
q^2)^{-1}$ (see Fig. \[fig:1\](c)), which is equivalent to the effective diffraction introduced in the context of light beam propagation in optically-induced photonic lattices [@Sukhorukov]. In our experiment only the lowest band is populated, which is characterized by two dispersion regimes, normal and anomalous dispersion, corresponding to positive and negative effective mass. A pathological situation arises at the quasimomentum $q_\infty^{\pm}$, where the group velocity $v_g(q)$ is extremal, $|m_{eff}|$ diverges and thus the dispersion vanishes.
 Band structure for atoms in an optical lattice with $V_0=1.2\,E_{rec}$ (solid), parabolic approximation to the lowest energy band at $q=\pi/d=G/2$ (dashed), corresponding group velocity (b) and effective mass (c) in the lowest energy band. The vertical dashed lines at $q=q^\pm_\infty$ indicate where $|m_\mathrm{eff}|=\infty$. The shaded region shows the range of quasimomenta where the effective mass is negative.](fig1){width="8cm"}
In the following we will show that the two preparation schemes employed in the experiment lead to a continuous change of the quasimomentum distribution, and thus to a continuous change of dispersion. One of the preparation schemes allows to switch periodically from positive to negative mass values and thus a slowly spreading wave packet is formed. This is an extension of the experiment reporting on dispersion management [@Eiermann1]. The second preparation gives further insight into the ongoing nonlinear dynamics for the initial propagation.
Experimental Setup {#ExperimentalSetup}
==================
The wave packets in our experiments have been realized with a $^{87}$Rb Bose-Einstein condensate (BEC). The atoms are collected in a magneto-optical trap and subsequently loaded into a magnetic time-orbiting potential trap. By evaporative cooling we produce a cold atomic cloud which is then transferred into an optical dipole trap realized by two focused Nd:YAG laser beams with $60\,\mu m $ waist crossing at the center of the magnetic trap (see Fig.\[fig:2\](a)). Further evaporative cooling is achieved by lowering the optical potential leading to pure Bose-Einstein condensates with $1 \cdot 10^4$ atoms in the $|F=2, m_F=+2\rangle$ state. By switching off one dipole trap beam the atomic matter wave is released into a trap acting as a one-dimensional waveguide with radial trapping frequency $\omega_\perp = 2 \pi \cdot 100
\,Hz$ and longitudinal trapping frequency $\omega_\parallel = 2
\pi \cdot 1.5 \,Hz$. It is important to note that the dipole trap allows to release the BEC in a very controlled way leading to an initial mean velocity uncertainty smaller than 1/10 of the photon recoil velocity.
![\[fig:2\] Scheme for wave packet preparation (a-d). (a) initial wave packet is obtained by condensation in a crossed dipole trap. (b) A stationary periodic potential is ramped up adiabatically preparing the atoms at quasimomentum $q_c=0$ in the lowest band. (c),(d) The periodic potential is accelerated to a constant velocity. (e) shows the numerically deduced quasimomentum shift for the preparation method I described in the text. (f) The motion of the center quasimomentum for the preparation method II described in the text. The additional shift to higher quasimomenta for long times results from the residual trap in the direction of the waveguide. The shaded area represents the quasimomenta corresponding to negative effective mass.](fig2){width="8cm"}
The periodic potential is realized by a far off-resonant standing light wave with a single beam peak intensity of up to $1\,W/cm^2$. The chosen detuning of 2nm to the blue off the D2 line leads to a spontaneous emission rate below $1\, Hz$. The standing light wave and the waveguide enclose an angle of $\theta=21^\circ$ (see Fig. \[fig:2\](b)). The frequency and phase of the individual laser beams are controlled by acousto-optic modulators driven by a two channel arbitrary waveform generator allowing for full control of the velocity and amplitude of the periodic potential. The light intensity and thus the absolute value of the potential depth was calibrated independently by analyzing results on Bragg scattering [@Bragg] and Landau Zener tunneling [@Kasevich; @Morsch01b; @Raizen97].
The wave packet evolution inside the combined potential of the waveguide and the lattice is studied by taking absorption images of the atomic density distribution after a variable time delay. The density profiles along the waveguide, $n(x,t)$, are obtained by integrating the absorption images over the transverse dimension.
Dynamics in reciprocal space {#preparation}
============================
In our experimental situation an acceleration of the periodic potential to a constant velocity leads to a collective transverse excitation as indicated in Fig. \[fig:2\](d). Since the transverse motion in the waveguide has a non vanishing component in the direction of the periodic potential due to the angle $\theta$, a change of the transverse velocity leads to a shift of the central quasimomentum of the wave packet. The coupling between the transverse motion in the waveguide and the motion along the standing light wave gives rise to a nontrivial motion in reciprocal (see Fig. \[fig:2\](e,f)) and real space.
The appropriate theoretical description of the presented experimental situation requires the solution of the three dimensional nonlinear Schrödinger equation (NLSE) and thus requires long computation times. In order to understand the basic physics we follow a simple approach which solves the problem approximately and explains all the features observed in the experiment. For that purpose we first solve the semiclassical equations of motion of a particle which obeys the equation $\vec{F} = M^* \ddot{\vec{x}} $ where $M^*$ is a mass tensor describing the directionality of the effective mass. We deduce the time dependent quasimomentum $q_c(t)$ in the direction of the periodic potential by identifying $\hbar \dot{q_c} = F_{\hat x}$ and $\dot{\hat{x}} = v_g(q_c)$ (definition of $\hat{x}$ see Fig. \[fig:2\](b)). Subsequently we can solve the one dimensional NPSE (non-polynomial nonlinear Schrödinger equation)[@Salasnich] where the momentum distribution is shifted in each integration step according to the calculated $q_c(t)$. Thus the transverse motion is taken into account properly for [*narrow*]{} momentum distributions. We use a split step Fourier method to integrate the NPSE where the kinetic energy contribution is described by the numerically obtained energy dispersion relation of the lowest band $E_0(q)$. It is important to note, that this description includes all higher derivatives of $E_0(q)$, and thus goes beyond the effective mass approximation.
In the following we discuss in detail the employed preparation schemes:
[*Acceleration scheme I*]{}: After the periodic potential is adiabatically ramped up to $V_0=6 E_{rec}$ it is accelerated within $3\,$ms to a velocity $v_{pot}=\cos^2(\theta)1.5v_{rec}$. Then the potential depth is lowered to $V_0=0.52 E_{rec}$ within $1.5\,$ms and the periodic potential is decelerated within $3\,$ms to $v_{pot}=\cos^2(\theta)v_{rec}$ subsequently. $V_0$ and $v_{pot}$ are kept constant during the following propagation. The calculated motion in reciprocal space $q_c(t)$ is shown in Fig. \[fig:2\](e).
[*Acceleration scheme II*]{}: The periodic potential is ramped up adiabatically to $V_0=0.37\,E_{rec}$ and is subsequently accelerated within $3\,$ms to a final velocity $v_{pot}=\cos^2(\theta) \times 1.05\, v_{rec}$. The potential depth is kept constant throughout the whole experiment. Fig. \[fig:2\](f) reveals that in contrast to the former acceleration scheme the quasimomentum for the initial propagation is mainly in the negative effective mass regime.
Experimental and Numerical Results {#ExperimentalResults}
==================================
In this section we compare the experimental results with the predictions of our simple theoretical model discussed above. The numerical simulation reveal all the experimentally observed features of the dynamics such as linear slowly spreading oscillating wave packets, nonlinear wave packet compression and splitting of wave packets. The observed nonlinear phenomena can be understood by realizing that in the negative effective mass regime the repulsive atom-atom interaction leads to compression of the wave packet in real space and to a broadening of the momentum distribution. An equivalent picture borrowed from nonlinear photon optics [@Agrawal01] is the transient formation of higher order solitons, which show periodic compression in real space with an increase in momentum width and vice versa.
Preparation I
-------------
The experimental results for the first acceleration scheme discussed in section [\[preparation\]]{} are shown in Fig. \[fig:3\]. Clearly we observe that a wave packet with reduced density is formed which spreads out slowly and reveals oscillations in real space. This wave packet results from the initial dynamics characterized by two stages of compression which lead to radiation of atoms [@scott]. The observed behavior is well described by our numerical simulation which allows further insight into the ongoing physics.
![\[fig:3\]Wave packet dynamics for preparation I. (a) Experimental observation of wave packet propagation. (b) Result of the numerical simulation as discussed in the text. The data is convoluted with the optical resolution of the experiment. The obtained results are in good agreement with the experimental observations. The theoretically obtained (c) quasimomentum distribution and (d) real space distribution are given for the initial 14ms of propagation. The inset reveals the phase of the observed slowly spreading wave packet.](fig3){width="8cm"}
In Fig. \[fig:3\](c,d) we show the calculated momentum and real space distribution for the first 14ms of propagation. As can be seen the acceleration of the standing light wave leads to a oscillatory behavior in momentum space. For the chosen parameters the wave packet is initially dragged with a tight binding potential ($V_0=6 E_{rec}$) over the critical negative mass regime. While the real space distribution does not change during this process, the momentum distribution broadens due to self phase modulation [@Agrawal01]. The subsequent propagation in the positive mass regime leads to a further broadening in momentum space and real space (t=4-9ms).
The dynamics changes drastically as soon as a significant part of the momentum distribution populates quasimomenta in the negative mass regime (t=10ms). There the real space distribution reveals nonlinear compression as known from the initial dynamics of higher order solitons. This compression leads to a significant further broadening in momentum space and thus to population of quasimomenta corresponding to positive mass. This results in a spreading in real space due to the different group velocities involved and leads to the observed background. The change of the quasimomentum due to the transverse motion prohibits a further significant increase in momentum width, since the whole momentum distribution is shifted out of the critical negative mass regime at t=14ms.
The long time dynamics of the slowly spreading wave packet is mainly given by the momentum distribution marked with the shaded area for t=14ms in Fig. \[fig:3\](c). The subsequent motion is dominated by the change of the quasimomentum due to the transverse motion. This leads to a periodic change from normal to anomalous dispersion and thus the linear spreading is suppressed. This is an extension of our previous work on dispersion management for matter waves - continuous dispersion management.
Preparation II
--------------
This preparation scheme reveals in more detail the transient solitonic propagation leading to the significant spreading in momentum space. This results in a splitting of the wave packet which cannot be understood within a linear theory. The results are shown in Fig. \[fig:4\] and the observed splitting is confirmed by our numerical simulations.
![\[fig:4\]Wave packet dynamics for preparation II. (a) Experimental results on wave packet propagation. (b) Result of the numerical simulation as discussed in the text. The simulation reproduces the observed wave packet splitting. The theoretically obtained (c) quasimomentum distribution and (d) real space distribution are given for the initial 14ms of propagation. The inset reveals that the transient formed wave packet has a flat phase indicating solitonic propagation.](fig4){width="8cm"}
In contrast to the former preparation scheme the momentum distribution is prepared as a whole in the critical negative mass regime. Our numerical simulations reveal that the wave packet compresses quickly in real space after t=4ms which is accompanied by an expansion in momentum space. The momentum distribution which stays localized in the negative mass regime reveals further solitonic propagation characterized by an expansion in real space and narrowing of the momentum distribution (t=5-10ms). The transverse motion shifts this momentum distribution into the normal dispersion regime after 11ms of propagation resulting in a wave packet moving with positive group velocity (i.e. moving to the right in fig. [\[fig:4\]]{}(b)). The initial compression at t=4ms even produces a significant population of atoms in the normal mass regime which subsequently move with negative group velocity showing up as a wave packet moving to the left in Fig. [\[fig:4\]]{}(b). Thus the splitting in real space is a consequence of the significant nonlinear broadening in momentum space.
Conclusion {#Conclusion}
==========
In this paper we report on experimental observations of nonlinear wave packet dynamics in the regime of positive and negative effective mass. Our experimental setup realizing a BEC in a quasi-one dimensional situation allows the observation of wave dynamics for short times, where the nonlinearity due to the atom-atom interaction dominates and also for long times, where linear wave propagation is revealed.
We have shown that a slowly spreading wave packet can be realized by changing the quasimomentum periodically from the normal to anomalous dispersion regime. This can be viewed as an implementation of continuous dispersion management. We further investigate in detail the formation process of these packets, which are a result of the initial spreading in momentum space due to nonlinear compression. A second experiment investigates in more detail the nonlinear dynamics in the negative mass regime where the solitonic propagation leads to a significant broadening in momentum space. This shows up in the experiment as splitting of the condensate into two wave packets which propagate in opposite directions.
The developed theoretical description utilizing the effective mass tensor models the experimental system in one dimension and can explain all main features observed in the experiment.
Acknowledgment
==============
This work was supported by Deutsche Forschungsgemeinschaft, Emmy Noether Program, by the European Union, Contract No. HPRN-CT-2000-00125, and the Optik Zentrum University of Konstanz.
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abstract: 'We propose a framework for synthesis of geological images based on an exemplar image (a.k.a. training image). We synthesize new realizations such that the discrepancy in the *patch distribution* between the realizations and the exemplar image is minimized. Such discrepancy is quantified using a kernel method for two-sample test called maximum mean discrepancy. To enable fast synthesis, we train a generative neural network in an offline phase to sample realizations efficiently during deployment, while also providing a parametrization of the synthesis process. We assess the framework on a classical binary image representing channelized subsurface reservoirs, finding that the method reproduces the visual patterns and spatial statistics (image histogram and two-point probability functions) of the exemplar image.'
author:
- 'Shing Chan[^1]'
- 'Ahmed H. Elsheikh'
bibliography:
- 'biblio.bib'
title: 'Exemplar-based synthesis of geology using kernel discrepancies and generative neural networks'
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[^1]: Corresponding author.\
E-mail addresses: `[email protected]` (Shing Chan), `[email protected]` (Ahmed H. Elsheikh).
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author:
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title: 'Human-level performance in first-person multiplayer games with population-based deep reinforcement learning'
---
Recent progress in artificial intelligence through reinforcement learning (RL) has shown great success on increasingly complex single-agent environments [@MnihDQN; @MnihA3C; @SchulmanPPO; @LillicrapDDPG; @JaderbergUnreal] and two-player turn-based games [@TesauroTDGammon; @silver2017mastering; @MoravcikDeepStack]. However, the real-world contains multiple agents, each learning and acting independently to cooperate and compete with other agents, and environments reflecting this degree of complexity remain an open challenge. In this work, we demonstrate for the first time that an agent can achieve human-level in a popular 3D multiplayer first-person video game, Quake III Arena Capture the Flag [@QuakeThree], using only pixels and game points as input. These results were achieved by a novel two-tier optimisation process in which a population of independent RL agents are trained concurrently from thousands of parallel matches with agents playing in teams together and against each other on randomly generated environments. Each agent in the population learns its own internal reward signal to complement the sparse delayed reward from winning, and selects actions using a novel temporally hierarchical representation that enables the agent to reason at multiple timescales. During game-play, these agents display human-like behaviours such as navigating, following, and defending based on a rich learned representation that is shown to encode high-level game knowledge. In an extensive tournament-style evaluation the trained agents exceeded the win-rate of strong human players both as teammates and opponents, and proved far stronger than existing state-of-the-art agents. These results demonstrate a significant jump in the capabilities of artificial agents, bringing us closer to the goal of human-level intelligence.
We demonstrate how intelligent behaviour can emerge from training sophisticated new learning agents within complex multi-agent environments. End-to-end reinforcement learning methods [@MnihDQN; @MnihA3C] have so far not succeeded in training agents in multi-agent games that combine team and competitive play due to the high complexity of the learning problem [@BernsteinDecPomdp; @MatignonIndependentLearners] that arises from the concurrent adaptation of other learning agents in the environment. We approach this challenge by studying team-based multiplayer 3D first-person video games, a genre which is particularly immersive for humans [@ermi2005fundamental] and has even been shown to improve a wide range of cognitive abilities [@green2015action]. We focus specifically on a modified version [@beattie2016deepmind] of Quake III Arena [@QuakeThree], the canonical multiplayer 3D first-person video game, whose game mechanics served as the basis for many subsequent games, and which has a thriving professional scene [@QuakeCon]. The task we consider is the game mode Capture the Flag (CTF) on per game randomly generated maps of both indoor and outdoor theme ([Figure \[fig:one\]]{} (a,b)). Two opposing teams consisting of multiple individual players compete to capture each other’s flags by strategically navigating, tagging, and evading opponents. The team with the greatest number of flag captures after five minutes wins. CTF is played in a visually rich simulated physical environment (Supplementary Video <https://youtu.be/dltN4MxV1RI>), and agents interact with the environment and with other agents through their actions and observations. In contrast to previous work [@silver2017mastering; @MoravcikDeepStack; @foerster2017learning; @LoweMADDPG; @mordatch2017emergence; @NIPS2016_6398; @riedmiller2007experiences; @stone2000layered; @LNAI17-MacAlpine2], agents do not have access to models of the environment, other agents, or human policy priors, nor can they communicate with each other outside of the game environment. Each agent acts and learns independently, resulting in decentralised control within a team.
![[**CTF task and computational training framework.**]{} Shown are two example maps that have been sampled from the distribution of outdoor maps (a) and indoor maps (b). Each agent in the game only sees its own first-person pixel view of the environment (c). Training data is generated by playing thousands of CTF games in parallel on a diverse distribution of procedurally generated maps (d), and used to train the agents that played in each game with reinforcement learning (e). We train a population of 30 different agents together, which provides a diverse set of teammates and opponents to play with, and is also used to evolve the internal rewards and hyperparameters of agents and learning process (f). Game-play footage and further exposition of the environment variability can be found in Supplementary Video <https://youtu.be/dltN4MxV1RI>.[]{data-label="fig:one"}](figure1){width="\textwidth"}
Since we wish to develop a learning agent capable of acquiring generalisable skills, we go beyond training fixed teams of agents on a fixed map, and instead devise an algorithm and training procedure that enables agents to acquire policies that are robust to the variability of maps, number of players, and choice of teammates, a paradigm closely related to ad-hoc team play [@stone2010ad]. The proposed training algorithm stabilises the learning process in partially observable multi-agent environments by concurrently training a diverse population of agents who learn by playing with each other, and in addition the agent population provides a mechanism for meta-optimisation. We solve the prohibitively hard credit assignment problem of learning from the sparse and delayed episodic team win/loss signal (optimising thousands of actions based on a single final reward) by enabling agents to evolve an internal reward signal that acts as a proxy for winning and provides denser rewards. Finally, we meet the memory and long-term temporal reasoning requirements of high-level, strategic CTF play by introducing an agent architecture that features a multi-timescale representation, reminiscent of what has been observed in primate cerebral cortex [@chen2015processing], and an external working memory module, broadly inspired by human episodic memory [@hassabis2017neuroscience]. These three innovations, integrated within a scalable, massively distributed, asynchronous computational framework, enables the training of highly skilled CTF agents through solely multi-agent interaction and single bits of feedback about game outcomes.
In our formulation, the agent’s policy $\pi$ uses the same interface available to human players. It receives raw RGB pixel input $\vec{x}_t$ from the agent’s first-person perspective at timestep $t$, produces control actions $a_t\sim \pi$ simulating a gamepad, and receives game points $\rho_t$ attained – the points received by the player for various game events which is visible on the in-game scoreboard. The goal of reinforcement learning (RL) is to find a policy that maximises the expected cumulative $\gamma$-discounted reward ${\mathbb{E}}_{\pi}[\sum_{t=0}^T \gamma^{t} r_t]$ over a CTF game with $T$ time steps. The agent’s policy $\pi$ is parameterised by a multi-timescale recurrent neural network with external memory [@graves2016hybrid] ([Figure \[fig:two\]]{} (a), [Figure \[fig:arch\]]{}). Actions in this model are generated conditional on a stochastic latent variable, whose distribution is modulated by a more slowly evolving prior process. The variational objective function encodes a trade-off between maximising expected reward and consistency between the two timescales of inference (more details are given in Supplementary Materials Section \[sec:ftwagent\]). Whereas some previous hierarchical RL agents construct explicit hierarchical goals or skills [@SuttonOptions; @VezhnevetsFun; @BaconOptionCritic], this agent architecture is conceptually more closely related to work on building hierarchical temporal representations [@clockwork; @chung2016hierarchical; @schmidhuber1992learning; @el1996hierarchical] and recurrent latent variable models for sequential data [@chung2015recurrent; @fraccaro2016sequential]. The resulting model constructs a temporally hierarchical representation space in a novel way to promote the use of memory (Figure \[fig:ext\_dnc\]) and temporally coherent action sequences.
For ad-hoc teams, we postulate that an agent’s policy $\pi_0$ should maximise the probability of winning for its team, $\{\pi_0, \pi_1, \ldots, \pi_{\frac{N}{2}-1}\}$, which is composed of $\pi_0$ itself, and its teammates’ policies $\pi_1, \ldots, \pi_{\frac{N}{2}-1}$, for a total of $N$ players in the game: $$\mathbbm{P}(\text{$\pi_0$'s team wins}| \omega, ( \pi_n )_{n=0}^{N-1} ) =
{\mathbb{E}}_{\vec{a} \sim ( \pi_n)_{n=0}^{N-1}}\left[
\{ \pi_0, \pi_1, \ldots, \pi_{\frac{N}{2}-1} \}
\overset{\text{\faFlagO}}{>}
\{ \pi_{\frac{N}{2}}, \ldots, \pi_{N-1}\}
\right].
\label{eqn:winning}$$
The winning operator $\overset{\text{\faFlagO}}{>}$ returns 1 if the left team wins, 0 for losing, and randomly breaks ties. $\omega \sim \Omega$ represents the specific map instance and random seeds, which are stochastic in learning and testing. Since game outcome as the only reward signal is too sparse for RL to be effective, we require rewards $r_t$ to direct the learning process towards winning yet are more frequently available than the game outcome. In our approach, we operationalise the idea that each agent has a dense internal reward function [@singh2009rewards; @singh2010intrinsically; @wolpert1999introduction], by specifying $r_t = \vec{w}(\rho_t)$ based on the available game points signals $\rho_t$ (points are registered for events such as capturing a flag), and, crucially, allowing the agent to learn the transformation $\vec{w}$ such that policy optimisation on the internal rewards $r_t$ optimises the policy [**F**]{}or [**T**]{}he [**W**]{}in, giving us the *FTW agent*.
Training agents in multi-agent systems requires instantiations of other agents in the environment, like teammates and opponents, to generate learning experience. A solution could be self-play RL, in which an agent is trained by playing against its own policy. While self-play variants can prove effective in some multi-agent games [@SilverAlphaGo; @silver2017mastering; @MoravcikDeepStack; @BansalEmergentComplexity; @BrownFictitiousPlay; @LanctotPSRO; @HeinrichDeepFictitious], these methods can be unstable and in their basic form do not support concurrent training which is crucial for scalability. Our solution is to train a population of $P$ different agents $\vec{\pi} = (\pi_p)_{p=1}^{P}$ in parallel that play with each other, introducing diversity amongst players to stabilise training [@rosin1997new]. Each agent within this population learns from experience generated by playing with teammates and opponents sampled from the population. We sample the agents indexed by $\iota$ for a training game using a stochastic matchmaking scheme $m_p(\vec{\pi})$ that biases co-players to be of similar skill to player $p$. This scheme ensures that – a priori – the outcome is sufficiently uncertain to provide a meaningful learning signal, and that a diverse set of teammates and opponents are seen during training. Agents’ skill levels are estimated online by calculating Elo scores (adapted from chess [@ELO]) based on outcomes of training games. We also use the population to meta-optimise the internal rewards and hyperparameters of the RL process itself, which results in the joint maximisation of: $$\begin{aligned}
J_\mathrm{inner}( \pi_p | \vec{w}_p) &= \mathbb{E}_{\iota \sim m_p(\vec{\pi}), \omega \sim \Omega}
\; \mathbb{E}_{\vec{a}\sim \vec{\pi}_{\iota}} \left [ \sum_{t=0}^T \gamma^t \vec{w}_p(\rho_{p,t}) \right ] \;\;\; \forall \pi_p \in \vec{\pi}\\
J_\mathrm{outer}( \vec{w}_p, \vec{\phi}_p | \vec{\pi} ) &= \mathbb{E}_{\iota \sim m_p(\vec{\pi}), \omega \sim \Omega}
\; {\mathbb{P}\left( \pi_p^{\vec{w},\vec{\phi}}\text{'s team wins}|\omega, \vec{\pi}^{\vec{w},\vec{\phi}}_{\iota} \right)}
\end{aligned}$$ $$\pi^{\vec{w},\vec{\phi}}_p = \text{optimise}_{\pi_p}(J_\mathrm{inner}, \vec{w}, \vec{\phi}).$$ This can be seen as a two-tier reinforcement learning problem. The inner optimisation maximises $J_\mathrm{inner}$, the agents’ expected future discounted internal rewards. The outer optimisation of $J_\mathrm{outer}$ can be viewed as a meta-game, in which the meta-reward of winning the match is maximised with respect to internal reward schemes $\vec{w}_p$ and hyperparameters $\vec{\phi}_p$, with the inner optimisation providing the meta transition dynamics. We solve the inner optimisation with RL as previously described, and the outer optimisation with Population Based Training (PBT) [@jaderberg2017population]. PBT is an online evolutionary process which adapts internal rewards and hyperparameters and performs model selection by replacing under-performing agents with mutated versions of better agents. This joint optimisation of the agent policy using RL together with the optimisation of the RL procedure itself towards a high-level goal proves to be an effective and generally applicable strategy, and utilises the potential of combining learning and evolution [@ackley1991interactions] in large scale learning systems.
![[**Agent architecture and benchmarking.**]{} (a) Shown is how the agent processes a temporal sequence of observations $\vec{x}_t$ from the environment. The model operates at two different time scales, faster at the bottom, and slower by a factor of $\tau$ at the top. A stochastic vector-valued latent variable is sampled at the fast time scale from distribution $\mathbb{Q}_t$ based on observations $\vec{x}_t$. The action distribution $\pi_t$ is sampled conditional on the latent variable at each time step $t$. The latent variable is regularised by the slow moving prior $\mathbb{P}_t$ which helps capture long-range temporal correlations and promotes memory. The network parameters are updated using reinforcement learning based on the agent’s own internal reward signal $r_t$, which is obtained from a learnt transformation $\vec{w}$ of game points $\rho_{t}$. $\vec{w}$ is optimised for winning probability through population based training, another level of training performed at yet a slower time scale than RL. Detailed network architectures are described in Figure \[fig:arch\]. (b) Top: Shown are the Elo skill ratings of the FTW agent population throughout training (blue) together with those of the best baseline agents using hand tuned reward shaping (RS) (red) and game winning reward signal only (black), compared to human and random agent reference points (violet, shaded region shows strength between 10th and 90th percentile). It can be seen that the FTW agent achieves a skill level considerably beyond strong human subjects, whereas the baseline agent’s skill plateaus below, and does not learn anything without reward shaping (see Supplementary Materials for evaluation procedure). (b) Bottom: Shown is the evolution of three hyperparameters of the FTW agent population: learning rate, KL weighting, and internal time scale $\tau$, plotted as mean and standard deviation across the population.[]{data-label="fig:two"}](figure2.pdf){width="\textwidth"}
To assess the generalisation performance of agents at different points during training, we performed a large tournament on procedurally generated maps with ad-hoc matches involving three types of agents as teammates and opponents: ablated versions of FTW (including state-of-the-art baselines), Quake III Arena scripted bots of various levels [@waveren2001quakebots], and human participants with first-person video game experience. [Figure \[fig:two\]]{} (b) and Figure \[tab:leaderboard\_proc\] show the Elo scores and derived winning probabilities for different ablations of FTW, and how the combination of components provide superior performance. The FTW agents clearly exceeded the win-rate of humans in maps which neither agent nor human had seen previously, zero-shot generalisation, with a team of two humans on average capturing 16 flags per game less than a team of two FTW agents (Figure \[tab:leaderboard\_proc\] Bottom, FF vs hh). Interestingly, only as part of a human-agent team did we observe a human winning over an agent-agent team (5% win probability). This result suggests that trained agents are capable of cooperating with never seen before teammates, such as humans. In a separate study, we probed the exploitability of the FTW agent by allowing a team of two professional games testers with full communication to play continuously against a fixed pair of FTW agents. Even after twelve hours of practice the human game testers were only able to win 25% (6.3% draw rate) of games against the agent team.
Interpreting the difference in performance between agents and humans must take into account the subtle differences in observation resolution, frame rate, control fidelity, and intrinsic limitations in reaction time and sensorimotor skills (Figure \[fig:humanagentdiff\] (a), Supplementary Materials Section \[sec:humandiff\]). For example, humans have superior observation and control resolution – this may be responsible for humans successfully tagging at long range where agents could not (humans: 17% tags above 5 map units, agents: 0.5%). In contrast, at short range, agents have superior tagging reaction times to humans: by one measure FTW agents respond to newly appeared opponents in 258ms, compared with 559ms for humans (Figure \[fig:humanagentdiff\] (b)). Another advantage exhibited by agents is their tagging accuracy, where FTW agents achieve 80% accuracy compared to humans’ 48%. By artificially reducing the FTW agents’ tagging accuracy to be similar to humans (without retraining them), agents’ win-rate was reduced, though still exceeded that of humans (Figure \[fig:humanagentdiff\] (c)). Thus, while agents learn to make use of their potential for better tagging accuracy, this is only one factor contributing to their overall performance.
![[**Knowledge representation and behavioural analysis.**]{} (a) The 2D t-SNE embedding of an FTW agent’s internal states during game-play. Each point represents the internal state $(\vec{h}^p, \vec{h}^q)$ at a particular point in the game, and is coloured according to the high-level game state at this time – the conjunction of four basic CTF situations (b). Colour clusters form, showing that nearby regions in the internal representation of the agent correspond to the same high-level game state. (c) A visualisation of the expected internal state arranged in a similarity-preserving topological embedding (Figure \[fig:ext\_neural\_response\]). (d) We show distributions of situation conditional activations for particular single neurons which are distinctly selective for these CTF situations, and show the predictive accuracy of this neuron. (e) The true return of the agent’s internal reward signal and (f) the agent’s prediction, its value function. (g) Regions where the agent’s internal two-timescale representation diverges, the agent’s surprise. (h) The four-step temporal sequence of the high-level strategy [*opponent base camping*]{}. (i) Three automatically discovered high-level behaviours of agents and corresponding regions in the t-SNE embedding. To the right, average occurrence per game of each behaviour for the FTW agent, the FTW agent without temporal hierarchy (TH), self-play with reward shaping agent, and human subjects (more detail in Figure \[fig:ext\_behvaiours\]).[]{data-label="fig:three"}](figure3.pdf){width="\textwidth"}
We hypothesise that trained agents of such high skill have learned a rich representation of the game. To investigate this, we extracted ground-truth state from the game engine at each point in time in terms of 200 binary features such as “Do I have the flag?”, “Did I see my teammate recently?”, and “Will I be in the opponent’s base soon?”. We say that the agent has knowledge of a given feature if logistic regression on the internal state of the agent accurately models the feature. In this sense, the internal representation of the agent was found to encode a wide variety of knowledge about the game situation (Figure \[fig:ext\_knowledge\]). Interestingly, the FTW agent’s representation was found to encode features related to the past particularly well: the FTW agent was able to classify the state *both flags are stray* (flags dropped not at base) with 91% AUCROC (area under the receiver operating characteristic curve), compared to 70% with the self-play baseline. Looking at the acquisition of knowledge as training progresses, the agent first learned about its own base, then about the opponent’s base, and picking up the flag. Immediately useful flag knowledge was learned prior to knowledge related to tagging or one’s teammate’s situation. Note that agents were never explicitly trained to model this knowledge, thus these results show the spontaneous emergence of these concepts purely through RL-based training.
A visualisation of how the agent represents knowledge was obtained by performing dimensionality reduction of the agent’s activations using t-SNE [@maaten2008visualizing]. As can be seen from [Figure \[fig:three\]]{}, internal agent state clustered in accordance with conjunctions of high-level game state features: flag status, respawn state, and room type. We also found individual neurons whose activations coded directly for some of these features, a neuron that was active if and only if the agent’s teammate was holding the flag, reminiscent of concept cells [@quiroga2012concept]. This knowledge was acquired in a distributed manner early in training (after 45K games), but then represented by a single, highly discriminative neuron later in training (at around 200K games). This observed disentangling of game state is most pronounced in the FTW agent (Figure \[fig:ext\_tsnes\]).
One of the most salient aspects of the CTF task is that each game takes place on a randomly generated map, with walls, bases, and flags in new locations. We hypothesise that this requires agents to develop rich representations of these spatial environments to deal with task demands, and that the temporal hierarchy and explicit memory module of the FTW agent help towards this. An analysis of the memory recall patterns of the FTW agent playing in indoor environments shows precisely that: once the agent had discovered the entrances to the two bases, it primarily recalled memories formed at these base entrances ([Figure \[fig:four\]]{}, Figure \[fig:ext\_dnc\]). We also found that the full FTW agent with temporal hierarchy learned a coordination strategy during maze navigation that ablated versions of the agent did not, resulting in more efficient flag capturing (Figure \[tab:leaderboard\_fetch\_2vs2\]).
![[**Progression of agent during training.**]{} Shown is the development of knowledge representation and behaviours of the FTW agent over the training period of 450K games, segmented into three phases (Supplementary Video <https://youtu.be/dltN4MxV1RI>). [**Knowledge:**]{} Shown is the percentage of game knowledge that is linearly decodable from the agent’s representation, measured by average scaled AUCROC across 200 features of game state. Some knowledge is compressed to single neuron responses (Figure \[fig:three\] (a)), whose emergence in training is shown at the top. [**Relative Internal Reward Magnitude:**]{} Shown is the relative magnitude of the agent’s internal reward weights of three of the thirteen events corresponding to game points $\rho$. Early in training, the agent puts large reward weight on picking up the opponent flag, whereas later this weight is reduced, and reward for tagging an opponent and penalty when opponents capture a flag are increased by a factor of two. [**Behaviour Probability:**]{} Shown are the frequencies of occurrence for three of the 32 automatically discovered behaviour clusters through training. [*Opponent base camping*]{} (red) is discovered early on, whereas [*teammate following*]{} (blue) becomes very prominent midway through training before mostly disappearing. The [*home base defence*]{} behaviour (green) resurges in occurrence towards the end of training, in line with the agent’s increased internal penalty for more opponent flag captures. [**Memory Usage:**]{} Shown are heat maps of visitation frequencies for locations in a particular map (left), and locations of the agent at which the top-ten most frequently read memories were written to memory, normalised by random reads from memory, indicating which locations the agent *learned* to recall. Recalled locations change considerably throughout training, eventually showing the agent recalling the entrances to both bases, presumably in order to perform more efficient navigation in unseen maps, shown more generally in Figure \[fig:ext\_dnc\]. []{data-label="fig:four"}](figure4.pdf){width="\textwidth"}
Analysis of temporally extended behaviours provided another view on the complexity of behavioural strategies learned by the agent [@krakauer2017neuroscience]. We developed an unsupervised method to automatically discover and quantitatively characterise temporally extended behaviour patterns, inspired by models of mouse behaviour [@wiltschko2015mapping], which groups short game-play sequences into behavioural clusters (Figure \[fig:ext\_behvaiours\], Supplementary Video <https://youtu.be/dltN4MxV1RI>). The discovered behaviours included well known tactics observed in human play, such as [*waiting in the opponents base for a flag to reappear (opponent base camping)*]{} which we only observed in FTW agents with a temporal hierarchy. Some behaviours, such as [*following a flag-carrying teammate*]{}, were discovered and discarded midway through training, while others such as [*performing home base defence*]{} are most prominent later in training ([Figure \[fig:four\]]{}).
In this work, we have demonstrated that an artificial agent using only pixels and game points as input can learn to play highly competitively in a rich multi-agent environment: a popular multiplayer first-person video game. This was achieved by combining a number of innovations in agent training – population based training of agents, internal reward optimisation, and temporally hierarchical RL – together with scalable computational architectures. The presented framework of training populations of agents, each with their own learnt rewards, makes minimal assumptions about the game structure, and therefore should be applicable for scalable and stable learning in a wide variety of multi-agent systems, and the temporally hierarchical agent represents a sophisticated new architecture for problems requiring memory and temporally extended inference. Limitations of the current framework, which should be addressed in future work, include the difficulty of maintaining diversity in agent populations, the greedy nature of the meta-optimisation performed by PBT, and the variance from temporal credit assignment in the proposed RL updates. Trained agents exceeded the win-rate of humans in tournaments, and were shown to be robust to previously unseen teammates, opponents, maps, and numbers of players, and to exhibit complex and cooperative behaviours. We discovered a highly compressed representation of important underlying game state in the trained agents, which enabled them to execute complex behavioural motifs. In summary, our work introduces novel techniques to train agents which can achieve human-level performance at previously insurmountable tasks. When trained in a sufficiently rich multi-agent world, complex and surprising high-level intelligent artificial behaviour emerged.
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Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Matt Botvinick, Simon Osindero, Volodymyr Mnih, Alex Graves, Nando de Freitas, Nicolas Heess, and Karl Tuyls for helpful comments on the manuscript; Simon Green and Drew Purves for additional environment support and design; Kevin McKee and Tina Zhu for human experiment assistance; Amir Sadik and Sarah York for exploitation study participation; Adam Cain for help with figure design; Paul Lewis, Doug Fritz, and Jaume Sanchez Elias for 3D map visualisation work; Vicky Holgate, Adrian Bolton, Chloe Hillier, and Helen King for organisational support; and the rest of the DeepMind team for their invaluable support and ideas.
Supplementary Materials {#supplementary-materials .unnumbered}
=======================
Task
====
Rules of Capture the Flag
-------------------------
CTF is a team game with the objective of scoring more flag captures than the opposing team in five minutes of play time. To score a *capture*, a player must navigate to the opposing team’s base, *pick up* the flag (by touching the flag), *carry* it back to their own base, and capture it by running into their own flag. A capture is only possible if the flag of the scoring player’s team is safe at their base. Players may *tag* opponents which teleports them back to their base after a delay (*respawn*). If a *flag carrier* is tagged, the flag they are carrying drops on the ground and becomes *stray*. If a player on the team that owns the dropped flag touches the dropped flag, it is immediately returned back to their own base. If a player on the opposing team touches the dropped flag, that player will pick up the flag and can continue to attempt to capture the flag.
Environment
-----------
The environment we use is DeepMind Lab [@beattie2016deepmind] which is a modified version of Quake III Arena [@QuakeThree]. The modifications reduce visual connotations of violence, but retain all core game mechanics. Video games form an important domain for research [@laird2001human]. Previous work on first-person games considers either much simpler games [@MnihA3C; @JaderbergUnreal; @wu2016training; @lample2017playing], simplified agent interfaces [@van2009hierarchical], or non-learning systems [@orkin2006three; @waveren2001quakebots], and previously studied multi-agent domains often consist of discrete-state environments [@leibo2017multi; @NIPS2016_6398; @foerster2017learning], have simplified 2D dynamics [@riedmiller2007experiences; @LoweMADDPG; @hausknecht2015deep] or have fully observable or non-perceptual features [@LoweMADDPG; @mordatch2017emergence; @NIPS2016_6398; @foerster2017learning; @riedmiller2007experiences; @hausknecht2015deep] rather than pixel observations. As an example, the RoboCup simulation league [@kitano1997robocup] is a multi-agent environment that shares some of the same challenges of our environment, and successful work has included RL components [@stone2000layered; @LNAI17-MacAlpine2; @riedmiller2007experiences], however these solutions use a combination of hand-engineering, human-specified task decompositions, centralised control, and low-dimensional non-visual inputs, compared to our approach of end-to-end machine learning of independent reinforcement learners.
CTF games are played in an artificial environment referred to as a *map*. In this work we consider two themes of procedurally generated maps in which agents play, indoor maps, and outdoor maps, example schematics of which are shown in Figure \[fig:ext\_maps\]. The *procedural indoor maps* are flat, maze-like maps, rotationally symmetric and contain rooms connected by corridors. For each team there is a base room that contains their flag and player spawn points. Maps are contextually coloured: the red base is coloured red, the blue base blue. The *procedural outdoor maps* are open and hilly naturalistic maps containing randomly sized rocks, cacti, bushes, and rugged terrain that may be impassable. Each team’s flag and starting positions are located in opposite quadrants of the map. Both the procedural indoor maps and the procedural outdoor maps are randomly generated each episode (some random seeds are not used for training and reserved for performance evaluation), providing a very large set of environments. More details can be found in Section \[sec:procmap\]. Every player carries a disc gadget (equivalent to the railgun in Quake III Arena) which can be used for tagging, and can see their team, shield, and flag status on screen.
![Shown are schematics of samples of procedurally generated maps on which agents were trained. In order to demonstrate the robustness of our approach we trained agents on two distinct styles of maps, procedural outdoor maps (top) and procedural indoor maps (bottom).[]{data-label="fig:ext_maps"}](procmaps.png){width="90.00000%"}
Agent
=====
FTW Agent Architecture {#sec:ftwagent}
----------------------
The agent’s policy $\pi$ is represented by a neural network and optimised with reinforcement learning (RL). In a fully observed Markov Decision Process, one would aim at finding a policy that maximises expected $\gamma$-discounted return ${\mathbb{E}}_{\pi(\cdot|\vec{s}_t)} [R_t]$ in game state $\vec{s}_t$, where $R_t = \sum_{k=0}^{T-t} \gamma^{k} r_{t+k}$. However, when an agent does not have information about the entire environment (which is often the case in real world problems, including CTF), it becomes a Partially-Observed Markov Decision Process, and hence we instead seek to maximise ${\mathbb{E}}_{\pi(\cdot|\vec{x}_{\leq t})} [R_t]$, the expected return under a policy conditioned on the agent’s history of individual observations. Due to the ambiguity of the true state given the observations, ${\mathbb{P}\left( \vec{s}_t | \vec{x}_{\leq t} \right)}$, we represent the current value as a random variable, $V_t = {\mathbb{E}}_{\pi(\cdot|\vec{x}_{\leq t})} [R_t] = \sum_{\vec{s}} {\mathbb{P}\left( \vec{s}|\mathbf{x}_{<t} \right)}{\mathbb{E}}_{\pi(\cdot|\vec{s})} [R_t]$. We follow the idea of RL as probabilistic inference [@weber2015reinforced; @levine2013variational; @vlassis2009learning] which leads to a Kullback-Leibler divergence (KL) regularised objective in which the policy $\mathbb{Q}$ is regularised against a prior policy $\mathbb{P}$. We choose both to contain a latent variable $\vec{z}_t$, the purpose of which is to model the dependence on past observations. Letting the policy and the prior differ only in the way this dependence on past observations is modelled leads to the following objective: $${\mathbb{E}}_{{\mathbb{Q}\left( \vec{z}_{t} | \mathrm{C}_t^q \right)}} \left [ R_t\right ] - D_\text{KL}[{\mathbb{Q}\left( \vec{z}_{t} | \mathrm{C}_t^q \right)} || {\mathbb{P}\left( \vec{z}_{t}| \mathrm{C}_t^p \right)} ],
\label{eqn:rlcost}$$ where ${\mathbb{P}\left( \vec{z}_{t}| \mathrm{C}_t^p \right)}$ and ${\mathbb{Q}\left( \vec{z}_{t} | \mathrm{C}_t^q \right)} $ are the prior and variational posterior distributions on $\vec{z}_t$ conditioned on different sets of variables $\mathrm{C}_t^p$ and $\mathrm{C}_t^q$ respectively, and $D_\text{KL}$ is the Kullback-Leibler divergence. The sets of conditioning variables $\mathrm{C}_t^p$ and $\mathrm{C}_t^q$ determine the structure of the probabilistic model of the agent, and can be used to equip the model with representational priors. In addition to optimising the return as in Equation \[eqn:rlcost\], we can also optimise extra modelling targets which are conditional on the latent variable $\vec{z}_t$, such as the value function to be used as a baseline [@MnihA3C], and pixel control [@JaderbergUnreal], whose optimisation positively shapes the shared latent representation. The conditioning variables $\mathrm{C}_t^q$ and $\mathrm{C}_t^p$ and the associated neural network structure are chosen so as to promote forward planning and the use of memory. We use a hierarchical RNN consisting of two recurrent networks (LSTMs [@hochreiter1997long]) operating at different timescales. The hierarchical RNN’s fast timescale core generates a hidden state $\vec{h}_t^q$ at every environment step $t$, whereas its slow timescale core produces an updated hidden state every $\tau$ steps $\vec{h}_t^p = \vec{h}^p_{\tau \lfloor \frac{t}{\tau} \rfloor}$. We use the output of the fast ticking LSTM as the variational posterior, ${\mathbb{Q}\left( \mathbf{z}_{t} | {\mathbb{P}\left( \mathbf{z}_t \right)}, \mathbf{z}_{<t}, \mathbf{x}_{\leq t}, a_{<t}, r_{<t} \right)} = \mathcal{N}(\mu^q_t, \Sigma_t^q)$, where the mean $\mu^q_t$ and covariance $\Sigma_t^q = (\sigma_t^q \mathbf{I})^2$ of the normal distribution are parameterised by the linear transformation $(\mu^q_t, \log \sigma_t^q) = f_q(\vec{h}_t^q)$, and at each timestep take a sample of $\vec{z}_t \sim \mathcal{N}(\mu^q_t, \Sigma_t^q)$. The slow timescale LSTM output is used for the prior of ${\mathbb{P}\left( \mathbf{z}_{t}| \mathbf{z}_{< \tau \lfloor \frac{t}{\tau} \rfloor }, \mathbf{x}_{\leq \tau \lfloor \frac{t}{\tau} \rfloor }, a_{< \tau \lfloor \frac{t}{\tau} \rfloor }, r_{< \tau \lfloor \frac{t}{\tau} \rfloor } \right)} = \mathcal{N}(\mu^p_t, \Sigma_t^p)$ where $\Sigma_t^p = (\sigma_t^p \mathbf{I})^2$ , $(\mu^p_t, \log \sigma_t^p) = f_p(\vec{h}_t^p)$ and $f_p$ is a linear transformation. The fast timescale core takes as input the observation that has been encoded by a convolutional neural network (CNN), $\vec{u}_t = \text{CNN}(\vec{x}_t)$, the previous action $a_{t-1}$, previous reward $r_{t-1}$, as well as the prior distribution parameters $\mu^p_t$ and $\Sigma_t^p$, and the previous sample of the variational posterior $\vec{z}_{t-1} \sim \mathcal{N}(\mu^q_{t-1}, \Sigma_{t-1}^q)$. The slow core takes in the fast core’s hidden state as input, giving the recurrent network dynamics of
$$\vec{h}_t^q = g_q(\vec{u}_t, a_{t-1}, r_{t-1}, \vec{h}_t^p, \vec{h}_{t-1}^q, \mu^p_t, \Sigma_t^p, \vec{z}_{t-1})$$
$$\vec{h}^p_t = \begin{cases}
g_p(\vec{h}^q_{t-1}, \vec{h}^p_{t-1}) & \mbox{if}~ t \bmod \tau = 0 \\
\vec{h}^p_{\tau \lfloor \frac{t}{\tau} \rfloor} & \mbox{otherwise}\\
\end{cases}
\label{eqn:hierrnn}$$
where $g_q$ and $g_p$ are the fast and slow timescale LSTM cores respectively. Stochastic policy, value function, and pixel control signals are obtained from the samples $\vec{z}_t$ using further non-linear transformations. The resulting update direction is therefore: $$\nabla \big( {\mathbb{E}}_{\mathbf{z}_t \sim \mathbb{Q}} \left[-\mathcal{L}(\vec{z}_t, \vec{x}_t)\right] - D_\text{KL}\big[\mathbb{Q}(\mathbf{z}_{t} | \underset{\mathrm{C}_t^q}{\underbrace{{\mathbb{P}\left( \mathbf{z}_t \right)}, \mathbf{z}_{<t}, \mathbf{x}_{\leq t}, a_{<t}, r_{<t}}}) || \mathbb{P}(\mathbf{z}_{t}| \underset{\mathrm{C}_t^p}{\underbrace{\mathbf{z}_{< \tau \lfloor \frac{t}{\tau} \rfloor }, \mathbf{x}_{\leq \tau \lfloor \frac{t}{\tau} \rfloor }, a_{< \tau \lfloor \frac{t}{\tau} \rfloor }, r_{< \tau \lfloor \frac{t}{\tau} \rfloor }}})\big] \big).
\label{eqn:hiervar}$$ where $\mathcal{L}(\cdot, \cdot)$ represents the objective function composed of terms for multi-step policy gradient and value function optimisation [@MnihA3C], as well as pixel control and reward prediction auxiliary tasks [@JaderbergUnreal], see Section \[sec:supp\_train\]. Intuitively, this objective function captures the idea that the slow LSTM generates a prior on $\vec{z}$ which predicts the evolution of $\vec{z}$ for the subsequent $\tau$ steps, while the fast LSTM generates a variational posterior on $\vec{z}$ that incorporates new observations, but adheres to the predictions made by the prior. All the while, $\vec{z}$ must be a useful representation for maximising reward and auxiliary task performance. This architecture can be easily extended to more than two hierarchical layers, but we found in practice that more layers made little difference on this task. We also augmented this dual-LSTM agent with shared DNC memory [@graves2016hybrid] to further increase its ability to store and recall past experience (this merely modifies the functional form of $g_p$ and $g_q$). Finally, unlike previous work on DeepMind Lab [@JaderbergUnreal; @espeholt2018impala], the FTW agent uses a rich action space of 540 individual actions which are obtained by combining elements from six independent action dimensions. Exact agent architectures are described in Figure \[fig:arch\].
Internal Reward and Population Based Training
---------------------------------------------
We wish to optimise the FTW agent with RL as stated in [Equation \[eqn:hiervar\]]{}, using a reward signal that maximises the agent team’s win probability. Reward purely based on game outcome, such as win/draw/loss signal as a reward of $r_T=1$, $r_T=0$, and $r_T=-1$ respectively, is very sparse and delayed, resulting in no learning (Figure \[fig:two\] (b) Self-play). Hence, we obtain more frequent rewards by considering the game points stream $\rho_t$. These can be used simply for reward shaping [@ng1999policy] (Figure \[fig:two\] (b) Self-play + RS) or transformed into a reward signal $r_t = \vec{w}(\rho_t)$ using a learnt transformation $\vec{w}$ (Figure \[fig:two\] (b) FTW). This transformation is adapted such that performing RL to optimise the resulting cumulative sum of expected future discounted rewards effectively maximises the winning probability of the agent’s team, removing the need for manual reward shaping [@ng1999policy]. The transformation $\vec{w}$ is implemented as a table look-up for each of the 13 unique values of $\rho_t$, corresponding to the events listed in Section \[sec:events\]. In addition to optimising the internal rewards of the RL optimisation, we also optimise hyperparameters $\vec{\phi}$ of the agent and RL training process automatically. These include learning rate, slow LSTM time scale $\tau$, the weight of the $D_\text{KL}$ term in [Equation \[eqn:hiervar\]]{}, and the entropy cost (full list in Section \[sec:supp\_train\]). This optimisation of internal rewards and hyperparameters is performed using a process of population based training (PBT) [@jaderberg2017population]. In our case, a population of $P=30$ agents was trained in parallel. For each agent we periodically sampled another agent, and estimated the win probability of a team composed of only the first agent versus a team composed of only the second from training matches using Elo scores. If the estimated win probability of an agent was found to be less than 70% then the losing agent copied the policy, the internal reward transformation, and hyperparameters of the better agent, and explored new internal rewards and hyperparameters. This exploration was performed by perturbing the inherited value by $\pm20$% with a probability of 5%, with the exception of the slow LSTM time scale $\tau$, which was uniformly sampled from the integer range $[5, 20)$. A burn-in time of 1K games was used after each exploration step which prevents further exploration and allows learning to occur.
Training Architecture
---------------------
We used a distributed, population-based training framework for deep reinforcement learning agents designed for the fast optimisation of RL agents interacting with each other in an environment with high computational simulation costs. Our architecture is based on an actor-learner structure [@espeholt2018impala]: a large collection of 1920 *arena* processes continually play CTF games with players sampled at the beginning of each episode from the live training population to fill the $N$ player positions of the game (Section \[sec:traingames\] for details). We train with $N=4$ (2 vs 2 games) but find the agents generalise to different team sizes (Figure \[tab:leaderboard\_proc\_nsvsn\]). After every 100 agent steps, the trajectory of experience from each player’s point of view (observations, actions, rewards) is sent to the *learner* responsible for the policy carried out by that player. The learner corresponding to an agent composes batches of the 32 trajectories most recently received from arenas, and computes a weight update to the agent’s neural network parameters based on [Equation \[eqn:hiervar\]]{} using V-Trace off-policy correction [@espeholt2018impala] to account for off-policy drift.
Performance Evaluation
======================
An important dimension of assessing the success of training agents to play CTF is to evaluate their skill in terms of the agent team’s win probability. As opposed to single-agent tasks, assessing skill in multi-agent systems depends on the teammates and opponents used during evaluation. We quantified agent skill by playing evaluation games with players from the set of all agents to be assessed. Evaluation games were composed using ad-hoc matchmaking in the sense that all $N$ players of the game, from both teams, were drawn at random from the set of agents being evaluated. This allowed us to measure skill against any set of opponent agents and robustness to any set of teammate agents. We estimate skill using the Elo rating system [@ELO] extended to teams (see Section \[sec:supp\_elo\] for exact details of Elo calculation).
We performed evaluation matches with snapshots of the FTW agent and ablation study agents through training time, and also included *built-in bots* and *human participants* as reference agents for evaluation purposes only. Differences between these types of players is summarised in Figure \[fig:humanagentdiff\].
The various ablated agents in experiments are (i) UNREAL [@JaderbergUnreal] trained with self-play using game winning reward – this represents the state-of-the-art naive baseline – (ii) Self-play with reward shaping (RS) which instead uses the Quake default points scheme as reward, (iii) PBT with RS, which replaces self-play with population based training, and (iv) FTW without temporal hierarchy which is the full FTW agent but omitting the temporal hierarchy (Section \[sec:ablation\] for full details).
The built-in bots were scripted AI bots developed for Quake III Arena. Their policy has access to the entire game engine, game state, and map layout, but have no learning component [@waveren2001quakebots]. These bots were configured for various skill levels, from Bot 1 (very low skill level) to Bot 5 (very high skill level, increased shields), as described fully in Section \[sec:botdetails\].
The human participants consisted of 40 people with first-person video game playing experience. We collected results of evaluation games involving humans by playing five tournaments of eight human players. Players were given instructions on the game environment and rules, and performed two games against Bot 3 built-in bots. Human players then played seven games in ad-hoc teams, being randomly matched with other humans, FTW agents, and FTW without a temporal hierarchy agents as teammates and opponents. Players were not told with which agent types they were playing and were not allowed to communicate with each other. Agents were executed in real-time on the CPUs of the same workstations used by human players (desktops with a commodity GPU) without adversely affecting the frame-rate of the game.
Figure \[fig:ext\_human\_win\_prob\] shows the outcome of the tournaments involving humans. To obtain statistically valid Elo estimates from the small number of games played among individuals with high skill variance, we pooled the humans into two groups, top 20% (strong) and remaining 80% (average), according to their individual performances.
![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](winprob.png "fig:"){height="1cm"}\
![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](eval1.png "fig:"){height="9cm"} ![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](eval2.png "fig:"){height="9cm"} ![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](eval3.png "fig:"){height="9cm"}\
![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](eval4.png "fig:"){height="4.125cm"} ![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](eval5.png "fig:"){height="4.125cm"} ![ Shown are win probabilities of different agents, including bots and humans, in evaluation tournaments, when playing on procedurally generated maps of various sizes (13–21), team sizes (1–4) and styles (indoor/outdoor). On indoor maps, agents were trained with team size two on a mixture of $13\times 13$ and $17 \times 17$ maps, so performance in scenarios with different map and team sizes measures their ability to successfully generalise. Teams were composed by sampling agents from the set in the figure with replacement. [**Bottom:** ]{} Shown are win probabilities, differences in number of flags captured, and number of games played for the human evaluation tournament, in which human subjects played with agents as teammates and/or opponents on indoor procedurally generated $17 \times 17$ maps.[]{data-label="tab:leaderboard_proc"}](eval6.png "fig:"){height="4.125cm"}
\[tab:leaderboard\_natlab\] \[tab:leaderboard\_proc\_nsvsn\] \[fig:ext\_human\_win\_prob\]
[cc]{}
[lr]{}\
Agent & Flags\
Bot & 14\
Self-play + RS & 9\
PBT + RS & 14\
FTW w/o TH & 23\
FTW & 37\
Fetch-trained FTW w/o TH & 30\
Fetch-trained FTW & 44\
&
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Average number of flags scored per match for different CTF-trained agents playing two-player *fetch* (CTF without opponents) on indoor procedurally generated maps of size 17. This test provides a measure of agents’ ability to cooperate while navigating in previously unseen maps. Ten thousand matches were played, with teams consisting of two copies of the same agent, which had not been trained on this variant of the CTF task. All bot levels performed very similarly on this task, so we report a single number for all bot levels. In addition we show results when agents are trained solely on the fetch task (+1 reward for picking up and capturing a flag only). [**Right:**]{} Heatmaps of the visitation of the FTW agent during the second half of several episodes while playing fetch..[]{data-label="fig:fetch"}](fetch1.png "fig:"){width="19.50000%"} ![ Average number of flags scored per match for different CTF-trained agents playing two-player *fetch* (CTF without opponents) on indoor procedurally generated maps of size 17. This test provides a measure of agents’ ability to cooperate while navigating in previously unseen maps. Ten thousand matches were played, with teams consisting of two copies of the same agent, which had not been trained on this variant of the CTF task. All bot levels performed very similarly on this task, so we report a single number for all bot levels. In addition we show results when agents are trained solely on the fetch task (+1 reward for picking up and capturing a flag only). [**Right:**]{} Heatmaps of the visitation of the FTW agent during the second half of several episodes while playing fetch..[]{data-label="fig:fetch"}](fetch2.png "fig:"){width="19.50000%"} ![ Average number of flags scored per match for different CTF-trained agents playing two-player *fetch* (CTF without opponents) on indoor procedurally generated maps of size 17. This test provides a measure of agents’ ability to cooperate while navigating in previously unseen maps. Ten thousand matches were played, with teams consisting of two copies of the same agent, which had not been trained on this variant of the CTF task. All bot levels performed very similarly on this task, so we report a single number for all bot levels. In addition we show results when agents are trained solely on the fetch task (+1 reward for picking up and capturing a flag only). [**Right:**]{} Heatmaps of the visitation of the FTW agent during the second half of several episodes while playing fetch..[]{data-label="fig:fetch"}](fetch3.png "fig:"){width="19.50000%"}
![ Average number of flags scored per match for different CTF-trained agents playing two-player *fetch* (CTF without opponents) on indoor procedurally generated maps of size 17. This test provides a measure of agents’ ability to cooperate while navigating in previously unseen maps. Ten thousand matches were played, with teams consisting of two copies of the same agent, which had not been trained on this variant of the CTF task. All bot levels performed very similarly on this task, so we report a single number for all bot levels. In addition we show results when agents are trained solely on the fetch task (+1 reward for picking up and capturing a flag only). [**Right:**]{} Heatmaps of the visitation of the FTW agent during the second half of several episodes while playing fetch..[]{data-label="fig:fetch"}](fetch4.png "fig:"){width="19.50000%"} ![ Average number of flags scored per match for different CTF-trained agents playing two-player *fetch* (CTF without opponents) on indoor procedurally generated maps of size 17. This test provides a measure of agents’ ability to cooperate while navigating in previously unseen maps. Ten thousand matches were played, with teams consisting of two copies of the same agent, which had not been trained on this variant of the CTF task. All bot levels performed very similarly on this task, so we report a single number for all bot levels. In addition we show results when agents are trained solely on the fetch task (+1 reward for picking up and capturing a flag only). [**Right:**]{} Heatmaps of the visitation of the FTW agent during the second half of several episodes while playing fetch..[]{data-label="fig:fetch"}](fetch5.png "fig:"){width="19.50000%"} ![ Average number of flags scored per match for different CTF-trained agents playing two-player *fetch* (CTF without opponents) on indoor procedurally generated maps of size 17. This test provides a measure of agents’ ability to cooperate while navigating in previously unseen maps. Ten thousand matches were played, with teams consisting of two copies of the same agent, which had not been trained on this variant of the CTF task. All bot levels performed very similarly on this task, so we report a single number for all bot levels. In addition we show results when agents are trained solely on the fetch task (+1 reward for picking up and capturing a flag only). [**Right:**]{} Heatmaps of the visitation of the FTW agent during the second half of several episodes while playing fetch..[]{data-label="fig:fetch"}](fetch6.png "fig:"){width="19.50000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[tab:leaderboard\_fetch\_2vs2\]
We also performed another study with human players to find out if human ingenuity, adaptivity and teamwork would help humans find exploitative strategies against trained agents. We asked two professional games testers to play as a team against a team of two FTW agents on a fixed, particularly complex map, which had been held out of training. After six hours of practice and experimentation, the human games testers were able to consistently win against the FTW team on this single map by employing a high-level strategy. This winning strategy involved careful study of the preferred routes of the agents on this map in exploratory games, drawing explicit maps, and then precise communication between humans to coordinate successful flag captures by avoiding the agents’ preferred routes. In a second test, the maps were changed to be procedurally generated for each episode as during training. Under these conditions, the human game testers were not able to find a consistently winning strategy, resulting in a human win-rate of only 25% (draw rate of 6.3%).
Human-Agent Differences {#sec:humandiff}
-----------------------
It is important to recognise the intrinsic differences between agents and humans when evaluating results. It is very difficult to obtain an even playing ground between humans and agents, and it is likely that this will continue to be the case for all human/machine comparisons in the domain of action video games. While we attempted to ensure that the interaction of agents and humans within their shared environment was as fair as possible, engineering limitations mean that differences still exist. Figure \[fig:humanagentdiff\] (a) outlines these, which include the fact that the environment serves humans a richer interface than agents: observations with higher visual resolution and lower temporal latency, and a control space of higher fidelity and temporal resolution.
However, in spite of these environmental constraints, agents have a set of advantages over humans in terms of their ultimate sensorimotor precision and perception. Humans cannot take full advantage of what the environment offers: they have a visual-response feedback loop far slower than the 60Hz observation rate [@castel2005effects]; and although a high fidelity action space is available, humans’ cognitive and motor skills limit their effective control in video games [@berard2011limits].
One way that this manifests in CTF games is through reaction times to salient events. While we cannot measure reaction time directly within a full CTF game, we measure possible proxies for reaction time by considering how long it takes for an agent to respond to a newly-appeared opponent (Figure \[fig:humanagentdiff\] (b)). After an opponent first appears within a player’s (90 degree) field-of-view, it must be become “taggable”, positioned within a 10 degree cone of the player’s centre of vision. This occurs very quickly within both human and agent play, less than 200ms on average (though this does not necessarily reflect intentional reactions, and may also result from some combination of players’ movement statistics and prior orientation towards opponent appearance points). However, the time between first seeing an opponent and attempting a tag (the opponent is taggable and the tag action is emitted) is much lower for FTW agents (258ms on average) compared to humans (559ms), and when a successful tag is considered this gap widens (233ms FTW, 627ms humans). Stronger agents also had lower response times in general than weaker agents, but there was no statistically significant difference in strong humans’ response times compared to average humans.
The tagging accuracy of agents is also significantly higher than that of humans: 80% for FTW agents compared to 48% for humans. We measured the effect of tagging accuracy on the performance of FTW agents playing against a Bot 3 team by artificially impairing agents’ ability to fire, without retraining the agents (Figure \[fig:humanagentdiff\] (c)). Win probability decreased as the accuracy of the agent decreased, however at accuracies comparable to humans the FTW agents still had a greater win probability than humans (albeit with comparable mean flag capture differences). We also used this mechanism to attempt to measure the effect of successful tag time on win probability (Figure \[fig:humanagentdiff\] (d)), and found that an average response time of up to 375ms did not effect the win probability of the FTW agent – only at 448ms did the win rate drop to 85%.
Analysis
========
Knowledge Representation
------------------------
We carried out an analysis of the FTW agent’s internal representation to help us understand how it represents its environment, what aspects of game state are well represented, and how it uses its memory module and parses visual observations.
We say that the agent had game-state related knowledge of a given piece of information if that information could be decoded with sufficient accuracy from the agent’s recurrent hidden state $(\vec{h}_t^p, \vec{h}_t^q)$ using a linear probe classifier. We defined a set of 40 binary features that took the form of questions (found in Figure \[fig:ext\_knowledge\]) about the state of the game in the distant and recent past, present, and future, resulting in a total of 200 features. Probe classifiers were trained for each of the 200 features using balanced logistic regression on 4.5 million game situations, with results reported in terms of AUCROC evaluated with 3-fold episode-wise cross validation. This analysis was performed on the agent at multiple points in training to show what knowledge emerges at which point in training, with the results shown in Figure \[fig:ext\_knowledge\].
![Shown is prediction accuracy in terms of percent AUCROC of linear probe classifiers on 40 different high-level game state features (columns) for different agents (rows), followed by their averages across features, for five different temporal offsets ranging from -20 to +20 frames (top to bottom). Results are shown for the baseline self-play agent with reward shaping as well as the FTW agent after different numbers of training games, and an untrained randomly initialised FTW agent.[]{data-label="fig:ext_knowledge"}](ctf_knowledge.png){width="\textwidth"}
Further insights about the geometry of the representation space were gleaned by performing a t-SNE dimensionality reduction [@maaten2008visualizing] on the recurrent hidden state of the FTW agent. We found strong evidence of cluster structure in the agent’s representation reflecting conjunction of known CTF game state elements: flag possession, location of the agent, and the agent’s respawn state. Furthermore, we introduce *neural response maps* which clearly highlight the differences in co-activation of individual neurons of the agent in these different game states (Figure \[fig:ext\_neural\_responses\]). In fact, certain aspects of the game, such as whether an agent’s flag is held by an opponent or not, or whether the agent’s teammate holds the opponents flag or not, are represented by the response of single neurons.
![ Shown are [*neural response maps*]{} for the FTW agent for game state features used in the knowledge study of Extended Data Figure \[fig:ext\_knowledge\]. For each binary feature $y$ we plot the response vector $\mathbb{E}[(\vec{h}^p, \vec{h}^q) | y=1] - \mathbb{E}[(\vec{h}^p, \vec{h}^q) | y=0]$. [**Bottom:**]{} Process for generating similarity based topological embedding of the elements of vector $\vec{x} \in \mathbb{R}^H$ given a dataset of other $X \in \mathbb{R}^{T \times H}$. Here we use two independent t-SNE embeddings, one for each of the agent’s LSTM hidden state vectors at the two timescales.[]{data-label="fig:ext_neural_response"}](ctf_neural_responses.png "fig:"){width="90.00000%"} ![ Shown are [*neural response maps*]{} for the FTW agent for game state features used in the knowledge study of Extended Data Figure \[fig:ext\_knowledge\]. For each binary feature $y$ we plot the response vector $\mathbb{E}[(\vec{h}^p, \vec{h}^q) | y=1] - \mathbb{E}[(\vec{h}^p, \vec{h}^q) | y=0]$. [**Bottom:**]{} Process for generating similarity based topological embedding of the elements of vector $\vec{x} \in \mathbb{R}^H$ given a dataset of other $X \in \mathbb{R}^{T \times H}$. Here we use two independent t-SNE embeddings, one for each of the agent’s LSTM hidden state vectors at the two timescales.[]{data-label="fig:ext_neural_response"}](ctf_nrp.pdf "fig:"){width="60.00000%"}
\[fig:ext\_neural\_responses\]
Finally, we can decode the sensitivity of the agent’s value function, policy, and internal single neuron responses to its visual observations of the environment through gradient-based saliency analysis [@simonyan2013deep] (Figure \[fig:ext\_saliency\]). Sensitivity analysis combined with knowledge classifiers seems to indicate that the agent performed a kind of task-based scene understanding, with the effect that its value function estimate was sensitive to seeing the flag, other agents, and elements of the on-screen information. The exact scene objects which an agent’s value function was sensitive to were often found to be context dependent (Figure \[fig:ext\_attention\] bottom).
![ Selected saliency analysis of FTW agent. Contours show sensitivity $\left \| \tfrac{\partial f_t}{\partial \vec{x}_{t,ij}} \right \|_1$, where $f_t$ is instantiated as the agent’s value function at time $t$, its policy, or one of four highly selective neurons in the agent’s hidden state, and $\vec{x}_{t,ij}$ represents the pixel at position $ij$ at time $t$. Brighter colour means higher gradient norm and thus higher sensitivity to given pixels. [**Bottom:**]{} Saliency analysis of a single neuron that encodes whether an opponent is holding a flag. Shown is a single situation from the perspective of the FTW agent, in which attention is on an opponent flag carrier at time $t$, on both opponents at time $t+2$, and switches to the on-screen information at time $t+4$ once the flag carrier has been tagged and the flag returned.[]{data-label="fig:ext_saliency"}](ctf_saliency.pdf "fig:"){width="92.00000%"}\
![ Selected saliency analysis of FTW agent. Contours show sensitivity $\left \| \tfrac{\partial f_t}{\partial \vec{x}_{t,ij}} \right \|_1$, where $f_t$ is instantiated as the agent’s value function at time $t$, its policy, or one of four highly selective neurons in the agent’s hidden state, and $\vec{x}_{t,ij}$ represents the pixel at position $ij$ at time $t$. Brighter colour means higher gradient norm and thus higher sensitivity to given pixels. [**Bottom:**]{} Saliency analysis of a single neuron that encodes whether an opponent is holding a flag. Shown is a single situation from the perspective of the FTW agent, in which attention is on an opponent flag carrier at time $t$, on both opponents at time $t+2$, and switches to the on-screen information at time $t+4$ once the flag carrier has been tagged and the flag returned.[]{data-label="fig:ext_saliency"}](ctf_attention_switch.pdf "fig:"){width="80.00000%"}
\[fig:ext\_attention\]
![ Shown are Hinton diagrams representing how often the FTW agent reads memory slots written to at different locations, which are represented in terms of distance to home and opponent base, on 1000 procedurally generated maps, at different points during training. The size of each square represents the difference between the probability of reading from the given location compared to randomly reading from one of the locations visited earlier in the episode. Red indicates that the agent reads from this position more often than random, and blue less. At 450K the agent appears to have learned to read from near its own base and just outside the opponent base. [**Bottom:**]{} Shown are memory recall patterns for an example episode. The heatmap plot on the left shows memory recall frequency averaged across the episode. Shown on the right are the recall patterns during the agent’s first exploration of a newly encountered map. Early in the episode, the agent simply recalls its own path. In almost the same situation later in the episode, the agent recalls entering the opponent base instead. []{data-label="fig:ext_dnc"}](ctf_dnc.pdf "fig:"){width="\textwidth"} ![ Shown are Hinton diagrams representing how often the FTW agent reads memory slots written to at different locations, which are represented in terms of distance to home and opponent base, on 1000 procedurally generated maps, at different points during training. The size of each square represents the difference between the probability of reading from the given location compared to randomly reading from one of the locations visited earlier in the episode. Red indicates that the agent reads from this position more often than random, and blue less. At 450K the agent appears to have learned to read from near its own base and just outside the opponent base. [**Bottom:**]{} Shown are memory recall patterns for an example episode. The heatmap plot on the left shows memory recall frequency averaged across the episode. Shown on the right are the recall patterns during the agent’s first exploration of a newly encountered map. Early in the episode, the agent simply recalls its own path. In almost the same situation later in the episode, the agent recalls entering the opponent base instead. []{data-label="fig:ext_dnc"}](ctf_dnc_example.pdf "fig:"){width="\textwidth"}
![Shown is a side-by-side comparison of the internal representations learned from playing CTF for the FTW and Self-play + RS agent, visualised using t-SNE and single neuron activations (Figure \[fig:three\] for more information). The self-play agent’s representation is seen to be significantly less coherently clustered by game state, especially with respect to flag possessions. Furthermore, it appears to have developed only two highly selective neurons compared to four for the FTW agent.[]{data-label="fig:ext_tsnes"}](ctf_tsne_comp.png){width="\textwidth"}
Agent Behaviour
---------------
The CTF games our agents played were five minutes long and consisted of 4500 elemental actions by each player. To better understand and interpret the behaviour of agents we considered modelling temporal chunks of high-level game features. We segmented games into two-second periods represented by a sequence of game features ( distance from bases, agent’s room, visibility of teammates and opponents, flag status, see Section \[sec:behanalysis\]) and used a variational autoencoder (VAE) consisting of an RNN encoder and decoder [@bowman2015generating] to find a compressed vector representation of these two seconds of high-level agent-centric CTF game-play. We used a Gaussian mixture model (GMM) with 32 components to find clusters of behaviour in the VAE-induced vector representation of game-play segments (Section \[sec:behanalysis\] for more details). These discrete cluster assignments allowed us to represent high-level agent play as a sequence of clusters indices (Figure \[fig:ext\_behvaiours\] (b)). These two second behaviour prototypes were interpretable and represented a wide range of meaningful behaviours such as home base camping, opponents base camping, defensive behaviour, teammate following, respawning, and empty room navigation. Based on this representation, high-level agent behaviour could be represented by histograms of frequencies of behaviour prototypes over thousands of episodes. These behavioural fingerprints were shown to vary throughout training, differed strongly between hierarchical and non-hierarchical agent architectures, and were computed for human players as well (Figure \[fig:ext\_behvaiours\] (a)). Comparing these behaviour fingerprints using the Hellinger distance [@hellinger1909neue] we found that the human behaviour was most similar to the FTW agent after 200K games of training.
![ Shown is a collection of bar plots, one for each of 32 automatically discovered behaviour clusters, representing the number of frames per episode during which the behaviour has been observed for the FTW agent at different points in training, the FTW agent without the temporal hierarchy (TH), the Self-play + RS agent, and human players, averaged over maps and episodes. The behavioural fingerprint changes significantly throughout training, and differs considerably between models with and without temporal hierarchy. [**(b)**]{} Shown is the multi-variate time series of active behaviour clusters during an example episode played by the trained FTW agent. Shown are three particular situations represented by the behaviour clusters: [*following your teammate*]{}, [*enemy base camping*]{}, and [*home base defence*]{}.[]{data-label="fig:ext_behvaiours"}](behaviours.pdf){width="\textwidth"}
Experiment Details
==================
Elo Calculation {#sec:supp_elo}
---------------
We describe the performance of both agents (human or otherwise) in terms of Elo ratings[@ELO], as commonly used in both traditional games like chess and in competitive video game ranking and matchmaking services. While Elo ratings as described for chess address the one-versus-one case, we extend this for CTF to the $n$-versus-$n$ case by making the assumption that the rating of a team can be decomposed as the sum of skills of its team members.
Given a population of $M$ agents, let $\psi_i \in \mathbb{R}$ be the rating for agent $i$. We describe a given match between two teams, blue and red, with a vector $\vec{m} \in \mathbb{Z}^M$, where $m_i$ is the number of times agent $i$ appears in the blue team less the number of times the agent appears in the red team. Using our additive assumption we can then express the standard Elo formula as: $$\mathbb{P}(\text{blue wins against red} | \vec{m}, \vec{\phi}) = \frac{1}{1 + 10^{-\vec{\psi}^T\vec{m}/400}}.$$
To calculate ratings given a set of matches with team assignments $\vec{m}_i$ and outcomes $y_i$ ($y_i=1$ for “blue beats red” and $y_i=\frac{1}{2}$ for draw), we optimise $\vec{\psi}$ to find ratings $\vec{\psi}^\ast$ that maximise the likelihood of the data. Since win probabilities are determined only by absolute differences in ratings we typically anchor a particular agent (Bot 4) to a rating of 1000 for ease of interpretation.
For the purposes of PBT, we calculate the winning probability of $\pi_i$ versus $\pi_j$ using $\vec{m}_i = 2$ and $\vec{m}_j = -2$ (and $\vec{m}_k=0$ for $k \notin \{i,j\}$), we assume that both players on the blue team are $\pi_i$ and similarly for the red team.
Environment Observation and Action Space {#sec:supp_env_spec}
----------------------------------------
DeepMind Lab[@beattie2016deepmind] is capable of rendering colour observations at a wide range of resolutions. We elected to use a resolution of 84$\times$84 pixels as in previous related work in this environment[@A3C; @JaderbergUnreal]. Each pixel is represented by a triple of three bytes, which we scale by $\tfrac{1}{255}$ to produce an observation ${{\mathbf{x}}}_t \in [0, 1]^{84\times84\times3}$.
The environment accepts actions as a composite of six types of partial actions: change in yaw (continuous), change in pitch (continuous), strafing left or right (ternary), moving forward or backwards (ternary), tagging and jumping (both binary). To further simplify this space, we expose only two possible values for yaw rotations (10 and 60) and just one for pitch (5). Consequently, the number of possible composite actions that the agent can produce is of size $5 \cdot 3 \cdot 3 \cdot 3 \cdot 2 \cdot 2 = 540$.
Procedural Environments {#sec:procmap}
-----------------------
#### Indoor Procedural Maps
The procedural indoor maps are flat, point-symmetric mazes consisiting of rooms connected by corridors. Each map has two base rooms which contain the team’s flag spawn point and several possible player spawn points. Maps are contextually coloured: the red base is red, the blue base is blue, empty rooms are grey and narrow corridors are yellow. Artwork is randomly scattered around the map’s walls.
The procedure for generating an indoor map is as follows:
1. Generate random sized rectangular rooms within a fixed size square area ( $13 \times 13$ or $17 \times 17$ cells). Room edges were only placed on even cells meaning rooms always have odd sized walls. This restriction was used to work with the maze backtracking algorithm.
2. Fill the space between rooms using the backtracking maze algorithm to produce corridors. Backtracking only occurs on even cells to allow whole cell gaps as walls.
3. Remove dead ends and horseshoes in the maze.
4. Searching from the top left cell, the first room encountered is declared the base room. This ensures that base rooms are typically at opposite ends of the arena.
5. The map is then made to be point-symmetric by taking the first half of the map and concatenating it with its reversed self.
6. Flag bases and spawn points are added point-symmetrically to the base rooms.
7. The map is then checked for being solveable and for meeting certain constraints (base room is at least 9 units in area, the flags are a minimum distance apart).
8. Finally, the map is randomly rotated (to prevent agents from exploiting the skybox for navigation).
#### Outdoor Procedural Maps
The procedural outdoor maps are open and hilly naturalistic maps containing obstacles and rugged terrain. Each team’s flag and spawn locations are on opposite corners of the map. Cacti and boulders of random shapes and sizes are scattered over the landscape. To produce the levels, first the height map was generated using the diamond square fractal algorithm. This algorithm was run twice, first with a low variance and then with a high variance and compiled using the element-wise max operator. Cacti and shrubs were placed in the environment using rejection sampling. Each plant species has a preference for a distribution over the height above the water table. After initial placement, a lifecycle of the plants was simulated with seeds being dispersed near plants and competition limiting growth in high-vegetation areas. Rocks were placed randomly and simulated sliding down terrain to their final resting places. After all entities had been placed on the map, we performed pruning to ensure props were not overlapping too much. Flags and spawn points were placed in opposite quadrants of the map. The parameters of each map (such as water table height, cacti, shrub and rock density) were also randomly sampled over each individual map. 1000 maps were generated and 10 were reserved for evaluation.
Training Details {#sec:supp_train}
----------------
Agents received observations from the environment 15 times (steps) per second. For each observation, the agent returns an action to the environment, which is repeated four times within the environment[@A3C; @JaderbergUnreal]. Every training game lasts for five minutes, or, equivalently, for 4500 agent steps. Agents were trained for two billion steps, corresponding to approximately 450K games.
Agents’ parameters were updated every time a batch of 32 trajectories of length 100 had been accumulated from the arenas in which the respective agents were playing. We used RMSProp[@RMSPROP] as the optimiser, with epsilon $10^{-5}$, momentum $0$, and decay rate $0.99$. The initial learning rate was sampled per agent from LogUniform$(10^{-5},5\cdot 10^{-3})$ and further tuned during training by PBT, with a population size of 30. Both V-Trace clipping thresholds $\bar \rho, \bar c$ were set to 1. RL discounting factor $\gamma$ was set to 0.99.
All agents were trained with at least the components of the UNREAL loss[@JaderbergUnreal]: the losses used by A3C[@A3C], plus pixel control and reward prediction auxiliary task losses. The baseline cost weight was fixed at $0.5$, the initial entropy cost was sampled per agent from LogUniform$(5 \cdot 10^{-4},10^{-2})$, the initial reward prediction loss weight was sampled from LogUniform$(0.1,1)$, and the initial pixel control loss weight was sampled from LogUniform$(0.01,0.1)$. All weights except the baseline cost weight were tuned during training by PBT.
Due to the composite nature of action space, instead of training pixel control policies directly on 540 actions, we trained independent pixel control policies for each of the six action groups.
The reward prediction loss was trained using a small replay buffer, as in UNREAL[@JaderbergUnreal]. In particular, the replay buffer had capacity for 800 non-zero-reward and 800 zero-reward sequences. Sequences consisted of three observations. The batch size for the reward prediction loss was 32, the same as the batch size for all the other losses. The batch consisted of 16 non-zero-reward sequences and 16 zero-reward sequences.
For the FTW agent with temporal hierarchy, the loss includes the KL divergence between the prior distribution (from the slow-ticking core) and the posterior distribution (from the fast-ticking core), as well as KL divergence between the prior distribution and a multivariate Gaussian with mean 0, standard deviation 0.1. The weight on the first divergence was sampled from $\text{LogUniform}(10^{-3},1)$, and the weight on the second divergence was sampled from $\text{LogUniform}(10^{-4},10^{-1})$. A scaling factor on the gradients flowing from fast to slow ticking cores was sampled from $\text{LogUniform}(0.1,1)$. Finally, the initial slower-ticking core time period $\tau$ was sampled from Categorical$([5,6,\dots, 20])$. These four quantities were further optimised during training by PBT.
### Training Games {#sec:traingames}
Each training CTF game was started by randomly sampling the level to play on. For indoor procedural maps, first (with 50% probability) the size of map (13 or 17) and its geometry were generated according to the procedure described in Section \[sec:procmap\]. For outdoor procedural maps one of the 1000 pre-generated maps was sampled uniformly. Next, a single agent $\pi_p$ was randomly sampled from the population. Based on its Elo score three more agents were sampled without replacement from the population according to the distribution $$\forall_{\pi \in \vec{\pi}_{-p}} \mathbb{P}(\pi|\pi_p) \propto \tfrac{1}{\sqrt{2\pi \sigma^2}} \exp\left ( - \frac{
(\mathbb{P}({\pi_p\text{ beats }\pi} | \phi) - 0.5)^2
}{2\sigma^2} \right )\;\;\;\;\;\text{ where } \sigma = \tfrac{1}{6}$$ which is a normal distribution over Elo-based probabilities of winning, centred on agents of the same skill. For self-play ablation studies agents were paired with their own policy instead. The agents in the game pool were randomly assigned to the red and blue teams. After each 5 minute episode this process was repeated.
![Shown are network architectures of agents used in this study. All agents have the same high-level architecture [**(a)**]{}, using a decomposed policy [**(b)**]{} (see Section \[sec:supp\_env\_spec\]), value function [**(c)**]{}, and convolutional neural network (CNN) visual feature extractor [**(d)**]{}. The baseline agents and ablated FTW without temporal hierarchy agents use an LSTM for recurrent processing [**(e)**]{}. The FTW agent uses a temporal hierarchy for recurrent processing [**(f)**]{} which is composed of two variational units [**(g)**]{}. All agents use reward prediction [**(h)**]{} and independent pixel control [**(i)**]{} auxiliary task networks.[]{data-label="fig:arch"}](architecture.png){width="\textwidth"}
Game Events {#sec:events}
-----------
There are 13 binary game events with unique game point values $\rho_t$. These events are listed below, along with the default values $\vec{w}_\text{quake}$ from the default Quake III Arena points system used for manual reward shaping baselines (Self-play + RS, PBT + RS): $$\begin{aligned}
\rho^{(1)}_t &= \text{I am tagged with the flag} & & p^{(1)} = -1 & & \vec{w}^{(1)}_\text{quake} = 0 \\
\rho^{(2)}_t &= \text{I am tagged without the flag} & & p^{(2)} = -1 & & \vec{w}^{(2)}_\text{quake} = 0 \\
\rho^{(3)}_t &= \text{I captured the flag} & & p^{(3)} = 1 & & \vec{w}^{(3)}_\text{quake} = 6 \\
\rho^{(4)}_t &= \text{I picked up the flag} & & p^{(4)} = 1 & & \vec{w}^{(4)}_\text{quake} = 1 \\
\rho^{(5)}_t &= \text{I returned the flag} & & p^{(5)} = 1 & & \vec{w}^{(5)}_\text{quake} = 1 \\
\rho^{(6)}_t &= \text{Teammate captured the flag} & & p^{(6)} = 1 & & \vec{w}^{(6)}_\text{quake} = 5 \\
\rho^{(7)}_t &= \text{Teammate picked up the flag} & & p^{(7)} = 1 & & \vec{w}^{(7)}_\text{quake} = 0 \\
\rho^{(8)}_t &= \text{Teammate returned the flag} & & p^{(8)} = 1 & & \vec{w}^{(8)}_\text{quake} = 0 \\
\rho^{(9)}_t &= \text{I tagged opponent with the flag} & & p^{(9)} = 1 & & \vec{w}^{(9)}_\text{quake} = 2 \\
\rho^{(10)}_t &= \text{I tagged opponent without the flag} & & p^{(10)} = 1 & & \vec{w}^{(10)}_\text{quake} = 1 \\
\rho^{(11)}_t &= \text{Opponents captured the flag} & & p^{(11)} = -1 & & \vec{w}^{(11)}_\text{quake} = 0 \\
\rho^{(12)}_t &= \text{Opponents picked up the flag} & & p^{(12)} = -1 & & \vec{w}^{(12)}_\text{quake} = 0 \\
\rho^{(13)}_t &= \text{Opponents returned the flag} & & p^{(13)} = -1 & & \vec{w}^{(13)}_\text{quake} = 0
\end{aligned}$$ Agents did not have direct access to these events. FTW agents’ initial internal reward mapping was sampled independently for each agent in the population according to $$\vec{w}^{(i)}_\text{initial} = \epsilon \cdot p^{(i)}\;\;\;\;\;\;\;\; \epsilon \sim \text{LogUniform}(0.1, 10.0).$$ after which it was adapted through training with reward evolution.
Ablation {#sec:ablation}
--------
We performed two separate series of ablation studies, one on procedural indoor maps and one on procedural outdoor maps. For each environment type we ran the following experiments:
- [**Self-play:**]{} An agent with an LSTM recurrent processing core (Figure \[fig:arch\] (e)) trained with the UNREAL loss functions described in Section \[sec:supp\_train\]. Four identical agent policies played in each game, two versus two. Since there was only one agent policy trained, no Elo scores could be calculated, and population-based training was disabled. A single reward was provided to the agent at the end of each episode, +1 for winning, -1 for losing and 0 for draw.
- [**Self-play + Reward Shaping:**]{} Same setup as Self-play above, but with manual reward shaping given by $\vec{w}_\text{quake}$.
- [**PBT + Reward Shaping**]{}: Same agent and losses as Self-play + Reward Shaping above, but for each game in each arena the four participating agents were sampled without replacement from the population using the process described in Section \[sec:supp\_train\]. Based on the match outcomes Elo scores were calculated for the agents in the population as described in Section \[sec:supp\_elo\], and were used for PBT.
- [**FTW w/o Temporal Hierarchy**]{}: Same setup as PBT + Reward Shaping above, but with Reward Shaping replaced by an internal reward signals evolved by PBT.
- [**FTW**]{}: The FTW agent, using the recurrent processing core with temporal hierarchy (Figure \[fig:arch\] (f)), with the training setup described in Methods: matchmaking, PBT, and internal reward signal.
![ The differences between the environment interface offered to humans, agents, and bots. [**(b)**]{} Humans and agents are in addition bound by other sensorimotor limitations. To illustrate we measure humans’ and agents’ response times, when playing against a Bot 3 team on indoor procedural maps. Time delays all measured from first appearance of an opponent in an observation. Left: delay until the opponent becoming taggable ( lies within a 10 degree visual cone). Middle: delay until an attempted tag ( opponent lies within a 10 degree visual cone and tag action emitted). Right: delay until a successful tag. We ignore situations where opponents are further than 1.5 map units away. The shaded region represents values which are impossible to obtain due to environment constraints. [**(c)**]{} Effect of tagging accuracy on win probability against a Bot 3 team on indoor procedural maps. Accuracy is the number of successful tags divided by valid tag attempts. Agents have a trained accuracy of 80%, much higher than the 48% of humans. In order to measure the effect of decreased accuracy on the FTW agent, additional evaluation matches were performed where a proportion of tag events were artificially discarded. As the agent’s accuracy increases from below human (40%) to 80% the win probability increases from 90% to 100% which represents a significant change in performance. [**(d)**]{} Effect of successful tag time on win probability against a Bot 3 team on indoor procedural maps. In contrast to (c), the tag *actions* were artificially discarded $p$% of the time – different values of $p$ result in the spectrum of response times reported. Values of $p$ greater than 0.9 did not reduce response time, showing the limitations of $p$ as a proxy. Note that in both (c) and (d), the agents were not retrained with these $p$ values and so obtained values are only a lower-bound of the potential performance of agents – this relies on the agents generalising outside of the physical environment they were trained in. []{data-label="fig:humanagentdiff"}](humanagent.pdf){width="\textwidth"}
Distinctly Selective Neurons
----------------------------
For identifying the neuron in a given agent that is most selective for a game state feature $y$ we recorded 100 episodes of the agent playing against Bot 3. Given this dataset of activations $\vec{h}_i$ and corresponding labels $y_i$ we fit a Decision Tree of depth 1 using Gini impurity criterion. The decision tree learner selects the most discriminative dimension of $\vec{h}$ and hence the neuron most selective for $y$. If the accuracy of the resulting stump exceeds 97% over $100 \cdot 4500$ steps we consider it to be a *distinctly selective neuron*.
Behavioural Analysis {#sec:behanalysis}
--------------------
For the behavioural analysis, we model chunks of two seconds (30 agent steps) of gameplay. Each step is represented by 56 agent-centric binary features derived from groundtruth game state:
- (3 features) Thresholded shortest path distance from other three agents.
- (4 features) Thresholded shortest path distance from each team’s base and flags.
- (4 features) Whether an opponent captured, dropped, picked up, or returned a flag.
- (4 features) Whether the agent captured, dropped, picked up, or returned a flag.
- (4 features) Whether the agent’s teammate captured, dropped, picked up, or returned a flag.
- (4 features) Whether the agent was tagged without respawning, was tagged and must respawn, tagged an opponent without them respawning, or tagged an opponent and they must respawn.
- (4 features) What room an agent is in: home base, opponent base, corridor, empty room.
- (5 features) Visibility of teammate (visible and not visible), no opponents visible, one opponent visible, two opponents visible.
- (5 features) Which other agents are in the same room: teammate in room, teammate not in room, no opponents in room, one opponent in room, two opponents in room.
- (4 features) Each team’s base visibility.
- (13 features) Each team’s flag status and visibility. Flags status can be either at base, held by teammate, held by opponent, held by the agent, or stray.
- (2 features) Whether agent is respawning and cannot move or not.
For each of the agents analysed, 1000 episodes of pairs of the agent playing against pairs of Bot 3 were recorded and combined into a single dataset. A variational autoencoder (VAE)[@rezende2014stochastic; @kingma2013auto] was trained on batches of this mixed agent dataset (each data point has dimensions 30$\times$56) using an LSTM encoder (256 units) over the 30 time steps, whose final output vector is linearly projected to a 128 dimensional latent variable (diagonal Gaussian). The decoder was an LSTM (256 units) which took in the sampled latent variable at every step.
After training the VAE, a dataset of 400K data points was sampled, the latent variable means computed, and a Gaussian mixture model (GMM) was fit to this dataset of 400K$\times$128, with diagonal covariance and 32 mixture components. The resulting components were treated as behavioural clusters, letting us characterise a two second clip of CTF gameplay as one belonging to one of 32 behavioural clusters.
Bot Details {#sec:botdetails}
-----------
The bots we use for evaluation are a pair of Tauri and Centuri bots from Quake III Arena as defined below.
Bot Personality
------------------ ------ ------ ------ ---------- --------- ------ ------ ------ ---------- ----------
Bot Level 1 2 3 4 5 1 2 3 4 5
ATTACK\_SKILL 0.0 0.25 0.5 1.0 1.0 0.0 0.25 0.5 1.0 1.0
AIM\_SKILL 0.0 0.25 0.5 1.0 1.0 0.0 0.25 0.5 1.0 1.0
AIM\_ACCURACY 0.0 0.25 0.5 1.0 1.0 0.0 0.25 0.5 1.0 1.0
VIEW\_FACTOR 0.1 0.35 0.6 0.9 1.0 0.1 0.35 0.6 0.9 1.0
VIEW\_MAXCHANGE 5 90 120 240 360 5 90 120 240 360
REACTIONTIME 5.0 4.0 3.0 1.75 0.0 5.0 4.0 3.0 1.75 0.0
CROUCHER 0.4 0.25 0.1 0.1 0.0 0.4 0.25 0.1 0.1 0.0
JUMPER 0.4 0.45 0.5 1.0 1.0 0.4 0.45 0.5 1.0 1.0
WALKER 0.1 0.05 0.0 0.0 0.0 0.1 0.05 0.0 0.0 0.0
WEAPONJUMPING 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
GRAPPLE\_USER 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
AGGRESSION 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
SELFPRESERVATION 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
VENGEFULNESS 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
CAMPER 0.0 0.25 0.5 0.5 0.0 0.0 0.25 0.5 0.5 0.0
EASY\_FRAGGER 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
ALERTNESS 0.1 0.3 0.5 1.0 1.0 0.1 0.3 0.5 1.0 1.0
AIM\_ACCURACY 0.0 0.22 0.45 **0.75** 1.0 0.0 0.22 0.45 **0.95** 1.0
FIRETHROTTLE 0.01 0.13 0.25 **1.0** **1.0** 0.01 0.13 0.25 **0.1** **0.01**
|
---
abstract: |
Let ${\mathsf{B}}_1$ be the polynomial ring ${\mathbb{C}}[a^{\pm1},b]$ with the structure of a complex Hopf algebra induced from its interpretation as the algebra of regular functions on the affine linear algebraic group of complex invertible upper triangular 2-by-2 matrices of the form $\left(
\begin{smallmatrix}
a&b\\0&1
\end{smallmatrix}\right)$. We prove that the universal invariant of a long knot $K$ associated to ${\mathsf{B}}_1$ is the reciprocal of the canonically normalised Alexander polynomial $\Delta_K(a)$. Given the fact that ${\mathsf{B}}_1$ admits a $q$-deformation ${\mathsf{B}}_q$ which underlies the (coloured) Jones polynomials, our result provides another conceptual interpretation for the Melvin–Morton–Rozansky conjecture proven by Bar-Nathan and Garoufalidis, and Garoufalidis and Lê.
address: 'Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève 4, Suisse\'
author:
- Rinat Kashaev
date: 'July 21, 2020.'
title: The Alexander polynomial as a universal invariant
---
Introduction
============
The universal quantum knot invariants introduced and studied in a number of works [@MR1025161; @MR1124415; @MR1153694; @MR1227011; @MR1324033; @MR2186115; @MR2253443; @MR2251160] is a convenient tool allowing to encode the multitude of quantum invariants associated to a given Hopf algebra into a single algebraic object in completely representation independent way. As a result, the universal invariants are of great potential for conceptual understanding and organisation of the diversity of quantum invariants though the increased computational complexity makes them perhaps less useful for practical calculations. Nonetheless, as shows the example of the logarithmic invariants of Murakami–Nagatomo [@MR2466562], invariants associated to non semi-simple representations are sometimes better accessible through the universal invariants than directly from the R-matrix calculations.
In this paper we address the problem of identification of the universal invariant of long knots in one of the simplest cases of non-trivial Hopf algebras, namely the commutative but non co-commutative complex Hopf algebra ${\mathsf{B}}_1:={\mathbb{C}}[a^{\pm1},b]$ with the group-like element $a$ and the element $b$ with the co-product $$\label{eq:coprod-b}
\Delta(b)=a\otimes b+b\otimes 1.$$ More abstractly, one can think of ${\mathsf{B}}_1$ as the algebra of regular functions ${\mathbb{C}}[\operatorname{Aff}_1({\mathbb{C}})]$ on the affine linear algebraic group $\operatorname{Aff}_1({\mathbb{C}}):={\mathbb{G}}_a({\mathbb{C}})\rtimes {\mathbb{G}}_m({\mathbb{C}})$ of invertible upper triangular complex 2-by-2 matrices of the form $\left(
\begin{smallmatrix}
a&b\\0&1
\end{smallmatrix}\right)$ where the Hopf algebra structure is canonically induced by the group structure of $\operatorname{Aff}_1({\mathbb{C}})$, see [@MR547117].
The (maximal) Drinfeld’s quantum double of ${\mathsf{B}}_1$ is a Hopf algebra $D({\mathsf{B}}_1)$ which contains two Hopf sub-algebras: ${\mathsf{B}}_1$ and the universal enveloping algebra of the Lie algebra of $\operatorname{Aff}_1({\mathbb{C}})$ generated by two primitive elements $\phi$ and $\psi$ subject to the commutation relation $$\label{eq:cr-lie-alg}
\phi\psi-\psi\phi=\phi.$$ Within the algebra $D({\mathsf{B}}_1)$, the element $a$ is central while the element $b$ interacts with $\phi$ and $\psi$ through the commutation relations $$\phi b-b\phi=1-a ,\quad\psi b-b\psi=b.$$ The formal universal R-matrix of $D({\mathsf{B}}_1)$ $$\label{eq:un-r-mat}
R=(1\otimes a)^{\psi\otimes 1}e^{\phi\otimes b}=\sum_{m,n\ge0}\frac 1{n!}\binom{\psi}{m}\phi^n\otimes (a-1)^m b^n,$$ finds itself behind the associated universal invariant $Z_{{\mathsf{B}}_1}(K)$ of a long knot $K$ which is a central element of a specific “profinite completion” of $D({\mathsf{B}}_1)$ obtained as the convolution algebra $(D({\mathsf{B}}_1)^o)^*$ of the co-algebra structure of the restricted dual Hopf algebra $D({\mathsf{B}}_1)^o$. The following main result of this paper was conjectured in [@Kashaev2019].
\[thm-1\] The universal invariant of a long knot $K$ associated to the Hopf algebra ${\mathsf{B}}_1$ is of the form $
Z_{{\mathsf{B}}_1}(K)=(\Delta_K(a))^{-1}
$ where $\Delta_K(t)$ is the (canonically normalised) Alexander polynomial of $K$.
The reciprocal of the Alexander polynomial in this theorem should be thought of as an element of the ring of formal power series ${\mathbb{C}}[[a-1]]$ which naturally arises upon consideration of all finite dimensional representations of $D({\mathsf{B}}_1)$ where the element $a$ is always unipotent (that is $a-1$ is nilpotent). Taking into account the close relationship of the Alexander polynomial with the Burau representation of the braid groups [@MR3069652], it is interesting to note that Salter in a recent work [@Salter2019] considers the Burau representation over the ring ${\mathbb{Z}}[[t-1]]$ in order to show that the Burau images of the braid groups are dense in Squier’s unitary groups relative to the topology induced by the formal power series.
The algebra ${\mathsf{B}}_1$ can be $q$-deformed to a non-commutative Hopf algebra ${\mathsf{B}}_q$ with the same co-algebra structure but with a $q$-commutative relation $ab=qba$. For $q$ not a root of unity, the quantum double $D({\mathsf{B}}_q)$ contains the quantum group $U_q(sl_2)$ as a Hopf sub-algebra. In particular, for each $n\in{\mathbb{Z}}_{>0}$, it admits an $n$-dimensional irreducible representation corresponding to the $n$-th coloured Jones polynomial. In the large $n$ limit with $q=t^{1/n}$ and fixed $t$, one recovers an infinite-dimensional representation of the Hopf algebra $D({\mathsf{B}}_1)$ where the central element $a$ is realised by the scalar $t$. From that standpoint, Theorem \[thm-1\] is consistent with the Melvin–Morton–Rozansky conjecture proven by Bar-Nathan and Garoufalidis in [@MR1389962] and by Garoufalidis and Lê in [@MR2860990].
Theorem \[thm-1\], in conjunction with the group-like nature of the element $a$, makes absolutely transparent a result of Burau [@MR3069587; @MR3069631] on the property of the Alexander polynomial related to cables: if one takes the $n$-th cable of a knot $K$ and composes it with the braid of $n$ strands which brings the first strand under all others to $n$-th position, then the standard closure of the obtained string link gives a knot whose Alexander polynomial is $\Delta_K(t^n)$.
The main tool of the proof of Theorem \[thm-1\] is the use of a specific infinite dimensional representation of $D({\mathsf{B}}_1)$ on a dense vector subspace $A^1$ of a complex Hilbert space of square integrable holomorphic functions on ${\mathbb{C}}$ considered over the algebra of formal power series ${\mathbb{C}}[[\hbar]]$. The evaluation of the formal universal R-matrix under this representation is a well defined element of the algebra $(\operatorname{End} (A^1))^{\otimes2}[[\hbar]]$, and it is this property of the R-matrix which, from one hand side, makes the corresponding Reshetikhin–Turaev functor well defined as formal power series in $\hbar$ despite the infinite dimensionality of the representation, and from the other hand, it allows us to use the Gaussian integrals to express the quantum invariant in terms of a minor of the unreduced Burau representation matrix. Similarly to the work [@MR1612375], the identification of the quantum invariant with the Alexander polynomial is accomplished through a direct relationship of the latter to a minor of the unreduced Burau representation matrix.
\[thm-2\] Let a knot $K$ be the closure of a braid $\beta\in B_n$ and $\psi_n(\beta)\in \operatorname{GL}_n({\mathbb{Z}}[t^{\pm1}])$ the image of $\beta$ under the unrestricted Burau representation (where the images of the standard Artin generators are linear in $t$). Let $\hat{\beta}_n$ be the $(n-1)\times(n-1)$ matrix obtained from $\psi_n(\beta)$ by throwing away the $n$-th column and the $n$-th row. Then, the Alexander polynomial of $K$ is given by the formula $$\label{eq:re-alex-det-bur}
\Delta_K(t)=t^{\frac{1-n-g(\beta)}2}\det(I_{n-1}-\hat\beta_n)$$ where $I_k$ denotes the identity $k\times k$ matrix and $g\colon B_n\to {\mathbb{Z}}$ is the group homomorphism that sends the Artin generators to 1.
Notice that the exponent of $t$ in the front factor of is always an integer due to a specific parity property of the number $g(\beta)$. A proof of Theorem \[thm-2\], based on the Alexander–Conway skein relation, is outlined in [@MR1133269]. As an independent proof, we directly relate to another known determinantal formula for the Alexander polynomial [@MR0375281; @MR2435235] that uses the reduced Burau representation and where the correcting multiplicative factor is slightly more complicated.
Outline {#outline .unnumbered}
-------
Section \[sec:ifha\] starts with a concise review of the definition of the universal invariants from [@Kashaev2019], and then describes the center of $D({\mathsf{B}}_1)$. Remark that the universal invariant takes its values in a certain “profinite completion” of this center. Section \[sec-schrodinger\] introduces few algebraic and analytic tools used in the subsequent sections: the Hilbert spaces $H^n$ of holomorphic functions on ${\mathbb{C}}^n$ together with a particular class of elements called Schrödinger’s coherent states (which are just linear exponential functions), the dense subspaces $A^n\subset H^n$ generated by products of coherent states and polynomials, and the Gaussian integration formula. The latter is the standard tool in quantum field theory which can also be thought of as an analytic version of MacMahon’s Master theorem [@MR0141605]. In Section \[sec-repr\] the representation of $D({\mathsf{B}}_1)$ in the space $A^1[[\hbar]]$ is introduced, the evaluation of the formal universal R-matrix is shown to be well defined and related to the basic building $2\times2$ matrix of the Burau representation of the braid groups, and the diagrammatic rules for calculation of the Reshetikhin–Turaev functor are specified. The last two Sections \[sec-thm1\] and \[sec-thm2\] contain the proofs of Theorems \[thm-1\] and \[thm-2\] respectively.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Louis-Hadrien Robert and Roland van der Veen for useful discussions. This work is partially supported by the Swiss National Science Foundation, the subsidy no $200020\_192081$.
Universal invariants of long knots from Hopf algebras {#sec:ifha}
=====================================================
In this section, based on the construction of R-matrix invariants of long knots in [@MR1036112; @MR1025161; @MR2796628], we briefly describe the definition of the universal invariants of long knots given in [@Kashaev2019], see also [@MR1324033; @MR2251160] for an approach through the co-end.
Consider the category $\mathbf{Hopf}_{\mathbb{K}}$ of Hopf algebras over a field ${\mathbb{K}}$ with invertible antipode. The restricted dual of an algebra provides us with a contravariant endofunctor $(\cdot)^o\colon \mathbf{Hopf}_{\mathbb{K}}\to \mathbf{Hopf}_{\mathbb{K}}$ which associates to a Hopf algebra $H$ with multiplication $\nabla$ the Hopf algebra $$H^o:=(\nabla^*)^{-1}(H^*\otimes H^*)\subset H^*$$ whose underlying vector space is the vector subspace of the algebraic dual $H^*$ generated by all matrix coefficients of all finite dimensional representations of $H$ [@MR1786197].
Drinfeld’s quantum double of $H\in \operatorname{Ob}\mathbf{Hopf}_{\mathbb{K}}$ (see, for example, [@MR1381692]) is a Hopf algebra $D(H)\in \operatorname{Ob}\mathbf{Hopf}_{\mathbb{K}}$ uniquely determined by the property that there are two Hopf algebra inclusions $$\imath\colon H\to D(H),\quad \jmath\colon H^{o,\text{op}}\to D(H)$$ such that $D(H)$ is generated by their images subject to the commutation relations $$\label{eq:comm-rel-j-i}
\jmath(f)\imath(x)= \langle f_{(1)},x_{(1)}\rangle \langle f_{(3)},S(x_{(3)})\rangle\imath(x_{(2)})\jmath(f_{(2)})\quad \forall (x,f)\in H\times H^o$$ where we use Sweedler’s notation for the co-multiplication $$\Delta(x)=x_{(1)}\otimes x_{(2)},\quad (\Delta\otimes\operatorname{id})(\Delta(x))=x_{(1)}\otimes x_{(2)}\otimes x_{(3)},\ \dots$$
The restricted dual of the quantum double $D(H)^o$ is a dual quasi-triangular Hopf algebra with the dual universal R-matrix $$\varrho\colon D(H)^o\otimes D(H)^o\to {\mathbb{K}},\quad x\otimes y\mapsto \langle x,\jmath(\imath^o(y))\rangle$$ which, among other things, satisfies the Yang–Baxter relation $$\varrho_{1,2}*\varrho_{1,3}*\varrho_{2,3}=\varrho_{2,3}*\varrho_{1,3}*\varrho_{1,2}$$ in the convolution algebra $((D(H)^o)^{\otimes3})^*$. If $\{e_i\}_{i\in I}$ is a linear basis of $H$ and $\{e^i\}_{i\in I}$ is the associated set of canonical (dual) linear forms, then one can write a formal universal R-matrix $$\label{eq.un-R-mat}
R:=\sum_{i\in I}\jmath(e^i)\otimes\imath(e_i)$$ as the formal conjugate of the dual universal R-matrix in the sense of the equality $$\langle x\otimes y,R\rangle=\langle \varrho, x\otimes y\rangle\quad\forall x,y\in D(H)^o.$$ Furthermore, for any finite-dimensional right co-module $$V\to V\otimes D(H)^o,\quad v\mapsto v_{(0)}\otimes v_{(1)},$$ the dual universal R-matrix gives rise to a rigid R-matrix $$r_V\colon V\otimes V\to V\otimes V,\quad u\otimes v\mapsto v_{(0)}\otimes u_{(0)}\langle \varrho,v_{(1)}\otimes u_{(1)}\rangle.$$ This implies that there exists a *universal invariant* of long knots $Z_H(K)$ taking its values in the center of the convolution algebra $(D(H)^o)^*$ such that $$J_{r_V}(K)v=v_{(0)}\langle Z_H(K),v_{(1)}\rangle\quad \forall v\in V$$ where $J_{r_V}(K)\in\operatorname{End}(V)$ is the invariant of long knots associated to $r_V$, see [@Kashaev2019] for details.
The Hopf algebra $D({\mathsf{B}}_1)$ and its center
---------------------------------------------------
Recall that ${\mathsf{B}}_1$ is the polynomial algebra ${\mathbb{C}}[a^{\pm1},b]$ provided with the structure of a Hopf algebra where $a$ is a group-like element and the co-product of $b$ is given in .
The opposite ${\mathsf{B}}_1^{o,\text{op}}$ of the restricted dual Hopf algebra ${\mathsf{B}}_1^o$ is composed of two Hopf sub-algebras: the group algebra ${\mathbb{C}}[\operatorname{Aff}_1({\mathbb{C}})]$ generated by group-like elements $$\chi_{u,v},\quad (u,v)\in {\mathbb{C}}\times{\mathbb{C}}_{\ne0},\quad \chi_{u,v}\chi_{u',v'}=\chi_{u+vu',vv'},$$ and the universal enveloping algebra $U(\operatorname{Lie}\operatorname{Aff}_1({\mathbb{C}}))$ generated by two primitive elements $\psi$ and $\phi$ satisfying the relation . The relations between the generators of ${\mathbb{C}}[\operatorname{Aff}_1({\mathbb{C}})]$ and $U(\operatorname{Lie}\operatorname{Aff}_1({\mathbb{C}}))$ are of the form $$[\chi_{u,v},\psi]=u\phi \chi_{u,v},\quad \chi_{u,v} \phi =v\phi\chi_{u,v}\quad \forall (u,v)\in {\mathbb{C}}\times{\mathbb{C}}_{\ne0}$$ where $[x,y]:=xy-yx$. As linear forms on ${\mathsf{B}}_1$, they are defined by the relations $$\begin{gathered}
\langle\chi_{u,v},b^ma^n\rangle=u^mv^{-m-n},\\
\langle\phi,b^ma^n\rangle=\delta_{m,1},
\quad \langle\psi,b^ma^n\rangle=\delta_{m,0}n,\quad \forall(m,n)\in{\mathbb{Z}}_{\ge0}\times {\mathbb{Z}}.\end{gathered}$$
The commutation relations in the case of the quantum double $D({\mathsf{B}}_1)$ take the form $$[\psi,b]=b,\quad
[\phi,b]=1-a,\quad b\chi_{u,v} =\chi_{u,v}(bv+(a-1)u)\quad \forall (u,v)\in {\mathbb{C}}\times{\mathbb{C}}_{\ne0}$$ and $a$ is central. The formal universal R-matrix is given by formula .
Any finite dimensional right co-module $V$ over $D({\mathsf{B}}_1)^o$ is canonically a left module over $D({\mathsf{B}}_1)$ defined by $$xw=w_{(0)}\langle w_{(1)},x\rangle,\quad \forall (x,w)\in D({\mathsf{B}}_1)\times V,$$ where the elements $a-1$, $b$ and $\phi$ are necessarily nilpotent, so that the formal infinite double sum in truncates to a well defined finite sum.
The center of the algebra $D({\mathsf{B}}_1)$ is the polynomial sub-algebra ${\mathbb{C}}[a^{\pm1},c]$ where $$\label{eq:cent-el-c}
c:=\phi b+(a-1)\psi.$$
It is easily verified that $c$ is central. Any element $x\in D({\mathsf{B}}_1)$ can uniquely be written in the form $$x=\sum_{(u,v,m)\in {\mathbb{C}}\times {\mathbb{C}}_{\ne0}\times {\mathbb{Z}}}\chi_{u,v} e_m p_{u,v,m}(a,c,\psi),$$ where $$e_m:=\left\{
\begin{array}{cc}
b^m & \text{if }\ m>0; \\
1 & \text{if }\ m=0; \\
\phi^{-m} & \text{if }\ m<0
\end{array}
\right.$$ and $ p_{u,v,m}(a,c,\psi)\in{\mathbb{C}}[a^{\pm1},c,\psi]$ is non-zero for only finitely many triples $(u,v,m)$.
Assume that $x$ is central. Then, for any $s\in{\mathbb{C}}_{\ne0}$, we have the equality $$\begin{gathered}
x=\chi_{0,s}^{-1} x \chi_{0,s}=\sum_{(u,v,m)\in {\mathbb{C}}\times {\mathbb{C}}_{\ne0}\times {\mathbb{Z}}}\chi_{u/s,v} e_m s^m p_{u,v,m}(a,c,\psi)\\
=\sum_{(u,v,m)\in {\mathbb{C}}\times {\mathbb{C}}_{\ne0}\times {\mathbb{Z}}}\chi_{u,v} e_m s^m p_{us,v,m}(a,c,\psi)\end{gathered}$$ which implies that for any fixed triple $(u,v,m)\in {\mathbb{C}}\times {\mathbb{C}}_{\ne0}\times {\mathbb{Z}}$, one has the family of equalities $$p_{u,v,m}=s^m p_{us,v,m}\quad \forall s\in{\mathbb{C}}_{\ne0}.$$ This means that $p_{u,v,m}$ can only be non-zero if $u=m=0$. Thus, the element $x$ takes the form $$x=\sum_{v\in{\mathbb{C}}_{\ne0}}\chi_{0,v} p_{0,v,0}(a,c,\psi).$$ The equality $$bx=xb=b\sum_{v\in{\mathbb{C}}_{\ne0}}\chi_{0,v} v^{-1}p_{0,v,0}(a,c,\psi+1).$$ is equivalent to the equalities $$p_{0,v,0}(a,c,\psi+1)=v^{-1}p_{0,v,0}(a,c,\psi) \quad \forall v\in {\mathbb{C}}_{\ne0}$$ which imply that the polynomial $p_{0,v,0}(a,c,\psi)$ can be non-zero only if $v=1$ and if it does not depend on $\psi$. We conclude that $x\in {\mathbb{C}}[a^{\pm1},c]$.
Schrödinger’s coherent states {#sec-schrodinger}
=============================
Here we briefly review the theory of standard Schrödinger’s coherent states (see for example [@MR858831]).
For any $n\in{\mathbb{Z}}_{>0}$, let $H^n\subset L^2({\mathbb{C}}^n,\mu_n)$ be the complex Hilbert space of square integrable holomorphic functions $f\colon {\mathbb{C}}^n \to {\mathbb{C}}$ with the scalar product $$\langle f\vert g\rangle :=\int_{{\mathbb{C}}^n}\overline{f(z)} g(z)\operatorname{d}\!\mu_n(z)$$ where the measure $\mu_n$ on ${\mathbb{C}}^n$ is absolutely continuous with respect to the Lebesgue measure $\lambda_{2n}$ on ${\mathbb{C}}^n\simeq {\mathbb{R}}^{2n}$ with the Radon–Nikodym derivative $$\frac{\operatorname{d}\!\mu_n}{\operatorname{d}\!\lambda_{2n}}(z)=\frac1{\pi^n} e^{-\|z\|^2},\quad \|z\|:=\sqrt{\sum\nolimits_{i=0}\nolimits^{n-1}|z_i|^2}.$$ By direct calculation, one verifies that the monomials $$e_k(z):=\prod_{i=0}^{n-1}\frac{z_i^{k_i}}{\sqrt{k_i!}},\quad k\in {\mathbb{Z}}_{\ge0}^n$$ form an orthonormal family in $H^n$ which is, in fact, a Hilbert basis due to the validity of Taylor’s (multivariable) expansion for holomorphic functions: $$\label{eq:taylor-exp}
f(z)=\sum_{k\in {\mathbb{Z}}_{\ge0}^n}\prod_{i=0}^{n-1}\frac{z_i^{k_i}}{k_i!}\frac{\partial^{k_i}f(w)}{\partial w_i^{k_i}}\biggr\vert_{w=0}=\sum_{k\in {\mathbb{Z}}_{\ge0}^n}e_k(z)\prod_{i=0}^{n-1}\frac{1}{\sqrt{k_i!}}\frac{\partial^{k_i}f(w)}{\partial w_i^{k_i}}\biggr\vert_{w=0}$$ which, in the case when $f\in H^n$, implies that $$\label{eq:fourier-coeff-hol}
\int_{{\mathbb{C}}^n}\overline {e_k(z)}f(z)\operatorname{d}\!\mu_n(z)=\prod_{i=0}^{n-1}\frac{1}{\sqrt{k_i!}}\frac{\partial^{k_i}f(w)}{\partial w_i^{k_i}}\biggr\vert_{w=0}\quad \forall k\in {\mathbb{Z}}_{\ge0}^n.$$ For any $u\in{\mathbb{C}}^n$, multiplying both sides of by $e_k(u)$, summing over all $k\in {\mathbb{Z}}_{\ge0}^n$ and, using the Fubini (or dominant convergence) theorem in the left hand side for exchanging the integration and summation, and the Taylor formula in the right hand side, we obtain $$\label{eq:rep-prop-cs}
\int_{{\mathbb{C}}^n}\varphi_u(\bar z)f(z)\operatorname{d}\!\mu_n(z)=f(u)\quad \forall f\in H^n$$ where the holomorphic function $$\varphi_u\colon {\mathbb{C}}^n\to{\mathbb{C}},\quad z\mapsto \sum_{k\in {\mathbb{Z}}_{\ge0}^n}e_k(u)e_k(z)=e^{\sum_{i=0}^{n-1}u_i z_i}$$ determines an element $
\varphi_u\in H^n$ called *(Schrödinger’s) coherent state*. By treating elements of ${\mathbb{C}}^n$ as column vectors we can write $\varphi_u(z)=e^{{{u}^{\top}}z}$. Let us also adopt the notation $w^*:={{\bar w}^{\top}}$ for the Hermitian conjugation, i.e. the transposition combined with the complex conjugation. With this notation we have the equalities $$\|w\|^2=w^*w,\quad \varphi_u(\bar z)=e^{z^*u}.$$
The integral formula expresses the reproducing property of coherent states $$\langle \varphi_{\bar u}\vert f\rangle=f(u)\quad \forall (f,u)\in H^n\times{\mathbb{C}}^n.$$ The choice $f=\varphi_v$ in the last formula gives the scalar product between the coherent states $$\label{eq:sc-pr-coh-st}
\langle \varphi_{\bar u} | \varphi_v\rangle=\varphi_{v}(u)=\varphi_u(v)=e^{{{u}^{\top}}v}.$$ In particular, the norm of a coherent state $\varphi_v$ is determined by the Euclidean norm of $v$ through the formula $$\|\varphi_v\|=e^{\|v\|^2/2}.$$
A dense subspace of $H^n$
-------------------------
Another useful property of the coherent states is that the (dense) vector subspace $A^n$ of $H^n$ generated by products of coherent states and polynomials is stable under the multiplication of elements of $A^n$ as functions so that $A^n$ carries the additional structure of a commutative algebra, and it is in the domain of any linear differential operator with coefficients in $A^n$. For example, when $n=1$, the Hilbert basis of $H^1$ given by the monomials $\{e_k\}_{k\in{\mathbb{Z}}_{\ge0}}\subset A^1$ is the eigenvector basis of the 1-dimensional quantum harmonic oscillator with the (self-adjoint) Hamiltonian operator $z\frac{\partial}{\partial z}$.
Gaussian integration formula
----------------------------
Writing out explicitly the scalar product as an integral in , we obtain an integral identity $$\int_{{\mathbb{C}}^n} e^{{{v}^{\top}}z+z^*u}\operatorname{d}\!\mu_n(z)=e^{{{v}^{\top}}u}$$ which is a special case of the general Gaussian integration formula $$\label{eq:gauss-int-form}
\int_{{\mathbb{C}}^n} e^{v^*z+z^*u + z^*Mz}\operatorname{d}\!\mu_n(z)=\frac{e^{v^*W^{-1}u}}{\det(W)},\quad W:=I_n-M$$ where $M$ is an arbitrary complex $n$-by-$n$ matrix sufficiently close to zero so that the integral is absolutely convergent. Furthermore, the expansion of in power series in $M$ with $u=v=0$ corresponds to the purely combinatorial MacMahon Master theorem [@MR0141605].
Representations of $D({\mathsf{B}}_1)$ in $A^1[[\hbar]]$ {#sec-repr}
========================================================
Recall that $A^1$ is the vector subspace of $H^1$ generated by products of coherent states with polynomials. For any $\lambda\in {\mathbb{C}}$, the mappings $$\label{eq:rho-hom}
a\mapsto 1+\hbar,\quad b\mapsto \frac{\partial}{\partial z},\quad \phi\mapsto \hbar z,\quad \psi\mapsto \lambda-z\frac{\partial}{\partial z}$$ and the action $$\chi_{u,v}f(z)=e^{\hbar uz}f(vz)$$ determine a homomorphism of algebras $$\label{eq:rst-rep}
\rho_\lambda\colon D({\mathsf{B}}_1)\to \operatorname{End}(A^1[[\hbar]])$$ which sends the central element $c$ defined in to $\lambda\hbar$.
An important property of the representation $\rho_\lambda$ is that the image under $ \rho_\lambda^{\otimes 2}$ of the formal R-matrix is a well defined element of the algebra $ \operatorname{End}(A^1)^{\otimes 2}[[\hbar]]$:
$$\label{eq:eval-r-mat}
\rho_\lambda^{\otimes 2}(R)=(1+\hbar)^{\lambda-z_0\frac{\partial}{\partial z_0}}e^{\hbar z_0\frac{\partial}{\partial z_1}}=\sum_{m,n\ge0}\frac{\hbar^{m+n}}{n!}\binom{\lambda-z_0\frac{\partial}{\partial z_0}}{m} \Big(z_0\frac{\partial}{\partial z_1}\Big)^n.$$
In particular, the double sum in truncates to a finite sum if the indeterminate $\hbar$ is nilpotent. Thus, despite the fact that the representation $\rho_\lambda$ is infinite dimensional, the corresponding R-matrix is well suited for calculation of the image under $\rho_\lambda$ of the universal invariant $Z_{{\mathsf{B}}_1}(K)$. Moreover, as the parameter $\lambda$ enters only through the overall normalisation factor $(1+\hbar)^\lambda$ of the R-matrix, the associated invariant is independent of $\lambda$. For that reason, in what follows, we put $\lambda=0$ and work only with the representation $\rho:=\rho_0$.
In order to apply the construction of [@Kashaev2019], we define the input R-matrix $$r:=\rho^{\otimes 2}(R)P$$ where $P\in \operatorname{Aut}(A^2)$ is the permutation operator acting by exchanging the arguments. By using , we obtain the following explicit action of $r$: $$\begin{gathered}
rf(z)= rf(z_0,z_1)=(1+\hbar)^{-z_0\frac{\partial}{\partial z_0}}f(z_1+\hbar z_0,z_0)\\
=f\Big(z_1+\frac{\hbar}{1+\hbar}z_0,\frac{1}{1+\hbar}z_0\Big)
=f({{U}^{\top}}z)$$ where $$\label{eq:U-matrix}
U :=\left(\begin{matrix}
\frac{\hbar}{1+\hbar}&\frac1{1+\hbar}\\
1&0
\end{matrix}\right)=\left(\begin{matrix}
1-t&t\\
1&0
\end{matrix}\right),\quad t:=\frac1{1+\hbar},$$ is the 2-by-2 matrix entering the definition of the (unrestricted) Burau representation of the braid groups [@MR3069652]. The action of $r$ on the coherent states is realized by the action of the transposed matrix on the space of parameters: $$\label{eq:r-acts-on-phi}
r\varphi_{v}(z)=\varphi_v({{U}^{\top}} z) =\varphi_{U v}(z).$$ In what follows, we use the indeterminate $t$ defined in terms of $\hbar$ through the formula in .
The diagrammatic rules for the Reshetikhin–Turaev functor
---------------------------------------------------------
From the formula , one calculates the integral kernel of $r$ with respect to the coherent states $$\langle \varphi_{w}|r|\varphi_{v}\rangle=\langle \varphi_{w_0,w_1}|r|\varphi_{v_0,v_1}\rangle=e^{w^*U v}$$ which corresponds to the value of the Reshetikhin–Turaev functor associated to positive crossings of all orientations in normal long knot diagrams with edges coloured by complex numbers: $$\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,<-] (0,1) to [out=-90,in=90] (1,0);
\draw[line width=3pt,white] (1,1) to [out=-90,in=90] (0,0);
\draw[thick,<-] (1,1) to [out=-90,in=90] (0,0);
\node (sw) at (0,-.1){\tiny $v_0$};\node (se) at (1,-.1){\tiny $v_1$};
\node (nw) at (0,1.1){\tiny $w_0$};\node (ne) at (1,1.1){\tiny $w_1$};
\end{tikzpicture},
\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,->] (1,1) to [out=-90,in=90] (0,0);
\draw[line width=3pt,white] (0,1) to [out=-90,in=90] (1,0);
\draw[thick,<-] (0,1) to [out=-90,in=90] (1,0);
\node (sw) at (0,-.1){\tiny $w_0$};\node (se) at (1,-.1){\tiny $v_0$};
\node (nw) at (0,1.1){\tiny $w_1$};\node (ne) at (1,1.1){\tiny $v_1$};
\end{tikzpicture},
\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,->] (0,1) to [out=-90,in=90] (1,0);
\draw[line width=3pt,white] (1,1) to [out=-90,in=90] (0,0);
\draw[thick,->] (1,1) to [out=-90,in=90] (0,0);
\node (sw) at (0,-.1){\tiny $w_1$};\node (se) at (1,-.1){\tiny $w_0$};
\node (nw) at (0,1.1){\tiny $v_1$};\node (ne) at (1,1.1){\tiny $v_0$};
\end{tikzpicture},
\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,<-] (1,1) to [out=-90,in=90] (0,0);
\draw[line width=3pt,white] (0,1) to [out=-90,in=90] (1,0);
\draw[thick,->] (0,1) to [out=-90,in=90] (1,0);
\node (sw) at (0,-.1){\tiny $v_1$};\node (se) at (1,-.1){\tiny $w_1$};
\node (nw) at (0,1.1){\tiny $v_0$};\node (ne) at (1,1.1){\tiny $w_0$};
\end{tikzpicture}\ \xmapsto{RT_r}\ \langle \varphi_{w_0,w_1}|r|\varphi_{v_0,v_1}\rangle$$ Likewise, the integral kernel of $r^{-1}$ given by the formula $$\langle \varphi_{w}|r^{-1}|\varphi_{v}\rangle=\langle \varphi_{w_0,w_1}|r^{-1}|\varphi_{v_0,v_1}\rangle=e^{w^*U^{-1} v}$$ is associated to negative crossings of all orientations: $$\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,<-] (1,1) to [out=-90,in=90] (0,0);
\draw[line width=3pt,white] (0,1) to [out=-90,in=90] (1,0);
\draw[thick,<-] (0,1) to [out=-90,in=90] (1,0);
\node (sw) at (0,-.1){\tiny $v_0$};\node (se) at (1,-.1){\tiny $v_1$};
\node (nw) at (0,1.1){\tiny $w_0$};\node (ne) at (1,1.1){\tiny $w_1$};
\end{tikzpicture},
\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,<-] (0,1) to [out=-90,in=90] (1,0);
\draw[line width=3pt,white] (1,1) to [out=-90,in=90] (0,0);
\draw[thick,->] (1,1) to [out=-90,in=90] (0,0);
\node (sw) at (0,-.1){\tiny $w_0$};\node (se) at (1,-.1){\tiny $v_0$};
\node (nw) at (0,1.1){\tiny $w_1$};\node (ne) at (1,1.1){\tiny $v_1$};
\end{tikzpicture},
\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,->] (1,1) to [out=-90,in=90] (0,0);
\draw[line width=3pt,white] (0,1) to [out=-90,in=90] (1,0);
\draw[thick,->] (0,1) to [out=-90,in=90] (1,0);
\node (sw) at (0,-.1){\tiny $w_1$};\node (se) at (1,-.1){\tiny $w_0$};
\node (nw) at (0,1.1){\tiny $v_1$};\node (ne) at (1,1.1){\tiny $v_0$};
\end{tikzpicture},
\begin{tikzpicture}[scale=1,baseline=10]
\draw[thick,->] (0,1) to [out=-90,in=90] (1,0);
\draw[line width=3pt,white] (1,1) to [out=-90,in=90] (0,0);
\draw[thick,<-] (1,1) to [out=-90,in=90] (0,0);
\node (sw) at (0,-.1){\tiny $v_1$};\node (se) at (1,-.1){\tiny $w_1$};
\node (nw) at (0,1.1){\tiny $v_0$};\node (ne) at (1,1.1){\tiny $w_0$};
\end{tikzpicture}\ \xmapsto{RT_r}\ \langle \varphi_{w_0,w_1}|r^{-1}|\varphi_{v_0,v_1}\rangle.$$ We complete the list of the diagrammatic rules by adding the rules for vertical segments and local extrema $$\begin{tikzpicture}[yscale=.5,baseline]
\draw[thick,->] (0,0) to [out=90,in=-90] (0,1);
\node (n) at (0,1.2){\tiny $w$};\node (s) at (0,-.2){\tiny $v$};
\end{tikzpicture}\ ,
\begin{tikzpicture}[yscale=.5,baseline]
\draw[thick,<-] (0,0) to [out=90,in=-90] (0,1);
\node (n) at (0,1.2){\tiny $v$};\node (s) at (0,-.2){\tiny $w$};
\end{tikzpicture}\ ,
\begin{tikzpicture}[xscale=1,baseline=0]
\draw[thick,<-] (0,0) to [out=90,in=90] (1,0);
\node (w) at (0,-.1){\tiny $w$};\node (e) at (1,-.1){\tiny $v$};
\end{tikzpicture}\ ,
\begin{tikzpicture}[xscale=1,baseline=25]
\draw[thick,<-] (0,1) to [out=-90,in=-90] (1,1);
\node (w) at (0,1.1){\tiny $w$};\node (e) at (1,1.1){\tiny $v$};
\end{tikzpicture}\ \xmapsto{RT_r} \ e^{\bar w v}$$ where $e^{\bar w v}$ is the integral kernel of the identity operator $\operatorname{id}_{A^1}$: $$\langle \varphi_{w}|\operatorname{id}_{A^1}|\varphi_{v}\rangle=\langle \varphi_{w}|\varphi_{v}\rangle=e^{\bar w v}.$$
For later use, we calculate the following two Reshetikhin–Turaev images $$\begin{gathered}
\label{eq:+cup}
\langle\varphi_w|RT_r\Big(
\begin{tikzpicture}[yscale=1,baseline=18]
\coordinate (a1) at (0,1);
\coordinate (a2) at (.75,.5);
\coordinate (a3) at (0.25,.5);
\coordinate (a4) at (1,1);
\draw[thick] (a2) to [out=-90,in=-90] (a3);
\draw[thick,->] (a3) to [out=90,in=-135] (a4);
\draw[line width=3pt,white] (a1) to [out=-45,in=90] (a2);
\draw[thick] (a1) to [out=-45,in=90] (a2);
\end{tikzpicture}
\Big)|\varphi_v\rangle=
RT_r\Big(
\begin{tikzpicture}[yscale=1,baseline=18]
\coordinate (a1) at (0,1);
\coordinate (a2) at (.75,.5);
\coordinate (a3) at (0.25,.5);
\coordinate (a4) at (1,1);
\node at (0,1.1) {\tiny$v$};
\node at (1,1.1) {\tiny$w$};
\draw[thick] (a2) to [out=-90,in=-90] (a3);
\draw[thick,->] (a3) to [out=90,in=-135] (a4);
\draw[line width=3pt,white] (a1) to [out=-45,in=90] (a2);
\draw[thick] (a1) to [out=-45,in=90] (a2);
\end{tikzpicture}
\Big)
=\int_{{\mathbb{C}}}\langle\varphi_{w,u}|r|\varphi_{v,u}\rangle\operatorname{d}\!\mu_1(u)\\
=\int_{{\mathbb{C}}}e^{\left(\begin{smallmatrix}
\bar w&\bar u
\end{smallmatrix}\right)
\left(\begin{smallmatrix}
1-t&t\\
1&0
\end{smallmatrix}\right)
\left(\begin{smallmatrix}
v\\
u
\end{smallmatrix}\right)
}\operatorname{d}\!\mu_1(u)
=\int_{{\mathbb{C}}}e^{
\bar w(1-t)v+\bar w tu+\bar uv
}\operatorname{d}\!\mu_1(u)=e^{\bar wv}\end{gathered}$$ and $$\begin{gathered}
\label{eq:-cup}
\langle\varphi_w|RT_r\Big(
\begin{tikzpicture}[yscale=1,baseline=18]
\coordinate (a1) at (0,1);
\coordinate (a2) at (.75,.5);
\coordinate (a3) at (0.25,.5);
\coordinate (a4) at (1,1);
\draw[thick] (a1) to [out=-45,in=90] (a2);
\draw[thick] (a2) to [out=-90,in=-90] (a3);
\draw[line width=3pt,white] (a3) to [out=90,in=-135] (a4);
\draw[thick,->] (a3) to [out=90,in=-135] (a4);
\end{tikzpicture}
\Big)|\varphi_v\rangle=
RT_r\Big(
\begin{tikzpicture}[yscale=1,baseline=18]
\coordinate (a1) at (0,1);
\coordinate (a2) at (.75,.5);
\coordinate (a3) at (0.25,.5);
\coordinate (a4) at (1,1);
\node at (0,1.1) {\tiny$v$};
\node at (1,1.1) {\tiny$w$};
\draw[thick] (a1) to [out=-45,in=90] (a2);
\draw[thick] (a2) to [out=-90,in=-90] (a3);
\draw[line width=3pt,white] (a3) to [out=90,in=-135] (a4);
\draw[thick,->] (a3) to [out=90,in=-135] (a4);
\end{tikzpicture}
\Big)
=\int_{{\mathbb{C}}}\langle\varphi_{w,u}|r^{-1}|\varphi_{v,u}\rangle\operatorname{d}\!\mu_1(u)\\
=\int_{{\mathbb{C}}}e^{\left(\begin{smallmatrix}
\bar w&\bar u
\end{smallmatrix}\right)
\left(\begin{smallmatrix}
0&1\\
t^{-1}&1-t^{-1}
\end{smallmatrix}\right)
\left(\begin{smallmatrix}
v\\
u
\end{smallmatrix}\right)
}\operatorname{d}\!\mu_1(u)
=\int_{{\mathbb{C}}}e^{
\bar w u+\bar ut^{-1}v+\bar u(1-t^{-1})u
}\operatorname{d}\!\mu_1(u)=te^{\bar wv}\end{gathered}$$ where the integrals are calculated by using the Gaussian integration formula .
Proof of Theorem \[thm-1\] {#sec-thm1}
==========================
Let $K$ be represented by the closure of a braid $\beta\in B_n$. Let us choose a normal long knot diagram $D_\beta$ representing $K$ according to the picture $$\label{pic:long-knot}
D_\beta=
\begin{tikzpicture}[baseline=-2]
\node (a) [hvector] at (0,0) {$D_\beta$};
\draw[thick,->] (a)--(0,0.5);
\draw[thick] (a)--(0,-0.5);
\end{tikzpicture}
=\
\begin{tikzpicture}[yscale=.5,baseline]
\draw [hvector](.3,0) rectangle (1.7,1);
\coordinate (a1) at (-.4,.5);
\coordinate (a2) at (.4,-.7);
\coordinate (a3) at (0.1,-.7);
\node (center) at (1,.5){ $\beta$};
\node (udots) at (.8,1.1){\tiny$\ldots$};
\node (bdots) at (.8,-.1){\tiny$\ldots$};
\node (cdots) at (-.7,.5){\tiny$\ldots$};
\draw[thick,->] (.5,1) to [out=120,in=90] (a1);
\draw[thick] (a2) to [out=-90,in=-90] (a3);
\draw[thick] (a3) to [out=90,in=-120] (0.5,0);
\draw[line width=3pt,white] (a1) to [out=-90,in=90] (a2);
\draw[thick] (a1) to [out=-90,in=90] (a2);
\coordinate (b1) at (-1,.5);
\coordinate (b2) at (.6,-2);
\coordinate (b3) at (0.3,-2);
\draw[thick,->] (1.3,1) to [out=120,in=90] (b1);
\draw[thick] (b2) to [out=-90,in=-90] (b3);
\draw[thick] (b3) to [out=90,in=-120] (1.3,0);
\draw[line width=3pt,white] (b1) to [out=-90,in=90] (b2);
\draw[thick] (b1) to [out=-90,in=90] (b2);
\draw[thick,->] (1.5,1) to [out=90,in=-90] (1.5,1.5);
\draw[thick] (1.5,-2) to [out=90,in=-90] (1.5,0);
\end{tikzpicture}$$ with the writhe $g(D_\beta)=g(\beta)+n-1$ which is an even number. Taking into account the value , writing the matrix $\psi_n(\beta)$ in the block form $$\psi_n(\beta)=
\begin{pmatrix}
\hat\beta_n&b_\beta\\
c_\beta& d_\beta
\end{pmatrix},$$ and using the general Gaussian integration formula , we calculate $$\begin{gathered}
\label{eq:rt_r-d_beta}
\langle \varphi_w| RT_r(D_\beta)| \varphi_v\rangle=RT_r\bigg(\begin{tikzpicture}[baseline=-3]
\node (a) [hvector] at (0,0) {$D_\beta$};
\node (up) [] at (0,0.65){\tiny$w$};
\node (do) [] at (0,-0.6){\tiny$v$};
\draw[thick,->] (a)--(up);
\draw[thick] (a)--(do);
\end{tikzpicture}\bigg)
=\int_{{\mathbb{C}}^{n-1}}e^{\left(\begin{smallmatrix}
u^*&\bar w
\end{smallmatrix}\right)\psi_n(\beta)\left(\begin{smallmatrix}
u\\
v
\end{smallmatrix}\right)}\operatorname{d}\!\mu_{n-1}(u)\\
=\int_{{\mathbb{C}}^{n-1}}e^{\left(\begin{smallmatrix}
u^*&\bar w
\end{smallmatrix}\right)\left(\begin{smallmatrix}
\hat\beta_n&b_\beta\\
c_\beta& d_\beta
\end{smallmatrix}\right)\left(\begin{smallmatrix}
u\\
v
\end{smallmatrix}\right)}\operatorname{d}\!\mu_{n-1}(u)\\
=
\int_{{\mathbb{C}}^{n-1}}e^{\bar wd_\beta v+\bar wc_\beta u+u^*b_\beta v+u^*\hat\beta_n u}\operatorname{d}\!\mu_{n-1}(u)
=\frac{e^{\bar wd_\beta v+\bar wc_\beta(I_{n-1}-\hat\beta_n)^{-1}b_\beta v}}{\det(I_{n-1}-\hat\beta_n)}.\end{gathered}$$ On the other hand, given the fact that we are calculating a central element realised by a scalar so that on à priori grounds the result should be proportional to the integral kernel of the identity operator $e^{\bar w v}$, we conclude that the identity $$d_\beta+c_\beta(I_{n-1}-\hat\beta_n)^{-1}b_\beta=1$$ is satisfied, a property of $\psi_n(\beta)$ which does not look to be easy to prove without passing through the Gaussian integration and referring to the universal invariant.
Finally, it remains to take into account the writhe correction, which, according to the values in and is given by the formula $$\label{eq:writhe-cor}
\langle \varphi_w| RT_r\big(\xi^{-g(D_\beta)/2}\big)| \varphi_v\rangle e^{-\bar w v} =t^{g(D_\beta)/2}=t^{(g(\beta)+n-1)/2}$$ where we use the notation $\xi^k$ from [@Kashaev2019] for a specific class of long knot diagrams used to compensate the writhe of the diagram. Putting together and , the result for the invariant $J_r(K)$ reads $$\langle \varphi_w|J_r(K)| \varphi_v\rangle e^{-\bar w v}=\frac{t^{(g(\beta)+n-1)/2} }{\det(I_{n-1}-\hat\beta_n)}=\frac1{\Delta_K(t)}$$ where the last equality is due to formula . Taking into account the relation between $\hbar$, $t$ and the realisation of the central element $a$ of $D({\mathsf{B}}_1)$ as well as the symmetry of the Alexander polynomial under the substitution $t\mapsto t^{-1}$, we conclude the proof.
Proof of Theorem \[thm-2\] {#sec-thm2}
==========================
In this section, we adopt the notation of [@MR2435235] and first briefly describe the unreduced and reduced Burau representations of the braid groups $B_n$ for $n\ge2$.
For any $k\ge 1$, denote by $I_k$ the identity $k\times k$ matrix. Let $$\psi_n\colon B_n\to\operatorname{GL}_n(\Lambda),\quad \Lambda:={\mathbb{Z}}[t^{\pm1}],$$ be the unrestricted Burau representation where Artin’s standard generators $\sigma_i$, $1\le i< n$ , are realised by the matrices $$\psi_n(\sigma_i)= U_i:= I_{i-1}\oplus U\oplus I_{n-i-1}.$$ For any $k\ge 1$, define the invertible upper triangular $k\times k$ matrix $$C_k=\sum_{1\le i\le j\le k}E_{i,j}=I_k+\sum_{1\le i< j\le k}E_{i,j}$$ where $E_{i,j}$ is the matrix with the only non-zero element 1 at the place $(i,j)$. Its inverse has the form $$C_k^{-1}=I_k-\sum_{i=1}^{n-1}E_{i,i+1}.$$ Indeed, one easily calculates $$C_k(I_k-\sum_{i=1}^{n-1}E_{i,i+1})=C_k-\sum_{1\le i< j\le k}E_{i,j}=I_k.$$ We remark on the block structure of $C_k^{\pm1}$: $$\label{eq:block-c_n}
C_k=\begin{pmatrix}
C_{k-1}&1_{k-1}\\
0_{k-1}^\top &1
\end{pmatrix},\quad C_k^{-1}=\begin{pmatrix}
C_{k-1}^{-1}&-C_{k-1}^{-1}1_{k-1}\\
0_{k-1}^\top &1
\end{pmatrix}=
\begin{pmatrix}
C_{k-1}^{-1}&
\begin{smallmatrix}
0_{k-2}\\
-1
\end{smallmatrix}
\\
0_{k-1}^\top &1
\end{pmatrix}$$ where $0_i$ (respectively $1_i$) is the column of length $i$ composed of $0$’s (respectively of $1$’s) and, in the last equality, we use the relation $$C_{k}^{-1}1_k=
\begin{pmatrix}
0_{k-1}\\
1
\end{pmatrix}.$$
As is shown in [@MR2435235], for any $\beta\in B_n$, one has the the equality $$C_n^{-1}\psi_n(\beta)C_n=
\begin{pmatrix}
\psi_n^r(\beta)&0_{n-1}\\
*_\beta &1
\end{pmatrix}$$ where $\psi_n^r\colon B_n\to \operatorname{GL}_{n-1}(\Lambda)$ is the reduced Burau representation, and $*_\beta$ is a row of length $n-1$ over $\Lambda$ linearly depending on the rows $a_i$, $1\le i\le n-1$, of the matrix $\psi_n^r(\beta)-I_{n-1}$ through the formula[^1] $$\label{eq:lin-dep-row}
(1-t^n)*_\beta=\sum_{i=1}^{n-1}(t^i-1) a_i.$$
\[lem-2\] Let $\hat\beta_n$ be the $(n-1)\times(n-1)$ matrix obtained from $\psi_n(\beta)$ by throwing away the $n$-th row and the $n$-th column. Then, one has the following equality in $\Lambda$: $$(t^{-n}-1) \det(\hat\beta_n-I_{n-1})=(t^{-1}-1)\det(\psi_n^r(\beta)-I_{n-1}).$$
We have the following equality of matrices: $$\begin{gathered}
\hat\beta_n=(C_{n-1}\psi_n^r(\beta)+1_{n-1}*_\beta)C_{n-1}^{-1}
\\
\Leftrightarrow\quad C_{n-1}^{-1}\hat\beta_nC_{n-1}=\psi_n^r(\beta)+C_{n-1}^{-1}1_{n-1}*_\beta
=
\psi_n^r(\beta)+\left(\begin{smallmatrix}
0_{n-2}\\
1
\end{smallmatrix}\right)*_\beta.
$$ One can verify this by explicit calculation based on the block structure : $$\begin{gathered}
\psi_n(\beta)=
\begin{pmatrix}
C_{n-1}&1_{n-1}\\
0_{n-1}^\top&1
\end{pmatrix}
\begin{pmatrix}
\psi_n^r(\beta)&0_{n-1}\\
*_\beta &1
\end{pmatrix} C_{n}^{-1}
\\
=\begin{pmatrix}
C_{n-1}\psi_n^r(\beta)+1_{n-1}*_\beta&*\\
*_\beta &1
\end{pmatrix}
\begin{pmatrix}
C_{n-1}^{-1}&*\\
0_{n-1}^\top &1
\end{pmatrix}\\
=\begin{pmatrix}
(C_{n-1}\psi_n^r(\beta)+1_{n-1}*_\beta)C_{n-1}^{-1}&*\\
* &1
\end{pmatrix}.\end{gathered}$$ Thus, $$\label{eq:det-id-1}
\det(\hat\beta_n-I_{n-1})=\det\left(\psi_n^r(\beta)-I_{n-1}+\left(\begin{smallmatrix}
0_{n-2}\\
1
\end{smallmatrix}\right)*_\beta\right)
=\det
\left(\begin{smallmatrix}
a_1\\
\vdots\\
a_{n-2}\\
a_{n-1}+*_\beta
\end{smallmatrix}
\right).$$ By multiplying both sides of by $(1-t^n)$ and using , we obtain $$\begin{gathered}
(1-t^n) \det(\hat\beta_n-I_{n-1})=\det\!\!
\left(\begin{smallmatrix}
a_1\\
\vdots\\
a_{n-2}\\
(1-t^n)( a_{n-1}+*_\beta)
\end{smallmatrix}
\right)
=\det\!\!
\left(\begin{smallmatrix}
a_1\\
\vdots\\
a_{n-2}\\
(1-t^n)a_{n-1}+\sum_{i=1}^{n-1}(t^i-1)a_i
\end{smallmatrix}
\right)\\
=\det\!\!
\left(\begin{smallmatrix}
a_1\\
\vdots\\
a_{n-2}\\
(t^{-1}-1)t^na_{n-1}
\end{smallmatrix}
\right)
=(t^{-1}-1)t^n\det(\psi_n^r(\beta)-I_{n-1})\end{gathered}$$ where in the third equality we dropped from the sum all the terms proportional to the rows different from $n-1$.
The formula for the Alexander polynomial proven in [@MR2435235 Theorem 3.13] is of the form $$\Delta_K(t)=(-1)^{n-1}t^{(n-1-g(\beta))/2}\frac{t-1}{t^n-1}\det(\psi_n^r(\beta)-I_{n-1})$$ which is equivalent to due to Lemma \[lem-2\].
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[^1]: This is the content of Lemma 3.10 of [@MR2435235] where the formula is written with a typo.
|
---
abstract: 'We consider configurations of $N$ charged particles on the interval with nearest neighbour Coulomb interaction and constant external force. For different values of external force we find 4 different phases of the asymptotic particle density for the configuration corresponding to the minimum of the energy.'
author:
- 'Malyshev V. A. [^1]'
title: 'Phase transitions in one-dimensional static Coulomb media'
---
Introduction
============
The problem of finding $N$ point particle configurations on a manifold having minimal energy (or even fixed configurations) was claimed important already long ago [@berkenbusch]. That is why we shall say shortly about the history of this question. We consider systems of particles with equal charges and Coulomb interaction. Immediately the problem is separated into two cases: when $N$ is small, where one should find such configurations explicitely, and the case of large $N$, where the asymptotics is of main interest. Already J. J. Thomson (discovering electron in 1897) suggested the problem of finding such configurations on the sphere, and the answer has been known for $N=2,3,4$ for more than 100 years, but for $N=5$ the solution was obtained only quite recently [@schwartz]. In one-dimensional case T, J, Stieltjss studied the problem with logarithmic interaction and found its connection with zeros of orthogonal polynomials on the corresponding interval, see [@chaitanya], [@ismail]. However, the problem of finding minimal energy configurations on two-dimensional sphere for any $N$ and power interaction (sometimes it is called the seventh problem of S. Smale, it is also connected with the names of F. Risz and M. Fekete) was completely solved only for quadratic interaction (see [@smale], [@dimitrov], [@kuijlaars] and review [@nerattini]). For more general compacts see review [@korevaar].
Here we follow alternative direction: namely, we study how the configuration could change in the presence of weak or strong external force. It appears that even in the simplified one-dimensional model with nearest neighbour interaction there is an interesting structure of fixed points (more exactly, fixed configurations), rich both in the number and in the charge distribution. For the constant force case we find 4 phases of the charge density, with respect to the parameter - the ratio of the constant of interaction strength and the value of external force.
We call Coulomb media the space of configurations $$-L\leq x_{N}<...<x_{1}<x_{0}\leq0$$ of $N+1$ point particles with equal charges on the segment $[-L,0]$. Here $N$ is assumed to be sufficiently large, however some results are valid for any $N\geq2$. We assume repulsive Coulomb interaction of nearest neighbours, and external force $\alpha_{ext}F_{0}(x)$, that is the potential energy is
$$U=\sum_{i=1}^{N}V(x_{i-1}-x_{i})-\sum_{i=0}^{N}\int_{-L}^{x_{i}}\alpha_{ext}F_{0}(x)dx,V(x)=\frac{\alpha_{int}}{|x|}\label{energy_U}$$
where $\alpha_{ext},\alpha_{int}$ are positive constants. This defines the dynamics of the system of charges, if one defines exactly what occurs with particles $0$ and $N$ in the points $0$ and $-L$ correspondingly. Namely, we assume completely inelastic boundary conditions. More exactly, when particle $x_{0}(t)$ at time $t$ reaches point $0$, having some velocity $v_{0}(t-0)\geq0$, then its velocity $v_{0}(t)$ immediately becomes zero, and the particle itself stays at point $0$ until the force acting on it (which varies accordingly to the motion of other particles) becomes negative. Similarly for the particle $x_{N}(t)$ at point $-L$.
To discover phase transitions it is common to consider asymptotics $N\to\infty$, with the parameters $L,l,F_{0}(x)$ being fixed. Then the fixed points will depend only on the “renormalized force” $F=\frac{\alpha_{ext}}{\alpha_{int}}F_{o}$, and we assume that the renormalized constant $\alpha_{ren}=\frac{\alpha_{ext}}{\alpha_{int}}$ can tend to infinity together with $N$, namely as $\alpha_{ren}=cN^{\gamma}$, where $c,\gamma>0$. It is eviodent that if $F_{0}\equiv0$, then for all $k=1,...,N$ $$\delta_{k}=x_{k-1}-x_{k}=\frac{L}{N}\label{F_ravno_0}$$ The case when $\alpha_{ren}$ does not depend on $N$ was discussed in detail in [@Mal-1], there are no phase transitions but it is discovered that the structure of the fixed configuration differs from (\[F\_ravno\_0\]) only on the sub-micro-scale of the order $N^{-2}$.
The necessity to consider cases when $\alpha_{ren}$ depends on $N$, issues from concrete examples where $\alpha_{ren}\gg N$. E.g. the linear density of electrons in some conductors, see [@Ashcroft], is of the order $N\approx10^{9}m^{-1}$, $\alpha_{int}=\frac{e^{2}}{\epsilon_{0}}\approx10^{-28}$ and $\alpha_{_{ext}}=220\frac{volt}{meter}e=220\times10^{-19}$ (in SI system). Thus $\alpha_{ren}$ has the order $10^{11}$. This is close to the critical point of our model, which, as will shown, is asymptotically $c_{cr}N$.
We study the density $\rho(x)$ (proving its existence), defined so that for any subintervals $I\subset[-L,0]$ there exist the limits $$\rho(I)=\int_{I}\rho(x)dx=\lim_{N\to\infty}\frac{\#\{i:x_{i}\in I\}}{N}$$ We find four phases: 1) uniform (constant) density, 2) nonuniform but positive smooth density, 3) continuous density, zero on some subinterval, 4) density of $\delta$-function type.
One-dimensional case shows what can be expected in multi-dimensional case, which is more complicated but has great interest in connection to the static charge distrubution in the atmosphere or in the live organism. For example case 4) of the theorem 2 is related to the discharge possibility, as after disappearance of large external force, the big concentration of charged particles can produce strong discharge.
### Main results {#main-results .unnumbered}
Assume that $F_{0}(x)$ is continuous, nonnegative and does not increase, that is $F(x)\leq F(y)$ if $x>y$. Then for any $N,L,\alpha_{ren}$ the fixed point exists and is unique. If $y$ is such that $F(x)=0,x\geq y,$ and $F(x)>0,x<y,$ then $\delta_{k+1}>\delta_{k}$, if $x_{k+1}<y$.
Further on we assume for simplicity that $F_{0}>0$ is uniform (constant in $x$).
(critical force)
For any $N,L$ there exists $F_{cr}$ such that for the fixed point the following holds: $x_{N}>-L$ for $F>F_{cr}$ and $x_{N}=-L$ for $F\leq F_{cr}$. If $F=cN^{\gamma},\gamma>1,$ then for any $c>0$ we have $x_{N}\to0$. If $F=cN$ then $F_{cr}\sim_{N\to\infty}c_{cr}N$, where $$c_{cr}=\frac{4}{L^{2}}\label{c_critical}$$
(four phases)
1. If $F=o(N)$, then the density exists and is strictly uniform, that is for all $k=1,...,N$ as $N\to\infty$ $$\max_{k}|(x_{k-1}-x_{k})-\frac{L}{N}|=o(\frac{1}{N})\label{th_2}$$
2. If $F=cN$ and $0<c\leq c_{cr}$, then $x_{N}=-L$ and the density of particles exists, is nowhere zero, but is not uniform (not constant in $x$);
3. If $F=cN$ and $c>c_{cr}$, then as $N\to\infty$ $$-L<x_{N}\to-\frac{2}{\sqrt{c}}\label{c_crit}$$ and the density on the interval $(-\frac{2}{\sqrt{c}},0)$ is not uniform;
4. If $F=cN^{\gamma},\gamma>1,$ then the density $\rho(x)\to\delta(x)$ in the sense of distributions.
### Uniqueness - proof of lemma 1 {#uniqueness---proof-of-lemma-1 .unnumbered}
Put $$f_{k}=\delta_{k}^{-2},k=1,;..,N.$$ At least one fixed point exists because the minimum of $U$ evidently exists. Any fixed point satisfies the following conditions $$x_{0}=0$$ - $$f_{k+1}+F(x_{k})=f_{k},k=1,...,N-1\label{fixed_point_cond_k}$$ However, for tha particle $N$ there are two possibilities: $$f_{N}\geq F(x_{N})\label{fixed_point_cond_N_more}$$ if $x_{N}=-L$, and $$f_{N}=F(x_{N})\label{f_equals_F}$$ if $x_{N}>-L$.
Forgetting for a while about fixed points, we will consider equations (\[fixed\_point\_cond\_k\]) as the equations uniquely defining (by induction in $k$) the functions $f_{k}$ of $\delta_{1}$, and thus $\delta_{k}=\frac{1}{\sqrt{f_{k}}}$ and also $x_{k}=-(\delta_{1}+...+\delta_{k})$. It is evident that $f_{k}$ and $x_{k}$ are decreasing, and $\delta_{k}$ are increasing functions of $\delta_{1}$. Moreover, if $\delta_{1}\to0$ then all $f_{k}\to\infty$, and $\delta_{k}$ and $x_{k}$ tend to $0$, then for $\delta_{1}$ sufficiently small the inequality (\[fixed\_point\_cond\_N\_more\]) holds. Thus, if $\delta_{1}$ increases, two cases are possible: 1) there exists $\delta_{1,final}$ such that $$F(x_{N})=f_{N},x_{N}>-L,$$ At the same time if $\delta_{1}>\delta_{1,final}$ then $F(x_{N})$ and $\delta_{N}$ increase as functions $\delta_{1}$, and $f_{N}$ decreases. that is why $F(x_{N})>f_{N}$. It follows that in this case there are no other fixed points; 2) such $\delta_{1}$ does not exist, but then for some $\delta_{1}$ we have $$x_{N}=-L,F(x_{N})\leq f_{N}$$ This defines the unique fixed point.
### Note about nonuniqueness {#note-about-nonuniqueness .unnumbered}
The monotonicity assumption in the uniqueness lemma is very essential. One can give an example of nonuniqueness, for a function $F_{0}(x)$ with the only maximum, where the number of fixed points is of the order of $N$ or more. Namely, on the interval $[-1,1]$ put for $b>a>0$ $$F_{0}(x)=a-2ax,x\geq0$$ $$F_{0}(x)=a+2bx,x\leq0$$ Then there exists $C_{cr}>0$ such that for all sufficiently large $N$ and $\alpha_{ren}=cN,c>C_{cr}$, one can show using similar techniques that for any odd $N_{1}<N$ there exists fixed point such that $$-1=x_{N}<...x_{N_{1}}<0<x_{N_{1}-1}<...<x_{\frac{N_{1}+!}{2}}=\frac{1}{2}<...<x_{0}<1$$ Moreover, any such point will give local minimum of the energy.
### Critical force - proof of theorems 1 and 2.4 {#critical-force---proof-of-theorems-1-and-2.4 .unnumbered}
In case of constant positive force it follows from (\[fixed\_point\_cond\_k\]) that $$f_{i}>f_{i+1}\Longleftrightarrow\delta_{i}<\delta_{i+1},i=1,...,N-1,\label{increase-1}$$ that is the lengths $\delta_{i}$ of intervals strictly increase with $i$. That is why $$\delta_{1}<\frac{L}{N}\label{delta_1_less}$$ Summation (\[fixed\_point\_cond\_k\]) over $i=1,...,k-1$ gives that for any $k=1,...,N$, $$f_{k}=f_{1}-(k-1)F,k=1,...,N\label{f_k_1-1}$$ Similarly to (\[f\_k\_1-1\]), summing over $i=N-1,...k-1,$ we get $$f_{k}=f_{N}+(N-k)F\label{f_k_N-1}$$ Then from (\[f\_k\_1-1\]) we get $$\delta_{k}=(\delta_{1}^{-2}-(k-1)F)^{-\frac{1}{2}}=\delta_{1}(1-\delta_{1}^{2}(k-1)F)^{-\frac{1}{2}}\label{delta_k-1}$$ and (as the fixed point exists) $$1-\delta_{1}^{2}(k-1)F>0\label{bound_1}$$ or $$\delta_{1}<(\frac{1}{(N-1)F})^{\frac{1}{2}}\label{delta_1_2-1}$$
To prove theorem 1 consider a simpler auxiliary model with $L=\infty$. That is we assume that the particles are situated on the interval $(-\infty,0]$, and the force $F$ is constant on all $(-\infty,0]$. In this model for any $F>0$ there is unique fixed point, given explicitely $$f_{N}=F,f_{k}=(N-k+1)F,k=N-1,...,1$$ which follows from (\[f\_k\_N-1\]). From this we get $$\delta_{N}=F^{-\frac{1}{2}},\delta_{k}=\frac{1}{\sqrt{(N-k+1)F}}\label{delta_k_simpler-1}$$ and $$-x_{N}=\sum_{k=1}^{N}\delta_{k}=\frac{1}{\sqrt{F}}\sum_{k=1}^{N}\frac{1}{\sqrt{k}}$$ The relation of this model to the initial is quite simple. If $S\leq L$, then the fixed ploints for both models coincide. If $S\geq L$, then $x_{N}=-L$. In fact, assuming that for the critical point in the main model $x_{N}>-L$, we get contradiction with the auxiliary model. That is why the critical force can be found from the condition that $x_{N}=-L$ in the auxiliary model, that is $$F=F_{cr}=(\frac{1}{L}\sum_{k=1}^{N}\frac{1}{\sqrt{k}})^{2}\sim_{N\to\infty}(\frac{2}{L})^{2}N$$ One can also say that for any $x<0$ there exists unique $F=F_{x}$ such that $x_{N}=x$.
For $F=cN,c>c_{cr},$ we have $$-x_{N}=L=\sum_{k=1}^{N}\delta_{k}=\frac{1}{\sqrt{F}}\sum_{k=1}^{N}\frac{1}{\sqrt{k}}\sim\frac{2}{\sqrt{c}}$$ from where (\[c\_critical\]) follows, this gives Theorem 1. Similarly the theorem 2.4 follows as $x_{N}\to0$ if $F=cN^{\gamma},\gamma>1$.
### Nonuniform density - proof of theorems 2.2 and 2.3 {#nonuniform-density---proof-of-theorems-2.2-and-2.3 .unnumbered}
Firstly, consider the case $F=cN,c>c_{cr}$. Then for $k=aN$ we have from (\[delta\_k\_simpler-1\]) $$\delta_{k}=\frac{1}{\sqrt{(N-k+1)F}}\sim\frac{1}{\sqrt{(1-a)c}}\frac{1}{N}$$ That is why the density exists, and moreover it equals zero on $[-L,x_{N}]$ and is nonuniform $[x_{N},0]$.
Let now $c\leq c_{cr}$. One can assume $\delta_{1}=b\frac{L}{N},0<b=b(N)\leq1$. Then for $k=aN,a<1,$ we get from (\[delta\_k-1\]) $$\delta_{k}=b\frac{L}{N}(1-L^{2}cb^{2}\frac{k-1}{N})^{-\frac{1}{2}}\sim b\frac{L}{N}(1-b^{2}cL^{2}a)^{-\frac{1}{2}}$$ Thus, the density is not uniform.
### Uniform density - proof of theorem 2.1 {#uniform-density---proof-of-theorem-2.1 .unnumbered}
From (\[delta\_1\_less\]) we have
$$\delta_{1}^{2}(k-1)F\leq\delta_{1}^{2}(N-1)F=o(1)\label{delta_1_cube}$$
Then $$L=\sum_{k=1}^{N}\delta_{k}=\delta_{1}\sum_{k=1}^{N}(1-\delta_{1}^{2}(k-1)F)^{-\frac{1}{2}}=\delta_{1}\sum_{k=1}^{N}(1+\frac{1}{2}\delta_{1}^{2}(k-1)F+O((\delta_{1}^{2}(k-1)F)^{2})=$$ $$=N\delta_{1}+\frac{1}{4}\delta_{1}^{3}FN^{2}+o(\delta_{1}^{3}FN^{2})\label{L_sum_1-1}$$ But by (\[delta\_1\_cube\]) we have $$\delta_{1}^{3}FN^{2}=o(N\delta_{1})$$ and that is why $$\delta_{1}=\frac{L}{N}+o(\frac{L}{N})$$ The result for all $k$ follows from (\[delta\_k-1\]).
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[^1]: Faculty of Mechanics and Mathematics, Moscow State University
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---
abstract: 'Alfvén waves are a particular class of magnetohydrodynamic waves relevant in many astrophysical and laboratory plasmas. In partially ionized plasmas the dynamics of Alfvén waves is affected by the interaction between ionized and neutral species. Here we study Alfvén waves in a partially ionized plasma from the theoretical point of view using the two-fluid description. We consider that the plasma is composed of an ion-electron fluid and a neutral fluid, which interact by means of particle collisions. To keep our investigation as general as possible we take the neutral-ion collision frequency and the ionization degree as free parameters. First, we perform a normal mode analysis. We find the modification due to neutral-ion collisions of the wave frequencies and study the temporal and spatial attenuation of the waves. In addition, we discuss the presence of cut-off values of the wavelength that constrain the existence of oscillatory standing waves in weakly ionized plasmas. Later, we go beyond the normal mode approach and solve the initial-value problem in order to study the time-dependent evolution of the wave perturbations in the two fluids. An application to Alfvén waves in the low solar atmospheric plasma is performed and the implication of partial ionization for the energy flux is discussed.'
author:
- 'R. Soler$^{1}$, M. Carbonell$^{2}$, J. L. Ballester$^{1}$, & J. Terradas$^{1}$'
bibliography:
- 'refs.bib'
title: 'Alfvén waves in a partially ionized two-fluid plasma'
---
Introduction
============
Alfvén waves are a particular class of magnetohydrodynamic (MHD) waves driven by magnetic tension [@alfven1942]. In a uniform and infinite plasma the motions of Alfvén waves are incompressible and polarized perpendicularly to the direction of the magnetic field [see, e.g., @hasegawa1982; @cramer2001; @goossens2003]. Alfvén waves are found in both laboratory and astrophysical plasmas [see review by @gekelman11].
Since the pioneering works by, e.g., @piddington1956 and @kulsrud1969 it is known that partial ionization of the plasma affects the dynamics of Alfvén waves. The feature most extensively investigated in the literature is the wave damping due to collisions between ions and neutrals, although other effects as, e.g., the existence of cut-off values of the wavelength are also an important consequence of partial ionization [@kulsrud1969].
Most of the works that studied Alfvén waves in partially ionized plasmas adopted the so-called single-fluid approximation [see, e.g., @brag]. The single-fluid approximation assumes a strong coupling between ions and neutrals. For an MHD wave, this condition means that the wave frequency has to be much lower than the frequency at which ion and neutrals collide. In other words, it is necessary that in one period there are enough collisions for ions and neutrals to behave as one fluid. This restriction is fulfilled in, e.g., the partially ionized solar plasma and so the single-fluid approximation is usually adopted in that case [see, e.g., @depontieu2001; @khodachenko2004; @forteza2007; @soler2009PI among others].
An alternative approach is the multi-fluid theory [see, e.g., @zaqarashvili2011a], which considers the various species in the plasma as separate fluids. In the multi-fluid description no restriction is imposed on the relative values of the wave frequency and the collision frequency, although the mathematical treatment is usually more complicated than in the single-fluid case. However the multi-fluid theory has the advantage that it is more general than the single-fluid approximation and can be used regardless the value of the collision frequency. Hence the multi-fluid theory is adequate to study MHD waves in those situations where the single-fluid approximation does not apply. This may be the case of molecular clouds [see, e.g., @pudritz1990; @balsara1996; @mouschovias2011]. A particular form of the multi-fluid theory is the two-fluid theory in which ions and electrons are considered together as an ion-electron fluid, while neutrals form another fluid that interacts with the ion-electron fluid by means of collisions. For the investigation of MHD waves this approach was followed by, e.g., @kumar2003 [@zaqarashvili2011a; @mouschovias2011; @soler2012].
Despite the existing literature on this topic (see the references in the above paragraphs), the purpose of the present article is to revisit the theoretical investigation of Alfvén waves in partially ionized plasmas using the two-fluid theory. Our reasons for tackling this task are the following.
First of all, the existing papers in the literature often focus on very specific situations as, e.g., the interstellar medium [e.g., @kulsrud1969], molecular clouds [see, e.g., @pudritz1990; @balsara1996; @mouschovias2011], and solar plasmas [@kumar2003; @zaqarashvili2011a; @soler2012] among other cases. Here our aim is to keep the investigation as general as possible. To do so we take the neutral-ion collision frequency and the plasma ionization degree as free parameters. This makes the results of the present article to be widely applicable. We put emphasis on the mathematical transparency and on the finding of approximate analytic solutions.
In addition, there has been recently some confusion about the existence of cut-off wavelengths for Alfvén waves in a partially ionized plasma [see @zaqarashvili2011a; @zaqarashvili2012]. While @zaqarashvili2012 have shown that the presence of cut-offs in the single-fluid approximation is a mathematical artifact, the existence of cut-offs in the two-fluid case is a real physical phenomenon [e.g., @kulsrud1969; @pudritz1990; @kamaya1998; @mouschovias2011]. An important goal of the present article is to stress the existence of physical cut-offs for Alfvén waves in a two-fluid plasma.
Finally, unlike previous works that are restricted to the normal mode analysis, here we combine results of normal modes with the solution of the initial-value problem. This procedure allows us to investigate how strong is the coupling between the perturbations in the ionized fluid and the neutral fluid depending on the relative values of the wave frequency and the neutral-ion collision frequency.
This paper is organized as follows. Section \[sec:equi\] contains the description of the equilibrium configuration and the basic equations of the two-fluid theory. Alfvén waves are investigated following a normal mode analysis in Section \[sec:normal\], while the initial-value problem is solved in Section \[sec:initial\]. Later, Section \[sec:app\] contains an application to the solar atmospheric plasma and Section \[sec:energy\] discusses the implications of partial ionization for the energy flux of Alfvén waves. Finally the conclusions of this work are given in Section \[sec:con\].
Equilibrium and basic equations {#sec:equi}
===============================
We consider a partially ionized medium composed of ions, electrons, and neutrals. We use the two-fluid theory in which ions and electrons are considered together as an ion-electron fluid, i.e., the ionized fluid, while neutrals form another fluid that interacts with the ionized fluid by means of collisions [see, e.g., @zaqarashvili2011a; @soler2012]. In all the following expressions, the subscripts ‘i’ and ‘n’ refer to the ionized fluid and the neutral fluid, respectively.
The equilibrium is made of a uniform and unbounded partially ionized plasma. We use Cartesian coordinates. The equilibrium magnetic field is straight and constant along the $z$-direction, namely ${\bf B}=B\, \hat{z}$. We ignore the effect of gravity. We also assume that the equilibrium is static so that there are no equilibrium flows. The governing equations for the various species composing the plasma can be found in, e.g., @zaqarashvili2011a. Here we restrict ourselves to the study of linear perturbations superimposed on the equilibrium state. Hence, the governing equations are linearized. The resulting equations are $$\begin{aligned}
{\rho_{\rm i}}\frac{{\partial}{\bf v}_{\rm i}}{\partial t} &=& - \nabla p_{\rm i} + \frac{1}{\mu}\left( \nabla \times {\bf b} \right) \times {\bf B} -{\alpha_{\rm in}}\left( {\bf v}_{\rm i} - {\bf v}_{\rm n}\right), \label{eq:momlinion}\\
{\rho_{\rm n}}\frac{{\partial}{\bf v}_{\rm n}}{\partial t} &=& - \nabla p_{\rm n} -{\alpha_{\rm in}}\left( {\bf v}_{\rm n} - {\bf v}_{\rm i}\right), \label{eq:momlinneu}\\
\frac{{\partial}{\bf b}}{\partial t} &=& \nabla \times \left( {\bf v}_{\rm i} \times {\bf B} \right), \label{eq:inductionlin} \\
\frac{\partial p_{\rm i}}{\partial t} &=& - \gamma P_{\rm i} \nabla \cdot {\bf v}_{\rm i}, \label{eq:presslinion} \\
\frac{\partial p_{\rm n}}{\partial t} &=& - \gamma P_{\rm n} \nabla \cdot {\bf v}_{\rm n}, \label{eq:presslin} \end{aligned}$$ where ${\bf v}_{\rm i}$, $p_{\rm i}$, $P_{\rm i}$ and ${\rho_{\rm i}}$ are the velocity perturbation, pressure perturbation, equilibrium pressure, and equilibrium density of the ionized fluid, ${\bf v}_{\rm n}$, $p_{\rm n}$, $P_{\rm n}$, and ${\rho_{\rm n}}$ are the respective quantities but for the neutral fluid, ${\bf b}$ is the magnetic field perturbation, $\mu$ is the magnetic permeability, $\gamma$ is the adiabatic index, and ${\alpha_{\rm in}}$ is the ion-neutral friction coefficient. In the specific case of a hydrogen plasma, the expression of ${\alpha_{\rm in}}$ is given by @brag, namely $${\alpha_{\rm in}}= \frac{1}{2} \frac{{\rho_{\rm i}}{\rho_{\rm n}}}{m_{\rm n}}\sqrt{\frac{16 k_{\rm B} T}{\pi m_{\rm i}}}\sigma_{\rm in}, \label{eq:ainhydrogen}$$ where $m_{\rm i}$ and $m_{\rm n}$ are the ion and neutral masses, respectively ($m_{\rm i}\approx m_{\rm n}$ for hydrogen), $k_{\rm B}$ is Boltzmann’s constant, $T$ is the plasma temperature, and $\sigma_{\rm in}$ is the collision cross section. In the following analysis we do not use this expression of ${\alpha_{\rm in}}$ since we take ${\alpha_{\rm in}}$ as a free parameter. We do so to conveniently control the strength of the ion-neutral friction force. Equation (\[eq:ainhydrogen\]) is used in the application to solar plasmas done in Section \[sec:app\].
We perform a Fourier analysis of the perturbations in space. In linear theory, an arbitrary perturbation can be represented by the superposition of Fourier components, so that we can restrict ourselves to study particular Fourier components. Therefore, the spatial dependence of perturbations is put proportional to $\exp \left(i k_x x + ik_y y + i k_z z \right)$, where $k_x$, $k_y$, and $k_z$ are the components of the wavenumber in the $x$-, $y$-, and $z$-directions, respectively. In a uniform and infinite plasma Alfvén waves are the only MHD modes that propagate vorticity perturbations [see, e.g., @cramer2001; @goossens2003]. In addition, Alfvén waves are incompressible and their motions are confined to perpendicular planes to the magnetic field, i.e., $v_{{\rm i}, z} = v_{{\rm n}, z} = 0$. Therefore, an appropriate quantity to describe Alfvén waves is the vorticity component along the magnetic field direction. By working with vorticity perturbations we are able to decouple Alfvén waves from magnetoacoustic waves. We define ${\Gamma_{\rm i}}$ and ${\Gamma_{\rm n}}$ as the $z$-components of vorticity of the ionized fluid and the neutral fluid, respectively, $$\begin{aligned}
{\Gamma_{\rm i}}&=& \left( \nabla \times {\bf v}_{\rm i} \right) \cdot \hat{z} = i k_x v_{{\rm i}, y} - i k_y v_{{\rm i}, x}, \label{eq:defgi} \\
{\Gamma_{\rm n}}&=& \left( \nabla \times {\bf v}_{\rm n} \right) \cdot \hat{z} = i k_x v_{{\rm n},y} - i k_y v_{{\rm n},x}.\label{eq:defgn}\end{aligned}$$ Note that in the reference frame in which $k_y=0$, ${\Gamma_{\rm i}}$ and ${\Gamma_{\rm n}}$ are proportional to $v_{{\rm i}, y}$ and $v_{{\rm n}, y}$, respectively. We combine Equations (\[eq:momlinion\])–(\[eq:presslin\]) and after some algebraic manipulations we obtain the two following equations involving ${\Gamma_{\rm i}}$ and ${\Gamma_{\rm n}}$ only, namely $$\begin{aligned}
{\rho_{\rm i}}\frac{{\partial}^2 {\Gamma_{\rm i}}}{{\partial}t^2} + {\alpha_{\rm in}}\frac{{\partial}{\Gamma_{\rm i}}}{{\partial}t} + {\rho_{\rm i}}k_z^2 {c_{\mathrm{A}}}^2 {\Gamma_{\rm i}}&=& {\alpha_{\rm in}}\frac{{\partial}{\Gamma_{\rm n}}}{{\partial}t}, \label{eq:vorti} \\
{\rho_{\rm n}}\frac{{\partial}{\Gamma_{\rm n}}}{{\partial}t} + {\alpha_{\rm in}}{\Gamma_{\rm n}}&=& {\alpha_{\rm in}}{\Gamma_{\rm i}}, \label{eq:vortn}\end{aligned}$$ where ${c_{\mathrm{A}}}= B/\sqrt{\mu {\rho_{\rm i}}}$ is the Alfvén velocity. Note that the Alfvén velocity is here defined using the density of the ionized fluid only. Equations (\[eq:vorti\]) and (\[eq:vortn\]) are the governing equations for linear vorticity perturbations and, therefore, they are the governing equations of Alfvén waves.
For the subsequent analysis we define the ionization fraction, $\chi$, the ion-neutral collision frequency, $\nu_{\rm in}$, and the neutral-ion collision frequency, $\nu_{\rm ni}$, as follows $$\chi = \frac{{\rho_{\rm n}}}{{\rho_{\rm i}}}, \qquad \nu_{\rm in} = \frac{{\alpha_{\rm in}}}{{\rho_{\rm i}}}, \qquad \nu_{\rm ni} = \frac{{\alpha_{\rm in}}}{{\rho_{\rm n}}}.$$ Since the collision frequencies are related by ${\rho_{\rm i}}\nu_{\rm in} = {\rho_{\rm n}}\nu_{\rm ni}$, we use $\nu_{\rm ni}$ in all the following expressions for simplicity. Note that when ${\rho_{\rm i}}\neq {\rho_{\rm n}}$, $\nu_{\rm in} \neq \nu_{\rm ni}$, meaning that the ion-neutral and neutral-ion collision frequencies are different [see a discussion on this issue in @zaqarashvili2011helium].
Normal Mode Analysis {#sec:normal}
====================
Here we perform a normal mode analysis. The temporal dependence of the perturbations is put proportional to $\exp\left( -i\omega t \right)$, where $\omega$ is the angular frequency. From Equation (\[eq:vortn\]) we express ${\Gamma_{\rm n}}$ in terms of ${\Gamma_{\rm i}}$ and insert the expression in Equation (\[eq:vorti\]). We arrive at an equation involving ${\Gamma_{\rm i}}$ only, namely $$\mathcal{D}\left( \omega, k_z \right) {\Gamma_{\rm i}}= 0, \label{eq:gammai0}$$ with $$\mathcal{D}\left( \omega, k_z \right) = \omega^3 + i \left( 1+\chi \right) {\nu_{\mathrm{ni}}}\omega^2 - k_z^2 c_{\rm A}^2 \omega - i {\nu_{\mathrm{ni}}}k_z^2 c_{\rm A}^2. \label{eq:relalfpre}$$ For ${\Gamma_{\rm i}}\neq 0$, the solutions to Equation (\[eq:gammai0\]) must satisfy $\mathcal{D}\left( \omega, k_z \right) = 0$, i.e., $$\omega^3 + i \left( 1+\chi \right) {\nu_{\mathrm{ni}}}\omega^2 - k_z^2 c_{\rm A}^2 \omega - i {\nu_{\mathrm{ni}}}k_z^2 c_{\rm A}^2=0. \label{eq:relalf}$$ Equation (\[eq:relalf\]) is the dispersion relation of Alfvén waves. Although with different notations, Equation (\[eq:relalf\]) is equivalent to the dispersion relations previously found by, e.g., @piddington1956 [@kulsrud1969; @pudritz1990; @martin1997; @kamaya1998; @kumar2003; @zaqarashvili2011a; @mouschovias2011]. In the absence of collisions, ${\nu_{\mathrm{ni}}}=0$ and Equation (\[eq:relalf\]) becomes $$\omega \left( \omega^2-k_z^2{c_{\mathrm{A}}}^2 \right) = 0. \label{eq:relalfideal}$$ From Equation (\[eq:relalfideal\]) we get the classic dispersion relation of Alfvén waves in an ideal plasma, namely $ \omega^2=k_z^2{c_{\mathrm{A}}}^2 $, and an additional mode with $\omega=0$. The general situation ${\nu_{\mathrm{ni}}}\neq 0$ is investigated next.
Standing waves
--------------
We focus first on standing waves. Hence we assume a real wavenumber, $k_z$, and solve the dispersion relation (Equation (\[eq:relalf\])) to obtain the complex frequency, $\omega= \omega_{\rm R} + i \omega_{\rm I}$, with $\omega_{\rm R}$ and $\omega_{\rm I}$ the real and imaginary parts of $\omega$, respectively. Since $\omega$ is complex the amplitude of perturbations is multiplied by the factor $\exp(\omega_{\rm I} t)$, with $\omega_{\rm I} < 0$. Therefore the perturbations are damped in time.
Equation (\[eq:relalf\]) is a cubic equation so it has three solutions. Unfortunately the exact analytic solution to Equation (\[eq:relalf\]) is too complicated to shed any light on the physics. However we can investigate the nature of the solutions using the concept of the polynomial discriminant. We perform the change of variable $\omega = -i s$, so that Equation (\[eq:relalf\]) becomes $$s^3 + \left( 1+\chi \right) {\nu_{\mathrm{ni}}}s^2 + k_z^2 c_{\rm A}^2 s + {\nu_{\mathrm{ni}}}k_z^2 c_{\rm A}^2 = 0. \label{eq:relalf1}$$ Equation (\[eq:relalf1\]) is a cubic equation and all its coefficients are real. From Equation (\[eq:relalf1\]) we compute the discriminant, $\Lambda$, namely [see, e.g., @cohen2000] $$\begin{aligned}
\Lambda = &-& k_z^2 c_{\rm A}^2 \left[ 4\left(1+\chi \right)^3 {\nu_{\mathrm{ni}}}^4 \right. \nonumber \\
&-& \left. \left( \chi^2 + 20\chi -8\right){\nu_{\mathrm{ni}}}^2 k_z^2 c_{\rm A}^2 +4 k_z^4 c_{\rm A}^4\right],
\label{eq:relalf2u}\end{aligned}$$ The discriminant, $\Lambda$, is defined so that (i) Equation (\[eq:relalf1\]) has one real zero and two complex conjugate zeros when $\Lambda <0$, (ii) Equation (\[eq:relalf1\]) has a multiple zero and all the zeros are real when $\Lambda = 0$, and (iii) Equation (\[eq:relalf1\]) has three distinct real zeros when $\Lambda >0$. This classification is very relevant because the complex zeros of Equation (\[eq:relalf1\]) result in damped oscillatory solutions of Equation (\[eq:relalf\]) whereas the real zeros of Equation (\[eq:relalf1\]) correspond to evanescent solutions of Equation (\[eq:relalf\]).
It is instructive to consider again the paradigmatic situation in which there are no collisions between the two fluids, so we set ${\nu_{\mathrm{ni}}}=0$. The discriminant becomes $\Lambda= -4k_z^6 c_{\rm A}^6<0$, which means that Equation (\[eq:relalf1\]) has one real zero and two complex conjugate zeros. Indeed, when ${\nu_{\mathrm{ni}}}=0$ the zeros of Equation (\[eq:relalf1\]) are $$s = \pm i k_z {c_{\mathrm{A}}}, \qquad s=0,$$ which correspond to the following values of $\omega$, $$\omega = \pm k_z{c_{\mathrm{A}}}, \qquad \omega=0. \label{eq:uncoupled}$$ The two non-zero solutions correspond to the ideal Alfvén frequency, as expected.
We go back to the general case ${\nu_{\mathrm{ni}}}\neq 0$. To determine the location where the nature of the solutions changes we set $\Lambda = 0$ and find the corresponding relation between the various parameters. For given ${\nu_{\mathrm{ni}}}$ and $\chi$ we find two different values of $k_z$, denoted by $k_z^+$ and $k_z^-$, which satisfy $\Lambda = 0$, namely $$k_z^\pm = \frac{\nu_{\rm ni}}{c_{\rm A}}\left[\frac{\chi^2+20\chi-8}{8\left( 1+\chi \right)^3} \pm \frac{\chi^{1/2} \left(\chi-8 \right)^{3/2}}{8\left( 1+\chi \right)^3}\right]^{-1/2}. \label{eq:rmasmenos}$$ Since $k_z$ must be real, Equation (\[eq:rmasmenos\]) imposes a condition on the minimum value of $\chi$ which allows $\Lambda = 0$. This minimum value is $\chi = 8$ and the corresponding critical $k_z$ is $k_z^+ = k_z^- = 3\sqrt{3} {\nu_{\mathrm{ni}}}/{c_{\mathrm{A}}}$. When $\chi > 8$, Equation (\[eq:rmasmenos\]) gives $k_z^+ < k_z^-$. For $k_z$ outside the interval $(k_z^+,k_z^-)$ we have $\Lambda <0$ so that there are two propagating Alfvén waves and one evanescent solution. For $k_z\in(k_z^+,k_z^-)$ we have $\Lambda >0$ and so all three zeros of Equation (\[eq:relalf1\]) are real, i.e., they correspond to purely imaginary solutions of Equation (\[eq:relalf\]). There is no propagation of Alfvén waves for $k_z\in(k_z^+,k_z^-)$. We call this interval the cut-off region. To the best of our knowledge, @kulsrud1969 were the first to report on the existence of a cut-off region of wavenumbers for Alfvén waves in a partially ionized two-fluid plasma, when studying the propagation of cosmic rays. These cut-offs also appear in the works by, e.g., @pudritz1990 [@kumar2003; @mouschovias2011]. The cut-off wavenumbers were ignored by @zaqarashvili2011a, who stated that there is always a solution of Equation (\[eq:relalf\]) with a non-zero real part. @zaqarashvili2011a probably reached this wrong conclusion because they never took $\chi > 8$ in their computations. Here we clearly see that the three solutions of Equation (\[eq:relalf\]) are purely imaginary when $\chi>8$ and $k_z\in(k_z^+,k_z^-)$.
@kamaya1998 discussed the physical reason for the existence of a range of cut-off wavenumbers in weakly ionized plasmas [see also @mouschovias1987]. When $k_z > k_z^-$ magnetic tension drives ions to oscillate almost freely, since the friction force is not strong enough to transfer significant inertia to neutrals. In this case, disturbances in the magnetic field affect only the ionized fluid as happens for classic Alfvén waves in fully ionized plasmas. Conversely, when $k_z < k_z^+$ the ion-neutral friction is efficient enough for neutrals to be nearly frozen into the magnetic field. After a perturbation, neutrals are dragged by ions almost instantly and both species oscillate together as a single fluid. The intermediate situation occurs when $k_z\in(k_z^+,k_z^-)$. In this case, a disturbance in the magnetic field decays due to friction before the ion-neutral coupling has had time to transfer the restoring properties of magnetic tension to the neutral fluid. In other words, neutral-ion collisions are efficient enough to dissipate perturbations in the magnetic field but, on the contrary, they are not efficient enough to transfer significant inertia to neutrals before the magnetic field perturbations have decayed. Hence, oscillations of the magnetic field are suppressed when $k_z\in(k_z^+,k_z^-)$. Additional insight on the physical behavior of the perturbations near the cut-off region is given in Section \[sec:compar\] by analyzing the forces acting on the fluids.
To avoid confusion we must inform the reader that the existence of cut-off wavenumbers of Alfvén waves discussed above is a purely two-fluid effect. These cut-off wavenumbers are not the cut-offs obtained in the single-fluid approximation [see, e.g., @balsara1996; @forteza2007; @soler2009PI; @barcelo2011]. @zaqarashvili2012 have shown that the cut-off wavenumbers found in the single-fluid approximation are a mathematical artifact, i.e., they are caused by the approximations made when proceeding from the multi-fluid equations to single-fluid equations and are not connected to any real physical process. On the contrary, the cut-off wavenumbers found in the two-fluid case are physically and mathematically real and are caused by the two-fluid interaction between ions and neutrals [see @pudritz1990; @kamaya1998; @mouschovias2011].
### Approximate analytic solutions
We look for approximate analytic solutions to Equation (\[eq:relalf\]) corresponding to standing modes. We assume that $k_z$ is outside the cut-off interval $(k_z^+,k_z^-)$, so that Equation (\[eq:relalf\]) has two complex solutions and one purely imaginary solution. This is the most interesting situation for the study of standing Alfvén waves since no oscillatory modes exist when $k_z$ is within the cut-off interval. First we look for an approximate expression for the two oscillatory solutions. To do so we write $\omega = \omega_{\rm R} + i \omega_{\rm I}$ and insert this expression in Equation (\[eq:relalf\]). We assume $|\omega_{\rm I}|\ll |\omega_{\rm R}|$ and neglect terms with $\omega_{\rm I}^2$ and higher powers. Hence, it is crucial for the validity of this approximation that $k_z$ is not within or close to the cut-off region where $\omega_{\rm R} =0$. After some algebraic manipulations we derive approximate expressions for $\omega_{\rm R}$ and $\omega_{\rm I}$. For simplicity we omit the intermediate steps and give the final expressions, namely $$\begin{aligned}
\omega_{\rm R} &\approx & \pm k_z {c_{\mathrm{A}}}\sqrt{\frac{k_z^2 c_{\rm A}^2+\left( 1 +\chi \right){\nu_{\mathrm{ni}}}^2}{ k_z^2 c_{\rm A}^2+\left( 1 +\chi \right)^2{\nu_{\mathrm{ni}}}^2}}, \label{eq:wr} \\
\omega_{\rm I}&\approx & -\frac{\chi {\nu_{\mathrm{ni}}}}{2\left[ k_z^2 c_{\rm A}^2 +\left( 1+\chi \right)^2 {\nu_{\mathrm{ni}}}^2\right]} k_z^2 c_{\rm A}^2. \label{eq:wi}\end{aligned}$$ On the other hand, the remaining purely imaginary, i.e., evanescent, solution is $\omega = i \epsilon$, with the approximation to $\epsilon$ given by $$\epsilon \approx - {\nu_{\mathrm{ni}}}\frac{k_z^2 c_{\rm A}^2+\left( 1 +\chi \right)^2{\nu_{\mathrm{ni}}}^2}{k_z^2 c_{\rm A}^2+\left( 1 +\chi \right){\nu_{\mathrm{ni}}}^2}. \label{eq:gamma}$$ When ${\nu_{\mathrm{ni}}}=0$, we find $\omega_{\rm R} = \pm k_z c_{\rm A}$, $\omega_{\rm I} = 0$ and $\epsilon=0$, hence we recover the solutions in the uncoupled case (Equation (\[eq:uncoupled\])).
It is useful to investigate the behavior of the solutions in the various limits of ${\nu_{\mathrm{ni}}}$. First we consider the limit ${\nu_{\mathrm{ni}}}\ll k_z {c_{\mathrm{A}}}$, i.e., the case of low collision frequency, which means that the coupling between fluids is weak. Equations (\[eq:wr\]) and (\[eq:gamma\]) simplify to $$\begin{aligned}
\omega_{\rm R} &\approx & \pm k_z {c_{\mathrm{A}}}, \label{eq:wrlow} \\
\omega_{\rm I}&\approx & -\frac{\chi {\nu_{\mathrm{ni}}}}{2}, \label{eq:wilow} \\
\epsilon & \approx & -{\nu_{\mathrm{ni}}}.\end{aligned}$$ In this limit $\omega_{\rm R}$ coincides with its value in the ideal, uncoupled case and $\omega_{\rm I}$ is independent of $k_z$. Hence, the the damping of Alfvén waves does not depend on the wavenumber.
On the other hand, when ${\nu_{\mathrm{ni}}}\gg k_z {c_{\mathrm{A}}}$, i.e., the case of strong coupling between fluids, we find $$\begin{aligned}
\omega_{\rm R} &\approx & \pm \frac{k_z {c_{\mathrm{A}}}}{\sqrt{1+\chi}}, \label{eq:wrhigh} \\
\omega_{\rm I}&\approx & -\frac{\chi}{2\left( 1+\chi \right)^2 } \frac{k_z^2 c_{\rm A}^2 }{{\nu_{\mathrm{ni}}}}, \label{eq:wihigh} \\
\epsilon & \approx & -(1+\chi){\nu_{\mathrm{ni}}}.\end{aligned}$$ Now the expression of $\omega_{\rm R}$ involves the factor $\sqrt{1+\chi}$ in the denominator, so that the larger the amount of neutrals, the lower $\omega_{\rm R}$ compared to the value in the fully ionized case [see also @kumar2003; @soler2012]. Now $\omega_{\rm I}$ is proportional to $k_z^2$, meaning that the shorter the wavelength, the more efficient damping.
### Comparison with numerical results {#sec:compar}
Here we solve the full dispersion relation (Equation (\[eq:relalf\])) numerically and compare the numerical solutions with the previous approximations (Equations (\[eq:wr\])–(\[eq:gamma\])). First we set $\chi=2$ and vary the ratio ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}$ between $10^{-2}$ and $10^2$. We compute $\omega_{\rm R}/k_z {c_{\mathrm{A}}}$ and $\omega_{\rm I}/k_z {c_{\mathrm{A}}}$ (see Figure \[fig:compara\]). The agreement between numerical and analytic results is very good. There is no cut-off region for this choice of parameters because we have taken $\chi<8$. Regarding the real part of the frequency, we obtain that the oscillatory modes have $\omega_{\rm R}/k_z {c_{\mathrm{A}}}\approx \pm 1 $ when ${\nu_{\mathrm{in}}}/ k_z {c_{\mathrm{A}}}\ll 1$. When the ratio ${\nu_{\mathrm{ni}}}/ k_z {c_{\mathrm{A}}}$ increases, $\omega_{\rm R}/k_z {c_{\mathrm{A}}}$ decreases until de value $\omega_{\rm R}/ k_z {c_{\mathrm{A}}}\approx \pm 1/\sqrt{1+\chi}$ is reached. This behavior is consistent with the analytic Equation (\[eq:wr\]). The evanescent mode has $\omega_{\rm R} = 0$ regardless the value of ${\nu_{\mathrm{ni}}}/ k_z {c_{\mathrm{A}}}$. The imaginary part of the frequency of the oscillatory modes tend to zero in both limits ${\nu_{\mathrm{ni}}}/ k_z {c_{\mathrm{A}}}\ll 1$ and ${\nu_{\mathrm{ni}}}/ k_z {c_{\mathrm{A}}}\gg 1$, while the damping is most efficient when ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}\sim 1$. This result is also consistent with the analytic Equation (\[eq:wi\]), although the approximation underestimates the actual damping rate when ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}\sim 1$. This discrepancy is a consequence of the weak damping approximation, which assumes $|\omega_{\rm I}|\ll |\omega_{\rm R}|$. However when ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}\sim 1$, the numerical results show that $|\omega_{\rm I}|$ and $|\omega_{\rm R}|$ are of the same order, hence the damping is strong. The imaginary part of the frequency of the evanescent mode is very well approximated by Equation (\[eq:gamma\]).
Next we increase ionization ratio to $\chi = 20$ and compute the same results as before (Figure \[fig:compara2\]). Now there is a cut-off region because $\chi > 8$. The cut-off region is correctly described by Equation (\[eq:rmasmenos\]). At the cut-off the oscillatory and evanescent modes interact and they become three purely imaginary solutions. Propagation is forbidden in this interval. As expected, the agreement between numerical and analytic results is not good near the cut-off region, but both results are in reasonably agreement far from the cut-off interval.
![Results for standing waves. (a) $\omega_{\rm R}/k_z {c_{\mathrm{A}}}$ and (b) $\omega_{\rm I}/k_z {c_{\mathrm{A}}}$ as functions of ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}$. We have used $\chi=2$. Solid and dashed lines correspond to the numerical results of the oscillatory and evanescent modes, respectively, while the symbols correspond to the analytic expressions in the weak damping approximation (Equations (\[eq:wr\])–(\[eq:gamma\])). []{data-label="fig:compara"}](f01a.eps "fig:"){width=".65\columnwidth"} ![Results for standing waves. (a) $\omega_{\rm R}/k_z {c_{\mathrm{A}}}$ and (b) $\omega_{\rm I}/k_z {c_{\mathrm{A}}}$ as functions of ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}$. We have used $\chi=2$. Solid and dashed lines correspond to the numerical results of the oscillatory and evanescent modes, respectively, while the symbols correspond to the analytic expressions in the weak damping approximation (Equations (\[eq:wr\])–(\[eq:gamma\])). []{data-label="fig:compara"}](f01b.eps "fig:"){width=".65\columnwidth"}
![Same as Figure \[fig:compara\] but with $\chi = 20$. The shaded zone denotes the cut-off region according to Equation (\[eq:rmasmenos\]).[]{data-label="fig:compara2"}](f02a.eps "fig:"){width=".65\columnwidth"} ![Same as Figure \[fig:compara\] but with $\chi = 20$. The shaded zone denotes the cut-off region according to Equation (\[eq:rmasmenos\]).[]{data-label="fig:compara2"}](f02b.eps "fig:"){width=".65\columnwidth"}
To explore the physical behavior of the solutions near the cut-off region, we rewrite the momentum equations of ions (Equation (\[eq:momlinion\])) and neutrals (Equation (\[eq:momlinneu\])) in the following forms, $$\begin{aligned}
{\rho_{\rm i}}\frac{{\partial}{\bf v}_{\rm i}}{\partial t} &=& {\bf T} - {\bf R}, \label{eq:momion2} \\
{\rho_{\rm n}}\frac{{\partial}{\bf v}_{\rm n}}{\partial t} &=& {\bf R}, \label{eq:momneu2}\end{aligned}$$ where $\bf T$ and $\bf R$ are the magnetic tension force and the friction force, respectively, given by $$\begin{aligned}
{\bf T} &=& - i {\rho_{\rm i}}\frac{k_z^2 {c_{\mathrm{A}}}^2}{\omega} {\bf v}_{\rm i}, \\
{\bf R} &=& {\rho_{\rm n}}\frac{{\nu_{\mathrm{ni}}}\omega}{\omega + i{\nu_{\mathrm{ni}}}} {\bf v}_{\rm i}.\end{aligned}$$ In Equations (\[eq:momion2\]) and (\[eq:momneu2\]) we have not included magnetic pressure and gas pressure forces because they do not affect Alfvén waves. Now we use the numerically obtained solutions for $\chi = 20$ (Figure \[fig:compara2\]) to compute the moduli of $\bf T$ and $\bf R$, namely $||{\bf T}||$ and $||{\bf R}||$, as functions of ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}$. Figure \[fig:modulus\] displays the ratio $||{\bf T}||/||{\bf R}||$ versus ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}$ near the cut-off region. We have selected some locations in Figure \[fig:modulus\], denoted by letters from [*a*]{} to [*e*]{}, to support the following discussion on the importance of the two forces.
We start by analyzing the solutions on the left-hand side to the cut-off region. There are an oscillatory solution, [*a*]{}, and an evanescent solution, [*b*]{}. We find that $||{\bf T}|| \gg ||{\bf R}||$ for the oscillatory solution [*a*]{}, so that there is a net restoring force for ions in Equation (\[eq:momion2\]). Magnetic tension is the dominant force and drives ions to oscillate almost freely, whereas neutrals are only slightly perturbed by the weak friction force in Equation (\[eq:momneu2\]). In the case of the evanescent solution [*b*]{} we obtain that $||{\bf T}|| \approx ||{\bf R}||$. This means that there is no net force acting on ions. The evanescent solution [*b*]{} only produces perturbations in the neutral fluid.
We turn to location [*c*]{} in Figure \[fig:modulus\], i.e., within the cut-off interval. Here all the solutions are evanescent. The ratio $||{\bf T}||/||{\bf R}||$ of the solution that was previously oscillatory decreases and becomes $||{\bf T}||/||{\bf R}|| < 1$ before reaching the cut-off region. Now friction is the dominant force. Friction acts very efficiently in dissipating perturbations in the plasma before ions (and neutrals indirectly) have had time to feel the restoring force of magnetic tension. As a consequence, oscillatory modes are suppressed.
Finally, we analyze the forces acting on the solutions on the right-hand side to the cut-off region. Again, there are an oscillatory solution, [*d*]{}, and an evanescent solution, [*e*]{}. As happened for the oscillatory solution [*a*]{}, we find that $||{\bf T}|| > ||{\bf R}||$ for the oscillatory solution [*d*]{}, although in this case the friction force is not negligible. Magnetic tension provides now the necessary restoring force for the oscillations of ions, while the friction force is responsible for dragging neutrals when ions move. Hence, both species tend to oscillate together. Solution [*d*]{} represents a collective oscillation of the whole plasma. On the contrary, magnetic tension is negligible for the evanescent solution, [*e*]{}. This mode is governed by the friction force alone and simply causes the decay of perturbations.
![Ratio $||{\bf T}||/||{\bf R}||$ versus ${\nu_{\mathrm{ni}}}/k_z {c_{\mathrm{A}}}$ for the solutions displayed in Figure \[fig:compara2\] near the cut-off region (shaded zone). Solid and dashed lines correspond to oscillatory and evanescent solutions in time, respectively.[]{data-label="fig:modulus"}](f03.eps){width=".65\columnwidth"}
Propagating waves
-----------------
We move to the study of propagating waves. In the ideal fully ionized case the study of propagating waves is equivalent to that of standing waves. Here we shall see that ion-neutral collisions break this equivalence and propagating waves are worth being studied separately. For propagating waves we assume a real $\omega$ and solve the dispersion relation (Equation (\[eq:relalf\])) to find the complex wavenumber, $k_z = k_{z,\rm R}+i k_{z,\rm I}$, with $k_{z,\rm R}$ and $k_{z,\rm I}$ the real and imaginary parts of $k_z$, respectively. Then the amplitude of perturbations is multiplied by the factor $\exp(-k_{z,\rm I} z)$, so that the perturbations are spatially damped. For real and positive $\omega$, $k_{z,\rm R} > 0$ corresponds to waves propagating towards the positive $z$-direction. Conversely, $k_{z,\rm R} < 0$ corresponds to waves propagating towards the negative $z$-direction. In both situations, the sign of $k_{z,\rm I}$ is the same as that of $k_{z,\rm R}$.
Equation (\[eq:relalf\]) is a quadratic equation in $k_z$. The solution to Equation (\[eq:relalf\]) is $$k_z^2 = \frac{\omega^2}{{c_{\mathrm{A}}}^2} \frac{\omega + i (1+\chi){\nu_{\mathrm{ni}}}}{\omega+i{\nu_{\mathrm{ni}}}}. \label{eq:kz2}$$ When ${\nu_{\mathrm{ni}}}=0$ we recover the ideal result $k_z^2 = \omega^2/{c_{\mathrm{A}}}^2$. For ${\nu_{\mathrm{ni}}}\neq 0$ we write $k_z = k_{z,\rm R}+i k_{z,\rm I}$ and insert this expression in Equation (\[eq:kz2\]). Then, it is possible to obtain the exact expression of $k_{z,\rm R}^2$, namely $$\begin{aligned}
k_{z,\rm R}^2 &=& \frac{1}{2} \frac{\omega^2}{{c_{\mathrm{A}}}^2} \frac{\omega^2 + (1+\chi){\nu_{\mathrm{ni}}}^2}{\omega^2 + {\nu_{\mathrm{ni}}}^2} \nonumber \\
&\times & \left[ 1 + \left( 1 + \frac{\chi^2{\nu_{\mathrm{ni}}}^2\omega^2}{(\omega^2 + (1+\chi){\nu_{\mathrm{ni}}}^2)^2} \right)^{1/2} \right],\label{eq:kzr2}\end{aligned}$$ while the exact expression of $k_{z,\rm I}^2$ is $$k_{z,\rm I}^2 = k_{z,\rm R}^2 - \frac{\omega^2}{{c_{\mathrm{A}}}^2} \frac{\omega^2 + (1+\chi){\nu_{\mathrm{ni}}}^2}{\omega^2 + {\nu_{\mathrm{ni}}}^2}.\label{eq:kzi2}$$
Contrary to the case of standing waves there is no cut-off region for propagating Alfvén waves. It is always found that $k_{z,\rm R} \neq 0$ regardless the value of $\omega$. This apparent contradiction between the standing and propagating cases can be understood as follows. An oscillatory standing wave can be interpreted as the superposition of two propagating waves with the same frequency running in opposite directions. However such a representation does not work for a perturbation fixed in space and evanescent in time. The presence of static evanescent perturbations is a peculiar result only obtained when $k_z$ is real and $\omega$ is purely imaginary, a situation that cannot be described with propagating waves. For this reason evanescent solutions are absent from the present study of propagating waves.
### Approximate expressions and comparison with numerical results
Going back to Equations (\[eq:kzr2\]) and (\[eq:kzi2\]), we realize that it is possible to obtain simpler expressions of $k_{z,\rm R}^2$ and $k_{z,\rm I}^2$ by taking advantage of the fact that the second term within the square root of Equation (\[eq:kzr2\]) is always much smaller than unity. We can approximate Equations (\[eq:kzr2\]) and (\[eq:kzi2\]) as $$\begin{aligned}
k_{z,\rm R}^2 &\approx& \frac{\omega^2}{{c_{\mathrm{A}}}^2} \frac{\omega^2 + (1+\chi){\nu_{\mathrm{ni}}}^2}{\omega^2 + {\nu_{\mathrm{ni}}}^2} \nonumber \\ &+& \frac{1}{4} \frac{\omega^2}{{c_{\mathrm{A}}}^2} \frac{\chi^2{\nu_{\mathrm{ni}}}^2\omega^2}{\left(\omega^2 + {\nu_{\mathrm{ni}}}^2\right)\left(\omega^2 + (1+\chi){\nu_{\mathrm{ni}}}^2\right)}, \label{eq:appkr} \\
k_{z,\rm I}^2 &\approx& \frac{1}{4} \frac{\omega^2}{{c_{\mathrm{A}}}^2} \frac{\chi^2{\nu_{\mathrm{ni}}}^2\omega^2}{\left(\omega^2 + {\nu_{\mathrm{ni}}}^2\right)\left(\omega^2 + (1+\chi){\nu_{\mathrm{ni}}}^2\right)}.\label{eq:appki}\end{aligned}$$ As for standing waves we evaluate the analytic solutions (Equations (\[eq:appkr\]) and (\[eq:appki\])) in the various limits of ${\nu_{\mathrm{ni}}}$. When ${\nu_{\mathrm{ni}}}\ll \omega$, Equations (\[eq:appkr\]) and (\[eq:appki\]) simplify to $$\begin{aligned}
k_{z,\rm R} &\approx & \pm \sqrt{ \frac{\omega^2}{{c_{\mathrm{A}}}^2} + \frac{\chi^2{\nu_{\mathrm{ni}}}^2}{4{c_{\mathrm{A}}}^2}} \approx \pm \frac{\omega}{{c_{\mathrm{A}}}}, \label{eq:krlow} \\
k_{z,\rm I} &\approx & \pm \frac{\chi{\nu_{\mathrm{ni}}}}{2{c_{\mathrm{A}}}}. \label{eq:kilow} \end{aligned}$$ Hence we find that $ k_{z,\rm R}$ takes the same value as in the fully ionized case, and $k_{z,\rm I}$ is independent of $\omega$. Conversely when ${\nu_{\mathrm{ni}}}\gg \omega$ we find $$\begin{aligned}
k_{z,\rm R} &\approx & \pm \sqrt{ \frac{\omega^2}{{c_{\mathrm{A}}}^2}(1+\chi) + \frac{1}{4}\frac{\omega^2}{{c_{\mathrm{A}}}^2}\frac{\chi^2\omega^2}{(1+\chi)^2{\nu_{\mathrm{in}}}^2} } \nonumber \\ &\approx& \pm \frac{\omega}{{c_{\mathrm{A}}}}\sqrt{1+\chi}, \label{eq:krhigh} \\
k_{z,\rm I} &\approx & \pm \frac{\chi\omega^2}{2(1+\chi){c_{\mathrm{A}}}{\nu_{\mathrm{in}}}}. \label{eq:kihigh}\end{aligned}$$ We find the presence of the factor $\sqrt{1+\chi}$ in the approximate expression of $k_{z,\rm R}$, while now $k_{z,\rm I}$ is proportional to $\omega^2$. Hence, high-frequency waves are more efficiently damped than low-frequency waves.
Now we solve the dispersion relation (Equation (\[eq:relalf\])) numerically and compare the numerical solutions with the analytic approximations of $k_{z,\rm R}$ and $k_{z,\rm I}$ (Equations (\[eq:appkr\]) and (\[eq:appki\])). These results and shown in Figure \[fig:compara3\] for a particular set of parameters given in the caption of the Figure. As in the standing case, the agreement between numerical and analytic results is very good. We have replied these computations using various values of $\chi$ (not shown here for simplicity) and the analytic results are always in accordance with the numerical ones.
![Results for propagating waves. (a) $k_{z,\rm R}{c_{\mathrm{A}}}/\omega$ and (b) $k_{z,\rm I}{c_{\mathrm{A}}}/\omega$ as functions of ${\nu_{\mathrm{ni}}}/\omega$. Solid lines correspond to the numerical results while symbols correspond to the analytic approximations (Equations (\[eq:appkr\]) and (\[eq:appki\])). We have used $\chi=2$. []{data-label="fig:compara3"}](f04a.eps "fig:"){width=".65\columnwidth"} ![Results for propagating waves. (a) $k_{z,\rm R}{c_{\mathrm{A}}}/\omega$ and (b) $k_{z,\rm I}{c_{\mathrm{A}}}/\omega$ as functions of ${\nu_{\mathrm{ni}}}/\omega$. Solid lines correspond to the numerical results while symbols correspond to the analytic approximations (Equations (\[eq:appkr\]) and (\[eq:appki\])). We have used $\chi=2$. []{data-label="fig:compara3"}](f04b.eps "fig:"){width=".65\columnwidth"}
### Phase velocity and group velocity
Here we compute the phase and group velocities of the propagating Alfvén waves. The phase velocity, ${\bf v}_{\rm ph}$, is defined as $${\bf v}_{\rm ph} = \frac{\omega}{k} \hat{e}_k,$$ with $\hat{e}_k$ denoting the unit vector in the direction of the wave vector. From the dispersion relation (Equation (\[eq:relalf\])) we get $$\frac{\omega}{k} = {c_{\mathrm{A}}}\cos\theta \sqrt{\frac{\omega+i{\nu_{\mathrm{ni}}}}{\omega+i(1+\chi){\nu_{\mathrm{ni}}}}}. \label{eq:phasevel}$$ The phase velocity is a complex quantity. Its real part is related to the propagation speed of the wave while its imaginary part is related to the wave attenuation. Both real and imaginary parts of ${\bf v}_{\rm ph}$ have the same dependence on $\theta$, meaning that the damping rate is independent of the direction of propagation. Ion-neutral collisions do not affect the dependence of ${\bf v}_{\rm ph}$ on $\theta$. From Equation (\[eq:phasevel\]) we can define the effective Alfvén velocity, $\tilde{c}_{\rm A}$, as $$\tilde{c}_{\rm A} = {c_{\mathrm{A}}}\sqrt{\frac{\omega+i{\nu_{\mathrm{ni}}}}{\omega+i(1+\chi){\nu_{\mathrm{ni}}}}}. \label{eq:effva}$$
The group velocity, ${\bf v}_{\rm gr}$, is the propagation velocity of a wave packet and, therefore, of the wave energy. It is defined as $${\bf v}_{\rm gr} = \nabla_{\bf k} \omega = \frac{\partial \omega}{\partial k_x} \hat{e}_x + \frac{\partial \omega}{\partial k_y} \hat{e}_y + \frac{\partial \omega}{\partial k_z} \hat{e}_z.$$ Using the dispersion relation (Equation (\[eq:relalf\])) the group velocity can be more easily computed as $${\bf v}_{\rm gr} = - \frac{1}{\partial \mathcal{D} / \partial \omega} \left( \frac{\partial \mathcal{D}}{\partial k_x} \hat{e}_x + \frac{\partial \mathcal{D}}{\partial k_y} \hat{e}_y + \frac{\partial \mathcal{D}}{\partial k_z} \hat{e}_z \right),$$ that gives $$\begin{aligned}
{\bf v}_{\rm gr} &=& \frac{2 k_z {c_{\mathrm{A}}}^2 (\omega + i{\nu_{\mathrm{ni}}})}{3\omega^2 - k_z^2{c_{\mathrm{A}}}^2 + i2(1+\chi){\nu_{\mathrm{ni}}}\omega} \hat{e}_z \nonumber \\
&=& 2 {c_{\mathrm{A}}}\frac{\left(\omega+i{\nu_{\mathrm{ni}}}\right)^{3/2}\left( \omega + i(1+\chi){\nu_{\mathrm{ni}}}\right)^{1/2}}{2\omega^2-2(1+\chi){\nu_{\mathrm{ni}}}^2+i(1+\chi){\nu_{\mathrm{ni}}}\omega} \hat{e}_z.\end{aligned}$$ As it happens for the phase velocity, the group velocity is also complex. Since the damping of propagating Alfvén waves depends on $\omega$, the components of the wave packet with higher $\omega$ are more damped than the components with low $\omega$. Hence the concept of the group velocity as the velocity at which the whole wave packet propagates becomes obsolete when there is dissipation. However, following @muschietti1993 it is still possible to define an effective group velocity as the velocity at which the center of the wave packet propagates. This effective group velocity is a time-dependent combination of the real and imaginary parts of ${\bf v}_{\rm gr}$ and depends on the particular form of the wave packet [see extensive details in @muschietti1993].
In the limits of low and high collision frequency the damping is weak, so that the imaginary part of ${\bf v}_{\rm gr}$ can be neglected compared to the real part. When ${\nu_{\mathrm{ni}}}\ll \omega$, we find that ${\bf v}_{\rm gr} \approx {c_{\mathrm{A}}}\hat{e}_z$ as in the ideal case, while when ${\nu_{\mathrm{ni}}}\gg \omega$, the group velocity is ${\bf v}_{\rm gr} \approx {c_{\mathrm{A}}}/ \sqrt{1+\chi}\hat{e}_z$. Note again the presence of the factor $\sqrt{1+\chi}$.
Initial-value problem {#sec:initial}
=====================
In Section \[sec:normal\] we have followed a normal mode approach and have assumed that the temporal dependence of the perturbations is of the form $\exp \left( -i\omega t \right)$. Here we go beyond the normal mode analysis and look for general time-dependent solutions to Equations (\[eq:vorti\]) and (\[eq:vortn\]). To this end we follow two different approaches. First we solve the time dependent problem analytically by means of the Laplace transform. Later we use the numerical code MolMHD [@bona2009] to numerically evolve the full set of basic Equations (\[eq:momlinion\])–(\[eq:presslin\]). Finally, these two independent results are compared. In this section we restrict ourselves to standing waves and assume a real and positive $k_z$. We refer the reader to @roberge2007 for the problem of driven propagating waves in a two-fluid plasma.
Analytic solution
-----------------
Here we look for general time-dependent solutions to the coupled Equations (\[eq:vorti\]) and (\[eq:vortn\]) using the Laplace transform. We compute the Laplace transform in time of the vorticity perturbations ${\Gamma_{\rm i}}$ and ${\Gamma_{\rm n}}$ as $$\tilde{\Gamma}_{\beta} ( s ) = \mathcal{L}\left[ \Gamma_\beta \right] = \int_0^\infty \Gamma_\beta (t) \exp\left( -s t \right) {{\rm d}}t,$$ with $s$ is the transformed variable and $\beta$ denotes either ‘i’ or ‘n’. The Laplace transforms of the temporal derivatives of vorticity are given by $$\begin{aligned}
\mathcal{L}\left[ \frac{\partial \Gamma_\beta}{\partial t} \right] &=& s \tilde{\Gamma}_{\beta} ( s ) - \Gamma_{0,\beta}, \\
\mathcal{L}\left[ \frac{\partial^2 \Gamma_\beta}{\partial t^2} \right] &=& s^2 \tilde{\Gamma}_{\beta} ( s ) - s \Gamma_{0,\beta} - \Gamma'_{0,\beta},\end{aligned}$$ where $\Gamma_{0,\beta}$ and $\Gamma'_{0,\beta}$ denote the value of $\Gamma_\beta$ and its temporal derivative at $t=0$, respectively. From Equations (\[eq:vorti\]) and (\[eq:vortn\]) we deduce that the temporal derivatives of the vorticity perturbations at $t=0$ satisfy $$\begin{aligned}
\Gamma'_{0,\rm i} &=& -\chi {\nu_{\mathrm{ni}}}\left( \Gamma_{0,\rm i} - \Gamma_{0,\rm n} \right), \\
\Gamma'_{0,\rm n} &=& {\nu_{\mathrm{ni}}}\left( \Gamma_{0,\rm i} - \Gamma_{0,\rm n} \right).\end{aligned}$$ At the present stage, we leave $\Gamma_{0, \rm i}$ and $\Gamma_{0,\rm n} $ unspecified.
We apply the Laplace transform to Equations (\[eq:vorti\]) and (\[eq:vortn\]). From the transformed Equation (\[eq:vortn\]) we can express ${\tilde{\Gamma}_{\rm n}}$ in terms of ${\tilde{\Gamma}_{\rm i}}$ as $${\tilde{\Gamma}_{\rm n}}(s) = \frac{1}{s + {\nu_{\mathrm{ni}}}} \left( {\nu_{\mathrm{ni}}}{\tilde{\Gamma}_{\rm i}}(s) + \Gamma_{0, \rm n} \right). \label{eq:tgamman}$$ We insert this expression in the transformed Equation (\[eq:vorti\]) and obtain the expression for ${\tilde{\Gamma}_{\rm i}}$ as $${\tilde{\Gamma}_{\rm i}}(s) = \frac{\mathcal{B}(s)}{\mathcal{D}^*(s)}, \label{eq:tgammai}$$ where the functions $\mathcal{B}(s)$ and $\mathcal{D}^*(s)$ are defined as $$\begin{aligned}
\mathcal{B}(s) &=& s \left[\left( s + {\nu_{\mathrm{ni}}}\right)\Gamma_{0, \rm i} + \chi {\nu_{\mathrm{ni}}}\Gamma_{0, \rm n} \right], \\
\mathcal{D}^*(s) &=& s^3 + \left( 1+\chi \right) {\nu_{\mathrm{ni}}}s^2 + k_z^2{c_{\mathrm{A}}}^2 s + {\nu_{\mathrm{ni}}}k_z^2{c_{\mathrm{A}}}^2, \label{eq:disp2}\end{aligned}$$ The function $\mathcal{B}(s)$ depends on the initial conditions and so it contains information about how the waves are excited. Conversely, the function $\mathcal{D}^*(s)$ is independent on the initial conditions. The function $\mathcal{D}^*(s)$ plays a very important role because it corresponds to the dispersion function. It tells us how the plasma behaves after the excitation has taken place. We note that the expression of $\mathcal{D}^*(s)$ (Equation (\[eq:disp2\])) coincides with the left-hand side of Equation (\[eq:relalf1\]), which was obtained from the normal mode dispersion relation (Equation (\[eq:relalf\])) after performing the change of variable $\omega = -i s$. Hence the normal modes are consistently recovered from the present Laplace transform approach by setting $\mathcal{D}^*(s) = 0$.
To compute the vorticity perturbation, ${\Gamma_{\rm i}}$, in the actual temporal domain we perform the inverse Laplace transform of Equation (\[eq:tgammai\]), namely $${\Gamma_{\rm i}}(t) = \mathcal{L}^{-1} \left[ {\tilde{\Gamma}_{\rm i}}(s) \right] = \mathcal{L}^{-1} \left[ \frac{\mathcal{B}(s)}{\mathcal{D}^*(s)} \right]. \label{eq:tgammaiinv}$$ The inverse Laplace transform of Equation (\[eq:tgammaiinv\]) can be evaluated by using the technique of partial fraction decomposition [see, e.g., @dyke1999] if the roots of the equation $\mathcal{D}^*(s) = 0$ are known. The zeros of $\mathcal{D}^*(s)$ are precisely the normal modes studied in Section \[sec:normal\], so that we can take advantage of the approximate expressions found in Section \[sec:normal\]. Once ${\Gamma_{\rm i}}(t)$ is known, the corresponding expression of ${\Gamma_{\rm n}}(t)$ can be obtained using Equation (\[eq:tgamman\]) and applying the convolution theorem, namely $${\Gamma_{\rm n}}(t) = \Gamma_{0, \rm n} e^{-{\nu_{\mathrm{ni}}}t} + {\nu_{\mathrm{ni}}}\int_0^t {\Gamma_{\rm i}}(\tau) \exp\left[-{\nu_{\mathrm{ni}}}\left( t-\tau \right) \right] {{\rm d}}\tau.\label{eq:tgammaninv}$$ From Equations (\[eq:tgammaiinv\]) and (\[eq:tgammaninv\]) we obtain ${\Gamma_{\rm i}}(t)$ and ${\Gamma_{\rm n}}(t)$, namely $$\begin{aligned}
{\Gamma_{\rm i}}(t) &=& A_1 \exp\left( \epsilon t \right) \nonumber \\
&+& \left[ A_2 \cos\left(\omega_{\rm R} t\right) + A_3 \sin\left(\omega_{\rm R} t\right) \right]\exp\left( \omega_{\rm I} t \right),\label{eq:finalgi}\\
{\Gamma_{\rm n}}(t) &=& C_1 \exp\left( \epsilon t \right) + C_2 \exp\left( -{\nu_{\mathrm{ni}}}t \right) \nonumber \\
& +& \left[ C_3 \cos\left(\omega_{\rm R} t\right) + C_4 \sin\left(\omega_{\rm R} t\right) \right]\exp\left( \omega_{\rm I} t \right),\label{eq:finalgn}\end{aligned}$$ with the approximate expressions of $\omega_{\rm R}$, $\omega_{\rm I}$, and $\epsilon$ given in Equations (\[eq:wr\])–(\[eq:gamma\]) and the coefficients $A_1$–$A_3$ and $C_1$–$C_4$ given in the Appendix. In the expression of $\omega_{\rm R}$ (Equation (\[eq:wr\])) the $+$ sign has to be taken. We must recall that the expressions of $\omega_{\rm R}$, $\omega_{\rm I}$, and $\epsilon$ given in Equations (\[eq:wr\])–(\[eq:gamma\]) are computed in the weak damping approximation.
In this analysis we have worked with the vorticity perturbations. However we can also write the solution to the initial-value problem using velocity perturbations. To do so we move, for convenience, to a reference frame in which $k_y = 0$. Thus, the motions of Alfvén waves are polarized in the $y$-direction and from Equations (\[eq:defgi\]) and (\[eq:defgn\]) we get that the vorticity perturbations are proportional to the $y$-component of velocities, namely $v_{{\rm i},y}$ and $v_{{\rm n},y}$. Therefore the expressions for the temporal evolution of $v_{{\rm i},y}$ and $v_{{\rm n},y}$ are the same as those given in Equations (\[eq:finalgi\]) and (\[eq:finalgn\]) for ${\Gamma_{\rm i}}$ and ${\Gamma_{\rm n}}$, respectively. We just need to replace $\Gamma_{0, \rm i}$ by $ v_{{\rm i},0}$ and $\Gamma_{0, \rm n}$ by $v_{{\rm n},0}$, where $ v_{{\rm i},0}$ and $v_{{\rm n},0}$ denote the values of the ion and neutral velocities at $t=0$, respectively.
### Solution in the absence of magnetic field
We start the study of some interesting cases by considering the situation in which there is no magnetic field. We set $B=0$, and therefore ${c_{\mathrm{A}}}=0$. This situation was studied by @vranjes2008. Equations (\[eq:finalgi\]) and (\[eq:finalgn\]) become $$\begin{aligned}
{\Gamma_{\rm i}}(t) &=& \frac{\Gamma_{0, \rm i} + \chi \Gamma_{0, \rm n}}{1+\chi} \nonumber \\ &+& \frac{\chi \left( \Gamma_{0, \rm i} - \Gamma_{0, \rm n} \right) }{1+\chi}\exp \left[ - \left( 1 + \chi \right) {\nu_{\mathrm{ni}}}t \right], \label{eq:solb0i} \\
{\Gamma_{\rm n}}(t) &=& \frac{\Gamma_{0, \rm i} + \chi \Gamma_{0, \rm n}}{1+\chi} \nonumber \\ &-& \frac{\left( \Gamma_{0, \rm i} - \Gamma_{0, \rm n} \right) }{1+\chi}\exp \left[ - \left( 1 + \chi \right) {\nu_{\mathrm{ni}}}t \right]. \label{eq:solb0n}\end{aligned}$$ These solutions agree with those found by @vranjes2008. After an initial phase, i.e., when $\left( 1 + \chi \right) {\nu_{\mathrm{ni}}}t \gg 1$, both ${\Gamma_{\rm i}}$ and ${\Gamma_{\rm n}}$ relax to the same constant value, which corresponds to a weighted average of the initial conditions, namely $${\Gamma_{\rm i}}(t) \approx {\Gamma_{\rm n}}(t) \approx \frac{\Gamma_{0, \rm i} + \chi \Gamma_{0, \rm n}}{1+\chi} = \frac{{\rho_{\rm i}}\Gamma_{0, \rm i} + {\rho_{\rm n}}\Gamma_{0, \rm n}}{{\rho_{\rm i}}+{\rho_{\rm n}}} . \label{eq:amplitude}$$ In the absence of magnetic field no oscillatory solutions are found. Vorticity perturbations remain constant after the relaxation phase.
### Low collision frequency
Here we incorporate again the magnetic field but we consider the case of low collision frequency compared to the oscillation frequency, i.e., ${\nu_{\mathrm{ni}}}\ll \omega_{\rm R}$. Equations (\[eq:finalgi\]) and (\[eq:finalgn\]) simplify to $$\begin{aligned}
{\Gamma_{\rm i}}(t) & \approx & \Gamma_{0, \rm i} \cos \left( k_z {c_{\mathrm{A}}}t \right) \exp \left( - \frac{\chi {\nu_{\mathrm{ni}}}}{2} t \right), \label{eq:gilow} \\
{\Gamma_{\rm n}}(t) & \approx & \Gamma_{0, \rm n} \exp \left( - {\nu_{\mathrm{ni}}}t \right). \label{eq:gnlow}\end{aligned}$$ When ${\nu_{\mathrm{ni}}}\ll \omega_{\rm R}$ the vorticity perturbations of the ionized fluid and the neutral fluid are essentially decoupled from each other. After the excitation, vortical motions in the ionized fluid oscillate at the Alfvén frequency, $k_z{c_{\mathrm{A}}}$, and with a damping rate proportional to ${\nu_{\mathrm{ni}}}$. These are weakly damped Alfvén waves. Conversely, vortical disturbances in the neutral fluid are evanescent in time. There is no driving force for vortical motions in the neutral fluid, and so there is no oscillatory behavior.
### High collision frequency
Now we turn to the limit of high collision frequency compared to the oscillation frequency. This is the realistic case for many astrophysical applications and deserves special attention. We compute the limit of Equations (\[eq:finalgi\]) and (\[eq:finalgn\]) when ${\nu_{\mathrm{ni}}}\gg \omega_{\rm R}$. We obtain $$\begin{aligned}
{\Gamma_{\rm i}}(t) & \approx & \frac{ \Gamma_{0, \rm i} + \chi \Gamma_{0, \rm n}}{1+\chi} \cos\left( \frac{k_z{c_{\mathrm{A}}}}{\sqrt{1+\chi}} t \right) \nonumber \\ &\times& \exp\left( -\frac{\chi}{2\left( 1+\chi \right)^2 } \frac{k_z^2 c_{\rm A}^2 }{{\nu_{\mathrm{ni}}}} t \right) \nonumber \\
&+& \frac{\chi \left( \Gamma_{0, \rm i} - \Gamma_{0, \rm n} \right) }{1+\chi}\exp \left[ - \left( 1 + \chi \right) {\nu_{\mathrm{in}}}t \right], \label{eq:gilimit} \\
{\Gamma_{\rm n}}(t) & \approx & \frac{ \Gamma_{0, \rm i} + \chi \Gamma_{0, \rm n}}{1+\chi}\cos\left( \frac{k_z{c_{\mathrm{A}}}}{\sqrt{1+\chi}} t \right) \nonumber \\ &\times& \exp\left( -\frac{\chi}{2\left( 1+\chi \right)^2 } \frac{k_z^2 c_{\rm A}^2 }{{\nu_{\mathrm{ni}}}} t \right) \nonumber \\
&-& \frac{\Gamma_{0, \rm i} - \Gamma_{0, \rm n} }{1+\chi}\exp \left[ - \left( 1 + \chi \right) {\nu_{\mathrm{in}}}t \right]. \label{eq:gnlimit}\end{aligned}$$
By comparing Equations (\[eq:gilimit\]) and (\[eq:gnlimit\]) with the equivalent solutions in the absence of magnetic field (Equations (\[eq:solb0i\]) and (\[eq:solb0n\])) we find that in both cases there is a relaxation phase whose time scale is determined by the collision frequency. In the absence of magnetic field (Equations (\[eq:solb0i\]) and (\[eq:solb0n\])) the vorticity perturbations remain constant after the relaxation phase, while in the presence of magnetic field (Equations (\[eq:gilimit\])–(\[eq:gnlimit\])) the perturbations of both ions and neutrals oscillate together as a single fluid at the modified Alfvén frequency $k_z {c_{\mathrm{A}}}/\sqrt{1+\chi}$. The oscillations are exponentially damped in time.
We define $\Delta(t)$ as the difference of ${\Gamma_{\rm i}}(t)$ and ${\Gamma_{\rm n}}(t)$ computed from Equations (\[eq:gilimit\]) and (\[eq:gnlimit\]), namely $$\Delta(t) \equiv {\Gamma_{\rm i}}(t) - {\Gamma_{\rm n}}(t) \approx \left( \Gamma_{0, \rm i} -\Gamma_{0, \rm n} \right) \exp\left(- \frac{t}{\tau_{\rm rel}}\right),$$ with $\tau_{\rm rel}$ the relaxation time scale given by $$\tau_{\rm rel} = \frac{1}{\left(1+\chi\right){\nu_{\mathrm{ni}}}} = \frac{1}{{\nu_{\mathrm{in}}}+ {\nu_{\mathrm{ni}}}} . \label{eq:timerel}$$ The function $\Delta(t)$ informs us about the relaxation phase. The function $\Delta(t) \to 0$ when $ t \gg \tau_{\rm rel}$. Importantly, $\Delta(t) = 0$ when $\Gamma_{0, \rm i} =\Gamma_{0, \rm n} $, i.e., there is no relaxation phase when the initial disturbance perturbs ions and neutrals in the same way. In such a case, both ions and neutrals oscillate together with amplitude equal to the initial condition. When $\Gamma_{0, \rm i} \neq \Gamma_{0, \rm n} $ the amplitude of the Alfvénic oscillations is the weighted average of the initial conditions (Equation (\[eq:amplitude\])). The weights correspond to the densities of each fluid.
Numerical solution
------------------
Here we use the numerical code MolMHD to evolve in time the basic Equations (\[eq:momlinion\])–(\[eq:presslin\]) in a fully numerical way. The numerical code [see @bona2009 for details about the scheme] uses the method of lines for the discretization of the variables, and the time and space variables are treated separately. For the temporal part, a 4th order Runge-Kutta method is used, whereas for the space discretization a finite difference scheme with a 4th order centered stencil is used. For a given spatial resolution, the time step is selected so as to satisfy the Courant condition.
The MolMHD code evolves in time the plasma and magnetic field perturbations after an initial condition. In the present application the perturbations are assumed invariant in the $x$- and $y$-directions, so that perturbations only depend on the $z$-direction and Alfvén waves are strictly polarized in the $y$-direction. The only non-zero perturbations are the components of velocity and magnetic field in the $y$-direction. The numerical integration of Equations (\[eq:momlinion\])–(\[eq:presslin\]) is done in the interval $z\in[-10 L, 10 L]$, where $L$ is an arbitrary length scale. We use a uniform grid with 1001 grid points. Since we study standing waves the boundary conditions used at $z=\pm 10L$ are that the velocity perturbations are fixed to zero. The boundary conditions for the other wave perturbations are that their spatial derivatives are set to zero. The initial condition at $t=0$ for the velocity of ions, $v_{{\rm i},y}$, is $$v_{{\rm i},y}= v_{{\rm i},0} \cos\left( \frac{\pi}{20L}z \right), \label{eq:vinitial}$$ which corresponds to the fundamental standing mode in our domain with dimensionless wavenumber $k_zL = \pi/20$. In the fully ionized case, the dimensionless frequency of the fundamental mode is $\omega L / {c_{\mathrm{A}}}= \pi/20 \approx 0.157$. Since we are dealing with linear perturbations we express the velocity perturbation in arbitrary units and set $v_{{\rm i},0} = 1$.
Figure \[fig:mol\] shows the velocity perturbation of ions and neutrals at $z=0$ computed numerically with the MolMHD code for various values of the neutral-ion collision frequency. We compare the numerical results with the analytic ones (Equations (\[eq:finalgi\]) and (\[eq:finalgn\])). The results in the top row of Figure \[fig:mol\] (panels a, b, and c) are obtained when the velocity of neutrals at $t=0$ is the same as that of ions (Equation (\[eq:vinitial\])), while in the bottom row of Figure \[fig:mol\] (panels d, e, and f) neutrals are initially at rest. In the following paragraphs we discuss the results displayed in Figure \[fig:mol\] and relate the time-dependent solutions with the normal modes investigated in Section \[sec:normal\].
{width=".99\columnwidth"}
\(i) In Figure \[fig:mol\](a,d) we use ${\nu_{\mathrm{ni}}}L/{c_{\mathrm{A}}}=0.01$ so that the collision frequency is an order of magnitude lower than the ideal Alfvén frequency. Ions and neutrals behave almost independently. Regardless the initial condition for neutrals, ions oscillate at the Alfvén frequency $k_z {c_{\mathrm{A}}}$ and are damped due to collisions. The oscillations of the ionized fluid act as a periodic forcing on neutrals, although the oscillations generated in the neutral fluid are of much lower amplitude compared to those in the ionized fluid. The dominant behavior in neutrals when the initial perturbation is nonzero is the exponential decay of the initial perturbation. This behavior is consistent with Equations (\[eq:gilow\]) and (\[eq:gnlow\]). In relation with the normal modes of Section \[sec:normal\], we see that when the collision frequency is low the oscillatory mode is mostly excited in the ionized fluid, while the evanescent mode dominates the neutral fluid behavior. Hence we can understand the physical reason for the existence of the evanescent mode as the way in which vorticity perturbations in the neutral fluid decay in time due to collisions [see also @zaqarashvili2011a].
\(ii) In Figure \[fig:mol\](b,e) we increase the collision frequency to ${\nu_{\mathrm{ni}}}L/{c_{\mathrm{A}}}=0.1$. The collision frequency and the ideal Alfvén frequency are of the same order of magnitude. Now both ions and neutrals display strongly damped oscillations and it is not possible to relate the behavior of each fluid with one particular normal mode. Both oscillatory and evanescent normal modes are excited in both fluids and the observed behavior is the result of the joint effect of both oscillatory and evanescent modes. We notice that in Figure \[fig:mol\](b,e) the analytic solution of the time-dependent problem does not exactly follow the full numerical result. The source of the discrepancy is in the approximate expressions of $\omega_{\rm R}$, $\omega_{\rm I}$, and $\epsilon$ (Equations (\[eq:wr\])–(\[eq:gamma\])) used in the analytic solution of the initial-value problem (Equations (\[eq:finalgi\]) and (\[eq:finalgn\])). These approximate expressions were obtained in the case of weak damping, which obviously does not apply to the situation of Figure \[fig:mol\](b,e). However, it is remarkable that the analytic solution still captures the overall behavior correctly even beyond the weak damping limit. It is worth noting that instead of using the approximate values of $\omega_{\rm R}$, $\omega_{\rm I}$, and $\epsilon$ we could use their exact values obtained by numerically solving the dispersion relation. In such a case the agreement between analytic and numerical results is excellent (not shown here for simplicity).
\(iii) Finally we use ${\nu_{\mathrm{ni}}}L/{c_{\mathrm{A}}}=1$ in Figure \[fig:mol\](c,f) so that the collision frequency is an order of magnitude higher than the ideal Alfvén frequency. We find that ions and neutral behave as a single fluid and oscillate together at the modified Alfvén frequency $k_z {c_{\mathrm{A}}}/\sqrt{1+\chi}$. Again, both numerical and analytic results are in excellent agreement. The common amplitude of the $y$-components of velocity of ions and neutrals, namely $\hat{v}_{y}$, is $$\hat{v}_{y} = \frac{{\rho_{\rm i}}v_{{\rm i},0} + {\rho_{\rm n}}v_{{\rm n},0}}{{\rho_{\rm i}}+ {\rho_{\rm n}}}. \label{eq:relvel}$$ The oscillations are weakly damped. When the initial velocity of neutrals is the same as that of ions (Figure \[fig:mol\](c)) both fluids oscillate together from $t=0$. The evanescent mode is not excited when $v_{{\rm i},0} = v_{{\rm n},0}$. Conversely, when neutrals are initially at rest (Figure \[fig:mol\](f)), the single-fluid behavior begins after a short relaxation phase. Although very brief, the relaxation phase can be seen in Figure \[fig:mol\](f) near $t=0$ as sudden decrease of $v_{{\rm i},y}$ and increase of $v_{{\rm n},y}$. Figure \[fig:mol2\] shows the same results displayed in Figure \[fig:mol\](f) but near $t=0$ in order to observe the relaxation phase in detail. The behavior of the velocity perturbations is governed by the evanescent normal mode during the relaxation phase and by the oscillatory normal mode afterwards. This is consistent with Equations (\[eq:gilimit\]) and (\[eq:gnlimit\]). In this case and since $v_{{\rm n},0} = 0$, the amplitude of the joint oscillations after the relaxation phase (Equation (\[eq:relvel\])) is $$\hat{v}_{y} = \frac{{\rho_{\rm i}}}{{\rho_{\rm i}}+ {\rho_{\rm n}}} v_{{\rm i},0} = \frac{1 }{1 + \chi} v_{{\rm i},0}.$$ Hence, the larger the ionization ratio $\chi$, the lower the amplitude.
 but near $t=0$.[]{data-label="fig:mol2"}](f06.eps){width=".65\columnwidth"}
APPLICATION TO THE LOW SOLAR ATMOSPHERE {#sec:app}
=======================================
In this Section we perform an application of the previous theoretical analysis to a particular astrophysical plasma, namely the solar atmosphere. We specialize in solar atmospheric plasma because of two main reasons. The plasma in the coolest parts of the solar atmosphere, i.e., the photosphere and the low chromosphere, is weakly ionized. In addition, recent observations have shown the ubiquitous presence of Alfvénic waves in the solar atmosphere [e.g.., @depontieu2007; @tomczyk2007; @mcintosh2011; @okamoto2011; @depontieu2012]. It is therefore interesting to apply the theory developed in the present paper to the solar case. Many works have focused on the damping of chromospheric waves due to ion-neutral collisions and its importance for plasma heating [see, e.g., @haerendel1992; @depontieu2001; @khodachenko2004; @leake2005; @soler2012; @zaqarashvili2013 among others]. In the present application we do not discuss damping but investigate other effects on Alfvén waves caused by partial ionization, namely the presence of cut-off wavelengths of standing waves and the modification of the effective Alfvén velocity. For results about wave damping, the reader is referred to the papers cited above.
We consider a simplified model for the solar atmospheric plasma. The variation of physical parameters with height from the photosphere to the base of the corona is taken from the quiet sun model C of @vernazza1981, hereafter VALC model. To compute the friction coefficient, ${\alpha_{\rm in}}$, we ignore the influence of heavier species and consider only hydrogen. Hence, the expression of ${\alpha_{\rm in}}$ is given in Equation (\[eq:ainhydrogen\]), where we take $\sigma_{\rm in}\approx 10^{-20}$ m$^2$ according to the estimation by @zaqarashvili2013 for the case of direct elastic collisions [see also @brag]. Figure \[fig:reltime\] shows the inverse of the relaxation time of the ion-neutral coupling, $\tau_{\rm rel}^{-1}$ (Equation (\[eq:relvel\])), as a function of height in the low solar atmosphere. The very large values of $\tau_{\rm rel}^{-1}$, i.e., very short $\tau_{\rm rel}$, point out that in the solar atmosphere ions and neutrals are very efficiently coupled and, for practical purposes, they can be considered as a single fluid. The single-fluid approach is usually followed in studies focused on wave damping (see the references in the previous paragraph). For comparison, we also plot in Figure \[fig:reltime\] the ion-neutral, ${\nu_{\mathrm{in}}}$, and neutral-ion, ${\nu_{\mathrm{ni}}}$, collision frequencies [see also similar plots in @depontieu2001]. We find that the value of $\tau_{\rm rel}$ is mainly determined by ${\nu_{\mathrm{in}}}$. Note that in Figure \[fig:reltime\] the solid and dotted lines are superimposed except at large heights.
![Inverse of the relaxation time of the ion-neutral coupling (solid line) in the low solar atmosphere according to the VALC model. The ion-neutral (dotted line) and neutral-ion (dashed line) collision frequencies are shown for comparison.[]{data-label="fig:reltime"}](f07.eps){width=".65\columnwidth"}
{width=".65\columnwidth"} {width=".65\columnwidth"}
Concerning the magnetic field strength, we consider the model by @leake2006 of a vertical chromospheric magnetic flux tube expanding with height, whose form is given by $$B = B_{\rm ph} \left( \frac{\rho}{\rho_{\rm ph}} \right)^\beta,$$ where $\rho = {\rho_{\rm i}}+ {\rho_{\rm n}}$ is the total density, $B_{\rm ph}$ and $\rho_{\rm ph}$ are the magnetic field strength and the total density, respectively, at the photospheric level, and $\beta=0.3$ is an empirical exponent. We use $\rho_{\rm ph} = 2.74\times10^{-4}$ kg m$^{-3}$ from the VALC model and $B_{\rm ph} = 1.5$ kG. The dependence of $\rho$ with height is determined by the VALC model. With this choice of parameters the magnetic field strength decreases with height so that $B\approx 100$ G at 1,000 km and $B\approx 20$ G at 2,000 km above the photosphere.
The physical parameters in the simplified model of the low solar atmosphere used here depend on the vertical direction, $z$, while in the theoretical analysis of the previous Sections all the parameters are taken constant in $z$. However, it is possible to apply the expressions derived before for a homogeneous plasma to the present stratified case if the wavelength in the $z$-direction, $\lambda_z = 2\pi/k_z$, is much shorter than the length scale of the variation of the effective Alfvén velocity. If such a restriction is fulfilled, we can perform a local analysis and use the physical parameters at a given height in the expressions derived for homogeneous plasma. We follow this approach.
Another assumption made in this Section is that the amplitudes of the waves are small enough for the linear analysis to remain valid. This is a reasonable assumption for chromospheric waves. A parameter that can quantify nonlinearity of the waves is the ratio of the wave velocity amplitude to the local Alfvén velocity. The median of the velocity amplitudes of the chromospheric waves detected by @okamoto2011 is 7.4 km s$^{-1}$. This value is, at least, an order of magnitude lower than the expected value of the Alfvén velocity in the chromosphere.
We start by computing the cut-off region of wavenumbers, $k_z$, of standing waves (Equation (\[eq:rmasmenos\])) as a function of height. Instead of $k_z$, we show in Figure \[fig:valc\](a) the corresponding wavelength, $\lambda_z$. The shaded area in Figure \[fig:valc\](a) indicates the range of wavelengths where oscillatory standing modes are not possible. At low heights above the photosphere the cut-off wavelengths are very short. The cut-off wavelengths increase with height and become of the order of a few kilometers at heights between 500 km and 1,500 km, approximately. Then the cut-off region disappears at a height of 1,600 km above the photosphere, approximately, because the ionization ratio, $\chi$, decreases with height as the plasma becomes more and more ionized and the threshold value $\chi = 8$ is reached at that height.
Next we turn to the computation of the effective Alfvén velocity (Equation (\[eq:effva\])). Since this quantity depends on the wave frequency we use a frequency of 22 mHz, which corresponds to the dominant frequency of the chromospheric Alfvénic waves detected by @okamoto2011. We display in Figure \[fig:valc\](b) the real part of the effective Alfvén velocity as a function of height. In Figure \[fig:valc\](b) we also show the corresponding Alfvén velocity computed using its classic definition that depends on the ion density only. We find that in the low chromosphere neutral-ion collisions are crucial to decrease the effective Alfvén velocity between one and two orders of magnitude with respect to the expected value taking into account the ion density only.
Now we can check the assumption that $\lambda_z$ is much shorter than the length scale of the variation of the effective Alfvén velocity. On the one hand from Figure \[fig:valc\](b) we get that the effective Alfvén velocity varies about an order of magnitude in 2,000 km, approximately. On the other hand the largest cut-off wavelengths displayed in Figure \[fig:valc\](a) are of the order of a few tens of kilometers. Hence the use of a local analysis in the present application is justified.
Implication for energy estimates {#sec:energy}
================================
In the solar context, the dissipation of Alfvénic waves may play an important role for the heating of the atmospheric plasma [see, e.g., @depontieu2007; @mcintosh2011]. The implications of partial ionization for the calculations of energy carried by Alfvén waves were discussed by, e.g., @vranjes2008 and @tsap2011. Both papers give expressions for the energy flux of Alfvén waves in a partially ionized plasmas. However the equations provided by @vranjes2008 and @tsap2011 are different. The expression of @vranjes2008 includes the factor $\left( {\rho_{\rm i}}/ {\rho_{\rm n}}\right)^2$, which is absent from the expression of @tsap2011. Hence the conclusions of these two papers regarding the impact of partial ionization are in apparent contradiction. On the one hand, @vranjes2008 explained that due to the factor $\left( {\rho_{\rm i}}/ {\rho_{\rm n}}\right)^2$ the energy flux in weakly ionized plasmas becomes orders of magnitude smaller compared to the fully ionized case. On the other hand, since @tsap2011 lacked that factor, they argued that the energy flux is independent of the ionization degree and obtained the same expression as for fully ionized plasmas. Here we shall try to solve this apparent contradiction between the results of @vranjes2008 and @tsap2011.
For application to the solar atmosphere we take the limit of high collision frequency. We choose a reference frame in which $k_y = 0$. After the relaxation phase, i.e., for $ t \gg \tau_{\rm rel}$, and neglecting the weak damping due to ion-neutral collisions, the $y$-components of velocity of ions and neutrals oscillate together with frequency, $k_z {c_{\mathrm{A}}}/\sqrt{1+\chi}$, and amplitude, $\hat{v}_{y}$, given in Equation (\[eq:relvel\]). The energy flux along magnetic field lines, $ S_z $, calculated using the Poynting vector [see, e.g., @walker2005] is $$S_z= \frac{1}{2 \mu} \hat{E}_{x} \hat{b}_{y},$$ where $\hat{E}_{x}$ and $\hat{b}_{y}$ denote the amplitudes of the $x$-component of the electric field and the $y$-component of the magnetic field perturbation. The electric field is $${\bf E} = - {\bf v}_{\rm i} \times {\bf B},$$ so that $\hat{E}_{x}\sim\hat{v}_{y} B$. On the other hand, the magnetic field perturbation is governed by Equation (\[eq:inductionlin\]), from where we get $\hat{b}_{y}\sim \hat{v}_{y} B \sqrt{1+\chi}/ {c_{\mathrm{A}}}$. Hence the energy flux is $$S_z = \frac{\sqrt{1+\chi}}{{c_{\mathrm{A}}}}\frac{B^2}{2\mu} \hat{v}_{y}^2= \frac{B}{2\sqrt{\mu}} \sqrt{\rho} \, \hat{v}_{y}^2, \label{eq:energy1}$$ with $\rho = {\rho_{\rm i}}+{\rho_{\rm n}}$ the total plasma density. Consistently we find the classical expression of the energy flux of Alfvén waves [see, e.g., @walker2005], with the only difference that the total plasma density replaces ion density and that the velocity amplitude after the relaxation phase, $\hat{v}_{y}$, is used. Since the relaxation phase is extremely short for realistic collision frequencies (Figure \[fig:reltime\]), $\hat{v}_{y}$ is the velocity amplitude that an observer would actually measure. In a fully ionized plasma, $\hat{v}_{y} = v_{{\rm i},0}$ so that $$S_z = \frac{B}{2\sqrt{\mu}} \sqrt{\rho} \, v_{{\rm i},0}^2. \label{eq:energy12}$$
Up to here there is no discrepancy between the analysis of @vranjes2008 and @tsap2011. The discrepancy arises when the energy flux is written in terms of the initial velocity amplitudes of ions and neutrals. We insert Equation (\[eq:relvel\]) in Equation (\[eq:energy1\]) and rewrite the energy flux in terms of the initial velocity amplitudes, namely $$S_z = \frac{B}{2\sqrt{\mu}} \sqrt{\rho} \left( \frac{{\rho_{\rm i}}v_{{\rm i},0} + {\rho_{\rm n}}v_{{\rm n},0}}{{\rho_{\rm i}}+ {\rho_{\rm n}}} \right)^2,$$ In weakly ionized plasmas ${\rho_{\rm n}}\gg {\rho_{\rm i}}$ and the expression simplifies to $$S_z \approx \frac{B}{2\sqrt{\mu}} \sqrt{\rho} \left( \frac{{\rho_{\rm i}}}{{\rho_{\rm n}}} v_{{\rm i},0} + v_{{\rm n},0}\right)^2. \label{eq:energy2}$$ In the analysis of @vranjes2008 the initial disturbance is localized in the ionized fluid only. So @vranjes2008 took $v_{{\rm n},0}=0$. In this case Equation (\[eq:energy2\]) becomes $$S_z \approx \frac{B}{2\sqrt{\mu}} \sqrt{\rho} \left( \frac{{\rho_{\rm i}}}{ {\rho_{\rm n}}} \right)^2v_{{\rm i},0}^2. \label{eq:vran}$$ Due to the presence of the factor $\left( {\rho_{\rm i}}/ {\rho_{\rm n}}\right)^2$ the energy flux computed by @vranjes2008 in a weakly ionized plasma is much lower than the corresponding value in a fully ionized plasma (Equation (\[eq:energy12\])). The physical reason is that significant portion of the wave energy is used to set into motion the neutral fluid, which is initially at rest in @vranjes2008.
On the contrary, @tsap2011 imposed $\hat{v}_{y} = v_{{\rm i},0}$, which can be satisfied only if $v_{{\rm i},0} = v_{{\rm n},0}$ according to Equation (\[eq:relvel\]). This means that @tsap2011 implicitly assumed that both ions and neutrals are initially perturbed with the same velocity, although this condition is not explicitly stated in the paper. The energy flux is then $$S_z \approx \frac{B}{2\sqrt{\mu}} \sqrt{\rho} \, v_{{\rm i},0}^2. \label{eq:tsap}$$ The expression found by @tsap2011 is the same as Equation (\[eq:energy12\]) for a fully ionized plasma. This is so because in @tsap2011 neutrals have initially the same velocity as ions and no wave energy needs to be used in setting neutrals into motion. @tsap2011 claimed that the expression of @vranjes2008 is incorrect. Here we clearly see that the discrepancy between Equations (\[eq:vran\]) and (\[eq:tsap\]) is caused by a different choice of the initial conditions and that both expressions are indeed correct. The source of the discrepancy was expressing the energy flux in terms of the initial velocity of ions. This is a velocity definitely not easy to measure observationally. Instead, the velocity that an observer can measure is the velocity amplitude after the relaxation phase, $\hat{v}_{y}$. Then, Equation (\[eq:energy1\]) should be used to estimate the energy flux.
Finally we recall that the present analysis, and so Equation (\[eq:energy1\]), is valid in the case of Alfvén waves propagating in a uniform medium so that the energy flux is uniform. As commented in Section \[sec:app\], the application of Equation (\[eq:energy1\]) to the case of Alfvénic waves in the solar atmosphere must be done with caution since neither the density nor the wave velocity amplitude are uniform in the solar plasma [see @goossens2013].
Conclusions {#sec:con}
===========
In this paper we have studied theoretically Alfvén waves in a plasma composed of an ion-electron fluid and a neutral fluid interacting by means of neutral-ion collisions. This configuration is relevant in many astrophysical and laboratory plasmas. To keep our investigation as general as possible we have taken the neutral-ion collision frequency and the ionization degree as free parameters.
First we have performed a normal mode analysis and have derived the dispersion relation of linear Alfvén waves. The dispersion relation agrees with the equations previously found by, e.g., @kulsrud1969 [@pudritz1990; @martin1997; @kumar2003; @zaqarashvili2011a; @mouschovias2011]. As in previous studies, we find that Alfvén waves are damped by neutral-ion collisions. The damping is most efficient when the wave frequency and the collision frequency are of the same order of magnitude. This conclusion applies to both standing and propagating waves. The effective Alfvén velocity of the plasma is modified by collisions, so that for high collision frequencies compared to the wave frequency the effective Alfvén velocity depends on the total density of the plasma and not on the ion density only [see also @kumar2003].
In addition, an important result is that when $\chi > 8$, i.e., when the plasma is weakly ionized, there is a range of wavenumbers (or, equivalently, of wavelengths) for which oscillatory standing Alfvén waves are not possible. These cut-off wavenumbers are physical and are caused by a purely two-fluid effect [see @kulsrud1969; @pudritz1990; @kamaya1998; @mouschovias2011]. We investigated the physical reason for the existence of this cut-off region by analyzing the relative importance of the magnetic tension force and the neutral-ion friction force. We found that friction becomes the dominant force in the cut-off region, while tension is more important outside the cut-off interval. Hence, a disturbance in the magnetic field whose wavenumber is within the cut-off range decays due to collisions before the plasma is able to feel the restoring force of magnetic tension. As a consequence, oscillations of the magnetic field are effectively suppressed. The cut-off wavenumbers investigated here do not appear in the single-fluid approximation and are different from the unphysical cut-offs found in the single-fluid approximation [@zaqarashvili2012].
The solution of the initial-value problem is fully consistent with the normal modes analysis, and shows a growing strength of the interaction between ions and neutrals as the collision frequency increases. For high collision frequencies both ions and neutrals behave as a single fluid. Here a ‘high collision frequency’ means that the neutral-ion collision frequency is at least an order of magnitude higher than the wave frequency. This condition is fulfilled in many astrophysical plasmas which makes the single-fluid approximation appropriate in those situations for the computations of periods/wavelengths and damping times/damping lengths. However, as explained above the single-fluid limit does not fully capture the details of the interaction between ions and neutrals and, for example, the existence of a range of cut-off wavenumbers is a two-fluid result.
As an example, we have considered Alfvén waves in a plasma with physical condition akin to those in the low solar atmosphere. As a matter of fact, due to the large values of the collision frequency this plasma could be studied using the single-fluid approximation instead of the more general two-fluid theory. In that respect, the effective Alfvén velocity is found to depend on the total density of the plasma. However, the presence of a certain range of cut-off wavelengths that can constrain the existence of oscillatory standing modes in the chromosphere is a result absent from those previous studies that use the limit of strong coupling [e.g., @haerendel1992; @depontieu1998; @khodachenko2004; @leake2005]. There are other astrophysical situations in which the ion-neutral coupling is not so strong as in the solar atmosphere and, therefore, a two-fluid treatment is needed. This may be the case of, e.g., protostellar discs [see a table with values of some physical parameters realistic of protostellar discs in @malyshkin2011] and molecular clouds [see, e.g., @balsara1996; @mouschovias2011].
In the present work we have restricted ourselves to Alfvén waves. Compressional magnetoacoustic waves have not been investigated. It is expected that both types of MHD waves are simultaneously present in a plasma. Magnetoacoustic waves in a two-fluid plasma will be investigated in a forthcoming work.
We acknowledge the support from MINECO and FEDER Funds through grant AYA2011-22846 and from CAIB through the ‘grups competitius’ scheme and FEDER Funds. JT acknowledges support from MINECO through a Ramón y Cajal grant. RS thanks Ramon Oliver for reading the manuscript and for his constructive criticism.
Expressions of Coefficients {#app}
===========================
The expressions of the coefficients of Equations (\[eq:finalgi\]) and (\[eq:finalgn\]) are as follows, $$\begin{aligned}
A_1 &=& \epsilon\frac{\left( \epsilon + {\nu_{\mathrm{ni}}}\right)\Gamma_{0, \rm i} + \chi {\nu_{\mathrm{ni}}}\Gamma_{0, \rm n}}{\omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2}, \\
A_2 &=& \frac{\left[ \omega_{\rm R}^2 + \omega_{\rm I}^2 - \epsilon \left( {\nu_{\mathrm{ni}}}+ 2 \omega_{\rm I} \right) \right]\Gamma_{0, \rm i} + \chi {\nu_{\mathrm{ni}}}\epsilon \Gamma_{0, \rm n}}{\omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2}, \\
A_3 &=& \frac{\omega_{\rm R}^2 \left( \epsilon + {\nu_{\mathrm{ni}}}+ \omega_{\rm I} \right) - \omega_{\rm I}\left( \epsilon - \omega_{\rm I} \right) \left( {\nu_{\mathrm{ni}}}+ \omega_{\rm I} \right)}{\omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2} \Gamma_{0, \rm i} + \frac{\chi{\nu_{\mathrm{ni}}}\left( \omega_{\rm R}^2 + \omega_{\rm I}^2 - \epsilon \omega_{\rm I} \right)}{\omega_{\rm R} \left[ \omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2 \right]} \Gamma_{0, \rm n}, \\
C_1 &=& \frac{{\nu_{\mathrm{ni}}}\epsilon}{\epsilon + {\nu_{\mathrm{ni}}}} \frac{\left( \epsilon + {\nu_{\mathrm{ni}}}\right)\Gamma_{0, \rm i} + \chi {\nu_{\mathrm{ni}}}\Gamma_{0, \rm n}}{\omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2}, \\
C_2 &=& \left( 1 + \frac{\chi{\nu_{\mathrm{ni}}}^3}{\left[ \omega_{\rm R}^2 + \left( {\nu_{\mathrm{ni}}}+ \omega_{\rm I} \right)^2 \right] \left( \epsilon + {\nu_{\mathrm{ni}}}\right)} \right) \Gamma_{0, \rm n}, \\
C_3 &=& - \frac{\epsilon {\nu_{\mathrm{ni}}}}{\omega_{\rm R}^2 + \left( \epsilon - {\nu_{\mathrm{ni}}}\right)^2} \Gamma_{0, \rm i} - \frac{\chi{\nu_{\mathrm{ni}}}^2 \left( \omega_{\rm R}^2 + \omega_{\rm I}^2 + \epsilon{\nu_{\mathrm{ni}}}\right)}{\left[ \omega_{\rm R}^2 + \left( {\nu_{\mathrm{ni}}}+ \omega_{\rm I} \right)^2 \right] \left[ \omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2\right]} \Gamma_{0, \rm n}, \\
C_4 &=& \frac{{\nu_{\mathrm{ni}}}\left( \omega_{\rm R}^2 + \omega_{\rm I}^2 - \epsilon \omega_{\rm I} \right)}{\omega_{\rm R}^2 + \left( \epsilon - {\nu_{\mathrm{ni}}}\right)^2} \Gamma_{0, \rm i} + \frac{\chi{\nu_{\mathrm{ni}}}^2 \left[ \omega_{\rm R}^2 \left( \omega_{\rm I} - \epsilon +{\nu_{\mathrm{ni}}}\right) + \omega_{\rm I} \left( {\nu_{\mathrm{ni}}}+ \omega_{\rm I} \right) \left( \omega_{\rm I} - \epsilon \right) \right]}{\omega_{\rm R}\left[ \omega_{\rm R}^2 + \left( {\nu_{\mathrm{ni}}}+ \omega_{\rm I} \right)^2 \right] \left[ \omega_{\rm R}^2 + \left( \epsilon - \omega_{\rm I} \right)^2\right]} \Gamma_{0, \rm n},\end{aligned}$$ with the expressions of $\omega_{\rm R}$, $\omega_{\rm I}$, and $\epsilon$ given in Equations (\[eq:wr\])–(\[eq:gamma\]). In the expression of $\omega_{\rm R}$ (Equation (\[eq:wr\])) the $+$ sign has to be taken.
|
---
author:
- Mai Gehrke
- Tomáš Jakl
- Luca Reggio
title: 'A Cook’s tour of duality in logic: from quantifiers, through Vietoris, to measures[^1]'
---
Algebras from logic {#s:algebras-from-logic}
===================
Boole wanted to view propositional logic as arithmetic. This idea, of seeing logic as a kind of algebra, reached a broader and more foundational level with the work of Tarski and the Polish school of algebraic logicians. The basic concept is embodied in what is now known as the *Lindenbaum-Tarski algebra* of a logic. In the classical cases, this algebra is obtained by quotienting the set of all formulas F by logical equivalence, that is, $${\ensuremath{\mathcal L}} = {\ensuremath{\mathcal F}}/_{\approx} {{\quad\text{where}\quad}} {\varphi}\approx \psi {{\enspace\text{if, and only if,}\enspace}} {\varphi}{{\enspace\text{and}\enspace}} \psi {{\enspace\text{are logically equivalent.}\enspace}}$$ When the equivalence relation $\approx$ is a congruence for the connectives of the logic, L may be seen as an algebra in the signature given by the connectives. This is the case for many propositional logics as well as for first-order logic. There is, however, a fundamental difference in how well this works at these two levels of logic.
For example, for Classical Propositional Logic (CPL), Intuitionistic Propositional Calculus (IPC) and modal logics, the Lindenbaum-Tarski algebra is the *free algebra* over the set of primitive propositions of the appropriate variety. In the above mentioned cases, these are Boolean algebras, Heyting algebras, and modal algebras of the appropriate signature, respectively. Further, for algebras in these varieties, congruences are given by the equivalence classes of the top elements which, logically speaking, are the theories of the corresponding logics. Consequently, we have that the Lindenbaum-Tarski algebras of theories, in which one quotients out by logical equivalence modulo the theory, account for the full varieties of Boolean algebras, Heyting algebras and modal algebras.
The picture is not always quite this simple, even at the propositional level. E.g. the Lindenbaum-Tarski algebra of positive propositional logic (i.e. the fragment of CPL without negation, which we will denote PPL) is indeed the free bounded distributive lattice over the set of primitive propositions. However, since there are lattices with multiple congruences giving the same filter, we do not have the same natural correspondence between the full variety of distributive lattices and the theories of PPL. This sort of problem can be dealt with and this is the subject of the far-reaching theory of Abstract Algebraic Logic, see [@Font91] for the example of PPL.
Let us now consider (classical) first-order logic. Here also, logical equivalence is a congruence for the logical connectives. We have the Boolean connectives, and unary connectives $\exists x$ and $\forall x$, a pair for each individual variable $x$ of the logical language.[^2] The latter give rise to pairs of unary operations that are inter-definable by conjugation with negation. Thus, in the Boolean setting, it is enough to consider the $\exists x$ operations. These are (unary) *modal operators*.
In its most basic form, modal propositional logic corresponds to the variety of modal algebras (MAs), which are Boolean algebras augmented by a unary operation that preserves finite joins. The algebraic approach is a powerful tool in the study of modal logics, see e.g. [@RWZ06] for a survey. In particular, the Lindenbaum-Tarski algebra for this logic is the free modal algebra over the propositional variables, the normal modal logic extensions correspond to the subvarieties of the variety of MAs, and theories in these logics correspond to the individual algebras in the corresponding varieties.
The Lindenbaum-Tarski algebra of first-order formulas modulo logical equivalence is a multimodal algebra, with modalities $\Diamond_x$, one for each variable $x$ in the first-order language. These modalities satisfy some equational properties such as[^3] $$\varphi\leq\Diamond_x\varphi\quad\quad \Diamond_x (\varphi \wedge \Diamond_x \psi) = \Diamond_x \varphi\wedge\Diamond_x\psi \quad\quad \Diamond_x \Diamond_y \varphi = \Diamond_y \Diamond_x\varphi.$$ A fundamental problem, as compared with the propositional examples given above, is that these Lindenbaum-Tarski algebras *are not free* in any reasonable setting. Tarski and his students introduced the variety of cylindric algebras of which these are examples, see [@Mo86] for an overview. However, not all cylindric algebras occur as Lindenbaum-Tarski algebras for first-order theories. For one, when we have an infinite set of variables, and thus of modalities, for every element $\varphi$ in the algebra there is a finite set $V_\varphi$ of variables such that $\Diamond_x\varphi = \varphi$ for all $x\not\in V_\varphi$.
Even though cylindric algebras have been extensively studied, little is known specifically about the ones arising as Lindenbaum-Tarski algebras of first-order theories. A notable exception is the paper [@Myers76] characterising the algebras for first-order logic over empty theories. Another important insight, due to Rasiowa and Sikorski, is the fact that the completeness theorem for first-order logic may be obtained using the Lindenbaum-Tarski construction [@RS50]. Their proof uses the famous Rasiowa-Sikorski Lemma. This lemma, which may be seen as a consequence of the Baire Category Theorem in topology, states that, given a specified countable collection of subsets with suprema in a Boolean algebra, one can separate the elements of the Boolean algebra with ultrafilters that are inaccessible by these suprema.
The lack of freeness of the Lindenbaum-Tarski algebras of first-order logic is overcome by moving from lattices with operators to categories and categorical logic. In the equational setting, algebraic theories can equivalently be described as Lawvere theories, i.e. categories with finite products and a distinguished object $X$ such that every object is a finite power of $X$.[^4] Similarly, theories in a given fragment of first-order logic correspond to a certain class of categories. For instance, theories in the positive existential fragment of first-order logic, also called coherent theories, correspond to coherent categories. Every coherent theory $T$ yields a coherent category, the *syntactic category* of $T$, which may be seen as a generalisation of the Lindenbaum-Tarski construction, and which is free in an appropriate sense. Central to this construction is the fundamental insight, of Lawvere, that quantifiers are adjoints to substitution maps. Thus, existential quantifiers are encoded in coherent categories as lower adjoints to certain homomorphisms between lattices of subobjects. Further, there is some sense in which the correspondence between theories and quotients is regained (at the level of so-called classifying toposes of the theories). See [@MR1977]. Other fragments of first-order logic can be dealt with in a similar fashion, e.g. intuitionistic first-order theories correspond to Heyting categories, and classical first-order theories to Boolean coherent categories. See [@ElephantV2] for a thorough exposition.
To make the relation between syntactic categories and Lindenbaum-Tarski algebras more explicit, we recall the notion of Boolean hyperdoctrines, tightly related to Boolean coherent categories. Consider the category ${\ensuremath{\mathbf{Con}}}$ of contexts and substitutions. A context is a finite list of variables $\overline{x}$, and a substitution from $\overline{x}$ to a context $\overline{y}=y_1,\ldots,y_n$ is a tuple $\langle t_1,\ldots, t_n\rangle$ of terms with free variables in $\overline{x}$. Given a first-order theory $T$, let $P(\overline{x})$ be the Lindenbaum-Tarski algebra of first-order formulas with free variables in $\overline{x}$, up to logical equivalence modulo $T$. A substitution $\langle t_1,\ldots, t_n\rangle\colon \overline{x}\to\overline{y}$ induces a Boolean algebra homomorphism $P(\overline{y})\to P(\overline{x})$ sending a formula ${\varphi}(\overline{y})$ to ${\varphi}(\langle t_1,\ldots, t_n\rangle/\overline{y})$.[^5] This yields a functor $$P\colon {\ensuremath{\mathbf{Con}}}^{\ensuremath{\mathrm{op}}}\to {\ensuremath{\mathbf{BA}}}.$$ The product projection $\pi_y\colon\overline{x},y\to \overline{x}$ in ${\ensuremath{\mathbf{Con}}}$ induces the Boolean algebra embedding $P(\pi_y)\colon P(\overline{x})\hookrightarrow P(\overline{x},y)$, which admits both lower and upper adjoints: $$\begin{gathered}
\exists_{y}\dashv P(\pi_y), \ \ \exists_{y}({\varphi}(\overline{x},y))=\exists y.{\varphi}(\overline{x},y), \\
P(\pi_y)\dashv \forall_{y}, \ \ \forall_{y}({\varphi}(\overline{x},y))=\forall y.{\varphi}(\overline{x},y).\end{gathered}$$ This accounts for the *Boolean hyperdoctrine* structure of $P$. The syntactic category of the theory $T$ can be obtained from $P$ by means of a 2-adjunction between Boolean hyperdoctrines and Boolean categories, cf. [@Pitts1983] or [@Coumans2012 Chapter 5].
While the categorical perspective solves a number of problems, it is not easily amenable to the inductive point of view that we want to highlight here. We will get back to this in Section \[s:three-ex-spaces\].
Topological methods in logic {#s:top-methods-in-logic}
============================
Topological methods in logic have their origin in the work of M. H. Stone. The paper [@Stone1936] established what is nowadays presented as a dual equivalence between the category ${\ensuremath{\mathbf{BA}}}$ of Boolean algebras with homomorphisms and a full subcategory ${\ensuremath{\mathbf{BStone}}}$ of the category of topological spaces with continuous maps. The objects of ${\ensuremath{\mathbf{BStone}}}$ are the so-called *Boolean (Stone) spaces*, i.e. compact Hausdorff spaces whose collection of *clopen* (simultaneously closed and open) subsets forms a basis for the topology. Usually referred to as *Stone duality for Boolean algebras*, this is the prototypical example of a dual equivalence induced by a dualizing object, i.e. an object sitting at the same time in two categories. In fact, the quasi-inverse functors providing the equivalence between ${\ensuremath{\mathbf{BA}}}^{\ensuremath{\mathrm{op}}}$ and ${\ensuremath{\mathbf{BStone}}}$ are given by enriching the set of homomorphisms into the appropriate structure on the two-element set ${\ensuremath{\mathbf{2}}}=\{0,1\}$, which can be seen either as the two-element Boolean algebra or as the two-element Boolean space when equipped with the discrete topology.
Given a Boolean algebra $B$, the space $X_B$ obtained by equipping the set of homomorphisms $$\hom_{{\ensuremath{\mathbf{BA}}}}(B,{\ensuremath{\mathbf{2}}})$$ with the subspace topology induced by the product topology on ${\ensuremath{\mathbf{2}}}^B$ is a Boolean space, the *(Stone) dual space* of $B$. Under the correspondence sending a Boolean algebra homomorphism $h\colon B\to{\ensuremath{\mathbf{2}}}$ to the subset $h^{-1}(1)\subseteq B$, the points of $X_B$ can be identified with the *ultrafilters* on $B$. In logical terms, these are the complete consistent theories over $B$. Conversely, given a Boolean space $X$, the set of continuous maps $$\hom_{{\ensuremath{\mathbf{BStone}}}}(X,{\ensuremath{\mathbf{2}}})$$ forms a Boolean subalgebra $B_X$ of the product algebra ${\ensuremath{\mathbf{2}}}^X$, where ${\ensuremath{\mathbf{2}}}$ is now viewed as a Boolean algebra. When equipped with the induced Boolean operations, $B_X$ is called the *dual algebra* of $X$. Upon identifying a continuous function $f\colon X\to{\ensuremath{\mathbf{2}}}$ with the clopen subset $f^{-1}(1)\subseteq X$, the Boolean algebra $B_X$ can be described as the field of clopen subsets of $X$ with the set-theoretic Boolean operations. Stone duality states that these object assignments extend to functors, and there are isomorphisms $B\cong B_{X_{B}}$ and $X\cong X_{B_{X}}$ (natural in $B$ and $X$, respectively). Throughout, the element of $B_{X_{B}}$ corresponding to $a\in B$ will be denoted by ${\widehat}{a}$.
Shortly after his seminal work in 1936, Stone generalised the duality to bounded distributive lattices [@Stone1938]; there, the relevant category of spaces consists of spectral spaces with perfect maps. A different formulation of the duality for distributive lattices, induced by the dualizing object ${\ensuremath{\mathbf{2}}}$ regarded either as a lattice or as a discrete *ordered* space where $0<1$, was later introduced in [@Priestley1970].
When combined with the algebraic semantics, as outlined in the previous section, Stone duality yields a powerful framework for developing and applying topological methods in logic. The potential advantages of applying duality are of two types. For one, duality theory often connects syntax and semantics. To wit, in the case of CPL, the Lindenbaum-Tarski algebra is the free Boolean algebra on the set $V$ of propositional variables, and its dual space is the Cantor space ${\ensuremath{\mathbf{2}}}^V$ of all valuations over $V$. The second type of advantage is that it *often is easier*, technically, to solve a problem on the dual side.
The use of duality is not restricted to the Boolean setting. Indeed, generalisations and extensions of Stone duality have been exploited to study fragments and extensions of CPL. Many other special cases have since been developed based on Stone’s and Priestley’s dualities for bounded distributive lattices (corresponding to PPL). Here we just mention the duality for Heyting algebras, the algebraic semantics of IPC, mainly developed by Leo Esakia [@Esakia1974; @Esakia2019]. Stone duality was also extended by Jónsson and Tarski to Boolean algebras with operators by introducing the powerful framework of canonical extensions [@JT1; @JT2]. This was a crucial step for many applications, e.g. in modal logic.
In theoretical computer science, the link between syntax and semantics provided by Stone-type dualities is particularly central as the two sides correspond to specification languages and to spaces of computational states, respectively. The ability to translate faithfully between these two worlds has often proved itself to be a powerful theoretical tool as well as a handle for solving problems. A prime example is Abramsky’s seminal work [@Abramsky87; @Abramsky91] linking program logic and domain theory via Stone duality for bounded distributive lattices, which was awarded the IEEE LICS “Test of Time” Award in 2007. Other examples include large parts of modal and intuitionistic logics, where Jónsson-Tarski duality yields Kripke semantics [@BlackburnDeRijkeVenema2001]. For a particular example, see Ghilardi’s work in modal and intuitionistic logic on unification [@Ghi2004] and normal forms [@Ghi1995].
By contrast, Stone duality has not played a significant role, at least overtly, in more algorithmic areas of theoretical computer science until recently. In the theory of regular languages, finite and profinite monoids are an important tool, in particular for proving decidability, ever since their introduction in the 1960s and 1980s, respectively, see [@Pin09] for a survey. While it was observed as early as 1937 by Birkhoff that profinite topological algebras are based on Boolean spaces [@Birkhoff1937], the connection with Stone duality was not used in automata theory until much more recently. It was exploited first in an isolated case by [@Pippenger97], and then more structurally by [@GGP2008]. Further, realising that these methods are instances of Stone duality provides an opportunity to generalise them to the setting of computational complexity and the search for lower bounds [@GK2017]. This line of work connects tools from semantics, such as Stone duality, with problems and methods on the algorithmic side of computer science, such as decidability and Eilenberg-Reiterman theory. Similarly, recent work of Samson Abramsky and co-workers connects categorical tools from semantics, such as comonads, with concepts from finite model theory, such as tree-width and tree-depth [@Abramsky2017b; @AbramskyShah2018].
Finite model theory, computational complexity theory and the theory of regular languages all belong to the branch of computer science where the use of resources in computing is the main focus, whereas category theory and Stone duality have long been central tools in semantics of programming languages. While the trend of making connections and seeking unifying results that bridge the gap between semantics and algorithmic issues has long been on the way (e.g. in the form of semantic work on resource sensitive logics), making this overt and placing it front and center stage is a recent phenomenon in which Samson Abramsky has played a central role. In particular, one may mention the 2017 semester-long program at The Simons Institute for the Theory of Computing on Logical Structures in Computation of which he was a co-organiser, and the ensuing work and ongoing project with Anuj Dawar focussing on bridging what they aptly call the *Structure versus Power* gap in theoretical computer science. The 2014 ERC project Duality in Formal Languages and Logic – a unifying approach to complexity and semantics (DuaLL), in which our recent work has taken place, shares these goals. In Section \[s:modal-Vietoris\], we highlight some of the ideas and concepts from Samson Abramsky’s work in semantics that are playing an important role in our recent work on the DuaLL project, which we will describe in Section \[s:Quant\]. In Section \[s:three-ex-spaces\], we briefly review two settings from logic pertinent to our work, and give a duality-centric description of the treatment of the function space construction in Abramsky’s Domain Theory in Logical Form. This allows us to make a connection to the profinite methods in automata theory.
Modal logic and the Vietoris functor {#s:modal-Vietoris}
------------------------------------
An important contribution of Samson Abramsky’s is to use the duality between syntax and semantics, *combined with a step-wise description of connectives* in logic applications. This phenomenon is the driving force behind his sweeping and elegant general solution to domain equations in the paper Domain Theory in Logical Form (DTLF), [@Abramsky91]. We will get back to this with a few more details in Section \[s:three-ex-spaces\]. In [@Abramsky1988], which is the published version of various talks given during the genesis of DTLF, Abramsky gives a simpler example of this general idea. The setting is non-well-founded sets, and the object he considers is the free modal algebra (over the empty set). Other early uses of similar methods are due to Ghilardi [@Ghilardi1992; @Ghi1995]. Subsequently, the treatment of the free modal algebra given in Abramsky’s talks, in particular his talk at the 1988 British Colloquium on Theoretical Computer Science in Edinburgh, has been identified as an important contribution to modal logic in its own right, see e.g. [@Ruttenetal93; @KuKuVe2004; @VV2014], and it is also very pertinent to the duality theoretic treatment of quantifiers which we will discuss in Section \[s:Quant\].
The step-wise description of an algebra from a set of generators is what is often called *Noetherian induction* in algebra and *induction on the complexity of a formula* in logic: The algebra is generated layer by layer, starting with the generators — which are said to be of rank $0$ — by adding consecutive layers of the operations to obtain higher rank elements. Also, instead of doing this with all the operations, we may do it relative to a fragment. In the case of modal algebras, for example, we may consider as rank $0$ all Boolean combinations of generators, rank less than or equal to $1$ any element which may be expressed as a Boolean combination of rank $0$ and diamonds of rank $0$ elements, and so on. This is a fine tool for the purpose of induction, but it is not a good tool for constructing algebras in general. However, if the operation is freely added modulo some equations which are of pure rank $1$, then it is in fact a powerful method of *construction*. This is exactly the situation for free modal algebras, which are Boolean algebras with an additional operation satisfying the equations $$\Diamond 0 \approx 0 {{\quad\text{and}\quad}} \Diamond(x\vee y) \approx \Diamond x \vee \Diamond y.$$ These equations are both of pure rank $1$. That is, in each equation, all occurrences of each variable are in the scope of exactly *one* layer of modal operators.
From a categorical point of view, one may see algebras in a variety as Eilenberg-Moore algebras for a finitary monad, but having a pure rank $1$ axiomatisation means that these are also presentable as the *algebras for an endofunctor*, see [@KurzRosicky12] where this is studied in greater generality. In the case of MAs, define the endofunctor ${\ensuremath{\mathbb{M}}}$ on Boolean algebras which takes a Boolean algebra $B$ to the Boolean algebra freely generated by elements $\Diamond a$, for every $a\in B$, subject to the equations for modal algebras viewed as relations on these generators: $$\Diamond 0 \approx 0 {{\quad\text{and}\quad}} \Diamond(a\vee b) \approx \Diamond a \vee \Diamond b \quad(\forall a,b\in B).$$ Then $B$, equipped with a unary operation $f\colon B\to B$, is a modal algebra if and only if the map $\Diamond a\mapsto f(a)$ extends to a Boolean algebra homomorphism $h\colon {\ensuremath{\mathbb{M}}}(B)\to B$. It also follows that the free modal algebra over a Boolean algebra $B$ may be *constructed inductively*, as the colimit of the sequence $$\begin{tikzcd}
B_0 \arrow[hookrightarrow]{r}{i_0} & B_1 \arrow[hookrightarrow]{r}{i_1} & B_2\arrow[hookrightarrow]{r}{i_2} & \dots
\end{tikzcd}$$ where $B_0=B$, $B_{n+1}$ is the coproduct $B\oplus{\ensuremath{\mathbb{M}}}(B_n)$, the map $i_0$ is the embedding of $B$ in the coproduct, and $i_{n+1}={\mathrm{id}}_{B} \oplus {\ensuremath{\mathbb{M}}}(i_n)$. Note that, if $B$ is finite, then so are all the algebras in the sequence. Moreover, if we start with the free Boolean algebra on a set $V$, then the colimit of the sequence is the free modal algebra over $V$, and $B_n$ is the Boolean subalgebra consisting of all formulas of rank at most $n$.
Further, we may of course dualize ${\ensuremath{\mathbb{M}}}$ to get a functor on ${\ensuremath{\mathbf{BStone}}}$ and a co-inductive description of the dual of free modal algebras. This dual endofunctor is the Vietoris functor. Recall that, given a Boolean space $X$, the *Vietoris hyperspace of $X$* is the collection ${\ensuremath{\mathcal{V}}}(X)$ of closed subsets of $X$ equipped with the topology generated by the sets of the form $$\Diamond U = \{ C \in {\ensuremath{\mathcal{V}}}(X) \mid C \cap U \neq \emptyset \} {{\quad\text{and}\quad}} (\Diamond U)^c$$ for $U$ a clopen subset of $X$. With respect to this topology, ${\ensuremath{\mathcal{V}}}(X)$ is again a Boolean space. See [@Vietoris1923; @Michael1951]. Furthermore, for every continuous map $f\colon X\to Y$, the forward-image map $f({\mkern1.5mu\text{-}\mkern1.5mu})\colon {\ensuremath{\mathcal{V}}}(X)\to{\ensuremath{\mathcal{V}}}(Y)$ is continuous. Hence, we obtain a functor $${\ensuremath{\mathcal{V}}}\colon {\ensuremath{\mathbf{BStone}}}\to{\ensuremath{\mathbf{BStone}}}.$$ Abramsky showed that the dual Stone space of the free modal algebra on no generators coincides with the final coalgebra for the functor ${\ensuremath{\mathcal{V}}}$. In general, the dual of the sequence of embeddings given above is $$\begin{tikzcd}[column sep=4.6em]
X \arrow[twoheadleftarrow]{r}{\pi_X} & X\times{\ensuremath{\mathcal{V}}}(X)=X_1 \arrow[twoheadleftarrow]{r}{\ {\mathrm{id}}_X\times{\ensuremath{\mathcal{V}}}(\pi_X)} & X\times{\ensuremath{\mathcal{V}}}(X_1)=X_2 \arrow[twoheadleftarrow]{r} & \dots
\end{tikzcd}$$ This result provides also a coalgebraic perspective on the duality between modal algebras and descriptive general Kripke frames. As such, it has had a strong influence on the very active coalgebraic approach to modal logic. The Vietoris hyperspace construction also appeared earlier in modal logic in the work (published in Russian) of Leo Esakia, cf. [@Esakia1974]. See also [@Esakia2019] for the recent English translation of Esakia’s 1985 book.
Three examples of dual spaces in logic {#s:three-ex-spaces}
--------------------------------------
In this section we discuss duality methods in logic in three settings: classical first-order logic, B[ü]{}chi’s logic on words, and Domain Theory in Logical Form.
#### First-order logic and spaces of types.
For classical first-order logic, the dual space of the Lindenbaum-Tarski algebra of formulas is fairly easy to describe. Fix a countably infinite set of first-order variables $v_1,v_2,\ldots$ and a first-order signature ${\ensuremath{\sigma}}$, i.e. ${\ensuremath{\sigma}}$ may contain relation symbols as well as function symbols and constants. Denote by ${\ensuremath{\mathrm{FO}_\omega}}$ the set of all first-order formulas in the signature ${\ensuremath{\sigma}}$ over the set of variables. Given a theory $T$, that is, any set of first-order sentences in the signature ${\ensuremath{\sigma}}$, consider the collection $${\ensuremath{\mathrm{Mod}_\omega}}(T)=\{(A,\alpha\colon \omega\to A)\mid A \ \text{is a ${\ensuremath{\sigma}}$-structure and} \ A\models T\}$$ of models of $T$ equipped with an assignment of the variables. The satisfaction relation ${\models}\subseteq {\ensuremath{\mathrm{Mod}_\omega}}\times{\ensuremath{\mathrm{FO}_\omega}}$ induces the equivalence relations of elementary equivalence and logical equivalence on these sets, respectively: $$(A,\alpha)\equiv (A',\alpha') \ \ \text{ iff } \ \ \forall {\varphi}\in{\ensuremath{\mathrm{FO}_\omega}}\ \ A,\alpha\models {\varphi}\ \Longleftrightarrow \ A',\alpha'\models {\varphi}$$ and $${\varphi}\approx \psi \ \ \text{ iff } \ \ \forall (A,\alpha)\in {\ensuremath{\mathrm{Mod}_\omega}}(T) \ \ A,\alpha\models {\varphi}\ \Longleftrightarrow \ A,\alpha\models \psi.$$ The quotient ${\ensuremath{\mathrm{FO}_\omega}}(T)={\ensuremath{\mathrm{FO}_\omega}}/{\approx}$, i.e. the Lindenbaum–Tarski algebra of $T$, carries a natural Boolean algebra structure. On the other hand, ${\ensuremath{\mathrm{Typ}_\omega}}(T)={\ensuremath{\mathrm{Mod}_\omega}}/{\equiv}$ is naturally equipped with a topology, generated by the sets $${\ensuremath{\llbracket {\varphi}\rrbracket}}=\{[(A,\alpha)]\mid A,\alpha\models {\varphi}\}$$ for ${\varphi}\in {\ensuremath{\mathrm{FO}_\omega}}$, and is known as the *space of types* of $T$. G[ö]{}del’s completeness theorem may now be stated as follows:
the space ${\ensuremath{\mathrm{Typ}_\omega}}(T)$ is the Stone dual of ${\ensuremath{\mathrm{FO}_\omega}}(T)$.
For every $n\in{\mathbb{N}}$, we can consider the Boolean subalgebra ${\ensuremath{\mathrm{FO}}}_n(T)$ of ${\ensuremath{\mathrm{FO}_\omega}}(T)$ consisting of the equivalence classes of formulas with free variables in $v_1,\ldots,v_n$. The dual space of ${\ensuremath{\mathrm{FO}}}_n(T)$ is then the space of $n$-types of $T$. In particular, for $n=0$, we see that the dual space of the Lindenbaum-Tarski algebra of sentences ${\ensuremath{\mathrm{FO}}}_0(T)$ is the space of elementary equivalence classes of models of $T$. Methods based on spaces of types play a central role in model theory. Their use can be traced back to Tarski’s work, but the functorial nature of the construction was brought out and exploited nearly thirty years later by Morley in [@Morley1974]. In fact, it has been suggested that the notion of type space may be more fundamental than the notion of model [@Macintyre2003]. This point of view is related to the categorical approach, as the type space functor of a theory $T$ can be essentially identified with the (pointwise) dual of the hyperdoctrine associated with $T$.
This approach relies on the presentation of the algebra ${\ensuremath{\mathrm{FO}_\omega}}(T)$ as the colimit of the following diagram of Boolean algebra embeddings: $$\begin{tikzcd}
{\ensuremath{\mathrm{FO}}}_0(T) \arrow[hookrightarrow]{r} & {\ensuremath{\mathrm{FO}}}_1(T) \arrow[hookrightarrow]{r} & {\ensuremath{\mathrm{FO}}}_2(T) \arrow[hookrightarrow]{r} & \dots
\end{tikzcd}$$ Interestingly, this presentation does not fit with the inductive treatment of modal logic in Section \[s:modal-Vietoris\], as the sentences, which is what we want to understand, belong to all the algebras in the chain. If we want to construct the Lindenbaum-Tarski algebra ${\ensuremath{\mathrm{FO}_\omega}}(T)$ inductively, by adding a layer of quantifier $\exists$ at each step, we should start from the Boolean subalgebra ${\ensuremath{\mathrm{FO}}}^0(T)$ of ${\ensuremath{\mathrm{FO}_\omega}}(T)$ consisting of the *quantifier-free* formulas. The algebra ${\ensuremath{\mathrm{FO}}}^0(T)$ sits inside the algebra ${\ensuremath{\mathrm{FO}}}^1(T)$ of formulas with quantifier rank at most $1$, and so forth. The colimit of the diagram $$\begin{tikzcd}
{\ensuremath{\mathrm{FO}}}^0(T) \arrow[hookrightarrow]{r} & {\ensuremath{\mathrm{FO}}}^1(T) \arrow[hookrightarrow]{r} & {\ensuremath{\mathrm{FO}}}^2(T) \arrow[hookrightarrow]{r} & \dots
\end{tikzcd}$$ is again the algebra ${\ensuremath{\mathrm{FO}_\omega}}(T)$. In Section \[s:Quant\], we will illustrate how the inductive methods used in B[ü]{}chi’s logic apply in the general first-order setting (and beyond) using the ideas set forth in Section \[s:modal-Vietoris\].
#### B[ü]{}chi’s logic on words and profinite monoids.
The connection between logic and automata goes back to the work of B[ü]{}chi, Elgot, Rabin and others in the 1960s. In particular, B[ü]{}chi’s logic on words provides a powerful tool for the study of formal languages. The basic idea consists in regarding words on a finite alphabet $A$, i.e. elements of the free monoid $A^*$, as finite models for so-called *logic on words*. That is, a word $w\in A^*$ is seen as a relational structure on the initial segment of the natural numbers $$\{1,\ldots, |w|\},$$ where $|w|$ is the length of $w$, equipped with a unary relation $P_a$ for each $a\in A$ which singles out the positions in $w$ where the letter $a$ appears. B[ü]{}chi’s theorem states that the Lindenbaum-Tarski algebra of monadic second-order sentences for logic on words with the successor relation (interpreted over finite words) is isomorphic to the Boolean subalgebra of ${\ensuremath{\mathcal{P}}}(A^*)$ consisting of the regular languages [@Buchi1966].
Since we are beyond first-order logic, and we have restricted to the finite models, the dual of the Lindenbaum-Tarski algebra is *not* $A^*$, i.e. the collection of (elementary equivalence classes of) finite models. For the FO fragment of logic on words we can identify the dual with a space of models provided we allow for pseudofinite words. See e.g. [@vanGoolSteinberg]. However, this is not the case for monadic second-order logic and duality guides the right choice for the space of generalised models as the dual of the Lindenbaum-Tarski algebra. The latter coincides with (the underlying space of) the profinite completion ${\widehat}{A^*}$ of the monoid $A^*$, or equivalently, the free profinite monoid on the set $A$.
The observation that the space underlying the free profinite monoid is the dual of the Boolean algebra of languages recognised by finite monoids essentially goes back to [@Birkhoff1937], and was rediscovered by Almeida in the setting of automata theory [@Almeida1989]. Further, the fact that the monoid multiplication of ${\widehat}{A^*}$ also arises from duality for Boolean algebras with operators as the dual of certain quotienting operations on regular languages was shown in [@GGP2008].
This type space tells us what generalised models for these logics should be, namely the points of the free profinite monoids. The realisation that these are an important tool in automata theory came in the 1980s [@Reiterman1982; @Almeida94]. However, it was introduced, not via logic and duality, but rather via the connection between automata and finite semigroups, where the multiplication available on the profinite monoid also plays a fundamental role.
An essential insight in the proof of B[ü]{}chi’s theorem is the fact that every monadic second-order formula is equivalent on words to an existential monadic second-order formula, and thus the iterative approach is not relevant as the hierarchy collapses. See [@GhivGo16] for a duality and type-theoretic approach via model companions. However, for the first-order fragment the iterative approach is very powerful. The first, and still prototypical application, is Sch[ü]{}tzenberger’s theorem which applies an iterative method, similar to the one of Section \[s:modal-Vietoris\], to characterise the first-order fragment via duality. To be more precise, [@Schutzenberger65] shows that the star-free languages are precisely those recognised by (finite) aperiodic monoids. To prove this, Sch[ü]{}tzenberger identified a semidirect product construction which captures dually the application of concatenation product on languages. The fact that star-free languages are precisely those given by first-order sentences of B[ü]{}chi’s logic was subsequently shown in [@MP1971], though some passages in the introduction of [@Schutzenberger65] suggest that Sch[ü]{}tzenberger was aware of this connection when he proved his result.
#### Domain Theory in Logical Form.
In denotational semantics one seeks mathematical models of programs, which should be assigned in a compositional way. The compositionality means that program constructors should correspond to type constructors, and solutions to domain equations should correspond to program specifications. Scott’s original solution to the domain equation $$X\cong [X,X],$$ seeking a domain $X$ which is isomorphic to the domain of its endomorphisms, was obtained by constructing a profinite poset, that is, a spectral space. Much further work confirmed that categorical methods, topology and in particular duality are central to the theory, cf. [@ScSt71; @Plo76; @SP82; @Smyth83; @LW91]. Rather than seeing Stone duality and its variants as useful technical tools for denotational semantics, Abramsky put Stone duality front and center stage: A *program logic* is given in which denotational types correspond to theories and the ensuing Lindenbaum-Tarski algebras of the theories are bounded distributive lattices, whose dual spaces yield the domains as types. The constructors involved in the domain equations thus have duals under Stone duality, and solutions are obtained as duals of the solutions of the corresponding equation on the lattice side. In [@Abramsky87] Stone duality is restricted to the so-called Scott domains. That is, algebraic domains that are consistently complete. These are fairly simple and are closed under many constructors, including function space. In [@Abramsky91] the larger category of bifinite domains, which, in addition, is closed under powerdomain constructions, is used. We will say a bit more about bifinite domains later, but for now, we illustrate with a simple example at the level of spectral spaces.
The Smyth powerdomain, $\mathbb S(X)$, is the space whose points are the compact and saturated[^6] subsets of $X$ equipped with the upper Vietoris topology [@Smyth83]. That is, the topology is generated by the subbasis given by the sets $$\Box U = \{K\in \mathbb S(X)\mid K\subseteq U\}, \ \ \text{for $U\subseteq X$ open}.$$ At first sight, this may seem like quite an exotic object to pull out of a hat to study non-determinism. However, in Abramsky’s duality with program logic, this construct is the Stone dual of adding a layer of (demonic) non-determinism. Indeed, if $X$ is a spectral space, then so is $\mathbb S(X)$, and if $L$ is the dual of $X$, then $\mathbb S(X)$ is the dual of $$F_\Box(L)=\mathbb F_{DL}(\Box L)/_{\approx}.$$ Here, $\mathbb F_{DL}(\Box L)$ denotes the free distributive lattice[^7] on the set of formal generators $\Box L=\{\Box a\mid a\in L\}$, and $\approx$ is the congruence given by the following scheme of relations between the generators: $$\Box(\bigwedge G)\ \approx \ \bigwedge\Box G \ \ \text{for $G\subseteq L$ finite}.$$ Note that the Smyth powerdomain generalises the Vietoris hyperspace construction for Boolean spaces and, indeed, when $L=B$ is a Boolean algebra, the Booleanization of the lattice $F_\Box(B)$ coincides with the Boolean algebra ${\ensuremath{\mathbb{M}}}(B)$ from Section \[s:modal-Vietoris\].
Now the domain equation $X=\mathbb S(X)$ is solved by the final coalgebra for $\mathbb S$. However, a priori, there is no guarantee that it exists. On the other hand, the dual equation $L=F_\Box(L)$ is solved by the initial algebra, i.e. the free $\Box$-algebra over the empty set. As explained in Section \[s:modal-Vietoris\], the latter algebra is guaranteed to exist since algebraic varieties are closed under filtered colimits.
Even though the duality theoretic paradigm supplied by the program logic makes it clearer why $\mathbb S(X)$ is the right object, one may still wonder how difficult it is to discover that $F_\Box(L)$ and $\mathbb S(X)$ are dual to each other. But this also is made quite algorithmic by duality: The dual of a free distributive lattice, such as $\mathbb F_{DL}(\Box L)$, is simply the Sierpinski cube ${\ensuremath{\mathbf{2}}}^{\Box L}$.[^8] Indeed, a subset $S\subseteq \Box L$ corresponds to the unique homomorphism $h_S\colon \mathbb F_{DL}(\Box L)\to {\ensuremath{\mathbf{2}}}$ extending the characteristic map $\chi_S\colon \Box L\to {\ensuremath{\mathbf{2}}}$. Viewed as a theory (or prime filter) it is $F_S=\{{\varphi}\mid \exists S'\subseteq S \text{ finite with }\bigwedge S'\leq{\varphi}\}$. Also, a quotient of $\mathbb F_{DL}(\Box L)$ such as $F_\Box(L)$ is dual to a subspace of ${\ensuremath{\mathbf{2}}}^{\Box L}$, namely the one consisting of all those $S\subseteq \Box L$ such that $$\Box(\bigwedge G)\in F_S\ \ \iff \ \ \bigwedge\Box G\in F_S, \ \ \text{ for $G\subseteq L$ finite}.$$ By the definition of $F_S$, this is equivalent to $$\Box(\bigwedge G)\in S \ \ \iff \ \ \Box G\subseteq S, \ \ \text{ for $G\subseteq L$ finite}.$$ Note that ${\ensuremath{\mathbf{2}}}^{\Box L}$ is homeomorphic to ${\ensuremath{\mathcal{P}}}(L)$ with the topology generated by the sets $\widetilde{a}=\{S\in {\ensuremath{\mathcal{P}}}(L)\mid a\in S\}$ for $a\in L$. Viewed as subsets of $L$, the elements that belong to the dual of $F_\Box(L)$ are precisely the filters of $L$. That is, $\mathbb S(X)$ is homeomorphic to the space ${\rm Filt}(L)$ equipped with the topology generated by the sets $\widetilde{a}$ for $a\in L$. This algorithmic method, using duality for quotients of free algebras and then inductively adding layers of a connective, has been applied widely in the setting of propositional logics, see e.g. [@Ghilardi1992; @BeGe10; @Ghilardi10; @CovG12].\
In [@Abramsky91] a large number of constructors such as $\mathbb S$ are treated, including the function space which, given two spaces $X$ and $Y$, yields the space $[X,Y]$ of all continuous functions $X\to Y$ in the compact-open topology. This case is more subtle, but it is closely related to the one above, and to the duality between lattices with residuation and Stone topological algebras, which is at the heart of the duality theory of profinite methods in automata theory. For these reasons, we go in a bit more detail. The following are extracts of a book in preparation [@GvGprepub].
Consider the duality as above but for the operator type of implication. That is, given distributive lattices (DLs) $L$ and $M$, define $$F_\to(L\times M)=\mathbb F_{DL}(\to(L\times M))/_{\approx},$$ where $\to(L\times M)=\{a\to b\mid a\in L, \, b\in M\}$ are the formal generators and $\approx$ is the congruence given by the following two schemes of relations between the generators:
1. $a\to\bigwedge G=\bigwedge\{a\to b\mid b\in G\}$ for $a\in L$ and $G\subseteq M$ finite;
2. $\bigvee F\to b=\bigwedge\{a\to b\mid a\in F\}$ for $F\subseteq L$ finite and $b\in M$.
Going through the same exercise as outlined above to identify the elements of ${\ensuremath{\mathbf{2}}}^{L\times M}$ which are compatible with the schemes (i) and (ii), one obtains the following result.
Let $L$ and $M$ be DLs, and let $X$ and $Y$ be their respective dual spaces. The dual of $F_\to(L\times M)$ is the space $[X,\mathbb S(Y)]$ of continuous functions from $X$ to the Smyth powerspace of $Y$, in the compact-open topology.
This provides a dual description of $[X,\mathbb S(Y)]$, but we are interested in $[X,Y]$ which is a subspace of $[X,\mathbb S(Y)]$. However, it is not in general a closed subspace in the patch topology, reflecting the fact that $[X,Y]$ is not in general a spectral space. One would need to move to frames, sober spaces and geometric theories to describe $[X,Y]$ as the dual of a quotient. However, we have the following approximation.
\[prop:preserves-joins-at-primes\] Let $L$ and $M$ be DLs, and $X,Y$ their respective dual spaces. The dual of the quotient of $F_\to(L\times M)$ by a congruence $\theta$ is a subspace of $[X,Y]$ if and only if for all $x\in X$, $a\in F_x$, and finite subset $G\subseteq M$, there is $a'\in F_x$ such that $$[a\to(\bigvee G)]_\theta\ \leq\ [\bigvee\{a'\to b\mid b\in G\}]_\theta.$$ Here, $F_x$ denotes the prime filter of $L$ corresponding to the point $x\in X$.
The above property may be thought of as saying that the operations $x\to({\mkern1.5mu\text{-}\mkern1.5mu})$, for $x\in X$, preserve finite joins. For this reason, it has been called ‘preserving joins at primes’. Cf. Section 3.2 of [@Geh16], where it is used to characterise the lattices with residuation that are dual to topological algebras based on Boolean spaces.
There is a special case in which we can get our hands on the property of preserving joins at primes with a finitary scheme of relations between generators. This is the case where the lattice $L$ has enough join prime elements, i.e. every $a\in L$ is a finite join of join prime elements of $L$. This is for example true in free distributive lattices (where the meets of finite sets of generators are join prime), and it is intimately related to the interaction of domain theory and Stone duality as we have the following theorem.
[@Abramsky91 Theorem 2.4.5] A lattice has enough join primes if, and only if, its dual space endowed with the Scott topology is a domain.
Let $L$ be a lattice with enough join primes, and $X$ its dual space. If $P=J(L)$ is the subposet of join prime elements of $L$, the free distributive lattice on the *poset* $P$ is isomorphic to $L$. Further, $X\cong{\rm Idl}(P^{{\ensuremath{\mathrm{op}}}})$, the free directed join completion of $P^{{\ensuremath{\mathrm{op}}}}$ in the Scott topology, while $P^{{\ensuremath{\mathrm{op}}}}\cong{\rm Comp}(X)$, the set of compact elements of $X$. In particular, $X$ is an algebraic domain. Accordingly, we see that everything, i.e. $L$, $X$, and the compact elements of $X$, is determined by $P$. The posets $P$ that occur in this way were described already in [@Plo76], where the profinite domains were characterised as those algebraic domains for which the set of compact elements form a ‘MUB-complete poset’ in the nomenclature of [@AbrJung]. We now have a corollary of Proposition \[prop:preserves-joins-at-primes\].
Let $L$ and $M$ be DLs with dual spaces $X$ and $Y$, respectively. Suppose $L$ has enough join primes and let $P=J(L)$. Then the quotient of $F_\to(L\times M)$ by the congruence $\theta$ given by the following scheme is dual to the function space $[X,Y]$: $$p\to\bigvee G\approx\bigvee\{p\to b\mid b\in G\} \ \ \text{for $p\in P$ and $G\subseteq M$ finite}.$$
In the above, we have just talked about spectral spaces and domains, but in order to have a class of spectral domains not only closed under function spaces and products, but also under the various versions of powerdomain, one must restrict oneself to the so-called bifinite domains. These were introduced (in the setting of domains with a least element) in [@Plo76] as generated by special MUB-complete posets $P$ now known as Plotkin orders [@AbrJung Definition 4.2.1]. These also have a beautiful very self-dual description relative to Stone duality.
The following definition applies to categories concrete over the category $\textbf{Pos}$ of posets and monotone maps, such as the category of DLs or that of spectral spaces and spectral maps (w.r.t. the specialization order) with the obvious forgetful functors.
\[def:embedding-retraction-pair\] Let $\mathcal C$ be a category equipped with a faithful functor $U\colon \mathcal{C}\to\mathbf{Pos}$. A pair of morphisms $C\xrightarrow{f}D\xrightarrow{g}C$ in $\mathcal C$ is an *embedding-retraction-pair (e-r-p)* provided $(U(f),U(g))$ is an adjoint pair, and $U(f)$ is injective.[^9] Further, such an e-r-p is said to be *finite* if $U(C)$ is finite.
We have the following easy duality result.
\[cor:e-r-p-duality\] In Stone duality, the dual of a (finite) embedding-retraction-pair on either side of the duality is a (finite) embedding-retraction-pair on the other side.
We may then define bifiniteness in the setting of spectral spaces, rather than in the setting of domains as it is customarily done.
\[def:spectral-bif\] Let $X$ be a spectral space, and $L$ its dual lattice. We say that $X$ and $L$ are *bifinite* provided the following two equivalent conditions are satisfied:
1. $X$ is the cofiltered limit of the retractions of its finite e-r-p’s;
2. $L$ is the filtered colimit of the embeddings of its finite e-r-p’s.
The following proposition, which clearly implies that a bifinite lattice must have enough join primes, allows us to conclude that bifinite spectral spaces are bifinite domains. Thus, the above definition is no more general than the standard one.
\[prop:fin-lat-e-r-p\] Let $L$ be a distributive lattice and $K\subseteq L$ a finite sublattice. Then the following conditions are equivalent:
1. There is a lattice homomorphism $h\colon L\to K$ making $(i,h)$ an embedding-retraction-pair, where $i\colon K\to L$ is the inclusion;
2. 1. For all $b\in L$, ${\downarrow}b\cap K$ is a principal downset;
2. $J(K)\subseteq J(L)$.
Quantifiers, free constructions and duality {#s:Quant}
===========================================
In the categorical logic approach, cf. Sections \[s:algebras-from-logic\] and \[s:three-ex-spaces\], the stratification of the algebra of formulas (up to logical equivalence modulo $T$) provided by the hyperdoctrine $P\colon {\ensuremath{\mathbf{Con}}}^{\ensuremath{\mathrm{op}}}\to{\ensuremath{\mathbf{BA}}}$ is in a sense impredicative. Indeed, it starts from the algebra of sentences $P(\emptyset)$, which is what we ultimately want to understand, to build all formulas on a countably infinite set of variables. This contrasts with the step-wise construction of algebras of formulas outlined in Section \[s:modal-Vietoris\].
We want to understand quantification as a step-by-step construction. To this end, in this section we analyse from a duality theoretic viewpoint the inductive process of applying a layer of quantifiers in three settings. First, we focus on existential quantification in first-order logic over arbitrary structures. Then, on semiring and probabilistic quantifiers in first-order logic over finite structures.
As explained in Section \[s:algebras-from-logic\], Lindenbaum-Tarski algebras of predicate logics typically fail to be free algebras. The challenge then consists, in a sense, in building free objects which approximate the Lindenbaum-Tarski algebra we are interested in. We illustrate this idea in the following examples.
Existential quantification and Vietoris {#s:exists-vietoris}
---------------------------------------
For existential quantification in first-order logic, the framework can be loosely described as follows. Assume we are given a Boolean algebra of formulas $B$, and we build a new Boolean algebra $B_{\exists x}$ by adding a layer of the quantifier $\exists x$ to the formulas in $B$. We then have a quotient map $$\begin{tikzcd}
{\ensuremath{\mathbb{M}}}(B) \arrow[twoheadrightarrow]{r} & {B_{\exists x}}
\end{tikzcd}$$ sending $\Diamond {\varphi}$ to $\exists x.{\varphi}$, where ${\ensuremath{\mathbb{M}}}(B)$ is the Boolean algebra obtained by freely adding one layer of modality as described in Section \[s:modal-Vietoris\]. Dually, we get a continuous embedding $$\begin{tikzcd}
{\ensuremath{\mathcal{V}}}(X) \arrow[hookleftarrow]{r} & {X_{\exists x}}
\end{tikzcd}$$ where $X$ and $X_{\exists x}$ are the dual spaces of $B$ and $B_{\exists x}$, respectively. We have approximated the space $B_{\exists x}$ by means of the Vietoris space ${\ensuremath{\mathcal{V}}}(X)$, whose dual is a *free* object (namely, the free modal algebra on $B$). The problem then consists in characterising $X_{\exists x}$ as a subspace of ${\ensuremath{\mathcal{V}}}(X)$. This is addressed by observing that $X_{\exists x}$ is the image of a continuous map into ${\ensuremath{\mathcal{V}}}(X)$ constructed in a canonical way. In the remaining of this section we provide the necessary details.
Recall from Section \[s:three-ex-spaces\] that a first-order formula ${\varphi}\in {\ensuremath{\mathrm{FO}_\omega}}(T)$ can be identified with the set ${\ensuremath{\llbracket {\varphi}\rrbracket}}\subseteq {\ensuremath{\mathrm{Mod}_\omega}}/{\equiv}$ consisting of the (equivalence classes of) models with assignments satisfying ${\varphi}$. If the free variables of ${\varphi}$ are contained in $v_1,\dots,v_n$, we can restrict the variable assignments accordingly. Write $${\ensuremath{\mathrm{Mod}}}_n=\{[(A,\alpha\colon \{v_1,\ldots,v_n\}\to A)]\mid A \ \text{is a ${\ensuremath{\sigma}}$-structure and} \ A\models T\},$$ where $[(A,\alpha)]=[(A',\alpha')]$ if and only if $A,\alpha\models {\varphi}\Leftrightarrow A',\alpha'\models {\varphi}$ for every ${\varphi}\in {\ensuremath{\mathrm{FO}}}_n(T)$. Henceforth, we abuse notation and denote an arbitrary element of ${\ensuremath{\mathrm{Mod}}}_n$ by $(A,\alpha)$ instead of $[(A,\alpha)]$. Then, ${\ensuremath{\mathrm{FO}}}_n(T)$ embeds into ${\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$ via the map $$\begin{aligned}
{\ensuremath{\mathrm{FO}}}_n(T) \hookrightarrow {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n), \ \ [{\varphi}]\mapsto {\ensuremath{\llbracket {\varphi}\rrbracket}}_n=\{(A,\alpha)\in {\ensuremath{\mathrm{Mod}}}_n \mid A,\alpha\models {\varphi}\}.\end{aligned}$$ The projection map $$\pi_i\colon {\ensuremath{\mathrm{Mod}}}_n{\twoheadrightarrow}{\ensuremath{\mathrm{Mod}}}_{n\setminus i}$$ which forgets the value of the assignments on the variable $v_i$ induces a Boolean algebra embedding $$\pi_i^{-1}\colon {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i}) {\hookrightarrow}{\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$$ by applying the contravariant power-set functor. As in the hyperdoctrine approach, the homomorphism $\pi_i^{-1}$ has a lower adjoint and it is given by taking direct images under $\pi_i$. $$\begin{tikzcd}[column sep=2.0em]
{\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i}) \arrow[bend left=35, looseness=1]{rr}[description]{\pi_i^{-1}} & {\footnotesize{\text{$\top$}}} & {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n) \arrow[bend left=35, looseness=1]{ll}[description]{\pi_i({\mkern1.5mu\text{-}\mkern1.5mu})}
\end{tikzcd}$$ This lower adjoint map can be thought of as the quantifier $\exists v_i$. Indeed, it is readily seen that $\pi_i({\ensuremath{\llbracket {\varphi}\rrbracket}}_n)={\ensuremath{\llbracket \exists v_i. {\varphi}\rrbracket}}_{n\setminus i}$. More generally, abstracting away from the Boolean subalgebra ${\ensuremath{\mathrm{FO}}}_n(T) {\hookrightarrow}{\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$, we can consider any Boolean algebra embedding $$j\colon B\hookrightarrow {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$$ and regard it as a ‘semantically given logic’. The Boolean algebra obtained by adding a layer of the quantifier $\exists v_i$ to $B$ can be identified with the Boolean subalgebra $B_{\exists}^i$ of ${\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i})$ generated by the set of direct images $$\{\pi_i(j({\varphi}))\mid {\varphi}\in B\}.$$
We now focus on the dual of the transformation $B\leadsto B_{\exists}^i$. Let $f\colon \beta({\ensuremath{\mathrm{Mod}}}_n){\twoheadrightarrow}X$ be the continuous map dual to $j\colon B\hookrightarrow {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$. Here, $\beta({\ensuremath{\mathrm{Mod}}}_n)$ denotes the Čech-Stone compactification of ${\ensuremath{\mathrm{Mod}}}_n$ regarded as a discrete space, and is the dual Stone space of ${\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$. We obtain a continuous map $$\begin{tikzcd}[column sep=3.5em]
R\colon \beta({\ensuremath{\mathrm{Mod}}}_{n\setminus i}) \arrow{r}{\beta(\pi_i)^{-1}} & {\ensuremath{\mathcal{V}}}(\beta({\ensuremath{\mathrm{Mod}}}_n)) \arrow{r}{{\ensuremath{\mathcal{V}}}(f)} & {\ensuremath{\mathcal{V}}}(X).
\end{tikzcd}$$ The first component of $R$ is the preimage map $x\mapsto \beta(\pi_i){^{-1}}(x)$, where the function $\beta(\pi_i)\colon \beta({\ensuremath{\mathrm{Mod}}}_n) \to \beta({\ensuremath{\mathrm{Mod}}}_{n\setminus i})$ is the Stone dual of $\pi_i^{-1}\colon {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i}) \to {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$. The map $\beta(\pi_i)^{-1}$ is continuous because $\pi_i^{-1}$ has a lower adjoint. Indeed, the join-semilattice homomorphism $\pi_i({\mkern1.5mu\text{-}\mkern1.5mu})\colon {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)\to {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i})$ induces a Boolean algebra homomorphism ${\ensuremath{\mathbb{M}}}({\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n))\to {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i})$, whose dual map is precisely $\beta(\pi_i)^{-1}$.
We then have the following result.
\[p:exists-Vietoris\] The image of the continuous map $R\colon \beta({\ensuremath{\mathrm{Mod}}}_{n\setminus i}) \to {\ensuremath{\mathcal{V}}}(X)$ is the dual space of $B_{\exists}^i$.
It is not difficult to verify that $R^{-1}(\Diamond {\widehat}{{\varphi}})={\widehat}{\pi_i(j({\varphi}))}$ for every ${\varphi}\in B$, see e.g. Corollary 3.2 of [@BG2019]. Consequently, the Boolean algebra dual to the image of $R$ can be identified with the subalgebra of ${\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_{n\setminus i})$ generated by the elements of the form $\pi_i(j({\varphi}))$ for ${\varphi}\in B$, which is precisely $B_{\exists}^i$.
To sum up, the transformation $B\leadsto B_{\exists}^i$ which adds one layer of quantifier $\exists v_i$ dually corresponds to taking the image of the continuous map $R\colon \beta({\ensuremath{\mathrm{Mod}}}_{n\setminus i})\to {\ensuremath{\mathcal{V}}}(X)$, canonically constructed from the continuous function $f\colon \beta({\ensuremath{\mathrm{Mod}}}_n){\twoheadrightarrow}X$. For a step-by-step treatment of quantifiers, we now want to add to $B_{\exists}^i$ the formulas which were already in $B$. Hence, we take the Boolean subalgebra of ${\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$ generated by the union $B\cup B_{\exists}^i$, which coincides with the image of the obvious Boolean algebra homomorphism $B+B_{\exists}^i\to {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Mod}}}_n)$. This corresponds, dually, to taking the image of the continuous product map $$\begin{tikzcd}[column sep=5.5em]
\beta({\ensuremath{\mathrm{Mod}}}_n) \arrow{r}{(R\circ \beta(\pi_i))\times f} & {\ensuremath{\mathcal{V}}}(X)\times X.
\end{tikzcd}$$ An essential obstacle to a two-sided duality theory for quantifiers is the lack of a characterisation of the continuous maps $\beta({\ensuremath{\mathrm{Mod}}}_{n})\to {\ensuremath{\mathcal{V}}}(X)\times X$ arising this way. We will return to this point in Section \[s:outlook\].
Semiring quantifiers and measures {#s:semiring-quant}
---------------------------------
The existential quantifier $\exists$ captures the existence, or non-existence, of an element satisfying a property. As such, it is a two-valued query. Semiring quantifiers, as studied for instance in logic on words, generalise $\exists$ by allowing us to count the number of witnesses in a given semiring.[^10] Recall that a *semiring* is a tuple $(S,+,\cdot,0,1)$ where $(S,+,0)$ is a commutative monoid, $(S,\cdot,1)$ is a monoid, the operation $\cdot$ distributes over $+$, and $0\cdot s=0=s\cdot 0$ for all $s\in S$. If $S$ is a fixed finite semiring, every element $k\in S$ determines a quantifier $\exists_{k}$. Given a first-order formula ${\varphi}$ with one free variable $v$ and a finite structure $A$, the semantics of the sentence $\exists_{k}v.{\varphi}(v)$ is given as follows: $$\begin{aligned}
A \models \exists_{k}v.{\varphi}(v)
& {{\quad\text{iff}\quad}} 1+\cdots+1 \text{ (repeated $m$-times) is equal to $k$ in $S$} \\
& \text{where $m$ is the number of elements $a\in A$ such that $A\models {\varphi}(a)$.}\end{aligned}$$ Notice that $A$ must be finite, for otherwise the set $\{a\in A\mid A\models {\varphi}(a)\}$ may be infinite and the sum $1+\cdots+1$ undefined. This problem could be overcome by requiring that $S$ be complete in an appropriate sense. The existential quantifier $\exists$ is recovered by letting $S={\ensuremath{\mathbf{2}}}$ be the two-element Boolean ring and $k=1$.
Let ${\ensuremath{\mathrm{Fin}}}_n$ be the subset of ${\ensuremath{\mathrm{Mod}}}_n$ consisting of the finite models with assignments. Given a Boolean algebra embedding $j\colon B\hookrightarrow {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Fin}}}_n)$ we can construct, akin to the case of $\exists$, a Boolean algebra $B_{\exists_S}^i$ obtained by adding a layer of semiring quantifiers $\exists_{k}v_i$ for $k\in S$. For every ${\varphi}\in B$ and $(A,\alpha)\in {\ensuremath{\mathrm{Fin}}}_{n\setminus i}$, write $m_{{\varphi},(A,\alpha)}$ for the number of elements $a$ in $A$ such that $(A, \alpha\cup \{v_i \mapsto a\})$ belongs to $j({\varphi})$. Then, $B_{\exists_S}^i$ can be defined as the Boolean subalgebra of ${\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Fin}}}_{n\setminus i})$ generated by the sets $$\{(A,\alpha) \in {\ensuremath{\mathrm{Fin}}}_{n\setminus i} \mid 1 + \cdots + 1 \text{ ($m_{{\varphi},(A,\alpha)}$-times) is equal to $k$} \}, \ \text{for} \ {\varphi}\in B \ \text{and} \ k\in S.$$
In order to describe the dual of the transformation $B\leadsto B_{\exists_S}^i$, we need to understand which construction plays the role of the Vietoris hyperspace in the case of semiring quantifiers. For this purpose, notice that the Vietoris space ${\ensuremath{\mathcal{V}}}(X)$ can be identified with a space of two-valued finitely additive measures on $X$, whenever $X$ is a Boolean space.[^11] Regard $X$ as a measurable space where the measurable subsets are precisely the clopens, i.e. the elements of the Boolean algebra $B$ dual to $X$. A finitely additive ${\ensuremath{\mathbf{2}}}$-valued measure on $X$ is then a function $\mu\colon B\to {\ensuremath{\mathbf{2}}}$ satisfying $$\mu(0)=0 {{\quad\text{and}\quad}} \mu(a\vee b)\vee \mu(a\wedge b)=\mu(a)\vee \mu(b) \ \ \forall a,b\in B.$$ Denote by ${\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{2}}})$ the collection of all finitely additive ${\ensuremath{\mathbf{2}}}$-valued measures on $X$, and equip it with the subspace topology induced by the product topology on ${\ensuremath{\mathbf{2}}}^B$.
For every Boolean space $X$, the Vietoris hyperspace ${\ensuremath{\mathcal{V}}}(X)$ is homeomorphic to ${\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{2}}})$ via the map $${\ensuremath{\mathcal{V}}}(X)\to {\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{2}}}), \ \ C\mapsto \mu_C, \ \ \text{where} \ \ \mu_C(a)=\begin{cases} 1 & \mbox{if \ ${\widehat}{a}\cap C\neq \emptyset$,} \\ 0 & \mbox{otherwise.} \end{cases}$$
It is straightforward to verify that the map in the statement is a continuous bijection, with inverse ${\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{2}}})\to{\ensuremath{\mathcal{V}}}(X)$, $\mu\mapsto \bigcap{\{{\widehat}{a}\subseteq X\mid \mu(\neg a)=0\}}$. Every continuous bijection between compact Hausdorff spaces is a homeomorphism, hence the statement follows.
For semiring quantifiers, the hyperspace ${\ensuremath{\mathcal{V}}}(X)$ will thus be replaced by ${\ensuremath{\mathcal{M}}}(X,S)$, the space of finitely additive $S$-valued measures on $X$. An element of ${\ensuremath{\mathcal{M}}}(X,S)$ is a function $\mu\colon B\to S$ satisfying $$\label{eq:fin-add}
\mu(0)=0 {{\quad\text{and}\quad}} \mu(a\vee b)+ \mu(a\wedge b)=\mu(a)+ \mu(b) \ \ \forall a,b\in B,$$ and the set ${\ensuremath{\mathcal{M}}}(X,S)$ is equipped with the subspace topology induced by the product topology on $S^B$. The equations in , encoding finite additivity, translate into equaliser diagrams in the category of Boolean spaces. Hence, the resulting space ${\ensuremath{\mathcal{M}}}(X,S)$ is again Boolean. Explicitly, the topology of ${\ensuremath{\mathcal{M}}}(X,S)$ is generated by the (clopen) subsets of the form $$[a,k]=\{\mu\in{\ensuremath{\mathcal{M}}}(X,S)\mid \mu(a)=k\}, \ \ \text{for} \ a\in B \ \text{and} \ k\in S.$$
In order to describe the dual of the construction $B\leadsto B_{\exists_S}^i$, we perform two steps. First, given a finite model with assignment $(A,\alpha)\in {\ensuremath{\mathrm{Fin}}}_{n\setminus i}$, let $$\label{eq:fsp-semiring}
{\ensuremath{\delta}}_{(A,\alpha)}\colon {\ensuremath{\mathrm{Fin}}}_n \to S$$ be the ‘$S$-valued characteristic function’ of $\pi_i^{-1}(A,\alpha)$, where $\pi_i\colon {\ensuremath{\mathrm{Fin}}}_n \to {\ensuremath{\mathrm{Fin}}}_{n\setminus i}$ is the map which forgets the assignment of the $i$th variable. That is, ${\ensuremath{\delta}}^i_{(A,\alpha)}(A',\alpha')$ is $1$ if $A = A'$ and $\alpha$ agrees with $\alpha'$ on the variables $v_1, \dots, v_{i-1}, v_{i+1}, \dots, v_n$, and $0$ otherwise. Since $A$ is finite, ${\ensuremath{\delta}}^i_{(A,\alpha)}$ belongs to the set ${\ensuremath{\mathbf{S}}}({\ensuremath{\mathrm{Fin}}}_n)$ of finitely supported $S$-valued functions on ${\ensuremath{\mathrm{Fin}}}_n$. In the second step, in order to construct a measure, we extend the function ${\ensuremath{\delta}}^i_{(A,\alpha)}$ to subsets of ${\ensuremath{\mathrm{Fin}}}_n$ by adding up all the non-zero values in a given subset. More generally, if $T$ is a set and $g\colon T\to S$ is a finitely supported function, the map $$\int g\colon {\ensuremath{\mathcal{P}}}(T) \to S,\quad P \mapsto \int_P g {{\quad\text{computed as}\quad}} \sum_{x\in P} g(x)$$ is a finitely additive $S$-valued measure on $\beta(T)$. We obtain an integration map[^12] $$\int\colon {\ensuremath{\mathbf{S}}}(T) \to {\ensuremath{\mathcal{M}}}(\beta(T), S).$$
Now, let $f\colon \beta({\ensuremath{\mathrm{Fin}}}_n) \to X$ be the dual of the embedding $j\colon B {\hookrightarrow}{\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Fin}}}_n)$. Consider the composite $$\label{eq:Fin-to-meas-semirings}
{\ensuremath{\mathrm{Fin}}}_{n\setminus i} \xrightarrow{{{\enspace{\ensuremath{\delta}}^i_{({\mkern1.5mu\text{-}\mkern1.5mu})}\enspace}}} {\ensuremath{\mathbf{S}}}({\ensuremath{\mathrm{Fin}}}_n) \xrightarrow{{{\enspace\int\enspace}}} {\ensuremath{\mathcal{M}}}(\beta({\ensuremath{\mathrm{Fin}}}_n), S) \xrightarrow{{{\enspacef_*\enspace}}} {\ensuremath{\mathcal{M}}}(X,S)$$ where $f_*$ sends a measure to its pushforward along $f$, i.e. $f_*(\mu)(a)=\mu(f^{-1}({\widehat}{a}))$ for every $\mu\in {\ensuremath{\mathcal{M}}}(\beta({\ensuremath{\mathrm{Fin}}}_n),S)$ and $a\in B$. The space ${\ensuremath{\mathcal{M}}}(X,S)$ is compact and Hausdorff, whence the above composition extends to a (unique) continuous function $$\label{eq:map-R}
R \colon \beta({\ensuremath{\mathrm{Fin}}}_{n\setminus i}) \to {\ensuremath{\mathcal{M}}}(X, S).$$ The following result generalises Proposition \[p:exists-Vietoris\] and can be proved in a similar manner (we omit the details here).
\[t:semiring-quant-measures\] The image of the continuous map $R\colon \beta({\ensuremath{\mathrm{Mod}}}_{n\setminus i}) \to {\ensuremath{\mathcal{M}}}(X,S)$ is the dual space of $B_{\exists_S}^i$.
The connection between semiring quantifiers and spaces of finitely additive measures was first explored, in the context of logic on words, in [@GPR2017]. The treatment in this section could be adapted to deal with any profinite semiring, such as the *tropical semiring* $(\mathbb{N}\cup\{\infty\},\min,+,\infty,0)$, and not just the finite ones. See [@R2020].
Probabilistic quantifiers and structural limits
-----------------------------------------------
Topological methods are also employed in the study of structural limits in finite model theory. A systematic investigation of limits of finite structures has been developed by Ne[š]{}et[ř]{}il and Ossona de Mendez and is based on an embedding, called the *Stone pairing*, of the collection of finite structures into a space of probability measures [@NO2012; @NOdM2020]. The latter space is complete, thus it provides the limit objects for those sequences of finite structures which embed as Cauchy sequences. Although this space of measures and the Stone pairing embedding did not originate from duality, in recent work we showed that a closely related version of the Stone pairing can be understood — via duality — as the embedding of finite structures into a space of types. Namely, the space of $0$-types of an extension of first-order logic obtained by adding a layer of certain probabilistic quantifiers [@GJR2020]. In the following, we highlight the similarities between the Stone pairing embedding and the space-of-measures construction introduced above in the context of existential and semiring quantification.
For every first-order formula ${\varphi}$ with free variables contained in $v_1,\dots,v_n$, and finite structure $A$, the *Stone pairing* of ${\varphi}$ and $A$ is defined as $${\ensuremath{\left<{\varphi},A\right>}} \ = \ \frac{|\{ \overline a \in A^n \mid A \models {\varphi}(\overline a)\}|}{|A|^n}.$$ In other words, ${\ensuremath{\left<{\varphi},A\right>}}$ is the probability that a random assignment of the variables $v_1,\dots,v_n$ in $A$ satisfies the formula ${\varphi}$. Upon fixing the second coordinate, the map ${\ensuremath{\left<{\mkern1.5mu\text{-}\mkern1.5mu},A\right>}}$ is a finitely additive measure on the dual space of the Lindenbaum-Tarski algebra of all first-order formulas ${\ensuremath{\mathrm{FO}_\omega}}$, with values in the unit interval $[0,1]$. I.e., $${\ensuremath{\left<\bot,A\right>}}=0 {{\quad\text{and}\quad}} {\ensuremath{\left<{\varphi}\vee \psi,A\right>}}+ {\ensuremath{\left<{\varphi}\wedge \psi,A\right>}}={\ensuremath{\left<{\varphi},A\right>}}+ {\ensuremath{\left<\psi,A\right>}} \ \ \forall {\varphi},\psi \in{\ensuremath{\mathrm{FO}_\omega}}.$$ Since the Boolean algebra ${\ensuremath{\mathrm{FO}_\omega}}$ is dual to the space of models and valuations ${\ensuremath{\mathrm{Mod}_\omega}}$, we obtain an embedding $$\begin{aligned}
{\ensuremath{\left<{\mkern1.5mu\text{-}\mkern1.5mu},{\mkern1.5mu\text{-}\mkern1.5mu}\right>}}\colon {\ensuremath{\mathrm{Fin}}}\longrightarrow {\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},[0,1]),\quad A \mapsto {\ensuremath{\left<{\mkern1.5mu\text{-}\mkern1.5mu},A\right>}}\end{aligned}$$ where ${\ensuremath{\mathrm{Fin}}}$ is the collection of finite structures, up to isomorphism (with the notation of Section \[s:semiring-quant\], ${\ensuremath{\mathrm{Fin}}}={\ensuremath{\mathrm{Fin}}}_0$). This is the *Stone pairing* embedding introduced by [Nešetřil and Ossona de Mendez]{}.
By restricting ${\ensuremath{\left<{\mkern1.5mu\text{-}\mkern1.5mu},A\right>}}$ to suitable fragments of first-order logic, [Nešetřil and Ossona de Mendez]{} obtained a unifying framework that captures various notions of convergence of finite structures, such as Lovasz–Szegedy convergence, Benjamini–Schramm convergence, elementary convergence, etc.[^13] Their insight was that each of these notions of convergence corresponds to a fragment of first-order logic. Further, since the ensuing spaces of finitely additive measures are complete, they admit a limit for every sequence of finite structures which embeds as a Cauchy sequence.
In section \[s:semiring-quant\], we defined a map from a set of finite structures with evaluations into a space of finitely additive measures, see equation , and showed that it dually captures the adding of a layer of semiring quantifiers. By analogy, we may ask if the Stone pairing also corresponds to applying a layer of quantifiers. One immediate obstacle is that the spaces $[0,1]$ and ${\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},[0,1])$ are not Boolean, whence not amenable to the methods of Stone duality for Boolean algebras.
We can overcome this problem by replacing $[0,1]$ with a profinite version of the unit interval obtained from a codirected system of finitary approximations of real numbers in $[0,1]$. This profinite space ${\ensuremath{\mathbf{\Gamma}}}$ is naturally equipped with a Priestley space structure and can therefore be studied using Stone-Priestley duality for distributive lattices. To define ${\ensuremath{\mathbf{\Gamma}}}$, we divide the unit interval into $n$ segments of equal length, i.e. $${\ensuremath{\mathbf{\Gamma}}}_n {{\enspace=\enspace}} \{ 0 {{\enspace<\enspace}} \tfrac{1}{n} {{\enspace<\enspace}} \tfrac{2}{n} {{\enspace<\enspace}} \dots {{\enspace<\enspace}} 1\}.$$ The chain ${\ensuremath{\mathbf{\Gamma}}}_n$ provides a finite approximation of $[0,1]$. The higher the value of $n\in{\mathbb{N}}$, the better the approximation is. Whenever $n\mid m$, we consider the flooring function ${\ensuremath{\mathbf{\Gamma}}}_m\to {\ensuremath{\mathbf{\Gamma}}}_n$ sending $\frac{a}{m}$ to the largest $\frac{b}{n} \in {\ensuremath{\mathbf{\Gamma}}}_n$ such that $\frac{b}{n} \leq \frac{a}{m}$. Note that the finite chains ${\ensuremath{\mathbf{\Gamma}}}_n$ with flooring functions between them form a codirected diagram in the category ${\ensuremath{\mathbf{Pos}_f}}$ of finite posets with monotone maps. The limit of this diagram is an object ${\ensuremath{\mathbf{\Gamma}}}$ of the pro-completion of ${\ensuremath{\mathbf{Pos}_f}}$, which is the category of Priestley spaces with continuous monotone maps.[^14] See e.g. Corollary VI.3.3 in [@Johnstone1986].
Concretely, the elements of ${\ensuremath{\mathbf{\Gamma}}}$ are the sequences of approximations $(x_n)_n\in\prod_{n\in{\mathbb{N}}}{{\ensuremath{\mathbf{\Gamma}}}_n}$ which are compatible with the flooring functions. Every $q\in (0,1]$ determines an element $q{^{-}}\in{\ensuremath{\mathbf{\Gamma}}}$, namely the sequence $$q{^{-}}=(q_1{^{-}},q_2{^{-}},q_3{^{-}},\ldots) {{\quad\text{where}\quad}} q_n{^{-}}=\max \{ \tfrac{a}{n} \in {\ensuremath{\mathbf{\Gamma}}}_n \mid \tfrac{a}{n} < q \}$$ which approximates $q$ from below while never reaching it. Further, if $q$ is rational, we also get a lower approximating sequence $q{^\circ}\in{\ensuremath{\mathbf{\Gamma}}}$ which eventually stabilises at $q$: $$q{^\circ}=(q_1{^\circ},q_2{^\circ},q_3{^\circ},\ldots) {{\quad\text{where}\quad}} q_n{^\circ}=\max \{ \tfrac{a}{n} \in {\ensuremath{\mathbf{\Gamma}}}_n \mid \tfrac{a}{n} \leq q \}.$$ In fact, any point of ${\ensuremath{\mathbf{\Gamma}}}$ is of one of these two types. We can thus think of ${\ensuremath{\mathbf{\Gamma}}}$ as a copy of the unit interval where all the non-zero rationals are doubled (in the picture, $q$ is rational while $r$ is irrational):
at (6.15,0) (1cc) ; at (6,0) (1mm) ; at (0,0) (0cc) ; at (4.4,0) (r) ; at (1.75,0) (qc) ; at (1.60,0) (qm) ; at ($(r) -(-0.1,0.5)$) (rmm) [$r{^{-}}$]{}; at ($(qc)+(0.15,0.5)$) (qcc) [$q{^\circ}$]{}; at ($(qm)-(0.05,0.5)$) (qmm) [$q{^{-}}$]{}; at ($(1cc)+(0.1,0.5)$) [$1{^\circ}$]{}; at ($(1mm)-(0.05,0.5)$) [$1{^{-}}$]{}; at ($(0cc)+(0.15,0.5)$) [$0{^\circ}$]{};
(1mm.center) – (qc.center); (qm.center) – (0cc.center);
in [1cc,1mm,0cc,r,qm,qc]{} [ ($(\pt)-(0,0.1)$) – ($(\pt)+(0,0.1)$); ]{} at (-1.1, 0) [${\ensuremath{\mathbf{\Gamma}}}\enspace =$]{};
Equivalently, ${\ensuremath{\mathbf{\Gamma}}}$ is a copy of the Cantor space with an extra top element which is topologically isolated (corresponding to $1{^\circ}$). The natural order of ${\ensuremath{\mathbf{\Gamma}}}$, illustrated in the previous picture, is the total order defined by the two conditions
- $r{^\circ}< s{^{-}}$ if and only if $r < s$ in $[0,1]$, and
- $q{^{-}}< q{^\circ}$ for every $q\in (0,1]$,
and its topology is the interval topology. Note that ${\ensuremath{\mathbf{\Gamma}}}$ retracts onto $[0,1]$. Indeed, the continuous surjection $$\gamma\colon {\ensuremath{\mathbf{\Gamma}}}\to[0,1], \ \ q{^{-}},q{^\circ}\mapsto q$$ has a (lower semicontinuous) section $$\iota\colon [0,1]\to {\ensuremath{\mathbf{\Gamma}}}, \ \ \iota(q)=\begin{cases}
q{^\circ}& \text{if $q$ is rational} \\
q{^{-}}& \text{otherwise}.
\end{cases}$$
The additive structure of $[0,1]$ lifts to ${\ensuremath{\mathbf{\Gamma}}}$ (as can be derived by duality for additional operators) so that it makes sense to consider the set ${\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{\Gamma}}})$ of finitely additive probability measures on a Boolean space $X$ with values in ${\ensuremath{\mathbf{\Gamma}}}$. This construction can be generalised to any Priestley space $X$, and it turns out that the assignment $X\mapsto {\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{\Gamma}}})$ is an endofunctor on the category of Priestley spaces. In particular, a continuous monotone map of Priestley spaces $f\colon X\to Y$ is sent to the map $$f_*\colon {\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{\Gamma}}})\to {\ensuremath{\mathcal{M}}}(Y,{\ensuremath{\mathbf{\Gamma}}})$$ taking a measure to its pushforward along $f$. Furthermore, the retraction-section pair $\gamma\colon {\ensuremath{\mathbf{\Gamma}}}\leftrightarrows [0,1]{ \nobreak
\mskip6mu plus1mu
\mathpunct{} \nonscript
\mkern-\thinmuskip
{:} \mskip2mu
\relax
}\iota$ lifts to a retraction-section pair $$\gamma^\#\colon {\ensuremath{\mathcal{M}}}(X,{\ensuremath{\mathbf{\Gamma}}})\leftrightarrows {\ensuremath{\mathcal{M}}}(X,[0,1]){ \nobreak
\mskip6mu plus1mu
\mathpunct{} \nonscript
\mkern-\thinmuskip
{:} \mskip2mu
\relax
}\iota^\#, {{\quad\text{where}\quad}} \gamma^\#(\mu)=\gamma \circ \mu {{\quad\text{and}\quad}} \iota^\#(\mu)=\iota\circ\mu.$$
Now we define a ${\ensuremath{\mathbf{\Gamma}}}$-valued variant of the Stone pairing by following the strategy set out in Section \[s:semiring-quant\] in the case of semiring quantifiers. Fix $n\in{\mathbb{N}}$, and let ${\ensuremath{\mathcal{F}}}({\ensuremath{\mathrm{Fin}}}_n, {\ensuremath{\mathbf{\Gamma}}})$ be the set of finitely supported functions ${\ensuremath{\mathrm{Fin}}}_n\to{\ensuremath{\mathbf{\Gamma}}}$ with total value $1{^\circ}$. We get a map $\delta_{({\mkern1.5mu\text{-}\mkern1.5mu})}\colon {\ensuremath{\mathrm{Fin}}}\to {\ensuremath{\mathcal{F}}}({\ensuremath{\mathrm{Fin}}}_n, {\ensuremath{\mathbf{\Gamma}}})$ sending a finite structure $A$ to $$\delta_A\colon {\ensuremath{\mathrm{Fin}}}_n \to {\ensuremath{\mathbf{\Gamma}}}, {{\quad\text{where}\quad}} \delta_A(A',\alpha') =
\begin{cases}
\left(\frac{1}{|A|^n}\right){^\circ}& \text{if } A' = A \\[0.7em]
0{^\circ}& \text{otherwise}.
\end{cases}$$ The map $\delta_{({\mkern1.5mu\text{-}\mkern1.5mu})}$ is the (normalized) ${\ensuremath{\mathbf{\Gamma}}}$-valued version of the function introduced in for semiring quantifiers. In a similar way, to move from finitely supported functions to measures, for every set $T$ we consider the integration map $$\int\colon {\ensuremath{\mathcal{F}}}(T,{\ensuremath{\mathbf{\Gamma}}}) \to {\ensuremath{\mathcal{M}}}(\beta(T),{\ensuremath{\mathbf{\Gamma}}}),\quad f\mapsto \int f.$$ Lastly, define the following composition $$\begin{tikzcd}
R_n\colon {\ensuremath{\mathrm{Fin}}}\arrow{r}{\delta_{({\mkern1.5mu\text{-}\mkern1.5mu})}} & {\ensuremath{\mathcal{F}}}({\ensuremath{\mathrm{Fin}}}_n,{\ensuremath{\mathbf{\Gamma}}}) \arrow{r}{\int} & {\ensuremath{\mathcal{M}}}(\beta({\ensuremath{\mathrm{Fin}}}_n),{\ensuremath{\mathbf{\Gamma}}}) \arrow{r}{f_*} & {\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}}}_n, {\ensuremath{\mathbf{\Gamma}}})
\end{tikzcd}$$ where $f\colon \beta({\ensuremath{\mathrm{Fin}}}_n) \to {\ensuremath{\mathrm{Mod}}}_n$ is the dual map of the Boolean algebra homomorphism $${\ensuremath{\mathrm{FO}}}_n \to {\ensuremath{\mathcal{P}}}({\ensuremath{\mathrm{Fin}}}_n), \ \ {\varphi}\mapsto {\ensuremath{\llbracket {\varphi}\rrbracket}}\cap {\ensuremath{\mathrm{Fin}}}_n.$$ The map $R_n$ can be extended to a continuous function $\widetilde{R}_n\colon \beta({\ensuremath{\mathrm{Fin}}})\to {\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}}}_n,{\ensuremath{\mathbf{\Gamma}}})$, corresponding to the map in . Using the fact that the space ${\ensuremath{\mathrm{Mod}_\omega}}$ is the codirected limit of the ${\ensuremath{\mathrm{Mod}}}_n$’s for $n\in{\mathbb{N}}$, and the functor ${\ensuremath{\mathcal{M}}}({\mkern1.5mu\text{-}\mkern1.5mu},{\ensuremath{\mathbf{\Gamma}}})$ preserves codirected limits, we can ‘glue’ the maps $\widetilde{R}_n$ to get a continuous function $\widetilde{R}\colon \beta({\ensuremath{\mathrm{Fin}}}) \to {\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},{\ensuremath{\mathbf{\Gamma}}})$. The restriction $R\colon {\ensuremath{\mathrm{Fin}}}\to{\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},{\ensuremath{\mathbf{\Gamma}}})$ of $\widetilde{R}$ is an equivalent ${\ensuremath{\mathbf{\Gamma}}}$-valued version of the Stone pairing, as expressed by the commutativity of the following diagram. $$\begin{tikzcd}[row sep=2em]
{} & {\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},{\ensuremath{\mathbf{\Gamma}}})\ar[bend left=25]{dd}{\gamma^\#} \\
{\ensuremath{\mathrm{Fin}}}\ar{ru}{R}\ar[swap]{rd}{{\ensuremath{\left<{\mkern1.5mu\text{-}\mkern1.5mu},{\mkern1.5mu\text{-}\mkern1.5mu}\right>}}} & \\
& {\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},[0,1])\ar[bend left=25]{uu}{\iota^\#}
\end{tikzcd}$$ The map $R$, and more precisely the way it is constructed, provides an interesting link between the theory of structural limits and the inductive study of semiring quantifiers. Further, the duality approach allows us to see (the ${\ensuremath{\mathbf{\Gamma}}}$-valued version of) the Stone pairing as an embedding of the finite structures into a space of types. This is the content of the following theorem, which is a special case of more general results in [@GJR2020].
The Boolean space ${\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},{\ensuremath{\mathbf{\Gamma}}})$ is dual to the Lindenbaum-Tarski algebra of the propositional logic having as atoms ${\mathbf{p}_{\geq q}\,} {\varphi}$ and ${\mathbf{p}_{< q}\,} {\varphi}$, for each ${\varphi}\in {\ensuremath{\mathrm{FO}_\omega}}$ and $q\in [0,1]\cap \mathbb Q$, and the following inference rules (along with the usual ones for the Boolean connectives): $$\addtolength{\fboxsep}{4pt}
\boxed{
\begin{gathered}
\infer[{\scriptstyle(\mathrm{if} \ p\, \leq \, q)}]
{{\mathbf{p}_{\geq p}\,} {\varphi}}{{\mathbf{p}_{\geq q}\,} {\varphi}}
\hspace{1.5em}
\infer[{\scriptstyle(\mathrm{if} \ {\varphi}\, \vdash \, \psi)}]
{{\mathbf{p}_{\geq q}\,} \psi}{{\mathbf{p}_{\geq q}\,} {\varphi}}
\hspace{1.5em}
\infer
{{\mathbf{p}_{\geq 0}\,} \bot}{}
\hspace{1.5em}
\infer[{\scriptstyle(\mathrm{if} \ q \, > \, 0)}]
{{\mathbf{p}_{< q}\,} \bot}{}
\hspace{1.5em}
\infer
{{\mathbf{p}_{\geq q}\,} \top}{}
\hspace{1.5em}
\infer=
{\neg {\mathbf{p}_{< q}\,} {\varphi}}{{\mathbf{p}_{\geq q}\,} {\varphi}} \\[2ex]
\infer
{{\mathbf{p}_{\geq p+q-r}\,}({\varphi}\vee \psi) \,\vee\, {\mathbf{p}_{\geq r}\,}({\varphi}\wedge \psi)}{{\mathbf{p}_{\geq p}\,} {\varphi}\,\wedge\, {\mathbf{p}_{\geq q}\,} \psi}
\hspace{1em}
\infer[{\hspace{0.4em}\scriptstyle(\mathrm{if} \ 0 \, \leq \, p+q-r \, \leq \, 1)}]
{{\mathbf{p}_{\geq p}\,} {\varphi}\,\vee\, {\mathbf{p}_{\geq q}\,} \psi}{{\mathbf{p}_{\geq p+q-r}\,}({\varphi}\vee \psi)\,\wedge\,{\mathbf{p}_{\geq r}\,}({\varphi}\wedge \psi)}
\end{gathered}
}$$
The intended models for this extension of FO are the measures $\mu\in{\ensuremath{\mathcal{M}}}({\ensuremath{\mathrm{Mod}_\omega}},{\ensuremath{\mathbf{\Gamma}}})$, and the probabilistic quantifiers ${\mathbf{p}_{\geq q}\,}$ and ${\mathbf{p}_{< q}\,}$ are interpreted as follows: $$\mu\models {\mathbf{p}_{\geq q}\,} {\varphi}\ \Leftrightarrow \ \mu({\varphi})\geq q{^\circ}{{\quad\text{and}\quad}} \mu\models {\mathbf{p}_{< q}\,} {\varphi}\ \Leftrightarrow \ \mu({\varphi})< q{^\circ}.$$ In particular, if $A$ is a finite structure, ${\ensuremath{\left<{\mkern1.5mu\text{-}\mkern1.5mu},A\right>}}\models {\mathbf{p}_{\geq q}\,} {\varphi}$ if and only if ${\varphi}$ is satisfied in $A$ with probability at least $q$. Similarly for ${\mathbf{p}_{< q}\,} {\varphi}$. Note that these probabilistic quantifiers bind all free variables in a formula. Thus, once applied a layer of quantifiers to ${\ensuremath{\mathrm{FO}_\omega}}$, we obtain an algebra of *sentences*. These sentences are seen as propositional atoms for a new logic and, by the previous theorem, the Stone pairing can be seen as embedding the collection of finite structures (up to isomorphism) into the space of $0$-types for this logic.
Therefore, we see that [Nešetřil and Ossona de Mendez]{}’s Stone pairing dually corresponds to adding a layer of probabilistic quantifiers. As such, it can be regarded as an instance of the inductive approach described in Section \[s:modal-Vietoris\].
Outlook {#s:outlook}
=======
We saw in Section \[s:exists-vietoris\] that adding a layer of existential quantifier $\exists$ to a Boolean algebra $B$ of first-order formulas (with free variables in $v_1,\ldots,v_n$) dually corresponds to taking the image of a continuous map $\beta({\ensuremath{\mathrm{Mod}}}_{n})\to {\ensuremath{\mathcal{V}}}(X)\times X$, where $X$ is the dual Stone space of $B$. A similar statement holds for semiring quantifiers, cf. Section \[s:semiring-quant\]. This continuous map is defined in a canonical way, and ensures the *soundness* of the construction. But we do not know, so far, how to characterise the continuous maps $\beta({\ensuremath{\mathrm{Mod}}}_{n})\to {\ensuremath{\mathcal{V}}}(X)\times X$ arising in this manner, which would establish the *completeness* of the construction. This is a notable obstacle to a full duality theoretic understanding of step-by-step quantification in predicate logics. On the other hand, such a completeness result is available for semiring quantifiers in logic on words, and makes use of the richer structure of the spaces of models (in the form of monoid actions). See Proposition VI.7 and Theorem VI.8 of [@GPR2017], where this is called a ‘Reutenauer-type theorem’. A question arises, whose answer would significantly further the use of topological methods in logic: *Is there a Reutenauer-type result for first-order logic over arbitrary structures?*
In this paper we have discussed several examples of topological methods in logic and computer science, highlighting their duality theoretic nature. However, there are topological methods in logic which have been successfully developed and applied, but for which no duality theoretic explanation is available so far. An appealing example is the theory of limits of schema mappings as developed in database theory by Kolaitis and his collaborators [@Kolaitis2018]. Understanding these tools and results from a duality theoretic perspective may yield new useful insights and is an exciting venue for future investigations. Another example are 0–1 laws in finite model theory, illustrating the limits of the expressive power of first-order logic over finite structures, see e.g. [@Fagin1976]. These are only some of the many opportunities for further development of the duality approach, which would contribute to unify the ‘structure’ and ‘power’ strands in theoretical computer science.
One of the main themes of our present contribution has been the analysis of step-by-step constructions in logic, which yield *free* objects on the algebra side and *co-free* objects on the space side. Note that, even though the step-wise process of adding a layer of connectives yields a *monad* in the (co)limit, the one-step functor is typically a *comonad*. For instance, the functor on Boolean algebras which adds one layer of modality $\Diamond$ is a comonad, whose dual is the Vietoris monad on Boolean spaces.
The recent work of Samson Abramsky and his coauthors on comonads for model-theoretic games [@Abramsky2017b; @AbramskyShah2018] is tightly related to this viewpoint. The connection between the comonadic approach and the duality one remains to be explored, and is an interesting avenue of research. In this direction, one may point out that the Ehrenfeucht-Fra[ï]{}ss[é]{} comonad introduced by Abramsky and Shah arises as the density comonad for a certain (contravariant) realization functor from a category of primitive positive sentences into the category of structures.
Besides the inductive treatment of quantifiers, another important theme of this paper has been the lack of freeness of Lindenbaum-Tarski algebras of first-order theories. Indeed, we pointed out that this is one of the main obstacles to a satisfactory algebraic and duality theoretic approach to predicate logics.
Another place where the lack of freeness plays an important role is quantum information and computation, to which Samson Abramsky has greatly contributed. There, as recently observed by Abramsky, the lack of freeness (of certain Boolean subalgebras of partial Boolean algebras) can be regarded as an obstruction to classicality. In fact, in the presence of freeness, the Kochen-Specker theorem does not apply. See [@AB2020]. Interestingly, in this context, this obstruction represents a (quantum) advantage.
We conclude with a question concerning a wider issue, which is instrumental in addressing the divide between structure and power, one of the main focuses of Samson Abramsky’s recent research. A difference between general model theory and finite model theory which is often emphasised is the fact that the major structure theorems such as compactness, L[ö]{}wenheim-Skolem, etc. do not carry over to the finite setting. Rossman’s Finite Homomorphism Preservation Theorem is a major advance because it provides such a theorem which does persist in the finite setting. Another take on this would be to conjecture that *topological variants of all the classical structure theorems* hold in the finite setting. A first result in this direction is Reiterman’s theorem for finite algebras, which shows that Birkhoff’s variety theorem has a finite variant once we topologize. *In weaker logics of resources, as studied for example in finite model theory, is there a topological component missing at the level of the associated Lawvere theories/categorical semantics?*
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[^1]: This project has been funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.670624). Tomáš Jakl has received partial support from the EPSRC grant EP/T007257/1. Luca Reggio has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement No.837724.
[^2]: Typically one also considers some named constants, which we are not mentioning here.
[^3]: Throughout, if no confusion arises, we write ${\varphi}$ for the corresponding element of the Lindenbaum-Tarski algebra, i.e. the logical equivalence class $[{\varphi}]_{\approx}$ of the formula ${\varphi}$.
[^4]: For a variety of algebras $\mathscr{V}$, the associated Lawvere theory is the *dual* of the category of finitely generated free $\mathscr{V}$-algebras with homomorphisms; the distinguished object is the free algebra on one generator.
[^5]: More precisely, the morphisms in ${\ensuremath{\mathbf{Con}}}$ are defined as equivalence classes of substitutions, by identifying two tuples $\langle s_1,\ldots, s_n\rangle$ and $\langle t_1,\ldots, t_n\rangle$ if they give rise to the same homomorphism.
[^6]: A subset $K\subseteq X$ is saturated provided it is an intersection of opens, or equivalently, it is an up-set in the specialisation order of the space $X$.
[^7]: All distributive lattices are assumed to be bounded, and lattice homomorphisms preserve these bounds.
[^8]: In this section, the dualizing object ${\ensuremath{\mathbf{2}}}$ is regarded as either a distributive lattice, or a spectral space by equipping the two-element set with the Sierpinski topology.
[^9]: It follows from these two conditions that $U(f)$ is an embedding with left inverse $U(g)$.
[^10]: A particular class of semiring quantifiers is given by the modular quantifiers, which count in a finite cyclic ring $\mathbb{Z}/q\mathbb{Z}$. These were introduced in logic on words in [@STRAUBINGTT].
[^11]: Perhaps more natural would be to first identify ${\ensuremath{\mathcal{V}}}(X)$ with the space of filters on the dual Boolean algebra of $X$, as explained towards the end of Section \[s:three-ex-spaces\] in the case of the Smyth powerspace, and then observe that filters can be seen as two-valued finitely additive measures.
[^12]: In fact, the construction $X\mapsto {\ensuremath{\mathcal{M}}}(X,S)$ yields a monad on ${\ensuremath{\mathbf{BStone}}}$ and the integration map can be upgraded to a monad morphism $\int\colon {\ensuremath{\mathbf{S}}}\circ U\to U\circ {\ensuremath{\mathcal{M}}}(-,S)$, where ${\ensuremath{\mathbf{S}}}$ is the semiring monad on ${\ensuremath{\mathbf{Set}}}$ and $U\colon {\ensuremath{\mathbf{BStone}}}\to {\ensuremath{\mathbf{Set}}}$ is the forgetful functor. Cf. [@GPR2017]. While, for the purpose of this section, we may assume $S$ is any pointed monoid, the monadic treatment requires the full semiring structure.
[^13]: Note that the restriction of the Stone pairing embedding to a fragment of FO may fail to be injective.
[^14]: A *Priestley space* is a pair $(X,\leq)$ where $X$ is a compact space and $\leq$ is a partial order such that, whenever $x\not\leq y$, there is a clopen subset $C\subseteq X$ which is upward closed and satisfies $x\in C$ and $y\notin C$.
|
---
abstract: 'We present a theoretical analysis of an electron confined by a Penning trap, also known as *geonium*, that is affected by gravity. In particular, we investigate the gravitational influence on the electron dynamics and the electromagnetic field of the trap. We consider the special case of a homogeneous gravitational field, which is represented by Rindler spacetime. In this spacetime the Hamiltonian of an electron with anomalous magnetic moment is constructed. Based on this Hamiltonian and the exact solution to Maxwell equations for the field of a Penning trap in Rindler spacetime, we derive the transition energies of geonium up to the relativistic corrections of $1/{\mathrm{c}}^2$. These transition energies are used to obtain an extension of the well known ${\mathrm{g}_s}$-factor formula introduced by L. S. Brown and G. Gabrielse \[Rev. Mod. Phys. 58, 233 1986\].'
author:
- 'S. Ulbricht'
- 'R. A. Müller'
- 'A. Surzhykov'
date: '\'
title: |
Gravitational effects on geonium and free electron ${\mathrm{g}_s}$-factor\
measurements in a Penning trap
---
Introduction {#section:I}
============
One way to study the properties of a single electron is to analyze the trapped electron in a well known electromagnetic field configuration. However, extracting characteristics of a free particle from transitions of a trapped one requires a deep understanding of the trapping conditions. For this purpose, commonly a Penning trap is used in modern high precision experiments [@Gab08; @Gab06; @Sturm14; @BASE]. Such a trap weakly confines the particle under usage of an electric quadrupole and a constant magnetic field. For the case of an electron, this leads to bound states with discrete energy levels [@Brow86; @Dehm88]. The transitions in such an *artificial atom*, called *geonium*, are used, for example, to determine the free electron ${\mathrm{g}_s}$-factor. This quantity is a dimensionless measure of the electron’s magnetic moment ${\bm}{\mu}$ in the unit of Bohr magnetons $|{\bm}{\mu}_B|=e\hbar/(2m)$ $${\bm}{\mu}={\mathrm{g}_s}\,{\bm}{\mu}_B\quad.$$ While in Dirac’s theory [@Dirac28] the ${\mathrm{g}_s}$-factor is ${{\mathrm{g}_s}}_{\mathrm{Dirac}}=2$, in practice QED effects lead to deviations from this value. A few years ago, D. Hanneke *et al.* have reported high accuracy Penning trap measurements, which determine ${\mathrm{g}_s}=2.002\,319\,304\,361\,46(56)$ [@Gab08; @Gab06]. This experimental result is in outstanding accordance with the calculations of T. Aoyama *et al.* [@Aoy18]. Such an interplay of theory and experiment can help to test fundamental properties of quantum field theory and to search for physics beyond the standard model, see for example [@Flam03; @Flam08] and references therein. The ${\mathrm{g}_s}$-factor experiments in Penning traps, as they are carried out by [@Gab08; @Gab06], are not performed in an isolated environment, but in the presence of the gravitational field of the Earth. This gravitational field distorts both, the electron dynamics and the electromagnetic field configuration of the trap. In this contribution, therefore, we perform the theoretical analysis of the effects of gravity on the result of Penning trap experiments. In particular, we also take into account gravitational effects on the electromagnetic field of the Penning trap, which in turn affects the motion of the electron. While gravitational effects on the bound electron ${\mathrm{g}_s}$-factor [@Jent13] and the cyclotron motion of the electron [@Moris15] have been considered, to the best of our knowledge, an analysis of the gravitational influence on Penning trap experiments have not been reported before. In order to understand, how gravity influences Penning trap experiments, it is natural to describe both, the electron and the electromagnetic field of the Penning trap, in curved spacetime. In our study, we will consider the case of a homogeneous gravitational field, which is a good approximation for gravity at the surface of the Earth, as we will discuss in Sec. \[section:IIa\]. The Dirac Hamiltonian, which describes the dynamics of an electron with anomalous magnetic moment in this spacetime, is obtained in Sec. \[section:IIb\]. While this Hamiltonian can be applied for any electron velocities and gravitational field strengths, we aim to use it to describe Penning trap experiments, which are performed in the non-relativistic regime. Therefore, in Sec. \[section:IIc\] we perform a Foldy-Wouthuysen transformation to obtain the non-relativistic Hamiltonian and its $1/{\mathrm{c}}^2$-corrections. Of course, this Hamiltonian accounts not only for gravitational effects, but also for the coupling to the electromagnetic field of a Penning trap. In Sec. \[section:IIIa\] this field is presented as an exact solution to Maxwell equations in the spacetime of homogeneous gravity. Using first order perturbation theory, we determine the eigenenergies of geonium exposed to gravity up to order $1/{\mathrm{c}}^2$ in Sec. \[section:IIIb\] and Sec. \[section:IIIc\]. Finally, these energies are used to derive an expression for the free electron ${\mathrm{g}_s}$-factor, which generalises the well known results of L. S. Brown and G. Gabrielse. The summary of our results is given in Sec. \[section:IV\].
Electron in homogeneous gravitational field {#section:II}
===========================================
The homogeneous gravitational field\
in general relativity {#section:IIa}
------------------------------------
On Earth the biggest empirical effect of gravity is the acceleration of ${g}=9.81\, \mathrm{m}/\mathrm{s}^2$ pointing downwards. In the Newtonian theory of gravity, a vector field of constant acceleration ${\bm}{{g}}$ is a suitable approximation of the gravitational field perceived by this observer. Higher order effects, accounting for the Earth as a spherical body, can be neglected in a small environment of the observers position. The approximation of a homogeneous acceleration ${\bm}{{g}}$ also holds in general relativity in terms of the non-geodesic motion of an observer; Bound to Earth’s surface, the observer is not able to follow gravity in a *free fall*. In a general relativistic framework, the Newtonian gravitational field of the Earth is replaced by the famous Schwarzschild spacetime [@Schwarz16]. At the surface of the Earth, this spacetime can be approximated by so called Rindler spacetime [@Rind60; @Rind66], which is merely flat Minkowski spacetime, but seen by an accelerated observer. At this level of approximation, there is no spacetime curvature, but a *distortion* of spacetime by acceleration. In order to describe physics perceived by an accelerated observer, we start with the line element ${\mathrm{d}}s^2$ of Minkowski spacetime and perform a coordinate transformation towards a coordinate system, that describes the reference frame of the accelerated observer. The Minkowski line element, expressed in terms of Cartesian coordinates $\tilde{{\bm}{r}}=(\tilde x, \tilde y, \tilde z)$ and proper time $\tau$, reads $${\mathrm{d}}s^2 =\eta_{\mu\nu}\,d x^\mu {\mathrm{d}}x^\nu = {\mathrm{d}}({\mathrm{c}}\tau)^2 - {\mathrm{d}}{\tilde x}^2- {\mathrm{d}}{\tilde y}^2- {\mathrm{d}}{\tilde z}^2\quad, \label{eqn:Minkowski}$$ where we introduced the metric tensor $(\eta_{\mu\nu})=\mathrm{diag}(1,-1,-1,-1)$. In this sign-convention time-like distances are described by positive values of the line element ${\mathrm{d}}s^2>0$. Moreover, we use Einsteinian sum convention, which means that a sum is performed from $0$ to $4$ when paired Greek letters appear. The index $0$ is set to be the index of the time-like coordinate. The coordinates of Rindler spacetime are related to the coordinates of Minkowski spacetime by $$\begin{aligned}
x'&=&\tilde x \quad, \nonumber\\
y'&=&\tilde y \quad,\nonumber\\
z'&=&-\frac{{g}}{2}\,\tau^2+\tilde z \left(1+\frac{{g}\tilde z}{2 {\mathrm{c}}^2}\right) \quad,\label{eqn:RindlerTrafo}\\
{\mathrm{c}}t&=& \frac{{\mathrm{c}}^2}{2{g}}\,\mathrm{log}\left(\frac{{\mathrm{c}}^2 +{g}({\mathrm{c}}\tau+\tilde z)}{{\mathrm{c}}^2 -{g}({\mathrm{c}}\tau-\tilde z)}\right)\quad, \nonumber\end{aligned}$$ as discussed in [@Rind60]. Here $t$ is the proper time of the observer, located in the center of the accelerated frame. This frame is denoted by the primed coordinates ${\bm}{r}'=(x',y',z')$. In order to describe physical processes in the accelerated frame, we use the transformation (\[eqn:RindlerTrafo\]) to derive the line element of Rindler spacetime $$\begin{aligned}
{\mathrm{d}}s^2 &=& \left(1+\frac{2{g}z'}{{\mathrm{c}}^2}\right){\mathrm{d}}({\mathrm{c}}t)^2 \label{eqn:firstRindlerMetric}\\
& &\quad - {\mathrm{d}}{x'}^2- {\mathrm{d}}{y'}^2- \left(1+\frac{2{g}z'}{{\mathrm{c}}^2}\right)^{-1}{\mathrm{d}}{z'}^2\quad. \nonumber\end{aligned}$$ From now on, we will call $t$ the *time* and only perform spatial coordinate transformations, if required by the geometry of the problem under consideration. For our further investigation, it is useful to introduce the auxiliary coordinate $u(z')=(\sqrt{1+2{g}z'/{\mathrm{c}}^2}-1)\,{\mathrm{c}}^2/{g}$, as described in [@Rind66], such that the spatial part of the line element is isotropic $$\begin{aligned}
{\mathrm{d}}s^2 &=& \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^2{\mathrm{d}}({\mathrm{c}}t)^2 - {\mathrm{d}}{x'}^2- {\mathrm{d}}{y'}^2- {\mathrm{d}}u^2 \label{eqn:RindlerMetric}\\
&=& g_{\mu'\nu'}(u)\,{\mathrm{d}}x^{\mu'}{\mathrm{d}}x^{\nu'}\quad, \nonumber\end{aligned}$$ As seen from Eq. (\[eqn:RindlerMetric\]), the line element in the accelerated frame is coordinate dependent. Therefore, the measure of time is different at different heights and the factor $\left(1+{g}u/{\mathrm{c}}^2\right)$ in front of the infinitesimal time step ${\mathrm{d}}({\mathrm{c}}t)$ gives rise to the gravitational redshift [@Rind60; @Rind66]. For vanishing acceleration the Rindler line element (\[eqn:RindlerMetric\]) reduces to the Minkowski line element. The same holds in the $(x',y')$-plane, where we reach flat Minkowski spacetime asymptotically for $u\to 0$. Therefore, it is legitimate to apply methods of quantum mechanics in Minkowski spacetime in a small area around the coordinate center and treat the modifications, caused by deviation from Minkowski spacetime, as corrections later. In our case this assumption is valid, since we are interested in quantum objects bound close to ${\bm}{r}'=0$, where typical length scales $z'$ are in the micrometer domain and, therefore, much smaller than ${\mathrm{c}}^2/{g}\sim \mathrm{ly}$, which is the typical length scale for the considered gravitational effects.
Electron with anomalous magnetic moment\
in Rindler spacetime {#section:IIb}
----------------------------------------
In the previous section, we introduced the spacetime of a homogeneously accelerated observer, known as Rindler spacetime. Now we want to pay particular attention to the dynamics of an electron in this spacetime. Again we start our analysis in Minkowski spacetime, where the electron motion is described by the Dirac equation [@Dirac28] $$({i}\hbar \gamma^\mu{\partial}_\mu-m{\mathrm{c}})\psi(x^\nu)=0\quad. \label{eqn:Dirac}$$ Here $\psi(x^\nu)$ is the Dirac spinor, whose four components represent not only the electron, but also the positron in their two spin states. The Dirac matrices $\gamma^\mu$ are chosen such, that their anti-commutator generates the metric tensor of the line element (\[eqn:Minkowski\]) of Minkowski spacetime $$\{\gamma_\mu,\gamma_\nu\}=2\,\eta_{\mu \nu}\quad, \label{eqn:Clifford}$$ where the upper index of the Dirac matrices in Eq. (\[eqn:Dirac\]) is raised by the inverse metric tensor $\gamma^\mu=\eta^{\mu\nu}\gamma_\nu$. In the following, we transform the Dirac equation (\[eqn:Dirac\]) to the frame of an accelerated observer. For this purpose, it is convenient to start with the Dirac action in Minkowski spacetime $$S[\bar\psi,\psi]=\int\bar\psi(x^\nu)({i}\hbar \gamma^\mu{\partial}_\mu-m{\mathrm{c}})\psi(x^\nu)\,{\mathrm{d}}x^4\quad, \label{eqn:DiracAction}$$ where $\bar \psi(x^\nu)=\psi^\dagger(x^\nu) \gamma^{0}$ is the Dirac adjoint of $\psi(x^\nu)$ and $\int {\mathrm{d}}x^4$ is the integral over all four spacetime coordinates $(x^{\mu})=({\mathrm{c}}\tau, \tilde x,\tilde y,\tilde z )$. From action (\[eqn:DiracAction\]) the Dirac equation (\[eqn:Dirac\]) can be obtained by the principle of stationary action. The next step is to express the Dirac action in the coordinates $(x^{\mu'})=({\mathrm{c}}t, x', y', u)$ of the accelerated frame, in order to obtain Dirac equation in Rindler spacetime. This coordinate transformation of the action requires some attention [@Hehl90; @Lippoldt; @Weldon] and, therefore, is discussed in Appendix \[section:A\]. After the coordinate transformation (\[eqn:RindlerTrafo\]), the action (\[eqn:DiracAction\]) reads $$\begin{aligned}
S[\bar\psi',\psi']=\int\bar\psi'(x^{\nu'})({i}\hbar \gamma^{\mu'}(u){\partial}_{\mu'}-m{\mathrm{c}})\psi'(x^{\nu'}) \qquad\qquad \label{eqn:DiracAction1}& &\\
\times \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,{\mathrm{d}}{x'}^4\quad, \nonumber & &\end{aligned}$$ where the infinitesimal spacetime volume ${\mathrm{d}}{x}^4$ in now replaced by $\left(1+{g}u/{\mathrm{c}}^2\right)\, {\mathrm{d}}{x'}^4$. In addition the primed Dirac matrices in Eq. (\[eqn:DiracAction1\]) are spacetime dependent and have to obey the relation $$\{\gamma_{\mu'}(u),\gamma_{\nu'}(u)\}=2\,{g}_{{\mu'} {\nu'}}(u)\quad, \label{eqn:Clifford2}$$ instead of the relation (\[eqn:Clifford\]). Here ${g}_{{\mu'} {\nu'}}(u)$ is the metric tensor of Rindler spacetime, defined in Eq. (\[eqn:RindlerMetric\]). The Dirac adjoint spinor now reads $\bar \psi'(x^{\nu'})=(\psi'(x^{\nu'}))^\dagger \gamma^{0'}(u)$. Indeed, in the case of Penning trap experiments, the electron is not only exposed to gravity, but is located in an electromagnetic field. Therefore, we go the common way to introduce a minimal coupling to the electromagnetic field by the replacement of the partial derivative ${\partial}_{\mu'}\to {\partial}_{\mu'}+{i}\frac{e}{\hbar}A_{\mu'}$, which brings in the four potential $A_{\mu'}{=}(\Phi/c,-{\bm}{A})$, which contains the electric scalar potential $\Phi$ and the magnetic vector potential ${\bm}{A}$. In our considerations, we will treat these potentials as classical. With these alterations, the Dirac action (\[eqn:DiracAction1\]) becomes $$\begin{aligned}
S[\bar\psi',\psi']=\hspace{0.7\linewidth}\nonumber\\\int\bar\psi'(x^{\nu'})\left[{i}\hbar \gamma^{\mu'}(u)\left({\partial}_{\mu'}+{i}\frac{e}{\hbar}A_{\mu'}\right)-m{\mathrm{c}}\right]\psi'(x^{\nu'}) \qquad \label{eqn:DiracAction2}& &\\
\times \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,{\mathrm{d}}{x'}^4\quad, \nonumber & &\end{aligned}$$ which now is the action for a Dirac electron, in the presence of an electromagnetic field and seen by a homogeneously accelerated observer. However, an important feature of the system is still missing in Eq. (\[eqn:DiracAction2\]): the anomalous contribution to the magnetic moment of the electron. Therefore, we introduce the anomaly $a$, which accounts for the discrepancy between the gyromagnetic ratio ${{\mathrm{g}_s}}_{\mathrm{Dirac}}=2$ of Dirac theory and the measured value ${\mathrm{g}_s}=2(1+a)$, caused by interactions between the electron and the quantum vacuum. In order to account for this anomaly, we introduce a non-minimal coupling of the electron to the electromagnetic field strength tensor $F_{\mu'\nu'}={\partial}_{\mu'}A_{\nu'}-{\partial}_{\nu'}A_{\mu'}$. With this additional term the action reads $$\begin{aligned}
S[\bar\psi',\psi']=\hspace{0.7\linewidth}\nonumber\\\int\bar\psi'(x^{\nu'})\left[{i}\hbar \gamma^{\mu'}(u)\left({\partial}_{\mu'}+{i}\frac{e}{\hbar}A_{\mu'}\right)\right.\hspace{0.25\linewidth}& &\label{eqn:FinalAction}\\
\left.+\,\,\,a\,\frac{e\hbar}{2m{\mathrm{c}}}\frac{{i}}{4}[\gamma^{\mu'}(u),\gamma^{\nu'}(u)]F_{\mu'\nu'}-m{\mathrm{c}}\right]\psi'(x^{\nu'}) \quad & & \nonumber\\
\times \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,{\mathrm{d}}{x'}^4\quad, \nonumber & &\end{aligned}$$ where the commutators of the coordinate dependent Dirac matrices are the generators of local Lorentz transformations. The structure of the additional term in the action can be motivated by QED considerations [@Fold58]. The action (\[eqn:FinalAction\]) is discussed in detail for an electron in an inertial system in [@Brow86; @Grae69; @Bjor64] and references therein. Naturally, we recover their cases in the limit of vanishing acceleration $g$. As we discussed above, these steps were performed in order to derive Dirac equation in Rindler spacetime. Since we want to give this equation in the Hamiltonian representation, we separate the spacetime coordinates $(x^{\nu'})=({\mathrm{c}}t, x', y', u)$ into the time parameter $t$ and the spatial coordinates $(x^{i'})=(x', y', u)$, where $i'=1',\dots,3'$. Furthermore we redefine the Dirac matrices to be $$\gamma^{0'}(u)=\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1}\,\beta \quad,\quad \gamma^{i'}=\beta\alpha^{i'}\quad, \label{eqn:BetaAlpha1}$$ where the *constant* matrices $\beta$ and $\alpha^{i'}$ obey the following relations: $$\{\alpha^{i'},\alpha^{j'}\}=2\delta^{i'j'}\quad,\quad \beta\,\alpha^{i'}+\alpha^{i'}\beta=0\quad. \label{eqn:BetaAlpha2}$$ With these definitions (\[eqn:BetaAlpha1\]) and (\[eqn:BetaAlpha2\]) the relation (\[eqn:Clifford2\]) for the $u$-dependent Dirac matrices is satisfied. We combine the $\alpha^{i'}$ to be a vector ${\bm}{\alpha}{=}(\alpha^{x'},\alpha^{y'},\alpha^{u})$ and define the canonical momentum ${\bm}{\pi}{=}{\bm}{p}-e{\bm}{A}$, where ${\bm}{p}{=}-{i}\hbar \nabla$ is the momentum operator and $\nabla=({\partial}_{x'},{\partial}_{y'},{\partial}_u)$ is the gradient in the $(x',y',u)$-coordinate system. After these steps we rewrite the action $$S[\psi'^\dagger,\psi']=\int \Bigl(\langle \psi' |{i}\hbar {\partial}_t \psi'\rangle-\langle \psi' | H\psi'\rangle \Bigr)\, {\mathrm{d}}t\quad. \label{eqn:FinalAction2}$$ By variation of this action we directly obtain the Dirac equation in Schrödinger form $${i}\hbar {\partial}_t |\psi'\rangle= H|\psi'\rangle \quad.$$ Moreover, the structure of Eq. (\[eqn:FinalAction2\]) helps us to construct the scalar product $$\langle\psi_1'|\psi_2'\rangle{=}\int \psi_1'^\dagger \psi_2' \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1}{\mathrm{d}}x' {\mathrm{d}}y' {\mathrm{d}}u \quad. \label{eqn:ScalarProduct}$$ and the corresponding hermitian Hamiltonian. $$\begin{aligned}
H&{=}&\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\left({\mathrm{c}}{\bm}{\alpha}\cdot{\bm}{\pi}+m{\mathrm{c}}^2\beta-\,\frac{ae}{m}\beta\,{\bm}{B}\cdot{\bm}{s}\right)\label{eqn:Hamiltonian1}\\
& &\quad +\,e\Phi \,+\,\frac{{i}ae\hbar}{2m{\mathrm{c}}}\beta \,{\bm}{\alpha}\cdot{\bm}{E}\quad\nonumber \,.\end{aligned}$$ Here the electric and magnetic fields are defined by ${\bm}{E}{=}-\nabla \Phi-{\partial}_t {\bm}{A}$ and ${\bm}{B}{=}\nabla\times{\bm}{A}$. Moreover, ${\bm}{s}{=}-\frac{{i}}{4}\hbar\,{\bm}{\alpha}\times {\bm}{\alpha}$ is the four-spin operator. It is easy to check, that the Hamiltonian (\[eqn:Hamiltonian1\]) is hermitian with respect to the scalar product (\[eqn:ScalarProduct\]).
Non-relativistic reduction of the Hamiltonian {#section:IIc}
---------------------------------------------
In the last section, we derived the Hamiltonian of a relativistic spin 1/2 particle, that moves in Rindler spacetime in the presence of an electromagnetic field. In addition we introduced the anomalous magnetic moment of this particle, which accounts for the interaction of the electron with the quantum vacuum. The Hamiltonian (\[eqn:Hamiltonian1\]), in its general form, can be applied for any velocities ($v<{\mathrm{c}}$) of an electron. In this work, however, we concentrate on a scenario, where the electron is stored in a Penning trap in a laboratory on Earth. In this case, the velocity $v\ll {\mathrm{c}}$ is non-relativistic and the quantity $g L /c^2 \ll 1$ is a small parameter, where $L$ is a typical length scale of the experiment. Therefore, we can simplify (\[eqn:Hamiltonian1\]) to be the Hamiltonian of a non-relativistic particle and take into account correctional $1/{\mathrm{c}}^2$ effects only. All higher orders in $1/{\mathrm{c}}$ are collected in the Landau symbol $\mathrm{O}(1/{\mathrm{c}}^3)$ and will be neglected later. In what follows, we derive the non-relativistic reduction of the Hamiltonian (\[eqn:Hamiltonian1\]). There are many approaches to construct a non-relativistic Hamiltonian and its post-Newtonian corrections, see for example [@Giulini] and references therein. In this work, we derive these corrections by a Foldy-Wouthuysen transformation [@Bjor64; @Fold50], which is used to decouple the electronic and positronic sector of $H$. The starting point of a Foldy-Wouthuysen transformation is to rewrite the Hamiltonian in the form $$H = m{\mathrm{c}}^2 \beta + \mathcal{E} + \mathcal{O} \quad , \label{eqn:Hevenandodd}$$ where the part of the Hamiltonian, which acts on the electronic and positronic degrees of freedom separately, is called the *even* part of $H$: $$\begin{aligned}
\mathcal{E}&=& m{g}u\,\beta -\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\frac{ae}{m}\beta\,{\bm}{B}\cdot{\bm}{s}+\,e\Phi\end{aligned}$$ and the part of the Hamiltonian, which couples the electronic and positronic sector is denoted as its *odd* part: $$\begin{aligned}
\mathcal{O}&=& \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right){\mathrm{c}}{\bm}{\alpha}\cdot{\bm}{\pi}+\,\frac{{i}ae\hbar}{2m{\mathrm{c}}}\beta \,{\bm}{\alpha}\cdot{\bm}{E}\quad.\end{aligned}$$ In the next step, we minimize the contribution of the odd part by an unitary transformation $$\begin{aligned}
H' &=& {\mathrm{e}}^{\mathcal{W}}H{\mathrm{e}}^{-\mathcal{W}}\nonumber \\
&=& H +[\mathcal{W},H]+\frac{1}{2!}[\mathcal{W},[\mathcal{W},H]]+\dots \quad , \label{eqn:BCH}\end{aligned}$$ where the anti-hermitian operator $\mathcal{W}=\beta\, \mathcal{O}/(2m{\mathrm{c}}^2)$ is chosen, such that $[\mathcal{W},\beta\,m{\mathrm{c}}^2]=-\mathcal{O}$. Because of this choice, the odd part $\mathcal{O}$ is canceled out in the first two terms in the right hand side of Eq. (\[eqn:BCH\]). Considering all terms in (\[eqn:BCH\]), which enter $H'$ up to order $\mathrm{O}(1/c^3)$, the new Hamiltonian can be written as $$H' = m{\mathrm{c}}^2\beta + \mathcal{E}' + \mathcal{O}' \quad ,$$ in similarity to Eq. (\[eqn:Hevenandodd\]). The new even and odd parts read $$\begin{aligned}
\mathcal{E}'&=& \mathcal{E}+\frac{1}{2}[\mathcal{W},\mathcal{O}]+\frac{1}{2}[\mathcal{W},[\mathcal{W},\mathcal{E}]]\label{eqn:evenprime}\\
& &\qquad+\frac{1}{8}[\mathcal{W},[\mathcal{W},[\mathcal{W},\mathcal{E}]]]+\mathrm{O}(1/{\mathrm{c}}^3)\quad,\nonumber\\
\mathcal{O}'&=& [\mathcal{W},\mathcal{E}]+\frac{1}{3}[\mathcal{W},[\mathcal{W},\mathcal{O}]]+\mathrm{O}(1/{\mathrm{c}}^3)\quad.\end{aligned}$$ We see, that the former odd part $\mathcal{O}$ appears in the commutators with $\mathcal{W}$ only, while the new odd part $\mathcal{O}'$ is proportional to $1/{\mathrm{c}}$ in the leading order. To further minimize the order of $\mathcal{O}'$, we can iterate the Foldy-Wouthuysen transformation until $\mathcal{O}'''=\mathrm{O}(1/{\mathrm{c}}^3)$ is reached, such that $$\begin{aligned}
H'''&=&m{\mathrm{c}}^2 \beta +\mathcal{E}'''+\mathrm{O}(1/{\mathrm{c}}^3) \quad,\end{aligned}$$ where the further iterations do not affect $\mathcal{E}'''=\mathcal{E}'+\mathrm{O}(1/c^3)$. Therefore, by calculating (\[eqn:evenprime\]), we find the Hamiltonian $H'''$ of the electron with anomalous magnetic moment and its antiparticle in an accelerated frame and an arbitrary electromagnetic field with all its corrections in $1/{\mathrm{c}}^2$: $$\begin{aligned}
H'''&=&m {\mathrm{c}}^2\beta + um {g}\beta + e\Phi \nonumber \hspace{10.63em}\textnormal{\emph{(i)}} \\
& & -\,\frac{1}{2m^2{\mathrm{c}}^2}\,{\bm}{s}\cdot \Bigl((1+2a)e{\bm}{E}-m{\bm}{{g}}\beta\Bigr)\times{\bm}{\pi}\nonumber\hspace{2.61em}\textnormal{\emph{(ii)}}\\
& & + \beta\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\left(\frac{1}{2m} {\bm}{\pi}^{\,2}-(1+a)\frac{e}{m}\,{\bm}{B}\cdot{\bm}{s}\right)\nonumber\quad\textnormal{\emph{(iii)}}\\
& & -\,\frac{1}{8{\mathrm{c}}^2 m^3} \beta({\bm}{\pi}^2-2e {\bm}{B}\cdot{\bm}{s})^2 \nonumber\\
& & +\,(1+2a)\frac{e\hbar^2}{8c^2m^2}\,\Delta\Phi\label{eqn:newHamilton}\\
& & +\,\frac{ae}{8m^3c^2}\beta\Bigl(2\{{\bm}{s}\cdot{\bm}{\pi},{\bm}{B}\cdot{\bm}{\pi}\}
+[{\bm}{\pi}^2,{\bm}{B}\cdot{\bm}{s}]\Bigr)\nonumber\\
& & +\,\frac{ae\hbar^2}{8 m^3 c^2}\,\beta\, (\nabla\times{\bm}{B})\cdot{\bm}{\pi} \nonumber\\
& & +\,\frac{{i}ae \hbar}{4 m^3 c^2} \beta\, [\nabla({\bm}{B}\cdot{\bm}{s})]\cdot{\bm}{\pi}+\,\,\mathrm{O}(1/{\mathrm{c}}^3)\nonumber\quad,\end{aligned}$$ where we defined the acceleration vector ${\bm}{g}=(0,0,{g})$. In the absence of gravity, i.e. for ${\bm}{g}=0$, all terms in $H'''$ are well known and have been studied extensively, first and foremost [@Brow86; @Grae69] and references therein. The presence of gravity, however, gives rise to additional parts, which have to be discussed in more detail. As seen from Eq. (\[eqn:newHamilton\]), gravity enters this Hamiltonian at three points. *(i)* It gives the usual Newtonian potential, as it is known from classical mechanics, *(ii)* it acts as a correction to the spin orbit coupling therm and *(iii)* it induces a redshift of the non-relativistic kinetic energy and the coupling term between ${\bm}{B}$ and ${\bm}{s}$, which is important for our investigation of gravitational effects on free electron ${\mathrm{g}_s}$-factor measurements. The Hamiltonian (\[eqn:newHamilton\]) acts separately on the electronic and positronic degrees of freedom. Therefore, these two sectors are decoupled up to the desired order $\mathrm{O}(1/{\mathrm{c}}^3)$. The choice of sector is made by selecting the positive or negative eigenvalue of $\beta$. In our case we restrict ourselves to the discussion of the electron only, whose dynamics is described by $H'''$ after the replacement $\beta=+1$. The Hamiltonian $H'''$ now contains all effects on the electron caused by the homogeneous acceleration and the anomalous magnetic moment and all relativistic effects, up to the order of $1/{\mathrm{c}}^2$. It provides two particular limits, which are well known, either in the theory of Fermions in non-geodesic motion [@Fis81; @Hehl90; @Exp1; @Exp2], or in the physics of traps [@Brow86; @Grae69]. In the case of vanishing electromagnetic fields, for example, we get the Hamiltonian of a free electron in accelerated motion: $$\begin{aligned}
\lim \limits_{{\bm}{E},{\bm}{B}\to 0}
H'''&=&m {\mathrm{c}}^2 + um {g}+ \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\frac{1}{2m} {\bm}{p}^{\,2} \label{eqn:LimitOne}\\
& &\quad -\,\frac{1}{8{\mathrm{c}}^2 m^3} {\bm}{p}^4 +\,\frac{1}{2m{\mathrm{c}}^2}\,{\bm}{s}\cdot ( {\bm}{{g}}\times{\bm}{p})\nonumber\\
& &\hspace{11em}+\,\,\mathrm{O}(1/{\mathrm{c}}^3)\nonumber\quad.\end{aligned}$$ Here, the relativistic correction of the kinetic energy $\sim {\bm}{p}^4$, the gravitational redshift $(1+{g}u/{\mathrm{c}}^2)$ and the spin-gravity coupling $\sim {\bm}{s}\cdot({\bm}{g}\times{\bm}{p})$ show up clearly. A detailed discussion of (\[eqn:LimitOne\]) and the physical consequences of the single terms can be found in [@Fis81; @Hehl90; @Exp1; @Exp2]. On the other hand, in the absence of gravity the Hamiltonian (\[eqn:newHamilton\]) reduces to $$\begin{aligned}
\lim \limits_{{\bm}{g}\to 0}
H'''&=&m {\mathrm{c}}^2 + e\Phi \nonumber + \left(\frac{1}{2m} {\bm}{\pi}^{\,2}-(1+a)\frac{e}{m}\,{\bm}{B}\cdot{\bm}{s}\right)\nonumber\\
& & \quad -\,\frac{1}{8{\mathrm{c}}^2 m^3} ({\bm}{\pi}^2-2e {\bm}{B}\cdot{\bm}{s})^2\ \label{eqn:limit2}\\
& &\quad -\,(1+2a)\frac{e}{2m^2{\mathrm{c}}^2}\,{\bm}{s}\cdot( {\bm}{E}\times{\bm}{\pi})\nonumber\\
& &\quad +\,\frac{ae}{2 m^3c^2}({\bm}{s}\cdot{\bm}{\pi})({\bm}{B}\cdot{\bm}{\pi}) \,+\,\mathrm{O}(1/{\mathrm{c}}^3)\nonumber\quad,\end{aligned}$$ where we, moreover, assumed a constant magnetic and a source free electric field. Expression (\[eqn:limit2\]) recovers [@Grae69] and has been the starting point for the investigation of relativistic corrections in geonium by Brown and Gabrielse [@Brow86] and Dehmelt [@Dehm88], whose work we aim to extend by the consideration of a gravitational field.
The electron in a Penning Trap {#section:III}
==============================
The electromagnetic field of a Penning trap\
in Rindler spacetime {#section:IIIa}
--------------------------------------------
In the previous section, we derived the Hamiltonian (\[eqn:newHamilton\]) of an electron with anomalous magnetic moment, affected by a homogeneous gravitational and an arbitrary electromagnetic field. Although, this Hamiltonian can be applied to any kind of electric and magnetic fields, we want to apply it to the particular case of the electromagnetic field of a Penning trap. Therefore we assume an ideal trap potential, without any imperfections, consisting of a static homogeneous magnetic field and a static electric quadrupole field. Moreover, the Penning trap is placed in the spacetime of an accelerated observer. Therefore, the electromagnetic field of the trap is distorted. In order to account for this distortion, we need to formulate Maxwell equations in Rindler spacetime. These equations for a source free electromagnetic field in their covariant form read $${\partial}_{\mu'}F^{\mu'\nu'}+\Gamma^{\mu'}_{\mu'\rho'}F^{\rho'\nu'}=0 \quad, \label{eqn:Maxwell}$$ in terms of the electromagnetic field strength tensor $F_{\mu'\nu'}={\partial}_{\mu'}A_{\nu'}-{\partial}_{\nu'}A_{\mu'}$ and the Christoffel symbols $\Gamma^{\mu'}_{\mu'\rho'}=\frac{1}{2}g^{\mu'\sigma'}{\partial}_{\rho'}g_{\mu'\sigma'}=\frac{{g}}{{\mathrm{c}}^2}\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1}\, \delta^{3}_{\rho'}$. For more details and the definition of the Christoffel symbols, see Appendix B. Eq. (\[eqn:Maxwell\]) allows us to calculate the vector potential ${\bm}{A}$ and the scalar potential $\Phi$, needed in the Hamiltonian (\[eqn:newHamilton\]). These potentials enter into the four-potential $(A_{\mu'})=(\Phi/{\mathrm{c}},-{\bm}{A})$, which we assume to be static, i.e. ${\partial}_t A_{\mu'}=0$ and, moreover, to satisfy the gauge condition $${\partial}_{\mu'}A^{\mu'}+\Gamma^{\mu'}_{\mu'\rho'}A^{\rho'}=0 \quad. \label{eqn:gauge}$$ Under these assumptions, we can rewrite the spatial part of equation (\[eqn:Maxwell\]) in the form $$\begin{aligned}
\nabla\cdot\left[\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\nabla {\bm}{A}\right]&=&\frac{{g}}{c^4}\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1} {\bm}{g}\,A_{3} \label{eqn:MaxwellA}\quad,\end{aligned}$$ which determines the vector potential ${\bm}{A}=(A_1,A_2,A_3)$. In this expression, moreover, $\nabla=({\partial}_{x'},{\partial}_{y'},{\partial}_u)$ is the gradient in the coordinate system $(x',y',u)$. In order to solve Eq. (\[eqn:MaxwellA\]), one has to define explicit boundary conditions. In the case of a Penning trap configuration, for example, we demand that ${\bm}{A}$ is the vector potential of a constant magnetic field ${\bm}{B}^{(0)}=(B_1,B_2,B_3)$ in the center of the trap. For this requirement, the solution of (\[eqn:MaxwellA\]) is given by $$\begin{aligned}
{\bm}{A}&=&-\frac{1}{2}\left(\begin{array}{c}
x' \\
y' \\
w(u)
\end{array}\right)\times \left(\begin{array}{c}
B_1\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1}\\
B_2\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1}\\
B_3\\
\end{array}\right)\,, \label{eqn:RindlerA}\end{aligned}$$ where $$\begin{aligned}
w(u)& =& \frac{{\mathrm{c}}^2}{{g}}\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,\log \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,. \end{aligned}$$ For vanishing acceleration ${\bm}{g}=0$, Eq. (\[eqn:RindlerA\]) reduces to the well known vector potential of a constant magnetic field $-\frac{1}{2} \, {\bm}{r}'\times {\bm}{B}^{(0)}$ globally. However, the presence of gravity leads to a distortion of the magnetic field, which is characterized by the factor $(1+g u /{\mathrm{c}}^2)$. In the same way, we can find the scalar quadrupole potential $\Phi$ of the trap. This potential has to be a solution to the time-component of Eq. (\[eqn:MaxwellA\]), which under gauge condition (\[eqn:gauge\]) becomes $$\nabla\cdot\left[\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^{-1}\!\!\nabla\Phi\right]=0\quad. \label{eqn:MaxwellPhi}$$ Again the physically relevant boundary conditions have to be set here. They are chosen such that the potential $\Phi$ is determined by a constant, traceless quadrupole matrix $\hat Q$ in the coordinate center. The corresponding solution to Eq. (\[eqn:MaxwellPhi\]) is given by $$\begin{aligned}
\Phi&=& \Bigl(x',y',f(u)\Bigr)\cdot \hat{Q}\cdot \left(\begin{array}{c}
x'\\
y'\\
f(u)
\end{array}\right)+ Q_{33} \,h(u)\,,\label{eqn:RindlerPhi}\end{aligned}$$ where $$\begin{aligned}
f(u)&=&\left(1+\frac{{g}u}{2 {\mathrm{c}}^2}\right)\, u\,\end{aligned}$$ and $$\begin{aligned}
h(u)&=&\left(\frac{{\mathrm{c}}^2}{{g}}\right)^2\left\{\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)^2\log\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\right.\nonumber\\
& & \left.- \frac{{g}u}{{\mathrm{c}}^2}\left(1+\frac{{g}u}{2 {\mathrm{c}}^2}\right)\left[1+\frac{{g}u}{{\mathrm{c}}^2}\left(1+\frac{{g}u}{2 {\mathrm{c}}^2}\right)\right]\right\}\,.\end{aligned}$$ For vanishing acceleration ${\bm}{g}=0$, the potential (\[eqn:RindlerPhi\]) reduces to the ideal quadrupole potential ${\bm}{r}'\cdot (\hat Q \cdot {\bm}{r}')$. Having found the solutions (\[eqn:RindlerA\]) and (\[eqn:RindlerPhi\]) for the vector and scalar potential, we are ready now to set up the Penning trap field configuration. For this purpose, we need to specify the geometry of the trap. We adopt the coordinate system such that $B_2=0$ and introduce the angle $\theta$ between ${\bm}{B}^{(0)}$ and ${\bm}{g}$. For this choice, the constants ${\bm}{B}^{(0)}$ and $\hat Q$ are given by $${\bm}{B}^{(0)}=(-B \,\sin \theta\,,\,0\,,\,B\,\cos\theta)\quad$$ and $$\hat Q = -\frac{V}{4L^2}\left(\begin{array}{ccc}
\frac{1}{2}(3\cos(2\theta)-1)&0& 3 \cos\theta\sin\theta\\
0& 1 & 0 \\
3 \cos\theta\sin\theta & 0 & -\frac{1}{2}(3\cos(2\theta)+1)
\end{array}\right)\quad ,$$ where $B$ is the absolute value of ${\bm}{B}^{(0)}$ and the quantities $V$ and $L$ are the typical voltage and spatial length scale of the Penning trap. Having found the vector and scalar potentials ${\bm}{A}$ and $\Phi$, we can use them in the exact Dirac Hamiltonian (\[eqn:Hamiltonian1\]), or in its expanded form (\[eqn:newHamilton\]), as we will do in the next section.
The electron in a Penning trap in Newtonian gravity {#section:IIIb}
---------------------------------------------------
We derived the Hamiltonian $H'''$ of an electron with anomalous magnetic moment in an accelerated frame and an arbitrary electromagnetic field in Sec. \[section:IIc\]. Below, we want to use this Hamiltonian to describe the electron dynamics in a Penning trap, distorted by acceleration. Therefore, we insert the vector and scalar potentials (\[eqn:RindlerA\]) and (\[eqn:RindlerPhi\]) into Eq. (\[eqn:newHamilton\]). For the sake of brevity, we will not present this lengthy expression here, that contains all relativistic effects on both, the electron and the trap, up to the order of $1/{\mathrm{c}}^2$. In this section, we derive the exact solution of the eigenvalue problem of the Hamiltonian in the Newtonian limit of low velocities and weak gravitational fields. Treating $1/{\mathrm{c}}^2$-effects as first order perturbations, we find the solution of the whole eigenvalue problem of $H'''$, afterwards. Within the Newtonian limit, the non-relativistic Hamiltonian $H_0$ is obtained by considering the zeroth order in an $1/{\mathrm{c}}$-expansion of $H'''$, only: $$H'''=H_0+\,\mathrm{O}(1/{\mathrm{c}})\quad,$$ where $$\begin{aligned}
H_0&=&m{\mathrm{c}}^2+ m\,{\bm}{{g}}\cdot{\bm}{r}'+e\,{\bm}{r}'\cdot(\hat{Q}\cdot{\bm}{r}')\label{eqn:H0}\\
& & \qquad\qquad +\frac{1}{2m} {\bm}{\pi}^2-\frac{e{\mathrm{g}_s}}{2m} {\bm}{B}^{(0)}\cdot{\bm}{s}\quad. \nonumber\end{aligned}$$ In this expression we dropped the auxiliary coordinate $u$ in favour of the coordinate system ${\bm}{r}'=(x',y',z')$, given by Eq. (\[eqn:firstRindlerMetric\]). Since $u=z'+\mathrm{O}(1/{\mathrm{c}})$, this coordinate transformation allowed us to replace the canonical momentum by ${\bm}{\pi} = {\bm}{p}' + \frac{1}{2}\,e \, {\bm}{r}'\times {\bm}{B}^{(0)} $ and the electromagnetic potentials by $\Phi={\bm}{r}'\cdot (\hat Q \cdot {\bm}{r}')$ and ${\bm}{A} = - \frac{1}{2} \, {\bm}{r}'\times {\bm}{B}^{(0)} $.
![(Color online) from left to right: Change from laboratory frame with coordinates $(x',y',z')$, where ${\bm}{g}$ points into $z'$-direction, to the frame of trap geometry with $(x,y,z)$, where the $z$-axis and ${\bm}{B}^{(0)}$ are aligned. The angle ${\theta}$ is determined by the relation ${\bm}{g}\cdot{\bm}{B}^{(0)}={g}B \cos{\theta}$. []{data-label="Fig1"}](Bild3.jpg "fig:"){width="0.5\linewidth"}![(Color online) from left to right: Change from laboratory frame with coordinates $(x',y',z')$, where ${\bm}{g}$ points into $z'$-direction, to the frame of trap geometry with $(x,y,z)$, where the $z$-axis and ${\bm}{B}^{(0)}$ are aligned. The angle ${\theta}$ is determined by the relation ${\bm}{g}\cdot{\bm}{B}^{(0)}={g}B \cos{\theta}$. []{data-label="Fig1"}](Bild4.jpg "fig:"){width="0.5\linewidth"}
Eq. (\[eqn:H0\]) closely resembles the well known Hamiltonian of a non-relativistic electron in a Penning trap [@Brow86]. The essential difference is the presence of the Newtonian potential $m\,{\bm}{{g}}\cdot{\bm}{r}'$. The Hamiltonian $H_0$ can be further simplified by performing two additional transformations. First, we rotate the coordinate system, such that the $z$-axis is aligned with the direction of the magnetic field ${\bm}{B}^{(0)}$, see FIG. \[Fig1\]. This is conventional in the analysis of Penning trap experiments [@Brow86; @Dehm88]. As a second step, we shift the coordinate center by a constant vector, such that it coincides with the new equilibrium position of the electron motion, see FIG. \[Fig2\]. Moreover, we apply an unitary transformation $\tilde H_0= U^\dagger H_0 U$ in order to shift the momentum in $y$-direction by a constant value. While a detailed discussion of these transformations is given in Appendix \[section:C\], here we just present the obtained Hamiltonian
![(Color online) After the rotation, shown in FIG. \[Fig1\], the coordinate center is shifted into the new equilibrium position of the electron motion. []{data-label="Fig2"}](Bild4.jpg "fig:"){width="0.5\linewidth"}![(Color online) After the rotation, shown in FIG. \[Fig1\], the coordinate center is shifted into the new equilibrium position of the electron motion. []{data-label="Fig2"}](Bild5neu.jpg "fig:"){width="0.5\linewidth"}
$$\begin{aligned}
\tilde H_0&=& m{\mathrm{c}}^2+ \frac{m}{2}\omega_z^2\,\zeta^2 +\frac{m}{8}(\omega^2_c-2\omega_{z}^2)\,\rho^2 +\frac{1}{2m} {\bm}{P}^2 \qquad \label{eqn:hatH0}\\
& & \quad-\frac{\omega_c}{2}(L_3+{\mathrm{g}_s}\, s_3)+\frac{m{g}^2}{\omega_z^2}\left(1-\frac{3}{2}\cos^2\theta\right) \,.\nonumber \end{aligned}$$
Here, due to the axial symmetry of a Penning trap, it is convenient to use cylindrical coordinates ${\bm}{R} =
( \rho, \varphi, \zeta)$. The momentum operator in this coordinate system is denoted as ${\bm}{P}$. In Eq. (\[eqn:hatH0\]), moreover, $$\begin{aligned}
\omega_c & =& eB/m \quad, \label{eqn:cyclotron}\\
\omega_z & = & \sqrt{eV/(mL^2)}\label{eqn:axial}\quad.\end{aligned}$$ are the cyclotron frequency $\omega_c$ and the axial frequency $\omega_z$. The operator $\tilde H_0$ now is the well known Hamiltonian of an electron in a Penning trap [@Brow86], except for the very last term in Eq. (\[eqn:hatH0\]). This term describes the effect of Newtonian gravity on the electron and depends on the orientation of the Penning trap with respect to the acceleration ${\bm}{g}$. Since this term is constant, we are able to solve the eigenvalue problem $$\tilde H_0 \phi_{0}^{k,n,\ell,s} = E_{k,n,\ell,s} \phi_{0}^{k,n,\ell,s} \label{eqn:eigenvalueeq}$$ analytically. As the result, we get the well established energies of the eigenvalue problem of geonium [@Brow86], shifted by this constant gravitational term: $$\begin{aligned}
& & E_{k,n,\ell,s}=m{\mathrm{c}}^2\nonumber\\
& &\quad +\,\hbar \omega_z \left(k+\frac{1}{2}\right)+ \hbar \omega_{c'} \left(n+\frac{1}{2}\right)-\hbar \omega_{m}\left(\ell+\frac{1}{2}\right)\nonumber\\
& &\quad +
\,\frac{{\mathrm{g}_s}}{2}\,\hbar\omega_{c}\,s+\frac{m{g}^2}{\omega_z^2}\left(1-\frac{3}{2}\cos^2\theta\right)\,. \label{eqn:Energy}\end{aligned}$$ Here, following [@Brow86], $k$ and $n$ are the non-negative integer quantum numbers of axial and cyclotron oscillation, while $\ell=0,1,2,\dots$ and $s=\pm 1/2$ account for the angular momentum and spin projection of the electron onto the direction of the magnetic field. Moreover, the corrected or reduced cyclotron frequency $\omega_{c'}$ and the magnetron frequency $\omega_m$ are defined by $$\begin{aligned}
\omega_{c'} &=& (\omega_{c} + \sqrt{\omega^2_c-2\omega_{z}^2} \,)/2 \quad, \label{eqn:cprime}\\
\omega_m &=&(\omega_{c} - \sqrt{\omega^2_c-2\omega_{z}^2} \,)/2\quad . \label{eqn:magnetron}\end{aligned}$$ Together with the axial frequency $\omega_z$, these are standard observables in Penning trap experiments. As seen from (\[eqn:Energy\]), Newtonian gravity leads to a constant shift of energy levels, only. This shift is independent of the quantum numbers of the electron in the trap and, therefore, does not effect frequencies of bound-bound transitions in geonium. In the next section we will see, that this is not the case if we take into account relativistic effects.
Relativistic energy correction for a gravitationally influenced electron in a Penning trap {#section:IIIc}
------------------------------------------------------------------------------------------
In the previous section we considered the Hamiltonian of a non-relativistic electron in a Penning trap in the presence of a homogeneous Newtonian gravitational field. The eigenvalues of this Hamiltonian are given by (\[eqn:Energy\]), while the explicit form of corresponding eigenfunctions is given in Appendix \[section:C\]. In this section we will use these eigenfunctions as a basis for a perturbation analysis in order to account for relativistic effects. The perturbation $\tilde H_I$ of the Hamiltonian $\tilde H_0$ can be formally written as $$\tilde H_I=\tilde H'''-\tilde H_0+\,\mathrm{O}(1/{\mathrm{c}}^3)\quad.$$ Within first order perturbation theory, the energy corrections can be expressed by $$\delta E_{k,n,\ell,s}=\langle \phi^{k,n,\ell,s} |(\tilde H'''-\tilde H_0)|\phi^{k,n,\ell,s}\rangle +\,\mathrm{O}(1/{\mathrm{c}}^3)\,, \label{eqn:corr}$$ where similar to the steps, leading to Eq. (\[eqn:hatH0\]), we apply a transformation to cylindrical coordinates ${\bm}{R} = ( \rho, \varphi, \zeta)$ and perform the unitary transformation $\tilde H'''=\tilde H'''({\bm}{R},{\bm}{P})=U^\dagger H'''({\bm}{r}',{\bm}{p}')U$, afterwards. However, in contrast to the last section, the auxiliary coordinate $u=z'-{g}z'^2/(2c^2)+\,\mathrm{O}(1/{\mathrm{c}}^3)$ is now replaced by $z'$, taking into account the relativistic corrections of order $1/{\mathrm{c}}^2$. While the energy correction (\[eqn:corr\]) can be applied for any set of quantum numbers $n$, $k$, $\ell$, $s$, in the following we want to use these energy corrections in order to investigate the relativistic effects on ${\mathrm{g}_s}$-factor measurements, as they are discussed in [@Gab08]. In these experiments the transitions between the lowest energy levels in geonium are driven under the change of quantum numbers $n$ and $s$, while $k=\ell=0$. For this scenario the energy correction reads $$\begin{aligned}
\delta E_{0,n,0,s}/\hbar &=& -\frac{1}{8}(1+2n+2s)^2\delta \label{eqn:energyshifts}\\
& & -\frac{1}{2}(1+2n+{\mathrm{g}_s}s)\sigma_1(\theta)+(1+n)\sigma_2(\theta)\,,\quad\nonumber \end{aligned}$$ where $\delta$ is the special relativistic correction due to cyclotron motion: $$\begin{aligned}
\delta &{=}& \frac{\hbar\omega_c^2}{m c^2}\,. \label{eqn:delta}\end{aligned}$$ Moreover, the frequencies $$\begin{aligned}
\sigma_1(\theta) &{=}& -2\,\frac{{g}^2\omega_c}{{\mathrm{c}}^2\omega_z^2}\,\cos^2\theta\left(1-\frac{3}{2}\cos^2\theta\right)\,\label{eqn:sigma1}\end{aligned}$$ and $$\begin{aligned}
\sigma_2(\theta) &{=}& \frac{{g}^2}{2{\mathrm{c}}^2\omega_c}\,\sin^2\theta\left(1-\frac{9}{4}\sin^2(2\theta)\right)\, \label{eqn:sigma2}\end{aligned}$$ are related to the first order non-vanishing gravitational effects. In these expressions we assumed $\omega_c=\omega_{ c}'$ and neglected all higher orders of $\omega_z/\omega_c$. In order to investigate the corrections to the ${\mathrm{g}_s}$-factor formula introduced by [@Brow86] we will use the energy correction (\[eqn:energyshifts\]) in the next section.
Gravitational effect on free electron ${\mathrm{g}_s}$-factor measurements {#section:IIId}
--------------------------------------------------------------------------
Having derived the energy (\[eqn:Energy\]) of an electron in a Penning trap and its relativistic correction (\[eqn:energyshifts\]), we are prepared to discuss the effect of gravity on the result of free electron ${\mathrm{g}_s}$-factor measurements. Therefore, we follow the steps, performed by L. S. Brown and G. Gabrielse in order to obtain the ${\mathrm{g}_s}$-factor formula, presented in [@Brow86], which in our case will contain additional corrections. In their analysis, the ${\mathrm{g}_s}$-factor is extracted by the measurement of two frequencies of transitions in geonium, namely the anomalous frequency and the reduced cyclotron frequency. The first one we obtain from the spin-flip transition between the energy levels ($n=1$, $s=-1/2$) and ($n=0$, $s=+1/2$), see FIG. \[Fig4\]. By employing Eq. (\[eqn:Energy\]) and Eq. (\[eqn:energyshifts\]), this frequency can be calculated as
![(Color online) Level scheme for spin states and the lowest cyclotron oscillator states of an electron in a Penning trap. Here, $n$ is the quantum number of cyclotron oscillation and $s=\pm 1/2$ refers to the two spin states of the electron in the Penning trap. The spin-flip transition $\bar{\omega}_{a'}$ (\[eqn:anomalous\]) and the cyclotron transition $\bar{\omega}_{c'}$ (\[eqn:cyclo\]), used to determine the free electron ${\mathrm{g}_s}$-factor are marked in red. Due to the $n$,$s$-dependence of the relativistic corrections (\[eqn:energyshifts\]), the energy levels are not equidistant. []{data-label="Fig4"}](level.png){width="0.7\linewidth"}
$$\begin{aligned}
\bar{\omega}_{a'} &=& (E+\delta E)_{0,0,0,+1/2}/\hbar - (E+\delta E)_{0,1,0,-1/2}/\hbar \nonumber \\
&=& {\mathrm{g}_s}\omega_c/2 -\omega_{c'} - ({\mathrm{g}_s}/2-1) \sigma_1 (\theta) - \sigma_2 (\theta) \quad.\label{eqn:anomalous}\end{aligned}$$
In contrast to previous investigations [@Gab08; @Gab06; @Brow86], this expression now also contains the gravitational correction frequencies $\sigma_1 (\theta)$ and $\sigma_2 (\theta)$. In order to extract the free electron ${\mathrm{g}_s}$-factor from Eq. (\[eqn:anomalous\]), we can rearrange it $${\mathrm{g}_s}/2=1+\frac{{\bar{\omega}}_{ a'}-(\omega_c-\omega_{c'}) + \sigma_2(\theta)}{\omega_c-\sigma_1(\theta)}\quad \label{eqn:anomaly01}$$ and further simplify it, using $$\omega_c-\omega_{ c'} =\omega_m = \frac{\omega_z^2}{2\omega_{c'}} \quad, \label{eqn:simp}$$ which can be obtained from the definitions of the frequencies (\[eqn:cyclotron\]),(\[eqn:axial\]) and (\[eqn:cprime\]), (\[eqn:magnetron\]). With the help of Eq. (\[eqn:simp\]), we obtain $${\mathrm{g}_s}/2=1+\frac{{\bar{\omega}}_{ a'}-\frac{\omega_z^2}{2\omega_{c'}} + \sigma_2(\theta)}{\omega_{c'}+\frac{\omega_z^2}{2\omega_{c'}}-\sigma_1(\theta)} \label{eqn:anomaly02}\quad.$$ As seen from this expression, the ${\mathrm{g}_s}$-factor formula depends not only on ${\bar{\omega}}_{ a'}$, but also on $\omega_{c'}$ and $\omega_z$. Since the latter is obtained by tracking the mirror charge of the electron [@Brow86], we assume $\omega_z$ to be known and focus on the discussion of $\omega_{c'}$. In the non-relativistic limit, $\omega_{c'}$ is the frequency of transitions between the energy levels ($n=0$, $s=1/2$) and ($n=1$, $s=1/2$). In practice, the frequency of this transition is affected by relativistic effects, see FIG. \[Fig4\]. This actually measured frequency will be denoted as $\bar{\omega}_{c'}$. Using Eq. (\[eqn:Energy\]) and Eq. (\[eqn:energyshifts\]), we can express this frequency as $$\begin{aligned}
\bar{\omega}_{c'} &=& (E+\delta E)_{0,1,0,1/2}/\hbar - (E+\delta E)_{0,0,0,1/2}/\hbar \nonumber \\
&=&{\omega}_{c'} - \frac{3}{2}\delta -\sigma_1 (\theta) + \sigma_2 (\theta) \quad.\end{aligned}$$ By this relation, we can express ${\omega}_{c'}$ in Eq. (\[eqn:anomaly02\]) in terms of $\bar{\omega}_{c'}$ and the correction frequencies. In order to further simplify the ${\mathrm{g}_s}$-factor formula, we make a Taylor expansion of $$\begin{aligned}
\frac{\omega_z^2}{2\omega_{c'}} &=& \frac{\omega_z^2}{2({\bar{\omega}}_{c'}+3\delta/2+\sigma_1 (\theta) - \sigma_2 (\theta)) }\label{eqn:cyclo}\\
&\approx& \frac{\omega_z^2}{2\bar{\omega}_c'}- \frac{1}{2}\left(\frac{\omega_z}{\omega_c}\right)^2\left(\frac{3}{2}\delta +\sigma_1 (\theta) -\sigma_2 (\theta)\right) \quad. \nonumber\end{aligned}$$ In the last step, we insert this expansion in Eq. (\[eqn:anomaly02\]), where we consider only the leading contributions of $\delta$, $\sigma_1(\theta)$ and $\sigma_2(\theta)$. We finally obtain $$\begin{aligned}
{\mathrm{g}_s}/2 &=& 1+\frac{\bar{\omega}_{a'}+\sigma(\theta)-\omega_z^2/(2\bar{\omega}_{c'})}{\bar{\omega}_{c'}+\frac{3}{2}\delta-\sigma(\theta)+\omega_z^2/(2\bar{\omega}_{c'})}\quad, \label{eqn:final}\end{aligned}$$ where all gravitational correction frequencies enter the expression $$\sigma(\theta)= \frac{1}{2}\left(\frac{\omega_{z}}{\omega_{c}}\right)^2\sigma_1(\theta)+\sigma_2(\theta).$$ In the case of vanishing acceleration, i.e. ${\bm}{g}=0$, this equation recovers the known ${\mathrm{g}_s}$-factor formula by L. S. Brown and G. Gabrielse from [@Brow86], while the presence of gravity leads to additional contributions to this formula. In order to illustrate this, we can expand Eq. (\[eqn:final\]) into $${\mathrm{g}_s}/2={\mathrm{g}_s}^{(0)}/2+\delta_\sigma{\mathrm{g}_s}/2\quad,$$ where ${\mathrm{g}_s}^{(0)}$ is the known expression for the free electron ${\mathrm{g}_s}$-factor from [@Brow86], while the relative shift of ${\mathrm{g}_s}$-factor due to gravity is of the order $$\frac{\delta_\sigma {\mathrm{g}_s}}{{\mathrm{g}_s}}\sim \frac{1}{2} \frac{({g}/{\mathrm{c}})^2}{\omega_c^2}\quad.$$ In the recent Penning trap experiments of [@Gab08], a cyclotron frequency of $\omega_c=2\pi\cdot 149 \,\mathrm{GHz}$ is used. For such an experiment performed in a laboratory on Earth, i.e. ${g}=9.81\, \mathrm{m}/\mathrm{s}^2$, we obtain $\delta_\sigma {\mathrm{g}_s}/{\mathrm{g}_s}\sim 6.1\times 10^{-40} $.
Summary and Conclusion {#section:IV}
======================
In this work we presented a theoretical investigation of an electron in a Penning trap in the presence of a gravitational field. In this system we analyzed how the presence of gravity may affect the result of free electron ${\mathrm{g}_s}$-factor measurements. Therefore, we considered a single electron with anomalous magnetic moment in the presence of electromagnetic fields in the spacetime of homogeneous acceleration. For this scenario we derived the Hamiltonian (\[eqn:newHamilton\]), which accounts for the relativistic effects up to order $1/{\mathrm{c}}^2$. This Hamiltonian has been applied to the electron dynamics in a gravitational distorted Penning trap, whose electromagnetic field (\[eqn:RindlerA\]), (\[eqn:RindlerPhi\]) is given as an exact solution of Maxwell equations in Rindler spacetime. Making use of first order perturbation theory, we derived analytical expressions for the energy eigenvalues (\[eqn:Energy\]),(\[eqn:energyshifts\]) of that Hamiltonian up to order $1/{\mathrm{c}}^2$. A detailed analysis of these energies has shown, that Newtonian gravity only leads to constant shifts of the energy levels of geonium. Thus, Newtonian gravity has no effect on measured transition frequencies. In contrast, the relativistic effects of order $1/{\mathrm{c}}^2$ lead to relative shifts of the energy levels. We, therefore, argue that these relativistic corrections may affect the ${\mathrm{g}_s}$-factor measurements, which rely on transitions in geonium. In order to quantify the gravitational effects, we derived the expression (\[eqn:final\]), which for ${\bm}{{g}}=0$ recovers the known ${\mathrm{g}_s}$-factor formula introduced by L. S. Brown and G. Gabrielse, while for ${\bm}{{g}}\not=0$ it predicts a shift of the measured ${\mathrm{g}_s}$-factor of $\delta_\sigma {\mathrm{g}_s}/{\mathrm{g}_s}\sim 6.1\times 10^{-40}$. While this can not be measured in experiments of current accuracy, it can be enhanced in the case of lower frequencies and higher accelerations and, therefore, may be important for future studies.
Appendix {#appendix .unnumbered}
========
Coordinate transformation of Dirac action towards Rindler spacetime {#section:A}
===================================================================
In section \[section:IIb\] we pointed out, that some attention has to be drawn to the transformation of the Dirac action towards Rindler spacetime. The generalization of Dirac equation to curved spacetime or spacetime of non-geodesic motion is discussed in a wide range of publications, see for instance [@Hehl90; @Lippoldt; @Weldon]. Indeed there are many degrees of freedom – especially the freedom of an additional spin base transformation of the spinor and/or the Dirac matrices. In this Appendix, we show the way we have chosen to get the Dirac action in the form of (\[eqn:DiracAction1\]), where the spin base of the spinor and the Dirac matrices transforms under the spin representation of the coordinate transformation (\[eqn:RindlerTrafo\]). We want to emphasize, that this is a choice, that is of advantage in our case, and by no means an advice how to perform such a transformation in general. Starting with the Dirac action (\[eqn:DiracAction\]), we perform the coordinate transformation (\[eqn:RindlerTrafo\]) in the form of $x^{\mu}=\frac{{\partial}x^{\mu}}{{\partial}x^{\mu'}}\,x^{\mu'}$, where spacetime indices transform under the common properties of coordinate differentials. For the derivative and the volume of the spacetime integral, this means $${\partial}_{\mu}=\frac{{\partial}x^{\mu'}}{{\partial}x^{\mu}}\,{\partial}_{\mu'}\quad,\quad {\mathrm{d}}x^4=\left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,{\mathrm{d}}{x'}^4\quad. \label{eqn:A01}$$ In addition we want to allow, that the spinor and the Dirac matrices are affected by a spin base transformation ${\mathcal{S}}={\mathcal{S}}(x^{\mu'})$, which has to be specified later: $$\psi(x^{\mu})={\mathcal{S}}\psi'(x^{\mu'})\quad,\quad\gamma^\mu=\frac{{\partial}x^{\mu}}{{\partial}x^{\mu'}} {\mathcal{S}}\gamma^{\mu'}(u){\mathcal{S}}^{-1}\quad.$$ Under these assumptions, the Dirac adjoint spinor reads $$\begin{aligned}
\bar\psi(x^{\mu})&=&(\psi(x^{\mu}))^\dagger \gamma^0 \label{eqn:A03}\\
&=&(\psi'(x^{\mu'}))^{\dagger}{\mathcal{S}}^{\dagger}\frac{{\partial}x^{0}}{{\partial}x^{\mu'}} {\mathcal{S}}\gamma^{\mu'}(u){\mathcal{S}}^{-1} \,. \nonumber\end{aligned}$$ Applying the transformations (\[eqn:A01\]) - (\[eqn:A03\]) to the Dirac action, we obtain $$\begin{aligned}
S[\bar\psi',\psi']=\int(\psi'(x^{\nu'}))^\dagger {\mathcal{S}}^\dagger {\mathcal{S}}\frac{{\partial}x^{0}}{{\partial}x^{\mu'}}\gamma^{\mu'}(u) \hspace{0.15 \linewidth} & & \label{eqn:A04}\\ \times \,\left[{i}\hbar \gamma^{\mu'}(u)\left({\partial}_{\mu'}+S^{-1}{\partial}_{\mu'}{\mathcal{S}}\right)-m{\mathrm{c}}\right]\psi'(x^{\nu'}) & &\nonumber\\
\times \,\, \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,{\mathrm{d}}{x'}^4\quad, \nonumber & &\end{aligned}$$ where we used $\frac{{\partial}x^{\mu}}{{\partial}x^{\mu'}}\frac{{\partial}x^{\nu'}}{{\partial}x^{\mu}}=\delta^{\nu'}_{\mu'}$ and ${\mathcal{S}}{\mathcal{S}}^{-1}=1$. In similarity to $\bar\psi(x^{\mu})=(\psi(x^{\mu}))^\dagger \gamma^0 $, we want the Dirac adjoint spinor to have the form $\bar\psi'(x^{\nu'})=(\psi'(x^{\mu'}))^\dagger \gamma^{0'}(u)$ in the new coordinate system. This leads to the condition $${\mathcal{S}}^\dagger {\mathcal{S}}\frac{{\partial}x^{0}}{{\partial}x^{\mu'}}\gamma^{\mu'}(u) \stackrel{!}{=}
\gamma^{0'}(u)\quad, \label{eqn:determineS}$$ which is suitable to determine ${\mathcal{S}}^\dagger {\mathcal{S}}$, fixing ${\mathcal{S}}$ up to an unitary transformation. The used spin base transformation, which satisfies (\[eqn:determineS\]) is $$\begin{aligned}
{\mathcal{S}}=& &\sqrt{1+\frac{{g}u}{{\mathrm{c}}^2}}\,\cosh\left(\frac{{g}t}{2c}\right)\, \gamma^{0'}(u)\\
& &\qquad+ \,\frac{1}{\sqrt{1+\frac{{g}u}{{\mathrm{c}}^2}}}\,\sinh\left(\frac{{g}t}{2c}\right)\, \gamma^{3'} \quad.\nonumber\end{aligned}$$ In terms of the matrices ${\bm}{\alpha}$ and $\beta$, defined in (\[eqn:BetaAlpha1\]) we get $$\begin{aligned}
{\mathcal{S}}=\frac{1}{\sqrt{1+\frac{{g}u}{{\mathrm{c}}^2}}}\,\beta\, \mathrm{exp}\left({\frac{{\bm}{{g}}\cdot{\bm}{\alpha}\, t}{2c}}\right)\quad. \label{eqn:A8}\end{aligned}$$ where ${\bm}{{g}}=(0,0,{g})$. The action (\[eqn:A03\]) now reads $$\begin{aligned}
S[\bar\psi',\psi']=\int\bar\psi'(x^{\nu'}) \hspace{0.4 \linewidth} & &\\ \times \,\left[{i}\hbar \gamma^{\mu'}(u)\left({\partial}_{\mu'}+{\mathcal{S}}^{-1}{\partial}_{\mu'}{\mathcal{S}}\right)-m{\mathrm{c}}\right]\psi'(x^{\nu'}) & &\nonumber\\
\times \,\, \left(1+\frac{{g}u}{{\mathrm{c}}^2}\right)\,{\mathrm{d}}{x'}^4\quad. \nonumber & &\end{aligned}$$ Finally the additional term $\gamma^{\mu'}(u){\mathcal{S}}^{-1}{\partial}_{\mu'}{\mathcal{S}}=0$ turns out to be zero and, thus, Eq. (\[eqn:A8\]) results in Eq. (\[eqn:DiracAction1\])
Reminder on Covariant Derivatives {#section:B}
=================================
What follows is a short reminder on covariant derivatives, needed to formulate the covariant Maxwell equations in Rindler spacetime. For a detailed discussion of this topic, see for instance [@Carrol], [@Wald]. The covariant derivative $\nabla_\mu$, acting on a tensor $A^\nu$ with one upper or $A_\nu$ with one lower index, is connected to the partial derivative by $$\begin{aligned}
\nabla_\mu A^\nu={\partial}_\mu A^\nu +\Gamma^\nu_{\mu\rho} A^\rho\quad,\\
\nabla_\mu A_\nu={\partial}_\mu A_\nu -\Gamma^\rho_{\mu\nu} A_\rho\quad,\end{aligned}$$ where the Christoffel symbols $\Gamma^\rho_{\mu\nu}$ are constructed by first partial derivatives of the metric $$\begin{aligned}
\Gamma^\rho_{\mu\nu}=\frac{1}{2} g^{\rho\sigma}\left({\partial}_\mu g_{\sigma \nu}+{\partial}_\nu g_{\sigma \mu}-{\partial}_\sigma g_{\mu\nu}\right)\quad.\end{aligned}$$ Therefore the covariant Lorentz gauge condition for a four-vector potential $A_\mu=(\Phi/c,-{\bm}{A})$ is $$\begin{aligned}
\nabla_\mu A^\mu={\partial}_\mu A^\mu +\Gamma^\mu_{\mu\rho} A^\rho=0\quad,\end{aligned}$$ where the indices of $A^\mu=g^{\mu\nu} A_\nu$ are risen up with the inverse metric $g^{\mu\nu}$. The same way the electromagnetic field strength tensor with two upper indices $F^{\mu\nu}=g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma}$ is constructed from $F_{\mu\nu}={\partial}_{\mu} A_{\nu} - {\partial}_{\nu} A_{\mu}$. The covariant derivative of this quantity is $$\begin{aligned}
\nabla_\rho F^{\mu\nu}&=&{\partial}_\rho F^{\mu\nu} +\Gamma^\nu_{\rho\sigma} F^{\mu\sigma}+\Gamma^\mu_{\rho\sigma} F^{\sigma\nu}\quad,\\
\nabla_\nu F^{\mu\nu}&=&{\partial}_\nu F^{\mu\nu} +\Gamma^\nu_{\nu\sigma} F^{\mu\sigma} \quad, \label{eqn:AppMaxwell}\end{aligned}$$ where in the special case of the vacuum Maxwell equations $\nabla_\nu F^{\mu\nu}=0$, for $\rho=\nu$ in (\[eqn:AppMaxwell\]) the term $\Gamma^\mu_{\sigma\nu} F^{\nu\sigma}=0$ vanishes, which gives Maxwell equations in the form of Eq. (\[eqn:Maxwell\]).
Towards geonium in Newtonian gravity {#section:C}
====================================
In this Appendix we discuss the steps and transformations, leading to the Hamiltonian (\[eqn:hatH0\]), its energies $E_{n,k,\ell,s}$ and the corresponding eigenfunctions $\phi_{k,n,\ell,s}$. Therefore, the starting point is $H_0$, from (\[eqn:H0\]), which is the Hamiltonian of geonium, affected by a homogeneous gravitational field: $$\begin{aligned}
H_0&=&m{\mathrm{c}}^2+ m {\bm}{{g}}\cdot{\bm}{r}'+e\,{\bm}{r}'\cdot(\hat{Q}\cdot{\bm}{r}')\label{eqn:C1} \\
& & \qquad\qquad +\frac{1}{2m} {\bm}{\pi}^2-\frac{e{\mathrm{g}_s}}{2m} {\bm}{B}^{(0)}\cdot{\bm}{s}\,, \nonumber\end{aligned}$$ *(i)* In the first step, we rotate the coordinate system from the laboratory frame with the coordinates ${\bm}{r}'=(x',y',z')$, where the $z'$-axis points into direction of ${\bm}{g}$, to the frame of trap geometry with ${\bm}{r}=(x,y,z)$, where the $z$-axis and ${\bm}{B}^{(0)}$ are aligned: $$\begin{aligned}
x'=x \cos{\theta}- z\sin{\theta}\,,\,y'=y\,,\,z'=z \cos{\theta}+ x\sin{\theta}. \qquad\end{aligned}$$ This transformation introduces the angle $\theta$ between the direction of acceleration and the magnetic field, as it is shown in FIG. \[Fig2\]. The Hamiltonian (\[eqn:C1\]) in the rotated system reads $$\begin{aligned}
H_0=m{\mathrm{c}}^2+ m {g}(z\cos{\theta}+ x\sin{\theta})-\frac{eV}{4L^2}(x^2+y^2-2z^2)\quad\nonumber & & \\
+\frac{1}{2m} {\bm}{p}^2 + \frac{e^2 B^2 }{8m}(x^2+y^2) -\frac{e B }{2m} \,L_3-\frac{e{\mathrm{g}_s}B }{2m} \, s_3\,.\quad\qquad\label{eqn:C3}& &\end{aligned}$$ *(ii)* In the second step, the coordinate dependent Newtonian potential in (\[eqn:C3\]) is absorbed by an additional coordinate transformation to coordinates ${\bm}{R}=(X,Y,Z)$, that shifts the coordinate center by a constant vector, such that it coincides with the new equilibrium position of the electron motion, as it is shown in FIG. \[Fig2\]: $$\begin{aligned}
x=X + 2{g}\sin{\theta}/\omega_z^2 \,\,,\,\, y=Y\,\,,\,\, z=Z-{g}\cos{\theta}/\omega_z^2\,, \quad\quad\end{aligned}$$ Moreover, we have to absorb the upcoming constant shift of the linear momentum in $Y$-direction by an unitary transformation $U(Y)=\mathrm{exp}\left({i}m{g}\sin{\theta}\, \omega_c Y/(\hbar\omega_z^2)\right)$ of the Hamiltonian $\tilde H_0=U(Y)^{\dagger}H_0U(Y)$ and the electron wave function $\phi({\bm}{R})=U(Y)\phi_0({\bm}{R})$. After that, the coordinate dependent Newtonian potential in $\tilde H_0$ is replaced by an additive constant value: $$\begin{aligned}
\tilde H_0&=& m{\mathrm{c}}^2+ \frac{m}{2}\omega_z^2Z^2 +\frac{m}{8}(\omega^2_c-2\omega_{z}^2)(X^2+Y^2) \qquad \label{eqn:HGeo}\\
& & +\frac{1}{2m} {\bm}{P}^2 -\frac{\omega_c}{2}(L_3+{\mathrm{g}_s}\, s_3)+\frac{m{g}^2}{\omega_z^2}\left(1-\frac{3}{2}\cos^2{\theta}\right) \,,\nonumber\end{aligned}$$ where we replaced the electromagnetic quantities by the frequencies (\[eqn:cyclotron\]) and (\[eqn:axial\]). Expressing ${\bm}{R}$ in cylindrical coordinates $$\begin{aligned}
X&=&\rho\,\cos\varphi \quad,\quad Y=\rho\,\sin\varphi\quad,\quad Z=\zeta \,,\end{aligned}$$ we obtain the Hamiltonian $\tilde H_0$ as shown in (\[eqn:hatH0\]). The solution $\phi^{k,n,\ell,s}({\bm}{R})=U(Y)\phi^{k,n,\ell,s}_0({\bm}{R})$ to the eigenvalue problem (\[eqn:eigenvalueeq\]) is: $$\begin{aligned}
& &\phi^{k,n,\ell,s}({\bm}{R})=\nonumber\\[0.5em]
& & \frac{1}{\sqrt{2\pi}}U(\rho\sin\varphi){\mathrm{e}}^{{i}(\ell-n) \varphi}
R^{n,\ell}(\rho)W^k(\zeta)\,|s\rangle\quad, \label{eqn:wavefunctions}\end{aligned}$$ where $$\begin{aligned}
U(\rho\sin\varphi)&=&\mathrm{exp}\left(\frac{{i}m{g}\sin{\theta}\, \omega_c \,\rho\sin\varphi}{\hbar\omega_z^2}\right)\\
R^{n,\ell}(\rho)&=&\left(\frac{m\omega_{\bar c}}{2\hbar}\right)^{(1+\ell-n)/2}\sqrt{\frac{2\,n!}{\ell!}}\label{eqn:Radial}\\
& &\qquad \times {\mathrm{e}}^{-\frac{m\omega_{\bar c}\rho^2}{4\hbar} }\rho^{\ell-n}\mathcal{L}^{\ell-n}_{n}\left(\frac{m\omega_{\bar c}}{2\hbar}\rho^2\right)\nonumber\\
W^k(\zeta)&=& \left(\frac{m\omega_z}{\pi\hbar}\right)^{1/4}\frac{2^{-k/2}}{\sqrt{k!}}\\
& & \qquad \times \, {\mathrm{e}}^{-\frac{m\omega_z\zeta^2}{2\hbar} }\mathcal{H}_k\left(\sqrt{\frac{m\omega_z}{\hbar}}\zeta\right)\nonumber\end{aligned}$$ where we defined $\omega_{\bar c}=\sqrt{\omega^2_c-2\omega_{z}^2}$ and have used the Laguerre polynomials $\mathcal{L}^{\ell-n}_{n}$, the Hermite polynomials $\mathcal{H}_n$ and the spin basis $|s\rangle=|\pm1/2\rangle$. The wave functions (\[eqn:wavefunctions\]) are normalized with respect to the scalar product $\langle\phi_1|\phi_2\rangle_{0}{=}\int \phi_1^\dagger\phi_2 \,\rho\, {\mathrm{d}}\rho{\mathrm{d}}\varphi{\mathrm{d}}\zeta$. We are allowed to apply first order perturbation theory as usual in flat space, using the scalar product $\langle\phi_1|\phi_2\rangle_{0}$, since the result would coincide with the energy corrections, obtained from scalar product (\[eqn:ScalarProduct\]) to desired order. The justification of this procedure is motivated very detailed in [@Parker80].
[12]{}
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|
---
author:
- |
Robert R. Tucci\
P.O. Box 226\
Bedford, MA 01730\
[email protected]
date:
title: |
An Introduction to\
Cartan’s KAK Decomposition\
for QC Programmers
---
Abstract {#abstract .unnumbered}
========
This paper presents no new results; its goals are purely pedagogical. A special case of the Cartan Decomposition has found much utility in the field of quantum computing, especially in its sub-field of quantum compiling. This special case allows one to factor a general 2-qubit operation (i.e., an element of $U(4)$) into local operations applied before and after a three parameter, non-local operation. In this paper, we give a complete and rigorous proof of this special case of Cartan’s Decomposition. From the point of view of QC programmers who might not be familiar with the subtleties of Lie Group Theory, the proof given here has the virtues, that it is constructive in nature, and that it uses only Linear Algebra. The constructive proof presented in this paper is implemented in some Octave/Matlab m-files that are included with the paper. Thus, this paper serves as documentation for the attached m-files.
Introduction and Motivation
===========================
Cartan’s KAK Decomposition was discovered by the awesome mathematical genius, Elie Cartan (1869-1951). Henceforth, for succinctness, we will refer to his decomposition merely as KAK. The letters KAK come from the fact that in stating and proving KAK, one considers a group $\ul{G}=\exp(\ul{g})$ with a subgroup $\ul{K}=\exp(\ul{k})$ and a Cartan subalgebra $\ul{a}$, where $\ul{g} = \ul{k}\oplus \ul{k}^\perp$ and $\ul{a}\subset \ul{k}^\perp$. Then one shows that any $G\in \ul{G}$ can be expressed as $G=K_1 A K_2$, where $K_1, K_2\in \ul{K}$ and $A\in \exp(\ul{a})$. An authoritative discussion of KAK can be found in the book by Helgason[@Helg].
KAK was first applied to quantum computing (QC) by Khaneja and Glaser in Refs.[@KG]. Since we are using “KAK" to refer to the general theorem, we will use “KAK1" to refer to the special case of KAK used by Khaneja and Glaser. Besides KAK1, the Cosine-Sine Decomposition (CSD)[@Golub][@Paige] is another decomposition that is very useful[@Rudi] in QC. After Refs.[@KG] and [@Rudi], QC workers came to the realization[@SS-Gestapo] that CSD also follows from KAK , even though CSD was discovered[@Paige] quite independently from KAK.
This paper will only discuss KAK1. KAK1 is the assertion that: Given any $U\in SU(4)$, one can find $A_1, A_0, B_1, B_0\in SU(2)$ and $\vec{k}\in \RR^3$ so that
U= (A\_1A\_0)(B\_1B\_0) , where $\vec{\Sigma}$ is an operator that is independent of $U$ and will be defined later. Thus KAK1 parameterizes $SU(4)$, a 15-parameter Lie Group, so that 12 parameters characterize local operations, and only 3 parameters (the 3 components of $\vec{k}$) characterize non-local ones.
Ever since Refs.[@KG] appeared, many workers other than Khaneja and Glaser have used KAK1 in QC to great advantage (see, for example, Refs.[@VD], [@VW], [@Zhang]). Mainly, they have used KAK1 to compile 2-qubit operations. For instance, Vidal and Dawson used KAK1 to prove that any 2-qubit operation can be expressed with 3 or fewer CNOTs and some 1-qubit rotations.
This paper includes a complete, rigorous proof of KAK1 and related theorems. The proof of KAK1 presented here is based on the well known isomorphism $SO(4) =
\frac{SU(2)\times SU(2)}{\{(1,1),(-1-1)\}}$ and on a theorem by Eckart and Young (EY)[@EY]. The EY theorem gives necessary and sufficient conditions for simultaneous SVD (singular value decomposition) of two matrices. The relevance of the EY theorem to KAK1 was pointed out in Ref.[@Gsponer]. The proof of KAK1 given here is a constructive proof, and it uses only Linear Algebra. Contrast this to the proof of KAK given in Ref.[@Helg], which, although much more general, is a non-constructive (“existence") proof, and it uses advanced concepts in Lie Group Theory.
Octave is a programming environment and language that is gratis and open software. It copies most of Matlab’s function names and capabilities in Linear Algebra. A collection of Octave/Matlab m-files that implement the algorithms in this paper, can be found at ArXiv (under the “source" for this paper), and at my website (www.ar-tiste.com).
Notation and Other Preliminaries
================================
In this section, we will define some notation that is used throughout this paper. For additional information about our notation, see Ref.[@Paulinesia].
We will use the word “ditto" to mean likewise and respectively. For example, “$x$ (ditto, $y$) is in $A$ (ditto, $B$)", means $x$ is in $A$ and $y$ is in $B$.
As usual, $\RR, \CC$ will stand for the real and complex numbers. For any complex matrix $A$, the symbols $A^*, A^T, A^\dagger$ will stand for the complex conjugate, transpose, and Hermitian conjugate, respectively, of $A$. (Hermitian conjugate a.k.a. conjugate transpose and adjoint)
The Pauli matrices are defined by:
= (
[cc]{} 0 & 1\
1 & 0
) , = (
[cc]{} 0 & -i\
i & 0
) , = (
[cc]{} 1 & 0\
0 & -1
) . They satisfy = -= i , and the two other equations obtained from this one by permuting the indices $(x,y,z)$ cyclically. We will also have occasion to use the operator $\vec{\sigma}$, defined by:
= (, , ) .
Let $\sigma_{X_\mu}$ for $\mu \in\{0,1,2,3\}$ be defined by $\sigma_{X_0}=\sigma_1 = I_2$, where $I_2$ is the 2 dimensional identity matrix, $\sigma_{X_1}=\sigx$, $\sigma_{X_2}=\sigy$, and $\sigma_{X_3}=\sigz$. Now define
\_[X\_X\_]{}= \_[X\_]{}\_[X\_]{} for $\mu, \nu\in\{0,1,2,3\}$. For example, $\sigxy= \sigx\otimes \sigy$ and $\sigux = I_2\otimes\sigx$. The matrices $\sigma_{X_\mu X_\nu}$ satisfy = = -, \[eq-com-sigxx-sigyy\] and the two other equations obtained from this one by permuting the indices $(x,y,z)$ cyclically. We will also have occasion to use the operator $\vec{\Sigma}$, defined by:
= (, , ) .
Define
= (
[cccc]{} 1 & 0 & 0 & i\
0 & i & 1 & 0\
0 & i &-1 & 0\
1 & 0 & 0 &-i
) . It is easy to check that $\unor$ is a unitary matrix. The columns of $\unor$ are an orthonormal basis, often called the “magic basis" in the quantum computing literature. (That’s why we have chosen to call this matrix $\unor$, because of the “m" in magic).
In this paper, we often need to find the outcome $\unor^\dagger X \unor$ (or $\unor X \unor^\dagger$) of a similarity transformation ( equivalent to a change of basis) of a matrix $X\in \CC^{4\times 4}$ with respect to $\unor$. Since $X$ can always be expressed as a linear combination of the $\sigma_{X_\mu X_\nu}$, it is useful to know the outcomes $\unor^\dagger(\sigma_{X_\mu X_\nu})\unor$ (or $\unor(\sigma_{X_\mu X_\nu})\unor^\dagger$) for $\mu, \nu\in\{0,1,2,3\}$. One finds the following two tables:
\^(AB)=
[|rr|rrrr|]{}&&& B &&\
& &1 & & &\
&1 &1 &-& &-\
A &&-& &-&-\
&&-& &-&\
&&-& & &\
,\[tab-mh-m\]
(AB)\^=
[|rr|rrrr|]{}&&& B &&\
& &1 & & &\
&1 &1 & &-&-\
A && &-&-&-\
&&-&-& &\
&& & &-&\
.\[tab-m-mh\]
Proof of KAK1
=============
In this section, we present a proof of KAK1 and related theorems. The proofs are constructive in nature and yield the algorithms used in our software for calculating KAK1. Thus, even those persons that are not too enamored with mathematical proofs may benefit from reading this section.
\[claim-isom3\] Define a map $\phi$ by
: SU(2)SO(3), (A)= \^(AA\^\*) . Then $\phi$ is a well defined, onto, 2-1, homomorphism. Well-defined: For all $A\in SU(2)$, $\unor^\dagger (A \otimes A^*)\unor \in SO(3)$. Onto: For all $Q\in SO(3)$, there exist $A \in SU(2)$ such that $Q= \unor^\dagger (A \otimes A^*)\unor$. 2-1: $\phi$ maps exactly two elements ($A$ and $-A$) into one ($\phi(A)$). Homomorphism: $\phi$ preserves group operations.
\[claim-isom4\] Define a map $\Phi$ by
: SU(2)SU(2)SO(4) , (A, B)= \^(AB\^\*). Then $\Phi$ is a well defined, onto, 2-1, homomorphism. Well-defined: For all $
A, B\in SU(2)$, $\unor^\dagger (A \otimes B^*)\unor \in SO(4)$. Onto: For all $Q\in SO(4)$, there exist $A,B \in SU(2)$ such that $Q= \unor^\dagger (A \otimes B^*)\unor$. 2-1: $\phi$ maps exactly two elements ($(A,B)$ and $(-A, -B)$) into one ($\phi(A,B)$). Homomorphism: $\Phi$ preserves group operations.
Theorems \[claim-isom3\] and \[claim-isom4\] are proven in most modern treatises on quaternions, albeit using a different language, the language of quaternions. See Version 2 or higher of Ref.[@Paulinesia], for proofs of Theorems \[claim-isom3\] and \[claim-isom4\], given in the language favored here and within the quantum computing community.
\[claim-re-im-x\] Suppose $X$ is a unitary matrix and define $X_R = \frac{X + X^*}{2}$, $X_I = \frac{X - X^*}{2i}$. Then $Q=\left(
\begin{array}{cc}
X_R & X_I \\
-X_I & X_R
\end{array} \right)$ is an orthogonal matrix. Furthermore, $X_R$ and $X_I$ are real matrices satisfying $X_R X_R^T + X_I X_I^T =
X_R^T X_R + X_I^T X_I = 1$. Furthermore, $X_I X_R^T$ and $X_I^T X_R$ are both real, symmetric matrices.
1=X X\^=(X\_R + i X\_I)(X\_R\^T - i X\_I\^T) , so
\[eq-w-ortho1\] X\_R X\_R\^T + X\_I X\_I\^T = 1 , and
X\_I X\_R\^T - X\_R X\_I\^T=0 . \[eq-w-ortho-sym1\]
From $1=X^\dagger X$ we also get
\[eq-w-ortho2\] X\_R\^T X\_R + X\_I\^T X\_I = 1 , and
X\_I\^T X\_R - X\_R\^T X\_I=0 . \[eq-w-ortho-sym2\]
Note that Eqs.(\[eq-w-ortho1\]) and Eqs.(\[eq-w-ortho2\]) are identical except that in Eqs.(\[eq-w-ortho1\]), the second matrix of each product is transposed, whereas in Eqs.(\[eq-w-ortho2\]), the first is. $Q$ is clearly a real matrix, and Eqs.(\[eq-w-ortho1\]) imply that its columns are orthonormal. Hence $Q$ is orthogonal. Eq.(\[eq-w-ortho-sym1\]) (ditto, Eq.(\[eq-w-ortho-sym2\])) implies that $ X_I X_R^T$ (ditto, $ X_I^T X_R$) is symmetric.
The next theorem, due to Eckart and Young, gives necessary and sufficient conditions for finding a pair of unitary matrices $U, V$ that simultaneously accomplish the SVD (singular value decomposition) of two same-sized but otherwise arbitrary matrices $A$ and $B$. The proof reveals that the problem of finding simultaneous SVD’s can be reduced to the simpler problem of finding simultaneous diagonalizations of two commuting Hermitian matrices. The problem of simultaneously diagonalizing two commuting Hermitian operators (a.k.a. observables) is well known to physicists from their study of Quantum Mechanics.
\[claim-ey\] (Eckart-Young) Suppose $A, B$ are two complex (ditto, real) rectangular matrices of the same size. There exist two unitary (ditto, orthogonal) matrices $U,V$ such that $D_1 = U^\dagger A V$ and $D_2 = U^\dagger B V$ are both real diagonal matrices if and only if $AB^\dagger$ and $A^\dagger B$ are Hermitian (ditto, real symmetric) matrices.
($\Rightarrow$) $AB^\dagger = U D_1 D_2 U^\dagger$ and $A^\dagger B = V D_1 D_2 V^\dagger$ so they are Hermitian.
($\Leftarrow$)Let
A’ = U\_A\^A V\_A = (
[cc]{} D & 0\_2\
0\_3 & 0\_4
) be a SVD of $A$. Thus, $U_A, V_A$ are unitary matrices, $0_2, 0_3, 0_4$ are zero matrices, and $D$ is a square diagonal matrix whose diagonal elements are strictly positive. Let
B’ = U\_A\^B V\_A = (
[cc]{} G & K\
L & H
) , where $D$ and $G$ are square matrices of the same dimension, $rank(A)$. Note that
A’B\^[’]{}=B’ A\^[’]{}(
[cc]{} D G\^& D L\^\
0 & 0
)= (
[cc]{} G D & 0\
L D & 0
) , and
A\^[’]{}B’=B\^[’]{}A’ (
[cc]{} D G & D K\
0 & 0
) = (
[cc]{} G\^D & 0\
K\^D & 0
) . Therefore,
L=K=0 , and
D G\^= GD , D G = G\^D . \[eq-commute\] When written in index notation, Eqs.(\[eq-commute\]) become
d\_i g\^\*\_[ji]{} = g\_[ij]{}d\_j, d\_i g\_[ij]{}= g\^\*\_[ji]{}d\_j , \[eq-index1\] where the indices range over $\{1,2,\ldots,rank(A)\}$. Eqs.(\[eq-index1\]) imply
(d\_i + d\_j)(g\^\*\_[ji]{} - g\_[ij]{})=0 . Since $d_i>0$, we conclude that $G$ is a Hermitian matrix. $D$ is Hermitian too, and, by virtue of Eq.(\[eq-commute\]), $D$ and $G$ commute. Thus, these two commuting observables can be diagonalized simultaneously. Let $P$ be a unitary matrix that accomplishes this diagonalization:
D = P\^D P, D\_G = P\^G P . Let
D\_H = U\_H\^H V\_H be a SVD of $H$. $D_H$ is a diagonal matrix with non-negative diagonal entries and $U_H, V_H$ are unitary matrices. Now let
U\^= (
[cc]{} P\^& 0\
0 & U\^\_H
) U\^\_A , V = V\_A (
[cc]{} P & 0\
0 & V\_H
) . \[eq-u-v-defs\] The matrices $U$ and $V$ defined by Eq.(\[eq-u-v-defs\]) can be taken to be the matrices $U$ and $V$ defined in the statement of the theorem.
\[claim-odo\] If $X$ is a unitary matrix, then there exist orthogonal matrices $Q_L$ and $Q_R$ and a diagonal unitary matrix $e^{i\Theta}$ such that $X= Q_L e^{i\Theta} Q_R^T$.
Let $X_R$ and $X_I$ be defined as in Lemma \[claim-re-im-x\]. According to Lemma \[claim-re-im-x\], $X_I X_R^T$ and $X_I^T X_R$ are real symmetric matrices, so we can apply Theorem \[claim-ey\] with $A=X_R$ and $B=X_I$. Thus, there exist orthogonal matrices $Q_R$ and $Q_L$ such that
D\_R = Q\_L\^T X\_R Q\_R, D\_I = Q\_L\^T X\_I Q\_R , \[eq-dr-di\] where $D_R, D_I$ are real diagonal matrices. Since $X$ is unitary, $D_R + i D_I$ is too. Thus, we can define a diagonal unitary matrix $e^{i\Theta}$ by
e\^[i]{} = D\_R + i D\_I . \[eq-eit\] Combining Eqs.(\[eq-dr-di\]) and (\[eq-eit\]) finally yields
e\^[i]{} = Q\_L\^T X Q\_R .
Let $t=(\theta_1, \theta_2,
\theta_3, \theta_4)\in \RR^4$ and $\Theta = diag(t)$ so that
e\^[i]{} = diag( e\^[i\_1]{}, e\^[i\_2]{}, e\^[i\_3]{}, e\^[i\_4]{}) . Let $(k_0, \vec{k})\in \RR^4$. According to Eq.(\[tab-mh-m\]),
\^ =e\^[i(k\_0 + k\_1 - k\_2 + k\_3 )]{} . If we set
e\^[i ]{} =\^ , \[eq-eit-eik\] then each point $(\theta_1, \theta_2, \theta_3, \theta_4)\in \RR^4$ is mapped in a 1-1 onto fashion into each point $(k_0, \vec{k})\in \RR^4$. Using the explicit forms of $\sigzu, \siguz, \sigzz$, one finds that
(
[c]{} \_0\
\_1\
\_2\
\_3
) = (
[c]{} k\_0\
k\_1\
k\_2\
k\_3
) , = (
[cccc]{} &&&\
&&&\
&&&\
&&&
) . It is easy to check that
\^[-1]{} = .
\[claim-kak1\] (KAK1) If $X\in U(4)$, then $X = (A_1\otimes A_0)\eikks (B_1 \otimes B_0)$, where $A_1, A_0, B_1, B_0\in SU(2)$ and $(k_0, \vec{k})\in \RR^4$.
Let X’ = \^X . $X'$ is a unitary matrix, so, according to Collorary \[claim-odo\], we can find orthogonal matrices $Q_L, Q_R$ and a diagonal unitary matrix $e^{i \Theta}$ such that
X’ = Q\_L e\^[i ]{} Q\_R\^T . According to Theorem \[claim-isom4\], we can find $A_1, A_0, B_1, B_0\in SU(2)$ such that
Q\_L \^= A\_1A\_0 , and
Q\_R \^= B\_1B\_0 . As in Eq.(\[eq-eit-eik\]), set
e\^[i ]{} \^=. It follows that
X = (A\_1A\_0)(B\_1 B\_0) .
Canonical Class Vector
======================
In this section we discuss how KAK1 partitions $SU(4)$ into disjoint classes characterized by a 3d real vector $\vec{k}$.
We will say that $U,V\in SU(4)$ [**are equivalent up to local operations**]{} and write $U\sim V$ if $U =(R_1 \otimes R_0)
V (S_1 \otimes S_0)$ where $R_1, R_0, S_1, S_0 \in U(2)$. It is easy to prove that $\sim$ is an equivalence relation. Hence, it partitions $SU(4)$ into disjoint subsets (i.e., equivalence classes). If $X\in SU(4)$ and $\vec{k}\in \RR^3$ are related as in Collorary \[claim-kak1\], then $X\sim \eiks$. Henceforth, we will call this $\vec{k}$ a [**class vector**]{} of $X$. We will say that $\vec{k'}$ and $\vec{k}$ are [**equivalent class vectors**]{} and write $\vec{k'}\sim \vec{k}$ if $\eikps \sim
\eiks$.
Note that the following 3 operations map a class vector into another class vector of the same class; i.e., the operations are class-preserving.
1. (Shift) Suppose we shift $\vec{k}$ by plus or minus $\frac{\pi}{2}$ along any one of its 3 components. For example, a positive, $\frac{\pi}{2}$, X-shift would map (k\_x, k\_y, k\_z)(k\_x + , k\_y, k\_z) . This operation preserves $\vec{k}$’s class because
e\^[i\[(k\_x + )+ k\_y + k\_z \]]{} = e\^[i]{}\
= i.
2. (Reverse) Suppose we reverse the sign of any two components of $\vec{k}$. For example, an XY-reversal would map
(k\_x, k\_y, k\_z)(-k\_x, -k\_y, k\_z) . This operation preserves $\vec{k}$’s class because
= e\^[i(-k\_x, -k\_y, k\_z)]{} .
3. (Swap) Suppose we swap any two components of $\vec{k}$. For example, an XY-swap would map
(k\_x, k\_y, k\_z)(k\_y, k\_x, k\_z) . This operation preserves $\vec{k}$’s class because
e\^[-i(+ )]{} e\^[i(+ )]{} = e\^[i(k\_y, k\_x, k\_z)]{} .
Define ${\cal K}$ as the set of points $\vec{k}\in \RR^3$ such that
1. $\frac{\pi}{2}
>k_x \geq k_y \geq k_z \geq 0$
2. $k_x + k_y \leq \frac{\pi}{2}$
3. If $k_z =0$, then $k_x \leq \frac{\pi}{4}$.
${\cal K}$ is contained within the tetrahedral region $OA_1A_2A_3$ of Fig.\[fig-canon-vec\].
The 3 class-preserving operations given above generate a group $\ul{W}$. Given any class vector $\vec{k}\in \RR^3$, it is always possible to find an operation $G\in \ul{W}$ such that $G(\vec{k})\in{\cal K}$. Indeed, here is an algorithm, (implemented in the accompanying Octave software) that finds $G(\vec{k})\in{\cal K}$ for any $\vec{k}\in \RR^3$:
1. Make $k_x \in [0 \frac{\pi}{2})$ by shifting $k_x$ repeatedly by $\frac{\pi}{2}$. In the same way, shift $k_y$ and $k_z$ into $[0 \frac{\pi}{2})$.
2. Make $k_x\geq k_y \geq k_z$ by swapping the components of $\vec{k}$.
3. Perform this step iff at this point $k_x + k_y > \frac{\pi}{2}$. Transform $\vec{k}$ into $(\frac{\pi}{2}-k_y,
\frac{\pi}{2} - k_x, k_z)$ (This can be achieved by applying an XY-swap, XY-reverse, X-shift and Y-shift, in that order). At this point, $k_x\geq k_y$, but $k_z$ may be larger than $k_y$ or $k_x$, so finish this step by swapping coordinates until $k_x\geq k_y \geq k_z$ again.
4. Perform this step iff at this point $k_z=0$ and $k_x> \frac{\pi}{4}$. Transform $\vec{k}=(k_x, k_y, 0)$ into $(\frac{\pi}{2} - k_x, k_y , 0)$ (This can be achieved by applying an XZ-reverse and an X-shift, in that order).
We can find a subset $S$ of $\RR^3$ such that every equivalence class of $SU(4)$ is represented by one and only one point $\vec{k}$ of $S$. In fact, ${\cal K}$ defined above is one such $S$. We will refer to the elements of ${\cal K}$ as [**canonical class vectors**]{}.
We end this section by finding the canonical class vectors of some simple 2-qubit operations.
1. (CNOT): $CNOT(1\rarrow 0)$ is defined by
CNOT(10)= (0)\^[n(1)]{}= (
[cc]{} 1 & 0\
0 &
) . Since $n=\frac{1}{2}(1-\sigz)$, $n_X=\frac{1}{2}(1-\sigx)$, and $\sigx=(-1)^{n_X}=e^{i\pi n_X}$,
$$\begin{aligned}
\sigx(0)^{n(1)} &=& (-1)^{n_X(0) n(1)}\\
&=&e^{i \frac{\pi}{4}(1-\sigux)(1-\sigzu)}\\
&=&
e^{i \frac{\pi}{4}(1-\sigzu -\sigux)}
e^{i \frac{\pi}{4}\sigzx}\\
&=&
e^{i \frac{\pi}{4}(1-\sigzu -\sigux)}
e^{i\frac{\pi}{4}\sigyu}
e^{i \frac{\pi}{4}\sigxx}
e^{-i\frac{\pi}{4}\sigyu}\\
&\sim & e^{i\frac{\pi}{4}\sigxx}
\;.\end{aligned}$$
Therefore, the canonical class vector of CNOT is $(\frac{\pi}{4},0,0)$, which corresponds to the point $B$ in Fig.\[fig-canon-vec\].
2. ($\sqrt{CNOT}$) From the math just performed for CNOT, it is clear that
= e\^[i(1-)(1-)]{} \~e\^[i]{} . Therefore, the canonical class vector of $\sqrt{CNOT}$ is $(\frac{\pi}{8},0,0)$, which corresponds to the midpoint of the segment $OB$ in Fig.\[fig-canon-vec\].
3. (Exchanger, a.k.a. Swapper) As usual, the Exchanger is defined by
E= (
[cccc]{} 1&0&0&0\
0&0&1&0\
0&1&0&0\
0&0&0&1
) . (Note that $\det(E)=-1$). Using Eqs.(\[eq-com-sigxx-sigyy\]), it is easy to show that
E =e\^[-i ]{} e\^[i (++)]{} . Therefore, the canonical class vector of $e^{i \frac{\pi}{4}}E$ is $(\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4})$, which corresponds to the apex $A_3$ of the tetrahedron in Fig.\[fig-canon-vec\].
Software
========
A collection of Octave/Matlab m-files that implement the algorithms in this paper, can be found at ArXiv (under the “source" for this paper), and at my website (www.ar-tiste.com). These m-files have only been tested on Octave, but they should run on Matlab with few or no modifications. A file called “m-fun-index.html" that accompanies the m-files lists each function and its purpose.
[99]{}
S. Helgason, [*Differential Geometry, Lie Groups, and Symmetric Spaces*]{} (Am. Math. Soc., 2001 edition, corrected reprint of 1978 original edition)
Navin Khaneja, Steffen Glaser, “Cartan Decomposition of $SU(2^n)$, Constructive Controllability of Spin systems and Universal Quantum Computing", quant-ph/0010100 . Also “Cartan decomposition of $SU(2n)$ and control of spin systems", Chemical Physics, (2001), Pages 11-23.
G.H. Golub and C.F. Van Loan, [*Matrix Computations, Third Edition*]{} (John Hopkins Univ. Press, 1996).
C. C. Paige and M. Wei, “History and Generality of the CS Decomposition,” Linear Algebra and Appl. 208/209(1994)303-326.
R.R. Tucci, “A Rudimentary Quantum Compiler (2cnd Ed.)", quant-ph/9902062
SS Bullock, “Note on the Khaneja Glaser Decomposition", quant-ph/0403141
G. Vidal, C.M. Dawson, “A Universal Quantum Circuit for Two-qubit Transformations with 3 CNOT Gates", quant-ph/0307177
Farrokh Vatan, Colin Williams, “Optimal Quantum Circuits for General Two-Qubit Gates", quant-ph/0308006
Jun Zhang, Jiri Vala, K. Birgitta Whaley, Shankar Sastry, “A geometric theory of non-local two-qubit operations", quant-ph/0209120,Gsponer
C. Eckart, G. Young, Bull. Am. Math. Soc. (1939)118.
André Gsponer, “Explicit closed-form parametrization of $SU(3)$ and $SU(4)$ in terms of complex quaternions and elementary functions", math-ph/0211056
R.R.Tucci, “QC Paulinesia", quant-ph/0407215
|
---
abstract: 'In this talk, we discuss several topics related to the Abelian-projected $SU(3)$-QCD. First of them is the Aharonov-Bohm effect emerging during the extension of this theory by the introduction of the $\Theta$-term. Another topic is devoted to various consequences of screening of the dual vector bosons by electric vortex loops. In particular, it is demonstrated that this effect modifies significantly the interaction of quarks. Next, the influence of screening to electric and magnetic field correlators in the four-dimensional Abelian-projected $SU(3)$-QCD is studied. Finally, the bilocal correlator of electric field strengths in the three-dimensional gas of $SU(3)$ Abelian-projected monopoles is discussed.'
address: |
INFN-Sezione di Pisa, Universitá degli studi di Pisa,\
Dipartimento di Fisica, Via Buonarroti, 2 - Ed. B - I-56127 Pisa, Italy
author:
- 'D. Antonov'
title: 'Topological and confining properties of Abelian-projected SU(3)-QCD'
---
INTRODUCTION
============
In the present talk, we shall mostly discuss various nonperturbative properties of the effective low-energy theory of the $SU(3)$-QCD [@1], which models confinement of quarks as the dual Meissner effect [@2]. The partition function of this $[U(1)]^2$ magnetically gauge-invariant theory reads
$${\cal Z}=\int \left|\Phi_a\right|{\cal D}\left|\Phi_a\right|
{\cal D}\theta_a{\cal D}{\bf B}_\mu\delta\left(\sum\limits_{a=1}^{3}
\theta_a\right)\times$$
$$\times\exp\left\{-\int d^4x\left[\frac14\left({\bf F}_{\mu\nu}+
{\bf F}_{\mu\nu}^{(c)}\right)^2+\right.\right.$$
$$+\sum\limits_{a=1}^{3}\left(\frac12\left|\left(\partial_\mu-2ig_m{\bf e}_a
{\bf
B}_\mu\right)\Phi_a\right|^2+\right.$$
$$\label{1}
\left.\left.\left.+\lambda\left(\left|\Phi_a\right|^2-
\eta^2\right)^2\right)\right]\right\}.$$
Here, $g_m$ is the magnetic coupling constant, related to the QCD coupling constant $g$ as $g_m=\frac{4\pi}{g}$, and ${\bf e}_a$’s are the root vectors of $SU(3)$, whose explicit form is ${\bf e}_1=(1,0)$, ${\bf e}_2=\left(-\frac12,-\frac{\sqrt{3}}{2}\right)$, ${\bf e}_3=\left(-\frac12,\frac{\sqrt{3}}{2}\right)$. Next, in Eq. (\[1\]), ${\bf F}_{\mu\nu}$ stands for the field strength tensor of the field ${\bf B}_\mu$ dual to the field ${\bf A}_\mu\equiv\left(A_\mu^3,A_\mu^8\right)$, and $\Phi_a=\left|\Phi_a\right|{\rm e}^{i\theta_a}$ are the dual Higgs fields describing the condensation of Cooper pairs of Abelian-projected monopoles.
Note that the phases $\theta_a$’s are related to each other by the constraint $\sum\limits_{a=1}^{3}\theta_a=0$, imposed by the respective $\delta$-function on the right-hand side of Eq. (\[1\]). This constraint reflects the dependence of the monopoles of three kinds of each other. Such a dependence is inspired by the fact that the monopole magnetic charges are distributed over the lattice defined by the root vectors, whose sum vanishes.
Finally, in Eq. (\[1\]) we have introduced the field strength tensor ${\bf F}_{\mu\nu}^{(c)}$ of an external quark of the colour $c=R,B,G$ (red, blue, green, respectively), which obeys the equation $\partial_\mu\tilde{\bf F}_{\mu\nu}^{(c)}=g
{\bf Q}^{(c)}j_\nu$. Here, $\tilde{\cal O}_{\mu\nu}\equiv\frac12
\varepsilon_{\mu\nu\lambda\rho}{\cal O}_{\lambda\rho}$, $j_\nu(x)
\equiv\oint\limits_{C}^{}dx_\nu(\tau)\delta(x-x(\tau))$, and ${\bf Q}^{(c)}$’s are the charges of a quark of the colour $c$ with respect to the Cartan subgroup of $SU(3)$: ${\bf Q}^{(R)}=\left(\frac12,\frac{1}{2\sqrt{3}}\right)$, ${\bf Q}^{(B)}=\left(-\frac12,\frac{1}{2\sqrt{3}}\right)$, ${\bf Q}^{(C)}=\left(0,-\frac{1}{\sqrt{3}}\right)$. The present lattice data [@3] indicate that in the regime of the model (\[1\]) corresponding to the real QCD, the coupling constant $\lambda$ is much larger than one, namely $\lambda\simeq 65$. This makes it reasonable to consider this model in the London limit, $\lambda\to\infty$. In this limit, the model (\[1\]) allows for an exact string representation, and the resulting string effective action reads [@4], [@komarov]
$$S_c=\pi^2\int
d^4x\int d^4yD_m^{(4)}(x-y)\times$$
$$\label{2}
\times\left[\eta^2\bar\Sigma_{\mu\nu}^a(x)
\bar\Sigma_{\mu\nu}^a(y)+\frac{8}{3g_m^2}j_\mu(x)
j_\mu(y)\right].$$
Here, $m=g_H\eta$ is the mass of the dual vector bosons with $g_H\equiv\sqrt{6}g_m$ standing for their magnetic charge, which they acquire due to the Higgs mechanism. Next, $D_m^{(4)}(x)=\frac{m}{4\pi^2|x|}K_1(m|x|)$ is the propagator of these bosons, where from now on $K_\nu$’s stand for the modified Bessel functions. In Eq. (\[2\]), we have also introduced the notation $\bar\Sigma_{\mu\nu}^a=\Sigma_{\mu\nu}^a-2s_a^{(c)}
\Sigma_{\mu\nu}$. In this expression, $s_a^{(c)}$’s stand for certain numbers equal to $0$ and $\pm 1$, which obey the relation ${\bf Q}^{(c)}=\frac13s_a^{(c)}{\bf e}_a$, and $\Sigma_{\mu\nu}^a(x)\equiv\int\limits_{\Sigma_a}^{}d\sigma_{\mu\nu}
(x_a(\xi))\delta(x-x_a(\xi))$ is the vorticity tensor current defined on the closed dual string world sheet $\Sigma_a$ with $\xi\equiv\left(\xi^1,\xi^2\right)$. Note that owing to the one-to-one correspondence existing between $\Sigma_{\mu\nu}^a$’s and the multivalued parts of $\theta_a$’s, the three vorticity tensor currents are subject to the constraint $\sum\limits_{a=1}^{3}\Sigma_{\mu\nu}^a=0$, which stems from the analogous constraint imposed on $\theta_a$’s. Finally, in the definition of $\bar\Sigma_{\mu\nu}^a$, we have denoted by $\Sigma_{\mu\nu}$ the vorticity tensor current defined on an arbitrary surface $\Sigma$ bounded by the contour $C$, which is the world sheet of the open dual string, ending up at a quark and an antiquark.
The $\Sigma_{\mu\nu}\times\Sigma_{\mu\nu}$-interaction in the action (\[2\]) can be shown [@5] to describe confinement of quarks, whereas the $j_\mu\times j_\mu$-interaction clearly describes their Yukawa interaction at small distances. Notice also that in what follows we shall be interested in the string effective actions, rather than the measure of integration over world-sheet coordinates $x_a(\xi)$’s. The Jacobian emerging during the change of integration variables $\theta_a\to x_a(\xi)$, which should be accounted for in this measure, has been evaluated in Ref. [@prd].
INCLUDING THE $\Theta$-TERM
===========================
Let us now add to the Lagrangian of the model (\[1\]) the following $\Theta$-term:
$$\Delta{\cal L}=-\frac{i\Theta g_m^2}{4\pi^2}
\left({\bf F}_{\mu\nu}+{\bf F}_{\mu\nu}^{(c)}\right)\left(\tilde
{\bf F}_{\mu\nu}+\tilde {\bf F}_{\mu\nu}^{(c)}\right).$$ In the London limit, the string representation of such an extended partition function then reads [@6]
$${\cal Z}_{\Theta}^c=\exp\left[-\frac23\left(\frac{(2\pi)^2}{g_m^2}+
\frac{(\Theta g_m)^2}{\pi^2}\right)\times\right.$$
$$\times\oint\limits_{C}^{}dx_\mu
\oint\limits_{C}^{}dy_\mu D_m^{(4)}(x-y)-8(\pi\eta)^2\times$$
$$\left.\times\int d^4x\int d^4y\Sigma_{\mu\nu}(x)D_m^{(4)}(x-y)
\Sigma_{\mu\nu}(y)\right]\times$$
$$\times\left<\exp\left\{2s_a^{(c)}\left[\int d^4x\int d^4y\Biggl(
\frac{i\Theta}{3}\tilde{\bar\Sigma}_{\mu\nu}^a(x)j_\nu(y)\times
\right.\right.\right.$$
$$\times\partial_\mu D_m^{(4)}(x-y)+2(\pi\eta)^2\Sigma_{\mu\nu}^a(x)
D_m^{(4)}(x-y)\times$$
$$\label{t1}
\left.\left.\left.
\times\Sigma_{\mu\nu}(y)\Biggr)-\frac{i\Theta}{3}
\hat L(\Sigma_a, C)\right]\right\}\right>_{\Sigma_a}.$$
Here,
$$\hat L\left(\Sigma_a, C\right)\equiv$$
$$\equiv\int d^4x\int d^4y
\tilde\Sigma_{\mu\nu}^a(x)j_\nu(y)\partial_\mu D_0^{(4)}(x-y)$$ is the 4D Gauss linking number of the contour $C$ with the closed world sheet $\Sigma_a$, where $D_0^{(4)}(x)=\frac{1}{4\pi^2x^2}$. The average $\left<\ldots\right>_{\Sigma_a}$ is formally defined with respect to $S_c[\Sigma=0]$, but its exact meaning will be discussed below.
The first argument in the first exponential factor on the right-hand side of Eq. (\[t1\]) shows that due to the $\Theta$-term quarks acquire a nonvanishing magnetic charge [@wit], i.e. become dyons. We also see that there appear short- and long-ranged interactions of such dyons with closed dual strings, as well as the short-ranged dyon–open-string interaction. The above-mentioned long-ranged interaction, described by the linking number, can be viewed as a scattering of dyons by the closed dual strings, which thus play the rôle of solenoids carrying electric flux. According to Eq. (\[t1\]), such a scattering, which is nothing else but the four-dimensional analogue of the Aharonov-Bohm effect [@aha], takes place at $\Theta\ne 3\pi\times{\,}({\rm integer})$. Note that in the $SU(2)$-case, the respective critical values of $\Theta$ have been found [@emil] to be equal to $2\pi\times{\,}({\rm integer})$.
Let us now carry out the average $\left<\ldots\right>_{\Sigma_a}$, taking into account that at zero temperature closed dual strings with opposite winding numbers are known to form virtual bound states, called vortex loops [@popov]. The summation over the grand canonical ensemble of these objects has been performed in Ref. [@loops] and yields an effective sine-Gordon theory of two antisymmetric spin-1 tensor fields. In the dilute gas approximation, which is relevant to the reality since vortex loops are only virtual (and therefore small-sized) objects, such a theory enables one to calculate correlators of loops. Averaging the $\Sigma_a$-dependent exponential factor on the right-hand side of Eq. (\[t1\]) in the sense of this effective theory by making use of the cumulant expansion and accounting in this average only for the contribution of the dominant, bilocal, irreducible correlator of vortex loops, we get
$${\cal Z}_{\Theta}^c=\exp\Biggl\{-8\int d^4x\int d^4y\Biggl[
(\pi\eta)^2\Sigma_{\mu\nu}(x)\times$$
$$\times D_{M_1}^{(4)}(x-y)\Sigma_{\mu\nu}(y)+\frac{i\Theta}{3}
\tilde\Sigma_{\mu\nu}(x)j_\nu(y)\times$$
$$\times\partial_\mu
D_m^{(4)}(x-y)\Biggr]-
\frac23\left(\frac{(2\pi)^2}{g_m^2}+\frac{(\Theta g_m)^2}{\pi^2}\right)
\times$$
$$\times\frac{1}{g_H^2+g_D^2}
\oint\limits_{C}^{}dx_\mu\oint\limits_{C}^{}dy_\mu\times$$
$$\label{t2}
\times\left(g_H^2
D_{M_1}^{(4)}(x-y)+g_D^2D_0^{(4)}(x-y)\right)\Biggr\}.$$
Here, $g_D=\frac{2\pi\sqrt{\zeta}}{\Lambda^2}$ describes the contribution to the magnetic charge of the dual vector bosons, stemming from the Debye screening of those in the gas of electric vortex loops. In this formula, $\Lambda=\sqrt{\frac{L}{a^3}}$ is the ultraviolet momentum cutoff with $L$ standing for the typical distance between the neighbours in the gas of vortex loops and $a$ denoting the typical size of the loop, $a\ll L$. Next, $\zeta\propto {\rm e}^{-S_0}$ is the Boltzmann factor of a single vortex loop with the action $S_0$ equal to the string tension times the characteristic area of the loop.
Due to the Debye screening, the mass $m$ of the dual vector bosons increases. Since owing to the constraint $\sum\limits_{a=1}^{3}\Sigma_{\mu\nu}^a=0$ there exists the second independent type of vortex loops, there respectively appears also the second value of the Debye charge, equal to $g_D\sqrt{3}$. The two full masses then read $M_1=\eta\sqrt{g_H^2+g_D^2}$ and $M_2=\eta\sqrt{g_H^2+3g_D^2}$.
As it follows from Eq. (\[t2\]), the above-mentioned screening changes drastically the interaction of quarks (dyons). Namely, besides the modification of the massive propagator due to the additional contribution to the mass of the dual vector bosons, there also appears a novel massless interaction. It is also worth noting that, as it should be, when the effect of screening is disregarded, i.e. $g_D\ll g_H$, one recovers the classical result, which is nothing else, but the $\Sigma_a$-independent part of Eq. (\[t1\]).
APPLICATIONS TO THE STOCHASTIC VACUUM MODEL
===========================================
Let us now discuss in more details the confining properties of the model (\[1\]) in the London limit. Namely, let us consider the bilocal correlator of electric field strengths, ${\bf f}_{\mu\nu}=
\partial_\mu{\bf A}_\nu-\partial_\nu{\bf A}_\mu$, which plays the major rôle in the so-called stochastic vacuum model [@svm] (see Ref. [@rev1] for reviews). Within this model, such a correlator is parametrized by the two coefficient functions as follows:
$$\left<f_{\mu\nu}^i(x)f_{\lambda\rho}^j(0)\right>_{{\bf A}_\mu,
{\bf j}_\mu^{\rm m}}=\delta^{ij}\left\{\left(\delta_{\mu\lambda}
\delta_{\nu\rho}-\delta_{\mu\rho}\delta_{\nu\lambda}\right)\times
\right.$$
$$\times D\left(x^2
\right)+\frac12\left[\partial_\mu\left(x_\lambda\delta_{\nu\rho}-
x_\rho\delta_{\nu\lambda}\right)+\right.$$
$$\label{correl}
\left.\left.+\partial_\nu\left(
x_\rho\delta_{\mu\lambda}-x_\lambda\delta_{\mu\rho}\right)\right]
D_1\left(x^2\right)\right\}.$$
In this formula, $\left<\ldots\right>_{{\bf A}_\mu}$ stands for the average over free diagonal gluons, and $\left<\ldots\right>_{{\bf j}_\mu^{\rm m}}$ denotes a certain average over monopoles, which provides the condensation of their Cooper pairs. It is the coupling of the ${\bf B}_\mu$-field, dual to the field ${\bf A}_\mu$, to ${\bf j}_\mu^{\rm m}$’s, which makes both functions $D$ and $D_1$ nontrivial and finally adequate to those of the real QCD. Namely, these functions turn out to have the following form [@JH]:
$$D=\frac{m^2M_1}{4\pi^2}\frac{K_1(M_1|x|)}{|x|},$$
$$D_1=\frac{g_D^2}{\pi^2\left(g_H^2+g_D^2\right)|x|^4}+
\frac{m^2}{2\pi^2M_1x^2}\times$$
$$\times\left[\frac{K_1(M_1|x|)}{|x|}+\frac{M_1}{2}
\left(K_0(M_1|x|)+K_2(M_1|x|)\right)\right].$$ In the infrared limit, $|x|\gg M_1^{-1}$, the respective asymptotic behaviours read
$$\label{as1}
D\to\frac{(mM_1)^2}{4\sqrt{2}\pi^{\frac32}}
\frac{{\rm e}^{-M_1|x|}}{(M_1|x|)^{\frac32}},$$
$$\label{as2}
D_1\to\frac{g_D^2}{\pi^2\left(g_H^2+g_D^2\right)|x|^4}+
\frac{(mM_1)^2}{2\sqrt{2}\pi^{\frac32}}
\frac{{\rm e}^{-M_1|x|}}{(M_1|x|)^{\frac52}}.$$
One can see that according to the above-presented asymptotics, the inverse correlation length of the vacuum in the model under study is equal to the smallest of the two full masses, $M_1$. Besides that, owing to the screening, the function $D_1$ contains the novel nonperturbative $\frac{1}{|x|^4}$-term, which might be important for modelling the Lüscher term [@lu] in the quark-antiquark potential [@dosch]. Subtracting from the function $D_1$ this contribution, which has the same functional form as the pure perturbative contribution in the real QCD, we see that the remained part of the asymptotics (\[as2\]) together with the asymptotics (\[as1\]) are in a good agreement with the QCD lattice measurements of the functions $D$ and $D_1$ [@lattice] (see Ref. [@develop] for recent developments and Ref. [@rev2] for reviews). Namely, due to the preexponential behaviour, $D\gg D_1$ in the infrared limit, $|x|\gg M_1^{-1}$, whereas in the ultraviolet limit, $D\ll D_1$.
It is also worth noting that when the screening is disregarded, i.e. $g_D\ll g_H$, the inverse correlation length of the vacuum goes over to $m$, and the classical expressions [@4] (see Ref. [@class] for the $SU(2)$-case) for the functions $D$ and $D_1$ recover. In particular, the nonperturbative screening-motivated $\frac{1}{|x|^4}$-contribution to the function $D_1$ vanishes in this limit.
Besides the above-discussed modifications of the classical expressions for the correlators of electric field strengths inspired by screening, this effect changes also the classical expression for the propagator of the dual vector bosons. Indeed, the latter one reads
$$\label{BB}
\left<B_\mu^a(x)B_\nu^b(0)\right>=\delta^{ab}\delta_{\mu\nu}
D_m^{(4)}(x)$$
with $B_\mu^a\equiv {\bf e}_a{\bf B}_\mu$, whereas with the account for screening it changes to [@JH]
$$\delta_{\mu\nu}\left\{\frac13D_m^{(4)}(x)+\frac{g_D^2}{2}\left[
2\frac{g_H^2+2g_D^2}{\left(g_H^2+3g_D^2\right)\left(g_H^2+g_D^2\right)}
\times\right.\right.$$
$$\times
D_0^{(4)}(x)+\frac{g_H^2}{g_D^2}\left(\frac{1}{3\left(g_H^2+3g_D^2
\right)}D_{M_2}^{(4)}(x)+\right.$$
$$\label{diagon}
\left.\left.\left.+\frac{1}{g_H^2+g_D^2}
D_{M_1}^{(4)}(x)\right)\right]\right\}$$
for $a=b$ and
$$\delta_{\mu\nu}\left\{\frac13D_m^{(4)}(x)+\frac{g_D^2}{2}\left[
-\frac{2g_D^2}{\left(g_H^2+3g_D^2\right)\left(g_H^2+g_D^2\right)}
\times\right.\right.$$
$$\times D_0^{(4)}(x)+\frac{g_H^2}{g_D^2}\left(
\frac{1}{3\left(g_H^2+3g_D^2\right)}D_{M_2}^{(4)}(x)-\right.$$
$$\label{off}
\left.\left.\left.-\frac{1}{g_H^2+g_D^2}D_{M_1}^{(4)}(x)
\right)\right]\right\}$$
for $a\ne b$. We see that the screening makes the propagator of the dual vector bosons nonvanishing even for $a\ne b$. As it should be, in the limit when the screening is disregarded, $g_D\ll g_H$, these off-diagonal components of the propagator given by Eq. (\[off\]) vanish, whereas the diagonal ones given by Eq. (\[diagon\]) go over to the classical expression, so that Eq. (\[BB\]) recovers.
Contrary to the four-dimensional case, in three dimensions the present lattice data allow one to assume that Abelian-projected monopoles form a gas. Such a gas of $SU(3)$-monopoles has for the first time been considered in Ref. [@su3], and its partition function reads
$${\cal Z}=1+\sum\limits_{N=1}^{\infty}
\frac{\zeta^N}{N!}\left(\prod\limits_{n=1}^{N}
\int d^3z_n\sum\limits_{a_n=\pm 1,\pm 2, \pm 3}^{}
\right)\times$$
$$\label{mongas}
\times\exp\left[-\frac{g_m^2}{4\pi}\sum\limits_{n<k}^{}
\frac{{\bf e}_{a_n}{\bf e}_{a_k}}{|{\bf z}_n-{\bf z}_k|}
\right].$$
In this formula, $\zeta\propto\exp\left(-\frac{{\rm const}}{g^2}
\right)$ stands for the Boltzmann factor of a single monopole, and ${\bf e}_{-a}=-{\bf e}_a$. The string representation of the Wilson loop in this gas, constructed in Ref. [@epl], turned out to be alternative to the one of the $SU(2)$-case, found in Ref. [@su2]. (See Ref. [@nikita] for the discussion of Polyakov loops and their correlators in the $SU(2)$ monopole gas.) By virtue of this representation in the approximation when the monopole gas is so dilute that its density is much less than $\zeta$, one can deduce the respective expressions for the functions $D$ and $D_1$. In the model under study, those are defined by Eq. (\[correl\]) with the average $\left<\ldots
\right>_{{\bf j}_\mu^{\rm m}}$ replaced by the average with respect to the partition function (\[mongas\]). Together with the contribution of the free diagonal gluons to the function $D_1$, these two functions read
$$\label{d}
D=12\pi\zeta\frac{{\rm e}^{-m|{\bf x}|}}{|{\bf x}|},$$
$$\label{d1}
D_1=\frac{24\pi\zeta}{(m|{\bf x}|)^2}\left(m+\frac{1}{|{\bf x}|}
\right){\rm e}^{-m|{\bf x}|}.$$
Here, $m=g_m\sqrt{3\zeta}$ is the Debye mass of the two scalar bosons, dual to the diagonal gluons. Similarly to the above-considered four-dimensional case, we see that Eqs. (\[d\]) and (\[d1\]) well agree with the lattice calculations in the real QCD [@lattice], [@develop], [@rev2]. In particular, the inverse correlation length of the vacuum is now equal to $m$, and at the distances larger than this length, $D\gg D_1$ due to the preexponential behaviour.
CONCLUSIONS
===========
In the present talk, we have discussed various nonperturbative properties of Abelian-projected $SU(3)$-QCD in four and three dimensions. In the four-dimensional case, we have firstly considered the respective dual Abelian Higgs type model extended by the introduction of the $\Theta$-term. In this way, the string representation of such an extended model has been derived, which has in particular demonstrated how the $\Theta$-term leads to the appearance of the magnetic charge of external quarks, making out of them dyons. The critical values of $\Theta$, at which the Aharonov-Bohm scattering of dyons over the closed dual strings disappears, have been found. Next, the effect of the Debye screening of the dual vector bosons by virtual electric vortex loops, built out of the closed dual strings, has been taken into account.
Then, the confining properties of the four- and three-dimensional $SU(3)$ Abelian-projected theories within the stochastic vacuum model have been addressed. In particular, in the four-dimensional case the rôle of the above-mentioned screening has been discussed. The influence of this effect to the propagators of the dual vector bosons has also been considered. Finally, the bilocal correlator of electric field strengths in the dilute three-dimensional gas of $SU(3)$ Abelian-projected monopoles has been evaluated.
In conclusion, the performed investigations have shown that Abelian-projected theories are not only adequate to the description of confinement of quarks in QCD, but possess also a lot of interesting nonperturbative properties themselves.
ACKNOWLEDGMENTS
===============
The author is indebted to Prof. A. Di Giacomo for useful discussions and hospitality, and INFN for the financial support. He is also greatful to the organizers of the Euroconference “QCD 00” (Montpellier, France, 6-13th July 2000) for an opportunity to present these results in a very stimulating atmosphere.
DISCUSSION
==========
[**Dr. N. Brambilla (University of Heidelberg)**]{}: [*You have mentioned that the London limit is in the agreement with the lattice results, but I would like to point out that a lot of the most recent lattice data show that the QCD vacuum looks like a dual superconductor at the border of type-I and type-II. Could you please comment on this issue.*]{}
[**D. Antonov**]{}: [*My statement was based on the lattice data of Ref. [@3], which demonstrated that in the QCD-relevant regime of the effective dual theory considered, the coupling constant of the dual Higgs potential should be much larger than unity.*]{}
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|
---
abstract: |
The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo [@hunter-thoo] have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity.
The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation was shown in [@M-S-T:ONDE1; @M-S-T:ONDE2]. In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.
address: 'DICATAM, Sezione di Matematica, Università di Brescia,Via Valotti, 9, 25133 BRESCIA, Italy'
author:
- Alessandro Morando
- Paolo Secchi
- Paola Trebeschi
title: |
Data dependence of approximate current-vortex sheets\
near the onset of instability
---
Introduction {#sect1}
============
In the present paper we consider the following equation $$\label{equ1}
\varphi_{tt}-\mu\varphi_{xx}=\left(\frac12\mathbb H[\phi^2]_{xx}+\phi\varphi_{xx}\right)_{\!\!x}\,,\qquad\phi=\mathbb H[\varphi]\,,$$ where the unknown is the scalar function $\varphi=\varphi(t,x)$, where $t$ denotes the time, $x\in{{\mathbb R}}$ is the space variable and $\mathbb H$ denotes the Hilbert transform with respect to $x$, and $\mu$ is a constant. Hunter and Thoo [@hunter-thoo] have derived this asymptotic equation in order to study the dynamics of 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics, near the transition point between the linearized stability and instability.
Equation is an integro-differential evolution equation in one space dimension, with quadratic nonlinearity. This is a nonlocal equation of order two: in fact, it may also be written as $$\begin{array}{ll}\label{equ1bis}
\vphi_{tt}-\mu\vphi_{xx} = \left( [ {\mathbb H};\phi
]\partial_x \phi
_{x} + {\mathbb H}[\phi
^2_x]\right)_x \,,
\end{array}$$ where $[ {\mathbb H};\phi
]\partial_x$ is a pseudo-differential operator of order zero. This alternative form shows that is an equation of second order, due to a cancelation of the third order spatial derivatives appearing in .
Equation also admits the alternative spatial form $$\label{equ1ter}
\begin{split}
\varphi_{tt}-\left(\mu-2\phi_x\right)\varphi_{xx}+\mathcal Q\left[\varphi\right]=0\,,
\end{split}$$ where $$\label{termine_nl}
\mathcal Q\left[\varphi\right]:=-3\left[\mathbb H\,;\,\phi_x\right]\phi_{xx}-\left[\mathbb H\,;\,\phi\right]\phi_{xxx}\,.$$ The alternative form puts in evidence the difference $\mu-2\phi_x$ which has a meaningful role. In fact it can be shown that the linearized operator about a given basic state is elliptic and is locally linearly ill-posed in points where $$\mu-2\phi_x<0.$$ On the contrary, in points where $$\begin{array}{ll}\label{extstab}
\mu-2\phi
_{x}>0
\end{array}$$ the linearized operator is hyperbolic and is locally linearly well-posed, see [@hunter-thoo]. In this case we can think of as a nonlinear perturbation of the wave equation.
The local-in-time existence of smooth solutions to the Cauchy problem for , under the above stability condition , was shown in [@M-S-T:ONDE1; @M-S-T:ONDE2]. In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of in the classical sense of Hadamard, after existence and uniqueness.
As written in Kato’s paper [@kato75], this part may be the most difficult one, when dealing with hyperbolic problems. Our method is somehow inspired by Beir[ã]{}o da Veiga’s perturbation theory for the compressible Euler equations [@beirao92; @beirao93], and its application to the problem of convergence in strong norm of the incompressible limit, see [@beirao1994; @secchisingular]. Instead of directly estimating the difference between the given solution and the solutions to the problems with approximating initial data, the main idea is to use a triangularization with the more regular solution to a suitably chosen close enough problem.
Let us consider the initial value problem for the equation supplemented by the initial condition $$\label{id}
\varphi_{\vert\,t=0}=\varphi^{(0)}\,,\qquad \partial_t\varphi_{\vert\,t=0}=\varphi^{(1)}\,,$$ for sufficiently smooth initial data $\varphi^{(0)},\,\varphi^{(1)}$ satisfying the stability condition $$\label{stability_nl}
\mu-2\phi^{(0)}_{x}>0\,,\qquad\phi^{(0)}:=\mathbb H[\varphi^{(0)}]\,.$$
For the sake of convenience, in the paper the unknown $\varphi=\varphi(t,x)$ is a scalar function of the time $t\in\mathbb R^+$ and the space variable $x$, ranging on the one-dimensional torus $\mathbb T$ (that is $\varphi$ is periodic in $x$). For all notation we refer to the following Section \[prbt\].
In [@M-S-T:ONDE2] we prove the following existence theorem.
\[th\_esistenza\] Let $s\ge 3$ be a real number. Assume $\varphi^{(0)}\in H^{s}(\mathbb T)$, $\varphi^{(1)}\in H^{s-1}(\mathbb T)$, and let $\varphi^{(0)}$, $\varphi^{(1)}$ have zero spatial mean. Given $0<\delta<\mu$, there exist $0<R\le1$, and constants $C_1>0$, $C_2>0$ such that, if $$\label{ip_dati_iniziali}
\Vert\varphi^{(0)}_x\Vert^2_{H^2({{\mathbb T}})}+\Vert\varphi^{(1)}\Vert^2_{H^2({{\mathbb T}})}<R^2\,,$$ there exists a unique solution $\varphi\in C(I_0; H^{s}(\mathbb T))\cap C^1(I_0; H^{s-1}(\mathbb T))$ of the Cauchy problem , defined on the time interval $I_0=[0,T_0)$, where $$\begin{array}{ll}\label{tmax}
\ds T_0=C_1\left( \Vert\varphi^{(0)}_x\Vert^2_{H^2({{\mathbb T}})}+\Vert\varphi^{(1)}\Vert^2_{H^2({{\mathbb T}})} \right)^{-1/2} .
\end{array}$$ The solution $\varphi$ has zero spatial mean and satisfies, for all $t\in I_0$, $$\label{stab1}
\mu-2\phi_x\ge\delta\,,$$ $$\label{stima_Hs}
\Vert\varphi(t)\Vert^2_{H^s({{\mathbb T}})}+\Vert\varphi_t(t)\Vert^2_{H^{s-1}({{\mathbb T}})}\le {C_2} \left(\Vert\varphi^{(0)}\Vert^2_{H^{s}({{\mathbb T}})}+\Vert\varphi^{(1)}\Vert^2_{H^{s-1}({{\mathbb T}})}\right) .$$
Notice that the size of the existence time interval depends on the $H^3, H^2$ norms of the initial data, for all $s\ge3$. Every solution, with the regularity as in the statement of Theorem \[th\_esistenza\], has the additional regularity $$\varphi\in C^2(I_0; H^{s-2}(\mathbb T)),$$ following from equation and suitable commutator estimates, see [@M-S-T:ONDE2], Corollary 16.
The main result
---------------
Now we state the main result of this paper about the continuous dependence in strong norm of solutions on the initial data.
\[main\] Let $s\ge 3$ be a real number and $\mu>\delta>0$. Let us consider $\varphi^{(0)}\in H^{s}(\mathbb T)$, $\varphi^{(1)}\in H^{s-1}(\mathbb T)$ and sequences $\{\varphi^{(0)}_n\}_{n\in{{\mathbb N}}}\subset H^s({{\mathbb T}})$, $\{\varphi^{(1)}_n\}_{n\in{{\mathbb N}}}\subset H^{s-1}({{\mathbb T}})$, where all functions $\varphi^{(0)},\varphi^{(1)},\varphi^{(0)}_n,\varphi^{(1)}_n$ have zero spatial mean. Assume that $$\varphi^{(0)}_n\to\varphi^{(0)} \quad\mbox{strongly in }H^s({{\mathbb T}}), \qquad
\varphi^{(1)}_n\to\varphi^{(1)} \quad\mbox{strongly in }H^{s-1}({{\mathbb T}}),\;{\rm as }\; n\to+\infty.$$
Let $T>0$ and set $I=[0,T]$. Assume that there exists a unique solution $\varphi\in C(I;H^s({{\mathbb T}}))\cap C^1(I;H^{s-1}({{\mathbb T}}))$ of problem , with initial data $\varphi^{(0)},\varphi^{(1)}$, and that for all $n$ there exist unique solutions $\varphi_n\in C(I;H^s({{\mathbb T}}))\cap C^1(I;H^{s-1}({{\mathbb T}}))$ of , with initial data $\varphi^{(0)}_n,\varphi^{(1)}_n$. All solutions satisfy , on $I$ (with corresponding initial data in the r.h.s.) for a given constant $C_2$ independent of $n$.
Then $$\begin{array}{ll}\label{}
\varphi_n\to\varphi \qquad {\rm strongly\; in }\; \; C(I;H^s({{\mathbb T}}))\cap C^1(I;H^{s-1}({{\mathbb T}})),\;{\rm as }\; n\to+\infty.
\end{array}$$
Here we are assuming that all solutions $\vphi,\vphi_n$ are defined on the same time interval $I=[0,T]$, where $T$ is arbitrarily given, neither necessarily small nor given by .
Using the a priori estimate and standard arguments, it is rather easy to show the continuous dependence of solutions on the initial data in the topology of $C(I;H^{s-\eps}({{\mathbb T}}))\cap C^1(I;H^{s-1-\eps}({{\mathbb T}}))$, for all small enough $\eps>0$. Instead, in Theorem \[main\] we prove the continuous dependence precisely in the topology of $C(I;H^s({{\mathbb T}}))\cap C^1(I;H^{s-1}({{\mathbb T}}))$, i.e. in the same function space where we show the existence. From Theorem \[th\_esistenza\] and Theorem \[main\] we obtain that the initial value problem , is well-posed in $H^s$ in the classical sense of Hadamard.
The rest of the paper is organized as follows. In Section \[prbt\] we introduce some notation and give some preliminary technical results. In Section \[proof main\] we prove our main Theorem \[main\].
Notations and preliminary results {#prbt}
=================================
Notations
---------
In the paper we denote by $C$ generic positive constants, that may vary from line to line or even inside the same formula.
Let $\mathbb T$ denote the one-dimensional torus defined as $\mathbb T:=\mathbb R/(2\pi\mathbb Z).$ As usual, all functions defined on $\mathbb T$ can be considered as $2\pi-$periodic functions on $\mathbb R$.
All functions $f:\mathbb T\rightarrow\mathbb C$ can be expanded in terms of Fourier series as $$f(x)=\sum\limits_{k\in\mathbb Z}\widehat{f}(k)e^{ikx}\,,$$ where $\left\{\widehat{f}(k)\right\}_{k\in\mathbb Z}$ are the Fourier coefficients defined by $$\label{coeff_fourier}
\widehat{f}(k):=\frac1{2\pi}\int_{\mathbb T}f(x)e^{-ikx}\,dx\,,\qquad k\in\mathbb Z\,.$$
For positive real numbers $s$, $H^s=H^s(\mathbb T)$ denotes the Sobolev space of order $s$ on $\mathbb T$, defined to be the set of functions $f:\mathbb T\rightarrow\mathbb C$ such that $$\label{normaHs}
\Vert f\Vert_{H^s}^2:=\sum\limits_{k\in\mathbb Z}\langle k\rangle^{2s}\vert\widehat{f}(k)\vert^2<+\infty\,,$$ where it is set $$\label{bracket}
\langle k\rangle:=(1+\vert k\vert^2)^{1/2}\,.$$ The function $\Vert\cdot\Vert_{H^s}$ defines a norm on $H^s$, associated to the inner product $$(f,g)_{H^s}:=\sum\limits_{k\in\mathbb Z}\langle k\rangle^{2s}\widehat{f}(k)\overline{\widehat{g}(k)}\,,$$ which turns $H^s$ into a Hilbert space. For $s=0$ one has $H^0=L^2$. The $L^2$ norm will simply be denoted by $\|\cdot\|$. From we have $$\Vert f\Vert_{H^s}= \| \langle {\partial}_x\rangle^sf\|,$$ where $\langle\partial_x\rangle^s$ is the Fourier multiplier of symbol $\langle k\rangle^s$, defined by $$\label{ds}
\widehat{\langle\partial_x\rangle^s u}(k)=\langle k\rangle^s \widehat{u}(k)\,,\quad\forall\,k\in\mathbb Z\,.$$ We will work with functions with zero spatial mean, so that $\widehat{f}(0)=0$. For these functions, we easily obtain from the Poincaré inequality $$\begin{array}{ll}\label{poincare}
\Vert f\Vert_{H^s}\le \sqrt2\Vert f_x\Vert_{H^{s-1}} \qquad \forall s\ge1.
\end{array}$$ For $T>0$ and $j\in\mathbb N$, we denote by $C^j([0,T]; \mathcal X)$ the space of $j$ times continuously differentiable functions $f:\mathbb R\rightarrow \mathcal X$.
Preliminary results
-------------------
Let us consider the Cauchy problem $$\label{equ2}
\begin{cases}
\psi_{tt}-\left(\mu-2\phi_x\right)\psi_{xx}=F, &\qquad t\in I, \; x\in {{\mathbb T}},
\\
\psi_{|t=0}=\psi^{(0)}, \quad {\partial}_t\psi_{|t=0}=\psi^{(1)}&\qquad x\in {{\mathbb T}},
\end{cases}$$ with unknown the scalar function $\psi=\psi(t,x)$, and where $\phi=\mathbb H[\varphi]$ is the Hilbert transform of a given function $\varphi$, sufficiently smooth, with zero mean and such that $$\label{stab2}
\mu -2\phi_x\ge \delta \qquad t\in I,\; x\in {{\mathbb T}},$$ for given constants $\mu>\d>0$. Let us define $$\Ec(t):=\left( \|\psi_t(t)\|^2 + \int_{{\mathbb T}}(\mu -2\phi_x)|\psi_x(t)|^2 dx \right)^{1/2}.$$
\[lemmapsi0\] Let $\varphi\in C(I;H^3)\cap C^1(I;H^{2})$ be given and satisfying . For all $\psi^{(0)}\in H^1$, $\psi^{(1)}\in L^2$, both functions with zero mean, and $F\in L^1(I;L^2)$ there exists a unique solution $\psi\in C(I;H^1)\cap C^1(I;L^{2})$, of , with zero mean and such that $$\begin{array}{ll}\label{stimapsi0}
\ds \frac{d}{dt} \Ec
\le C \left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)\Ec + \| F\| ,
\end{array}$$ for every $t\in I$.
To obtain we multiply the equation in by $\psi_t$ and integrate over ${{\mathbb T}}$. Integrating by parts gives $$\begin{array}{ll}\label{}
\ds \frac12\frac{d}{dt} \left( \|\psi_t\|^2 + \int_{{\mathbb T}}(\mu -2\phi_x)|\psi_x|^2 dx \right)
=2\int_{{\mathbb T}}\phi_{xx}\psi_t\psi_x\, dx -\int_{{\mathbb T}}\phi_{xt}|\psi_x|^2\, dx + \int_{{\mathbb T}}F\psi_t\, dx.
\end{array}$$ Then, by the Cauchy-Schwarz inequality, the Sobolev imbedding $H^1\hookrightarrow L^\infty$ and the estimate $$\label{stima_hilbert}
\Vert\mathbb H[\varphi]\Vert_{H^s}\le\Vert \varphi\Vert_{H^s}\,,\qquad\forall\,\varphi\in H^s,\; s\in\mathbb R,$$ we obtain $$\begin{array}{ll}\label{stimapsi00}
\ds \frac12 \frac{d}{dt} \left( \|\psi_t\|^2 + \int_{{\mathbb T}}(\mu -2\phi_x)|\psi_x|^2 dx \right)
\le C \left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)\left( \|\psi_t\|^2 + \|\psi_x\|^2 \right) + \| F\| \|\psi_t\|.
\end{array}$$ From we obtain the estimate $$\label{stimadeltapsix}
\|\psi_x\| \le \frac1{\sqrt\d}\left( \int_{{\mathbb T}}(\mu -2\phi_x)|\psi_x|^2 dx \right)^{1/2} \le \frac1{\sqrt\d}\, \Ec,$$ and, substituting it in , we obtain the bound $$\begin{gathered}
\ds \frac12 \frac{d}{dt} \left( \|\psi_t\|^2 + \int_{{\mathbb T}}(\mu -2\phi_x)|\psi_x|^2 dx \right)
\\
\le C \left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)\left( \|\psi_t\|^2 + \int_{{\mathbb T}}(\mu -2\phi_x)|\psi_x|^2 dx \right) + \| F\| \|\psi_t\|,\end{gathered}$$ with a new constant $C$ also depending on $\d$. Since $\|\psi_t\|\le\Ec,$ the above inequality implies $$\label{stimapsi000}
\ds \frac{d}{dt} \Ec
\le C \left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)\Ec + \| F\| ,$$ that is . Applying the Gronwall lemma to gives the a priori estimate $$\label{}
\Ec(t)
\le e^{C T\max_{\tau\in I}\left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)} \left\{ \Ec(0) + \int^t_0\| F\|\, d\tau
\right\},$$ which yields $$\begin{gathered}
\label{apL2}
\left( \|\psi_t(t)\|^2 + \|\psi_x(t)\|^2 \right)^{1/2}
\\
\le Ce^{C T\max_{\tau\in I}\left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)} \left\{ \left( \|\psi^{(1)}\|^2 + (\mu +2\|\varphi^{(0)}\|_{H^2})\|\psi_x^{(0)}\|^2 \right)^{1/2} + \int^T_0\| F\|\, d\tau
\right\},\end{gathered}$$ for all $t\in I$, with $C$ also depending on $\d$. Using , the existence of one solution is obtained by standard arguments, e.g. see [@kato]. By linearity of the problem, the difference of any two solutions with the same data satisfies with zero right-hand side. This shows that the solution is defined up to an additive constant. Thus, requiring the solution to have zero mean gives one uniquely defined solution. Since such a solution has zero mean, we can apply the Poincaré inequality and obtain from $$\begin{gathered}
\label{}
\left( \|\psi_t(t)\|^2 + \|\psi(t)\|^2_{H^1} \right)^{1/2}
\\
\le Ce^{C T\max_{\tau\in I}\left( \|\varphi_t\|_{H^2} + \|\varphi\|_{H^3} \right)} \left\{ \left( \|\psi^{(1)}\|^2 + (\mu +2\|\varphi^{(0)}\|_{H^2})\|\psi_x^{(0)}\|^2 \right)^{1/2} + \int^T_0\| F\|\, d\tau
\right\}.\end{gathered}$$ for all $t\in I$, which shows that $\psi \in C(I;H^1)\cap C^1(I;L^{2})$.
The next lemma concerns the regularity of the solution $\psi$ to .
\[lemmapsi01\] Let $r\ge2$ and $s\ge\max\{3,r\}$. Let $\varphi\in C(I;H^s)\cap C^1(I;H^{2})$ satisfying . For all $\psi^{(0)}\in H^r$, $\psi^{(1)}\in H^{r-1}$ with zero mean, and $F\in L^2(I;H^{r-1})$ there exists a unique solution $\psi\in C(I;H^r)\cap C^1(I;H^{r-1})$ of , with zero mean and such that $$\begin{gathered}
\label{stimapsi01}
\|\psi(t)\|^2_{H^{r}} + \|\psi_t(t)\|^2_{H^{r-1}}
\\
\le Ce^{C T\max_{\tau\in I}\left( 1+ \|\varphi_t\|_{H^2} + \|\varphi\|_{H^{s}} \right)} \left\{ \|\psi^{(1)}\|^2_{H^{r-1}} + (\mu +2\|\varphi^{(0)}\|_{H^2})\|\psi^{(0)}\|^2_{H^{r}} + \|F \|_{L^2(0,t;H^{r-1})}^2
\right\}\end{gathered}$$ for every $t\in I$.
We apply $\langle\partial_x\rangle^{r-1}$ to the equation in , multiply by $\langle\partial_x\rangle^{r-1}\psi_t$ and integrate over ${{\mathbb T}}$. Integrating by parts gives $$\begin{array}{ll}\label{equpsi01}
\ds \frac12\frac{d}{dt} \left( \|\psi_t\|^2_{H^{r-1}} + \int_{{\mathbb T}}(\mu -2\phi_x)|\langle\partial_x\rangle^{r-1}\psi_x|^2 dx \right)
\\
\ds =2\int_{{\mathbb T}}\phi_{xx}\langle\partial_x\rangle^{r-1}\psi_t \, \langle\partial_x\rangle^{r-1}\psi_x\, dx -\int_{{\mathbb T}}\phi_{xt}|\langle\partial_x\rangle^{r-1}\psi_x|^2\, dx
\\
\ds -2\int_{{\mathbb T}}[\langle\partial_x\rangle^{r-1};\phi_x]\psi_{xx}\, \langle\partial_x\rangle^{r-1}\psi_t \, dx
+ \int_{{\mathbb T}}\langle\partial_x\rangle^{r-1}F \, \langle\partial_x\rangle^{r-1}\psi_t\, dx
\\
=\ds \sum_{k=1}^4I_k.
\end{array}$$ We estimate each term of this sum.
[*Estimate of $I_1$.*]{} We apply the Cauchy-Schwarz inequality, the Sobolev imbedding $H^1\hookrightarrow L^\infty$ and the estimate : $$\begin{array}{ll}\label{i1}
|I_1 | \le 2 \|\phi_{xx} \|_{L^\infty} \| \langle\partial_x\rangle^{r-1}\psi_t\| \| \langle\partial_x\rangle^{r-1}\psi_x\| \le C \|\varphi _x\|_{H^2}\|\psi_t\|_{H^{r-1}} \|\psi_x\|_{H^{r-1}} .
\end{array}$$ [*Estimate of $I_2$.*]{} In a similar way we obtain $$\begin{array}{ll}\label{i2}
|I_2| \le \|\phi_{xt} \|_{L^\infty} \| \langle\partial_x\rangle^{r-1}\psi_x\|^2
\le C \|\varphi _t\|_{H^2} \|\psi_x\|^2_{H^{r-1}} .
\end{array}$$ [*Estimate of $I_3$.*]{} First of all we write $$\begin{array}{ll}\label{I30}
|I_3| \le 2\| [\langle\partial_x\rangle^{r-1};\phi_x]\psi_{xx} \| \, \| \langle\partial_x\rangle^{r-1}\psi_t \| \, .
\end{array}$$ For the estimate of the commutator we need to distinguish between the different values of $r$. i) If $r>5/2$ we apply the estimate for commutators with $\t=r-1, \sigma=r-2>1/2$ and to obtain $$\begin{array}{ll}\label{I3i}
\| [\langle\partial_x\rangle^{r-1};\phi_x]\psi_{xx} \| \le C \left( \|\phi_x\|_{H^{r-1}} + \|\phi_{xx}\|_{H^{1}}
\right) \|\psi_{xx}\|_{H^{r-2}}
\\
\le C \left( \|\varphi_x\|_{H^{r-1}} + \|\varphi_{x}\|_{H^{2}} \right) \|\psi_{x}\|_{H^{r-1}}
\le C \|\varphi_x\|_{H^{s-1}} \|\psi_{x}\|_{H^{r-1}} .
\end{array}$$ ii) If $2\le r<5/2$ we apply with $\t=r-1, \sigma=3-r>1/2$ and to get $$\begin{array}{ll}\label{I3ii}
\| [\langle\partial_x\rangle^{r-1};\phi_x]\psi_{xx} \| \le C \left( \|\phi_x\|_{H^{2}} \|\psi_{xx}\| + \|\phi_{xx}\|_{H^{1}}
\|\psi_{xx}\|_{H^{r-2}} \right)
\\
\le C \|\varphi_{x}\|_{H^{2}} \left( \|\psi_{x}\|_{H^{1}} + \|\psi_{x}\|_{H^{r-1}} \right)
\le C \|\varphi_x\|_{H^{s-1}} \|\psi_{x}\|_{H^{r-1}} .
\end{array}$$ iii) Finally if $ r=5/2$ we apply with $\t=r-1$ and to obtain $$\begin{array}{ll}\label{I3iii}
\| [\langle\partial_x\rangle^{r-1};\phi_x]\psi_{xx} \| \le C \left( \|\phi_x\|_{H^{2}} \|\psi_{xx}\|_{H^{1/2}} + \|\phi_{xx}\|_{H^{1}}
\|\psi_{xx}\|_{H^{r-2}} \right)
\\
\le C \|\varphi_{x}\|_{H^{2}} \|\psi_{x}\|_{H^{3/2}}
\le C \|\varphi_x\|_{H^{s-1}} \|\psi_{x}\|_{H^{r-1}} .
\end{array}$$ Recalling , from – we have obtained, for all values of $r\ge2$, $$\begin{array}{ll}\label{i3}
|I_3| \le C \|\varphi_x\|_{H^{s-1}} \|\psi_{x}\|_{H^{r-1}} \|\psi_t\|_{H^{r-1}} .
\end{array}$$ [*Estimate of $I_4$.*]{} The Cauchy-Schwarz inequality gives $$\begin{array}{ll}\label{i4}
|I_4 | \le \| \langle\partial_x\rangle^{r-1}F \| \|\langle\partial_x\rangle^{r-1}\psi_t\|
= \| F\|_{H^{r-1}} \|\psi_t\|_{H^{r-1}} .
\end{array}$$ Estimating the right-hand side of by , , , yields $$\begin{gathered}
\label{27}
\ds \frac{d}{dt} \left( \|\psi_t\|^2_{H^{r-1}} + \int_{{\mathbb T}}(\mu -2\phi_x)|\langle\partial_x\rangle^{r-1}\psi_x|^2 dx \right)
\\
\ds \le C \left( \|\varphi_t\|_{H^2} + \|\varphi_x\|_{H^{s-1}} \right)\left( \|\psi_t\|^2_{H^{r-1}} + \|\psi_x\|^2_{H^{r-1}} \right)
\ds + 2\|F\|_{H^{r-1}} \|\psi_t\|_{H^{r-1}}
\\
\ds \le C \left(1+ \|\varphi_t\|_{H^2} + \|\varphi_x\|_{H^{s-1}} \right)\left( \|\psi_t\|^2_{H^{r-1}} + \|\psi_x\|^2_{H^{r-1}} \right)
+ \|F\|_{H^{r-1}} ^2,
\\
\ds \le C \left( 1+ \|\varphi_t\|_{H^2} + \|\varphi_x\|_{H^{s-1}} \right)\left( \|\psi_t\|^2_{H^{r-1}} +\int_{{\mathbb T}}(\mu -2\phi_x)|\langle\partial_x\rangle^{r-1}\psi_x|^2 dx \right)
+ \|F\|_{H^{r-1}} ^2,\end{gathered}$$ where in the last inequality we have used . Applying the Gronwall lemma to gives $$\begin{gathered}
\label{}
\|\psi_t(t)\|^2_{H^{r-1}} + \int_{{\mathbb T}}(\mu -2\phi_x)|\langle\partial_x\rangle^{r-1}\psi_x(t)|^2 dx
\\
\le e^{C T\max_{\tau\in I}\left( 1+ \|\varphi_t\|_{H^2} + \|\varphi_x\|_{H^{s-1}} \right)} \left\{ \|\psi^{(1)}\|^2_{H^{r-1}} + \int_{{\mathbb T}}(\mu -2\phi_x(0))|\langle\partial_x\rangle^{r-1}\psi_x^{(0)}|^2dx + \int^t_0\| F\|^2_{H^{r-1}}\, d\tau
\right\},\end{gathered}$$ which yields, by and the Poincaré inequality, $$\begin{gathered}
\label{apHs}
\|\psi_t(t)\|^2_{H^{r-1}} + \|\psi(t)\|^2 _{H^{r}}
\\
\le Ce^{C T\max_{\tau\in I}\left( 1+ \|\varphi_t\|_{H^2} + \|\varphi\|_{H^{s}} \right)} \left\{ \|\psi^{(1)}\|^2_{H^{r-1}} + (\mu +2\|\varphi^{(0)}\|_{H^2})\|\psi^{(0)}\|^2_{H^{r}} + \int^t_0\| F\|^2_{H^{r-1}}\, d\tau
\right\},\end{gathered}$$ for all $t\in I$, with $C$ also depending on $\d$, that is . provides the a priori estimate for $\psi$ in $ C(I;H^r)\cap C^1(I;H^{r-1})$.
Now we consider the quadratic operator $\mathcal Q\left[\varphi\right]$, defined in . First of all we recall the estimate proved in [@M-S-T:ONDE2].
\[lemma\_stima\_quadr\] There exists a positive constant $C$ such that for every real $s\ge 1$ and for all $\varphi\in H^s\cap H^3$ $$\label{stima_quadr}
\Vert \mathcal Q[\varphi]\Vert_{H^{s-1}}\le C\Vert\varphi_x\Vert_{H^2}\Vert\varphi_x\Vert_{H^{s-1}}\,.$$
For the proof see [@M-S-T:ONDE2].
Next we give an estimate for the difference of values of $\mathcal Q\left[\varphi\right]$.
\[\] The quadratic operator $\mathcal Q\left[\varphi\right]$, defined in , satisfies $$\begin{array}{ll}\label{diffQ}
\| \mathcal Q\left[\varphi \right] - \mathcal Q\left[\tilde\varphi \right] \| \le C \left( \|\varphi\|_{H^3}+ \|\tilde\varphi\|_{H^3} \right) \|(\varphi-\tilde\varphi)_x\|
\end{array}$$ for all functions $\varphi,\tilde\varphi \in H^3.$
We denote $\phi=\mathbb H[\varphi]$, $\tilde\phi=\mathbb H[\tilde\varphi]$, $\d\varphi=\varphi-\tilde\varphi$, $\d\phi=\phi-\tilde\phi$. Then we can write $$\begin{array}{ll}\label{}
\mathcal Q\left[\varphi \right] - \mathcal Q\left[\tilde\varphi \right] = -3\left[\mathbb H\,;\,\phi_x\right]\phi_{xx}-\left[\mathbb H\,;\,\phi\right]\phi_{xxx}
+3[\mathbb H\,;\,\tilde\phi_x]\tilde\phi_{xx}+[\mathbb H\,;\,\tilde\phi ]\tilde\phi_{xxx}
\\
=-3\left[\mathbb H\,;\,\d\phi_x \right]\phi_{xx}
-3\left[\mathbb H\,;\,\tilde\phi_x \right]\d\phi_{xx} - \left[\mathbb H\,;\,\d\phi\right]\phi_{xxx} - [\mathbb H\,;\,\tilde\phi ]\d\phi_{xxx}
\\
=\ds \sum_{k=1}^4J_k.
\end{array}$$ Now we estimate each term of the sum.
[*Estimate of $J_1$.*]{} We apply estimate , with $p=s=0$, and : $$\begin{array}{ll}\label{j1}
\|J_1\| \le C \| \d\phi_x \| \|\phi_{xx} \|_{H^1} \le C \| \d\varphi_x \| \|\varphi \|_{H^3}.
\end{array}$$ [*Estimate of $J_2$.*]{} We apply estimate , with $p=2, s=0$, and the Poincaré inequality: $$\begin{array}{ll}\label{j2}
\|J_2\| \le C \| \tilde\phi_{xxx} \| \|\d\phi \|_{H^1} \le C \|\tilde\varphi \|_{H^3} \| \d\varphi_x \| .
\end{array}$$ [*Estimate of $J_3$.*]{} We apply estimate , with $ s=1$, and the Poincaré inequality: $$\begin{array}{ll}\label{j3}
\|J_3\| \le C \| \d\phi \|_{H^1} \|\phi_{xxx} \| \le C \|\d\varphi_x \| \| \varphi \|_{H^3} .
\end{array}$$ [*Estimate of $J_4$.*]{} We apply estimate , with $p=3, s=0$, and the Poincaré inequality: $$\begin{array}{ll}\label{j4}
\|J_4\| \le C \| \tilde\phi_{xxx} \| \|\d\phi \|_{H^1} \le C \|\tilde\varphi \|_{H^3} \| \d\varphi_x \| .
\end{array}$$ Collecting – gives .
More generally we apply with different choices of $p$ to estimate the $H^{s-1}$-norm of $J_1-J_4$ and obtain
\[\] Let $s\ge3$. The quadratic operator $\mathcal Q\left[\varphi\right]$, defined in , satisfies $$\begin{array}{ll}\label{diffQs}
\| \mathcal Q\left[\varphi \right] - \mathcal Q\left[\tilde\varphi \right] \|_{H^{s-1}} \le C \left( \|\varphi\|_{H^s}+ \|\tilde\varphi\|_{H^s} \right) \| \varphi-\tilde\varphi\|_{H^s}
\end{array}$$ for all functions $\varphi,\tilde\varphi \in H^s.$
Proof of Theorem \[main\] {#proof main}
=========================
By assumption the sequence $\{\varphi^{(0)}_n\}_{n\in{{\mathbb N}}}$ is uniformly bounded in $H^s$, and the sequence $\{\varphi^{(1)}_n\}_{n\in{{\mathbb N}}}$ is uniformly bounded in $H^{s-1}$. Because of the uniform a priori estimate , from now on we may assume that the sequence of solutions $\{\varphi_n\}_{n\in{{\mathbb N}}}$ is uniformly bounded in $C(I;H^s)\cap C^1(I;H^{s-1})$ on the common time interval $I=[0,T]$, i.e. there exists $K>0$ such that $$\begin{array}{ll}\label{stimapsin}
\|\varphi_n(t)\|_{H^s}^2+\|(\varphi_n)_t(t)\|_{H^{s-1}}^2\le C_2\left( \|\varphi_n^{(0)}\|_{H^s}^2+\|\varphi_n^{(1)}\|_{H^{s-1}}^2 \right) \le K, \qquad \forall t\in I, \, \forall n.
\end{array}$$ Notice that the similar bound holds as well for the solution $\varphi$.
From now on $T$ will be usually included in the generic constant $C$, but sometimes not, when we prefer to emphasize its presence.
In the next proposition we prove the convergence of $\{\varphi_n\}_{n\in{{\mathbb N}}}$ in a weaker topology than in our main Theorem \[main\].
\[convs-1\] Let $s\ge3$. Under the assumptions of Theorem \[main\] the sequence of solutions $\{\varphi_n\}_{n\in{{\mathbb N}}}$ converges to $\varphi$ strongly in $C(I;H^{s-1})\cap C^1(I;H^{s-2})$.
We take the difference of equation for $\varphi$ and $\varphi_n$ and get $$(\varphi-\varphi_n)_{tt} -\left(\mu-2\phi_x\right)(\varphi-\varphi_n)_{xx}= - \mathcal Q\left[\varphi\right] + \mathcal Q\left[\varphi_n \right] +
2\left(\phi_{n,x}-\phi_{x}\right)\varphi_{n,xx}\, ,$$ (where $\phi_{n,x}=(\phi_n)_{x}, \varphi_{n,xx}=(\varphi_n)_{xx}$) which has the form of with $$\psi =\varphi-\varphi_n,$$ $$F=- \mathcal Q\left[\varphi\right] + \mathcal Q\left[\varphi_n \right] +
2\left(\phi_{n,x}-\phi_{x}\right)\varphi_{n,xx}.$$ Applying gives $$\begin{array}{ll}\label{f1}
\| \mathcal Q\left[\varphi \right] - \mathcal Q\left[\varphi_n \right] \| \le C \left( \|\varphi\|_{H^3}+ \|\varphi_n\|_{H^3} \right) \|\psi_x\|
.
\end{array}$$ We also have $$\begin{array}{ll}\label{f2}
\| 2\left( \phi_{n,x}-\phi_{x}\right)\varphi_{n,xx} \| \le 2 \| \phi_{n,x}-\phi_{x} \| \| \varphi_{n,xx} \|_{L^\infty} \le C\|\psi_x\| \|\varphi_n \|_{H^3}.
\end{array}$$ Thus, from , we obtain $$\begin{array}{ll}\label{stimaF}
\| F\| \le C \left( \|\varphi\|_{H^3}+ \|\varphi_n\|_{H^3} \right) \|\psi_x\|
.
\end{array}$$ From Lemma \[lemmapsi0\], , we get $$\label{stimapsi1}
\ds \frac{d}{dt} \Ec
\le C \left( \|\varphi_t\|_{H^2} +\|\varphi\|_{H^3} \right)\Ec + C \left( \|\varphi\|_{H^3}+ \|\varphi_n\|_{H^3} \right) \|\psi_x\|
\ds \le C \Ec ,$$ where we have used the uniform boundedness for $\varphi_n$ and $\varphi$. Applying Gronwall’s lemma to and using again yields $$\|\psi_t(t)\|^2 + \|\psi_x(t)\|^2 \le Ce^{CT}\left( \|\psi_t(0)\|^2 + \|\psi_x(0)\|^2 \right) \qquad t\in I,$$ that is $$\|(\varphi-\varphi_n)_t(t)\|^2 + \|(\varphi-\varphi_n)_x(t)\|^2 \le Ce^{CT}\left( \|\varphi^{(1)}-\varphi^{(1)}_n\|^2 + \|(\varphi^{(0)}-\varphi^{(0)}_n)_x\|^2 \right) \qquad t\in I,$$ which gives the strong convergence of $\varphi_n$ to $\varphi$ in $C(I;H^1)\cap C^1(I;L^2)$, when passing to the limit as $n\to+\infty$. Recall that, since we are working with functions with zero spatial mean, the Poincaré inequality holds. By interpolation and the uniform boundedness we get $$\begin{gathered}
\|(\varphi-\varphi_n)_t(t)\|^2_{H^{s-2}} + \|(\varphi-\varphi_n)(t)\|^2_{H^{s-1}}
\\
\le C \left( \|(\varphi-\varphi_n)_t(t)\|_{H^{s-1}}^{1-1/(s-1)} \|(\varphi-\varphi_n)_t(t)\|_{L^2}^{1/(s-1)} + \|(\varphi-\varphi_n)(t)\|_{H^{s}}^{1-1/(s-1)} \|(\varphi-\varphi_n)(t)\|_{H^1}^{1/(s-1)} \right)^2
\\
\le C \left( \|(\varphi-\varphi_n)_t(t)\|_{L^2}^{2/(s-1)} + \|(\varphi-\varphi_n)(t)\|_{H^1}^{2/(s-1)} \right), \qquad t\in I.\end{gathered}$$ Finally, passing to the limit as $n\to+\infty$ in the above inequality gives the thesis.
\[\] Obviously, by a similar argument with a finer interpolation we could prove the strong convergence of $\varphi_n$ to $\varphi$ in $C(I;H^{s-\eps})\cap C^1(I;H^{s-1-\eps})$, for all small enough $\eps>0$. However, this is useless for the following argument.
Now we take one spatial derivative of and get $$(\varphi_x)_{tt} -\left(\mu-2\phi_x\right)(\varphi_x)_{xx}= - \mathcal Q\left[\varphi\right]_x -
2 \phi_{xx}\varphi_{xx}\, ,$$ which has the form of with $$\psi =\varphi_x,
\qquad
F= - \mathcal Q\left[\varphi\right]_x -
2 \phi_{xx}\varphi_{xx}.$$ Using Proposition \[lemma\_stima\_quadr\], a Moser-type estimate, the Sobolev imbedding and we compute $$\begin{array}{ll}\label{}
\|F\|_{H^{s-2}} \le \| \mathcal Q\left[\varphi\right]\|_{H^{s-1}} + 2 \|\phi_{xx}\varphi_{xx}\|_{H^{s-2}}
\\
\le C \left(\|\varphi_x\|_{H^{2}}\|\varphi_x\|_{H^{s-1}} + \|\phi_{xx}\|_{L^\infty}\|\varphi_{xx}\|_{H^{s-2}} + \|\phi_{xx}\|_{H^{s-2}} \|\varphi_{xx}\|_{L^\infty} \right)
\\
\le C \|\varphi\|_{H^{3}}\|\varphi\|_{H^{s}} ,
\end{array}$$ and we deduce that $F\in L^\infty(I;H^{s-2})$. Moreover, the initial values of $\varphi_x,\, (\varphi_x)_t$ are $\varphi_{x}^{(0)} \in H^{s-1},\, \varphi_{x}^{(1)}\in H^{s-2}$, respectively.
We are going to introduce regularized approximations of the data $\varphi_{x}^{(0)},\, \varphi_{x}^{(1)},\, F$. Given any $\eps>0$, let us take functions $ \Psi^{(0)}_\eps\in H^{s}, \Psi^{(1)}_\eps\in H^{s-1}$ with zero mean, and $F^\eps \in L^2(I;H^{s-1})$ such that $$\begin{array}{ll}\label{stimaeps}
\| \Psi^{(0)}_\eps - \varphi_{x}^{(0)}\|_{H^{s-1}} + \| \Psi^{(1)}_\eps - \varphi_{x}^{(1)}\|_{H^{s-2}} + \|F^\eps - F \|_{L^2(I;H^{s-2})} \le \eps.
\end{array}$$
Let us consider the Cauchy problem with regularized data $$\label{equ3}
\begin{cases}
\Psi^\eps_{tt}-\left(\mu-2\phi_x\right)\Psi^\eps_{xx}=F^\eps, &\qquad t\in I, \; x\in {{\mathbb T}},
\\
\Psi^\eps_{|t=0}=\Psi^{(0)}_\eps, \quad {\partial}_t\Psi^\eps_{|t=0}=\Psi^{(1)}_\eps&\qquad x\in {{\mathbb T}}.
\end{cases}$$ Again this problem has the form .
\[\] Let $s\ge3$. For any $\eps>0$, the Cauchy problem has a unique solution $\Psi^\eps\in C(I;H^{s})\cap C^1(I;H^{s-1})$ and $$\begin{gathered}
\label{stimaPsi}
\|\Psi^\eps \|^2_{C(I;H^{s})} + \|\Psi^\eps _t\|^2_{C(I;H^{s-1})}
\\
\le Ce^{C T\max_{\tau\in I}\left( 1+ \|\varphi_t\|_{H^2} + \|\varphi\|_{H^{s}} \right)} \left\{ \|\Psi^{(1)}_\eps\|^2_{H^{s-1}} + (\mu +2\|\varphi^{(0)}\|_{H^2})\|\Psi^{(0)}_\eps\|^2_{H^{s}} + \|F^\eps \|_{L^2(I;H^{s-1})}^2
\right\}.\end{gathered}$$
The result follows from Lemma \[lemmapsi01\] with $r=s$.
Now we estimate the difference $\Psi^\eps - \varphi_{x}$, which solves the linear problem $$\begin{cases}\label{probdiff}
(\Psi^\eps - \varphi_{x})_{tt}-\left(\mu-2\phi_x\right)(\Psi^\eps - \varphi_{x})_{xx}=F^\eps-F, &\qquad t\in I, \; x\in {{\mathbb T}},
\\
(\Psi^\eps - \varphi_{x})_{|t=0}=\Psi^{(0)}_\eps- \varphi_{x}^{(0)}, \quad {\partial}_t(\Psi^\eps - \varphi_{x})_{|t=0}=\Psi^{(1)}_\eps- \varphi_{x}^{(1)}&\qquad x\in {{\mathbb T}}.
\end{cases}$$
\[\] Let $s\ge3$. For any $\eps>0$, the difference $\Psi^\eps - \varphi_{x}$ satisfies the estimate $$\begin{array}{ll}\label{stimadiff}
\ds \|(\Psi^\eps - \varphi_{x})(t) \|_{H^{s-1}} + \| (\Psi^\eps - \varphi_{x})_t(t) \|_{H^{s-2}} \le C\eps \qquad \forall t\in I.
\end{array}$$
We apply Lemma \[lemmapsi01\] with $r=s-1$ and get $$\begin{gathered}
\label{stimadiff2}
\|\Psi^\eps - \varphi_{x} \|^2_{C(I;H^{s-1})} + \|(\Psi^\eps - \varphi_{x})_t \|^2_{C(I;H^{s-2})}
\\
\le Ce^{C T\max_{\tau\in I}\left( 1+ \|\varphi_t\|_{H^2} + \|\varphi\|_{H^{s}} \right)} \Big\{ \| \Psi^{(1)}_\eps- \varphi_{x}^{(1)}\|^2_{H^{s-2}}
\\
+ (\mu +2\|\varphi^{(0)}\|_{H^2})\| \Psi^{(0)}_\eps- \varphi_{x}^{(0)} \|^2_{H^{s-1}} + \|F^\eps-F \|_{L^2(I;H^{s-2})}^2
\Big\}.\end{gathered}$$ Thus follows from .
Finally we estimate the difference between $\varphi_{n,x}=(\varphi_n)_{x}$ and $\Psi^\eps$. The difference of the corresponding problems reads $$\begin{cases}\label{probdiff2}
(\Psi^\eps - \varphi_{n,x})_{tt}-\left(\mu-2\phi_{n,x}\right)(\Psi^\eps - \varphi_{n,x})_{xx}=G^{n,\eps}, &\qquad t\in I, \; x\in {{\mathbb T}},
\\
(\Psi^\eps - \varphi_{n,x})_{|t=0}=\Psi^{(0)}_\eps- \varphi_{n,x}^{(0)}, \\ {\partial}_t(\Psi^\eps - \varphi_{n,x})_{|t=0}=\Psi^{(1)}_\eps- \varphi_{n,x}^{(1)}&\qquad x\in {{\mathbb T}}.
\end{cases}$$ where we have set $$\begin{array}{ll}\label{defGneps}
G^{n,\eps} =F^\eps-F^n + 2\left(\phi_{n,x}-\phi_{x}\right)\Psi^\eps_{xx}, \qquad F^n= - \mathcal Q\left[\varphi_n\right]_x -
2 \phi_{n,xx}\varphi_{n,xx},
\end{array}$$ $$\varphi_{n,x}^{(0)}=(\varphi_{n}^{(0)})_x, \qquad \varphi_{n,x}^{(1)}=(\varphi_{n}^{(1)})_x .$$
\[\] Let $s\ge3$. For any $\eps>0$, there exists $M(\eps)>0$ such that, for any $n$ the difference $\Psi^\eps - \varphi_{n,x}$ satisfies the estimate $$\begin{gathered}
\label{stimadiff6}
\|(\Psi^\eps - \varphi_{n,x})(t) \|^2_{H^{s-1}} + \|(\Psi^\eps - \varphi_{n,x})_t(t) \|^2_{H^{s-2}}
\\
\le C \Big\{ \eps^2 + \| \varphi^{(1)}- \varphi_{n}^{(1)}\|^2_{H^{s-1}}
+ \| \varphi^{(0)} - \varphi_{n}^{(0)} \|^2_{H^{s}} + \left| \int_0^t\|\varphi-\varphi_n\|_{H^{s}}^2 d\tau \right| + TM(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})}
\Big\}\end{gathered}$$ for any $t\in I$.
We apply Lemma \[lemmapsi01\] with $r=s-1$ and obtain for every $t\in I$ $$\begin{gathered}
\label{stimadiff4}
\|(\Psi^\eps - \varphi_{n,x})(t) \|^2_{H^{s-1}} + \|(\Psi^\eps - \varphi_{n,x})_t(t) \|^2_{H^{s-2}}
\\
\le Ce^{C T\max_{\tau\in I}\left( 1+ \|\varphi_{n,t}\|_{H^2} + \|\varphi_n\|_{H^{s}} \right)} \Big\{ \| \Psi^{(1)}_\eps- \varphi_{n,x}^{(1)}\|^2_{H^{s-2}}
\\
+ (\mu +2\|\varphi^{(0)}_n\|_{H^2})\| \Psi^{(0)}_\eps- \varphi_{n,x}^{(0)} \|^2_{H^{s-1}} + \|G^{n,\eps} \|_{L^2(0,t;H^{s-2})}^2
\Big\}.\end{gathered}$$ First of all, recalling the uniform boundedness w.r.t. $n$ , we may write as $$\begin{gathered}
\label{stimadiff5}
\|(\Psi^\eps - \varphi_{n,x})(t) \|^2_{H^{s-1}} + \|(\Psi^\eps - \varphi_{n,x})_t(t) \|^2_{H^{s-2}}
\\
\le C \Big\{ \| \Psi^{(1)}_\eps- \varphi_{n,x}^{(1)}\|^2_{H^{s-2}}
+ \| \Psi^{(0)}_\eps- \varphi_{n,x}^{(0)} \|^2_{H^{s-1}} + \|G^{n,\eps} \|_{L^2(0,t;H^{s-2})}^2
\Big\}.\end{gathered}$$ From the definition in we have, for every $\tau\in[0,t]$, $$\begin{array}{ll}\label{}
\|G^{n,\eps} \|_{H^{s-2}} \le \| F^\eps-F^n
\|_{H^{s-2}} + 2\|\left(\phi_{n,x}-\phi_{x}\right)\Psi^\eps_{xx} \|_{H^{s-2}} \\
\\
\le \| F^\eps-F
\|_{H^{s-2}}
+ \| F-F^n
\|_{H^{s-2}} + C\|\varphi_n -\varphi\|_{H^{s-1}} \|\Psi^\eps \|_{H^{s}}
.
\end{array}$$ Integrating this inequality in $\tau$ between $0$ and $t$ gives $$\begin{gathered}
\label{42}
\ds \left| \int_0^t\|G^{n,\eps} \|^2_{H^{s-2}} d\tau \right|
\le 3\left| \int_0^t \| F^\eps-F
\|^2_{H^{s-2}} d\tau \right|
+ 3\left| \int_0^t\| F-F^n
\|^2_{H^{s-2}} d\tau \right| + CTM(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})} \end{gathered}$$ where we have denoted $$\ds M(\eps) := Ce^{C T\max_{\tau\in I}\left( 1+ \|\varphi_t\|_{H^2} + \|\varphi\|_{H^{s}} \right)} \left\{ \|\Psi^{(1)}_\eps\|^2_{H^{s-1}} + (\mu +2\|\varphi^{(0)}\|_{H^2})\|\Psi^{(0)}_\eps\|^2_{H^{s}} + \|F^\eps \|_{L^2(I;H^{s-1})}^2
\right\},$$ that is the right-hand side of . On the other hand, for all $\tau$ we have $$\begin{array}{ll}\label{diffF1}
\| F-F^n
\|_{H^{s-2}} \leq \| \mathcal Q\left[\varphi\right] - \mathcal Q\left[\varphi_n\right] \|_{H^{s-1}} + 2 \| \phi_{xx}\varphi_{xx} -
\phi_{n,xx}\varphi_{n,xx} \|_{H^{s-2}}.
\end{array}$$ From we have $$\begin{array}{ll}\label{}
\| \mathcal Q\left[\varphi\right] - \mathcal Q\left[\varphi_n\right] \|_{H^{s-1}}
\le C (\|\varphi\|_{H^{s}}+ \|\varphi_n\|_{H^{s}}) \|\varphi-\varphi_n\|_{H^{s}}
.
\end{array}$$ Moreover, since $H^{s-2}$ is an algebra we can estimate $$\begin{gathered}
\label{diffF2}
2 \| \phi_{xx}\varphi_{xx} -
\phi_{n,xx}\varphi_{n,xx} \|_{H^{s-2}}
\\ \le
C \| \phi_{xx} -
\phi_{n,xx} \|_{H^{s-2}} \| \varphi_{xx}\|_{H^{s-2}}
+ C \| \phi_{n,xx}\|_{H^{s-2}} \|\varphi_{xx} -
\varphi_{n,xx} \|_{H^{s-2}}
\\
\le C (\|\varphi\|_{H^{s}}+ \|\varphi_n\|_{H^{s}}) \|\varphi-\varphi_n\|_{H^{s}}
.\end{gathered}$$ From – and the uniform boundedness we obtain $$\begin{array}{ll}\label{diffF}
\| F-F^n
\|_{H^{s-2}} \le C \|\varphi-\varphi_n\|_{H^{s}}
\qquad \forall \tau\in I,
\end{array}$$ and substituting it in gives $$\begin{gathered}
\label{stimaGneps}
\ds \left| \int_0^t\|G^{n,\eps} \|^2_{H^{s-2}} d\tau \right|
\le 3\left| \int_0^t \| F^\eps-F
\|^2_{H^{s-2}} d\tau \right|
+ C\left| \int_0^t\|\varphi-\varphi_n\|_{H^{s}}^2 d\tau \right| + CTM(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})} .\end{gathered}$$ Finally, from , , we get
$$\begin{gathered}
\label{}
\|(\Psi^\eps - \varphi_{n,x})(t) \|^2_{H^{s-1}} + \|(\Psi^\eps - \varphi_{n,x})_t(t) \|^2_{H^{s-2}}
\\
\le C \Big\{ \eps^2 + \| \varphi^{(1)}- \varphi_{n}^{(1)}\|^2_{H^{s-1}}
+ \| \varphi^{(0)} - \varphi_{n}^{(0)} \|^2_{H^{s}} + \left| \int_0^t\|\varphi-\varphi_n\|_{H^{s}}^2 d\tau \right| + TM(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})}
\Big\}\end{gathered}$$
for all $t\in I$, that is .
Adding , , and applying the Poincaré inequality gives $$\begin{gathered}
\label{}
\|(\varphi - \varphi_{n})(t) \|^2_{H^{s}} + \|(\varphi - \varphi_{n})_t(t) \|^2_{H^{s-1}}
\\
\le C \Big\{ \eps^2 + \| \varphi^{(1)}- \varphi_{n}^{(1)}\|^2_{H^{s-1}}
+ \| \varphi^{(0)} - \varphi_{n}^{(0)} \|^2_{H^{s}} + \left| \int_0^t\|\varphi-\varphi_n\|_{H^{s}}^2 d\tau \right| + TM(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})}
\Big\}\end{gathered}$$ for all $t\in I$. Then, applying the Gronwall lemma yields $$\begin{array}{ll}\label{51}
\ds \|\varphi - \varphi_{n} \|^2_{C(I;H^{s})} + \|(\varphi - \varphi_{n})_t \|^2_{C(I;H^{s-1})}
\\
\ds \le C_3 \Big\{ \eps^2 + \| \varphi^{(1)}- \varphi_{n}^{(1)}\|^2_{H^{s-1}}
+ \| \varphi^{(0)} - \varphi_{n}^{(0)} \|^2_{H^{s}} + M(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})}
\Big\}.
\end{array}$$ Given any $\eps'>0$, let $\eps=\eps(\eps')$ be such that $C_3\eps^2<\eps'/3$. With this fixed $\eps$ in $M(\eps)$, and taking account of Proposition \[convs-1\], let $n_0$ be such that, for any $n\ge n_0$, $$C_3\left\{\| \varphi^{(1)}- \varphi_{n}^{(1)}\|^2_{H^{s-1}}
+ \| \varphi^{(0)} - \varphi_{n}^{(0)} \|^2_{H^{s}} \right\} <\eps'/3,$$ $$C_3 M(\eps)\|\varphi_n -\varphi\|^2_{C(I;H^{s-1})}<\eps'/3.$$ It follows from that $$\ds \|\varphi - \varphi_{n} \|^2_{C(I;H^{s})} + \|(\varphi - \varphi_{n})_t \|^2_{C(I;H^{s-1})}
<\eps' \qquad \forall n\ge n_0.$$ This concludes the proof of Theorem \[main\].
Some commutator estimates {#stima_commutatore}
=========================
\[lemma\_comm\] For $\t>1/2$ there exists a constant $C_\t>0$ such that $$\begin{aligned}
\Vert \left[\mathbb H\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_\t\Vert v\Vert_{H^\t(\mathbb T)}\Vert f\Vert_{L^2(\mathbb T)}\,,\quad\forall\,v\in H^\t(\mathbb T)\,,\,\,\forall\,f\in L^2(\mathbb T)\,,
\label{stima_comm_1}\end{aligned}$$ where $\left[\mathbb H\,;\,v\right]$ is the commutator between the Hilbert transform $\mathbb H$ and the multiplication by $v$.
The proof can be found in [@M-S-T:ONDE1].
\[lemma\_comm\_ale\_2\] For every real $\t\ge 0$ and integer $p\ge 0$ there exists a constant $C_{\t,p}>0$ such that for all functions $v\in H^{\t+p}(\mathbb T)$ and $f\in H^1(\mathbb T)$ $$\label{stima_comm_p}
\Vert \left[\mathbb H\,;\,v\right]\partial^p_x f\Vert_{H^\t(\mathbb T)}\le C_{\t,p}\Vert \partial^{p}_x v\Vert_{H^\t(\mathbb T)}\Vert f\Vert_{H^1(\mathbb T)}\,.$$
The proof can be found in [@M-S-T:ONDE2].
\[commutatore\_ds\] For every real $\t\ge 1$ and $\sigma>1/2$ there exists a constant $C_{\t,\sigma}>0$ such that
- for all $f\in H^{\t-1}(\mathbb T)\cap H^\sigma(\mathbb T)$ and $v\in H^\t(\mathbb T)\cap H^2(\mathbb T)$ $$\label{stima_comm_ds}
\Vert \left[\langle\partial_x\rangle^\t\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_{\t,\sigma}\left\{\Vert v\Vert_{H^\t(\mathbb T)}\Vert f\Vert_{H^\sigma(\mathbb T)}+\Vert v_x\Vert_{H^1(\mathbb T)}\Vert f\Vert_{H^{\t-1}(\mathbb T)}\right\}\,;$$
- for all $f\in H^{\t-1}(\mathbb T)$ and $v\in H^{\t+\sigma}(\mathbb T)\cap H^2(\mathbb T)$ $$\label{stima_comm_ds1}
\Vert \left[\langle\partial_x\rangle^\t\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_{\t,\sigma}\left\{\Vert v\Vert_{H^{\t+\sigma}(\mathbb T)}\Vert f\Vert_{L^2(\mathbb T)}+\Vert v_x\Vert_{H^1(\mathbb T)}\Vert f\Vert_{H^{\t-1}(\mathbb T)}\right\}\,.$$
For all $\t\ge 1$ there exists a positive constant $C_\t$ such that for all $f\in H^{\t-1}(\mathbb T)\cap H^{1/2}(\mathbb T)$ and $v\in H^{\t+1/2}(\mathbb T)\cap H^2(\mathbb T)$ $$\label{stima_comm_ds2}
\Vert \left[\langle\partial_x\rangle^\t\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_{\t}\left\{\Vert v\Vert_{H^{\t+1/2}(\mathbb T)}\Vert f\Vert_{H^{1/2}(\mathbb T)}+\Vert v_x\Vert_{H^1(\mathbb T)}\Vert f\Vert_{H^{\t-1}(\mathbb T)}\right\}\,.$$
For all $k\in\mathbb Z$ we compute $$\label{coeff_fourier_comm}
\begin{split}
\widehat{\left[\langle\partial_x\rangle^\t\,;\,v\right]f}(k)&=\langle k\rangle^\t\widehat{vf}(k)-\widehat{v\langle\partial_x\rangle^\t f}(k)\\
&=\frac1{2\pi}\langle k\rangle^\t\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat f(\ell)-\frac1{2\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)\langle\ell\rangle^\t\widehat{f}(\ell)\\
&=\frac1{2\pi}\sum\limits_{\ell}\left(\langle k\rangle^\t-\langle\ell\rangle^\t\right)\widehat{v}(k-\ell)\widehat{f}(\ell)\,.
\end{split}$$ On the other hand we have $$\langle k\rangle^\t-\langle\ell\rangle^\t=\int_0^1\frac{d}{d\theta}\left(\langle \ell+\theta(k-\ell)\rangle^\t\right)\,d\theta
=(k-\ell)\int_0^1 D\left(\langle \cdot\rangle^\t\right)(\ell+\theta(k-\ell))d\theta\,,$$ where $D$ denotes the derivative of the function $\langle \cdot\rangle^\t$. Combining the preceding with the estimate $$\label{derivate_ds}
\left\vert\frac{d}{d\xi}\langle \xi\rangle^\t\right\vert\le C_\t\langle\xi\rangle^{\t-1}\,,\quad\forall\,\xi\in\mathbb R\,,$$ then gives $$\label{stima_diff_ds}
\vert\langle k\rangle^\t-\langle\ell\rangle^\t\vert\le\vert k-\ell\vert\int_0^1 \vert D\left(\langle \cdot\rangle^\t\right)(\ell+\theta(k-\ell))\vert d\theta\le C_\t\vert k-\ell\vert\int_0^1\langle \ell+\theta(k-\ell)\rangle^{\t-1}\,d\theta\,.$$ Using , from we get $$\label{stima_coeff_comm}
\begin{split}
\vert\widehat{\left[\langle\partial_x\rangle^\t\,;\,v\right]f}(k)\vert&\le\frac1{2\pi}\sum\limits_{\ell}\left\vert\langle k\rangle^\t-\langle\ell\rangle^\t\right\vert\vert\widehat{v}(k-\ell)\vert\vert\widehat{f}(\ell)\vert\\
&\le C_\t\sum\limits_{\ell}\int_0^1\vert k-\ell\vert\langle \ell+\theta(k-\ell)\rangle^{\t-1}\vert\widehat{v}(k-\ell)\vert\vert\widehat{f}(\ell)\vert\,d\theta\,.
\end{split}$$ Since the function $\langle\zeta\rangle^{\t-1}$ is sub-additive and $0\le\theta\le 1$, we have $$\label{sub-add}
\langle\ell+\theta(k-\ell)\rangle^{\t-1}\le C_\t\left\{\langle\theta(k-\ell)\rangle^{\t-1}+\langle\ell\rangle^{\t-1}\right\}\le C_\t\left\{\langle k-\ell\rangle^{\t-1}+\langle\ell\rangle^{\t-1}\right\}\,,$$ with positive constant $C_\t$ depending only on $\t$. Using to estimate the right-hand side of then gives $$\label{stima_coeff_comm1}
\begin{split}
\vert\widehat{\left[\langle\partial_x\rangle^\t\,;\,v\right]f}(k)\vert&
\le C_\t\sum\limits_{\ell}\left\{\int_0^1\langle k-\ell\rangle^{\t}\vert\widehat{v}(k-\ell)\vert\vert\widehat{f}(\ell)\vert\,d\theta+\int_0^1\vert k-\ell\vert \vert\widehat{v}(k-\ell)\langle\ell\rangle^{\t-1}\vert\vert\widehat{f}(\ell)\vert\,d\theta\right\}\\
&\le C^\prime_\t\left\{\left(\vert\widehat{\langle\partial_x\rangle^\t v}\vert\ast\vert\widehat{f}\vert\right) (k)+\left(\vert\widehat{v_x}\vert\ast\vert\widehat{\langle\partial_x\rangle^{\t-1}f}\vert\right)(k)\right\}\,.
\end{split}$$ Using Parseval’s identity and Young’s inequality with $\left\{\vert\widehat{\langle\partial_x\rangle^\t v}(k)\vert\right\}\in\ell^2$, $\left\{\vert\widehat{f}(k)\vert\right\}\in\ell^1$, $\left\{\vert\widehat{v_x}(k)\vert\right\}\in\ell^1$, $\left\{\vert\widehat{\langle\partial_x\rangle^{\t-1} f}(k)\vert\right\}\in\ell^2$, from we derive $$\label{stima_coeff_comm2}
\begin{split}
\Vert\left[\langle\partial_x\rangle^\t\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C^\prime_\t\left\{\Vert\{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}\Vert_{\ell^2}\Vert\{\vert\widehat{f}\vert\}\Vert_{\ell^1}+\Vert\{\vert\widehat{v_x}\vert\}\Vert_{\ell^1}\Vert\{\vert\widehat{\langle\partial_x\rangle^{\t-1}f}\vert\}\Vert_{\ell^2}\right\}\,.
\end{split}$$ We get the first inequality of Lemma \[commutatore\_ds\], by using once again Parseval’s identity and the estimates $$\label{stima_coeff_comm3}
\Vert\{\vert\widehat{f}\vert\}\Vert_{\ell^1}\le C_\sigma\Vert f\Vert_{H^\sigma(\mathbb T)}\,,\qquad \Vert\{\vert\widehat{v_x}\vert\}\Vert_{\ell^1}\le C\Vert v_x\Vert_{H^1(\mathbb T)}\, .$$ To get the second inequality it is sufficient to interchange the role of the sequences $\{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}$ and $\{\vert\widehat{f}\vert\}$ when we apply Young’s inequality to the first term in the right-hand side of , by taking the $\ell^1$-norm of $\{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}$ and the $\ell^2$-norm of $\{\vert\widehat{f}\vert\}$; then the $\ell^1-$norm of $\{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}$ is estimated again by the first inequality in $$\Vert\{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}\Vert_{\ell^1}\le C_\sigma\Vert \langle\partial_x\rangle^\t v\Vert_{H^\sigma(\mathbb T)}\le C_\sigma\Vert v\Vert_{H^{\t+\sigma}(\mathbb T)}\,.$$
To obtain the last inequality , the $\ell^2-$norm of the sequence $\{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\ast\vert\widehat{f}\vert\}$ in the right-hand side of is estimated by Young’s inequality as $$\label{stima_coeff_comm4}
\Vert \{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\ast\vert\widehat{f}\vert\} \Vert_{\ell^2}\le \Vert \{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}\Vert_{\ell^{4/3}}\Vert\{\vert\widehat{f}\vert\}\Vert_{\ell^{4/3}}\,;$$ then recalling that for every $p\in]1,2]$ a positive constant $C_p$ exists such that $$\label{stima_coeff_comm5}
\Vert \{\widehat{f}\}\Vert_{\ell^p}\le C_p\Vert f\Vert_{H^{1/2}(\mathbb T)}\,,\quad\forall\,f\in H^{1/2}(\mathbb T)\,,$$ the $\ell^{4/3}-$norms in the right-hand side of are estimated as $$\label{stima_coeff_comm6}
\Vert \{\vert\widehat{\langle\partial_x\rangle^\t v}\vert\}\Vert_{\ell^{4/3}}\le C\Vert \langle\partial_x\rangle^\t v\Vert_{H^{1/2}(\mathbb T)}\le C\Vert v\Vert_{H^{\t+1/2}(\mathbb T)}\,,\qquad \Vert \{\vert\widehat{f}\vert\}\Vert_{\ell^{4/3}}\le C\Vert f\Vert_{H^{1/2}(\mathbb T)}$$ (that is with $p=\frac{4}{3}$). Then follows from gathering the estimates , and repeating for the rest the same calculations as above.
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---
abstract: 'We reinvestigate the behavior of the conductivity of several disordered quantum lattice models at infinite temperature using exact diagonalization. Contrary to the conclusion drawn in a recent investigation of similar quantities in identical systems, we find evidence of a localized regime for strong random fields. We estimate the location of the critical field for the many-body localization transition for the random-field XXZ spin chain, and compare our findings with recent investigations in related systems.'
author:
- 'Timothy C. Berkelbach'
- 'David R. Reichman'
bibliography:
- 'mbl.bib'
title: |
Conductivity of disordered quantum lattice models at infinite temperature:\
Many-body localization
---
Introduction
============
Anderson localization in non-interacting systems is a well understood physical process whereby sufficiently strong disorder leads to localization of eigenstates and hence insulating behavior in systems that would otherwise be conductors.[@abr79] Extensive work performed over the last 50 years has established in great detail the nature of the Anderson transition in non-interacting systems, while much less work has been aimed at elucidating how short-ranged interactions modify the simple picture of transport that emerges in the non-interacting situation.[@lag09] This is somewhat surprising, given the fact that attention to the issue of interactions already appears in Anderson’s classic 1958 paper.[@and58]
Recently Basko, Aleiner and Altshuler (BAA) performed a detailed diagrammatic analysis demonstrating that weak, short-ranged electron-electron interactions generically lead to a finite temperature metal-insulator transition in systems that would be localized in the absence of interactions.[@bas06] In fact the analysis and implications of the work of BAA go beyond consideration of interacting electrons, suggesting that more general quantum entities (e.g. spins, bosons) with local interactions may generically fail to thermalize until a threshold energy is reached. The notion that such a “many-body” localization (MBL) transition may occur is at odds with the intuition that interactions should lead to a finite dc conductivity at all finite temperatures in analogy with the mechanism of phonon-mediated hopping conductivity. It is also at odds with, for example, the analysis of Fleishman and Anderson which argues that truly short-ranged interactions are insufficient to induce conductivity in an otherwise localized system at any temperature.[@fle80]
The work of BAA has motivated more recent investigations of the possibility of a finite temperature transition from a localized (non-ergodic) phase to a delocalized phase where interactions afford thermalization of the system. These works, both analytical and numerical, have reached somewhat conflicting conclusions.[@oga07; @zni08; @kar09; @mue09; @iof09] In this work, we reconsider the analysis of Karahalios et al.[@kar09] Via examination of the conductivity, these authors concluded that, in general, finite temperature systems of one-dimensional interacting spins are always conducting, thus contradicting the claim of BAA.
Imaginary frequency conductivity
================================
Here, as in previous work, we make use of the important observation of Oganesyan and Huse that the many-body localization transition may be probed at infinite temperature by varying the disorder strength.[@oga07] This simplifies the problem by reducing the number of control parameters that may be varied to tune the system from a delocalized to a localized phase. As in the work of Karahalios et al., we examine one-dimensional spin chains via exact diagonalization, calculating the conductivity via the Kubo formula. An important conclusion of our work is that sufficient care needs to be exercised in the interpretation of the zero frequency conductivity as a function of disorder strength and level broadening.
For a finite system of length $L$ at $T \rightarrow \infty$, the Kubo formula for the conductivity is given by $$\sigma(\omega) = \frac{\beta}{L} \lim_{\eta\to 0}
\int_0^\infty e^{i(\omega + i\eta)t} \langle j(t) j(0) \rangle dt,$$ where $\beta=1/k_BT$, $j(t)$ is the current operator at time $t$, and $\eta$ can be thought of as both a numerical tool for convergence as well as a phenomenological level broadening for discrete spectra.[@imr02] The real part of the conductivity may be decomposed as $\sigma^\prime(\omega)=D\delta(\omega) +
\sigma_{\rm reg}(\omega)$, where the Drude weight $D$ measures purely ballistic conduction and arises due to pairs of degenerate states connected by the current operator. However, in the systems studied here, all level degeneracies are lifted in the presence of disorder, and conductivity is purely diffusive. Thus we may take as our definition of the dc conductivity $\sigma_{\rm dc} = \sigma(\omega\rightarrow0)$ without concern for the Drude contribution. By employing the spectral representation of the Hamiltonian we arrive at the ‘imaginary frequency’ dc conductivity, $$\sigma(i\eta) = \frac{\beta}{ZL} \sum_{m,n}
|\langle m|j|n\rangle|^2
\frac{\eta}{\eta^2 + (\omega_{nm})^2},$$ where $H|n\rangle = E_n|n\rangle$, $\omega_{nm}=E_n-E_m$, and $Z$ is the partition function.
The authors of Ref.\[8\] calculate the full frequency-dependent (ac) conductivity spectrum using a level-broadening binning procedure and draw conclusions regarding dc conductivity based on the $\omega
\rightarrow 0$ behavior. However, as we will show, all finite-sized systems with level broadening will exhibit a non-zero dc conductivity, and thus such an analysis is inconclusive. Rather, it is the conductivity’s [*dependence*]{} on this level broadening which allows one to draw conclusions regarding conducting and insulating behaviors in the thermodynamic limit.
As discussed by Thouless and Kirkpatrick,[@tho81] finite systems should display a simple asymptotic behavior for the dc conductivity that scales as $\eta$ for small $\eta$ and $\eta^{-1}$ for large $\eta$. The distinction between conductor and insulator manifests in the behavior between these asymptotic regimes. In particular, we expect that a system with insulating behavior will exhibit an imaginary frequency dc conductivity with well resolved $\eta$ and $\eta^{-1}$ regimes separated by a simple maximum. Finite sized systems expected to behave as conductors in the $L \rightarrow \infty$ limit exhibit a broad crossover between these regimes, with a plateau signifying the onset of a true dc conductivity.[@tho81; @imr02] Although the dc conductivity is well defined only in the thermodynamic limit (specifically, $L\rightarrow\infty$, then $\eta\rightarrow0$), we see evidence for dc conductivity manifesting itself even at the small system sizes accessible by exact diagonalization.
It should be pointed out that although the analysis employed here was originally developed for non-interacting systems, our results empirically show that it is equally applicable to interacting ones, by replacing the single-particle levels with many-body levels. Specifically, we locate an insulating regime in which $\sigma(i\eta)\propto \eta$ when $\eta$ is less than the level spacing in the many-body localization volume, which does not scale with the size of the system. This behavior is to be contrasted with an observed metallic regime, which has $\sigma(i\eta) \propto \eta$ as long as $\eta$ is less than the many-body level spacing in the system volume – a spacing which vanishes in the thermodynamic limit yielding a dc conductivity plateau at small to intermediate $\eta$.
Quantum Lattice Models
======================
We study two quantum lattice models in the presence of disorder: the $XXZ$ spin chain and the ${t\mhyphen t^\prime\mhyphen V}$ model of spinless fermions, originally studied in its disordered form by Oganesyan and Huse[@oga07] and more recently by Monthus and Garel.[@mon10] The disordered $XXZ$ chain is given by the Hamiltonian $$\begin{aligned}
H_{XXZ} = & J \sum_{j=1}^{L} \left[ S^x_{j} S^x_{j+1} + S^y_{j} S^y_{j+1}
+ \Delta S^z_{j} S^z_{j+1}\right] \\
&+\sum_{j=1}^{L} w_j S^z_j, \notag\end{aligned}$$ where we choose the random fields $w_j$ uniformly from $[-W,W]$. The current operator for the $XXZ$ chain is given by $$j_{XXZ} = J\sum_{j=1}^{L} \left[ S^x_{j} S^y_{j+1}
- S^y_{j} S^x_{j+1} \right].$$
![Local spin-spin correlation function, $\langle S^z_j(t)S^z_j(0)\rangle$ for the disordered $XXZ$ chain with $W=5$. Results are shown for the non-interacting case, $\Delta=0.0$ (filled circles) and for the interacting case, $\Delta=0.5$ (solid line). Error bars for the non-interacting case (not shown for clarity) are of the same order as those shown for the interacting case. []{data-label="fig:spin"}](paper_spin.eps)
The disordered ${t\mhyphen t^\prime\mhyphen V}$ model is described by the Hamiltonian $$\begin{aligned}
H_{{t\mhyphen t^\prime\mhyphen V}} = \sum_{j=1}^{L} &\bigg[ -t\left(c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j\right) \\
&- t'\left(c_j^\dagger c_{j+2} + c_{j+2}^\dagger c_j\right) \notag\\
&+ V \left(n_j-\frac{1}{2}\right) \left(n_{j+1}-\frac{1}{2}\right) + w_j n_j \bigg], \notag\end{aligned}$$ where, following Oganesyan and Huse, the random on-site energies $w_j$ are chosen from a Gaussian distribution with mean $0$ and variance $W^2$. The ${t\mhyphen t^\prime\mhyphen V}$ model’s current operator is $$\begin{aligned}
j_{{t\mhyphen t^\prime\mhyphen V}} = i\sum_{j=1}^{L} &\left[
t\left(c_j^\dagger c_{j+1} - c_{j+1}^\dagger c_j\right) \right. \\
&+ \left. 2t^\prime\left(c_j^\dagger c_{j+2}
- c_{j+2}^\dagger c_j\right) \right]. \notag\end{aligned}$$
![The dc conductivity as a function of the imaginary frequency, $\eta$, for the disordered $XXZ$ chain ($J=1$) with $W = 1-6$ \[(a)$-$(f)\] for the non-interacting case, $\Delta=0.0$ (black line, circles) and for two interacting cases, $\Delta=0.5$ (red line, squares) and $\Delta=1.0$ (green line, diamonds). Error bars are smaller than the symbols. The blue line shows linear slope. []{data-label="fig:xxz"}](paper_dc_xxz_half.eps)
In what follows, we restrict our Hilbert space to $S^z_{tot}=0$ for the $XXZ$ chain and to half-filling for the ${t\mhyphen t^\prime\mhyphen V}$ model. We have explored other alternatives and find our results to be qualitatively similar. All results are presented for $L=14$ with periodic boundary conditions, although we have studied systems as large as $L=16$ and find our conclusions unaltered. We average over $N_r=100$ independent realizations of disorder and calculate error bars as the standard deviation of the mean, $\pm \sigma/\sqrt{N_r}$, where $\sigma$ (not to be confused with the conductivity) is the standard deviation across disorder realizations. Our results appear to be converged, although for such a small range of system sizes one should view such statements with care. This is especially true for smaller values of disorder, where finite localization lengths, if they exist, can clearly be larger than the system sizes accessible from exact diagonalization.
The aforementioned claim of Karahalios et al. is rather surprising in light of the fact that an earlier tDMRG calculation performed on strongly disordered spin chains found evidence of a localized phase, at least when viewed from the standpoint of the local spin-spin correlation function.[@zni08] In Fig. \[fig:spin\] we reproduce this result (for a smaller system and the shorter time scales accessible in exact diagonalization) and compare it with dynamics in the non-interacting system which is known to be localized. It is clear that, for the interacting system, the local spin-spin correlation exhibits quantitatively similar relaxation compared to the localized system and shows no sign of decay from a plateau (the analog of the Edwards-Anderson parameter), indicative of a glassy phase.
![System size dependence of the dc conductivity as a function of the imaginary frequency, $\eta$, for the disordered $XXZ$ chain ($J=1$) with $\Delta = 0.5$ and $W = 1$. Conductivities are presented for $L = 10$ (green plusses), 12 (blue diamonds), 14 (red squares), and 16 (black circles), showing the development of the dc conductivity plateau in the $L \rightarrow \infty$ limit. []{data-label="fig:scaling"}](paper_dc_xxz_scaling.eps)
Results and the many-body localization transition
=================================================
We turn next to an investigation of the conductivity in the above disordered lattice models. As discussed above, it is useful to compare the $\eta$ dependence of the dc conductivity in the interacting models directly with their noninteracting counterparts to set a baseline for localization. In Fig. \[fig:xxz\] we present results for two different values of the anisotropy $\Delta$ in the $XXZ$ spin chain. We extend the results of Karahalios et al., who examine disorder strengths only as high as $W=1$, by performing our calculations up to $W=6$. Clearly, by $W=4$ in both interacting cases the conductivity curves are essentially indistinguishable from the non-interacting localized case. Thus, at least with regard to exact diagonalization on these system sizes and based solely on examination of the conductivity, the interacting behavior is identical to the non-interacting, localized behavior. Using this condition, we can place an upper limit, $W_{c}\lesssim 4$ in both cases $\Delta=0.5$ and $\Delta=1.0$.[^1] It should be noticed as well that already at $W=1$ there is significant structure in $\sigma(i\eta)$, exhibiting a nearly flat region in between the small and large $\eta$ regimes. This suggests that these interacting data are in a conducting regime, although care must be used because this also might be an indication of localization behavior on length scales larger than we can access via exact diagonalization. One should also note that this conducting behavior is fully consistent with the conclusions of Karahalios et al. at $W = 1.0$. With the above caveats, we can place the critical value of $W$ for a many-body localization transition in the range $3 < W_{c} < 4$.[^2] It is thus unsurprising that Karahalios et al. found no evidence for the MBL transition, as we have shown that it occurs at disorder strengths larger than those investigated in Ref.\[8\]. We have additionally analyzed adjacent many-body level-spacings (not shown here), whose crossover from the Gaussian Orthogonal Ensemble to Poisson statistics[@shk93; @hof93; @hof94] occurs at this same critical strength of disorder, confirming the robustness of our approach.
![The same as in Fig.\[fig:xxz\] but for the disordered ${t\mhyphen t^\prime\mhyphen V}$ model ($t=t^\prime=1$) with $W = 3$ (a), $W = 5$ (b), $W=10$ (c), and $W = 16$ (d), for the non-interacting case, $V=0$ (black line, circles) and for the interacting case, $V=2$ (red line, squares). Where not shown, error bars are smaller than the symbols. The blue line shows linear slope. []{data-label="fig:ttv"}](paper_dc_ttv_half.eps)
In order to confirm our expectations of a conducting phase in the thermodynamic limit, $L \rightarrow \infty$, we have examined the effects of system size on the conductivity of the interacting $XXZ$ spin chain with $\Delta = 0.5$. We focus on the disorder strength $W = 1$ because of its apparent conducting behavior in Fig. \[fig:xxz\]. Furthermore, one is in danger of approaching localization lengths equal to the size of the sytem for values of disorder much smaller than this. We show in Fig. \[fig:scaling\] the conductivities for system sizes $L=$ 10, 12, 14, and 16. Results for $L=16$ are shown for $N_r = 50$ realizations of disorder. Clearly, the plateau becomes more resolved for larger system sizes, strongly suggesting a non-zero dc conductivity in the thermodynamic limit. Furthermore, the plateau’s growth extends toward smaller $\eta$ in agreement with the many-body level spacing discussion above.
We have also examined the behavior of the disordered ${t\mhyphen t^\prime\mhyphen V}$ model of Oganesyan and Huse, the results of which are presented in Fig. \[fig:ttv\]. The behavior is qualitatively the same as above, suggesting the existence of a MBL transition in this model as well. However, it should be pointed out that the crossover between apparent conducting and insulating behavior in $\sigma(i\eta)$ is significantly broader in the ${t\mhyphen t^\prime\mhyphen V}$ model than in the $XXZ$ systems, making a prediction of the critical disorder strength difficult. At larger system sizes this crossover should become more abrupt, but unfortunately such sizes are beyond the reach of exact diagonalization. Despite the above difficulties, we would expect, based on the same means of analysis presented above, that the critical value of $W$ in this model is higher than the $W_{c}\approx5$ range found in the real-space renormalization group calculation of Monthus and Garel. Although, it is not clear when studying finite systems that different quantities, such as those investigated here and by Monthus and Garel, should behave in a similar manner. However, our result does strongly suggest conducting behavior at $W\approx5$.
Conclusions
===========
To summarize, we have carried out a systematic investigation of the diffusive conductivities of two common disordered quantum lattice models: the $XXZ$ spin chain and the ${t\mhyphen t^\prime\mhyphen V}$ model of spinless fermions. We find that by studying the behavior of $\sigma(i\eta)$ we can place reasonable bounds on the location of the MBL transition. By examining disorder strengths higher than those explored in previous works, we find for the disordered $XXZ$ chain (both $\Delta=0.5$ and $\Delta=1.0$) that the finite size conductivity extrapolates quantitatively to the non-interacting values by $W\approx4$. Our results are also qualitatively consistent with the recent work of Monthus and Garel, but we would expect based on finite size studies of the conductivity, that the transition occurs at a disorder value larger than that found by their numerical renormalization group procedure.
We thank G. Biroli, V. Oganesyan, A. Millis, and F. Zamponi for useful discussions. We would also like to thank the NSF for financial support under Grant No. CHE-0719089.
[^1]: It should be noted that it is possible that a residual conductivity, impossible to resolve via an exact diagonalization study of $\sigma_{dc}$ on small systems, exists. We see no evidence for finite size effects here, though we are restricted to very small $L$.
[^2]: After this work was completed we became aware of the work of Pal and Huse (), who also investigate the $XXZ$ chain with $\Delta=1$. Their conclusions are quantitatively consistent with our results. It should be noted that their work goes beyond that presented here by connecting the MBL transition to the infinite randomness universality class.
|
---
abstract: 'We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symplectic quotient.'
address: 'Department of Mathematics, University of Illinois, Urbana, IL 61801'
author:
- Eugene Lerman
- Susan Tolman
title: Intersection cohomology of $S^1$ symplectic quotients and small resolutions
---
Introduction
============
Let a compact Lie group act effectively on a compact connected symplectic manifold $M$ with a moment map $\Phi: M \to {{\mathfrak g}}^*$. In the case that $0$ is a regular value of the moment map, the symplectic quotient $\red := \Phi \inv (0)/G$ is an orbifold, and its rational cohomology ring is fairly well understood [@Kibook; @Wi; @Wu; @Ka; @Ka-Gu; @JK; @TW].
However, many interesting spaces arise as reduced spaces at singular values of the moment map. Some examples include: the moduli space of flat connections, some polygon spaces, many physical systems, and projective toric varieties whose polytopes are not simple. Since the symplectic quotient at a singular value is a stratified space [@SjL], a natural invariant to compute is the intersection cohomology (with middle perversity). Less is known in this case. Kirwan has provided formulas to compute the Betti numbers in the algebraic case [@Kibook; @Ki4]; Woolf extended this work to the symplectic case. Moreover, Jeffrey and Kirwan computed the pairing in the intersection cohomology of particular symplectic quotients [@ktalk].
The main result of this paper is two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. More precisely, these formulas depend on the image of the restriction map in equivariant cohomology $H^*_{S^1}(M; \R) \to
H^*_{S^1}(M^{S^1}; \R)$ from the original manifold to the fixed point set.
Let the circle $S^1$ act on a compact connected symplectic manifold $M$ with moment map $\Phi: M \to \R$ so that $0$ is in the interior of $\Phi(M)$. Let $\red := \Phi\inv(0)/S^1$ denote the reduced space.
Then there exists a surjective map $\kappa$ from the equivariant cohomology ring $H^*_{S^1}(M;\R)$ to the intersection cohomology $IH^*(\red ;\R)$. Moreover, given any equivariant cohomology class $\alpha$ and $\beta$ in $H^*_{S^1}(M)$, the pairing of $\kappa(\alpha)$ and $\kappa(\beta)$ in $IH^*(\red)$ is given by the formula $$\left< \kappa(\alpha), \kappa(\beta) \right> =
\Res _0 \sum_{F \in \cF^+} \int_F \frac{ i_F^*(\alpha \beta) }{e_F} .$$ Here, $e_F$ denotes the equivariant Euler class of the normal bundle of $F$, and $\cF^+ $ denotes the set of components $F$ of the fixed point set ${S^1}$ such that either
1. $\Phi(F) > 0$ or
2. $\Phi(F) = 0$ and $ \index F \leq \frac{1}{2} (\dim M - \dim F)$,
where the index of $F$ is the dimension of the negative eigenspace of the Hessian of the moment map $\Phi$ at a point of $F$.
The meaning of the right hand side is as follows. The map $i_F^*$ is simply the restriction to $F$. The equivariant cohomology ring $H^*_{S^1}(F)$ is naturally isomorphic to the polynomial ring in one variable $H^*(F)[t]$. The equivariant Euler class $e_F$ is invertible in the localized ring $H^*(F)(t)$; thus, $\frac{i_F^*(\alpha)}{e_F}$ is an element of this ring. The integral $\int_F : H^*(F)(t) \to
\R(t)$ acts by integrating each coefficient in the series. Finally, $\Res_0$ denotes the operator which returns the coefficient of $t\inv$.
Our convention is that the pairing in intersection cohomology between two classes $\alpha \in IH^p(\red)$ and $\beta \in IH^q(\red)$ is zero if $p + q \neq \dim(\red)$.
Note that since $\kappa$ is surjective, this theorem determines the pairing for all pairs of elements in $IH^*(\red)$. Additionally, by Poincare duality in intersection cohomology, it determines the kernel of $\kappa$.
We now provide an alternative version of our main result:
Let the circle $S^1$ act on a compact connected symplectic manifold $M$ with moment map $\Phi: M \to \R$ so that $0$ is in the interior of $\Phi(M)$. Let $\red := \Phi\inv(0)/S^1$ denote the reduced space.
Then there exists a ring structure on the intersection cohomology $IH^*(\red;\R)$ so that
- The ring structure on $IH^*(\red)$ is compatible with the pairing, in the sense that their exists an isomorphism $\int$ from the top dimensional intersection cohomology to $\R$ so that $\int \alpha \cdot \beta = \left< \alpha, \beta \right>$.
- As a graded ring, $IH^*(M;\R)$ is isomorphic to $H^*_{S^1}(M;\R)/K$, where $$K:=
\{ \alpha \in H_{S^1}^*(M) \mid \alpha|_{F} = 0 \ \ \forall \ \
F \in {\cF}^+\} \oplus
\{ \alpha \in H_{S^1}^*(M) \mid \alpha|_{F} = 0 \ \ \forall \ \
F \in {\cF}^-\} .$$
Here, $\cF^+$ denotes the set of components $F$ of the fixed point set $M^{S^1}$ such that either
1. $\Phi(F) > 0$ or
2. $\Phi(F) = 0$ and $ \index F \leq \frac{1}{2} (\dim M - \dim F)$,
where the index of $F$ is the dimension of the negative eigenspace of the Hessian of the moment map $\Phi$ at a point of $F$. Additionally, $\cF^-$ denotes the set of all other components of the fixed point set.
In principle, these two formulas for the intersection cohomology give almost exactly the same information. We include both, because, in practice, one or the other might be better suited to tackle a particular problem.
We prove these two theorems simultaneously. First, we construct an orbifold $\tred$, which we call the [**perturbed quotient**]{}, The perturbed quotient is a small resolution of the symplectic quotient; thus. as a graded vector space with pairing, $IH^*(\Phi\inv (0)/S^1)$ is isomorphic to $H^*(\widetilde{\red})$. Moreover, even though the perturbed quotient is not symplectic, it is constructed in such a way that the standard techniques for computing the cohomology ring of a symplectic quotient can be applied to it, yielding the above formulae.
The construction of the perturbed quotient is fairly straightforward. The singularities of the reduced space $\red := \Phi\inv(0)/S^1$ correspond to components $Y$ of the fixed point set $M^{S^1}$ lying on the zero level set $\Phi\inv(0)$. In the setting projective varieties, it is known that the neighborhoods of these singularities have small resolutions [@Hu].[^1] Although these resolutions are only local, it is possible to piece them together into a global resolution.
We construct the perturbed quotient as the quotient of a fiber of a perturbation $\tPhi: M \to \R$ of the original moment map. Since this perturbed moment map $\tPhi$ is Bott-Morse, and since its critical points are exactly the fixed points of the action of $S^1$ on $M$, the standard techniques used to compute the cohomology of symplectic quotients can also be applied to compute the cohomology of the perturbed quotient.
Finally, we construct a pairing preserving isomorphism between the intersection cohohomology of the symplectic quotient and the cohomology of the perturbed quotient. In the algebraic case, this follows immediately from the fact that the perturbed quotient is a small resolution of the symplectic quotient (see §6.2 in [@GM2]). However, while it “appears to be clear that this theorem will also be valid in our case", we have decided to provide a direct proof.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The work in this paper was inspired by the lectures of Francis Kirwan at the Newton institute in the Fall of 1994.
We thank Reyer Sjamaar for many helpful discussion. In particular the idea that for $S^1$ quotients the intersection cohomology should be very simple to compute is due to him. We thank Sam Evens for a number of useful discussions.\
Simple stratified spaces and intersection cohomology
====================================================
In this section, we introduce the two main concepts that we will need in this paper: simple stratified spaces and intersection cohomology. The notion of a simple stratified space is not standard; it is, however, convenient for our purposes. The definition of intersection cohomology we use is essentially identical to the definition of the complex of intersection differential forms due to Goresky and MacPherson (see [@B]), except that we allow the strata to be orbifolds, and that we only consider simple stratified spaces.
Recall that an open [**cone**]{} on a topological space $L$ is $$\overset \circ c (L) := L\times [0, 1)/\sim,$$ where $(x,0) \sim (x',0)$ for all $x, x'\in L$. Equivalently $\conec
(L) = L\times [0, \infty)/\sim$.
\[def\_sss\] A [**simple stratified space**]{} is a Hausdorff topological space $X$ with the following properties:
- The space $X$ is a disjoint (set-theoretic) union of orbifolds, called [**strata**]{}.
- There exists an open dense oriented stratum $X^r$, called the [**top stratum**]{}.
- The complement of $X^r$ in $X$ is a disjoint union of connected orbifolds, $X \smallsetminus X^r = \coprod Y_i$, called the [**singular strata**]{}.
- For each singular stratum $Y$ there is a neighborhood $\tT$ of $Y$ in $X$ and a map $\pi :\tT \to Y$ which is a $C^0$ fiber bundle with a typical fiber $\overset \circ c (L)$ for some orbifold $L$, which depends on $Y$. (Thus $Y$ embeds into $\tT$ as the vertex section.)
- Their exists a diffeomorphism from the complement $\tT \smallsetminus Y$ to $Q \times (0,1)$, where $ Q\to Y$ is a $C^\infty$ fiber bundle with typical fiber $L$, such that the following diagram commutes: $$\begin{CD}
\tT \smallsetminus Y @>>> Q_i \times (0,1)\\
@V{\pi}VV @VVV\\
Y @= Y.
\end{CD}$$ In particular $\pi :\tT \smallsetminus Y \to Y$ is a smooth fiber bundle with a typical fiber $L \times (0,1).$
Thus a simple stratified space $X$ is a decomposition $X= X^r \sqcup
\coprod Y_i$ and a collection of maps $\{\pi_i : \tT_i \to Y_i \}$.
\[rmrk\_t\] Note that the composite $\tT\smallsetminus Y \to Q \times
(0,1)\to (0,1)$, where $Q\times (0,1)\to (0,1)$ is the obvious projection, extends to a continuous map $r: \tT\to [0, 1)$. In the definition of intersection cohomology of $X$ it will be convenient for us to consider smaller tubular neighborhoods $T$ defined by $$T = r\inv ([0, 1/2)).$$
Let $\pi : E\to B$ be a smooth submersion of orbifolds. The Cartan filtration ${\bF}_k\Omega^*(E)$ of the complex of forms $\Omega^* (E)$ on $E$ is given by $$\begin{split}
{\bF}_k\Omega^*(E)=\{ \omega \in \Omega^* (E) \mid & \ \ \text{for all
}\, e\in E \, \text{and for all
vectors } \xi_0, \cdots, \xi_{k} \in \ker d\pi _e \\
& \ \ i(\xi_0) \circ \cdots \circ i(\xi_{k})(\omega(e)) = 0 \\
& \ \ \text{and } i(\xi_0) \circ \cdots \circ i(\xi_{k}) (d\omega (e)) = 0 \} .
\end{split}$$
By convention, $i(\xi_0) \circ \cdots \circ i(\xi_{k})(\sigma)
= 0$ if $\deg \sigma \leq k$. Note that ${\bF}_0 \Omega ^*
(E) $ consists of basic forms.
Let $X= X^r \sqcup \coprod Y_i$ be a simple stratified space. A [**perversity**]{} $\bar{p} :\{Y_i\} \to \bN$ is a function that assigns a nonnegative integer to each singular stratum $Y_i$. The [**middle perversity**]{} $\bm$ is defined by $$\bar{m} (Y_i)= \lfloor \frac{1}{2} (\dim X^r -\dim Y_i) \rfloor -1.$$
Let $(X= X^r \sqcup \coprod Y_i,\{\pi_i : \tT_i \to
Y_i \})$ be a simple stratified space and let $\bp : \{Y_i\} \to \bN$ be a perversity. The [**complex of intersection differential forms**]{} $I\Omega_{\bar p}^* (X)$ is a sub-complex of the complex of differential forms on the top stratum: $$I\Omega_{\bar p}^* (X):= \{ \omega \in \Omega^* ( X^r) \mid \omega
|_{T_i \cap X^r} \in {\bF}_{\bar{p} (Y_i)} \Omega^*(T_i \cap
X^r)\}$$ where the filtration ${\bF}_{\bar{p} (Y_i)} \Omega^*(T_i \cap
X^r)$ is defined relative to the submersion $\pi_i: T_i \cap X^r = T_i
\smallsetminus Y_i \to Y_i$. The coboundary map is the exterior differentiation $d$.
The [**intersection cohomology**]{} $IH_{\bar{p}}^* (X)$ of the simple stratified space $X$ with perversity $\bar{p}$ is the cohomology of the complex $(I\Omega _{\bar{p}}^* (X), d)$.
If the strata of a simple stratified space $X$ are manifolds, then $X$ is a pseudomanifold. In this case our definition of intersection forms is exactly the Goresky-MacPherson definition of the complex of differential forms and our definition of intersection cohomology agrees with the standard definition (see § 1.2 in [@B]).
We now define the pairing on the middle perversity intersection cohomology of a compact simple stratified space $X$ with an oriented top stratum.
Note that if $q > \dim(Y_i) + \bp(Y_i)$, then every $\alpha \in I\Omega^q_\bp(X)$ vanishes on $T_i$. In particular, if $\dim X^r > \dim Y_i + \bp(Y_i)$ for all singular strata $Y_i$ then every $\alpha \in I\Omega_\bp^{\dim X^r} (X)$ is supported on the compact set $X \smallsetminus \bigcup T_i$. Therefore, if the top stratum $X^r$ is oriented, there is a well-defined integration map $\int : I\Omega_\bp^{\dim X^r} (X) \to
\R$, $\alpha \mapsto \int _{X^r} \alpha$. Similarly, if $\dim X^r - 1 > \dim Y_i + \bp(Y_i)$ for all $i$, then any $\beta \in I\Omega_\bp^{\dim X^r - 1}(X)$ is also supported in $X \smallsetminus \bigcup T_i$. Thus, integration descends to a well-defined map on cohomology $\int : IH_\bp^{\dim X^r} (X) \to \R$; we extend this by zero to a map $\int : IH_\bp^*(X) \to \R$.
Given $\alpha$ and $\beta$ in $I\Omega^*_\bm(X)$, notice that $\alpha \wedge \beta \in I\Omega_{2 \bm}(X)$, where $2
\bm$ is twice the middle perversity. This follows from the property of the Cartan filtration: for any $\alpha \in {\bF}_k\Omega^*(E)$ and $\beta \in {\bF}_l\Omega^*(E)$, $\alpha \wedge \beta \in
{\bF}_{k+l}\Omega^*(E)$. Moreover, $2 \bm(Y_i) \leq \dim X^r - \dim
Y_i - 2$ for all singular strata $Y_i$. Thus, there is a well-defined bilinear pairing $IH_\bm^p (X) \times IH_\bm^q
(X) \to \R$ which sends $[\alpha] \in IH_\bm^p (X)$ and $[\beta] \in
IH_\bm^q (X)$ to the integral $\int_{X^r} \alpha \wedge \beta$.
The structure of the symplectic quotient
========================================
In this section, we recall a normal form for the neighborhoods of fixed points on symplectic manifolds with Hamiltonian circle actions. Using this, we give a normal form for the neighborhoods of the singularities in a symplectic quotient. In particular, we show that the quotient is a simple stratified space.
This last statement is a special case of a theorem of Sjaamar and Lerman [@SjL], who show that every symplectic quotient by a compact Lie group is a stratified space. Note, however, that in [@SjL] the stratification is by orbit type, whereas here we use the slightly coarser stratification by infinitesimal orbit type.
Let a circle act on a symplectic manifold $M$ in a Hamiltonian fashion with a moment map $\Phi: M \to \R$. Recall that the [**symplectic quotient**]{} (a.k.a. the [**reduced space**]{}) is $M_{\text{red}}:= \Phi\inv(0)/S^1$. If $0$ is a regular value for $\Phi$, then the quotient is a symplectic orbifold. More generally, $\Phi$ is regular on $M\smallsetminus M^{S^1}$, and $\red^r := \left(\mu\inv (0) \cap (M\smallsetminus M^{S^1})\right)/S^1$ is an orbifold; this is the top stratum. Moreover, since the restriction of the symplectic form on $M$ to $\Phi\inv (0)\cap (M\smallsetminus M^{S^1})$ descends to a symplectic form on $\red^r$, $\red^r$ is naturally oriented.
Recall that the moment map is constant on each component of the fixed point set $M^{S^1}$, and that these components are isolated. Thus, every component $Y$ of the fixed point set which intersects the zero level set is entirely contained in the level set, and gives rise to a stratum of $\red$ diffeomorphic to $Y$. To see how these strata fit together, we need the following lemma.
\[lemma\_model\] Let $S^1$ act on a symplectic manifold $(M,\omega)$ in a Hamiltonian fashion with a moment map $\Phi: M \to \R$. Every connected component $Y$ of the fixed point set $M^{S^1}$ has even codimension, say $2n$. Moreover, there exists
- positive integers $n_1,\ldots,n_k$ such that $\sum_i n_i = n $,
- a principal $G := \prod U(n_i) \subset U(n)$ bundle $P$ over $Y$, and
- distinct non-zero weights $\kappa_1,\ldots,\kappa_k$,
such that:
- there is a diffeomorphism $\sigma$ from a neighborhood $U$ of $Y$ in $M$ to a neighborhood $U_0$ of the zero section in the associated bundle $ P \times _G \C^n \to Y$;
- this diffeomorphism is equivariant with respect to the circle action on $P \times_G \C^n$ defined by the weights $\kappa_i$;
- the diffeomorphism pulls back the moment map $\Phi$ to the map $\mu: P \times_G \C^n \to \R$ given below, i.e., $\Phi \circ \sigma = \mu,$ where $$\mu ([q, (\vec{z}_1, \ldots, \vec{z}_k)]) = \frac{1}{2} \sum \kappa _i
|\vec{z}_i|^2 + \Phi (Y), \quad \forall \ (\vec{z}_1, \ldots,
\vec{z}_k)\in \C^{n_1} \oplus \cdots \oplus \C^{n_k} = \C^n.$$
Consider the symplectic perpendicular bundle $E = TY^\omega$ of $Y$ in $(M, \omega)$. Since $Y$ is a symplectic submanifold of $M$ we have $TM|_Y = TY \oplus TY^\omega$. So $E$ is the normal bundle of $Y$, and $E$ is a symplectic vector bundle. The group $S^1$ acts on the bundle $E$ by fiber-preserving vector bundle maps. We may choose an $S^1$ invariant complex structure on $E$ compatible with the symplectic structure. Up to an equivariant homotopy, this complex structure is unique.
A fiber $\C^n$ of $E$ splits into the direct sum of isotypical representations of $S^1$, $\C^n = \C^{n_1}\oplus \cdots \C^{n_k}$, so that the action of $\lambda \in S^1$ on $\C^{n_i}$ is given by multiplication by $\lambda ^{\kappa_i}$ for some weight $\kappa_i \in
\Z$. Under the above identification of the fiber of $E$ with $\C^n$ the symplectic structure is the imaginary part of the standard Hermitian inner product. Hence a moment map for the $S^1$ action on the fiber is $$\C^{n_1}\oplus \cdots \oplus \C^{n_k}\ni (\vec{z}_1, \ldots,
\vec{z}_k) \mapsto \frac{1}{2} \sum \kappa _i |\vec{z}_i|^2.$$
The structure group of the vector bundle $E$ reduces to the subgroup of $U(n)$ consisting of transformations that commute with the action of $S^1$ described above, that is, it reduces to $G := \prod
U(n_i) \subset U(n)$. Consequently $E = P\times _G \C^n$ for some principal $G$ bundle $P$ over $Y$.
The equivariant symplectic embedding theorem (see, for example, Theorem 2.2.1 in [@GLS] and the subsequent discussion) implies that we may identify the neighborhood of the submanifold $Y$ in $M$ with a neighborhood of the zero section of $E$ in such a way that a moment map $\mu : E \to \R$ is given by $$\mu ([p, (\vec{z}_1, \ldots, \vec{z}_k)]) =
\frac{1}{2} \sum \kappa _i |\vec{z}_i|^2 + \text{a constant}$$ for all $([p, (\vec{z}_1, \ldots, \vec{z}_k)] \in P\times _G (\C^{n_1}\oplus
\cdots \oplus \C^{n_k}).$
\[split\] Let $V^+$ and $V^-$ be the the sum of the positive and negative weight spaces, respectively. That is, we may assume that the corresponding weights satisfy $\kappa_1, \ldots \kappa_s >0$ and $\kappa _{s+1}, \ldots, \kappa _k < 0$, and set $$V^+ = \bigoplus _{i=1}^s \C^{n_i}, \qquad
V^- = \bigoplus _{i=s+1}^k \C^{n_i}.$$ We then have linear representations of $G$ on $V^+$ and $V^-$ so that $E$ splits: $E = E^+ \oplus E^-$ where $E^\pm = P \times _G
V^\pm$. By Lemma \[lemma\_model\], the index of the moment map $\Phi :M\to \R$ at $Y$ is $\dim V^-$.
We now use Lemma \[lemma\_model\] above to show that the reduced space $\red$ is a simple stratified space.
\[prop\_simple\] Let the circle $S^1$ act effectively on a compact connected symplectic manifold $M$ in a Hamiltonian fashion with a moment map $\Phi: M \to
\R$. Assume that 0 is in the interior of the image of the moment map. The reduced space $\red = \Phi\inv (0)/S^1$ is a simple stratified space. The singular strata of $\red$ are connected components $Y$ of the fixed point set $M^{S^1}$ with $\Phi
(Y) = 0$. For each such stratum there exists:
- a faithful unitary representation $\rho: S^1 \to U(p) \times U(q)$, where $p$ and $q$ are positive integers whose sum is the codimension of $Y$; and
- a principal $G$ bundle $P \to Y$, where $G$ is a subgroup of $U(p) \times U(q)$ which commutes with $\rho(S^1)$;
so that a neighborhood $\tilde{T}$ of $Y$ in $M$ is the associated cone bundle $P \times_G \conec (S^{2p-1} \times_{S^1} S^{2q-1})$.
By “a neighborhood $\tilde{T}$ of $Y$ in $\red$ is the associated cone bundle $P \times_G \conec (S^{2p-1} \times_{S^1} S^{2q-1})$” we mean that there exists a stratum-preserving homeomorphism from a neighborhood $\tilde{T}$ of $Y$ in $\red$ to the associated bundle $P
\times_G \conec (S^{2p-1} \times_{S^1} S^{2q-1})$, which restricts to a diffeomorphism on each stratum. The stratification of $P \times_G
\conec (S^{2p-1} \times_{S^1} S^{2q-1})$ comes from the stratification of the cone $\conec (S^{2p-1} \times_{S^1} S^{2q-1})$ into the vertex and the complement of the vertex.
Let $Y$ be a component of the fixed point set with $\Phi (Y) = 0$. We use the the notation of Lemma \[lemma\_model\] and Remark \[split\].
By Lemma \[lemma\_model\] the zero level set $\Phi \inv (0)$ near $Y$ is isomorphic to $$\left\{ [q, (\vec{z}_1, \ldots, \vec{z}_k)] \left| \sum _{i=1}^{s}
\kappa _i |\vec{z}_i|^2 = \sum _{i=s+1}^{k} \kappa _i |\vec{z}_i|^2
\right. \right\} \simeq P\times _G \overset \circ c(S^+ \times S^-),$$ where $S^+ = \{z \in V^+ \mid \sum _{i=1}^{s} \kappa _i |\vec{z}_i|^2 =
1\}$, and $S^-$ is defined similarly. Therefore the reduced space $\Phi \inv (0) /S^1$ near the stratum $Y$ is $P\times _G \overset \circ c (S^+ \times_{S^1} S^-)$ where the action of $S^1$ on $S^+ \times S^- \subset V^+ \times V^-$ is defined by the weights $\kappa_1, \ldots \kappa _k$.
The perturbed quotient
======================
In this section we will construct an orbifold $\tred$, which we call the [**perturbed quotient**]{}, together with a map $f:
\tred \to \red$. The perturbed quotient has two key properties: it is straightforward to explicitly compute its cohomology ring; and $f$ induces a pairing preserving isomorphism between the cohomology ring of the perturbed quotient and the intersection cohomology (middle perversity) of the reduced space. While we do explain what $f$ looks like locally in this section, we defer showing that $f$ induces an isomorphism in cohomology to the last section.
The key idea {#the-key-idea .unnumbered}
------------
The key idea that makes this work is an observation due to Yi Hu (this observation was made in the context of algebraic actions on projective varieties [@Hu]): If $0$ is a singular value of an $S^1$ moment map $\Phi: M\to \R$ and $0$ lies in the interior of the image $\Phi (M)$ then for each component $Y$ of the fixed point set $M^{S^1}$ with $\Phi (Y) = 0$ there exists a regular value $\epsilon
\in \R$ of $\Phi$ and a neighborhood $U$ of $Y$ in $M$ so that there is a natural isomorphism $$IH_{\bm} ^* \left((\Phi
\inv (0) \cap U)/S^1\right) \simeq H^* \left((\Phi \inv (\epsilon)
\cap U)/S^1\right).$$
We use the notation of Lemma \[lemma\_model\], Remark \[split\] and Proposition \[prop\_simple\]. Fix a component $Y$ of the fixed point set. Consider the associated bundle $P\times _G (V^+\times V^-)$, together with the moment map $$\mu ([p, z^+, z^-]) = |z^+|^2 - |z^-|^2,$$ where $z^+ = \sum _{i=1}^{s}\vec{z}_i\in V^+$,$z^- = \sum
_{i=s+1}^{k}\vec{z}_i\in V^-$, $|z^+|^2 = \sum _{i=1}^{s}\kappa_i
|\vec{z}_i|^2$, and $|z^-|^2 = \sum _{i=s+1}^{k}\kappa_i
|\vec{z}_i|^2$.
For any $\epsilon > 0$, $\pm\epsilon$ are regular values of $\mu$ and $$\mu \inv (\pm \epsilon)/S^1 = P\times _G X^\pm$$ where $$X^+ \simeq S^+ \times _{S^1} V^-, \quad\text{a $V^-$ bundle over the
weighted projective space $S^+/S^1$}.$$ $$X^- \simeq V^+ \times _{S^1} S^-, \quad\text{a $V^+$ bundle over the
weighted projective space $S^-/S^1$}.$$ Here as in Proposition \[prop\_simple\], $S^\pm$ is the unit sphere in $V^\pm$, $S^\pm := \{ z\in V^\pm \mid |z^\pm|^2 = 1\}$, and the $S^1$ action on $V^+\oplus V^-$ is given by the weights $\kappa_1 ,
\ldots, \kappa_k$.
Note first that the only fixed point in $V^+\times V^-$ under the $S^1$ action is $(0,0)$, and that $\mu (0,0) = 0$. Therefore any $\alpha \not = 0$ is a regular value of $\mu$. Next assume that $Y$ is a point; in this case $P\times _G (V^+\times V^-)$ is simply $V^+\times
V^-$. Then $\mu \inv (\epsilon) = \left\{ (z^+, z^-)\mid |z^+|^2 -
|z^-|^2 = \epsilon \right\} = \left\{ (z^+, z^-)\mid |z^+|^2 = |z^-|^2
+ \epsilon \right\}$. If $\epsilon >0$, we have an $S^1$-equivariant diffeomorphism $S^+\times V^- \to \mu \inv (\epsilon)$ given by $(\zeta, w)\mapsto (\sqrt{\epsilon + |w|^2} \zeta, w)$. Therefore $\mu \inv (\epsilon)/S^1 = S^+ \times _{S^1} V^-$ for $\epsilon >0$. Since the norm $|(z^+, z^-)|^2 = |z^+|^2 + |z^-|^2$ is $G\times
S^1$-invariant by construction (see Lemma \[lemma\_model\] and Remark \[split\]) the claim follows.
Observe that $S^+\times _{S^1} S^-$ is the sphere bundle of the vector bundle $X^+ = S^+\times _{S^1} V^-$ over the weighted projective space $S^+/S^1$. Therefore, by collapsing the zero section of the bundle $X^+\to S^+/S^1$ to a point we obtain a map from $X^+$ to the cone $\overset \circ c (S^+ \times_{S^1} S^-)$, which is a diffeomorphism off the zero section onto the cone minus the vertex.
We will see later on that if $\dim V^+ \leq \dim V^-$, then $$IH_{\bm}^* (\mu \inv (0)/S^1) = IH_{\bm}^* (P\times _G \overset \circ
c (S^+ \times_{S^1} S^-)) \simeq H^* (P\times _G X^+) = H^* (\mu \inv
(\epsilon )/S^1)$$ for any $\epsilon >0$. Similarly, if $\dim V^+ \geq \dim V^-$ then $$IH_{\bm}^* (\mu \inv (0)/S^1) \simeq H^* (\mu \inv (\epsilon
)/S^1)\quad \text{for any }\quad \epsilon < 0.$$ In the algebraic category, these isomorphisms follow from the fact that the collapsing map is a [**small resolution**]{}, and a fact that small resolutions induce isomorphisms in cohomology. This isomorphism is also valid in the symplectic context, as we prove in the next section.
Note that $\dim V^+ \leq \dim V^-$ if and only if the index of $Y$ as a critical manifold of the Bott-Morse function $\Phi$ is at most $\frac{1}{2} (\dim M - \dim Y)$. Unfortunately we cannot expect such inequalities to hold globally, that is, if $0$ is a singular value of the moment map $\Phi: M\to \R$ we [**should not**]{} expect $IH_{\bm}^* (\Phi \inv (0)/S^1) = H^* (\Phi \inv (\epsilon
)/S^1)$ for some $\epsilon \not = 0$: it may well happen that at one component $Y$ we would need to shift the value of the moment map down and at another component to shift the value up in order to obtain a resolution of the singularities of the reduced space at zero.
The Construction
----------------
Let the circle $S^1$ act effectively on a compact connected symplectic manifold $M$ with a moment map $\Phi: M \to \R$ so that $0$ is in the interior of the image $\Phi (M)$. We will now construct a Morse-Bott function $\tilde\Phi: M \to \R$ and an $S^1$ equivariant map $f: \tPhi\inv (0)/{S^1} \to \Phi\inv (0)/{S^1}$ with the following properties.
- The critical points of $\tPhi$ are exactly the fixed points of $S^1$ on $M$.
- $0$ is a regular value of $\tPhi$.
- The map $f:\tPhi \inv (0)/S^1 \to \Phi \inv (0)/S^1$ induces an isomorphism in cohomology $IH^*_\bm(\Phi\inv(0)/S^1)
\cong H^*(\tPhi\inv(0)/S^1)$.
We call the subquotient $\tred := \tPhi \inv(0)/S^1$ the [**perturbed quotient**]{}.
The first two properties guarantee that $\tred$ is an orbifold, and that it is possible to compute the cohomology ring $H^*(\tred)$ in a fairly straightforward manner. This will be treated explicitly in the next subsection. As we mentioned earlier, the last property will not be proved in this section. However, we will prove Lemma \[lemma\_smallres\], which we will later see is sufficient to construct this isomorphism.
For each critical manifold $Y_i$ of $\Phi$ in $\Phi \inv (0)$, there is a neighborhood $U_i$ of $Y_i$ in $M$ which is equivariantly isomorphic to the model $$P_i \times_{G_i} (V_i^+\times V_i^-)$$ where the principal bundle $G_i \to P_i \to Y_i$ and the vector spaces $V_i^+$, $V_i^-$ are as in the preceding section. We may assume that the $U_i$’s for distinct critical manifolds do not intersect. There exists $\delta >0$ so that $0$ is the only critical value of $\Phi$ in $(-\delta, \delta)$ and $U_i$ is the image of the set $$P_i \times_{G_i} (\{ (z_i^+, z_i^-)\mid |z_i^+|^2 + |z_i^-|^2< 3\delta \}.$$
Therefore, we will simply give our construction on the vector space $V^+\times V^-$. As long as our definition of $\tPhi$ and $f$ are $G$-invariant, these construction can be naturally extended to the local model. Additionally, as long as $\tPhi = \Phi$ and $f$ is the identity outside the set $\{ (z_i^+, z_i^-)\mid |z_i^+|^2 + |z_i^-|^2< 3\delta \}$, they can be extended globally by taking $\tPhi = \Phi$ and $f = \id$ on $M \smallsetminus \cup U_i$.
Choose a smooth function $\rho: \R \to \R$ such that $\rho (t) = 1$ for all $t<\delta$, $\rho (t) = 0$ for all $t>2\delta$ and $\rho
'(t)\leq 0$ for all $t$. Let $C = \sup |\rho ' (t)|$, and choose $\epsilon \in \R$ so that $\epsilon \neq 0$, $|\epsilon | < C\inv$ and $|\epsilon | <\delta$. Moreover, choose $\epsilon$ so that $\epsilon > 0$ if and only if $\dim V^+ \leq \dim V^-$. We now define our new function $\tPhi$ : $$\tPhi (z^+, z^-) :=
\Phi (z^+, z^-) + \epsilon \rho (|(z^+, z^-)|^2) =
|z^+|^2 - |z^-|^2 + \epsilon \rho (|(z^+, z^-)|^2).$$
The norm $$|(z^+, z^-)|^2 = |z^+|^2 + |z^-|^2$$ is $G\times S^1$-invariant by construction (see Lemma \[lemma\_model\], Proposition \[prop\_simple\] and the subsequent discussion). Therefore the function $\rho (|(z^+,
z^-)|^2)$, and hence also the function $\tPhi$, is $G\times S^1$ invariant. Moreover, for $(z^+, z^-)$ with $|(z^+,z^-)|^2 >2\delta$, $ \tPhi(z^+, z^-)= |z^+|^2 - |z^-|^2 .$ Therefore $$\tPhi \inv (0) \cap \{|(z^+,z^-)|^2 > 2\delta \} = \Phi \inv (0)
\cap \{|(z^+,z^-)|^2 > 2\delta \}.$$ In contrast, for $(z^+, z^-)$ with $|(z^+,z^-)|^2 < \delta$, $ \tPhi(z^+, z^-)= |z^+|^2 - |z^-|^2 + \epsilon. $ Thus $(0,0)$ is a nondegenerate critical point, and $$\tPhi \inv (0) \cap \{|(z^+,z^-)|^2 < \delta \} = \Phi \inv (- \epsilon)
\cap \{|(z^+,z^-)|^2 < \delta\}.$$ (Note that $|\epsilon| < \delta$ guarantees that $\Phi \inv (-
\epsilon) \cap \{|(z^+,z^-)|^2 < \delta\} \not = \emptyset$.) Moreover, $(0,0)$ is the only critical point of $\tPhi$, because $$\begin{split}
d \tPhi & = d |z^+|^2 - d |z^-|^2 + \epsilon d\rho (|(z^+,
z^-)|^2) \\
& = \left(1 + \epsilon \rho' (|(z^+,z^-)|^2)\right)d |z^+|^2 -
\left(1 - \epsilon \rho' (|(z^+,z^-)|^2)\right)d |z^-|^2
\end{split}$$ and $|1 \pm \epsilon \rho'| \geq 1 - |\epsilon| (\sup |\rho ' (t)|) >
0$, since $|\epsilon| (\sup |\rho ' (t)|) < 1$ by the choice of $\epsilon$. It follows that $\tPhi$ is a Bott-Morse function, and that $0$ is a regular value of $\tPhi$ (since $\tPhi (0,0) = \epsilon
\not = 0$).
Let $X= X^r\sqcup \coprod Y_i$ be a simple stratified space. A [**resolution**]{} $h:\tilde X \to X$ is a continuous surjective map from a smooth orbifold $\tilde X $ such that $h\inv (X^r)$ is dense in $\tilde X$ and $h: f\inv (X^r)\to X^r$ is a diffeomorphism.
We will now construct a resolution $f : \tred \to \red$. We start by considering a $G\times S^1$-equivariant map $\psi : V^+\times V^- \to \Phi \inv (0)$ defined by $$\label{eq_fiber}
\begin{split}
\psi (z^+, z^-) &= ( \left(\frac{|z^-|^2}{|z^+|^2}\right)^{1/4} z^+,
\left(\frac{|z^+|^2}{|z^-|^2}\right)^{1/4} z^-) \quad\text{if} \quad
z^+, z^- \not = 0\\
\psi (0, z^-) &= \psi (z^+,0) = (0, 0).\\
\end{split}$$ We let $f: \tred \to \red$ be $G$-equivariant map induced by the restriction $\psi|_{\tPhi\inv(0)} : \tPhi\inv (0) \to \Phi \inv (0)$.
To prove that $f$ is a resolution, it is enough to show that $\psi |_{\tPhi\inv
(0) \smallsetminus \psi\inv (0,0)} : \tPhi\inv (0) \smallsetminus
\psi\inv (0,0) \to \Phi \inv (0) \smallsetminus \{(0,0)\}$ is a diffeomorphism. It follows from (\[eq\_fiber\]) that for $(0,0) \not
= (z^+, z^-) \in \Phi\inv (0)$, $$\label{eq_fiber_psi}
\psi \inv (z^+, z^-) = \{ (\lambda z^+, \lambda^{-1} z^-) \mid
\lambda >0 \}.$$ Consequently $\psi|_{\tPhi\inv (0) \smallsetminus \psi\inv (0,0)} :
\tPhi\inv (0) \smallsetminus \psi\inv (0,0) \to \Phi \inv (0)
\smallsetminus \{(0,0)\}$ is one-to-one and onto. Therefore it remains to prove that $d \psi |_{T (\tPhi\inv (0) \smallsetminus
\psi\inv (0,0)}$ is one-to-one, or, equivalently, that for any $(z^+,
z^-) \in \tPhi\inv (0) \smallsetminus \psi\inv (0,0)$ $$0 = \ker d\psi \cap T_{(z^+, z^-)} \tPhi\inv (0) = \ker d\psi \cap
\ker d\tPhi.$$ By (\[eq\_fiber\_psi\]), the kernel of $d\psi$ at $(z^+, z^-)$ is spanned by the vector $\left.\frac{d}{d\lambda}\right|_{\lambda = 1}
(\lambda z^+,\lambda^{-1} z^-)$. Thus it remains to show that for any $(z^+, z^-) \in \tPhi \inv (0) \smallsetminus \psi \inv (0,0)$ we have $$\left. \frac{d}{d\lambda}\right|_{\lambda = 1} \tPhi (\lambda
z^+,\lambda^{-1} z^-) \not = 0.$$ Now $$\begin{gathered}
\left. \frac{d}{d\lambda}\right|_{\lambda = 1} \left( |\lambda z^+|^2
- |\lambda \inv z^-|^2 + \epsilon \rho (|(\lambda z^+|^2+ |\lambda\inv
z^-|^2)\right) = \\
\left. \left( 2 \lambda |z^+|^2 + 2 \lambda ^{-3} |z^-|^2 + \epsilon
\rho' (|(\lambda z^+|^2+ |\lambda\inv z^-|^2)(2 \lambda |z^+|^2 - 2
\lambda ^{-3} |z^-|^2) \right) \right| _{\lambda =1} =
\\
2 \left(|z^+|^2 + |z^-|^2\right) \ + \
2 \epsilon \rho'(|z^+|^2 + |z^-|^2) (|z^+|^2 - |z^-|^2).
$$ For $(z^+,z^-) \in \tPhi\inv(0)$, we have $|z^+|^2 - |z^-|^2 = -\epsilon \rho(|(z^+,z^-)|) $. Since $ \rho'(t) \leq 0$ for all $t$, $-\epsilon ^2 \rho (t) \rho'
(t)\geq 0$ for all $t$. Moreover, $(z^+,z^-) \neq (0,0)$. Hence $$\begin{gathered}
\left. \frac{d}{d\lambda}\right|_{\lambda = 1} \tPhi (\lambda
z^+,\lambda^{-1} z^-) = \\
2 \left(|z^+|^2 + |z^-|^2\right)
- 2 \epsilon \rho'(|z^+|^2 + |z^-|^2) \epsilon \rho (|z^+|^2 + |z^-|^2)
\ \ \geq \ \
2 \left(|z^+|^2 + |z^-|^2 \right)
> 0.
$$ Thus, we have proved the following.
\[lemma\_smallres\] Let a circle $S^1$ act on a symplectic manifold $M$ with a moment map $\Phi: M \to \R$ so that $0$ is in the interior of the image $\Phi(M)$. Let $\tPhi: M \to \R$ and $f :\tred = \tPhi \inv (0)/S^1\to \red = \Phi \inv (0)/S^1$ be constructed as above.
Then $f$ is a resolution. Moreover, for each singular stratum $Y$ of $\red$ there exist:
- an even dimensional orbifold vector bundle $E\to N$ over a compact orbifold $N$ with a sphere bundle $L\to N$ such that $$\dim N \leq \frac{1}{2} \dim E - 1,$$
- a principal $G$ bundle $P \to Y$,
- an action of $G$ on $E$ by vector bundle maps
- an isomorphism from a neighborhood of the vertex section of the cone bundle $P\times _G \conec (L) \to Y$ to a neighborhood $U$ of $Y$ in $\red$,
- an isomorphism from a neighborhood of the zero section of the vector bundle $P\times _G E \to P\times _G N$ to the neighborhood $f\inv (U)$ of $f\inv (Y)$ in $\tred$ such that the diagram $$\begin{CD}
P\times _G E @<<< f\inv (U) @>>> \tred\\
@V{h}VV @VV{f}V @VV{f}V\\
P\times _G \conec (L) @<<< U @>>> \red
\end{CD}$$ commutes. Here the map $h$ is induced by the natural blow-down map $E\to \conec (L)$ taking the zero section to the vertex.
Notice that there is no reason to suspect that the perturbed quotient possesses a symplectic structure.
Morally, the above Lemma should be read as a claim that $f:\tred \to \red$ is a small resolution (cf. §6.2 of [@GM2]).
Computation of the cohomology of the perturbed quotient
-------------------------------------------------------
We can now compute the cohomology of the perturbed quotient by adapting techniques used to compute the cohomology ring of a symplectic quotient at a regular value.
We begin by reviewing those techniques. Let a circle $S^1$ act on a compact connected symplectic manifold $M$ with a moment map $\Phi$. Assume that $0$ is a regular value. There is a natural restriction from $H_{S^1}^*(M; \R)$, the equivariant cohomology of $M$, to $H_{S^1}^*(\Phi\inv(0); \R))$, the equivariant cohomology of the preimage of $0$. Since $0$ is a regular value, the stabilizer of every point in $\Phi\inv(0)$ is discrete. Therefore, there is a natural isomorphism from $H^*_{S^1}(\Phi\inv(0),\R)$ to the $H^*(\red)$, the ordinary cohomology of the symplectic quotient $\red
: \Phi\inv(0)/S^1$. The composition of these two maps gives a natural map, $\kappa : H_{S^1}^*(M) \to H^*(\red)$, called the [**Kirwan map**]{}.
(Kirwan, \[K\]) Let a circle $S^1$ act on a compact connected symplectic manifold $M$ with a moment map $\Phi$ so that $0$ is a regular value. The Kirwan map $\kappa : H_{S^1}^*(M; \R) \to H^*(\red ; \R)$ is surjective.
Thus, assuming we know the ring structure on $M$, the ring structure on $\red$ can be computed from the kernel of $\kappa$. By Poincare duality, to compute the kernel it is enough to compute the integral of $\kappa(\alpha)$ over the reduced spaces for every equivariant cohomology class $\alpha$ on $M$. We take one formula for this integral from Kalkman [@Ka]; slightly different but morally equivalant formulas were proved by Wu [@Wu] and a more general version by Jeffrey-Kirwan [@JK]. See also [@Ka-Gu]. All of these results were inspired by a paper of Witten [@Wi].
Let a circle $S^1$ act on a compact connected symplectic manifold $M$ with a moment map $\Phi$ so that $0$ is a regular value. Let $\cF^+$ denote the set of components $F$ of the fixed point set $M^{S^1}$ such that $\phi(F) > 0$.
Given an equivariant cohomology class $\alpha \in H^*_{S^1}(M)$, the integral of $\kappa(\alpha)$ over $\red$ is given by the formula $$\int_{\red} \kappa(\alpha) = \Res _0 \sum_{F \in \cF^+} \int_F \frac{
i_F^*(\alpha) }{e_F} ,$$ where $e_F$ denotes the equivariant Euler class of the normal bundle of $F$.
The right hand side of this formula requires some explanation. The map $i_F^*$ is simply the restriction to $F$. The equivariant cohomology ring $H^*_{S^1}(F)$, is naturally isomorphic to $H^*(F)[t]$. The equivariant Euler class $e_F$ is invertable in the localized ring $H^*(F)(t)$; thus, $\frac{i_F^*(\alpha)}{e_F}$ is an element of this ring. The integral $\int_F : H^*(F)(t) \to \R(t)$ acts by integrating each coefficient in the series. Finally, $\Res _0$ denotes the operator which returns the coefficient of $t\inv$.
An alternative way of computing the kernel is given by a theorem of Tolman and Weitsman.
Let the circle $S^1$ act on a compact connected symplectic manifold $M$ with a moment map $\Phi: M \to \R$ so that $0$ is a regular value.
Let $\cF^+$ denote the set of components $F$ of the fixed point set such that $\Phi(F) > 0$; let $\cF^-$ denote the set of components $F$ of the fixed point set such that $\Phi(F) < 0$. Define $$K_\pm := \{ \alpha \in H_{S^1}^*(M) \mid \alpha|_{F} = 0 \ \ \forall \ \
F \in {\cF}^\pm\}.$$
The kernel of the Kirwan map is $K_+ \oplus K_-$.
In our case, closely analogous propositions are true.
\[compute\] Let the circle $S^1$ act on a compact connected symplectic manifold $2n$ dimensional manifold $M$ with a moment map $\Phi: M \to
\R$. Assume that $0$ is in the interior of $\Phi(M)$. Let $\tred$ denote the perturbed quotient.
Then there is a surjective ring homomorphism $\kappa: H^*_{S^1}(M) \to H^*(\tred)$. Moreover,
- The kernel of $\kappa$ is $K_+ \oplus K_-$, where $K_\pm := \{ \alpha \in H_{S^1}^*(M) \mid \alpha|_{F} = 0 \ \forall \ \
F \in {\cF}^\pm\}. $
- Given an equivariant cohomology class $\alpha \in H^*_{S^1}(M)$, the integral of $\kappa(\alpha)$ over $\red$ is given by the formula $$\int_{\red} \kappa(\alpha) =
\Res_0 \sum_{F \in \cF^+} \int_F \frac{ i_F^*(\alpha) }{e_F} ,$$ where $e_F$ denotes the equivariant Euler class of the normal bundle of $F$.
Here, et $\cF^+$ denotes the set of components $F$ of the fixed point set $M^{S^1}$ such that either
1. $\Phi(F) > 0$ or
2. $\Phi(F) = 0 $ and $ 2 \index F + \dim F \leq \dim M$.
Additionally, $\cF^- $ denotes all other components of the fixed point set, $\cF^- = \cF \setminus \cF^+$.
The reason that this proposition is true is that the perturbed quotient $\tred$ is defined in a way very similar to the ordinary reduced space.
Thus, for example, Kalkman’s formula follows immediately from the fact that there exists a smooth invariant function $\Phi : M \to \R$ so that $0$ is regular and $\tred$ is defined to be $\tPhi\inv(0)/S^1$. His proof relies only on the fact that $\tPhi\inv([0,\infty))$ is a manifold with boundary. Thus, one only need note that $\cF^+$ does indeed correspond to the components $F$ of the fixed point set such that $\tPhi(F) > 0$.
To see that Kirwan’s surjectivity holds, and the Tolman-Weitsman formula for the kernel of $\kappa$, we must also use the fact that $\tPhi$ is a Morse-Bott function and that its critical points are exactly the fixed points of the action. This is sufficient to prove both theorems, as was pointed out in [@TW].
The isomorphism
===============
The goal of this section is to prove that the intersection cohomology of the symplectic quotient by a Hamiltonian circle actoin is isomorphic to the (ordinary) cohomology of the perturbed quotient. Because we have already computed the cohomology of the perturbed quotient in Proposition \[compute\], this will allow to obtain the description of the intersection cohomology of the symplectic quotient, and thus prove our main theorems. More precisely, we will be done once we have proved the following.
\[theorem-iso\] Let the circle $S^1$ act on a compact connected symplectic manifold $M$ with moment map $\Phi: M \to \R$ so that $0$ is in the interior of $\Phi(M)$. Let $\red := \Phi\inv(0)/S^1$ denote the reduced space and let $\tred$ denote the perturbed reduced space. There is a natural pairing preserving isomorphism between the intersection cohomology of the symplectic quotient $\red$ and the cohomology if the perturbed quotient $\tred$.
More precisely there exists an isomorphism $\psi: H^* (\tred) \to IH^*
(\red)$ of graded vector spaces such for any $\alpha \in H^p (\tred)$ and $\beta \in H^q (\tred)$ with $p+q = \dim \tred$ we have $$\int _{\tred} \alpha \cup \beta = \int _{\red} \langle \psi (\alpha),
\psi (\beta) \rangle .$$
Instead of trying to construct the isomorphism between the intersection cohomology of the reduced space and the (ordinary) cohomology of the perturbed quotient directly, we will introduce a new complex $A^*_\bm(\tred) = A^*_\bm(f:\tred \to \red) $ and show that the cohomology of $A^*_\bm (\tred \to \red) $ is naturally isomorphic to both $H^* (\tred )$ and $IH_\bm^* (\red)$.
Let $f: \tX \to X$ be a resolution of a simple stratified space. Let $X^r$ be the top stratum of $X$, $\tX^r$ be its preimage $f\inv(X^r$), and let $\iota: \tilde{X}^r \hookrightarrow \tilde{X}$ denote the inclusion. By construction, there are maps of complexes $f^* :
I\Omega _{\bm}^* (X) \to \Omega^* (\tilde{X}^r )$ and $\iota ^* :
\Omega ^*(\tilde{X}) \to \Omega^* (\tilde{X}^r )$. We define the complex of [**resolution forms**]{} $$\label{the_complex}
A_\bm^\bullet (\tX) = A_\bm^\bullet (f: \tilde{X} \to X) := f^*
\left( I\Omega _{\bm}^\bullet (X)\right ) \cap \iota^* \left(\Omega
^\bullet (\tilde{X}) \right)$$
Note that $f^*$ and $\iota^*$ are both injective. Therefore we may think of a resolution form as an intersection form on $X^r$ which extends to a globally defined form on $\tX$. This gives us the inclusions of complexes $A_\bm^\bullet (\tX) \to I\Omega _{\bm} ^\bullet (X)$ and $ A_\bm^\bullet (\tX) \to \Omega^\bullet(\tilde{X})$, which induce maps in cohomology $j:H^* (A_\bm^\bullet (\tX)) \to IH^*_\bm (X)$ and $i:H^* (A_\bm^\bullet (\tX)) \to H^* (\tX)$. Note that the graded vector space $H^*(A_\bm (\tX)$ has a pairing defined by taking the exterior product of the representatives of the classes and then integrating the product over $\tX$. Clearly the maps $i$ and $j$ are pairing preserving. Thus, to prove Theorem \[theorem-iso\] it is enough to show that the maps $i$ and $j$ are isomorphisms.
Local issues
------------
We start with a simple calculation.
\[lemma\_calculation\] Let $E\to N$ be an even dimensional orbifold vector bundle over an orbifold $N$, and let $L$ denote the sphere bundle of $E$. Then the obvious blow-down map $f: E \to \conec (L)$ is a resolution. If $\dim
N \leq \frac{1}{2} E -1$ then the maps $$A_\bm^\bullet (E) \hookrightarrow \Omega^\bullet (E)$$ and $$A_\bm^\bullet (E) \hookrightarrow I\Omega^\bullet_\bm (\conec (L))$$ induce isomorphisms in cohomology.
To prove the Lemma we will need the following technical observation.
\[prop\_cyl\] Let $L$ be an orbifold. Let $\alpha$ be a closed $k$ form on the cylinder $L\times (0,\infty)$ which vanishes on $L\times (0, a)$ for some $a$. Then $\alpha = d\beta$ for some $k-1$ form $\beta$ which also vanishes on $L\times (0, a)$.
Consider first the case where $L$ is a point and $\alpha = f(r) \,dr$ is a 1 form on $(0, \infty)$. Then $\beta = \int _0 ^r f (s)\, ds$.
In general, if $L$ is not a point, the $k$-form $\alpha$ has to be of the form $f(r)\wedge dr$ where $f(r)$ is in $\Omega^{k-1} (L)$ for each $r\in (0,\infty)$. Let $\beta = \int _0 ^r f(s) \, ds$.
Note first that the middle perversity of the vertex $*$ of the cone $\conec (L)$ is $\bm (*) = \frac{1}{2} \dim E -1$, since $E$ is even dimensional.
Recall that $\conec (L)$ is a stratified space with two strata: the vertex $*$ and the complement $ L\times (0, \infty)\simeq
E\smallsetminus N$. We may choose the tubular neighborhood $T$ to be any neighborhood of the vertex $*$ of the form $L\times (0,a)/\sim$ for some $a$.
It follows from the definitions that $$I\Omega^q_\bm (\conec (L)) =
\begin{cases}
\Omega ^q (E\smallsetminus N)&
\text{for $q< \bm (*)$}\\
\left\{ \alpha \in \Omega ^q (E\smallsetminus N) \mid d\alpha |_T =
0\right\} & \text{ for $q = \bm (*)$}\\
\left\{ \alpha \in \Omega ^q (E\smallsetminus N) \mid\alpha |_T = 0
\text{ and } d\alpha |_T = 0\right\} & \text{ for $q > \bm (*)$}
\end{cases}$$ Consequently $$A^q_\bm (E) =
\begin{cases}
\Omega^q (E)&
\text{for $q< \bm (*)$}\\
\left\{ \alpha \in \Omega ^q (E) \mid d\alpha |_T =
0\right\} & \text{ for $q = \bm (*)$}\\
\left\{ \alpha \in \Omega ^q (E) \mid\alpha |_T = 0
\text{ and } d\alpha |_T = 0\right\} & \text{ for $q > \bm (*)$}
\end{cases}$$ The map $A^q_\bm (E) \to I\Omega_\bm^q (\conec (L))$ is induced by the restriction from $\Omega ^* (E) $ to $\Omega ^* (E\smallsetminus N)$.
It follows from Lemma \[prop\_cyl\] that for $q\leq \bm (*)$ the map $H^q (A^\bullet _\bm (E)) \to H^q (E)$ is an isomorphism and that $H^q (A^\bullet _\bm (E)) =0$ for $q> \bm (*)$. Since $H^* (E) =
H^* (N)$ and since $\bm (*) \geq \dim N$ by assumption, the map $H^q
(A^\bullet _\bm (E)) \to H^q (E)$ is an isomorphism for all $q$.
Similarly, $$IH^q_\bm (\conec (L)) =
\begin{cases}
H^q (E\smallsetminus N)&
\text{for $q\leq \bm (*)$}\\
0 & \text{ for $q >\bm (*)$}
\end{cases}.$$ Consider the Gysin sequence $$\cdots \to H^{q- \lambda} (E) \to H^q (E) \to H^q (E\smallsetminus N)
\to H^{q -\lambda +1} (E) \to \cdots$$ where $\lambda = \dim E - \dim N$. Since for $q -\lambda + 1 \leq -1$ (i.e., for $q \leq \lambda -2$) we have $ H^{q -\lambda +1} (E) = 0 =
H^{q -\lambda } (E) $, the pull-back map $H^q (E) \to H^q
(E\smallsetminus N)$ is an isomorphism. In particular the pull-back is an isomorphism for for $q \leq \bm (*)= \frac{1}{2} \dim E - 1 =
\dim E - (\frac{1}{2} \dim E - 1) - 2 \leq \dim E - \dim N - 2 =
\lambda - 2$.
\[local-iso\] Let the circle $S^1$ act on a compact connected symplectic manifold $M$ with moment map $\Phi: M \to \R$. Assume that $0$ is in the interior of the image $\Phi(M)$. Let $\red := \Phi\inv(0)/S^1$ denote the reduced space and let $f: \tred \to \red$ denote its resolution by the perturbed quotient.
There exists a cover $\cU$ of $\red$ such that the natural inclusions $A_\bm^* (f\inv(U_{\alpha_1}) \cap \cdots \cap f\inv(U_{\alpha_k}) )
\to I\Omega ^*_{\bm} (U_{\alpha_1} \cap \cdots U_{\alpha_k})$ and $
A_\bm^* (f\inv (U_{\alpha_1}) \cap \cdots \cap f\inv(U_{\alpha_k}))
\to \Omega^*( f\inv(U_{\alpha_1}) \cap \cdots \cap f\inv(U_{\alpha
_k}))$ induce isomorphisms in cohomology for all $k$-tuples $\{ U_{\alpha_1}, \ldots, U_{\alpha_k} \}$ of elements of $\cU$.
We have seen in Proposition \[prop\_simple\] that $\red = \red^r
\coprod Y_i$ where $Y_i$ are compact manifolds. Further, for each singular stratum $Y$ there exists a tubular neighborhood $\tT$ of $Y$, the fiber bundle $\conec (L)\to \tT\stackrel{\pi}{\to} Y$, and the map $r:\tT\to [0,1)$ (c.f. Remark \[rmrk\_t\]). Recall also that by Lemma \[lemma\_smallres\] we may assume that $\tT = P\times
_G \conec (L)$, that $f\inv (\tT) = P\times _G E$ and that $f: P\times
_G E \to P\times _G \conec (L)$ is induced by the obvious blow-down map $E\to \conec (L)$.
We take $U_0 = \red \smallsetminus \cup r_i \inv ([0, 1/2]) = \red
\smallsetminus \overline{T}_i$. Since each singular stratum $Y$ is a compact manifold, it possesses a finite good cover $\{V_\alpha \}$. Moreover we may assume that $\pi \inv (V_\alpha ) \simeq \conec
(L)\times V_\alpha $. We take $U_\alpha := \pi \inv (V_\alpha )
\cap \tT \subset \tT$. This give us a cover $\cU$ of $\red$.
Note that by construction for a $k$-tuple $\{ U_{\alpha_1}, \ldots,
U_{\alpha_k} \}$ of elements of $\cU$ we either have that $U_{\alpha_1} \cap \cdots \cap U_{\alpha_k}$ does not intersect any singular stratum $Y$ (in which case $f\inv (U_{\alpha_1}) \cap \cdots
\cap f\inv(U_{\alpha_k})$ and $U_{\alpha_1} \cap \cdots \cap U_{\alpha_k}$ are diffeomorphic) or there is a unique stratum $Y$ such that $Y \cap
U_{\alpha_1} \cap \cdots \cap U_{\alpha_k} \not = \emptyset$. In the latter case $U_{\alpha_1} \cap \cdots \cap U_{\alpha_k} \simeq D \times
\conec (L)$, $f\inv (U_{\alpha_1}) \cap \cdots \cap f\inv(U_{\alpha_k})
\simeq D\times E$ and $f: f\inv (U_{\alpha_1}) \cap \cdots \cap
f\inv(U_{\alpha_k}) \to U_{\alpha_1} \cap \cdots U_{\alpha_k}$ is equivalent to the map $ h \times id: E\times D \to \conec (L)\times D$, where $D$ is a disk in $Y$ and $h: E \to \conec (L)$ is the resolution.
Given a disk $D$ and a set $X$ we have an inclusion $\iota : X
\hookrightarrow X \times D$, $\iota (x) = (x, 0)$. Clearly the diagram $$\begin{CD}
E @>{\iota}>> E \times D\\
@V{h}VV @VV{h\times id}V\\
\conec (L) @>{\iota}>> \conec (L) \times D
\end{CD}$$ commutes. Since $h: E\to \conec (L)$ and $h\times id : E \times D \to
\conec (L) \times D$ are resolutions, we have a commutative diagram of complexes $$\label{eq_diagram}
\begin{CD}
\Omega^* (E) @<{\iota^*}<<\Omega^* (E \times D)
@= \Omega^* (f\inv (U_{\alpha_1}) \cap \cdots \cap f\inv(U_{\alpha_k}))\\
@AAA @AAA\\
A^*_{\bm}(E) @<{\iota^*}<< A^*_{\bm}(E \times D)
@= A_\bm ^* (f\inv (U_{\alpha_1}) \cap \cdots \cap f\inv(U_{\alpha_k}))\\
@VVV @VVV\\
I\Omega_{\bm}^*(\conec (L)) @<{\iota^*}<< I\Omega_{\bm}^*(\conec
(L)\times D)
@= I\Omega^*_\bm (U_{\alpha_1} \cap \cdots \cap U_{\alpha_k}).
\end{CD}$$ Since the disk $D$ is contractible, the horizontal maps induce isomorphisms in cohomology. Since the left vertical maps induced isomorphisms in cohomology by Lemma \[lemma\_calculation\], the right vertical maps induce isomorphisms as well.
Let the circle $S^1$ act on a compact connected symplectic manifold $M$ with moment map $\Phi: M \to \R$. Assume that $0$ is in the interior of the image $\Phi(M)$. Let $\red := \Phi\inv(0)/S^1$ denote the reduced space and let $f: \tred \to \red$ denote its resolution by the perturbed quotient.
The inclusions $A_\bm^* (\tred) \to I\Omega _{\bm}^* (\red)$ and $
A_\bm^* (\tred) \to \Omega^*(\tred)$ induce isomorphisms in cohomology.
The proof is now a standard spectral sequence argument.
Let $\cU = \{U_\alpha \}$ be the cover of $\red$ constructed in the proof of Proposition \[local-iso\]. We now construct a continuous partition of unity $\rho_\alpha $ subordinate to the cover $\cU$ with the properties that
- the functions $\rho_\alpha $ restrict to smooth functions on $\red^r$,
- the functions $\rho_\alpha $ are constant along the fibers of $\pi :T\to Y$ for all the singular strata $Y$.
These properties ensure that
- $\{f^*\rho_\alpha \}$ is a partition of unity on $\tred$ subordinate to the cover $\{f\inv (U_\alpha)\}$ of $\tred$, and that
- for any intersection form $\gamma \in I\Omega _\bm^* (\red)$ or resolution form $\delta \in A_\bm^*$, the products $\rho _\alpha \gamma$ and $f^*\rho_\alpha \delta$ are also in $I\Omega _\bm^*(\red)$ and $A^*_\bm$, respectively.
We first consider the set $U_0$ which is entirely contained in the smooth part $\red^r$ of the quotient. We choose $\trho _0$ to be a smooth nonnegative function on $\red^r$ supported in $U_0$ with $\trho
_0 = 1$ in $\red \smallsetminus \cup r_i \inv ([0, 3/4))$.
By construction a set $U_\alpha$ with $U_\alpha \cap Y \not = \emptyset$ is of the form $\pi \inv (V_\alpha ) \cap \tT$ where $\{V_\alpha \}$ is a good cover of $Y$. We can choose a smooth partition of unity $\{\tau
_\alpha \}$ on $Y$ subordinate to $\{V_\alpha \}$ and also a nonnegative smooth function $\sigma$ on $\red^r$ supported in $ \tT$ with $\sigma$ identically 1 on the $\overline{T} \cap \red^r = r\inv
([0, 1/2]) \cap \red ^r$. Let $\trho _\alpha := \sigma (\pi^* \tau
_\alpha |_{\red})$; it extends to a continuous function on $\red$. The functions $\rho _\alpha := \frac{\trho _\alpha }{\sum_{\beta} \trho
_\beta }$ form the desired partition of unity.
Next we define three double complexes whose ${i,j}$’th terms are given as follows for $j \geq 0$: $$A^{i,j}_\bm(\cU) :=
\oplus A^i_\bm(f\inv(U_{\alpha _0}) \cap \cdots \cap f\inv(U_{\alpha _j}))$$ $$I\Omega^{i,j}_\bm(\cU) := \oplus I\Omega_\bm^i(U_{\alpha _0}
\cap \cdots \cap U_{\alpha _j}), \quad \text{ and}$$ $$\tilde{\Omega}^{i,j}(\cU) := \oplus \Omega^i(f\inv(U_{\alpha _0})
\cap \cdots \cap f\inv(U_{\alpha _j})),$$ where the sums are taken over all $j+1$-tuples $\{\alpha _0,
\ldots \alpha _j\}$. For all three complexes the differentials are given by the de Rham and Čech differentials.
First, in order to show that the cohomology of the double complexes are the intersection cohomology of $\red$, the cohomology of the complex $A^i_\bm(\tred)$ and the cohomology of $\tred$ respectively, we will consider the spectral sequences associated to the filtration by $i$. Since we constructed a nice partition of unity subordinate to the locally finite cover $\cU$, for any fixed $i$ the Čech cohomology of the sheaf $I\Omega^i_\bm$ is trivial for $j > 0$, and for $j = 0$ consists of the global forms $I\Omega^i_\bm(\red)$. Thus, the spectral sequence converges at the $E_2$ term. Moreover, $E_2^{i,j} = 0$ for $j > 0$, and $E_2^{i,0} = IH_\bm^i(\tred)$ for all $i$. Thus, the cohomology of the double complex $I\Omega_\bm^{i,j}(\cU)$ is the intersection cohomology $IH_\bm^i(\red)$. Virtually identical arguments show that the cohomology of the double complexes $A^{i,j}_\bm(\cU)$ and $\tilde{\Omega}^{i,j}(\cU)$ are the cohomology of the complexes $A^i_\bm(\tred)$ and $\Omega^i(\tred)$, respectively.
Next, in order to show that inclusions induce isomorphism from $H^*(A_\bm(\tred))$ to $IH^*_\bm(\red)$ and from $H^*(A_\bm(\tred))$ to $H^* (\tred)$ respectively, we will consider the spectral sequences associated with the filtration by $j$. The double complex $
A^{i,j}_\bm(\cU) := \oplus A^i _\bm(f\inv(U_{\alpha _1}) \cap \cdots
\cap f\inv(U_{\alpha _j}))$ includes naturally into the double complex $ I\Omega^{i,j}_\bm(\cU) := \oplus I\Omega_\bm^i(U_{\alpha _1} \cap
\cdots \cap U_{\alpha j})$. By Proposition \[local-iso\] this inclusion induces an isomorphism on the $E_1$ terms of these spectral sequences. This implies that the inclusion induces an isomorphism on every $E_k$. Hence, by the proceeding paragraph, inclusion induces an isomorphism from $H^*(A_\bm(\tred))$ to $IH^*_\bm(\red)$. An essentially identical argument shows that the inclusion $A^*_\bm(\tred) \to \Omega^*(\tred)$ induces an isomorphism in cohomology.
This completes the proof of Theorem \[theorem-iso\]. By combining Theorem \[theorem-iso\] with Proposition \[compute\] we now obtain the main result of the paper: Theorem 1 and Theorem $1'$.
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[^1]: It is claimed in [@Hu] that the resolutions are global, which need not be the case; see [@Hu2],
|
---
abstract: 'Cross sections for inclusive neutrino scattering off deuteron induced by neutral and charge-changing weak currents are calculated from threshold up to 150 MeV energies in a chiral effective field theory including high orders in the power counting. Contributions beyond leading order (LO) in the weak current are found to be small, and increase the cross sections obtained with the LO transition operators by a couple of percent over the whole energy range (0–150) MeV. The cutoff dependence is negligible, and the predicted cross sections are within $\sim 2$% of, albeit consistently larger than, corresponding predictions obtained in conventional meson-exchange frameworks.'
author:
- 'A. Baroni$^{\,{\rm a}}$ and R. Schiavilla$^{\,{\rm a,b}}$'
title: Inclusive neutrino scattering off deuteron at low energies in chiral effective field theory
---
Introduction {#sec:xsect}
============
A number of studies of neutrino-deuteron scattering were carried out in the past several decades, and work done up to the mid 1990’s is reviewed in Ref. [@Kubodera94]. In the early 2000’s, these efforts culminated in a set of predictions [@Nakamura01; @Nakamura02] for neutrino-deuteron cross sections induced by both neutral and charge-changing weak currents and incoming neutrino energies up to 150 MeV. The calculations were based on the conventional meson-exchange framework, and used last-generation realistic potentials available at the time and a realistic model for the nuclear weak currents, which included one- and two-body terms. The vector part of these currents was shown to provide an excellent description of the $np$ radiative capture cross section for neutron energies up to 100 MeV [@Nakamura01], while the axial part was constrained to reproduce the Gamow-Teller matrix element contributing to tritium $\beta$-decay [@Nakamura02]. The Nakamura [*et al.*]{} studies played an important role in the analysis and interpretation of the Sudbury Neutrino Observatory (SNO) experiments [@Ahmad02], which have established solar neutrino oscillations and the validity of the standard model for the generation of energy and neutrinos in the sun [@Bahcall04].
Concurrent with those studies was a next-to-next-to-leading order calculation of neutrino-deuteron cross sections at low energies ($\lesssim 20$ MeV) in an effective field theory in which pion degrees of freedom are integrated out and which is consequently parametrized in terms of contact terms [@Butler2001]. In the strong-interaction sector, the low-energy constants (LECs) multiplying these contact terms were fixed by fitting the effective range expansions in the $^1$S$_0$ and $^3$S$_1$ two-nucleon channels (which dominate the low-energy cross sections). The weak current included one-body terms with couplings (nucleon magnetic moments and axial coupling constant) taken from experiment as well as two-body terms. In the vector sector, the two LECs associated with these two-body terms were determined by reproducing the radiative capture rate of neutrons on protons at thermal energies and the deuteron magnetic moment. In the axial sector the two-body terms were characterized by a single LEC (labeled $L_{1,A}$), which however remained undetermined. Nevertheless, by fitting the results of Ref. [@Nakamura02], Butler [*et al.*]{} [@Butler2001] were able to show that the resulting value for $L_{1,A}$ was natural, and that the calculated cross sections reproduced well the energy dependence of those obtained by Nakamura [*et al.*]{}.
The energy range of the Nakamura [*et al.*]{} studies was extended up to 1 GeV in the more recent calculations by Shen [*et al.*]{} [@Shen2012]. These calculations too were based on the conventional framework, but included refinements in the modeling of the weak currents. However, they turned out to have only a minor impact on the predicted cross sections [@Shen2012]. The results have confirmed those of Nakamura [*et al.*]{} in the energy range up to 150 MeV, and have provided important benchmarks for the studies of the weak inclusive response in light nuclei, including $^{12}$C, with the Green’s function Monte Carlo method that have followed since [@Lovato2013; @Lovato2014; @Lovato2015; @Lovato2016]. They have also been useful in a recent analysis of the world data on neutrino-deuteron scattering aimed at constraining the isovector axial form factor of the nucleon [@Meyer2016], by supplying reliable estimates for the size of nuclear corrections.
The present study differs from all previous ones in one essential aspect: it is fully based on a chiral-effective-field-theory ($\chi$EFT) formulation of the nuclear potential [@Entem2003; @Machleidt2011] and weak currents [@Pastore2009; @Pastore2011; @Piarulli2013; @Baroni2016; @Baroni2016a] at high orders in the power counting. The potential and currents contain intermediate- and long-range parts mediated by one- and two-pion (and selected multi-pion) exchanges, and a short-range part parametrized in terms of contact interactions. The latter are proportional to LECs, which, in the case of the potential, have been constrained by fitting the nucleon-nucleon scattering database in the energy range extending up to the pion-production threshold [@Entem2003; @Machleidt2011] and, in the case of the current, by reproducing a number of low-energy electro-weak observables in the $A\,$=$\, 2$ and 3 nuclei [@Piarulli2013; @Baroni2016a] (specifically, the isoscalar and isovector magnetic moments of the deuteron and trinucleons, and the tritium Gamow-Teller matrix element).
The importance that accurate predictions for cross sections of neutrino-induced deuteron breakup into proton-proton and proton-neutron pairs have in the analysis of the SNO experiments, has prompted us to re-examine these processes in the context of $\chi$EFT. Because of its direct connection to the symmetries of quantum chromodynamics, this framework affords a more fundamental approach to low-energy nuclear dynamics and electro-weak interactions than the meson-exchange phenomenology adopted in the Nakamura [*et al.*]{} [@Nakamura01; @Nakamura02] and Shen [*et al.*]{} [@Shen2012] calculations. The remainder of this paper is organized as follows. In Secs. \[sec:form\] and \[sec:cnts\] we provide a succinct summary of the theoretical framework, including the cross section formalism and $\chi$EFT modeling of the nuclear weak currents, while in Sec. \[sec:res\] we present results for the deuteron disintegration cross sections by neutral and charge-changing weak currents. A summary and concluding remarks are given in Sec. \[sec:concl\].
Neutrino inclusive cross section {#sec:form}
================================
The differential cross section for neutrino ($\nu$) and antineutrino ($\overline{\nu}$) inclusive scattering off a deuteron, specifically the processes $^2$H($\nu_l,\nu_l$)$pn$ and $^2$H($\overline{\nu}_l,\overline{\nu}_l$)$pn$ induced by neutral weak currents (NC) and denoted respectively as $\nu_l$-NC and $\overline{\nu}_l$-NC, and the processes $^2$H($\nu_e,e^-$)$pp$ and $^2$H($\overline{\nu}_e,e^+$)$nn$ induced by charge-changing weak currents (CC) and denoted respectively as $\nu_l$-CC and $\overline{\nu}_l$-CC, can be expressed as [@Shen2012] $$\left(\frac{ {\rm d}\sigma}{ {\rm d}\epsilon^\prime {\rm d}\Omega}\right)_{\nu/\overline{\nu}}=
\frac{G^2}{8\,\pi^2}\, \frac{k^\prime}{ \epsilon} \,F(Z,k^\prime)\, \Bigg[ v_{00}\,
R_{00} +v_{zz} \, R_{zz} -v_{0z}\, R_{0z} +
v_{xx}\, R_{xx} \mp v_{xy} \, R_{xy}
\Bigg] \ ,
\label{eq:xswa}$$ where $G$=$G_F$ for the NC processes and $G$=$G_F \, {\rm cos}\, \theta_C$ for the CC processes, and the $-$ ($+$) sign in the last term is relative to the $\nu$ ($\overline{\nu}$) initiated reactions. Following Ref. [@Nakamura02], we adopt the value $G_F=1.1803\times 10^{-5}$ GeV$^{-2}$ as obtained from an analysis of super-allowed $0^+ \rightarrow 0^+$ $\beta$-decays [@Towner99]—this value includes radiative corrections—while ${\rm cos}\, \theta_C$ is taken as 0.97425 from Ref. [@PDG]. The initial neutrino four-momentum is $k^\mu=(\epsilon, {\bf k})$, the final lepton four momentum is $k^{\mu \,\prime}=(\epsilon^\prime,{\bf k}^\prime)$, and the lepton scattering angle is denoted by $\theta$. We have also defined the lepton energy and momentum transfers as $\omega=\epsilon-\epsilon^\prime$ and ${\bf q}={\bf k}-{\bf k}^\prime$, respectively, and the squared four-momentum transfer as $Q^2=q^2-\omega^2 > 0$. The Fermi function $F(Z,k^\prime)$ with $Z=2$ accounts for the Coulomb distortion of the final lepton wave function in the CC reaction, $$F(Z,k^\prime) = 2\, (1+\gamma)\, (2\, k^\prime\, r_d)^{2\,\gamma-2}\, {\rm exp} \left(\pi\, y\right)\,
\Bigg| \frac{\Gamma(\gamma+i\, y)}{\Gamma(1+2\,\gamma)} \Bigg|^2 \ , \qquad
\gamma=\sqrt{1-\left(Z\,\alpha\right)^2} \ ,$$ and it is set to one otherwise. Here $y = Z\, \alpha \, \epsilon^\prime/k^\prime$, $\Gamma(z)$ is the gamma function, $r_d$ is the deuteron charge radius ($r_d=1.97$ fm), and $\alpha$ is the fine structure constant.
The factors $v_{\alpha\beta}$ denote combinations of lepton kinematical variables including the final lepton mass, while the nuclear response functions are defined schematically as (explicit expressions for the $v_{\alpha\beta}$ and $R_{\alpha\beta}$ can be found in Ref. [@Shen2012]) $$\label{eq:r1}
R_{\alpha\beta}(q,\omega) \sim \frac{1}{3} \sum_{M } \sum_f \delta( \omega+m_d-E_f)\,
\langle f| j^\alpha({\bf q},\omega) |d, M \rangle\, \langle f| j^\beta({\bf q},\omega) |d, M \rangle^* \ ,$$ where $|d, M\rangle$ and $|f\rangle$ represent, respectively, the initial deuteron state in spin projection $M$ and the final two-nucleon state of energy $E_f$, and $m_d$ is the deuteron rest mass. The three-momentum transfer ${\bf q}$ is taken along the $z$-axis (i.e., the spin-quantization axis), and $j^\alpha({\bf q},\omega)$ is the time component (for $\alpha=0$) or space component (for $\alpha=x,y,z$) of the NC or CC, denoted, respectively, by $j^\alpha_{NC}$ or $j^\alpha_{CC}$. The former is given by $$j^\alpha_{NC}=-2\, {\rm sin}^2\theta_W\, j^\alpha_{\gamma, S} + (1-2\, {\rm sin}^2\theta_W) \, j^\alpha_{\gamma, z}
+\, j^{\alpha 5}_z \ ,$$ where $\theta_W$ is the Weinberg angle (${\rm sin}^2\theta_W=0.2312$ [@PDG]), $j^\alpha_{\gamma,S}$ and $j^\alpha_{\gamma,z}$ include, respectively, the isoscalar and isovector terms of the electromagnetic current, and $j^{\alpha 5}_z$ includes the isovector terms of the axial current (the subscript $z$ on these indicates that they transform as the $z$-component of an isovector under rotations in isospin space).
The charge-changing weak current is written as the sum of polar- and axial-vector components $$j^\alpha_{CC}=j^\alpha_{\pm}+j^{\alpha 5}_{\pm} \ , \qquad j_\pm = j_x \pm i\, j_y \ .$$ The conserved-vector-current (CVC) constraint relates the polar-vector components $j^\alpha_b$ of the charge-changing weak current to the isovector component $j^\alpha_{\gamma,z}$ of the electromagnetic current via $$\left[ \, T_a \, , \, j^\alpha_{\gamma,z} \, \right]=i\, \epsilon_{azb}\, j^\alpha_b \ ,$$ where $T_a$ are isospin operators. Before turning to a brief discussion of the one- and two-body $\chi$EFT contributions to the NC and CC, we note that, as described in considerable detail in Ref. [@Shen2012], we evaluate, by direct numerical integrations, the matrix elements of the weak current between the deuteron and the two-nucleon scattering states labeled by the relative momentum ${\bf p}$ and in given pair-spin and pair-isospin channels, thus avoiding cumbersome multipole expansions. Differential cross sections are then obtained by integrating over ${\bf p}$ and summing over the discrete quantum numbers the appropriate matrix-element combinations entering the response functions [@Shen2012].
Electro-weak current {#sec:cnts}
====================
The $\chi$EFT contributions up to one loop to the electromagnetic current [@Pastore2009; @Piarulli2013] and charge [@Pastore2011; @Piarulli2013] are illustrated diagrammatically in Figs. \[fig:f2\] and \[fig:f5\], while those to the weak axial current and charge [@Baroni2016; @Baroni2016a] in Figs. \[fig:f2a\] and \[fig:f5a\]. The former are denoted below as ${\bf j}_\gamma=j^i_\gamma$ and $\rho_\gamma=j^{0}_\gamma$, and the latter as ${\bf j}_5=j^{i 5}_z$ and $\rho_5=j^{0 5}_z$, respectively, and subscripts specifying isospin components are dropped for simplicity here.
![Diagrams illustrating one- and two-body electromagnetic currents entering at $Q^{-2}$ (LO), $Q^{-1}$ (N1LO), $Q^{\,0}$ (N2LO), and $Q^{\,1}$ (N3LO). Nucleons, pions, and photons are denoted by solid, dashed, and wavy lines, respectively. The square in panel (d) represents the $(Q/m)^2$ relativistic correction to the LO one-body current ($m$ is the nucleon mass); the solid circle in panel (j) is associated with the $\gamma \pi N$ coupling involving the LECs $d_8$, $d_9$, and $2\,d_{21}\,$–$\,d_{22}$ in the $\pi N$ chiral Lagrangian ${\cal L}^{(3)}_{\pi N}$ [@Fettes2000]; the solid circle in panel (k) denotes two-body contact terms of minimal and non-minimal nature, the latter involving two unknown LECs (see text). Only one among all possible time orderings is shown for the N1LO and N3LO currents, so that all direct- and crossed-box contributions are accounted for.[]{data-label="fig:f2"}](em_current){width="14cm"}
In these figures, the N$n$LO corrections are proportional to $Q^{\,n} \times \,Q^{\,\nu_0}$, where $Q$ denotes generically the low-momentum scale (the expansion parameter is $Q/\Lambda_\chi$, where $\Lambda_\chi\!\sim\! 1$ GeV is the chiral symmetry breaking scale) and $\nu_0$ characterizes the leading-order (LO) counting: $\nu_0\,$=$\,-2$ for the electromagnetic current and axial charge and $\nu_0=-3$ for the electromagnetic charge and axial current \[the chiral order in these operators is indicated by the superscript $(n)$\]. We begin by discussing the electromagnetic operators.
The electromagnetic currents from LO, N1LO, and N2LO terms and from N3LO loop corrections depend only on the nucleon axial coupling $g_A$ and and pion decay constant $f_\pi$ (N1LO and N3LO), and the nucleon magnetic moments (LO and N2LO). Unknown LECs enter the N3LO OPE contribution involving the $\gamma \pi N$ vertex from the chiral Lagrangian ${\cal L}^{(3)}_{\pi N}$ (see Ref. [@Fettes2000]) as well as the contact currents implied by non-minimal couplings, as discussed in Sec. \[sec:lecs\]. On the other hand, in the charge operator there are no unknown LECs up to one loop, and OPE contributions, illustrated in panels (c)-(e) of Fig. \[fig:f5\], only appear at N3LO.
![Diagrams illustrating one- and two-body electromagnetic charge operators entering at $Q^{-3}$ (LO), $Q^{-1}$ (N2LO), $Q^{0}$ (N3LO), $Q^{1}$ (N4LO). The square in panel (b) represents the $(Q/m)^2$ relativistic correction to the LO one-body charge operator, whereas panel (c) represents the charge operator $\rho^{(0)}_\gamma({\rm OPE})$ given in Eq. (\[eq:pich\]). As in Fig. \[fig:f2\], only a single time ordering is shown for the N3LO and N4LO contributions.[]{data-label="fig:f5"}](em_charge){width="12cm"}
The contributions in panels (d) and (e) involve non-static corrections [@Pastore2011], while those in panel (c) lead to the following operator, first derived by Phillips [@Phillips2003], $$\rho^{(0)}_\gamma({\rm OPE}) =\frac{e\,g_A^2}{8\, m\,f_\pi^2} \left( {\bm \tau}_1 \cdot {\bm \tau_2}
+ \tau_{2z}\right)\, \frac{{\bm \sigma}_1 \cdot {\bf q} \,\, {\bm \sigma}_2 \cdot {\bf k}_2}{k^2_2+m_\pi^2}
+ (1 \rightleftharpoons 2) \ ,
\label{eq:pich}$$ where ${\bf q}$ is the momentum imparted by the external field, ${\bf k}_i ={\bf p}_i^\prime -{\bf p}_i$ and ${\bf p}_i$ (${\bf p}_i^\prime$) is the initial (final) momentum of nucleon $i$ (with ${\bf k}_1+{\bf k}_2\,$=$\,{\bf q}$), ${\bm \sigma}_i$ and ${\bm \tau}_i$ are its Pauli spin and isospin operators, $m$ ($m_\pi$) is the nucleon (pion) mass. This operator plays an important role in yielding predictions for the $A\,$=$\,2$–4 charge form factors that are in excellent agreement with the experimental data at low and moderate values of the momentum transfer ($q \lesssim 1$ GeV/c) [@Piarulli2013; @Marcucci2016]. The calculations in Ref. [@Piarulli2013] also showed that the non-static corrections of pion range from panels (d) and (e) of Fig. \[fig:f5\] are typically an order of magnitude smaller than those generated by panel (c).
![Diagrams illustrating one- and two-body axial currents entering at $Q^{-3}$ (LO), $Q^{-1}$ (N2LO), $Q^{\,0}$ (N3LO), and $Q^{\,1}$ (N4LO). Nucleons, pions, and axial fields are denoted by solid, dashed, and wavy lines, respectively. The squares in panels (c) and (d) denote relativistic corrections to the one-body axial current, while the circles in panels (e) and (f) represent vertices implied by the ${\cal L}^{(2)}_{\pi N}$ chiral Lagrangian [@Fettes2000], involving the LECs $c_i$ (see Ref. [@Baroni2016] for additional explanations). As in Fig. \[fig:f2\], only a single time ordering is shown.[]{data-label="fig:f2a"}](ax_current){width="17cm"}
The axial current and charge operators illustrated in Figs. \[fig:f2a\] and \[fig:f5a\] include pion-pole contributions, which are crucial for the current to be conserved in the chiral limit [@Baroni2016] (obviously, these contributions are suppressed in low-momentum transfer processes). There are no direct couplings of the time-component of the external axial field to the nucleon, see panel (a) in Fig. \[fig:f5a\]. In the axial current pion-range contributions enter at N3LO, panels (e) and (f) of Fig. \[fig:f2a\], and involve vertices from the sub-leading ${\cal L}^{(2)}_{\pi N}$ chiral Lagrangian [@Fettes2000], proportional to the LECs $c_3$, $c_4$, and $c_6$. The associated operator is given by (the complete operator, including pion pole contributions, is listed in Ref. [@Baroni2016]) $$\begin{aligned}
\label{eq:opej1fin}
{\bf j}_{5,a}^{(0)}({\rm OPE})&=& \frac{g_A}{2\,f_\pi^2} \bigg\{
2\, c_3 \, \tau_{2,a}\, {\bf k}_2
+\left({\bm \tau}_1\times{\bm \tau}_2\right)_a
\bigg[ \frac{i}{2\, m} {\bf K}_1 - \frac{c_6+1}{4\, m} {\bm \sigma}_1\times{\bf q} \nonumber\\
&&+\left( c_4+\frac{1}{4\, m}\right) {\bm \sigma}_1\times{\bf k}_2 \bigg]
\bigg\} \frac{{\bm\sigma}_2\cdot{\bf k}_2} {k_2^2+m_\pi^2}+ (1\rightleftharpoons 2)\ ,\end{aligned}$$ where ${\bf K}_i=({\bf p}_i^\prime+{\bf p}_i)/2$. In contrast, the axial charge has a OPE contribution at N1LO, illustrated in panels (b) and (c) of Fig. \[fig:f5a\], which reads $$\begin{aligned}
\label{eq:6.61}
\rho^{(-1)}_{5,a}({\rm OPE})&=&
i\frac{g_A}{4\,f_\pi^2}\left({\bm \tau}_1\times{\bm \tau}_2\right)_a
\frac{{\bm \sigma}_2\cdot{\bf k}_2}{k_2^2+m_\pi^2} + (1\rightleftharpoons 2)\ .\end{aligned}$$ In fact, an operator of precisely this form was derived by Kubodera [*et al.*]{} [@Kubodera1978] in the late seventies, long before the systematic approach based on chiral Lagrangians now in use had been established. Corrections to the axial current at N4LO in panels (i)-(x) of Fig. \[fig:f2a\] have been included in a very recent calculation of the tritium Gamow-Teller matrix element [@Baroni2016a], while those to the axial charge at N3LO in panels (d)-(n) in Fig. \[fig:f5a\] are considered for the first time in the present study, to the best of our knowledge. It is worthwhile noting that vertices involving three or four pions, such as those, for example, occurring in panels (l), (p), (q) and (r) of Fig. \[fig:f2a\], depend on the pion field parametrization. This dependence must cancel out after summing the individual contributions associated with these diagrams, as indeed it does [@Baroni2016] (this and the requirement, remarked on below, that the axial current be conserved in the chiral limit provide useful checks of the calculation).
The loop integrals in the diagrams of Figs. \[fig:f2\]–\[fig:f5a\] are ultraviolet divergent and are regularized in dimensional regularization [@Pastore2009; @Pastore2011; @Baroni2016]. In the electromagnetic current the divergent parts of these loop integrals are reabsorbed by the LECs multiplying contact terms [@Pastore2009], while those in the electromagnetic charge cancel out, in line with the fact that there are no counter-terms at N4LO [@Pastore2011]. In the case of the axial operators [@Baroni2016], there are no divergencies in the current, while those in the charge lead to renormalization of the LECs multiplying contact-type contributions. In particular, the infinities in loop corrections to the OPE axial charge (not shown in Fig. \[fig:f5a\]) are re-absorbed by renormalization of the LECs $d_i$ in the ${\cal L}^{(3)}_{\pi N}$ chiral Lagrangian. For a discussion of these issues we defer to Ref. [@Baroni2016].
![Diagrams illustrating one- and two-body axial charge operators entering at $Q^{-2}$ (LO), $Q^{-1}$ (N1LO), and $Q^{\,1}$ (N3LO). Nucleons, pions, and axial fields are denoted by solid, dashed, and wavy lines, respectively. The diamonds in panels (l) and (m) indicate higher order $A\pi N$ vertices implied by the ${\cal L}^{(3)}_{\pi N}$ chiral Lagrangian [@Fettes2000], involving the LECs $d_i$ (see Ref. [@Baroni2016] for additional explanations). As in Fig. \[fig:f2\], only a single time ordering is shown.[]{data-label="fig:f5a"}](ax_charge){width="12cm"}
The two-nucleon chiral potentials used in the present study have been derived up to order $Q^4$ [@Entem2003; @Machleidt2011], requiring two-loop contributions. Conservation of the electromagnetic current ${\bf q}\cdot{\bf j}_\gamma=\left[\, H\, ,\, \rho_\gamma\,\right]$, where the two-nucleon Hamiltonian is given by $H=T^{(-1)}+v^{(0)}+v^{(2)}+v^{(3)}+v^{(4)}$ with the (two-nucleon) kinetic energy $T^{(-1)}$ being counted as $Q^{-1}$ and where the $v^{(n)}$’s are the potentials of order $Q^n$, implies [@Pastore2009], order by order in the power counting, a set of non-trivial relations between the ${\bf j}_\gamma^{(n)}$ and the $T^{(-1)}$, $v^{(n)}$, and $\rho_\gamma^{(n)}$. Since commutators implicitly bring in extra factors of $Q^{3}$, these relations couple different orders in the power counting of the operators, making it impossible to carry out a calculation, which at a given $n$ for ${\bf j}_\gamma^{(n)}$, $v^{(n)}$, and $\rho_\gamma^{(n)}$ (and hence “consistent” from a power-counting perspective) also leads to a conserved current. Similar considerations also apply to the conservation of the axial current in the chiral limit [@Baroni2016].
We conclude this section by noting that a number of independent derivations of nuclear electromagnetic and axial currents exists in the literature in the $\chi$EFT formulation adopted here, in which nucleons and pions are the explicit degrees of freedom. The early and pioneering studies by Park [*et al.*]{} [@Park1993; @Park1996; @Park2003] used heavy-baryon covariant perturbation (HBPT) theory, while the more recent ones by the Bochum-Bonn group [@Koelling2009; @Koelling2011; @Krebs2016] are based on time-ordered perturbation theory (TOPT) and a different prescription for isolating non-iterative pieces in reducible diagrams than adopted in Refs. [@Pastore2009; @Pastore2011; @Piarulli2013; @Baroni2016]. Detailed comparisons between the operators obtained in these latter papers and the HBPT ones of Park [*et al.*]{} can be found in Refs. [@Pastore2009] and [@Baroni2016]. It suffices to note here that Park [*et al.*]{} in their evaluation of two-nucleon amplitudes have only included irreducible diagrams and, for the case of the axial currents, did not concern themselves with pion-pole contributions. Because of these limitations, the electromagnetic current and axial current in the chiral limit are not conserved.
The two TOPT-based methods lead to formally equivalent operator structures for the nuclear potential, electromagnetic current and charge, and axial charge up to one-loop corrections included [@Piarulli2013]. However, some of the N4LO loop corrections to the axial current obtained by Krebs [*et al.*]{} [@Krebs2016] are different from those reported in Refs. [@Baroni2016; @Baroni2016a]. These differences seem to originate from the evaluation of box diagrams, panels (m) and (n) of Fig. \[fig:f2a\]. Additional differences result from the fact non-static corrections at N4LO have been neglected in Ref. [@Baroni2016], while they have been retained explicitly in Ref. [@Krebs2016].
Constraining the LECs in the electro-weak currents {#sec:lecs}
--------------------------------------------------
There is a total of ten LECs entering the two-body electro-weak currents discussed above, five of these are in the electromagnetic (vector) sector and the remaining five (in the limit of vanishing momentum transfer) in the axial sector. In the vector sector, contact terms originate from minimal and non-minimal couplings. The LECs multiplying the former are known from fits of the two-nucleon scattering database [@Piarulli2013]. Non minimal couplings enter through the electromagnetic field tensor, and it has been shown [@Pastore2009] that only two independent structures occur at order $Q^1$ (see panel (k) in Fig. \[fig:f2\]): $${\bf j}^{(1)}_{\gamma}({\rm CT})= - i\,e \Big[ \widetilde{c}_\gamma^{\, S}\, {\bm \sigma}_1
+\widetilde{c}_\gamma^{\, V} (\tau_{1,z} - \tau_{2,z})\,{\bm \sigma}_1 \Big]\times {\bf q}
+ \left(1 \rightleftharpoons 2\right)\ ,
\label{eq:gnm}$$ where $e$ is the electric charge, $\widetilde{c}_\gamma^{\,S}$ and $\widetilde{c}_\gamma^{\,V}$ are the two LECs, and the superscripts specify the isoscalar ($S$) and isovector ($V$) character of the associated operator. There is also a pion-range two-body operator resulting from sub-leading $\gamma \pi N$ couplings associated with the ${\cal L}^{(3)}_{\pi N}$ Lagrangian, and illustrated by panel (j) in Fig. \[fig:f2\]. It reads: $$\begin{aligned}
\label{eq:cdlt}
{\bf j}_\gamma^{(1)}({\rm OPE})&=& i\,e\, \frac{g_A}{4\,f_\pi^2} \,
\frac{{\bm \sigma}_2 \cdot {\bf k}_2}{k_2^2+m_\pi^2} \bigg[ \left(\widetilde{d}^{\, V}_{\gamma,1} \tau_{2,z}
+ \widetilde{d}_\gamma^{\, S} \, {\bm \tau}_1\cdot{\bm \tau}_2 \right){\bf k}_2\nonumber \\
&&-\widetilde{d}^{\, V}_{\gamma,2} ({\bm \tau}_1\times{\bm \tau}_2)_z\, {\bm \sigma}_1\times {\bf k}_2 \bigg] \times {\bf q} + (1\rightleftharpoons 2) \ ,\end{aligned}$$ where the LECs $\widetilde{d}_{\gamma,1}^{\, V}$, $\widetilde{d}_{\gamma,2}^{\, V}$ and $\widetilde{d}_\gamma^{\, S}$ are related [@Piarulli2013] to the LECs $d_8$, $d_9$, $d_{21}$, and $d_{22}$ in the original ${\cal L}^{(3)}_{\pi N}$ Lagrangian [@Fettes2000] in the following way $$\widetilde{d}_{\gamma}^{\, S}=-8\,d_9\ ,\qquad
\widetilde{d}_{\gamma,1}^{\, V}=-8\,d_8\ ,\qquad
\widetilde{d}_{\gamma,2}^{\, V}=2\,d_{21}-d_{22}\ .$$ As discussed below, these LECs have been determined by a combination of resonance saturation arguments and fits to photo-nuclear data in the two- and three-nucleon systems.
In the weak axial sector, there is a single contact term at order $Q^0$ (or N3LO, see panels (g) and (h) of Fig. \[fig:f2a\]) $$\label{eq:jctct}
{\bf j}_{5,a}^{(0)}({\rm CT})=\widetilde{c}^{\, V}_{5,1}\left({\bm \tau}_1\times{\bm \tau}_2\right)_a \left[{\bm \sigma}_1\times{\bm \sigma}_2-\frac{{\bf q}}{q^2+m_\pi^2}\, {\bf q}\cdot\left({\bm \sigma}_1\times{\bm \sigma}_2\right) \right]\ ,$$ where the second term of Eq. (\[eq:jctct\]) is the pion-pole contribution, and none at order $Q^1$ (or N4LO). The axial charge operators at N3LO from OPE \[panels (l) and (m) of Fig. \[fig:f5a\]\] and contact interactions \[panel (n)\] involve, in principle, nine LECs [@Baroni2016]. Since the processes of interest in the present work are relatively low-momentum transfer ones, however, we have considered here these operators in the limit $q \rightarrow 0$ (or ${\bf k}_1 \simeq -{\bf k}_2$), which leads to $$\begin{aligned}
\label{eq:6.66}
\!\!\!\!\!\!\! \rho^{(1)}_{5,a}({\mbox{OPE}})\!&=&\!
i\frac{g_A}{384\,\pi^2\,f_\pi^{4}}\left({\bm \tau}_1\times{\bm \tau}_2\right)_a
\bigg\{g_A^2 \left [\left(5\, k_2^2+8\, m_\pi^{2}\right)\frac{s_2}{k_2}
\ln\frac{s_2+k_2}{s_2-k_2}-\frac{13}{3}k_2^2+2\,m_\pi^2\right] \nonumber\\
\!\!\!\!\!\!\!\!&+& \!\!\left(\frac{s_2^3}{k_2}\ln\frac{s_2+k_2}{s_2-k_2}-\frac{5}{3}k_2^2-8\, m_\pi^{2} \right)
+\widetilde{d}^{\, V}_{5,1}\, k_2^2
+ \widetilde{d}^{\, V}_{5,2} \,m_\pi^{2}\bigg\}\frac{{\bm \sigma}_2\cdot{\bf k}_2}{k_2^2+m_\pi^2} + (1\rightleftharpoons 2)\ , \end{aligned}$$ $$\begin{aligned}
\rho^{(1)}_{5,a}({\rm CT}) &=& i\, \widetilde{c}^{\, V}_{5,2} \left({\bm\tau}_1\times{\bm\tau}_2\right)_a
\, {\bm \sigma}_1\cdot{\bf k}_1
+ i\, \widetilde{c}^{\, V}_{5,3}\, \tau_{1,a}\, \left({\bm \sigma}_1\times{\bm \sigma}_2\right)\cdot {\bf k}_2+ (1\rightleftharpoons 2)\ ,
\label{eq:4.22}\end{aligned}$$ where $s_j=\sqrt{k_j^2+4\, m_\pi^2}$. The LECs $\widetilde{d}^{\, V}_{5,i}$ denote the combinations [@Baroni2016] $$\widetilde{d}^{\, V}_{5,1}= 4 \,(d_1+d_2+d_3) \ , \qquad \widetilde{d}^{\, V}_{5,2} = 4\, ( d_1+d_2+d_3)+8 \, d_5 \ ,$$ in terms of the $d_i$’s in ${\cal L}^{(3)}_{\pi N}$ [@Fettes2000], and are taken from an analysis of $\pi N$ scattering data as reported in Ref. [@Machleidt2011]. (It should be noted that a new analysis of these data has become recently available [@Hoferichter2015].) The LECs $\widetilde{c}^{\, V}_{5,2}$ and $\widetilde{c}^{\, V}_{5,3}$ have yet to be determined.
Configuration-space representations of the $\chi$EFT operators in Figs. \[fig:f2\]–\[fig:f5a\] are required in the computer programs. Those for the one-body operators, illustrated in panels (a) and (d) in Fig. \[fig:f2\], (a) and (b) of Fig. \[fig:f5\], (a)-(d) of Fig. \[fig:f2a\], and (a) of Fig. \[fig:f5a\], follow directly from the momentum-space expressions listed in Refs. [@Piarulli2013; @Baroni2016] by simply multiplying each term in these expressions by ${\rm exp}(i{\bf q}\cdot{\bf r}_i)$ and by replacing ${\bf K}_i$ with $ -i\, {\bm \nabla}_i$ (and properly symmetrizing for hermiticity). The configuration-space representations of the two-body operators are strongly singular at short inter-nucleon separations and must be regularized before they can be sandwiched between nuclear wave functions. This is accomplished by insertion in the Fourier transforms of a regulator of the form $C_\Lambda (k)={\rm exp}[-(k/ \Lambda)^n]$ with $n\,$=$\,4$ and $\Lambda$ in the range (500–600) MeV. For processes involving low momentum and energy transfers one would expect predictions to be fairly insensitive to variations of $\Lambda$, and this expectation is indeed borne out by the calculations reported in the present work.
[c|ccc|cc||cc|ccc]{} $\Lambda$ (MeV) & $d_\gamma^S$ & $d^V_{\gamma,1}$ & $d^V_{\gamma,2}$ & $c_\gamma^S$ & $c_\gamma^V$ & $d^V_{5,1}$ & $d^V_{5,2}$ &$c_{5,1}^V$ & $c_{5,2}^V$ & $c_{5,3}^V$\
500 & 0.219 & 3.458 & 0.865 & 4.072 & –7.981 & –0.210 & 0.690 & 13.22 &0.062 & 0.062\
600 & 0.323 & 4.980 & 1.245 & 11.38 & –11.69 & –0.302 & 0.994 &25.07 & 0.130 & 0.130\
\[tb:t1\]
In the electromagnetic sector, the two isoscalar LECs $\widetilde{c}_\gamma^{\, S}$ and $\widetilde{d}_\gamma^{\, S}$ are fixed (for each $\Lambda$) by reproducing the deuteron and isoscalar trinucleon magnetic moments, while the two isovector LECs $\widetilde{d}^{\, V}_{\gamma,1}$ and $\widetilde{d}^{\, V}_{\gamma,2}$ are constrained by assuming $\Delta$-resonance saturation [@Piarulli2013], $$\widetilde{d}^{\, V}_{\gamma,1}=\frac{4\, \mu_{\gamma N\Delta} \, h_A }{9\, m\,( m_\Delta-m)} \ , \qquad
\widetilde{d}^{\, V}_{\gamma,2}=\frac{1}{4}\, \widetilde{d}^{\, V}_{\gamma,1}\ ,$$ where $m_\Delta\,$–$\,m\,$=$\, 294$ MeV, $h_A/(2 f_\pi)=f_{\pi N\Delta}/m_\pi$ with $f^2_{\pi N\Delta}/(4\,\pi)=0.35$ as obtained by equating the first-order expression of the $\Delta$-decay width to the experimental value, and the transition magnetic moment $\mu_{\gamma N\Delta}$ is taken as $3\, \mu_N$ [@Carlson1986]. The remaining LEC $\widetilde{c}_\gamma^{\, V}$ is determined by reproducing the isovector trinucleon magnetic moment [@Piarulli2013]. In the weak axial sector, the LEC $\widetilde{c}^{\, V}_{5,1}$ is fixed by reproducing the tritium Gamow-Teller matrix element [@Baroni2016a], while the other two LECs $\widetilde{c}^{\, V}_{5,2}$ and $\widetilde{c}^{\, V}_{5,3}$ in the axial charge are taken here to assume natural values $\widetilde{c}^{\, V}_{5,i} \simeq 1/\Lambda_\chi^4$, for $i\,$=$\,2,3$ and with $\Lambda_\chi\,$=$\,1$ GeV. However, cross sections results are insensitive to variations of $\widetilde{c}^{\, V}_{5,2}$ and $\widetilde{c}^{\, V}_{5,3}$ over a rather broad range (see Sec. \[sec:res\]). In Table \[tb:t1\] we list the values of all these LECs in units of the short-range cutoff $\Lambda$, namely $$\begin{aligned}
\label{eq:adi}
&&\widetilde{d}_\gamma^{\,S}=d^{\, S}_\gamma/\Lambda^2\ ,\qquad \widetilde{d}_{\gamma,i}^{\,V}=d^{\, V}_{\gamma,i} /\Lambda^2\ ,
\qquad \widetilde{c}_\gamma^{\, S}=c^{\, S}_\gamma/\Lambda^4\ ,\qquad \widetilde{c}_\gamma^{\, V}=c^{\,V}_\gamma/\Lambda^4 \ ,\nonumber \\
&& \widetilde{d}_{5,i}^{\, V}=d^{\,V}_{5,i}/\Lambda^2 \ ,\qquad
\widetilde{c}_{5,1}^{\, V}=c^{\,V}_{5,1}/\Lambda^3 \ ,\qquad
\widetilde{c}_{5,2}^{\, V}=c^{\,V}_{5,2}/\Lambda^4 \ ,
\qquad \widetilde{c}_{5,3}^{\, V}=c^{\,V}_{5,3}/\Lambda^4 \ .\end{aligned}$$
Finally, we note that, since the processes under consideration involve small but non-vanishing four-momentum transfers $Q^2$, hadronic electro-weak form factors need to be included in the $\chi$EFT operators. Some of these form factors have been calculated in chiral perturbation theory [@Kubis2001], but the convergence of this calculation in powers of the momentum transfer appears to be rather poor. For this reason, in the results reported below, the form factors in the electromagnetic current and charge are accounted for as in Ref. [@Piarulli2013], [*i.e.*]{}, the nucleon, pion, and $N\Delta$-transition electromagnetic form factors are taken from fits to available electron scattering data. For the case of the axial charge and current, the operators are simply multiplied by $G_A(Q^2)/g_A$, where $G_A(Q^2)$ is the nucleon axial form factor, parametrized as $G_A(Q^2)=g_A/(1+Q^2/\Lambda^2_A)^2$ with $\Lambda_A\,$=$\, 1$ GeV, consistently with available neutrino scattering data (see [@Shen2012] and references therein).
Cross-section predictions {#sec:res}
=========================
Total cross sections, integrated over the final lepton energy and scattering angle and obtained for the $\nu_e$-CC, $\overline{\nu}_e$-CC, $\nu_l$-NC, and $\overline{\nu}_l$-NC processes, are shown, respectively, in Figs. \[fig:nu-pp\]–\[fig:nub-np\], where they are compared to the corresponding predictions from Ref. [@Nakamura02] for incoming neutrino energies ranging from threshold up to 150 MeV. The present $\chi$EFT calculations are based on the Entem and Machleidt potentials of Refs. [@Entem2003; @Machleidt2011] corresponding to cutoffs $\Lambda\,$=$\, 500$ and 600 MeV, and weak (vector and axial) current and charge operators of Refs. [@Pastore2009; @Pastore2011; @Piarulli2013; @Baroni2016], as described in the previous section. Matrix elements of these operators, suitably regularized as in Sec. \[sec:lecs\], between the initial deuteron and final two-nucleon scattering states are evaluated with the methods developed in Ref. [@Shen2012]. In practice, this entails obtaining the two-nucleon radial wave functions from solutions of the Lippmann-Schwinger equation in pair spin-isospin $ST$ channels with total angular momentum $J \le J_{\rm max}$, and in approximating these radial wave functions by spherical Bessel functions in channels with $J > J_{\rm max}$. The full wave function, labeled by the relative momentum ${\bf p}$ (and corresponding energy $p^2/(2\mu)$, $\mu$ being the reduced mass) and discrete quantum numbers $ST$, is then reconstructed from its partial-wave expansion [@Shen2012]. Consequently, interaction (including Coulomb in the case of two protons) effects in the final scattering states are exactly accounted for only in channels with $J\le J_{\rm max}$. For the neutrino energies of interest here, however, we find that these effects are negligible when $J_{\rm max} \gtrsim 5$ [@Shen2012].
![(Color online) Total cross sections in ${\rm fm}^2$ for the $\nu_e$-CC induced process on the deuteron. The solid line corresponds to the $\chi$EFT calculation with cutoff $\Lambda=$ 500 MeV, based on the chiral potential of Ref. [@Entem2003] and including electro-weak contributions up to N3LO in the vector current and axial charge, and up to N4LO in the axial current and vector charge, see Figs. \[fig:f2\]–\[fig:f5a\]. The dashed line is obtained within the conventional meson-exchange picture of Ref. [@Nakamura02]. The inset shows the ratio of conventional to $\chi$EFT predictions.[]{data-label="fig:nu-pp"}](sigma_pp){width="16cm"}
![(Color online) Same as in Fig. \[fig:nu-pp\] but for the $\overline{\nu}_e$-CC induced process on the deuteron.[]{data-label="fig:nu-nn"}](sigma_nn){width="16cm"}
The cross sections increase rapidly, by over two orders of magnitude, as the neutrino energy increases from threshold to 150 MeV. Nevertheless, the present $\chi$EFT predictions remain close to, albeit consistently larger at the 1–2% level than, those obtained in the conventional frameworks of Refs. [@Nakamura02] and [@Shen2012], as shown explicitly for the case of Ref. [@Nakamura02] by the insets in Figs. \[fig:nu-pp\]–\[fig:nub-np\]. The present $\chi$EFT electro-weak current and the meson-exchange models adopted in Refs. [@Nakamura02] and [@Shen2012] provide an excellent description of low-energy observables in the two- and three-nucleon systems (see Refs. [@Piarulli2013; @Marcucci2016] and references therein). In particular, the axial current in both approaches ($\chi$EFT and meson-exchange) is constrained to reproduce the tritium Gamow-Teller matrix element.
![Same as in Fig. \[fig:nu-pp\] but for the $\nu_e$-NC induced process on the deuteron.[]{data-label="fig:nu-np"}](sigma_np){width="16cm"}
------------------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
$E_\nu$ (MeV) 10 50 100 150 10 50 100 150
LO 2.676(–16) 1.345(–14) 6.611(–14) 1.591(–13) 1.243(–16) 7.441(–15) 2.661(–14) 4.944(–14)
N($1|2$)LO 2.670(–16) 1.345(–14) 6.606(–14) 1.581(–13) 1.237(–16) 7.341(–15) 2.602(–14) 4.792(–14)
N($2|3$)LO 2.794(–16) 1.413(–14) 6.913(–14) 1.653(–13) 1.298(–16) 7.825(–15) 2.801(–14) 5.221(–14)
N($3|4$)LO 2.734(–16) 1.388(–14) 6.852(–14) 1.650(–13) 1.266(–16) 7.523(–15) 2.676(–14) 4.981(–14)
LO$^\star$ 2.666(–16) 1.342(–14) 6.593(–14) 1.588(–13) 1.239(–16) 7.417(–15) 2.653(–14) 4.925(–14)
N($3|4$)LO$^\star$ 2.729(–16) 1.388(–14) 6.858(–14) 1.656(–13) 1.263(–16) 7.520(–15) 2.679(–14) 4.998(–14)
IA Ref. [@Shen2012] 2.630(–16) 1.314(–14) 6.424(–14) 1.516(–13) 1.219(–16) 7.260(–15) 2.567(–14) 4.688(–14)
TOT Ref. [@Shen2012] 2.680(–16) 1.348(–14) 6.631(–14) 1.574(–13) 1.242(–16) 7.403(–15) 2.606(–14) 4.751(–14)
TOT Ref. [@Nakamura02] 2.708(–16) 1.376(–14) 6.836(–14) 1.641(–13) 1.242(–16) 7.372(–15) 2.618(–14) 4.871(–14)
------------------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
: Total cross sections in fm$^2$, corresponding to cutoff $\Lambda=\,$500 MeV, for the CC-induced processes on the deuteron at selected initial neutrino energies and at increasing orders in the chiral counting. Referring to Figs. \[fig:f2\]–\[fig:f5a\], the rows are labeled as follows: LO for the leading-order vector and axial current and charge; N($1|2$)LO including the vector current and axial charge at N1LO, and the axial current and vector charge at N2LO; N($2|3$)LO including the vector current at N2LO, and the axial current and vector charge at N3LO; N($3|4$)LO including the vector current and axial charge at N3LO, and the axial current and vector charge at N4LO. Also listed are the results at LO and N($3|4$)LO but $\Lambda=600$ MeV (labeled as LO$^\star$ and N($3|4$)LO$^\star$), and those obtained in the conventional frameworks of (i) Ref. [@Shen2012] in impulse approximation (IA) and with inclusion of two-body currents (TOT) and (ii) Ref. [@Nakamura02] with inclusion of two-body currents (TOT). The notation $(xx)$ means $10^{xx}$.
\[tab:cppp\]
The $\chi$EFT cross sections of Figs. \[fig:nu-pp\]–\[fig:nub-np\] correspond to cutoff $\Lambda\,$=$\, 500$, but their variation as $\Lambda$ is increased to 600 MeV remains well below 1% over the whole energy range, as can be seen in Tables \[tab:cppp\] and \[tab:cpnp\], rows labeled N($3|4$)LO and N($3|4$)LO$^\star$. The convergence of the chiral expansion is also shown in these tables, where the various rows are labeled in accordance with the power counting adopted in the present work, see Figs. \[fig:nu-pp\]–\[fig:nub-np\]. A graphical representation of this convergence is provided by Fig. \[fig:ratio\]. Overall, contributions beyond LO lead to a couple of % increase in the cross sections for both the CC and NC processes. A similar increase due to two-body terms in the weak current is obtained in the conventional calculations, see rows labeled IA and TOT in Tables \[tab:cppp\] and \[tab:cpnp\]. Note that the IA row corresponds to results obtained with one-body currents, including relativistic corrections [@Shen2012]. These IA currents are the same as the $\chi$EFT ones illustrated by panel (a) of Fig. \[fig:f2\], panels (a) and (b) of Fig. \[fig:f5\], panels (a)-(d) of Fig. \[fig:f2a\], and panel (a) of Fig. \[fig:f5a\]. Since the contributions due to the OPE two-body terms in the vector current, panels (b) and (c) of Fig. \[fig:f2\], and axial charge, panels (b) and (c) of Fig. \[fig:f5a\], are very small, then the difference between the IA and N($1|2$)LO results essentially reflects differences in the wave functions obtained from conventional and chiral potentials. Indeed, the overall $\sim 2$ % offset between the TOT and N($3|4$)LO predictions is primarily due to these differences.
------------------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
$E_\nu$ (MeV) 10 50 100 150 10 50 100 150
LO 1.101(–16) 5.872(–15) 2.660(–14) 5.991(–14) 1.050(–16) 4.554(–15) 1.664(–14) 3.175(–14)
N($1|2$)LO 1.097(–16) 5.856(–15) 2.644(–14) 5.912(–14) 1.045(–16) 4.505(–15) 1.631(–14) 3.076(–14)
N($2|3$)LO 1.151(–16) 6.178(–15) 2.789(–14) 6.250(–14) 1.097(–16) 4.793(–15) 1.752(–14) 3.347(–14)
N($3|4$)LO 1.124(–16) 6.032(–15) 2.740(–14) 6.176(–14) 1.069(–16) 4.625(–15) 1.684(–14) 3.214(–14)
LO$^\star$ 1.096(–16) 5.853(–15) 2.652(–14) 5.973(–14) 1.045(–16) 4.539(–15) 1.659(–14) 3.165(–14)
N($3|4$)LO$^\star$ 1.121(–16) 6.028(–15) 2.742(–14) 6.191(–14) 1.067(–16) 4.622(–15) 1.685(–14) 3.224(–14)
IA Ref. [@Shen2012] 1.084(–16) 5.747(–15) 2.577(–14) 5.720(–14) 1.033(–16) 4.449(–15) 1.604(–14) 3.003(–14)
TOT Ref. [@Shen2012] 1.104(–16) 5.892(–15) 2.657(–14) 5.935(–14) 1.053(–16) 4.546(–15) 1.640(–14) 3.075(–14)
TOT Ref. [@Nakamura02] 1.107(–16) 5.944(–15) 2.711(–14) 6.130(–14) 1.053(–16) 4.535(–15) 1.647(–14) 3.129(–14)
------------------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
: Same as in Table \[tab:cppp\] but for the NC-induced processes.
\[tab:cpnp\]
![Same as in Fig. \[fig:nu-pp\] but for the $\overline{\nu}_e$-NC induced process on the deuteron.[]{data-label="fig:nub-np"}](sigma_np_nbar){width="16cm"}
![(Color online) The convergence pattern as function of increasing order in the chiral expansion of the weak current. Ratios of corrections at a given order relative to the preceding order are shown. Note that the $y$-axis scale in the r.h.s. panels is doubled relative to that in the l.h.s. panels. []{data-label="fig:ratio"}](ratio){width="16cm"}
The cross sections for the $\nu_l$-NC and $\overline{\nu}_l$-NC processes only differ in the sign of the interference response function $R_{xy}$ in Eq. (\[eq:xswa\]). In the case $\nu_e$-CC and $\overline{\nu}_e$-CC processes, additional differences result from isospin-symmetry breaking terms in the final state interactions of $pp$ versus $nn$. At low energies ($E_\nu \lesssim 10$ MeV), cross sections are dominated by the axial current, the associated contributions being more than two orders of magnitude larger than those from the vector current. As the energy increases, vector-current contributions increase becoming comparable, albeit still significantly smaller by over a factor of five at $E_\nu=150$ MeV than, axial-current ones. As a consequence, the $\nu_l$-NC and $\overline{\nu}_l$-NC are fairly close at low energies, but diverge significantly from each other as the energy increases. Because of the aforementioned isospin-symmetry breaking effects (primarily induced by the Coulomb repulsion), the $\nu_e$-CC and $\overline{\nu}_e$-CC differ even at low energies. Finally, contributions from the axial charge are negligible at $E_\nu \sim 10$ MeV, since at those energies the cross section is dominated by the $^1$S$_0$ channel, to which axial-charge transitions from the $^3$S$_1$-$^3$D$_1$ state of the deuteron are strongly suppressed. However, these axial-charge contributions remain well below 1% even at the high end of the energy range studied in this work, $E_\nu=150$ MeV. At this latter energy, for example, ignoring these axial-charge contributions altogether would reduce the $\nu_l$-NC ($\overline{\nu}_l$-NC) cross section from the N($3|4$)LO value of 6.176 (3.214) listed in Table \[tab:cpnp\] to 6.157 (3.194) in units of $10^{-14}$ fm$^2$. Thus, uncertainties in the values of the LECs $c_{5,2}$ and $c_{5,3}$ in the contact axial charge do not have a significant impact on the present cross section predictions.
Finally, for the purpose of illustration, Fig. \[fig:diff\_sigma\] shows the double-differential cross sections for CC-$\nu_e$ and CC-$\overline{\nu}_e$ induced processes as function of the final lepton energy at a fixed scattering angle of $90^{\circ}$ and for incident neutrino energy of $10$ MeV. The deuteron wave functions are obtained from the N3LO chiral potential with cutoff $\Lambda=500$ MeV. The energy spectrum of the $\chi$EFT predictions closely matches that of Ref. [@Nakamura02]. We have not explicitly verified that this agreement persists for different combinations of final lepton scattering angles and incident neutrino energies. However, we expect this to be the case for both the CC and NC reactions.
![ (Color online) Double differential cross sections in ${\rm fm}^2/({\rm MeV}\,{\rm sr})$ for the $\nu_e$-CC and $\overline{\nu}_e$-CC induced process on the deuteron. The solid line corresponds to the $\chi$EFT calculation with cutoff $\Lambda=$ 500 MeV, based on the chiral potential of Ref. [@Entem2003] and including electro-weak contributions up to N3LO in the vector current and axial charge, and up to N4LO in the axial current and vector charge, see Figs. \[fig:f2\]–\[fig:f5a\]. The dashed line is obtained within the conventional meson-exchange picture of Ref. [@Nakamura02]. The inset shows the ratio of conventional to $\chi$EFT predictions.[]{data-label="fig:diff_sigma"}](plots_CC_ad2){width="16cm"}
Summary and Conclusions {#sec:concl}
=======================
Cross sections for the reactions $^2$H($\nu_e,e^-$)$pp$, $^2$H($\overline{\nu}_e,e^+$)$nn$, and $^2$H($\nu_l/\overline{\nu}_l,\nu^{\,\prime}_l/\overline{\nu}^{\,\prime}_l$)$np$ have been studied in $\chi$EFT with the chiral potentials of Refs. [@Entem2003; @Machleidt2011] and chiral electro-weak current of Refs. [@Pastore2009; @Pastore2011; @Piarulli2013; @Baroni2016; @Baroni2016a]. The potentials include intermediate- and long-range parts mediated by one- and multi-pion exchanges, and a short-range part parametrized in terms of contact interactions, whose LECs have been constrained by fits to the nucleon-nucleon database for energies ranging from zero up to the pion-production threshold. The vector- and axial-vector components of the weak current have been derived up to one loop and include primarily one- and two-pion exchanges. In addition to these loop corrections, a number of contact terms occur. The five LECs that multiply the contact currents in the vector sector have been determined by a combination of resonance-saturation arguments and fits to photo-nuclear data in the two- and three-nucleon systems. Five LECs also enter the axial sector (in the limit of low-momentum transfer processes). Four of these are in the charge operator: two are known from analyses of $\pi N$ data, while the remaining two have yet to be determined and, in the present work, have been assumed to be of natural size. However, it is worthwhile emphasizing that the neutrino cross sections under consideration are only marginally impacted by the axial-charge components in the weak current. The fifth and only LEC entering the axial current has been fixed by reproducing the tritium Gamow-Teller matrix element.
Higher order contributions beyond LO lead to an overall increase by about 2% in the cross sections obtained with LO transition operators. Predictions are also fairly insensitive to variations in the short-range cutoff $\Lambda$, and change by a few parts in a thousand as $\Lambda$ is changed from 500 to 600 MeV in both the potential and weak current. As illustrated by Fig. \[fig:ratio\], there is good convergence in the chiral expansion of the weak current. The $\chi$EFT cross-section predictions reported here are consistently larger by a couple of percent than corresponding results obtained in conventional formulations based on meson-exchange phenomenology [@Nakamura02; @Shen2012]. These conventional calculations too are based on a model for the electro-weak current that provides an excellent description of electromagnetic observables in the few-nucleon systems and the tritium $\beta$-decay rate; indeed, two-body meson-exchange terms in the axial current are constrained to reproduce the Gamow-Teller matrix element, just as in $\chi$EFT. The enhancement in the cross section due to (two-body) meson-exchange terms in the weak current is similar (about 2%) to that obtained in the $\chi$EFT calculations. Indeed, as noted in the previous section, the approximately 1–2% offset between the conventional (Refs. [@Nakamura02; @Shen2012]) and present $\chi$EFT results originates from differences in the deuteron and two-nucleon continuum wave functions obtained with the corresponding potentials rather than from the modeling of the weak current. To explore this point further, we have carried out preliminary calculations of the $\nu_e$-NC and $\overline{\nu}_e$-NC cross sections with the LO weak current using one of the recently developed “minimally non-local” configuration-space chiral potential of Ref. [@Piarulli2015]. We find that the LO $\nu_e$-NC ($\overline{\nu}_e$-NC) cross sections, in units of fm$^2$, are $1.101 (1.050) \times 10^{-16}$ at 10 MeV and $5.937(3.147) \times 10^{-14}$ at 150 MeV, to be compared, respectively, to $1.101(1.050) \times 10^{-16}$ and $5.991(3.175)\times 10^{-14}$ obtained with the chiral (and strongly non-local in configuration space) potentials of Refs. [@Entem2003; @Machleidt2011] adopted in the present work. This suggests that the cross-section predictions based on chiral potentials and currents may have a very small error ($< 1\%$) in the low-energy regime. A more rigorous way to estimate the theoretical uncertainty of the calculated cross-sections is described in Refs. [@Epelbaum16a; @Devries17].
Finally, we conclude by noting that radiative corrections for the CC and NC processes due to bremsstrahlung and virtual photon and $Z$ exchanges have been evaluated by the authors of Refs. [@Towner98; @Kurylov02] at the low energies ($\sim 10$ MeV) most relevant for the SNO experiment, which measured the neutrino flux from the $^8$B decay in the sun. In the case of the $^2$H($\nu_e,e^-$)$pp$, these corrections lead to an enhancement of the tree-level cross sections calculated in the present work (and in Refs. [@Nakamura02; @Shen2012]), which ranges from about 4% in the threshold region to about 3% at the endpoint of the $^8$B $\nu_e$-spectrum—this enhancement is in fact larger than that induced by contributions in the weak current of order higher than leading. While the present results are not expected to impact the $^8$B $\nu_e$-spectrum deduced by the SNO measurements (as the inset of Fig. \[fig:nu-pp\]), they should nevertheless be helpful in reducing the theoretical error in this inferred spectrum. The work of R.S. is supported by the U.S. Department of Energy, Office of Nuclear Science, under contract DE-AC05-06OR23177. The calculations were made possible by grants of computing time from the National Energy Research Supercomputer Center.
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abstract: 'Process-aware Recommender systems (PAR systems) are information systems that aim to monitor process executions, predict their outcome, and recommend effective interventions to reduce the risk of failure. This paper discusses monitoring, predicting, and recommending using a PAR system within a financial institute in the Netherlands to avoid faulty executions. Although predictions were based on the analysis of historical data, the most opportune intervention was selected on the basis of human judgment and subjective opinions. The results showed that, although the predictions of risky cases were relatively accurate, no reduction was observed in the number of faulty executions. We believe that this was caused by incorrect choices of interventions. Although a large body of research exists on monitoring and predicting based on facts recorded in historical data, research on fact-based interventions is relatively limited. This paper reports on lessons learned from the case study in finance and identifies the need to develop interventions based on insights from factual, historical data.'
author:
- |
Marcus Dees, Massimiliano de Leoni,\
Wil M.P. van der Aalst and Hajo A. Reijers
bibliography:
- 'paper.bib'
title: |
What if Process Predictions are not\
followed by Good Recommendations?\
(Technical Report)
---
Introduction {#sec:intro}
============
Process-aware Recommender systems (hereafter shortened as PAR systems) are a new breed of information systems. They aim to predict how the executions of process instances are going to evolve in the future, to determine those that have higher chances to not meet desired levels of performance (e.g., costs, deadlines, customer satisfaction). Consequently recommendations are provided on which effective contingency actions should be enacted to try to recover from risky executions. PAR systems are expert systems that run in the background and continuously monitor the execution of processes, predict their future, and, possibly, provide recommendations. Examples of PAR systems are discussed by Conforti et al. [@CONFORTI20151] and Schobel et al. [@SchobelR17]. A substantial body of research exists on evaluating risks , also known as process monitoring and prediction; see, e.g., the surveys by Márquez-Chamorro et al. [@MarquezChamorro18] and by Teinemaa et al. [@TeinemaaDRM17]. Yet, as also indicated in [@MarquezChamorro18], *“little attention has been given to providing recommendations”*. In fact, it has often been overlooked how process participants would use these predictions to enact appropriate actions to recover from those executions that have a higher risk of causing problems. It seems that process participants are tacitly assumed to take the “right decision” for the most appropriate corrective actions for each case. This also holds for approaches based on mitigation / flexibility “by design” [@6560815]. Unfortunately, the assumption of selecting an effective corrective action is not always met in reality. When selecting an intervention, this is mainly done based on human judgment, which naturally relies on the subjective perception of the process instead of being based on objective facts.
In particular, the PAR system should analyze the past process executions, and correlate alternative corrective actions with the likelihood of being effective; it should then recommend the actions that are most likely to decrease risks. Otherwise, even if the monitor is able to drive the attention of process participants to those executions that actually require support, the recommender system is destined to ultimately fail. The positive occurrence of correctly monitoring a process and making an accurate prediction can be nullified by an improper recovery or intervention. An organization will only profit from using a recommender system if the system is capable of making accurate decisions and the organization is capable of making effective decisions on the basis of this. Much attention is being paid to making accurate decisions, specifically to the proper use of data, measuring accuracy, etc. *In this work, we show that the analysis of making effective decisions is just as important. Both parts are essential ingredients of an overall solution.*
This paper reports on a field experiment that we conducted within UWV, a Dutch governmental agency. Among other things, UWV provides financial support to Dutch residents that lose their job and seek a new employment. Several subjects (hereafter often referred to as customers) receive more unemployment benefits than the amount they are entitled to. Although this is eventually detected, it may take several months. Using the UWV’s terminology, a *reclamation* is created when this happens, i.e. a reclamation event is raised when a reclamation is detected. To reclaim the amount of unlawfully provided support is very hard, time-consuming, and, often unsuccessful. In this context, an effective recommender system should be able to detect the customers who are more likely to get a reclamation and provide operational support to *prevent* the provision of benefits without entitlement. To follow up on this idea, we developed a predictor module that relies on machine-learning techniques to monitor and identify the subjects who are more likely to receive unlawful support. Next, various possible interventions to prevent reclamations were considered by UWV’s stakeholders. The intervention that was selected to be tested in a field experiment consists of sending a specific email to the subjects who were suspected of being at higher risk. *The results show that risky customers were detected rather well, but no significant reduction of the number of reclamations was observed.* This indicates that the intervention did not achieve the desired effect, which ultimately means that the action was not effective in preventing reclamations. Our findings show the importance of conducting research not only on prediction but also on interventions. This is to ensure that the PAR system will indeed achieve the improvements that it aims at, hence creating process predictions that are followed by good recommendations.
The remainder of this paper is structured as follows. Section \[sec:UWV\] introduces the situation faced at UWV and Section \[sec:research\_method\] shows which actions were taken, i.e., the building of a PAR and the execution of a field experiment. Section \[sec:results\_achieved\] discusses the results from the field experiment and Section \[sec:discussion\] elaborates on the lessons learned from it. Section \[sec:conclusion\] concludes the paper.
Situation Faced – The Unemployment-Benefits Process at UWV {#sec:UWV}
==========================================================
UWV is the social security institute of the Netherlands and responsible for the implementation of a number of employee-related insurances. One of the processes that UWV executes is the unemployment benefits process. When residents in the Netherlands become unemployed, they need to file a request at UWV, which then decides if they are entitled to benefits. When requests are accepted, the customers receive monthly benefits until they find a new job or until the maximum period for their entitlement is reached.
![An example scenario of the potential activities that are related to the provision of the unemployment benefits for a customer for the months June, July and August (the year is irrelevant). Each row is related to the activities needed to handle an income form for the month of the benefits. Each benefits month takes several calendar months to be handled, e.g., the benefits for the month of June are handled from June until August.[]{data-label="fig:unemployment_benefits_payment_process"}](PredMon_IKO_Process_C05.jpg){width="100.00000%"}
The unemployment benefit payment process is bound by legal rules. Customers and employees of UWV are required to perform certain steps for each specific month (hereafter income month) in which customers have an entitlement. Fig. \[fig:unemployment\_benefits\_payment\_process\] depicts a typical scenario of a customer who receives benefits, with the steps that are executed in each calendar month. Before a customer receives a payment of benefits for an income month, an *income form* has to be sent to UWV. Through this form customers specify whether or not they received any kind of income next to their benefits, and, if so, what amount. The benefits can be adjusted monthly as a function of any potential income, up to receiving no benefits if the income exceeds the amount of benefits to which the customer is entitled.
Fig. \[fig:unemployment\_benefits\_payment\_process\] clearly shows that, in October, when the reclamation is handled, two months of unemployment benefits have already been paid, possibly erroneously. Although this seems a limited amount (usually a few hundred Euros) if one looks at a single customer, it should be realized that this needs to be multiplied by tens of thousands of customers in the same situation. UWV has on average 300,000 customers with unemployment benefits of whom each month on average 4% get a reclamation.
The main cause for reclamations lie with customers not correctly filling in the amount of income earned next to their benefits on the income form. The correct amount can be obtained from the payslip. If the payslip is not yet received by the customer, they will have to fill in an estimate. However, even with a payslip it is not trivial to fill in the correct amount. The required amount is the *social security wages*, which is not equal to the gross salary and also is not equal to the salary after taxes. An other reason for not correctly filling in the income form occurs when a customer is paid every 4-weeks, instead of every month. In this case there is one month each year with two 4-weekly payments. The second payment in the month is often forgotten. Apart from the reasons mentioned, there exist many more situations in which it can be hard to determine the correct amount.
Since the reclamations are caused by customers filling in income forms incorrectly, the only thing that UWV can do is to try to prevent customers from making mistakes filling in the income form. Unfortunately, targeting all customers with unemployment benefits every month to prevent reclamations can become very expensive. Furthermore, UWV wants to limit communications to customers to only the necessary contact moments. Otherwise, communication fatigue can set in with the customers, causing important messages of UWV to have less impact with the customers. Only targeting customers with a high chance of getting a reclamation reduces costs and should not influence the effectiveness of messages of UWV. Because of all these reasons, a recommender system that could effectively identify customers with a high risk of getting a reclamation would be really helpful for UWV. That recommender system needs to be able to target risky customers and propose opportune interventions.
Action Taken – Build PAR System and Execute Field Experiment {#sec:research_method}
============================================================
Our approach for the development and test of a PAR system for UWV is illustrated in Fig. \[fig:research\_method\]. The first steps (Step 1a and 1b) of the approach are to analyze and identify the organizational issue. As described in Section \[sec:UWV\] the organizational issue at UWV is related to reclamations. The second step is to develop a recommender system, which consists of a predictor module (Step 2a) and a set of interventions (Step 2b). The predictor module is needed to identify the cases on which the interventions should be applied, namely the cases with the highest risk to have reclamations. Section \[subsec:prediction\] describes the predictor module setup.
Together with the predictor module, an appropriate set of interventions needs to be selected. Interventions need to be determined in concert with stakeholders. Only by doing this together, interventions that have the support of the stakeholders can be identified. Support for the interventions is needed to also get support for the changes necessary to implement the interventions in the process.
At UWV several possible interventions were put forward, from which one was chosen (Step 3). Only one intervention could be selected, due to the limited availability of resources at UWV to execute an experiment. Section \[subsec:mitigatingAction\] elaborates on the collecting of interventions and selection of the intervention for the field experiment.
![Overview of the steps that make up the research method. These steps correspond to one improvement cycle. The “I” is used as an abbreviation for “Intervention”.[]{data-label="fig:research_method"}](PredMon_ResearchMethod_C06_4_IST.jpg){width="100.00000%"}
The next step is to design a field experiment (Step 4). The field experiment was set up as an *A/B test* [@Kohavi2017]. In an A/B test, one or more interventions are tested under the same conditions, to find the alternative that best delivers the desired effect. In our field experiment, risk type combined with the intervention can be tested in the natural setting of the process environment. The objective of the field experiment is to determine the effect of applying an intervention for cases at a specific risk level, with respect to the specific process metrics of interest, i.e. whether or not a customer gets a reclamation. All other factors that can play a role in the field experiment are controlled, as far as this is possible in our business environment. Under these conditions, the field experiment will show if a causal relation exists between the intervention and the change in the values of the process metrics. Section \[subsec:setup\_abtest\] describes the setup for the UWV study.
The results of the field experiment are analyzed to determine if an effect can be detected from applying the intervention (Step 5). The desired effect is a reduction in the number of customers with a reclamation. Section \[subsec:intervention\] and Section \[subsec:predmod\] contain respectively the analysis of the intervention and the predictor module. If the intervention can be identified as having an effect, then both the direction of the effect, i.e. whether the intervention leads to better or worse performance, and the size of the effect need to be calculated from the data. When an intervention has the desired effect, it can be selected to become a regular part of the process. The intervention then needs to be implemented in the process (Step 6). Interventions together with the predictor module from Step 2a, make up the PAR system. After the decision to implement an intervention it is necessary to update the predictor module of the PAR system. Changing the process also implies that the situation under which the predictions are made has changed. Some period of time after the change takes effect, needs to be reserved to gather a new set of historic process data on which the predictor module can be retrained.
The final step (Step 7) is the reflective phase in which the lessons learned from the execution of the approach are discussed. Within this research method, many choices need to be made. For example, which organizational issue will be tackled and which interventions will be tested. Prior to making a choice, the research participants should be aware of any assumptions or bias that could influence their choices. Section \[sec:discussion\] contains the lessons learned for the UWV case.
Building the Predictor Module {#subsec:prediction}
-----------------------------
The prediction is based on training a predictor component which uses historical data. This component was implemented as a stand-alone application in Python and leveraged the *sci-kit learn* [@scikit-learn] library to access the data-mining functionality. For the UWV case, the historical data was extracted from the company’s systems. It relates to the execution of every activity for more than 73,000 customers who concluded the reception of unemployment benefits in the period from July 2015 until July 2017.
![An example of an event-log fragment for two of UWV’s customers. Each row refers to an event; events with the same *Customer ID* are grouped into traces and ordered by *Event date*.[]{data-label="fig:eventLog"}](PredMon_Eventlog_Example_C03_2.jpg){width="100.00000%"}
The collected information can be represented in a tabular form, of which an excerpt is presented in Fig. \[fig:eventLog\]. Each row in Fig. \[fig:eventLog\] corresponds to an event, namely the execution of an activity at a certain moment in time and refers to a customer with an identifier and other given characteristics. The table forms a classical event log [@procMiningBook]. Events referring to the same customer can be grouped by customer id and ordered by timestamp, thus obtaining a trace. For the UWV case study, the event log contained 5 million events (i.e. rows) for 73,153 customers, i.e. the event log contained 73,153 traces. Every trace refers to a complete execution of the process to provide benefits, which can end with finding a job or with reaching the end of the maximum benefit provision. A trace can contain zero, one, or more reclamation events.
The classifier of the prediction module is trained using the traces of the UWV’s event log as input. Similarly to what is proposed in [@TeinemaaDMF16; @TeinemaaDRM17], each trace $\sigma=\langle e_1, e_2, ..., e_m \rangle$ is encoded into a vector of variables that contains:
1. The number of executions of each process’ activity in $\sigma$ (one numeric variable per activity);
2. the number of months for which the unemployment benefit can be maximally given (one numeric variable);
3. the duration of the process execution in terms of number of months, i.e. the number of months existing between $e_1$ and $e_m$ (one numeric variable);
4. customer characteristics, such as age, gender, and marital status;
5. properties of the employment that triggered the unemployment benefits like the sector, the type of contract, working pattern and the reason for the dismissal;
6. the presence / absence of a reclamation (one Boolean variable) at the end of $\sigma$.
![Example of vectors that are used as instances to train the predictor. These vectors correspond to the excerpt of the event log in Fig. \[fig:eventLog\][]{data-label="fig:preprocessed_dataset"}](PredMon_Preprocessed_Dataset_C05.jpg){width="100.00000%"}
Since we want to predict running cases and the event log records completed cases, we need to consider prefixes of running process instances. Namely, if a trace $\sigma=\langle e_1, e_2, ..., e_m \rangle$ is composed by $m$ events, we build $m$ prefixes: $\langle e_1\rangle,\langle e_1, e_2, \rangle,\ldots$, $\langle e_1, e_2, ..., e_m \rangle$. These prefixes are treated as running cases, with the notable difference that the eventual, actual outcome is known. In this way, the prefixes are a suitable input for training the predictor. Fig. \[fig:preprocessed\_dataset\] shows the set of training vectors that are generated for the two traces depicted in Fig. \[fig:eventLog\]. Each prefix is encoded as mentioned above, which includes the Boolean variable about the presence / absence of a reclamation at the end of the whole $\sigma$, called *Indication of reclamation*. This variable is used as the dependent variable, where the others are used as independent variables to correlate with the dependent. As an example, the first instance refers to the execution for customer $25879$ when only the first activity *Initialize the Income Form* was considered; the second is about the same customers when the first and second activity were accounted.
Teinemaa et al. illustrate that several data-mining predictors can be used to predict the dependent variable that encodes the KPI outcome [@TeinemaaDRM17], ranging, e.g., from Decision Tree, Random Forest, and Support Vector Machine till Generalized Boosted Regression Models, Logistic Regression, and ADA Boost. For our experiments, we decided to opt for Logistic Regression and ADA Boost because they provide a predicting model that allows one to analyze which vector’s components are most heavily affecting the prediction (e.g., the beta value of Logistic Regression). As discussed in Section \[sec:UWV\], the frequency with which activities are executed for each customer is of the order of once a month. Therefore, it is not worthwhile predicting and recommending more than once a month. So only prefixes referring to entire months are retained; in other words, we train the predictor using the prefix $\langle e_1, e_2, ..., e_i \rangle$ of trace $\sigma=\langle e_1, e_2, ..., e_m \rangle$ if $e_{i+1}$ belongs to the month that follows that of $e_i$. E.g., looking at Fig. \[fig:preprocessed\_dataset\] for customer $25879$, we train on the prefixes of length 1, 4, and 7, because these represent the last prefixes before the next month starts.
The techniques based on Logistic Regression and ADA Boost were tuned through hyper-parameter optimization [@ClaesenM15]. To this end, the UWV’s event log was split into a training set with 80% of the traces and a test set with 20% of the traces. The models were learned through a 5-fold cross validation on the training set, using different configurations of the algorithm’s parameters. The models trained with different parameter configurations were tested on the second set with 20% of the traces and ranked using the area under the ROC curve (shortened as AUC) [@ROC]. AUC was chosen because it is the most suitable criterion in case of unbalanced classes: the number of customers with reclamation is just around 4% of the total number. When performing hyper-parameter optimization, we also tested two alternatives, as advised by Teinemaa et al. [@TeinemaaDMF16; @TeinemaaDRM17]. The first alternative is to train a single predictor with the vectors referring to all prefixes referring to whole months (see discussion above about the prefixes retained). The second alternative was to cluster the vectors according to the length of the corresponding prefixes in months, and to assign each vector cluster to a different predictor. Therefore, one predictor was trained of the vector of prefixes spanning over one month, one predictor with those spanning over two months, etc. The outcome of the hyper-parameter optimization was that the second alternative generally led to higher AUC values in combination with the ADA Boost technique.
Collecting and Selecting the Interventions {#subsec:mitigatingAction}
------------------------------------------
After three brainstorm sessions, with 15 employees and 2 team managers of UWV, the choice of the intervention was made by the stakeholders. As mentioned earlier, the choice of intervention was based on the experience and expectations of the stakeholders. The aim of the intervention is to prevent customers from incorrectly filling the income form. More specifically, to prevent the customer from filling in the wrong amount. The sessions initially put forward three potential types of interventions. The types are defined based on the actors that are involved in the intervention (the customer, the UWV employee, or the last employer):
1. the customer is supported in advance on how to fill the income form;
2. the UWV employee verifies the information provided by the customer in the income form, and, if necessary, corrects it after contacting the customer;
3. the last employer of the UWV customer is asked to supply relevant information more quickly, so as to be able to promptly verify the truthfulness of the information provided by the customer in the income form;
An intervention can only be executed once a month, namely between two income forms for two consecutive months. In the final brainstorming session, out of the three intervention types, the stakeholders finally opted for option 1 in the list above, i.e. supporting the customer to correctly fill the income form. Stakeholders stated that, according to their experience, their support with filling the form helps customers reduce the chance of incurring in reclamations. As mentioned earlier, only one specific intervention was selected for the experiment, due to the limited availability of resources at UWV.
The selected intervention entails pro-actively informing the customer about specific topics regarding the income form, which frequently lead to an incorrect amount. These topics relate to the definition of social security wages, financial unemployment and receiving 4-weekly payments instead of monthly payments. The UWV employees indicated that they found that most mistakes were made regarding these topics.
Next to deciding the action, the medium through which the customer would be informed, had to be determined. The options were: a physical letter, an email, or a phone call by the UWV employee. In the spirit of keeping costs low, it was decided to send the support information by email. An editorial employee of UWV designed the exact phrasing. The email contained hyperlinks to web pages of the UWV website to allow customers to obtain more insights into the support information provided in the email itself. The customers to whom the email was sent were not informed about the fact that they were targeted because they were expected to have a higher risk of getting a reclamation. A tool used by UWV to send emails to large numbers of customers at the same time provided functionality to check whether the email was received by the recipient, namely without a bounce, as well as whether the email is opened by the customer’s email client application. Since the timing of sending the message can influence the success of the action, it was decided to send it on the day preceding the last working day of the calendar month in which the predictor module marked the customer as risky. This ensured that the message could potentially be read by the customer before filling in the income form for the subsequent month.
Design and Execution of the Field Experiment {#subsec:setup_abtest}
--------------------------------------------
The experiment aims to determine whether or not the use of the PAR system would reduce the number of reclamations in the way it had been designed in terms of prediction and intervention. Specifically, we first determined the number and the nature of the customers who were monitored. Then, the involved customers were split into two groups: on one group the PAR system was applied, i.e. the experimental group, the second group was handled without the PAR system, i.e. the control group.
We conducted the experiment with 86,850 cases, who were handled by the Amsterdam branch of UWV. These were customers currently receiving benefits, and they are different from the 73,153 cases who were used to train the predictor module. Out of the 86,850 cases, 35,812 were part of the experimental group. The experiment ran from August 2017 until October 2017. On 30 August 2017, 28 September 2017 and 30 October 2017 the intervention of sending an email was executed. The predictor was used to compute the probability of having a reclamation for the 35,812 cases of the experimental group. The probability was higher than 0.8 for 6,747 cases, and the intervention was executed for those cases.
Results Achieved {#sec:results_achieved}
================
The intervention did not have a preventive effect even though the risk was predicted reasonably accurate. Sections \[subsec:intervention\] and \[subsec:predmod\] describe the details of the results achieved.
The Intervention Did Not Have a Preventive Effect {#subsec:intervention}
-------------------------------------------------
![The number of cases and percentage of cases having a reclamation for all groups. The results show that risky customers are identified, but the intervention does not really help.[]{data-label="fig:results_group"}](PredMon_Breakdown_of_results_C05_3.jpg){width="95.00000%"}
Fig. \[fig:results\_group\] shows the results of the field experiment, where the black triangles illustrate the percentage of reclamations observed in each group. The triangles at the left-most stacked bar show that the number of reclamations did not significantly decrease when the system was used, i.e. from 4.0% without using the system to 3.8% while using the system. The effectiveness of the system as a whole is therefore 0.2%.
The second bar from the left shows how the PAR system was used for the customers: 6,747 cases were deemed risky and were e-mailed. Out of these 6,747 cases, 4,065 received the emails with the links to access further information. The other 2,682 cases did not receive the email. As mentioned in Section \[subsec:mitigatingAction\] the tool that UWV uses for sending bulk email can detect whether an email is received and is opened, i.e. there was no bounce. Since there were almost no bounces, the cases that did not receive the email, actually did not open the message in their email client. From the customers who have received the email, only 294 actually clicked on the links and accessed the UWV’s web site. Remarkably, among the customers who clicked the link, 10.9% of those had a reclamation in the subsequent month: this percentage is more than 2.5 times the average. Also, it is around 1.7 times of the frequency among the customers who received the email but did not click the links.
We conducted a comparative analysis among the customers who did not receive the email, those who received it but did not click the links and, finally, those who reached the web site. The results of the comparative analysis are shown in Fig. \[fig:dif\]. The results indicate that 76.5% of the customers who clicked the email’s links had an income next to the benefits. Recall that it is possible to receive benefits even when one is employed: this is the situation when the income is reduced and the customer receives benefits for the difference. It is a reasonable result: mistakes are more frequent when filling the income form is more complex (e.g., when there is some income, indeed). Additional distinguishing features of the customers who clicked on the email’s link are that 50.3% of these customers have had a previous reclamation, as well as that these customers are on average 3.5 years older, which is a statistically significant difference.
The results even seem to suggest that emailing appears counterproductive or, at least, that there was a positive correlation between exploring the additional information provided and being involved in a reclamation in the subsequent month. To a smaller extent, if compared with the average, a higher frequency of reclamations is observed among the customers who received the email but did not click the links: 6.2% of reclamations versus a mean of 3.8-4%. A discussion on the possible reasons for these results can be found in Section \[sec:discussion\]. However, it is clear that the intervention did not achieve the intended goal.
![Comparison of characteristics of customers who did not receive the email, those who received it but did not click the link and who accessed UWV’s web site through the email’s link. []{data-label="fig:dif"}](PredMon_Difference_between_experimental_groups_C05.jpg){width="92.00000%"}
The Risk Was Predicted Reasonably Accurate {#subsec:predmod}
------------------------------------------
As already mentioned in Section \[sec:intro\] and Section \[subsec:intervention\], the analysis shows the experiment did not lead to an improvement. To understand the cause, we analyzed whether this was caused by inaccurate predictions or an ineffective intervention or both. In this section, we analyze the actual quality of the predictor module. We use the so-called *cumulative lift curve* [@DBLP:conf/kdd/LingL98] to assess the prediction model. This measure is chosen because of the imbalance in the data as advised in [@DBLP:conf/kdd/LingL98]. As mentioned before in Section \[sec:UWV\], only 4% of the customers are eventually involved in reclamations. In cases of unbalanced data sets (such as between customers with and those without reclamations), precision and recall are less suitable to assess the quality of predictors. Furthermore, because of the low cost of the intervention of sending an email, the presence of *false negatives*, here meaning those customers with undetected reclamations *during the subsequent month*, is much more severe than *false positives*, i.e. customers who are wrongly detected as going to have reclamations *during the subsequent month*.
![The cumulative lift curve shows that using the recommender system leads to a better selection of cases than using a random selection of cases.[]{data-label="fig:cum_lift_curve"}](PredMon_CumLiftCurve_C02.png){width="70.00000%"}
Fig. \[fig:cum\_lift\_curve\] shows the curve for the case study at UWV. The rationale is that, within a set of $x\%$ of randomly selected customers, one expects to observe $x\%$ of the total number of reclamations. This trend is shown as a dotted line in Fig. \[fig:cum\_lift\_curve\]. In our case, the predictions are better than random. For example, the 10% of customers with the highest risk of having a reclamation accounted for 19% of all reclamations, which is roughly twice as what can be expected in a random sample.
In summary, although the prediction technique can certainly be improved, a considerable prediction effectiveness can be observed (cf. Section \[subsec:prediction\]). However, as mentioned in Section \[subsec:intervention\], the system as a whole did not bring a significant improvement. This leads us to conclude that the lack of a significant effect should be mostly caused by the ineffectiveness of the intervention. In Section \[sec:discussion\], we discuss this in more detail.
Lessons Learned {#sec:discussion}
===============
The experiment proved to be unsuccessful. On the positive side, the predictions were reasonably accurate. However, the intervention to send an email to high risk customers did not lead to a reduction in the number of reclamations. There even was a group of customers who had twice as many reclamations as the average population. Section \[subsec:why\_not\_work\] elaborates on the reasons why the intervention did not work. Section \[subsec:what\_done\_different\] focuses on the lesson learned, delineating how the research methodology needs to be updated.
Why Did the Intervention Not Work? {#subsec:why_not_work}
----------------------------------
One of the reasons why the intervention was not successful might be related to the wrong timing of sending the email. A different moment within the month could have been more appropriate. However, this does not explain why of the 6,747 cases selected only 294 acted on the email by clicking the links. Other reasons may be that the customers might have found the email message unclear or that the links in the email body pointed to confusing information on the UWV website. In the group of 294 cases who clicked the links and who took notice of this information a reclamation actually occurred 2.5 times as much.
Also, the communication channel could be part of the cause. Sending the message by letter, or by actively calling the customer might have worked better. In fact, when discussing reasons of the failure of the experiment, we heard several comments from different stakeholders that they did not expect the failure because *“after speaking to a customer about how to fill in the income form, almost no mistakes are made by that customer”* (quoted from a stakeholder). This illustrates how the subjective feelings can be far from objective facts.
What Should be Done Differently Next Time? {#subsec:what_done_different}
------------------------------------------
We certainly learned that the A/B testing is really beneficial to assess the effectiveness of interventions. The involvement of stakeholders and other process participants, including, e.g., the UWV’s customers, is beneficial towards achieving the goal. However, the results did not achieve the expected results. We learned a number of lessons to adjust our approach that we will put in place for the next round of the experiments:
1. Creating a predictor module requires the selection of independent features as inputs to build the predictive model. From the reflection and the analysis of the reasons that caused the failure of an intervention, one can derive interesting insights into new features that should be incorporated when training the predictor. For instance, the features presented in Fig. \[fig:dif\] can be used to train a better predictor for the UWV case. These features could be, e.g., a boolean feature whether a customer has income next to the benefits.
2. The insights discussed in the previous point, which can be derived from the analysis, can also be useful to putting forward potential interventions. For instance, an intervention could be to perform a manual check of the income form when a customer has had a reclamation in the previous month. This intervention example is derived from the feature representing the number of executions of *Detect Reclamation* as discussed in Section \[subsec:intervention\].
3. Before the selection of the interventions for the A/B test (Step 3 in Fig. \[fig:research\_method\]), they need to be pre-assessed. The intervention used in our experiment is about providing information to the customers concerning specific topics related to filling the income form. In fact, before running the experiments, we could have already checked on the historical event data whether the reclamations were on average fewer when providing information and support to fill the income form. If this would had been observed, we could prevent ourselves from running experiments destined to fail.
4. Since a control group was compared with another group on which the system was employed and the comparison is measured end-to-end, it is impossible to state the reason of the failure of the intervention, beyond just observing it. For instance, we should have used questionnaires to assess the reasons of the failure: the customers that received the email should have been asked why they did not click on the links or, even if clicked, still were mistaken. Clearly, questionnaires are not applicable for all kinds of interventions. Different methods also have to be envisaged to acquire the information needed to analyze the ineffectiveness of an intervention.
5. It is unlikely that the methodology in Section \[sec:research\_method\] already provided satisfactory results because of the methodology needs to be iterated in multiple cycles. In fact, this finding is compliant with the principle of *Action Research*, which is based on idea of continuous improvement cycles [@CG@ECRM03; @Rowell2017].
6. The point above highlights the importance of having interaction cycles. However, one cycle took a few months to be carried out. This is certainly inefficient: the whole cycle needs to be repeated at high speed and multiple interventions need to be tested at each cycle. Furthermore, if an intervention is clearly ineffective, the corresponding testing needs to be stopped without waiting for the cycle to end.
All the lessons learned share one leitmotif: *accurate predictions are crucial, but their effect is nullified if it is not matched by effective recommendations, and effective recommendations must be based on evidence from historical and/or experimental data*.
![Overview of the steps that make up the *updated research method*. These steps correspond to one improvement cycle and are repeated in every cycle. The “I” is used as abbreviation for “Intervention”. The components that are changed relative to Fig. \[fig:research\_method\] have red dashed lines.[]{data-label="fig:par_system"}](PredMon_ResearchMethod_C06_5_SOL.jpg){width="100.00000%"}
In light of this, the methodology introduced in Section \[sec:research\_method\] needs to be adjusted; the resulting new methodology is shown in Fig. \[fig:par\_system\]. The changes relative to Fig. \[fig:research\_method\] are shown in red dashed lines. To show how the lessons learned have impacted the original methodology, the items of the previous list are mapped on Fig. \[fig:par\_system\] as numbers within a red circle.
The impact of adapting the research method according to the lessons learned is not limited to the identified components. For example, the second lesson has impact on the collection of interventions. Generating interventions in a data-driven manner is added to the stakeholder-driven approach. The third lesson adds the new pre-assessment step to the approach (Step 2c). The result of this step is the deselection of interventions collected in Step 2b. The fourth lesson introduces Step 3b, in which the information needed to understand the (in)effectiveness of an intervention is defined. Defining this information has an impact on the design of the A/B Test and the analysis of the results in Step 5. For example when questionnaires need to be deployed.
Lesson 5 and 6, i.e. repeat the cycle, speed it up and use multiple interventions, are not linked to one specific step. These lessons have impact on the whole approach. Since the updated approach is more elaborate than the original approach it will require more effort to execute one cycle of this method, let alone multiple cycles with multiple interventions at a high speed. Systematic support needs to be developed for all of the steps of the updated research methodology to allow for a smooth execution.
Conclusion {#sec:conclusion}
==========
When building a Process-aware Recommender system, both the predictor module and the recommender parts of the system must be effective in order for the whole system to be effective. In our case, the predictor module was accurate enough. However, the intervention did not have the desired effect. The lessons learned from the field experiment are translated into an updated research method. The updated approach asks for high speed iterations with multiple interventions. Systematic support will be needed for each step of the approach to meet these requirements.
As future work, we plan to improve the predictor module to achieve better predictions by using different techniques and leveraging on contextual information about the customer and its history, e.g., the presence of some monetary income next to the benefits is strongly causally related to reclamations. As described, we want to use evidence from the process executions, and insights from building the predictor module, to select interventions to be tested in a new experiment.
Orthogonally to a new field experiment, we aim to devise a new technique that adaptively finds the best intervention based on the specific case. Different cases might require different interventions, and the choice of the best intervention should be automatically derived from the historical facts recorded in the system’s event logs. In other words, the system will rely on machine-learning techniques that (1) reason on past executions to find the interventions that have generally been more effective in the specific cases, and (2) recommend accordingly.
|
---
author:
- 'Victoria L. Martin'
bibliography:
- 'SpecBib.bib'
title: Spectral weight in holography with momentum relaxation
---
Introduction
============
The AdS/CFT correspondence [@maldacena1999large] provides an avenue to indirectly study aspects of strongly interacting quantum field theories. A system of particular interest is the so-called non-Fermi liquid phase describing the normal state of high-temperature cuprate superconductors [@varma1989phenomenology]. Some properties of non-Fermi liquids have already been realized holographically, notably the famous linear scaling of resistivity and specific heat with temperature [@Davison:2013txa]. Another attribute endemic to non-Fermi liquids is that they form Fermi surfaces in momentum space at low temperatures, which can be seen for example by applying an external magnetic field that destroys the superconducting dome [@varma1989phenomenology].
A principal diagnostic for the presence of a Fermi surface is the low-energy spectral weight (see, for example, [@Hartnoll:2016apf]) $$\label{specweight}
\sigma(k)=\lim_{\omega\to 0}\frac{\text{Im}G^R_{\mathcal{O}\mathcal{O}}(\omega,k)}{\omega}.$$ Here the operator $\mathcal{O}$ can be, for example, the charge density $J^t$ or current $J^x$, but for our purposes we will be interested in $\mathcal{O}=J_{\parallel}$ and $\mathcal{O}=J_{\perp}$, corresponding to the transverse and longitudinal channels of the perturbed bulk fields (to be introduced in subsequent sections). There are two different senses in which (\[specweight\]) can indicate the presence of a Fermi surface. First, experimental techniques such as angle-resolved photoemission spectroscopy (ARPES) detect a Fermi surface via a pole in the retarded Green’s function of (\[specweight\]) at $k=k_F$, when $\mathcal{O}=\psi$ [@armitage2002doping]. The Green’s function in (\[specweight\]) is the UV Green’s function, and so holographically we need to consider the full bulk geometry to gain access to this pole.
Second, at low energies we have [@Hartnoll:2016apf; @Iqbal:2011ae] $$\label{proportion}
\text{Im}G^R_{\mathcal{O}\mathcal{O}}(\omega,k)\propto\text{Im}\mathcal{G}^R_{\mathcal{O}\mathcal{O}}(\omega,k).$$ While this expression allows us to directly relate the IR Green’s function $\mathcal{G}^R$ to the UV one, we lose all information about a possible pole, which is stored in the proportionality constant of (\[proportion\]). However, we can still infer the presence of low energy spectral weight via the spectral decomposition [@Hartnoll:2016apf]: $$\label{specdecom}
\text{Im}G^R_{JJ}(\omega, k)=\sum_{m,n}e^{-\beta E_m}\left|\langle n(k^{'})|J(k)|m(k^{''})\rangle\right|^2\delta(\omega-E_m+E_n).$$ The expression (\[specdecom\]) contains two delta functions, one in energy and one in momentum (resulting from the inner product). Thus we see that the spectral weight directly counts *charged* degrees of freedom (charged due to the presence of $J$) at a given frequency and momentum. In particular, for a field theory at zero temperature the presence of low (zero) energy (frequency) spectral weight at a finite momentum would suggest a remnant of the Pauli exclusion principle, even in the absense of single-particle excitations. Thus we can infer the presence or absense of a Fermi surface by considering IR data alone. A more comprehensive exposition of the two preceding paragraphs is given in the Introduction of [@Martin:2019sxc] and in Appendix \[appA\].
The spectral weight has been calculated in IR geometries in several holographic theories. For the Einstein-Maxwell-dilaton (EMD) theory in an IR hyperscaling violating geometry (characterized by dynamical critical exponent $z$ and hyperscaling violating exponent $\theta$), [@hartnoll2012spectral; @Keeler:2014lia] showed that low-energy spectral weight is exponentially suppressed. However, it was discovered that in the limit $z\rightarrow\infty$ with the ratio $\eta=-\theta/z$ held fixed the geometry develops fermionic properties. That is, low-energy spectral weight exists in these so-called semi-local quantum liquid geometries (or $\eta$ geometries for short)[^1] for EMD in $d=4$ [@anantua2012pauli] and $d>4$ [@Martin:2019sxc], the holographic superconductor [@hartnoll2008building; @Gouteraux:2016arz], and the holographic superfluid with an additional Chern-Simons term [@Martin:2019sxc]. The calculation of spectral weight in holographic superconductors [@Gouteraux:2016arz] led to some intriguing results:
1. There exists an instability at finite momentum.
2. There exists nonzero low energy spectral weight at finite momentum.
3. A Fermi shell exists[^2].
The interpretation of the first point put forward in [@Gouteraux:2016arz] is that, within a certain range of parameter space, the semi-local quantum liquid geometry is not the true ground state of this theory. Indeed, some high-temperature superconductors have been seen to exhibit a charge density wave phase that coexists with (or perhaps competes with) the superconducting phase [@wu2011magnetic]. Thus perhaps the true groundstate of our system is a spatially modulated phase[^3].
The second result is quite surprising. In the case of the holographic superconductor, the bulk charge density manifestly forms a condensate, and thus one should expect to find a corresponding vanishing spectral weight at finite momentum in the boundary field theory. However, this is not borne out in the holographic calculation of the retarded Green’s function. A clear interpretation of this seemingly paradoxical result is still an open problem. If the bulk charge density is indeed meant to correspond to the boundary charge density in a meaningful way, perhaps there are other unaccounted for bulk degrees of freedom responsible for the nonzero spectral weight.
For the third result, it has been shown more recently that these Fermi shells are more pervasive in holographic bottom-up calculations than was previously supposed [@Martin:2019sxc], at least when considering $\eta$ geometries. Fermi shells are known to appear in top-down constructions, for example in $\mathcal{N}=4$ supersymmetric Yang-Mills [@DeWolfe:2012uv] and in ABJM theory [@DeWolfe:2014ifa]. Unlike in bottom-up models, in these top-down constructions the dual field theory is explicitly known, and the Fermi shell is known to result from overlapping Fermi surfaces of two distinct species of fermions.
In this work, we investigate the extent to which the three phenomena described in the previous paragraphs (the finite $k$ instability, the nonzero low-energy spectral weight, and the presence of a Fermi shell) persist in the presence of explicitly broken translation invariance. We accomplish this by adding massless scalar fields proportional to one of the coordinates (so-called “axion" terms) $\psi_ix_i$ to the bottom-up model of the holographic superconductor $$S=\int d^4x \sqrt{-g}\left[R-\frac12\partial\phi^2-\frac14Z(\phi)F^2-\frac12Y(\phi)\sum_i\partial\psi_i^2-\frac{1}{2}W(\phi)A^2-V(\phi)\right].$$ Einstein-Maxwell-dilaton-axion (EMDA) theories have been studied previously in the contexts of neutral and charged transport [@Davison:2014lua; @Gouteraux:2014hca] and the study of shear viscosity [@Ling:2016yxy]. In this work, we study a toy model of a theory that exhibits both a spontaneously broken $U(1)$ symmetry (as in the holographic superconductor) and explicitly broken translational symmetry (by adding axion terms), Our motivation for breaking translation invariance in this way is that it provides a toy model for studying the effect analytically, subverting the need to construct more complicated phases, such as spatially modulated phases, numerically. We investigate the issue of anomalous low-energy spectral weight in the presence of a condensate found in [@Gouteraux:2016arz] by examining the effect of varying condensate charge and axion strength on the size of the Fermi surface, both separately and together. This should be regarded as a sister work to [@Martin:2019sxc].
In Section \[sec2\] we review the relevant spectral weight analysis of the holographic superconductor as carried out in [@Gouteraux:2016arz], and add to that work by addressing the effect of changing the condensate charge $W_0$ on the size of the Fermi surface. In Section \[sec3\] we compute the low energy spectral weight in the EMDA theory, and in Section \[sec4\] we put it all together and study a holographic superfluid model with explicitly broken translation invariance. We end with a discussion of our results and conclusions in Section \[sec 5\]. In Appendix \[appA\] we offer a more thorough review of the quantity (\[specweight\]) and the sense in which we use it to diagnose Pauli exclusion.
Holographic Superconductor {#sec2}
==========================
The low energy spectral weight of the holographic superconductor in the semi-local quantum liquid geometry was analyzed in [@Gouteraux:2016arz], and we refer the reader to this resource for a more detailed description. In this section we add to that work by addressing the effect of changing the condensate charge $W_0$ on the size of the Fermi surface, which we define below.
The Lagrangian describing this theory is given by $$\label{Holosup}
S=\int d^4x \sqrt{-g}\left[R-\frac12\partial\phi^2-\frac14Z(\phi)F^2-\frac{1}{2}W(\phi)A^2-V(\phi)\right],$$ Here, and in all of the theories that we will consider, we take the coefficient functions to have the following IR scaling behavior: $$\label{IRfields}
V(\phi)=V_0 e^{-\delta\phi}\,,\qquad Z(\phi)=Z_0 e^{\gamma\phi}\,,\qquad W(\phi)=W_0 e^{\epsilon\phi}\,.$$ This is to ensure that we have a scaling solution, which is motivated by top-down realizations of holographic superfluids from string theory [@Gubser:2009qm; @Gauntlett:2009dn; @Gauntlett:2009bh; @Bobev:2011rv; @DeWolfe:2015kma; @dewolfe2016gapped]. We consider a one parameter family of background geometries labeled by $\eta$: $$\label{met}
ds^2=r^{-\eta}\left(\frac{-dt^2+ dr^2}{r^2}+dx^2+dy^2\right).$$ This metric is a special limit of the hyperscaling violating geometries, labeled by dynamical critical exponent $z$ and hyperscaling violating exponent $\theta$ (see for example [@Huijse:2011ef]). The metric (\[met\]) is obtained from the hyperscaling violating one by taking $z\rightarrow\infty$ while holding $\eta\equiv-\theta/z$ fixed. Our background gauge and scalar fields have the following profiles $$A=A(r)dt, \qquad A(r)=r^{\zeta-1}, \qquad \phi(r)=\kappa\log r$$ where $\zeta$ is a constant, free parameter in the theory, and $\kappa$ is a constant that will be fixed by the background equations of motion. To recover the pure EMD theory (as studied in [@anantua2012pauli]), one fixes $\zeta=-\eta$ (this is equivalent to setting $W_0=0$).
Transverse Channel
------------------
All perturbations to the background ansatz take the plane wave form $\delta X=\delta X(r)e^{i(kx-\omega t)}$. The transverse channel is characterized by those perturbations which are odd under the transformation $y\rightarrow-y$: $$\{\delta A_y, \delta g_{ty}, \delta g_{xy}\}.$$ Throughout the paper we work in radial gauge $\delta g_{r\mu}=0$. Here we restate the result reported in [@Gouteraux:2016arz], which is the existence of low energy spectral weight below the critical momentum $k_{\star}$: $$\sigma(k)=\lim_{\omega\to 0}\frac{\text{Im}G^R_{JJ}(\omega,k)}{\omega}\propto\lim_{\omega\to 0}\omega^{2\nu_{-}-1}=\left\{
\begin{array}{ll}
\infty \qquad & k<{k_\star}\\ ~ 0\qquad & k>{k_\star}
\end{array}
\right.$$ where $$\nu_{-}=\frac{1}{2}\sqrt{5+2\eta+\eta^2+4k^2-4\sqrt{(1+\eta)^2+2(1-\zeta)k^2}}$$ and $$k_{\star}^2=\frac{1}{4} \left(-4 \zeta -\eta (\eta +2)+2 \sqrt{2 \left(2
\zeta ^2+\eta(\zeta+1) (\eta +2)\right)}\right).$$ Since $\nu_{-}$ is real in the allowed parameter space, there is no instability in the transverse channel. We say that $k_{\star}$ defines the *size* of the Fermi surface, since this is the critical momentum above which the spectral weight vanishes.
It is interesting to recast the analysis of the Fermi surface given by $k_{\star}$ in terms of the condensate charge $W_0$. This is because, from the original analysis of the holographic superconductor [@hartnoll2008holographic], we know that the critical temperature for condensation grows monotonically with the charge of the complex scalar, making it easier to condense at large charge. Thus we might expect the size of the Fermi surface $k_{\star}$ to decrease as a function of $W_0$. One caveat behind this intuition is that the presence of low energy spectral weight in the holographic superconductor is surprising in its own right, and may somehow be related to other degrees of freedom apart from the condensate. Nevertheless, we will see that the spectral weight in the transverse channel supports this naïve intuition.
The full reduced parameter space found in [@Gouteraux:2016arz] is: $$\label{initparamsp}
\left(0<\eta\leq\frac{1}{2}\left(\sqrt{5}-1\right)\text{and}-\eta<\zeta<\frac{\eta^2}{2}\right)~\text{or}~\left(\eta>\frac{1}{2}\left(\sqrt{5}-1\right)\text{and}-\eta<\zeta<\frac{1-\eta}{2}\right)$$ To translate this into a parameter space involving $W_0$, we note that the background equations of motion fix $W_0$ to be $W_0=(\zeta+\eta)(1-\zeta)$. Thus $\zeta$ has two roots: $$\label{zeta}
\zeta=\frac{1-\eta\pm\sqrt{(\eta+1)^2-4W_0}}{2}.$$ The positive root corresponds to $\zeta\rightarrow1$ as $W_0\rightarrow0$. Since $\zeta=1$ eliminates the radial scaling of the background gauge field and conflicts with much of the allowed parameter space in (\[initparamsp\]) we focus on the negative root, which recovers $\zeta\rightarrow-\eta$ as $W_0\rightarrow0$. In terms of $W_0$ the parameter space (\[initparamsp\]) is $$\begin{split}
&\left(0<\eta\leq\frac{1}{2}\left(\sqrt{5}-1\right)\text{and}~0<W_0<\frac{1}{4}\left((1+\eta)^2-(1-\eta-\eta^2)^2\right)\right)\\&\text{or}\\&\left(\eta>\frac{1}{2}\left(\sqrt{5}-1\right)~\text{and}~0<W_0<\frac{(1+\eta)^2}{4}\right).
\end{split}$$ We can now see in Figure \[fig:kstartranssup\] how $k_{\star}$ changes as a function of $W_0$. As expected, we see from the left plot that the Fermi surface is suppressed as $W_0$ increases.
Longitudinal Channel {#sec2.2}
--------------------
In this channel, the low energy spectral weight $$\sigma(k)=\lim_{\omega\to 0}\omega^{2\nu_--1}$$ becomes imaginary within a subregion of the allowed parameter space. This signals an instability, potentially toward a spatially modulated phase. We refer the reader to [@Gouteraux:2016arz] for the exact form of $\nu_-$. The region of instability is $$\label{instabreg}
\left[0<\eta \leq \frac{1}{2} \left(\sqrt{5}-1\right)\; \textrm{and}\; 0<\zeta <\frac{\eta ^2}{2}\right] \textrm{or} \left[\frac{1}{2} \left(\sqrt{5}-1\right)<\eta <1\;\textrm{and}\; 0<\zeta <\frac{1-\eta }{2}\right].$$ This region is plotted in terms of the broader allowed parameter space in Figure \[fig:parspacelong\]. Equation (\[instabreg\]) basically restricts $\zeta<0$. The instability region in terms of $W_0$ is $$\label{instabregw}
\begin{split}
&\left(0<\eta\leq\frac{1}{2}\left(\sqrt{5}-1\right)\text{and}~\eta<W_0<\frac{1}{4}\left((1+\eta)^2-(1-\eta-\eta^2)^2\right)\right)\\&\text{or}\\&\left(\eta>\frac{1}{2}\left(\sqrt{5}-1\right)~\text{and}~\eta<W_0<\frac{(1+\eta)^2}{4}\right).
\end{split}$$ Equation (\[instabregw\]) restricts $0<W_0<\eta$.
![In the green region, the exponent $\nu_-(k)$ is complex for a range of $k$, signaling a finite $k$ instability. The exponent $\nu_-(k)$ is real in the brown and blue regions for all values of $k$. In the brown region, $2\nu^--1<0$ for a range of wavevectors $k^\star_-<k<k^\star_+$, signaling the presence of a Fermi shell. In the blue region $2\nu^--1>0$, and thus no spectral weight exists in this region.[]{data-label="fig:parspacelong"}](ParSpaceLongitudinal-final){width=".45\textwidth"}
Figure \[fig:parspacelong\] also depicts the region in which a Fermi shell exists, meaning a region of low-energy spectral weight over the range of momenta $k_-<k<k_+$, as was reported in [@Gouteraux:2016arz]. This is the brown region in Figure \[fig:parspacelong\]. We are now ready to study how the size of the Fermi shell $\Delta k\equiv k_+-k_-$ changes as a function of $W_0$. We obtain different qualitative results from those found in the transverse channel. That is, the size of the Fermi shell is increasing with increasing charge $W_0$, rather than decreasing. This is shown in Figure \[longferm\]. We offer an interpretation for this in the Discussion.
{width="6.5cm"}
{width="6.5cm"}
. \[longferm\]
We note that $\Delta k$ is always finite within the brown stability region of Figure \[fig:parspacelong\].
Einstein-Maxwell-dilaton with Axions {#sec3}
====================================
In this section, we study the impact of broken translational invariance alone on the low energy spectral weight by adding so-called axion terms to the Einstein-Maxwell-dilaton theory. Specifically, we are interested in the following Lagrangian: $$\label{EMDaction}
S=\int d^4x \sqrt{-g}\left[R-\frac12\partial\phi^2-\frac14Z(\phi)F^2-\frac12Y(\phi)\sum_i\partial\psi_i^2-V(\phi)\right],$$ where $i$ runs over boundary spatial dimensions (in our case two of them, $x$ and $y$). This theory was studied in [@Gouteraux:2014hca] in the context of charge transport. To break translational invariance, we choose fields proportional to the coordinates $$\psi_i=m x_i,$$ and for simplicity we take the proportionality constant $m$ to be the same for each $x_i$. As before, we choose the following IR behavior that yields a scaling solution: $$\label{IRfields}
V(\phi)=V_0 e^{-\delta\phi}\,,\qquad Z(\phi)=Z_0 e^{\gamma\phi}\,,\qquad Y(\phi)=Y_0 e^{\lambda\phi}\,.$$ For the rest of the analysis we are free to set $Z_0=1$ and $Y_0=1$. Our background parameters obey the following constraints: $$\label{background param}
\begin{split}
&A=\frac{\sqrt{2 \eta -m^2+2}}{\eta +1} r^{-\eta-1}\,,\quad V_0=-(\eta +1)^2-\frac{m^2}{2}, \quad \kappa=\sqrt{\eta(2+\eta)}\,\\
& \lambda=0\,,\quad \kappa\delta=-\eta\,,\quad\kappa\gamma=\eta\,.
\end{split}$$ The resulting parameter space for this theory is $$\label{axpar}
\eta>0 \qquad \text{and} \qquad -\sqrt{2+2\eta}<m<\sqrt{2+2\eta}.$$ Radial deformations do not impose any further constraints on the parameter space.
Transverse Channel {#3.1}
------------------
We first consider the transverse channel, and include the following perturbations: $$\{\delta A_y, \delta g_{ty}, \delta g_{ry}, \delta g_{xy}, \delta \psi_y\}.$$ The $y$ in the scalar $\delta \psi_y$ is a distinguishing subscript and not meant to indicate a vector component. All perturbations take the plane wave form $\delta X=\delta X(r)e^{i(kx-\omega t)}$. We work in radial gauge $\delta g_{\mu r}=0$. We wish to calculate the scaling exponent $\nu_-$ of the spectral weight: $$\sigma(k)=\lim_{\omega\to 0}\frac{\text{Im}G^R_{JJ}(\omega,k)}{\omega}\propto\lim_{\omega\to 0}\omega^{2\nu_{-}-1}.$$ To achieve this, we define the following scaling behavior for the perturbations: $$\begin{aligned}
\delta A_y=a_0r^{a_1}, \qquad \delta g_{ty}=t_0r^{t_1}, \qquad \delta g_{xy}=x_0r^{x_1}, \qquad \delta \psi_y=\psi_0r^{\psi_1}.\end{aligned}$$ A scaling analysis of the perturbed equations of motion relate the above exponents, and the constants $x_0, \psi_0$ and $\psi_1$ drop out or decouple from the rest of the equations. Therefore, taking a coefficient array of the two remaining equations in terms of $a_0$ and $t_0$ and setting the determinant to zero allows us to solve for the radial scaling: $$\label{transscale}
a_1=\frac{1}{2} \left(1-\eta \pm \sqrt{5+\eta ^2+2 \eta +4 k^2\pm 4 \sqrt{\eta ^2+2 \eta \left(k^2+1\right)+k^2
\left(2-m^2\right)+1}}\right).$$ We are interested in $\nu_-$, which is given by $$2\nu_-=\sqrt{5+\eta ^2+2 \eta +4 k^2 - 4 \sqrt{\eta ^2+2 \eta \left(k^2+1\right)+k^2
\left(2-m^2\right)+1}}.$$ This exponent is always real within our parameter space. This means that there is no instability in the transverse channel, which was also the case for the holographic superconductor. The critical wave number is found by solving the equation $2\nu_--1=0$ for $k$: $$k_{\star}^2=\frac{1}{8} \left(-2 \eta ^2+4 \eta -4 m^2\pm 2 \sqrt{2} \sqrt{-4 \eta ^3+4 \eta ^2+6 \eta +2 m^4+\left(2 \eta ^2-4 \eta +1\right) m^2-2}-1\right).$$ We can see that $k_{\star}$ vanishes at the values $$\eta=\{-4, -2, 0, 2\}.$$ The parameter $\eta$ is constrained to be positive by the null energy condition, however.
In the transverse channel, we see that the larger $m$ gets, the more the spectral weight is suppressed. This is similar to the effect of the parameter $W_0$ that we saw previously for the holographic superconductor. Indeed, we will see just how similarly the effects of these two terms are on spectral weight in the next section. The spectral weight is never suppressed completely, as our parameter space (\[axpar\]) constrains us to consider only $|m|<\sqrt{2+2\eta}$.
Longitudinal channel {#sec3.2}
--------------------
In the longitudinal channel, the perturbation variables are: $$\{\delta A_t, \delta A_x, \delta g_{tt}, \delta g_{tx}, \delta g_{xx}, \delta g_{yy}, \delta \psi_x, \delta \phi\}.$$ We chose radial gauge $\delta g_{\mu r}=A_r=0$. The modes $\delta A_x$ and $\delta g_{tx}$ decouple from the rest, and thus we can set them to zero.
As in the transverse channel, all perturbations take the plane wave form $\delta X=\delta X(r)e^{i(kx-\omega t)}$, and we define the scaling behavior for the perturbations as: $$\begin{split}
&\delta A_t=a_0r^{a_1}, \qquad \delta g_{tt}=t_0r^{t_1}, \qquad \delta g_{xx}=x_0r^{x_1},\\
&\qquad \delta g_{yy}=y_0r^{y_1}, \qquad \delta \psi_x=\psi_0r^{\psi_1}, \qquad \delta \phi=\phi_0r^{\phi_1}.
\end{split}$$ As before, we can use a scaling analysis to obtain the radial scaling of interest: $$\begin{split}
&\nu_0=\frac{1}{2} \left(\sqrt{(1+\eta)^2 +4 k^2+4 m^2}\right)\\
&\nu_\pm=\frac{1}{2}\sqrt{\frac{ \left(\eta^3 +12 \eta ^2+21 \eta +10 +4k^2(\eta+2)-2 \eta m^2\pm 2 \sqrt{X}\right)}{(\eta +2)}}.
\end{split}$$ where $$X=8 k^2 (\eta +1) (\eta +2) \left(2 \eta -m^2+2\right)+\left(\eta \left(4 \eta -m^2+8\right)+4\right)^2.$$ There are two major differences in the effects of broken $U(1)$ symmetry (as in the holographic superconductor) and broken translation invariance (as in the EMD plus axion theory) on the longitudinal channel. First, unlike for the holographic superconductor, here we find no instability in the longitudinal channel (i.e. $\nu_-$ is always real). Second, in the EMD plus axion case, there is no low energy spectral weight for any $m$. This generalizes the result found in [@anantua2012pauli] for the pure EMD theory in four dimensions.
Axion with Massive Vector {#sec4}
=========================
We are now ready to consider the Einstein-Maxwell-dilaton theory with a massive vector that breaks $U(1)$ symmetry and a massless scalar that breaks translation invariance: $$\label{EMDaction}
S=\int d^4x \sqrt{-g}\left[R-\frac12\partial\phi^2-\frac14Z(\phi)F^2-\frac12Y(\phi)\sum_i\partial\psi_i^2-\frac{1}{2}W(\phi)A^2-V(\phi)\right].$$ As in Section \[sec3\], we choose the axion ansatz $\psi_i=m x_i$, and the following IR scaling behavior for the action: $$\label{IRfields}
V(\phi)=V_0 e^{-\delta\phi}\,,\qquad Z(\phi)=Z_0 e^{\gamma\phi}\,,\qquad W(\phi)=W_0 e^{\epsilon\phi}\,,\qquad Y(\phi)=Y_0 e^{\lambda\phi}\,.$$ Henceforth we set $Z_0=1$ and $Y_0=1$.
Our metric and fields take the form: $$\label{metric}
ds^2=r^{-\eta}\left(\frac{-dt^2+ dr^2}{r^2}+dx^2+dy^2\right),\qquad A=A(r)\text{d}t,\qquad \phi(r)=\kappa\log r$$ and our background parameters obey the constraints: $$\begin{split}
&A=\sqrt{\frac{m^2-2 (\eta +1)}{(\zeta -1) (\eta +1)}} r^{\zeta-1}\,,\quad \kappa=\sqrt{\frac{\zeta \left(m^2-2 (\eta +1)\right)+\eta \left(\eta ^2+\eta +m^2\right)}{\eta +1}}\, \\
&V_0=-\frac{2 (\eta +1) \left(-\zeta +\eta ^2+\eta +1\right)+m^2 (\zeta +2 \eta +1)}{2 (\eta +1)}, \quad \kappa\delta=-\eta\,,\quad\kappa\gamma=\eta,\, \\
&W_0=(1-\zeta ) (\zeta +\eta ),\, \quad \lambda=0\,.
\end{split}$$
The parameter space arising from the reality of these background quantities, imposing $V_0<0$ and $W_0>0$, and from the null energy condition is
\[1\][>m[\#1]{}]{}
[|c|c|c |@m[0pt]{}@]{} & $-\eta<\zeta\leq\frac{\eta^2}{2}$ & $-\sqrt{2+2\eta}\leq m\leq\sqrt{2+2\eta}$\
& $\frac{\eta^2}{2}<\zeta<1$ & $-\sqrt{2+2\eta}\leq m<-\sqrt{\frac{(1+\eta)(2\zeta-\eta^2)}{\zeta+\eta}}$, $\sqrt{\frac{(1+\eta)(2\zeta-\eta^2)}{\zeta+\eta}}<m\leq\sqrt{2+2\eta}$\
$\sqrt{2}\leq\eta$ & $-\eta<\zeta<1$ & $-\sqrt{2+2\eta}\leq m\leq\sqrt{2+2\eta}$\
\[paramspace\]
This is not the full parameter space, however. We must also consider radial deformations to the background (\[metric\]) of the form $$ds^2=-D(r)dt^2+ B(r)dr^2+C(r)(dx^2+dy^2),\qquad A=\tilde{A}(r)\text{d}t,\qquad \phi(r)=\tilde{\phi}(r)$$ with $$\begin{split}
&D(r)=r^{-\eta-2}(1+\epsilon D1 r^{\beta}), ~ B(r)=r^{-\eta-2}(1+\epsilon B1 r^{\beta}), ~ C(r)=r^{-\eta}(1+\epsilon C1 r^{\beta}),\\
&\tilde{A}(r)=r^{\zeta-1}(1+\epsilon A1r^{\beta}), \qquad \tilde{\phi}(r)=\log(r^{\kappa}(1+\epsilon\phi 1r^{\beta}))
\end{split}$$ and $D1$, $B1$, $C1$, $A1$, $\phi 1$ are constants. There are three pairs of radial deformations, each pair summing to $1+\eta$. One of the pairs is just $(0,1+\eta)$, while the other two are $$\label{bpmpm}
\beta_{\pm,\pm}=\frac{1}{2} \left(1+\eta \pm\sqrt{\frac{A\pm2 C}{(\eta +1)^2
S}}\right),$$ where $A(\eta, \zeta, m)$, $B(\eta, \zeta, m)$ and $S(\eta, \zeta, m)$ are given in the Appendix \[append\]. Note that in the case of the holographic superconductor, the mode $(0,1+\eta)$ is doubly degenerate. That is, we had the freedom to write two of the constants (say $D1$ and $\phi 1$) in terms of the other three. In particular, $C1$ was a free parameter. The axion term forbids us from choosing $C1$ independently of the other constants. This is because our ansatz $\psi=mx$ should be kept fixed. One might imagine that one could simply undo a rescaling of $x$ with an appropriate rescaling of $m$, but because the other constants $D1$, etc depend on $m$, this is not an independent rescaling. To analyze the parameter space resulting from these deformations, we first need to ensure that all of the $\beta$s are real. We then require that we have two irrelevant modes (corresponding to $\beta<0$). There are only two modes that have a possibility of being negative, namely $\beta_{-+}$ and $\beta_{--}$. Since $\beta_{-+}<\beta_{--}$, it is enough to require that $\beta_{--}<0$. The resulting parameter space is too complicated to write down in closed form, but a portion of it is rendered in Figure \[paramsp\]. This can be compared with the parameter space for the holographic superfluid ($m=0$) given in Figure \[fig:parspacelong\].
![**Left:** A representative portion of the allowed parameter space for the EMD theory with broken $U(1)$ and translation symmetries. **Right:** A subregion of the allowed parameter space (with $0<m^2<1$) that will be useful for comparisons below.[]{data-label="paramsp"}](Fullparam){width="7cm"}
![**Left:** A representative portion of the allowed parameter space for the EMD theory with broken $U(1)$ and translation symmetries. **Right:** A subregion of the allowed parameter space (with $0<m^2<1$) that will be useful for comparisons below.[]{data-label="paramsp"}](ZoomedFullparam){width="7cm"}
We see that the effect of $|m|>0$ is to increase our allowed parameter space to include larger positive values of $\zeta$ (although the bound $\zeta<1$ reported in the table above still holds).
Transverse Channel {#transverse-channel-1}
------------------
We first consider the transverse channel, with the following perturbations: $$\{\delta A_y, \delta g_{ty}, \delta g_{xy}, \delta \psi_y\}.$$ Again, the $y$ subscript in the scalar perturbation $\delta \psi_y$ is just a distinguishing subscript and not a vector index. All perturbations take the plane wave form $\delta X=\delta X(r)e^{i(kx-\omega t)}$. We endow the perturbations with scaling profiles: $$\begin{aligned}
\delta A_y=a_0r^{a_1}, \qquad \delta g_{ty}=t_0r^{t_1}, \qquad \delta g_{xy}=x_0r^{x_1}, \qquad \delta \psi_y=\psi_0r^{\psi_1}.\end{aligned}$$ Redoing the scaling analysis of Section \[3.1\], we obtain the scaling exponent for the holographic superfluid with broken translational symmetry: $$a_1=\frac{1}{2} \left(1+2 \zeta +\eta \pm\sqrt{\eta ^2+2 \eta +4 k ^2+5+\frac{ 2m^2 (\zeta +\eta )\pm 2 X_1}{\eta +1}}\right)$$ where $$X_1=\sqrt{(m^2(\zeta+\eta)-2(1+\eta)^2)^2-4k^2(1+\eta)(\zeta-1)(2+2\eta-m^2)}$$ which gives $$2\nu_-=\sqrt{\eta ^2+2 \eta +4 k ^2+5+\frac{ 2m^2 (\zeta +\eta )- 2X_1}{\eta +1}}.$$ The exponent $\nu_-$ is always real within our parameter space, signaling again that there are no instabilities in this channel. When $m=0$ we reproduce the result obtained in [@Gouteraux:2016arz]. The low energy spectral weight for the transverse channel is thus $$\sigma(k)=\lim_{\omega\to 0}\frac{\text{Im}G^R_{JJ}(\omega,k)}{\omega}=\left\{
\begin{array}{ll}
\infty \qquad & ~~k<{k_\star}\\ ~ 0\qquad & ~~k>{k_\star}
\end{array}
\right.$$ where $$k_{\star}^2=\frac{1}{4} \left(-4 \zeta -\eta (\eta +2)-2 m^2+2 \sqrt{2 \left(2
\zeta ^2+\eta(\zeta+1) (\eta +2)\right)+Y_m}\right)$$ and $$Y_m=m^4-\frac{m^2 ( \eta (\zeta+3) (\eta +2)+4\zeta)}{\eta +1}.$$ Unlike the holographic superfluid case [@Gouteraux:2016arz], a nonzero axion term forbids $k_{\star}$ from vanishing at $\eta=2$. However, it does vanish at the special value $$\zeta=\frac{\eta(2-4m^2+\eta-\eta^2)}{4m^2},$$ which is nonzero inside the parameter space.
In Figure \[fig:wchange\]
we again see that the axion term and the vector mass term affect the critical momentum $k_{\star}$ in much the same way, that is to suppress low-energy spectral weight as their magnitudes grow. Indeed, the effect of one term barely seems to influence the other: the two effects do not appreciably mix in this channel. The reader will find that this is not the case in the longitudinal channel, however.
Longitudinal channel {#sec4.2}
--------------------
Now we turn to the longitudinal channel. The perturbation variables are: $$\{\delta A_t, \delta A_x, \delta g_{tt}, \delta g_{tx}, \delta g_{xx}, \delta g_{yy}, \delta \psi_x, \delta \phi\}$$ The modes $\delta A_x$ and $\delta g_{tx}$ decouple from the rest, and thus we can set them to zero.
As in the transverse channel, all perturbations take the plane wave form $\delta X=\delta X(r)e^{i(kx-\omega t)}$, and we define the scaling behavior for the perturbations as: $$\begin{split}
&\delta A_t=a_0r^{a_1}, \qquad \delta g_{tt}=t_0r^{t_1}, \qquad \delta g_{xx}=x_0r^{x_1},\\
&\qquad \delta g_{yy}=y_0r^{y_1}, \qquad \delta \psi_x=\psi_0r^{\psi_1}, \qquad \delta \phi=\phi_0r^{\phi_1}.
\end{split}$$ As before, we can use a scaling analysis to obtain the radial scaling of interest. Setting $m=0$ reproduces the result found in [@Gouteraux:2016arz].
For the longitudinal channel we again expect three scaling exponents: $\nu_0$ and $\nu_{\pm}$. In this case the closed form of the $\nu$ exponents are too complicated to report here, but they are of the form: $$\nu_{Y_i}=\frac{1}{2}\Bigg(\sqrt{(1+\eta)^2+4k^2+Y_i}\Bigg)$$ where the $Y_i$ are solutions to the cubic equation $$\label{poly}
aY_i^3+bY_i^2+cY_i+d=0,$$ with $$\label{coeff}
\begin{split}
a=&(\eta +1)^2 \left(\zeta \left(m^2-2 (\eta +1)\right)+\eta \left(\eta ^2+\eta +m^2\right)\right)\\
b=&-16\zeta ^3 (\eta +1)^2 \left(m^2-2 (\eta +1)\right)-8\zeta ^2(1+\eta) \left(m^2-2 (\eta +1)\right) \left((\eta +1) (2 \eta -3)+m^2\right)\\
&-4\zeta(1+\eta) \left(3
\eta ^3+\eta \left(4 m^2-1\right)+2\right) \left(m^2-2 (\eta +1)\right)-8 \eta(1+\eta) \left(\eta \left(m^2-\eta \right)+1\right) \left(\eta ^2+\eta
+m^2\right)\\
c=& -32 \left(m^2-2 (\eta +1)\right)(\zeta -1) (\eta +1)^2 k^2 \left(2 \zeta \eta +\zeta (2 \zeta -1)+\eta ^2\right)\\
&+16 \left(m^2-2 (\eta +1)\right)(\eta +1) m^2 (\zeta +\eta ) \left((\eta
+1) \left(2 \zeta ^2+2 \zeta (\eta -1)+(\eta -1) \eta \right)-(\zeta -1) k^2\right)\\
&+16 \left(m^2-2 (\eta +1)\right)m^4 (\zeta +\eta )^3\\
d=&-64 k^2 m^2 (1-\zeta) (\zeta +\eta )^2 \left(m^2-2\eta-2\right) \left(2 \eta ^2+4 \eta +m^2+2\right).
\end{split}$$ It is the $d$ term in (\[poly\]) that complicates the scaling exponent solution substantially compared with our previous cases. We see that $d$ depends upon both of our main parameters of interest: the translation-breaking axion parameter $m$ and the condensate charge $W_0=(1-\zeta)(\zeta+\eta)$. Unlike in the transverse channel, here our scaling exponents $\nu_{Y_i}$ depend heavily on the how the effects of the axion and the condensate terms act together.
Nevertheless, we can still analyze the spectral weight in this channel numerically. We begin by determining how the instability region for the holographic superfluid reported in Figure \[fig:parspacelong\] changes in the presence of a symmetry breaking axion term. This new instability region is presented in Figure \[instaball\]. We see that the axion strength $m$ allows for a larger viable parameter space (as reported in Figure \[paramsp\]) and thus an augmented instability region is possible. Note however that this instability region still only exist for $\zeta>0$, as was the case for the holographic superfluid. Unlike for the holographic superfluid, though, we now see that the stable region is not restricted to $\zeta<0$. The new stability region that exists for $\zeta>0$, which is partially depicted in Figure \[instaball\], grows steeply with increasing $|m|$.
![**Left:** The instability region for the EMD theory with $U(1)$ and translational symmetries broken. Here $x=m^2$. The presence of the axion strength $m$ allows for an augmented parameter space, and thus a richer instability structure. **Right:** A new stability region that exists for $\zeta>0$, appearing for $m\neq0$.[]{data-label="instaball"}](instaball){width="7cm"}
![**Left:** The instability region for the EMD theory with $U(1)$ and translational symmetries broken. Here $x=m^2$. The presence of the axion strength $m$ allows for an augmented parameter space, and thus a richer instability structure. **Right:** A new stability region that exists for $\zeta>0$, appearing for $m\neq0$.[]{data-label="instaball"}](staball){width="7cm"}
We now turn to the question of whether there exists low-energy spectral weight at finite momentum $k$ in the longitudinal channel, either in the form of a smeared Fermi surface or a Fermi shell. As before, the condition for nonzero spectral weight is $2\nu_{-}-1<0$. We will begin by presenting our results for the spectral weight in the presence of both the translation symmetry breaking axion term and the $U(1)$ symmetry breaking massive vector, and then compare these results to those presented for the holographic superfluid (in Section \[sec2.2\] and in [@Gouteraux:2016arz]) and for the axion alone (in Section \[sec3.2\]). The main results for the holographic superfluid in Section \[sec2.2\] that we would like to keep in mind are:
1. A finite $k$ instability appears for $\zeta>0$, effectively restricting our analysis of low-energy spectral weight to the region $\zeta<0$ (\[instabreg\]).
2. For an appropriate region of the parameter space (Figure \[fig:parspacelong\]) we see a Fermi shell, rather than a smeared Fermi surface.
3. The Fermi shell width ($\Delta k\equiv k_+-k_-$) increases with decreasing charge $W_0$.
The main results for the EMD plus axion theory in Section \[sec3.2\] to remember are:
1. In the longitudinal channel there is no spectral weight for any $m$.
2. All values of $m$ in the region $0<|m|<\sqrt{2+2\eta}$ are allowed.
The results for the longitudinal low-energy spectral weight (for the representative value $\eta=1$) are presented in Figure \[kstarlast\].
![Low-energy spectral weight results for $\eta=1$. **Left:** Non-zero low-energy spectral weight corresponds to the exponent $\nu_-$ dipping below the $\nu_-=1/2$ plane. In this plot $\zeta=-.04$. For small $|m|$ we have a Fermi shell, and for large enough $|m|$ spectral wieght is suppressed. **Right:** Contour plots of the intersection of $\nu_-$ with the $\nu_-=1/2$ plane for various values of $\zeta$.[]{data-label="kstarlast"}](last3d){width="7.5cm"}
![Low-energy spectral weight results for $\eta=1$. **Left:** Non-zero low-energy spectral weight corresponds to the exponent $\nu_-$ dipping below the $\nu_-=1/2$ plane. In this plot $\zeta=-.04$. For small $|m|$ we have a Fermi shell, and for large enough $|m|$ spectral wieght is suppressed. **Right:** Contour plots of the intersection of $\nu_-$ with the $\nu_-=1/2$ plane for various values of $\zeta$.[]{data-label="kstarlast"}](kstarlast){width="7.5cm"}
Non-zero spectral weight corresponds to the scaling exponent $\nu_{-}$ dipping below the $\nu_{-}=1/2$ plane. For negative $\zeta$ there are two distinct regions of interest. For small enough $\zeta$ (approximately between $-1<\zeta<-.07$ for $\eta=1$) there is no spectral weight for any $m$. This generalizes the result (i) above (which corresponded to $\zeta=-1$, since $\eta=1$ and $\zeta=-\eta$ means $W_0=0$) to a range of $\zeta$. For larger $\zeta$, in the approximate region $-.07<\zeta<0$, we have a Fermi shell (as in point 2) that increases in size with decreasing $W_0$ (as in point 3) but decreases with increasing $m$. Comparing with Figure \[fig:parspacelong\] for the holographic superfluid, we see that the presence of the axion parameter $m$ does not significantly affect the parameter space region that supports low-energy spectral weight, despite the fact that increasing $m$ decreases the shell width.
In the holographic superfluid, positive $\zeta$ was not allowed due to the finite $k$ instability (point 1). However, the axion term allows for stable theories with positive $\zeta$, the price being that not all $m$ in the region given in point (ii) are allowed. For some values of $\zeta>0$ the spectral weight is still a Fermi shell, but when $\zeta$ gets large enough our contour becomes monotonic in $k$, and we have a smeared Fermi surface.
Discussion {#sec 5}
==========
Here we have examined the low-energy spectral weight and stability structure of three bottom-up models: the holographic superfluid characterized by broken $U(1)$ symmetry, the EMD plus axion theory which spontaneously breaks translation symmetry, and the holographic superfluid plus axion theory in which both symmetries are broken. We find that the results for the transverse channels of these theories are largely the same. There is never any instability in the transverse channel, and there is always a smeared Fermi surface. We also find that the condensate charge $W_0$ and the axion strength $m$ have the same effect: the Fermi surface size $k_{\star}$ decreases with increasing $W_0$ and $m$. As discussed in Section \[sec2\], this aligns with the naïve intuition that it should be easier for the scalar to condense at large charge.
The longitudinal channels give more diverse results. In the EMD plus axion theory of Section \[sec3.2\], there is no low-energy spectral weight for any $m$ (though this restriction may be lifted when considering a higher number of spacetime dimensions; see [@Martin:2019sxc] for an example). There is also no instability in this theory for any $m$. Thus it is the $U(1)$ symmetry breaking term $W_0$ that drives both the existence of Fermi shells and the presence of an instability at finite momentum $k$. However, once these phenomena are present, the axion strength $m$ affects the structure of the spectral weight and the instability region, as seen by comparing the results of Sections \[sec2.2\] and \[sec4.2\]. Namely, increasing $|m|$ augments the instability region that was present for the holographic superfluid to include $\zeta>0$ (Figure \[instaball\]) and suppresses low-energy spectral weight for each $\zeta$ (Figure \[kstarlast\]).
Note that our expectation that large charge $W_0$ should facilitate condensation, and thus shrink the size of the Fermi surface, was not borne out in the longitudinal channels of Sections \[sec2.2\] and \[sec4.2\] where Fermi shells are present. That is, we note from Figure \[longferm\] that only $k_+$ increases with $W_0$, while $k_-$ decreases as was naïvely anticipated. One possible explanation for this lies in fact that we think of these Fermi shells (or nested Fermi surfaces) as *smeared*. This is is contrast to the sharply defined Fermi surface that exists for free fermions at zero temperature. Perhaps this smearing is telling us that it is some intermediate value of $k$ between $k_+$ and $k_-$ that is of true physical interest. Consider Figure \[kstarlast\], for example. While it’s true that the Fermi shell width $\Delta k=k_+-k_-$ increases with each increasing $\zeta$ curve, the peak of each $\zeta$ curve shifts to the left, as one might anticipate according to the discussion above. In future work it will be desirable to formulate a connection between bottom-up models exhibiting Fermi shells and the top-down constructions containing Fermi shells, such as those mentioned in the Introduction [@DeWolfe:2012uv; @DeWolfe:2014ifa].
We would like to thank Blaise Goutéraux in particular for many useful discussions and contributions regarding this work. We also thank Sean Hartnoll for insightful comments and Nikhil Monga for helpful contributions.
Supplemental material {#append}
=====================
$$A=B+2 \zeta \eta R \left(3 \eta ^2+2 \eta +2 m^2-1\right)+S \left(\eta \left((\eta -1) \eta +4 m^2+3\right)+5\right)$$
$$B=4 \zeta ^2 R \left((\eta +1) (2 \eta -3)+m^2\right)+8 \zeta^3 (\eta
+1) R$$
$$C=\sqrt{D^2-4 m^2 R S (\zeta +\eta ) \left(\zeta ^2 \left(2 (\eta +1)^2+m^2\right)+(2 \zeta +\eta ) (-\eta -\zeta R+S-1)\right)}$$
$$D=\left(\frac{B}{2}+2 \eta \left(\eta \left(m^2-\eta \right)+1\right) \left(\eta ^2+\eta +m^2\right)+\zeta R \left(3 \eta ^3+\eta \left(4
m^2-1\right)+2\right)\right)$$
$$R= \left(m^2-2 (\eta +1)\right)$$
$$S= \left(\zeta R+\eta \left(\eta ^2+\eta +m^2\right)\right).$$
Review of spectral weight {#appA}
=========================
What is the spectral weight?
----------------------------
Here we motivate the quantity that we are calculating, the spectral weight: $$S(k)=\frac{\text{Im}G^R_{\mathcal{O}\mathcal{O}}(\omega,k)}{\omega}.$$ We reserve the symbol $\sigma$ to denote the *low energy* spectral weight: $$\sigma(k)=\lim_{\omega\rightarrow0}\frac{\text{Im}G^R_{\mathcal{O}\mathcal{O}}(\omega,k)}{\omega}.$$ Possible operators of interest are $\mathcal{O}=J^t$, in which case $G^R_{J^tJ^t}(\omega, k)$ is the density-density correlation function, and $\mathcal{O}=J^x$, in which case $G^R_{J^xJ^x}(\omega, k)$ is a current-current correlator. In a fermionic theory with $\mathcal{O}=\psi$, the Green’s function is the fermion propagator, and a Fermi surface corresponds to a pole in this quantity at the Fermi momentum $k=k_F$. In this work we compute the Green’s function for generalized current operators $\mathcal{O}=J^{\parallel}$ and $\mathcal{O}=J^{\perp}$.
What does ARPES measure?
------------------------
In this subsection we follow the discussion presented in [@Iqbal:2011in; @Iqbal:2011ae]. Angle-resolved photoemission spectroscopy (ARPES) is a measurement technique that directly probes the distribution of electrons in a medium. That is, by ejecting electrons from a sample, ARPES measures the density of single-particle electron excitations governed by the fermion propagator $G^R_{\psi\psi}(\omega,k)$, or more directly the *single-particle spectral function* $$A(\omega,k)\equiv-\frac{1}{\pi}\text{Im}G^R_{\psi\psi}(\omega,k).$$
A pole in the spectral function $A(\omega, k)$ as $\omega\rightarrow0$ signifies the presence of a Fermi surface. This is immediately clear in the case of free fermions, where the propagator is $$\label{freeprop}
G^R_{\psi\psi}=\frac{1}{\omega-\xi(k)+i\epsilon},$$ where $$\xi(k)=\frac{k^2}{2m}-\mu=\frac{k^2}{2m}-\frac{k^2_F}{2m}=v_F(k-k_F).$$ By examining equation (\[freeprop\]), we see that the low energy pole occurs at $k=k_F$. The correspondence between a pole in $G^R_{\psi\psi}$ and the existence of a Fermi surface also exists in interacting theories (even strongly interacting theories), in which the propagator becomes $$\label{intprop}
G^R_{\psi\psi}=\frac{Z}{\omega-v_F(k-k_F)+\Sigma(\omega,k)}.$$ In (\[intprop\]), $Z$ is called the quasi-particle weight and $$\Sigma(\omega,k)=\frac{i\Gamma}{2}$$ is the self-energy, with $\Gamma$ the particle decay rate. In fact, experiments have shown that (\[intprop\]) is the form that the propagator takes in the now famous “strange metal” phase of certain high $T_c$ cuprate superconductors [@abrahams2000angle], with $$\Sigma(\omega)=C\omega\log\omega+D\omega,$$ where $C$ is real and $D$ is complex. This matches a theoretical model known as a $\emph{marginal Fermi liquid}$ [@varma1989phenomenology]. For clarity, the scaling of the imaginary part of the self-energy with $\omega$ for various theories is given in Table \[scaltab\].
---------------------------------------------------- ---------------------------------------
Fermi liquid Im$\Sigma(\omega)\sim\omega^2$
Semi-local quantum liquid Im$\Sigma(\omega)\sim\omega^{2\nu_k}$
Strange metal (marginal Fermi liquid, $\nu_k=1/2$) Im$\Sigma(\omega)\sim\omega$.
---------------------------------------------------- ---------------------------------------
: The scaling of the imaginary part of the self-energy for Fermi liquid theory, the semi-local quantum liquid, and the marginal Fermi liquid. The exponent $\nu_k$ is related to the conformal dimension of the dual operator by $\delta_k=\nu_k+\frac{1}{2}$.[]{data-label="scaltab"}
What do we measure in this paper?
---------------------------------
In holographic calculations, there are at least two distinct ways to search for the presence of a Fermi surface (or, more generally, the presence of Pauli exclusion). The first method is to directly compute the single-particle spectral function $A(\omega, k)$ in the bulk and see if it has a pole at some momentum $k_F$ as $\omega\rightarrow0$. Calculating $A(\omega, k)$ requires knowledge of “UV” or near-boundary data ($G^R_{\psi\psi}$ is the UV propagator), and so in practice one must
1. Consider a theory with at least one bulk fermion $\psi$.
2. Linearly perturb the bulk fields (for example $\psi\rightarrow\psi+\delta\psi$).
3. Solve the Dirac equation for the perturbed fields over the entire spacetime (this can be done numerically if necessary).
4. Read off the IR propagator via the standard holographic relationship $$G^R_{\psi\psi}(\omega, k)\propto\frac{\psi_{(1)}}{\psi_{(0)}},$$ where $\psi_{(0)}$ and $\psi_{(1)}$ are obtained from the near boundary expansion of the perturbed field $$\delta\psi(z\rightarrow0)=\frac{\psi_{(0)}}{L^{d/2}}z^{d-1-\Delta_k}+...+\frac{\psi_{(1)}}{L^{d/2}}z^{\Delta_k}$$ for a $d+2$-dimensional bulk spacetime. $L$ is the AdS radius, and $\psi_{(0)}$ and $\psi_{(1)}$ are constants in the radial coordinate $z$ but depend upon $\omega$ and $k$ (see for example [@Hartnoll:2016apf] for a review of these concepts).
This was the approach taken in [@Lee:2008xf; @Liu:2009dm; @Cubrovic:2009ye; @Cremonini:2018xgj].
The second method differs from the preceding one in several ways. First,we do not include any explicit bulk fermions $\psi$. Second, instead of looking at propagators of our bulk fields, we are interested in more general correlation functions $G^R_{\mathcal{O}\mathcal{O}}(\omega,k)$ and their associated low energy spectral weight $$\sigma(k)=\lim_{\omega\rightarrow0}\frac{\text{Im}G^R_{\mathcal{O}\mathcal{O}}(\omega,k)}{\omega}.$$ The operators $\mathcal{O}$ that we consider are related for example to charge density $J^t$ and current $J^x$, but are not exactly these. Rather, we study operators that we can call $J^{\parallel}$ and $J^{\perp}$, arising from the decoupling of the perturbed fields into transverse and longitudinal channels. Finally, we restrict ourselves to the near-horizon IR geometry. We will always call the associated IR Green’s function $\mathcal{G}^R_{\mathcal{O}\mathcal{O}}$ to differentiate it from the UV one. In fact, at low energies (that is, $\omega<<\mu$) the IR and UV Green’s functions can be related through a matching argument [@Hartnoll:2016apf]: $$\label{matching}
G^R_{\mathcal{O}\mathcal{O}}(\omega,k)=\frac{b^1_{(1)}+b^2_{(1)}\mathcal{G}^R_{\mathcal{O}\mathcal{O}}(\omega,k)}{b^1_{(0)}+b^2_{(0)}\mathcal{G}^R_{\mathcal{O}\mathcal{O}}(\omega,k)}$$ where the $b$’s are real constants independent of $\omega$. On the right hand side of (\[matching\]), all of the UV data is stored in the real constants. Taking the imaginary part of (\[matching\]), we find, to leading order as $\omega\rightarrow0$ [@Iqbal:2011ae], $$\label{Im}
\text{Im}G^R_{\mathcal{O}\mathcal{O}}(\omega,k)\propto\frac{\text{Im}\mathcal{G}^R_{\mathcal{O}\mathcal{O}}(\omega,k)}{(b^1_{(0)})^2}.$$ We have kept the real constant explicit in (\[Im\]) rather than folding it into the proportionality to make a point. If the constant $b^1_{(0)}=0$, then we get a pole in the spectral function $A(\omega, k)\sim\text{Im}G^R_{\mathcal{O}\mathcal{O}}$, and this would indicate the presence of a Fermi surface. For our purposes, we are only calculating $\text{Im}\mathcal{G}^R_{\mathcal{O}\mathcal{O}}$, and so we do not have access to the UV data and thus cannot determine whether $A(\omega, k))$ possesses such a pole. *Nevertheless*, it turns out that there is a second indicator of a Fermi surface and Pauli exclusion apart from this pole. We now describe how this works.
The spectral weight $\sigma(k)$ is aptly named, as it admits a spectral decomposition [@Hartnoll:2016apf]: $$\label{specdecomp}
\text{Im}G^R_{JJ}(\omega, k)=\sum_{m,n}e^{-\beta E_m}\left|\langle n(k^{'})|J(k)|m(k^{''})\rangle\right|^2\delta(\omega-E_m+E_n).$$ The sums in (\[specdecomp\]) are sums over eigenstates. There are actually two delta functions in (\[specdecomp\]), one in the energy difference between states and one in the momentum difference, resulting from the inner product. The $J$ tells us, then, that the spectral weight counts *charged* degrees of freedom that exist at a given frequency and momentum. Therefore, if one takes the $\omega\rightarrow0$ limit of (\[specdecomp\]) and finds that there are low energy degrees of freedom at non-zero $k$, one can conclude that the charged particles have not condensed, and a phenomenon resembling Pauli exclusion is at work.
If we again take $\mathcal{O}=J$, then the spectral weight is also the real part of the electrical conductivity (see for example [@Ammon:2015wua]). One can see this by comparing Ohm’s law[^4] $$\label{ohm}
J(\omega)=\tilde{\sigma}(\omega)E(\omega)$$ to the linear response expression[^5] $$\label{linear}
\langle J(\omega)\rangle=G^R_{JJ}(\omega)A(\omega)=\frac{G^R_{JJ}(\omega)}{i\omega}i\omega A(\omega)= \frac{G^R_{JJ}(\omega)}{i\omega}E(\omega).$$ From (\[ohm\]) and (\[linear\]), we can see that $$\label{cond}
\Tilde{\sigma}=\frac{G^R_{JJ}(\omega)}{i\omega}$$ This motivates the division by $\omega$ in the definition of the spectral weight, and from (\[cond\]) we also see that Re$\Tilde{\sigma}(\omega)=$Im$G^R_{JJ}(\omega)$.
[^1]: See [@Iqbal:2011ae] for a beautiful review of semi-local quantum liquids. We will also define this geometry more fully in the main body of this paper.
[^2]: The two types of low-energy spectral weight that we will encounter are when $\sigma(k)\neq 0$ for $k<k_*$ (which we call a smeared Fermi surface) and $\sigma(k)\neq 0$ for $k_-<k<k_+$ (which we call a Fermi shell).
[^3]: A similar conclusion was reached in [@Nakamura:2009tf].
[^4]: The tilde over the conductivity is simply to differentiate it from the spectral weight, which is also referred to as $\sigma$ in the literature.
[^5]: Here $A(\omega)$ is the electric potential and should not be confused with the spectral function $A(\omega,k)$!
|
---
abstract: 'The generalized Bohr Hamiltonian was used to describe the low-lying collective excitations in even-even isotopes of Ru, Pd, Te, Ba and Nd. The Strutinsky collective potential and cranking inertial functions were obtained using the Nilsson potential. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not contain any free parameter.'
author:
- |
K. Pomorski, L. Próchniak and K. Zajac\
[*Institute of Physics, Maria Curie-Sk[ł]{}odowska University, Lublin, Poland*]{}\
S. G. Rohoziński and J. Srebrny\
[*Department of Physics, Warsaw University, Warsaw, Poland*]{}
title: |
Collective Quadrupole Excitations\
in Transitional Nuclei
---
[**PACS**]{} 21.60.Ev, 23.20.-g, 27.60.+j
Introduction
============
For a long time the generalized Bohr hamiltonian (GBH) \[1-3\] was used to describe the low lying quadrupole collective excitations in nuclei. Especially the Bohr hamiltonian with the collective inertial functions and potential evaluated microscopically (see e.g. \[2,3\]) was attractive as a model containing no free parameters. Unfortunately confrontation of theoretical predictions of such a model with the experimental data leads to the conclusion that the microscopic inertial functions, i.e. mass parameters and moments of inertia, are too small. One has to magnify them 2 to 3 times in order to obtain the collective energy levels in right positions \[2\]. In paper \[3\] it was suggested that the pairing correlations in the collective excited states are weaker than in the ground state. This effect could explain the growth of the inertial functions. It was shown in Ref. \[3\] that decrease of the pairing strength by only 17% could increase the magnitude of the mass parameters by a factor 2 to 3 and in consequence obtain the energies of collective levels relatively close to the experimental data.
A nice explanation of the origin of the decrease of the pairing correlations in the collective excited states offers the collective pairing hamiltonian first introduced by Bès and coworkers in Ref. \[4\] for the two–levels model and than elaborated in \[5\] for a more realistic case. It was shown in \[5\] that the growth of the mass parameter with decreasing pairing gap ($\Delta$) produces a significant collective effect, namely that the most probable $\Delta$ is smaller than that obtained from the BCS solution. The coupling of the collective pairing vibrations with the collective quadrupole excitations was discussed in Ref. \[6\] for the axially symmetric case. The spectrum of collective levels obtained in the model with the coupling was almost twice compressed in comparison with the spectrum given by the Bohr hamiltonian which does not contain the coupling with pairing vibrations. Encouraging by the results obtained in \[6\] we have modify in Refs. \[7,8\] the generalized Bohr hamiltonian taking into account the major effect of the coupling with the pairing vibrations. Namely, we have evaluated (in each $\beta,\gamma$ point) all inertial functions for the most probable $\Delta$ not for that which corresponds to the BCS minimum.
In the present paper we are going to describe briefly our model and present some typical results for the neutron–rich isotopes of Pd and Ru and the neutron–deficient isotopes of Te, Ba and Nd. These examples illustrate well, how does work the model for transitional nuclei.
The model
=========
It is rather difficult to solve the nine dimensional eigenproblem of the full collective hamiltonian containing quadrupole and pairing vibrations for neutrons and protons. But assuming that the coupling between quadrupole and pairing variables is weak one can neglect mixing terms and obtain an approximate solution. Such approximate collective hamiltonian consists of two known terms and an operator $\hat{\cal H}_{\rm
int}$ which mix quadrupole and pairing variables: $$\hat{\cal H}_{\rm CQP} = \hat{\cal H}_{\rm CQ}({\beta},{\gamma},\Omega; \Delta
^p,\Delta^n) + \hat{\cal H}_{\rm CP}(\Delta^p,\Delta^n; {\beta},{\gamma})
+ \hat{\cal H}_{\rm int}\,\,.$$ The last term will be neglected in further calculations. The operator $\hat{\cal H}_{\rm CQ}$ describes quadrupole oscillations and rotations of a nucleus and it takes the form of the generalized Bohr hamiltonian \[2,3\]: $$\hat{\cal H}_{\rm CQ} = \hat{\cal T}_{\rm vib}({\beta},{\gamma};\Delta^p,\Delta^n)
+ \hat{\cal T}_{\rm rot}({\beta},{\gamma},\Omega;\Delta^p,\Delta^n) + V_{\rm
coll}({\beta},{\gamma};\Delta^p,\Delta^n) \,\,.$$ Here $V_{\rm coll}$ is the collective potential, the kinetic vibrational energy reads $$\begin{aligned}
\nonumber
\hat{\cal T}_{\rm vib}=-{{\hbar}^2\over{2\sqrt{wr}}}\bigg\{
{1\over {\beta}^4}\bigg[ {\partial_{{\beta}}}\bigg( {\beta}^4{\sqrt{r\over w}}{B^{}_{{\gamma}{\gamma}}}{\partial_{{\beta}}}\bigg) -
{\partial_{{\beta}}}\bigg({\beta}^3{\sqrt{r\over w}}{B^{}_{{\beta}{\gamma}}}{\partial_{{\gamma}}}\bigg)\bigg]+ &&\\
+ {1\over {\beta}{\sin\!3{\gamma}}}\bigg[ -{\partial_{{\gamma}}}\bigg( {\sqrt{r\over w}}{\sin\!3{\gamma}}{B^{}_{{\beta}{\gamma}}}{\partial_{{\beta}}}\bigg) +
{1\over{\beta}}{\partial_{{\gamma}}}\bigg({\sqrt{r\over w}}{\sin\!3{\gamma}}{B^{}_{{\beta}{\beta}}}\bigg){\partial_{{\gamma}}}\bigg]\bigg\}\end{aligned}$$ and the rotational energy is $$\hat{\cal T}_{\rm rot}={1\over 2}\sum_{k=1}^{3} \hat{I}^2_k/{\cal J}_k \,\,.$$ The intrinsic components of the total angular momentum are denoted as $\hat{I}_k,\, (k=1,2,3)$, while $w$ and $r$ are the determinants of the vibrational and rotational mass tensors. The mass parameters (or vibrational inertial functions) ${B^{}_{{\beta}{\beta}}}$, ${B^{}_{{\beta}{\gamma}}}$ and ${B^{}_{{\gamma}{\gamma}}}$ together with moments of inertia ${\cal J}_k,\, (k=1,2,3)$ depend on intrinsic variables ${\beta},{\gamma}$ and pairing gap values $\Delta^p,\Delta^n$. All inertial functions are determined from a microscopic theory. We apply the standard cranking method to evaluate the inertial functions assuming that the nucleus is a system of nucleons moving in the deformed mean field (Nilsson potential) and interacting through monopole pairing forces. One has to stress that for $\Delta$ corresponding to the minimum of the BCS energy the operator $\hat{\cal H}_{\rm CQ}$ is exactly the same as the Bohr hamiltonian used in Ref. \[2,3\].
For a given nucleus the second term in Eq. (1) describes collective pairing vibrations of systems of $Z$ protons and $A-Z$ neutrons $$\hat{\cal H}_{\rm CP} = \hat{\cal H}^Z_{\rm CP} +
\hat{\cal H}^{A-Z}_{\rm CP}$$ and it can be expressed in the following form \[4,5\]: $$\hat{\cal H}^{\cal N}_{\rm CP}=-\frac{\hbar^2}
{2\sqrt{g(\Delta)}}\frac{\partial}
{\partial\Delta}\frac{\sqrt{g(\Delta)}}
{B_{\Delta\Delta}(\Delta)}
\frac{\partial}{\partial\Delta} + V_{\rm pair}(\Delta),$$ where ${\cal N}=Z$, $\Delta=\Delta^p$ for protons and, respectively, ${\cal N}=A-Z$, $\Delta=\Delta^n$ for neutrons. The functions appearing in the hamiltonian (6), namely the pairing mass parameter $B_{\Delta\Delta}(\Delta)$, the determinant of the metric tensor $g(\Delta)$ and the collective pairing potential $V_{\rm pair}(\Delta)$ are determined microscopically.
Solving the eigenproblem of the collective pairing hamiltonian (6) one can find the pairing vibrational ground-state wave function $\Psi_0$ and the ground-energy $E_0$ at each deformation point. The most probable value of the energy gap $\Delta_{vib}$ corresponds to the maximum of the probability of finding a given gap value in the collective pairing ground-state (namely the maximum of the function $g(\Delta )|\Psi_{0}(\Delta )|^2$). As it is shown in Fig. 1 the $\Delta_{vib}$ is shifted towards smaller gaps from the equilibrium point $\Delta _{eq}$ determined by the minimum of $V_{\rm
pair}$ (or by the BCS formalism). Such a behavior of the pairing ground state function $\Psi_0$ is due to the rapid increase of pairing mass parameter $B_{\Delta \Delta}$. In general the ratio of $\Delta _{vib}$ to $\Delta _{eq}$ is of about $0.7$.
All collective functions appearing in Eqs. (3,4) are calculated using the most probable pairing gap values for protons and for neutrons instead the equilibrium ones. The collective potential corresponds to the ground state of the $\hat{\cal H}_{CP}$ hamiltonian (5) and it is very close to the BCS energy in each $\beta,\gamma$ point.
The approximation described above is rather crude but it includes the main effect (at least on average) of the coupling with the pairing vibrational mode. This procedure improves significantly the accuracy in reproducing the experimental data and it introduces no additional parameters into the model. Our calculations were done using the standard Nilsson single particle potential with the shell dependent parametrization found in Ref. \[9\]. The pairing strength was fitted in Refs. \[7,8\] to the mass differences.
Results
=======
We present here only some examples of results for the neutron–rich isotopes of Ru and Pd and for three chains of isotopes (Te, Ba and Nd) from the neutron–deficient region of nuclei.
In order to illustrate the effect of the coupling of the quadrupole and pairing vibrations we have compared in Fig. 2 the energy levels obtained with the traditional GBH (“old”) with those evaluated within the present model (“new”). As one can learn from Fig. 2 the improvement in reproducing the experimental data caused by coupling with the pairing vibrations is really significant.
In Fig. 3 we present the theoretical (open symbols) and experimental (full symbols) energy levels of the ground state band and the $\gamma$ band for the even–even isotopes of Ru and Pd with $64 \leq N \leq 74$ neutrons. The agreement of theoretical predictions with the experimental data is here rather good. The situation in the neutron–deficient nuclei is not so optimistic. A typical sets of results is presented in Fig. 4, where the lowest levels of the ground state band of Te, Ba and Nd isotopes are plotted as a function of the neutron number. It is seen that a good agreement is obtained for the Nd nuclei and for the lightest Te and Ba isotopes only.
The electromagnetic transitions between the band members and between the states belonging to different bands are also relatively well reproduced by our model (see Refs. \[7,8\]).
All experimental data in Figs. 2-4 are taken from Ref. \[10\].
Summary and conclusions
=======================
The generalized Bohr Hamiltonian (GBH) \[1-3\] is used to describe the low-lying collective excitations in even-even isotopes of the neutron–deficient and neutron–rich regions of nuclei \[7,8\]. The collective potential and inertial functions are determined by means of the Strutinsky method and the cranking model, respectively. A shell-dependent parametrization of the Nilsson potential is used. There are no adjustable parameters in the calculation. The coupling of the quadrupole and pairing vibrations \[5-6\] is taken into account and it brings the energy levels down to the scale comparable with that characteristic for the experimental levels \[6\].
In the neutron–reach region we have performed calculations for chains of isotopes of Ru and Pd. In this case theoretical estimates of energies of low lying collective states and electromagnetic transitions within bands, as well as between members of different bands, are even closer to experimental data than for nuclei from the neutron-deficient region. In the region of neutron–deficient nuclei the GBH works better in the case of elements with larger $Z$, namely, Ce, Nd and Sm than for Xe, Ba and Te which have only two protons outside the closed shell $Z=50$. Energies are especially well reproduced by the calculation for isotopes with lower number of neutrons. For those with neutron number $N=78,\, 80$ the energy levels are, as a rule, too high. On the contrary, electromagnetic properties seem to be better reproduced just in the case of heavier isotopes.
Concluding, we may say that adding of the coupling with the pairing vibrations to the generalized Bohr Hamiltonian improves significantly the quality of theoretical estimates for nuclei from the both regions.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported in part by the Polish Committee for Scientific Research under Contract No. 2 P03B 068 13.
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[**Figure captions:**]{}\
[**Fig. 1**]{} The pairing vibration mass parameter ($B_{\Delta\Delta}$), and potential ($V_{\rm pair}$), and the ground-state function ($\Psi ^N_0$) as function of the pairing energy gap $\Delta$ for the system of $60$ neutrons at the deformation point $\beta = 0.2,\, \gamma = 20^{\circ}$. The equilibrium value of the energy gap is $\Delta_{eq}
\approx 0.14\hbar\omega_0$, the most probable one is $\Delta_{vib}\approx 0.09\hbar\omega_0$.\
[**Fig. 2**]{} The lowest experimental and the theoretical (connected by straight lines) excited levels in $^{104}$Ru versus angular momentum $J^{\pi}$. The theoretical values were calculated including the effect of coupling with the pairing vibrations (“new”) and without this coupling, i.e. within usual microscopic Bohr model (“old”).\
[**Fig. 3**]{} The lowest theoretical and experimental energy levels of the ground state band and the $\gamma$ band for the chains of Ru and Pd isotopes.\
[**Fig. 4**]{} The lowest theoretical and experimental energy levels of the ground state band for the chains of Te, Ba and Nd isotopes.
[]{}
[]{}
|
---
abstract: 'We describe Spitzer/MIPS observations of the double cluster, h and $\chi$ Persei, covering a $\sim$ 0.6 square-degree area surrounding the cores of both clusters. The data are combined with IRAC and 2MASS data to investigate $\sim$ 616 sources from 1.25-24 $\mu m$. We use the long-baseline $K_{s}$-\[24\] color to identify two populations with IR excess indicative of circumstellar material: Be stars with 24 $\mu m$ excess from optically-thin free free emission and 17 fainter sources (J$\sim$ 14-15) with \[24\] excess consistent with a circumstellar disk. The frequency of IR excess for the fainter sources increases from 4.5 $\mu m$ through 24 $\mu m$. The IR excess is likely due to debris from the planet formation process. The wavelength-dependent behavior is consistent with an inside-out clearing of circumstellar disks. A comparison of the 24 $\mu m$ excess population in h and $\chi$ Per sources with results for other clusters shows that 24 $\mu m$ emission from debris disks ’rises’ from 5 to 10 Myr, peaks at $\sim$ 10-15 Myr, and then ’falls’ from $\sim$ 15/20 Myr to 1 Gyr.'
author:
- 'Thayne Currie, Scott J. Kenyon, Zoltan Balog,George Rieke, Ann Bragg, & Benjamin Bromley'
title: 'The Rise and Fall of Debris Disks: MIPS Observations of h and $\chi$ Persei and the Evolution of Mid-IR Emission from Planet Formation'
---
Introduction
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Most 1-2 Myr-old stars are surrounded by massive (M$_{disk}$ $\sim$ 0.01-0.1 M$_{\star}$) optically-thick accretion disks of gas and dust. The disk produces near-to-mid infrared (IR) emission comparable in brightness to the stellar photosphere (L$_{disk}$ $\sim$ L$_{\star}$) (e.g. Kenyon & Hartmann 1995, Hillenbrand 1997). The evolution of these ’primordial’ disks has been studied extensively (e.g. Haisch, Lada, & Lada 2001; Lada et al. 2006; Dahm & Hillenbrand 2007). By 5-10 Myr, primordial disks disappear and less massive (M$_{disk}$ $\lesssim$ 1 M$_{\oplus}$) gas-poor, optically-thin ’debris disks’ with weaker emission (L$_{disk}$ $\lesssim$ 10$^{-3}$ L$_{\star}$) emerge (e.g. Hernandez et al. 2006). By $\sim$ 10-20 Myr, primordial disks are extremely rare: almost all disks are debris disks (Currie et al. 2007a, hereafter C07a; Gorlova et al. 2007; Sicilia-Aguilar et al. 2006).
Debris disks older than $\sim$ 20Myr are well studied. Rieke et al. (2005; hereafter R05) showed that the 24 $\mu m$ emission declines with time as t$^{-1}$ (see also Kalas 1998; Habing et al. 2001; Decin et al. 2003). This decay agrees with expectations for the gradual depletion of the reservoir of small planetesimals. With fewer parent bodies to initiate the collisional cascades that yield the infrared-emitting dust, the infrared excesses drop systematically with time (Kenyon & Bromley 2002; Dominik & Decin 2003; R05; Wyatt et al. 2007a). R05 also found a large range in the amount of infrared excess emission at each age, even for very young systems. Wyatt et al. (2007a) demonstrate that the first-order cause of this range is probably the large variation in protostellar disk masses and hence in the mass available to form planetesimals.
Because the R05 sample and other studies of individual stars (e.g., Chen et al. 2005a) include few stars younger than 20Myr, they do not probe the 5-20 Myr transitional period from primordial to debris disks well. This transition marks an important phase for planet formation and other physical processes in disks. Gas accretion onto most young stars ceases by $\approx$ 10 Myr (Sicilia-Aguilar et al. 2005). Planets acquire most of their mass by $\approx$ 5-20 Myr (Kenyon & Bromley 2006; Chambers 2001; Wetherill & Stewart 1993).
With an age of 13 $\pm$ 1 Myr and with over $\sim$ 5000 members (C07a), the double cluster, h and $\chi$ Persei (d=2.34 kpc, A$_{V}$$\sim$ 1.62; Slesnick et al. 2002, Bragg & Kenyon 2005), provides an ideal laboratory to study disk evolution during this critical age. Recent observations of h and $\chi$ Per with the Spitzer Space Telescope have demonstrated the utility of using the double cluster to investigate disk evolution after the primordial stage. C07a used 3.6-8$\mu m$ Spitzer data to show that disks last longer around less massive stars and at greater distances from the star. C07b analyzed a well-constrained subsample of h and $\chi$ Per sources and showed that at least some of the disk emission in them comes from warm dust in the terrestrial zones of disks as a byproduct of terrestrial planet formation.
In this paper, we use data obtained with the Multiband Imaging Photometer for Spitzer (MIPS) to extend the study of h and $\chi$ Per to 24$\mu$m. This band allows us to search for high levels of mid-IR excess associated with cool dust that orbits in a disk at $\sim$ 2-50 AU from the central star. Our survey covers a region containing $\sim$ 600 intermediate-to-high mass cluster members. In §2 we describe the MIPS observations, data reduction, and sample selection. We analyze the 24$\mu$m photometry in §3. The main results are: 1) there are two IR-excess populations, Be stars with optically thin free-free emission and intermediate mass stars likely harboring disks; 2) debris disk excesses are more common at 24$\mu$m than at shorter wavelengths; and 3) there are several extreme disks similar to the nearby young debris disks around $\beta$ Pic, HR 4796A, and 49 Cet. Finally, in §4 we place h and $\chi$ Per in the context of results for other open clusters/associations with optically-thin debris disk candidates. The flux from debris disks rises from $\sim$ 5 Myr (when they first emerge), peaks at $\sim$ 10-15 Myr, and then falls as t$^{-1}$ as described by R05. We conclude with a summary of our findings and discuss future observations that may place even stronger constraints on debris disk evolution by accounting for wide range of IR excesses at 10-15 Myr.
Observations
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MIPS and Ground-Based Spectroscopic Data
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We acquired MIPS 24 $\mu m$ data using 80-second exposures in scan mode, covering two 0.3 square-degree regions centered on the two clusters. The frames were processed using the MIPS Data Analysis Tool (Gordon et al. 2005). PSF fitting in the IRAF/DAOPHOT package was used to obtain photometry using a 7.3 Jy zero-point for the 24 $\mu$m magnitude scale. The typical errors for the MIPS sources are 0.2 mag ($\sim$ 5$\sigma$) at a 24$\mu m$ magnitude of \[24\] $\sim$ 10.5-11. The number counts for the MIPS data peak at \[24\] $\sim$ 10.5 and decline to zero by \[24\]$\sim$ 11.5 (Figure 1a). We detect 2,493 potential h and $\chi$ Per sources.
We combined the MIPS photometry with the 2MASS/IRAC catalogue of h and $\chi$ Persei from C07a. To minimize potential contamination of stellar sources by background PAH-emission galaxies and AGN, we used a small 1.25" matching radius (about half of a MIPS pixel; r$_{M}$) to merge the 2MASS/IRAC and the MIPS catalogues. Although the MIPS beam is 6 arcsec in diameter, the instrument delivers positions good to one arcsec even for faint sources in crowded fields (Bai et al. 2007). This procedure yielded 616 sources (N$_{MIPS}$) with high-quality 1–24 $\mu m$ photometry. Table 1 shows the 2MASS/IRAC + MIPS catalogue. Optical UBV photometry from Slesnick et al. (2002) is included where available.
To \[24\]=10.5, the probability of chance alignments between distant PAH-emission galaxies/AGN and our sources is low. Using the galaxy number counts from Papovich et al. (2004), N$_{G}$ $\sim$ 3.5$\times$10$^{6}$/sr, we derive a probability of $\sim$ 24.8% that one of our 616 sources is contaminated ($\pi$r$_{M}$$^{2}$$\times$N$_{G}$$\times$N$_{MIPS}$/(3282.8$\times$3600$^{2}$)). The likelihood that many of our sources are contaminated is then much smaller.
To estimate the completeness of the MIPS sample, we compare the fraction of J band sources detected with MIPS within either cluster. Figure 1b shows that $\gtrsim$ 90% of the 2MASS sources brighter than J=10.5 are also detected in MIPS. The completeness falls to $\sim$ 50% by J=11 and to $\sim$ 10% by J=12. The dip at J$\sim$8-9 occurs because many sources in this range are near the cluster centers, where the high density of even brighter sources (J $\sim$ 6-8) masks the presence of fainter objects.
To provide additional constraints on the 24$\mu m$ excess sources, we also obtained Hectospec (Fabricant et al. 2005) and FAST (Fabricant et al. 1998) spectra of selected MIPS sources on the 6.5m MMT and 1.5m Tillinghast telescopes at F. L. Whipple Observatory during September-November 2006. Spectra for bright sources (J $\le$ 13) were also cross referenced with the FAST archive. The FAST spectroscopy, described in detail by Bragg & Kenyon (2002), typically had $\sim$ 10 minute integrations using a 300 g mm$^{-1}$ grating blazed at 4750 $\AA$ and a 3$\arcsec$ slit. These spectra cover 3700–7500 Å at 6 Å resolution. The typical signal-to-noise ratios were $\gtrsim$ 25-30 at 4000 Å. For each Hectospec source, we took three, 10-minute exposures using the 270/mm grating. This configuration yields spectra at 4000-9000$\dot{A}$ with 3$\dot{A}$ resolution. The data were processed using standard FAST and Hectospec reduction pipelines (e.g. Fabricant et al. 2005).
We acquired additional spectra of h and $\chi$ Per sources with the Hydra multifiber spectrograph (Barden et al. 1993) on the WIYN 3.5 m telescope at the Kitt Peak National Observatory. Hydra spectra were obtained during two observing runs in November 2000 and October 2001 and include stars brighter than V=17.0. We used the 400 g mm$^{-1}$ setting blazed at 42 degrees, with a resolution of 7 Å and a coverage of 3600-6700 Å. The standard IRAF task *dohydra* was used to reduce the spectra. These spectra had high signal-to-noise with $\gtrsim$ 1000 counts over most of the wavelength coverage.
Spatial Distribution of MIPS sources
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To investigate the spatial distribution of the MIPS sources and the likelihood that they are cluster members, we compare the projected sky surface densities derived from MIPS and 2MASS. C07a showed that $\sim$ 47% of stars within 15’ of the cluster centers are cluster members. Between 15’ and 25’, $\sim$ 40% of the 2MASS sources are in a halo population with roughly the same age as bona fide cluster stars. Because the MIPS coverage is complete only out to $\sim$ 15’ away from each cluster center, we cannot identify MIPS sources with this halo population. We compare the spatial distribution of MIPS sources to those in 2MASS from C07a by calculating the number density of sources in 5’-wide half-annuli facing away from the midpoint of the two clusters.
Through 15’ away from either cluster center, the number counts of sources detected with both MIPS and 2MASS fall off about as steeply or slightly more steeply than the counts for 2MASS alone from C07a (Figure \[dens\]). Near the center of the clusters the density of MIPS sources is $\sim$ 0.4/sq. arc-minute, or about an order of magnitude lower than from 2MASS. For h Persei and $\chi$ Persei, respectively, this density falls off by 4 % and 25 % from 0-5’ to 5’-10’ away from the cluster centers and 22%-41% from 0-5’ to ’10-15’ away from the centers. The low counts through 5’ and more shallow drop in number density for $\chi$ Persei is due to crowding in the inner $\sim$ 1-2’ of the $\chi$ Persei core; the slope of the MIPS number density in $\chi$ Persei shown in Figure \[dens\] is most likely a lower limit. In contrast, the number counts for the 2MASS data from C07a fall off by 10% (20%) and 30% (32%) for h ($\chi$) Persei over the same 5’ intervals (the values in Figure \[dens\] are slightly different due to the larger annuli used here). The MIPS source counts appear to be about as centrally concentrated as the 2MASS counts.
General nature of the 24 $\mu m$ sources
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Figure \[k24dist\] shows the histogram of K$_{s}$-\[24\] colors for the MIPS detections with 2MASS counterparts. The histogram has a main peak at K$_{s}$-\[24\]$\sim$0-1 and two groups with K$_{s}$-\[24\] $\sim$1-2 and K$_{s}$-\[24\]$\sim$2-6. The sources with very red K$_{s}$-\[24\] colors ($\ge$ 2) are in two main groups (Figure \[kexc\]). A bright group of very red sources has K$_{s}$$\sim$9-11; a fainter population of red sources stretches from K$_{s}$$\sim$13.5-15. A population of 13 Myr-old stars in h & $\chi$ Per with spectral types later than B9 (M$\le$3.0 $M_{\odot}$) should have J, $K_{s}$ magnitudes $\gtrsim$ 13.3 (Siess et al. 2000). Thus, some of these fainter sources with red K$_{s}$-\[24\] colors are possibly pre-main sequence stars.
Some of the sources are very faint in the near infrared. To examine the nature of the MIPS sources without J counterparts, we first compared the MIPS mosaic and the 2MASS J mosaic by eye. Many sources appear scattered throughout the MIPS mosaic but do not appear in 2MASS, even at very high contrast. These sources are likely cluster stars with J $\gtrsim$ 16-17 and very red J-\[24\] colors or background galaxies with negligible near-IR emission. Using the number density of galaxies in the MIPS 24 $\mu m$ filter from Papovich et al. (2004), we expect $\gtrsim$ 600-700 galaxies in the 0.6 square-degree coverage area brighter than \[24\] $\sim$ 10.5. Thus, many of the sources without 2MASS counterparts are likely not h & $\chi$ Per members. From the C07a survey, there are $\sim$ 4700 stars with J $\sim$ 10-15.5 within either cluster or the surrounding halo population of comparable age. For a reasonable IMF (e.g. Miller & Scalo 1979), we expect $\sim$ 2800 ($\sim$ 8900) cluster/halo stars with J$\sim$16-17 (17-18). If $\sim$ 10%-20% of these stars have large 24$\mu m$ excesses, as predicted from an extrapolation of the C07a results to fainter J magnitudes, we expect 1170-2340 cluster/halo stars with MIPS detections and no 2MASS counterparts. Together with the 600-700 background galaxies, this population yields the observed number of MIPS detections without 2MASS counterparts.
A \[24\] IR excess population in h and $\chi$ Persei
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Groups in the J, J-H Color-Magnitude Diagram
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To identify the nature of the 24$\mu m$ emission in sources with red K$_{s}$-\[24\] colors in Figure 2, we refer to previous MIPS observations of very ’red’ sources and consider possible contaminants. MIPS observations of the Pleiades (Gorlova et al. 2006) guide our analysis of $K_{s}$-\[24\] colors for IR excess disk/envelope sources. While the stellar density in h and $\chi$ Per is larger than in the Pleiades, other possible contaminants are less important. The level of galactic cirrus for h and $\chi$ Persei is much lower than for the Pleiades: 17-27 MJy/sr versus 36-63 MJy/sr. Gorlova et al. found that disk-bearing candidate sources have dereddened $K_{s}$-\[24\] colors $\gtrsim$ 0.25. Because h & $\chi$ Persei has a low, uniform extinction of $A_{V}\sim 1.62$, E(B-V)$\sim$0.52 (Bragg & Kenyon 2005), we convert the dereddened $K_{s}$-\[24\] excess criterion into a reddened $K_{s}$-\[24\] criterion using the reddening laws from Indebetouw et al. (2005) and Mathis (1990). For $A_{V}$$\sim$1.62, 24$\mu m$ excess sources should have $K_{s}$-\[24\]$\gtrsim$0.45. Because the MIPS data have $\sigma$ $\lesssim$ 0.2, we round this limit up to $K_{s}$-\[24\] $\gtrsim$ 0.65.
Figure \[jjh\] shows the distribution of sources with and without $K_{s}$-\[24\] excess in J/J-H color-magnitude space. The IR excess population is clustered into two main groups. Asterisks (diamonds) denote sources brighter (fainter) than J=13. Larger asterisks/diamonds correspond to sources with $K_{s}$-\[24\] $\ge$ 2, while smaller asterisks identify sources with $K_{s}$-\[24\]=0.65-2 and J $\le$ 13. The excess sources with J$\le$13 typically have J-H colors $\sim$ 0.2 mag redder than a typical stellar photosphere. Many sources with weak excess lie well off the 14 Myr isochrone and may be consistent with foreground M stars or supergiants. About 17 out of 21 stars with 24$\mu m$ excess fainter than J=13 fall along the 14 Myr isochrone with J$\approx$ 14-15. At 14 Myr and a distance of 2.4 kpc, this J magnitude range corresponds to stars with masses $\sim$ 2.2-1.4 $M_{\odot}$ (B9/A0-G2) stars (Siess et al. 2000). We inspected each faint excess source on the MIPS mosaic for extended emission (indicative of galaxies) or ’excess’ due to source confusion/crowding. We found no evidence issues such as as large extended emission or source confusion that could compromise the photometry of any of the faint excess sources.
Two Populations of IR-Excess Sources: Be stars with Circumstellar Envelopes and Faint Pre-Main Sequence Stars with Disks
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There are three main possibilities for the source of 24$\mu m$ excess emission around h and $\chi$ Per stars. Red giants or supergiants not associated with the clusters or the halo produce IR excesses in massive stellar winds, and should have J-H $\gtrsim$ 0.5. Two such stars have large 24 $\mu m$ excesses and are not considered further. Be stars in the clusters/halo population have IR excesses from optically-thin free-free emission and should have J $\lesssim$ 13-13.5 and J-H $\lesssim$ 0.4. Many potential Be stars have $K_{s}$ -\[24\] $\sim$ 2 (Figure \[jjh\]; large asterisks) and clearly are an important part of the cluster population. Aside from Be stars, circumstellar disks around lower-mass cluster/halo stars can produce excess emission. Figure 5 shows a significant population of fainter stars (all with J $\gtrsim$ 13.5) on the 14 Myr isochrone with large K$_{s}$ -\[24\] excesses. To identify the nature of the 24$\mu m$ excess sources, we analyze the near-IR colors and selected spectra of the excess population. We begin with Be star candidates and then discuss the fainter population.
Be stars are massive and are evolving off the main sequence (McSwain and Gies 2005). The IR-excess emission from Be stars arises from an optically-thin, flattened, circumstellar shell of ionized gas ejected from the star (Woolf, Stein, and Strittmatter 1970; Dachs et al. 1988). We find 57 candidates – J $\lesssim$ 13.5 and J-H $\le$ 0.4 – with 24 $\mu m$ excess (K$_{s}$-\[24\] $\gtrsim$ 0.65). Twenty of these stars have been previously identified as Be stars by Bragg and Kenyon (2002), all with Oosterhoff (1937) numbers, and have spectral types from Strom and Wolff (2005) and Bragg and Kenyon (2002). Table 2 lists the properties of these 57 candidates.
We can estimate the ratio of Be stars to B stars over a narrow range of spectral types (earlier than B4). In our MIPS survey, there are $\sim$ 175 stars that are likely B type stars (J=8-13.5; J-H $\le$ 0.2) without excess. All of these stars are probably earlier than B4 based on their 2MASS J band photometry (J $\lesssim$ 11.75). There are 57 Be star candidates with 24$\mu m$ excess (K$_{s}$-\[24\]$\gtrsim$0.65): 51 of these stars probably have spectral types earlier than B4 based on their J-band photometry. The ratio of Be star candidates to main sequence B-type stars earlier than B4 in the MIPS survey is then $\sim$ 0.29. This estimate is larger than the ratio derived from optical and near-IR data ($\sim$ 0.14, Bragg & Kenyon 2002). We explored this difference as follows.
First, we analyze the Be star candidate population in high-density regions close to the cluster centers where spectroscopic data on bright stars in h and $\chi$ Per is complete. Our candidates were cross correlated with spectroscopically identified Be stars from Bragg and Kenyon within 5’ of the cluster centers.. Ten known Be stars and two candidate stars from the MIPS survey are in this region. These two candidate stars with 24 $\mu m$ excesses are not Be stars. One is a 5th magnitude B3 supergiant cluster member identified by Slesnick et al. (2002) and the other is a G1 star, lying well off the isochrone with J=11.67 and J-H=0.39.
Because our spectroscopic sample of Be stars is spatially limited, we turn to near-IR colors from 2MASS to investigate the nature of the bright MIPS excess sources. Dougherty et al. (1991, 1994) showed that Be stars follow a distinct locus in JHK$_{s}$ colors. This locus is characteristic of free-free emission from optically-thin ionized gas and is well separated from main sequence colors and the near-IR colors produced by warm dust. Thus, the J-H/H-K$_{s}$ color-color diagram (Figure 6) provides a clear way to distinguish Be stars from lower mass stars with circumstellar dust emission. From Figure 6, it is clear that the bright sources with K$_{s}$-\[24\]$\ge$2 follow a locus (dotted line) in J-H/H-$K_{s}$ from (0,0) to (0.3,0.4), a range consistent with known Be star colors (Dougherty et al. 1991, 1994). The bright sources with weaker excess (small asterisks) also appear to lie along the Be star locus or are clumped close to the red giant locus at J-H $\sim$ 0.6-0.8, H-$K_{s}$$\sim$ 0.2-0.3. The observed distribution of IR colors suggests $\approx$ 15 Be stars and $\approx$ 25 giants/supergiants. If this ratio is confirmed by optical spectroscopy, then the fraction of Be stars among all B-type stars is similar to the 14% derived by Bragg & Kenyon (2002).
Finally, we search the FAST archive at the Telescope Data Center at the Smithsonian Astrophysical Observatory and the Slesnick et al. catalogue for additional spectra of the 35 Be star candidates in lower-density regions. The FAST archive contains additional data for four candidates; we find one additional source from Slesnick et al. The Slesnick et al. source is a confirmed Be star (B1Ie). The FAST sources contain an A2, F7, G2, and B4 star. The first three of these are bright and likely either foreground or giants associated with the halo population of h and $\chi$ Per. Thus, the spectra support our conclusion from the color-color diagram (Figure 6) that many of the candidate Be stars are not true identifications. If none of the remaining candidate stars are true Be stars, then the ratio of Be stars to B stars is $\sim$ 0.12, close to the Bragg and Kenyon value.
Interestingly, the B4 star identified by Bragg & Kenyon (2002), has 24 $\mu m$ excess *and* has a J magnitude and J-H colors marginally consistent with an early B star in h and $\chi$ Per. The $K_{s}$-\[24\] excess for this source is $\sim$ 1.05, though unlike Be stars it lacks clear IR excess at JHK$_{s}$ and in the IRAC bands. We show its spectrum compared with that of a known Be star (Oosterhoff number 517) in Figure 7. This star is the earliest, highest mass star+circumstellar disk source known so far in h & $\chi$ Per.
Based on their near-IR colors and optical spectra, the faint excess sources in our survey are clearly distinguishable from Be stars. The near-IR colors of the faint excess sources are evenly distributed between J-H = 0.1-0.6 and H-$K_{s}$=0-0.2 (Figure 6, diamonds). The lack of very red H-K$_{s}$ colors for a typical faint excess source is consistent with a lack of warm (T$\sim$ 1000 K) circumstellar envelope emission. Nearly all (17/21) faint sources are photometrically consistent with h & $\chi$ Per membership, though high-quality spectroscopic data are currently limited to 8 sources (2 from Hydra, 6 from Hectospec). Figure 8 shows the spectra. Seven of the eight faint excess sources have spectra consistent with h and $\chi$ Per membership. The one non-member source in Figure 8 is an F5 star with J=15.9, beyond the 2MASS completeness limit and below the isochrone by $\sim$ 1.0 magnitude, and thus is one of the four faint sources that is also photometrically inconsistent with cluster membership. The spectral types for the 7 sources consistent with cluster membership range from A2 to F9. None show strong H$_{\alpha}$ emission which is a signature of accretion (e.g. White & Basri 2003) and thus a reservoir of circumstellar gas. These stars are therefore very similar to the nearby young (8-12 Myr old) debris disks $\beta$ Pic, HR 4796A, and 49 Cet in that they have comparable spectral types, have 24 $\mu m$ excess, and lack any signatures of gas accretion. The 13 sources without known spectral types are either unobserved (12) or had too low signal-to-noise to derive spectral types (1, this source is well off the isochrone).
Many of the fainter 24$\mu m$ sources have high-quality IRAC photometry. Of the 17 MIPS excess sources fainter than J=13.5 on the isochrone, 14 (12, 11) also have IRAC measurements at \[4.5\] (\[5.8\], \[8\]). We summarize the observed properties of the faint MIPS excess sources in Table 3. In the following two sections, we focus on these sources, comparing the MIPS photometry to IRAC/2MASS photometry from C07a and modeling the sources’ emission from 2.2 $\mu m$ through 24$\mu m$.
Nature of the Disk Population in Faint Pre-Main Sequence Stars
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### Mid-IR colors and the Wavelength-Dependent Frequency of Disks
To constrain the nature of the 17 faint MIPS excess sources that are (photometrically) consistent with cluster membership, we compare the K$_{s}$/$K_{s}$-\[24\] CMD with CMDs using three IRAC colors, $K_{s}$-\[4.5\], K$_{s}$-\[5.8\], and $K_{s}$-\[8\] in Figure \[cmdirac\] (diamonds). For reference, we also show the colors for bright MIPS sources without 24$\mu m$ excess (squares). Following C07a, we identify sources with $K_{s}$-\[IRAC\] colors $\ge$ 0.4 as IR excess sources; sources with $K_{s}$-\[24\]$\ge$0.65 are 24$\mu m$ excess sources. A vertical line in Figure \[cmdirac\] shows the division between excess and non excess sources.
The frequency of IR excess varies with wavelength. Only 1/14 faint 24$\mu m$ excess sources also have excess at \[4.5\]. The fraction of sources with \[5.8\] excess is 3/12. The 8$\mu m$ excess population has a larger fraction of excess sources, 5/11. While some of the ’photospheric’ sources, $K_{s}$-\[4.5, 5.8, 8\] $\le$ 0.4, may have weak excesses, many sources have $K_{s}$-\[IRAC\] $\le$ 0.2 (observed) and $\lesssim$ 0.1 (dereddened). These sources are unlikely to have any dust emission at \[4.5\], \[5.8\], or \[8\]. While the small sample of 24$\mu m$ excess sources precludes a strong statistical significance for any trend of IR excess emission, the wavelength-dependent frequency of excess emission is consistent with results from larger surveys (e.g. C07a; Su et al. 2006).
### Temperature and Location of Circumstellar Dust
Analyzing the strength of IR excess emission at multiple bands places constraints on the temperature and location of the dust. Just over half of the faint 24$\mu m$ excess sources have no excess emission in the IRAC bands, so these sources lack circumstellar material with temperatures $\gtrsim$ 400K. Because a blackbody that peaks at 24$\mu m$ has T $\sim$ 120 - 125 K, the dust temperature in most of the faint 24$\mu m$ excess sources is probably $\lesssim$ 100-200 K. We can put more quantitative constraints on the dust temperature with a flux ratio diagram. Flux ratio diagrams have been an important tool in analyzing accretion disks in unresolved cataclysmic variable systems (e.g. Berriman et al. 1985; Mauche et al. 1997). In this method, the ratio of fluxes (in this case, $\lambda$F$_{\lambda}$) at different wavelengths such as $\lambda_{4.5}$F$_{4.5}$/$\lambda_{8}$F$_{8}$ and $\lambda_{24}$F$_{24}$/$\lambda_{8}$F$_{8}$ is computed. The ratios for blackbody emission follow a curve in flux ratio space. Because disk-bearing sources should be, to first order, the sum of two blackbodies (a hot stellar component and a cooler circumstellar component), their positions in flux ratio space should lie on a line between the circumstellar dust temperature and the stellar temperature.
Figure \[fr\] shows the flux ratio diagram for our sample, and Table 4 lists the derived disk temperatures (labeled as T$_{D}$ FR). We restrict our sample to 10 sources with 5$\sigma$ detections from 4.5 through 24 $\mu m$[^1]. Five of these sources have \[8\] excess; one has \[4.5\] excess. For 13-14 Myr-old sources, the range of spectral types with J=14-15.5 is $\sim$ A0 to G8 (Siess et al. 2000). The flux ratios for blackbody emission from 10 K to 10000 K follow the solid line with the temperatures characteristic of disks ($\sim$ 10-1000 K) on the vertical part of the line and those for stellar photospheres on the horizontal part. Loci showing the locations for a stellar photosphere+disk of a given temperature are shown ranging from T$_{disk}$= 300 K to 100 K assuming a stellar temperature of $T_{e,\star}$$\sim$ 7250 K (about F0 spectral type). The sources without (with) IRAC excess emission, K$_{s}$-\[IRAC\] $\lesssim$ 0.4, are shown as diamonds (thick diamonds). The further away from the origin point of the loci a given source is, the more the disk contributes to the total flux. The derived dust temperatures are only weakly sensitive to $T_{e,\star}$ as flux ratios for 5250-10000 K ($\sim$ G9-B8) blackbodies occupy roughly the same place at $\lambda_{4.5}$F$_{4.5}$/$\lambda_{8}$F$_{8}$ ($\sim$ 5$\pm$0.15) and $\lambda_{24}$F$_{24}$/$\lambda_{8}$F$_{8}$ ($\sim$ 0). The line for the ice sublimation temperature is shown in bold. The source (diamond at $\sim$ 2, 0.5) with a disk component of $\approx$ 300 K has a strong \[8\] excess and was previously identified as having $\sim$ 300-350 K dust (Source 5 in C07b) using a single blackbody $\chi$$^{2}$ fit to the disk SED. Four other sources, also with 8 $\mu m$ emission, have dust temperatures between 230 K and 250 K. All the sources with 8$\mu m$ excess then have dust temperatures $\ge$ 230 K. While these sources may have cooler dust components, some of the dust emission must come from warmer disk regions closer to their parent stars.
The dust temperatures of sources without 8$\mu m$ excess, characteristic of a slight majority in our sample, are significantly lower. Three sources have slightly cooler temperatures of $\sim$ 170-185 K, comparable to the water ice sublimation temperature (Hayashi 1981). The remaining sources have much cooler dust temperatures ($\sim$ 100-150 K). This diagram demonstrates that many sources must have cold dust with temperatures of T$_{dust}$ $\lesssim$ 200K.
For sources with photospheric IRAC emission, using a single-temperature blackbody - calculated by matching the 24 $\mu m$ excess while not producing significant excess in the IRAC bands - should match the observed disk emission well. However, many sources also have IRAC excess, and modeling the disk emission as coming from two sources (e.g. warm *and* cold dust) may yield a significantly better fit (e.g. Augereau et al. 1999). As an alternate way to constrain the disk temperature(s) and estimate the disk luminosity and location of the dust, we now consider blackbody fits to the dereddened SEDs. For sources with IRAC excess we add sources of hot and cold dust emission with temperatures of 50-250 K and 250-700 K, respectively, to the stellar photosphere. Sources without IRAC excess are modeled by a stellar photosphere + single-temperature disk. For the stellar blackbody, we use the conversion from spectral type to effective temperature from Kenyon and Hartmann (1995). If the star has no spectral type, we use the dereddened J band flux as a proxy for spectral type as in C07a, assuming A$_{J}$ $\sim$ 0.45 and using the Kenyon and Hartmann (1995) conversion table, and add a question mark after the spectral type in Tables 3 and 4. We use the stellar luminosity, L$_{\star}$, for 13-14 Myr old stars of a given spectral type from Siess et al. (2000). A $\chi$$^{2}$ fit to the 3.6-24 $\mu m$ fluxes is performed to find the best-fit one or two dust blackbody + stellar blackbody model following Augereau et al. (1999). Following Habing et al. (2001), we derive the disk luminosity from blackbody fits. The integrated fluxes for each dust population of a given temperature are added and then divided by the stellar flux to obtain the fractional disk luminosity, L$_{D}$/L$_{\star}$. Finally, we estimate the location of the dust populations from simple radiative equilibrium: $$R(AU) \approx (T_{disk}/{280})^{-2}({L_{\star}}/{L_{\odot}})^{0.5}.$$
The sources without IRAC excess have nearly identical disk temperatures to those derived from the flux-ratio diagram (Table 3), ranging from $\sim$ 90 K to 185 K, and are similar to equilibrium temperatures just beyond the terrestrial zone into the gas giant regions of the solar system. These sources have $\chi$$^{2}$ values slightly less than or comparable to the number of observations ($\sim$ 1-6). The fractional disk luminosities range from $\sim$ 5.5$\times$10$^{-4}$ to 3.5$\times$10$^{-3}$, which is similar to dust luminosities for young stars surrounded by optically-thin debris disks (e.g. Meyer et al. 2007). Dust in these systems is probably confined to disk regions of $\sim$ 8-40 AU.
Sources with both IRAC and MIPS excess emission have disk temperatures substantially different from those inferred from the flux-ratio diagram and show evidence of terrestrial zone dust emission and colder dust. The two dust population fits for the IRAC+MIPS excess sources show evidence for a wide range of dust temperatures with warm terrestrial dust emission and cold dust emission similar to that from sources without IRAC excess. For instance, the SED of the A6 star with IRAC and MIPS excess is best fit ($\chi$$^{2}$ $\sim$ 7.6) by a hot dust component of 375 K coming from 1.8 AU and a cold component of 85 K at $\sim$ 37 AU. The faint F9 star, identified previously as ’Source 5’, is extremely well fit ($\chi$$^{2}$ $\sim$ 0.5) by dust populations of 240 and 330 K at 1.8 and 3.4 AU, respectively. Because these sources have both warm and cold dust, it is not surprising that their fractional disk luminosities are typically higher. The fractional luminosity of Source 5 ($\sim$ 6$\times$10$^{-3}$) is comparable to the most massive debris disks (e.g. HR 4796A), and in general the luminosity of the disk population is consistent with values for massive debris disks. The most luminous disk source (L$_{D}$/L$_{\star}$ $\sim$ 1.5$\times$10$^{-2}$) is the lone exception and has a luminosity halfway in between values expected for luminous debris disks ($\sim$ several$\times$10$^{-3}$) and long-lived T Tauri disks with inner holes (e.g. TW Hya; Low et al. 2005). We analyze this system further in §3.3.3.
In summary, the faint MIPS excess sources have dust with a range of temperatures and luminosities. Sources without IRAC excess are well fit by single-temperature blackbodies and have cold dust components with temperatures $\sim$ 90-185 K. Sources with IRAC excess are better fit by two dust components, a hotter, terrestrial zone component and a cooler component. The disks in h and $\chi$ Persei then show evidence of having inner regions of varying sizes cleared of dust. All but one source has a fractional disk luminosity $\lesssim$ 10$^{-2}$, consistent with optically-thin debris disks. In the next section, we investigate the evolutionary state of the faint MIPS excess population further by comparing their properties to other predicted properties for massive debris disks and T Tauri disks.
### Evolutionary State of the MIPS Disk Candidates: A Population of Luminous $\sim$ 13-14 Myr-old Debris Disks
We now consider the evolutionary state of the dust in the 24$\mu m$ excess sources. Although the relative luminosities (L$_{d}$/L$_{\star}$ $\sim$ 10$^{-3}$) and lack of accretion signatures suggest these h and $\chi$ Per sources are debris disks, some T Tauri stars (e.g. ’transition’ T Tauri stars; Kenyon & Hartmann 1995) may also have inner regions cleared of gas and dust. Thus, it is important to compare their disk properties to models of debris disks and T Tauri disks.
We first examine the nature of the h and $\chi$ Per disk population as a whole. Because our smallest disk luminosities, $\sim 5 \times 10^{-4}$, are larger than more than half of known $\gtrsim$ 10 Myr-old disks (e.g. Meyer et al. 2007), our MIPS sample probably misses lower luminosity sources with $L_d/L_{\star} \lesssim 10^{-4}$. Similarly, our lack of 70 $\mu$m detections limits our ability to detect and to evaluate disk emission from cooler dust – such as is observed in $\beta$ Pic and HR 4796A – with SEDs that peak at 40–100 $\mu$m. For example, the nearby, luminous disk around 49 Cet (spectral type A1V, 8 Myr old; @Wa07) has a 24$\mu$m excess of $\sim$ 2.5 magnitudes. We detect only one faint (J $\ge$ 13) MIPS excess sources with K$_{s}$-\[24\] $\le$ 2.5. Therefore, it is possible that there are additional very luminous young disks in h and $\chi$ Per just below our detection limit.
These limits and the rich nature of the Double Cluster allow us to estimate the prevalence of massive, luminous disks. The total number of A0 to early F stars (F2) in h and $\chi$ Per is $\approx$ 1000 (Currie et al. 2007 in prep.; cf. Currie et al. 2007a). Assuming that the disk fraction is $\sim$ 20%[^2], we detect 17/200, or $\approx$ 10% of all disks with strong emission at 24$\mu$m. Thus, this population is extreme and yields a better understanding of the evolutionary state of the most luminous disks in a populous star cluster.
To constrain the evolutionary state of the disks, we compare the near-to-mid infrared disk colors to those expected for two disk models: a flat, optically-thick disk around a Classical T Tauri star ($T_{disk}$$\sim$$r^{-0.75}$; Kenyon & Hartmann 1987) and an optically-thin disk model from Kenyon & Bromley (2004a) for debris emission produced by planet formation. Because only one of our sources has \[4.5\] excess emission and less than half have \[8\] excess emission, we match the data to models of planet formation not in the terrestrial zone (Kenyon & Bromley 2004a) but at 30-150 AU from a 2.0 $M_{\odot}$ primary star (Kenyon & Bromley 2004b). For a $\sim$ 2.0 $M_{\odot}$, 20 L$_{\odot}$ star, the temperature range from 30 to 150 AU is comparable to the outer gas/ice giant region in our solar system ($\sim$ 6.7-34 AU). We adopt a $\Sigma \propto$ $r^{-1.5}$ profile for the initial column density of planetesimals and an initial disk mass of 3$\times$ a scaled Minimum Mass Solar Nebula (Hayashi 1981): 3$\times$ 0.01 M$_{\star}/M_{\odot}$ (where M$_{\star}$ = 2 M$_{\odot}$). Emission from planetesimal collisions is tracked for $\sim$ 10$^{8}$ yr. Model predictions are reddened to values for h and $\chi$ Persei (reddening laws in the IRAC/MIPS bands are described in C07b).
Figure \[colcol\] shows the K$_{s}$-\[4.5, 5.8\]/K$_{s}$-\[24\] color-color diagrams for bright photospheric sources and the faint 24$\mu m$ excess sources. The debris disk locus is overplotted as a thin black line. Debris from planet formation produces a peak excess emission at $K_{s}$-\[24\] $\sim$ 3.6 at $\sim$ 10$^{7}$ years; the $K_{s}$-\[4.5\] and $K_{s}$-\[8\] colors peak at $\sim$ 0.4 at earlier times ($\sim$ 10$^{6}$ years). The debris disk locus tracks the colors for most of the sources in $K_{s}$-\[4.5\]/$K_{s}$-\[24\] space very well (Figure \[colcol\]a). While the locus underpredicts the \[8\] excess for about half of the sources (Figure \[colcol\]b), warmer regions of a debris disk not modeled here may produce this excess (e.g. KB04a). C07b showed that planet formation in the terrestrial zone can produce strong \[8\] emission characteristic of some h and $\chi$ Per sources at $\sim$ 10-15 Myr. Indeed, the source with $K_{s}$-\[8\]$\sim$ 1.3, $K_{s}$-\[24\]$\sim$ 4.4 is Source 5 from C07b which was one of eight modeled as having terrestrial zone debris disk emission. The warm dust temperature ($\sim$ 300 K) derived for this source in §4.2 is consistent with terrestrial zone emission.
Disk models corresponding to earlier evolutionary states fare worse in matching the observed mid-IR colors. The optically-thick flat disk model (the triangle in both plots) predicts $K_{s}$-\[5.8\] (\[8\]) $\sim$ 1.5 (2.9) and $K_{s}$-\[24\]$\sim$6, consistently 1-2 magnitudes redder than the data. To match the observed \[24\] excess, any optically-thick disk with an inner hole (cf. C07b) must be cleared of dust out to the distances probed by the MIPS bands: $\sim$ 25 AU for a 20 $L_{\odot}$ primary star. While inner hole models may be constructed to fit the SEDs of sources with only 24 $\mu m$ excess, these models predict nearly zero IRAC color even though about half of the sample has excess at \[8\]. Lack of gas accretion signatures, low fractional disk luminosities, and SED modeling then suggest that at least many faint h and $\chi$ Per sources with 24 $\mu m$ excess are stars surrounded by optically-thin debris disks. More sensitive spectroscopic observations are needed to verify the lack of gas in these systems.
Despite the general success of the debris disk models, at least one h and $\chi$ Per source may harbor a disk at an earlier evolutionary state. This source has a K$_{s}$-\[24\] color of $\sim$ 6, which is $\sim$ 1 mag redder than HR 4796A, the strongest 24$\mu m$ excess source in R05. This color is close to the optically-thick disk predictions, is extremely difficult to produce with a debris disk model, and is more similar to the level of excesses in older T Tauri stars like HD 152404 and TW Hya (Chen et al. 2005b; Low et al. 2005).
To explore this possibility, we overplot the K$_{s}$-\[4.5\], K$_{s}$-\[8\], and K$_{s}$-\[24\] colors of TW Hya from Hartmann et al. (2005) and Low et al. (2005) in Figure \[colcol\] (large cross, reddened to h and $\chi$ Per). The mid-IR colors of our brightest source are similar to the colors of TW Hya. While TW Hya’s disk has an optically-thin inner region where the early stages of planet formation may be commencing (Eisner et al. 2006), the disk is probably optically-thick at 24$\mu m$ (Low et al. 2005). TW Hya also has strong H$_{\alpha}$ emission which indicates accretion. On the other hand, the fractional disk luminosity in h and $\chi$ Per sources is much lower than that of TW Hya ($\sim$ 0.27, Low et al. 2005) and in between values for debris disks and transition disks. Thus, some lines of evidence suggest that this extreme h and $\chi$ Per source is at an evolutionary state earlier than the debris disk phase while others are more ambiguous. Obtaining optical spectra of this source, to search for accretion signatures, may allow us to make a better comparison between it and older T Tauri stars like TW Hya. The spectral energy distributions (SEDs) of the faint MIPS excess sources show evidence for a range of dust temperature distributions, which may be connected to a range of evolutionary states (Figure \[sed4\]). We select four sources, three with spectra and one without, that are representative of the range of mid-IR colors from our sample. The first three sources of Figure \[sed4\], dereddened to A$_{V}$=1.62 (E(B-V)=0.52), have been spectroscopically confirmed as F9 (source 1), F9 (source 2), and A6 (source 3) stars, respectively; the second source was mentioned in the previous paragraph (with K$_{s}$-\[24\]$\sim$ 4.4). The SEDs for the bottom left source was also dereddened to A$_{V}$=1.62, and a spectral type of A2 was chosen based on the conversion from absolute magnitude to spectral type for 14 Myr-old sources (from Siess et al. 2000; Kenyon & Hartmann (1995) color conversions). The source with photospheric emission at $\lambda$ $<$24$\mu m$ (source 4) has IRAC colors representative of just over half of the faint MIPS-excess sources in Figure \[colcol\]. The debris disk model accurately predicts the SEDs of the source with photospheric 8$\mu m$ emission and two sources with weak 8$\mu m$ excess emission. The remaining source is not fit well by the disk model and shows clear evidence for a large warm dust population (see C07b). The evolutionary states for the sources shown in Figure \[sed4\] and the 9 sources with complete IRAC and MIPS photometry are listed Table 2.
Thus, we conclude that emission from at least half of the 24$\mu m$ excess sources around pre-main sequence stars in h and $\chi$ Per is best explained by debris from planet formation at locations comparable to the gas/ice giant regions in the solar nebula. Some of the other pre-main sequence stars with 24 $\mu m$ excess may also have ongoing planet formation in the inner, terrestrial zone regions as indicated by their 8 $\mu m$ excesses. One of our sources may be a T Tauri star at a slightly earlier evolutionary state than the debris disk sources in our sample.
If most of the disk population is then interpreted as an early debris disk population (not a Class II/III transition T Tauri disk population), the wavelength-dependent frequency of IRAC/MIPS disk excess identified in §3.3.1 implies a location-dependent evolution of debris disks, specifically a clearing of warm dust from inner disk regions. This behavior is consistent with standard models of planet formation (KB04a), which predict that dust emission from the planet formation process disappears at shorter wavelengths (e.g. IRAC bands) faster than at longer wavelengths (e.g. MIPS bands). This result is expected if planet formation runs to completion in the innermost regions of protoplanetary disks before planets are formed in the outer disk.
Evidence for a Rise and Fall of Debris Disk Emission
====================================================
To place our results in context, we now compare the excesses observed in h and $\chi$ Per sources with measurements of other stars with roughly similar masses. We follow R05 and consider the magnitude of the 24$\mu m$ excess, \[24\]$_{obs}$-\[24\]$_{\star}$, as a function of time. Using a sample of early (A) type stars with ages $\gtrsim$ 5 Myr, R05 showed that stars have a wide range of excesses at all ages and that sources with the largest excesses define an envelope that decays slowly with time (Figure \[excvagegr\]). Although this envelope is consistent with a power-law decay, \[24\]$_{obs}$-\[24\]$_{\star}$ $\propto$ t$^{-1}$, the R05 sample has relatively few stars with ages $\sim$ 5-20 Myr where debris disk models predict large excesses. The sources with the largest excesses, HR 4796A and $\beta$ Pic, fall within this age range at 8 and 12 Myr[^3], respectively.
Together with our results for h and $\chi$ Per, several recent surveys in young clusters and associations identify debris disks with ages of $\sim$ 5-20 Myr (Chen et al. 2005b; Hernandez et al. 2006; Sicilia-Aguilar et al. 2006). As in R05, these surveys show a large range of 24 $\mu m$ excesses at each age. In the well-sampled Sco-Cen Association, for example, Chen et al. (2005b) identify many stars with photospheric emission (no excess) at 24$\mu m$ and several stars with excesses considerably larger than the typical excess observed in the R05 sample. Although our h and $\chi$ Per data do not provide any measure of the number of stars with photospheric emission at 24 $\mu m$, the survey yields a good sample of stars with excesses much larger than the typical R05 source.
We now combine our results with those from R05 and from more recent surveys of debris disks in young clusters. Specifically, we add data from Tr 37 (4 Myr) and NGC 7160 (11.8 Myr) (in Cepheus; Sicilia-Aguilar et al. 2006), Orion OB1a (10 Myr) and Orion OB1b (5 Myr; both from Hernandez et al. 2006), and Sco-Cen ($\sim$ 5, 16, and 17 Myr for Upper Sco, Lower Centaurus Crux, and Upper Centaurus Lupus, respectively; Chen et al. 2005b). For h and $\chi$ Per and Cepheus sources, we include only the IR-excess sources. The sensitivity of the Sco-Cen observations allows more precise determinations of the photospheric flux levels farther down the initial mass function of the cluster, so we include data for all sources earlier than G0 with or without excess in this cluster. For sources with no published estimate of the photospheric flux, we assume that K$_{s}$-\[24\]$_{\star}$ $\sim$ 0 (dereddened), which is valid for our sample of A and F stars.
Observed Mid-IR Emission vs. Age
--------------------------------
When data from h and $\chi$ Persei and other young clusters are added to R05, the evolution of 24 $\mu m$ excess with age shows an important new trend. *From $\sim$ 5-10 Myr, there is a clear rise in the magnitude of excess followed by a peak at $\sim$ 10-15 Myr, and a slow t$^{-1}$ decay after $\sim$ 15-20 Myr* (Figure \[excvage\]). All sources with very large ($\gtrsim$ 3 mag) excesses have ages between 8 and 16 Myr. The 24 $\mu m$ excess emission peaks at $\sim$ 12-16 Myr as indicated by strong excess sources in h and $\chi$ Persei (diamonds), NGC 7160 (squares), and Sco-Cen (asterisks). Data from 5 Myr-old Orion OB1b and Upper Sco to 10 Myr-old Orion OB1a to 12-17 Myr-old NGC 7160, h and $\chi$ Per, and the two older Sco-Cen subgroups shows a sequential rise in the median 24 $\mu m$ excess [^4]. A peak in the 24 $\mu m$ excess emission at $\sim$ 10 Myr is also visible in a plot from Hernandez et al. (2006), albeit at a lower statistical significance. The addition of several $\lesssim$ 20 Myr-old clusters more strongly constrains the time when debris emission peaks and maps out its evolution from 5-20 Myr in more detail.
Removing possible TW Hya-like sources from this diagram does not modify the trends. The ’TW Hya-like’ source in h and $\chi$ Per (\[24\] excess $\sim$ 5.5) and the strongest excess source in Sco-Cen (\[24\] excess $\sim$ 5.75) have the largest 24$\mu m$ excesses and may be at an evolutionary state prior to the debris disk phase. However, many sources in the 10-15 Myr age range have $\sim$ 2-3.5 magnitude excesses, including HR 4796A and many h and $\chi$ Per sources, which have disk luminosities and mid-IR colors inconsistent with an optically-thick disk. The second most luminous source in Sco-Cen, HD 113766A (F3 spectral type) with a 24 $\mu m$ excess of $\sim$ 4.7 magnitudes, has a fractional disk luminosity characteristic of a massive debris disk (Chen et al. 2005b). Sources with \[24\]$_{observed}$-\[24\]$_{\star}$ $\ge$ 2-3 are more common at $\sim$ 10-15 Myr than at much younger ($\sim$ 5 Myr) or older ($\gtrsim$ 20 Myr) ages.
Statistical Verification of a Peak in 24 $\mu m$ Emission at 10-15 Myr
----------------------------------------------------------------------
The peak at 10-15 Myr is statistically robust. To test it, we adopted the underlying approach that Wyatt et al. (2007a) demonstrate gives a good first-order description of debris disk behavior: debris disks all evolve in a similar fashion, with the variations among them arising primarily from differences in initial mass. This result has two important implications for us: 1.) it validates deducing evolution with time from the upper envelope of the infrared excesses, since similar high-mass disks of different ages define this envelope; and 2.) it allows us to estimate the distribution of excesses at any time by scaling the excesses at another time according to $t^{-1}$ (by one over the ratio of the source ages), the general time dependence of disk decay (R05). We use the second of these results to predict the distribution of excesses at 5Myr from measurements of the distribution at 10 - 30 Myr, where enough systems have been measured to define the distribution well. We use three samples (Sco-Cen, Orion Ob1, and R05), each of which includes the complete range of \[24\]$_{obs}$-\[24\]$_*$ down to zero, i.e., photospheric colors. If the scaled colors for sources predict much larger excesses than the 5Myr old Orion Ob1b excesses, then we can conclude that the mid-IR colors of our samples from 5 to 15 Myr do not follow a $t^{-1}$ decline.
Figure \[scaled\] shows the scaled Sco-Cen and Rieke et al. excesses compared to the observed 5 Myr excesses in Orion Ob1b normalized to the total number of sources in each sample. Many scaled excess sources ($\sim$ 20% of the total population) are $\gtrsim$ 1-3 mags redder than any in Orion Ob1b. The Kolmogorov-Smirnov test shows that the scaled Sco-Cen (Rieke et al.) sources have a probability of 0.07 (0.06) of being drawn from the same population as Orion Ob1b with a $\gtrsim$ 0.75 mag excess. If we instead compare the scaled sources to Orion Ob1b sources with a $\gtrsim$ 0.5 mag excess, where Orion and the other two populations begin to overlap in Figure \[scaled\], the probability is even lower: 2.3$\times$10$^{-5}$ (3.5$\times$10$^{-8}$) for the Sco-Cen (Rieke et al.) sample. Thus, the evolution of 24 $\mu m$ emission from Orion Ob1b, Sco-Cen, and Rieke et al. sources is not consistent with a t$^{-1}$ decay.
There are two main alternatives to the t$^{-1}$ decay of IR excess with time for the youngest debris disks. The IR excess could be constant for $\sim$ 20 Myr and then follow a t$^{-1}$ decay law. The IR excess could also increase with time to some peak value and then follow a t$^{-1}$ decay law. To test the constant emission possibility, we use the Wilcoxon Rank-Sum test. The Rank-Sum test allows us to evaluate whether or not the populations have the same mean value or have intrinsically larger/smaller excesses than another sample (Z parameter). The test also measures the probability that two samples are drawn from the same parent population by the Prob(RS) parameter (as in the K-S test).
Table 5 summarizes our results. Statistical tests show that the \[24\]$_{obs}$-\[24\]$_{\star}$ excesses cannot be constant with time and verify that emission rises from 5 to 10 Myr (Table 3). Sco-Cen has the largest mean excess ratio ($\sim$ 1.25) and has the widest range of colors with a \[24\]-\[24\]$_{\star}$ standard deviation of $\sim$ 1.4 compared to $\sim$ 0.6-0.8 for the Orion subgroups. The Wilcoxan Rank-Sum test reveals that Sco-Cen has $\lesssim$ 5% probability of being drawn from the same population as Orion Ob1b and has a statistically significant larger peak (Z $\sim$ -1.65). Orion Ob1a also has a significant larger peak than Orion Ob1b (Z $\sim$ -2.8, Prob (RS) $\sim$ 0.002), and Sco-Cen’s peak is marginally larger than Orion Ob1a’s (Z $\sim$ -0.05).
These results lead us to conclude that ***the evolution of mid-IR excess emission from planet formation in debris disks is best characterized as a rise in excess from $\sim$ 5-10 Myr, a peak at $\sim$ 10-15 Myr, and a fall in excess from $\sim$ 15/20 Myr-1 Gyr***. The rise to maximum excess from 5 to 10 Myr is steep: typical excesses increase from $\sim$ 1 mag to $\sim$ 3 mag by 11.8 Myr. The peak in excess amplitude is $\sim$ 5-10 Myr broad because the excesses in NGC 7160, h and $\chi$ Per, and Sco-Cen are all comparable. By $\sim$ 25 Myr the typical excesses decline with age as suggested by R05.
Comparison with Models of Emission from Planet Formation
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To investigate how the ’rise and fall’ trend of debris emission may be connected with physical processes producing the emission, we overplot the debris disk evolution tracks (dotted lines) from §3.3.3 and debris disk tracks for a low-mass disk (1/3$\times$MMSN$_{scaled}$). The KB04 calculations start at t=0 with an ensemble of $\sim$ km-sized planetesimals. To bracket the likely timescale for km-sized planetesimals to form at 30-150 AU (e.g. Dominik & Dullemond 2005; Weidenschilling 1997), we include a second locus shifted by 2 Myr (solid lines). The range in debris disk masses is about a factor of 10, comparable to the range of disk masses inferred from submillimeter observations of young stars [@Aw05].
The debris disk models from §3.3.3 show a steep increase in debris emission from 5-10 Myr, a plateau for the following $\sim$ 20 Myr, and then a shallow decline in debris emission. The massive disk locus (dotted line) yields a peak in emission at $\sim$ 7-8 Myr, or very close to the age of HR 4796A. The locus started at 2 Myr (solid line) peaks at $\sim$ 9-10 Myr, close to the ages of h and $\chi$ Persei and NGC 7160, and yields substantial emission through 20 Myr before emission declines. The lower-mass disk loci peak later at $\sim$ 40 Myr with excesses comparable to the majority of those in R05. Caution should be taken to avoid overinterpreting these similarities: the exact time of the debris disk emission peak as well as the amplitude of the peak depend on input parameters such as planetesimal disruption energy. Nevertheless, the massive debris disk model yields the same general trend in the maximum 24 $\mu m$ excess amplitude with time; the low-mass disk model reproduces the 24 $\mu m$ excesses of many $\gtrsim$ 30 Myr-old sources. The observed behavior of 24 $\mu m$ excess with time is then at least qualitatively consistent with our understanding of the processes associated with planet formation.
Discussion
==========
Summary of Results
------------------
Our analysis of MIPS data for the 13-14 Myr-old double cluster, h and $\chi$ Persei, shows two significant 24 $\mu m$ excess populations. Bright Be stars with J$\lesssim$ 12-13 have 1-2 mag excesses at 24 $\mu m$ and follow a clear Be star locus in the J-H/H-K$_{s}$ color-color diagram. Optical spectra confirm the Be star status for just under half of the candidates from the color-color diagram. We also detect a B4 star with a clear 24 $\mu m$ excess but without H$_{\alpha}$ emission or evidence for near-IR excess.
Fainter stars with J $\sim$ 14-15 fall on the 14 Myr isochrone in a J/J-H color-magnitude diagram. Optical spectra confirm that many of these stars have late A-type or F-type spectra, consistent with cluster membership. The IRAC and MIPS colors of these sources suggest that the frequency of excess at wavelengths which probe IR excess emission increases with increasing wavelength. The wavelength-dependent frequency of excess is consistent with the presence of inner holes devoid of dust.
Our analysis of the dust temperatures in the fainter excess sources suggest two groups. A smaller group of stars has emission from warmer dust with T $\sim$ 200-300 K. A larger group has emission from colder dust, T $\lesssim$ 200 K. In both groups, the dust luminosity is a small fraction of the stellar luminosity, L$_{d}$/L$_{\star}$ $\sim$ 10$^{-4}$-10$^{-3}$, typical of debris disks like HR 4796A (Low et al. 2005). The IR colors and spectral energy distributions of the latter group are consistent with predictions from cold debris disk models; sources with warmer dust may have terrestrial zone debris emission (see also C07b).
The MIPS data from h & $\chi$ Persei and other recently surveyed clusters yield a large sample of disks at 5-20 Myr, an age range critically important for understanding debris disk evolution and planet formation. This sample shows that debris disk emission rises from 5 Myr to $\sim$ 10 Myr, peaks at $\sim$ 10-15 Myr, and then fades on a $\sim$ 150 Myr timescale as t$^{-1}$ (R05). Numerous statistical tests verify the observed trend. Debris production from ongoing planet formation explains the general time evolution of this emission (e.g. KB04).
Future Modeling Work: Explaining the Range of MIPS Excesses
-----------------------------------------------------------
The debris disk models from Kenyon & Bromley (2004a) generally explain the peak excesses for sources in h & $\chi$ Per and other $\sim$ 10-15 Myr-old clusters. However, at a given age stars have a wide spread of IR excesses above and below the debris disk model predictions. The IR excesses far weaker than the model predictions have several tenable explanations. Low-mass disks modeled in §4 have weaker excesses. Disks in systems with binary companions close to the disk radius are probably disrupted quickly, although binaries with wider separations should have little effect, and very close separations may actually enhance infrared excesses (Bouwman et al. 2006; Trilling et al. 2007). Gas giant planets may also remove IR-emitting dust.
Reproducing the larger IR excesses (K$_{s}$-\[24\] $\gtrsim$ 4-5) is more difficult. The debris disk model used in this paper yields a peak K$_{s}$-\[24\] $\sim$ 3.5 (unreddened), but HR 4796A and several sources in h and $\chi$ Per and Sco-Cen have stronger excesses. More massive disks should yield stronger 24$\mu m$ excesses, but the disk mass cannot be increased indefinitely. A disk with mass M$_{d}$ $\gtrsim$ 0.1-0.15 M$_{\star}$ would have been initially gravitationally unstable and would form a low-mass companion. However, the debris disk status of one of these extreme cases, HR 4796A with K$_{s}$-\[24\] $\approx$ 5, has been confirmed by extensive disk SED modeling (e.g. @Au99 [@Cu03; @Wa05]) and strict gas mass upper limits of $\lesssim$ 1 M$_{Jupiter}$ [@Ck04]. There are several ways to account for these larger excesses. For example, dynamical processes that allow small grains to be retained (which produce larger opacity) in rings like that observed for HR 4796A may explain the large-amplitude excesses in some debris disks (e.g. Klahr and Lin 2005; Takeuchi and Artymowicz 2001). Whether or not the strong excess sources in h and $\chi$ Per can be explained by such grain confinement mechanisms is the subject of future work.
Comparison with Previous Spitzer Observations and Analysis of h and $\chi$ Persei from @Cu07a and @Cu07b
--------------------------------------------------------------------------------------------------------
This paper completes the first study of the circumstellar disk population of pre-main sequence stars in the massive double cluster, h and $\chi$ Persei. Together with Cu07a and Cu07b, this work provides new constraints on the frequency, lifetimes, and evolutionary states of circumstellar disks in 10-15 Myr old stars. Here we summarize the main results and conclusion of these studies.
Spitzer data for h and $\chi$ Per provide clear evidence that the frequency of IR excess emission depends on wavelength and on the mass of the star (C07a; this paper, §3). Stars in both clusters have a higher frequency of IR excess at longer wavelengths. Lower mass (1.4-2 M$_{\odot}$) stars have IR excesses more often than more massive ($\gtrsim$ 2 M$_{\odot}$) stars. Su et al. (2006) and @Gr07 derive similar results for other clusters. Taken together, these results are consistent with an inside-out clearing of dust from young circumstellar disks, as expected from theoretical models of planet formation (e.g. @KB04b).
To compare the completeness level of the MIPS sample with the IRAC sample from C07a, we derive the fraction of IRAC sources with MIPS detections at each IRAC band. The IRAC survey has 90% completeness levels of $\sim$ 14.5 at \[4.5\] and $\sim$ 13.75 at \[8\]. The MIPS sample includes 87% (3%) of the IRAC sources with \[3.6\] $<$ 10 (\[3.6\] $<$ 14.5) and 88% (11%) of the IRAC sources with \[8\] $<$ 10 (\[8\] $<$ 13.75) within 25’ of either cluster center. Because the MIPS survey detects such a small percentage of the IRAC sources in C07a, we cannot analyze a statistically significant population of IRAC IR excess sources with MIPS detections. However, the MIPS sources yield interesting constraints on the Be star population in h and $\chi$ Per (§3.2) and demonstrate a clear peak in the time evolution of the 24 $\mu$m excess of debris disks (§4.).
Detailed analyses of the IRAC/MIPS colors and the broadband SEDs demonstrate that warm dust (T $\sim$ 240–400 K) is visible in 11 cluster stars (C07b; this paper, §3). The dust luminosities of ten of these sources ($\sim$ 10$^{-4}$-6$\times$10$^{-3}$ L$_{\star}$) suggest this emission arises from optically thin dust in a debris disk. The IR excesses of these sources – which comprise the majority of known warm debris disks (see Hines et al. 2006, Wyatt et al. 2007b, and @Gr07 for others) – are consistent with detailed calculations of terrestrial planet formation around $\sim$ 2 $M_{\odot}$ stars [@KB04a].
Most of the IRAC/MIPS IR excess sources show evidence for cooler dust with T $\sim$ 100–200 K (this paper; §3, 4). Although the lack of 70 $\mu$m detections prevents us from deriving precise limits on the dust temperatures and luminosities, the Spitzer data suggest that most (perhaps all) of these sources are debris disks with SEDs similar to those observed in Sco-Cen, the TW Hya Association and other young clusters. (@Ch05b [@Lo05]). When combined with data from the literature, these data provide clear evidence for a rise in the magnitude of the IR excesses from debris disks from $\sim$ 5 Myr to $\sim$ 10–15 Myr followed by a fall from $\sim$ 20 Myr onward.
Although theory provides a reasonably good first-order explanation for the time evolution of the IR excesses in $\gtrsim$ 1.4 M$_{\odot}$ stars, some aspects of the observations remain challenging. A large range of initial disk masses and binary companions can probably explain the large range in IR excesses at a given stellar age, but these explanations require further testing. Current theory does not explain the largest IR excesses observed in the 10–20 Myr old stars in h and $\chi$ Persei, Sco-Cen, and the TW Hya Association (specifically HR 4796A). Dynamical, radiative, and stochastic processes not included in the numerical calculations are possible solutions to this failure. Increasing the sample size of this extreme population would provide better constraints on these processes.
Finally, this survey has provided us with several interesting sources that warrant more detailed investigation. For instance, ’Source 5’ – discussed here and in @Cu07b – is probably an extremely massive debris disk. With a fractional disk luminosity of $\sim$ 6$\times$ 10$^{-3}$, its emission rivals that of HR 4796A, $\beta$ Pic and other massive, nearby debris disks. However, this source differs from these other sources in at least two important ways. First, its spectral type is later (F9) than most stars with massive debris disks. Second, it harbors far warmer dust (T$_{d}$ $\sim$ 240-330 K) than most massive debris disks like HR 4796A (T$_{d}$ $\sim$ 110 K; @Lo05). This feature may make it more similar to the warm debris disk of HD 113766A in Sco-Cen, the second most luminous source in Sco-Cen shown in Figure \[excvage\] [@Ch05b], than to HR 4796A and $\beta$ Pic. Longer wavelength observations of this h and $\chi$ Per source (e.g. 30-100 $\mu m$) will better constrain its SED and thus its dust population(s). Mid-IR spectroscopy of this source may also provide clues to the chemical composition of its circumstellar dust to compare with models of cometary and asteroidal material.
Future Observations
-------------------
Future observations of h and $\chi$ Persei will provide stronger constraints on debris disk evolution and the possibilities for producing the wide range of debris disk emission. A deeper MIPS survey (approved for Spitzer cycle 4) of the double cluster will identify $\gtrsim$ 1000-2000 cluster sources with \[24\] $\lesssim$ 12.25, the brightness of an 1.3 M$_{\odot}$ G9 (1.7 M$_{\odot}$ A8) star with a 3 (2) magnitude excess. This $\sim$ 2 magnitude increase in sensitivity ($\sim$ 2260 seconds/pixel integration) should yield a larger sample of 24 $\mu m$ excess sources which will better map out the distribution of mid-IR excesses during the primordial-to-debris disk transition. If the correlation of massive, high fractional luminosity-disks with early A stars is purely a selection effect, then this deeper survey of h and $\chi$ Persei should reveal many massive debris disks around slightly later-type stars like ’Source 5’. This survey will be complemented by a deeper IRAC survey (also approved for cycle 4) of the double cluster, which will identify h and $\chi$ Per sources with \[5.8\] (\[8\]) $\lesssim$ 15.9 (15.2), the brightess of a $\sim$ 0.8 M$_{\odot}$ M0 (1.0 M$_{\odot}$ K6) photosphere. The $\sim$ 1.5-2 magnitude increase in sensitivity ($\sim$ 120 seconds/pixel integration vs. 20.8 seconds/pixel from the C07a observations) will likely result in photometry for $\gtrsim$ 10,000-15,000 cluster stars through \[8\], assuming a typical cluster initial mass function (e.g. Miller and Scalo 1979). These two surveys will likely detect hundreds of debris disk (and perhaps transition disk) candidates and yield extremely strong constraints on evolution of dust in circumstellar disks from warm, inner regions (IRAC) to cooler regions (MIPS) at a critical age for planet formation. Ground-based surveys of h and $\chi$ Persei may also provide important clues about the evolution of disks around young stars. For instance, the ability of binary companions to affect the mid-IR excesses from disks can also be tested. At $\sim$ 2.34 kpc, a binary system with separation of $\sim$ 100 AU (and thus able to truncate debris disks) has an angular separation of $\approx$ 0.04". Such systems can be resolved by long-baseline interferometers such as the Keck Interferometer. Comparing the IR excesses from single and binary systems can then determine if weaker IR excess sources are binaries. A large-scale spectroscopic survey of all sources in h & $\chi$ Per brighter than V $\sim$ 21 ($\gtrsim$ 10,000) is underway (Currie et al. 2007, in prep.). This survey will identify sources most likely to be h & $\chi$ Per members as well as those with strong H$_{\alpha}$ emission indicate of gas accretion. Preliminary work indicates that the population of accreting h and $\chi$ Per sources is non negligible ($\gtrsim$ 20-30; @Cu07c). Comparing the IR excesses of accreting sources with those that are not accreting may examine the role of residual circumstellar gas in affecting the mid-IR excesses from disks. We thank the referee for a thorough review and suggestions which improved this manuscript. We also thank Michael Meyer, John Carpenter, Christine Chen, and Nadya Gorlova for useful discussions regarding debris disks in other clusters; Matt Ashby, Rob Gutermuth, and Anil Seth provided valuable advice regarding galaxy contamination. We acknowledge from the NASA Astrophysics Theory Program grant NAG5-13278, TPF Grant NNG06GH25G, and the Spitzer GO program (Proposal 20132). T.C. is supported by a SAO predoctoral fellowship; Z.B. received support from Hungarian OTKA Grants TS049872 and T049082. This work was partially supported by contract 1255094, issued by JPL/Caltech to the University of Arizona.
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[^1]: The first source in Table 3, with J=13.84, has a \[8\] flux that has a negative K$_{s}$-\[8\] color and thus is unphysically faint. An unphysically large ratio of the \[4.5\] to \[8\] flux cannot be interpreted with a flux-ratio diagram.
[^2]: disk fractions quoted by Chen et al. (2005b) range from 9% to 46%
[^3]: While $\beta$ Pictoris was given an age of 20 Myr in R05, derived from Barrado y Navasceus et al. (1999), recent work suggests a slightly younger age of $\sim$ 12 Myr (e.g. Zuckerman et al. 2001; Ortega et al. 2002).
[^4]: The debris disk candidates in Tr 37 have larger excesses than those in Orion OB1a. However, the strong excess may be explained by differences in stellar properties: 2/3 of the debris disk systems in Tr 37 are B3/B5 and B7 stars, which are far more massive than 10 Myr-old A/F stars ($\sim$ 3.5-6 M$_{\odot}$ vs. 1.5-2.5 M$_{\odot}$; cf. Siess et al. 2000). If typical disk masses scale with the stellar mass, then these much more massive stars should have more massive, more strongly emitting disks. The disk mass-dependent amplitude of excess is discussed in §4.3.
|
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abstract: 'In a recent paper Richard Gill has criticized an experimental proposal published in a journal of physics which describes how to detect a macroscopic signature of spinorial sign changes. Here I point out that Gill’s worries stem from his own elementary algebraic and conceptual mistakes, and presentseveral event-by-event numerical simulations which bring out his mistakes by explicit computations.'
author:
- Joy Christian
title: 'Macroscopic Observability of Spinorial Sign Changes: A Reply to Gill'
---
In a recent paper a mechanical experiment has been proposed to test possible macroscopic observability of spinorial sign changes under ${2\pi}$ rotations [@IJTP]. The proposed experiment is a variant of the local model for the spin-1/2 particles considered by Bell [@Bell], which was later further developed by Peres providing pedagogical details [@Peres][@Can]. Our experiment differs, however, from the one considered by Bell and Peres in one important respect. It involves measurements of the actual spin angular momenta of two fragments of an exploding bomb rather than their normalized spin values, ${\pm 1}$.
It is well known that angular momenta are best described, not by ordinary polar vectors, but by pseudo-vectors, or bivectors, that change sign upon reflection [@Can][@Christian]. One only has to compare a spinning object, like a barber’s pole, with its image in a mirror to appreciate this elementary fact. The mirror image of a polar vector representing the spinning object is not the polar vector that represents the mirror image of the spinning object. In fact it is the negative of the polar vector that does the job. Therefore the spin angular momenta ${{\bf L}({\bf a},\,\lambda)}$ in the theoretical analysis of the experiment proposed in the paper [@IJTP] have been represented by bivectors using the powerful language of geometric algebra [@GA]. They can be expressed in terms of the bivector basis (which are [*graded*]{} basis) satisfying the sub-algebra $$L_{\mu}(\lambda)\,L_{\nu}(\lambda) \,=\,-\,g_{\mu\nu}\,-\,\sum_{\rho}\,\epsilon_{\mu\nu\rho}\,L_{\rho}(\lambda)\,, \label{wh-o8899}$$ where ${\lambda=\pm\,1}$ represents a choice of orientation of a unit 3-sphere [@IJTP]. This brings us to the first of several elementary mistakes made by Gill. In his paper [@Gillprint] he claims that, from the above equation, using ${L_{\mu}(\lambda=-1)=-\,L_{\mu}(\lambda=+1)}$, $$\begin{aligned}
&L_{\mu}(\lambda=+1)\,L_{\nu}(\lambda=+1)\,-\,L_{\mu}(\lambda=-1)\,L_{\nu}(\lambda=-1) \,=\,-\,\sum_{\rho}\,\epsilon_{\mu\nu\rho}\,L_{\rho}(\lambda=+1)\,+\,\sum_{\rho}\,\epsilon_{\mu\nu\rho}
\,L_{\rho}(\lambda=-1) \label{2}\\
&\text{implies}\;\;\;\;\;\;
0\,=\,-2\sum_{\rho}\,\epsilon_{\mu\nu\rho}\,L_{\rho}(\lambda=+1)\,=\,+2\sum_{\rho}\,\epsilon_{\mu\nu\rho}\,L_{\rho}(\lambda=-1),
\label{notwh-o8899} \\
&\text{which in turn implies}\;\;\;\;\;\;
L_{\rho}(\lambda=+1)\,=\,L_{\rho}(\lambda=-1) \,=\,0\,.\label{yeswh-o8899}\end{aligned}$$ It is not difficult to see, however, that this claim is manifestly false. Gill’s mistake here is to miss the summation over the index ${\rho}$. And since the basis elements ${L_{\rho}(\lambda=+1)}$ and ${L_{\rho}(\lambda=-1)}$ do not vanish in general, contrary to his claim neither the spin angular momentum ${{\bf L}({\bf a},\,\lambda)}$ about the direction ${\bf a}$ nor the spin angular momentum ${{\bf L}({\bf b},\,\lambda)}$ about the direction ${\bf b}$ vanish in general. Only the spin angular momentum ${{\bf L}({\bf a}\times{\bf b},\,\lambda)}$ about the mutually orthogonal direction ${{\bf a}\times{\bf b}}$ vanish, as in Eq.${\,}$(110) of the paper [@IJTP]. This is of course consistent with the physical fact that there is no third fragment of the bomb spinning about the direction ${{\bf a}\times{\bf b}}$ exclusive (as well as orthogonal) to both directions ${\bf a}$ and ${\bf b}$. To see that only the spin angular momentum ${{\bf L}({\bf a}\times{\bf b},\,\lambda)}$ about the direction ${{\bf a}\times{\bf b}}$ vanishes, all one has to do is to contract Eq.${\,}$(\[wh-o8899\]) above with the vector components ${a_{\mu}}$ and ${b_{\nu}}$, on both sides, and then follow through the steps in Eqs.${\,}$(\[2\]) and (\[notwh-o8899\]). It is also crucial to appreciate that the spin angular momenta ${{\bf L}({\bf s},\,\lambda)}$ ([*i.e.*]{}, the bivectors) trace out an su(2) 2-sphere within the group manifold ${{\rm SU(2)}\sim S^3}$, not a round ${S^2}$ within ${{\rm I\!R^3}}$ as Gill has incorrectly assumed.
Unfortunately the conceptual mistakes made by Gill are even more serious than the above elementary mathematical mistakes [@Gillprint][@refute]. As already noted, what are supposed to be measured in the proposed experiment are the spin angular momenta ${{\bf L}({\bf a},\,\lambda)}$ and ${{\bf L}({\bf b},\,\lambda)}$ themselves, not their normalized spin values ${\pm 1}$ about some directions ${\bf a}$ and ${\bf b}$, where $$\begin{aligned}
{\bf L}({\bf a},\,{\lambda})\,&\equiv\,A({\bf a},\,\lambda)\,\equiv\,\lambda({\bf e}_x\wedge{\bf e}_y\wedge{\bf e}_z)\cdot{\bf a}\;=\,\pm 1 \;\,\text{spin about}\;\,
{\bf a} \\
\text{and}\;\;\;{\bf L}({\bf b},\,{\lambda})\,&\equiv\,B({\bf b},\,\lambda)\,\equiv\,\lambda({\bf e}_x\wedge{\bf e}_y\wedge{\bf e}_z)\cdot{\bf b}\,=\,\pm 1 \;\,\text{spin about}\;\, {\bf b}\,.\end{aligned}$$ This is in sharp contrast to what are measured as dynamical variables in the model considered by Bell and Peres [@Can].
In practice the above dynamical variables are supposed to be measured in the proposed experiment by directly observing the polar vectors ${{\bf s}^k}$ [*dual*]{} to the random bivectors ${{\bf L}({\bf s},\,\lambda^k)}$. It is only after all the runs of the experiment are completed and the vectors ${{\bf s}^k}$ are fully recorded, the traditional dynamical variables ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ supposed by Bell are to be calculated, by an algorithm extraneous to the actual experiment. These calculations may be done, for example, years after the experiment has been completed. Thus they are not supposed to be an integral part of the physical experiment itself, and this fact must be accounted for in the ensuing statistical analysis of data.
Since Gill has failed to appreciate the difference between the actual dynamical variables in the proposed experiment, namely the bivectors ${{\bf L}({\bf a},\,\lambda^k)}$ and ${{\bf L}({\bf b},\,\lambda^k)}$, and the subsequently calculated traditional variables ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$, he has ended up misapplying his calculation, which leads him to a result not relevant to the problem at hand. In order to correctly calculate the correlation between the variables ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ one must first specify their relation to the actual dynamical variables ${{\bf L}({\bf a},\,\lambda^k)}$ and ${{\bf L}({\bf b},\,\lambda^k)}$ to be observed in the experiment: $$\begin{aligned}
S^3\ni\pm1\,=\,{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})\,\equiv\,
{\mathscr A}({\bf a},\,{\lambda^k})\,&=\,\lim_{{\bf s}\,\rightarrow\,{\bf a}}\left\{\,-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\right\}
=\,
\begin{cases}
+\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,+\,1 \\
-\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,-\,1
\end{cases} \label{88-oi}\end{aligned}$$ and $$\begin{aligned}
S^3\ni\mp1\,=\,{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})\,\equiv\,
{\mathscr B}({\bf b},\,{\lambda^k})\,&=\,\lim_{{\bf s}\,\rightarrow\,{\bf b}}\left\{\,+\,{\bf D}({\bf b})\,{\bf L}({\bf s},\,\lambda^k)\right\}
=\,
\begin{cases}
-\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,+\,1 \\
+\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,-\,1\,,
\end{cases} \label{99-oi}\end{aligned}$$ where the orientation ${\lambda}$ of ${S^3}$ is assumed to be a random variable with 50/50 chance of being ${+1}$ or ${-\,1}$ at the moment of the bomb-explosion, making the spinning bivector ${{\bf L}({\bf a},\,\lambda)}$ a random variable [*relative*]{} to the detector bivector ${{\bf D}({\bf a})}$: $${\bf L}({\bf a},\,\lambda)
\,\equiv\,\{\,a_{\mu}\;L_{\mu}(\lambda)\,\}\,=\,\lambda\,\{\,a_{\nu}\;D_{\nu}\,\}\,\equiv\,\lambda\,{\bf D}({\bf a}). \label{OJS}$$
From the above discussion it should be clear that the raw scores ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ would be generated in the experiment with [*different*]{} bivectorial scales of dispersion, or [*different*]{} standard deviations, as explained between Eqs.${\,}$(102) and (109) in Ref.${\,}$[@IJTP]. Therefore the calculation of the correlation between these raw scores, as well as the derivation of the Tsirel’son’s bound on the strength of possible correlations, must be carried out with some care. In fact there are at least four physical considerations one must be mindful of before proceeding further: (1) the spin angular momenta are represented, not by polar vectors, but by bivectors that change sign upon reflection; (2) scalars and bivectors, despite being elements of different grades, are treated on equal footing in geometric algebra; (3) what would be actually observed in the proposed experiment are not the raw scores ${{\mathscr A}({\bf a},\,{\lambda}^k)\equiv{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{\mathscr B}({\bf b},\,{\lambda}^k)\equiv{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ but the standard scores ${A({\bf a},\,\lambda^k)\equiv{\bf L}({\bf a},\,\lambda^k)}$ and ${B({\bf b},\,\lambda^k)\equiv{\bf L}({\bf b},\,\lambda^k)}$; and (4) the correct association between the raw scores ${{\mathscr A}({\bf a},\,{\lambda}^k)}$ and ${{\mathscr B}({\bf b},\,{\lambda}^k)}$ can be inferred only by calculating the covariance between the corresponding standard scores ${A({\bf a},\,\lambda^k)}$ and ${B({\bf b},\,\lambda^k)}$. With these in mind, let us consider four reference vectors ${\bf a}$,${\bf a'}$, ${\bf b}$, and ${\bf b'}$. Then the bound on the corresponding CHSH string of expectation values [@Christian], namely, on the coefficient $${\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'})\,+\,
{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})\,, \label{B1-11}$$ can be derived using the four joint expectation values of the raw scores — such as ${{\mathscr A}({\bf a},\,{\lambda})}$ and ${{\mathscr B}({\bf b},\,{\lambda})}$ — defined as $${\cal E}({\bf a},\,{\bf b})\,=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\mathscr A}({\bf a},\,{\lambda}^k)\;{\mathscr B}({\bf b},\,{\lambda}^k)\right]\,\equiv\,\Bigl\langle\,{\mathscr A}_{\bf a}({\lambda})\,{\mathscr B}_{\bf b}({\lambda})\,\Bigr\rangle\,.\label{exppeu}$$ This allows us to express the above CHSH string of expectation values simply as a string of four averages as follows: $$\Bigl\langle\,{\mathscr A}_{\bf a}({\lambda})\,{\mathscr B}_{\bf b}({\lambda})\,\Bigr\rangle\,+\, \Bigl\langle\,{\mathscr A}_{\bf a}({\lambda})\,{\mathscr B}_{\bf b'}({\lambda})\,\Bigr\rangle\,+\, \Bigl\langle\,{\mathscr A}_{\bf a'}({\lambda})\,{\mathscr B}_{\bf b}({\lambda})\,\Bigr\rangle\,-\, \Bigl\langle\,{\mathscr A}_{\bf a'}({\lambda})\,{\mathscr B}_{\bf b'}({\lambda})\,\Bigr\rangle\,.\label{four}$$ It is in the next step that Gill makes his gravest mistake. He surreptitiously replaces the above string of four separate averages of numbers that are generated with [*different*]{} bivectorial scales of dispersion with the following single average: $$\Bigl\langle\,{\mathscr A}_{\bf a}({\lambda})\,{\mathscr B}_{\bf b}({\lambda})\,+\,
{\mathscr A}_{\bf a}({\lambda})\,{\mathscr B}_{\bf b'}({\lambda})\,+\, {\mathscr A}_{\bf a'}({\lambda})\,{\mathscr B}_{\bf b}({\lambda})\,-\,
{\mathscr A}_{\bf a'}({\lambda})\,{\mathscr B}_{\bf b'}({\lambda})\,\Bigr\rangle\,.\label{one}$$ As innocuous as this step may seem, it is in fact an illegitimate mathematical step within the context of the experiment proposed in Ref.${\,}$[@IJTP]. But this illegitimate maneuver does allow Gill to reduce the above average at once to the average $$\Bigl\langle\,{\mathscr A}_{\bf a}({\lambda})\,\big\{{\,\mathscr B}_{\bf b}({\lambda})+{\mathscr B}_{\bf b'}({\lambda})\,\big\}\,+\,{\mathscr A}_{\bf a'}({\lambda})\,\big\{\,{\mathscr B}_{\bf b}({\lambda})-{\mathscr B}_{\bf b'}({\lambda})\,\big\}\,\Bigr\rangle\,.$$ And since ${{\mathscr B}_{\bf b}({\lambda})=\pm1}$, if ${|{\mathscr B}_{\bf b}({\lambda})+{\mathscr B}_{\bf b'}({\lambda})|=2}$, then ${|{\mathscr B}_{\bf b}({\lambda})-{\mathscr B}_{\bf b'}({\lambda})|=0}$, and vice versa. Consequently, using ${{\mathscr A}_{\bf a}({\lambda})=\pm1}$, it is easy to conclude that the absolute value of the above average cannot exceed 2, as Gill has concluded.
As compelling as this conclusion by Gill may seem at first sight, it is entirely false. It is based on his illegitimate and careless maneuver of replacing the string of four separate averages of random variables (\[four\]) with a single average (\[one\]). Such a move is justified on mathematical grounds [*only*]{} if all four random variables ${{\mathscr A}_{\bf a}({\lambda})}$, ${{\mathscr B}_{\bf b}({\lambda})}$, ${{\mathscr A}_{\bf a'}({\lambda})}$, and ${{\mathscr B}_{\bf b'}({\lambda})}$ are on equal statistical and geometrical footings. But as we noted earlier, each one of these variables is generated with a [*different*]{} bivectorial scale of dispersion, or a [*different*]{} standard deviation, as explained between equations (102) and (109) in Ref.${\,}$[@IJTP]. Therefore the [*theoretical*]{} prediction of the correlation (\[exppeu\]) between these raw scores, as well as the [*theoretical*]{} derivation of the Tsirel’son’s bound on the strength of possible correlations, [*must*]{} proceed as follows.
Since the correct association between the raw scores ${{\mathscr A}({\bf a},\,{\lambda})}$ and ${{\mathscr B}({\bf b},\,{\lambda})}$ can be inferred only by calculating the covariance between the corresponding standardized variables ${A({\bf a},\,\lambda)}$ and ${B({\bf b},\,\lambda)}$ as calculated in Ref.${\,}$[@IJTP], namely $$\begin{aligned}
A_{\bf a}({\lambda})\,\equiv\,A({\bf a},\,\lambda)\,&\equiv\,{\bf L}({\bf a},\,\lambda) \label{dumtit-1} \\
\text{and}\;\;\;B_{\bf b}({\lambda})\,\equiv\,B({\bf b},\,\lambda)\,&\equiv\,{\bf L}({\bf b},\,\lambda)\,, \label{dumtit-2}\end{aligned}$$ the correlation between the raw scores ${{\mathscr A}({\bf a},\,{\lambda})}$ and ${{\mathscr B}({\bf b},\,{\lambda})}$ must be obtained by evaluating their product moment $${\cal E}({\bf a},\,{\bf b})\,=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{A}({\bf a},\,{\lambda}^k)\;{B}({\bf b},\,{\lambda}^k)\right].\label{stand-exppeu}$$ The numerical value of this coefficient is then necessarily equal to the value of the correlation calculated by Eq.${\,}$(\[exppeu\]).
Using the above expression for ${{\cal E}({\bf a},\,{\bf b})}$ the string of expectation values (\[B1-11\]) can now be rewritten as a single average in terms of the standard scores ${A_{\bf a}(\lambda)}$ and ${B_{\bf b}(\lambda)}$, because now they are on equal statistical and geometrical footings: $$\lim_{\,n\,\gg\,1}\Bigg[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\big\{
A_{\bf a}({\lambda}^k)\,B_{\bf b}({\lambda}^k)\,+\,
A_{\bf a}({\lambda}^k)\,B_{\bf b'}({\lambda}^k)\,+\, A_{\bf a'}({\lambda}^k)\,B_{\bf b}({\lambda}^k)\,-\,
A_{\bf a'}({\lambda}^k)\,B_{\bf b'}({\lambda}^k)\big\}\Bigg]. \label{probnonint}$$ But since ${A_{\bf a}({\lambda})\equiv{\bf L}({\bf a},\,\lambda)}$ and ${B_{\bf b}({\lambda})\equiv{\bf L}({\bf b},\,\lambda)}$ are two independent equatorial points of ${S^3}$, we can take them to belong to two disconnected “sections” of ${S^3}$ \[[*i.e.*]{}, two disconnected su(2) 2-spheres within ${S^3\sim {\rm SU}(2)}$\], satisfying $$\left[\,A_{\bf n}({\lambda}),\,B_{\bf n'}({\lambda})\,\right]\,=\,0\,
\;\;\;\forall\;\,{\bf n}\;\,{\rm and}\;\,{\bf n'}\,\in\,{\rm I\!R}^3,\label{com}$$ which is equivalent to anticipating a null outcome along the direction ${{\bf n}\times{\bf n'}}$ exclusive to both ${\bf n}$ and ${\bf n'}$. If we now square the integrand of equation (\[probnonint\]), use the above commutation relations, and use the fact that all unit bivectors square to ${-1}$, then the absolute value of the Bell-CHSH string (\[B1-11\]) leads to the following variance inequality [@Christian]: $$|{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'})\,+\,
{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})|\,\leqslant\sqrt{\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
\big\{\,4\,+\,4\,{\mathscr T}_{\,{\bf a\,a'}}({\lambda}^k)\,{\mathscr T}_{\,{\bf b'\,b}}({\lambda}^k)\,\big\}\right]},\label{yever}$$ where the classical commutators $${\mathscr T}_{\,{\bf a\,a'}}(\lambda):=\frac{1}{2}\left[\,A_{\bf a}(\lambda),\,A_{\bf a'}(\lambda)\right]
\,=\,-\,A_{{\bf a}\times{\bf a'}}(\lambda) \label{aa-potorsion-666}$$ and $${\mathscr T}_{\,{\bf b'\,b}}(\lambda)
:=\frac{1}{2}\left[\,B_{\bf b'}(\lambda),\,B_{\bf b}(\lambda)\right]\,=\,-\,B_{{\bf b'}\times{\bf b}}(\lambda)\label{bb-potor}$$ are the geometric measures of the torsion within ${S^3}$ [@Christian]. Thus, as discussed in the paper, it is the non-vanishing torsion ${\mathscr T}$ within the 3-sphere—the parallelizing torsion which makes its Riemann curvature vanish—that is responsible for the stronger-than-linear correlation. We can see this from Eq.${\,}$(\[yever\]) by setting ${{\mathscr T}=0}$, and in more detail as follows.
Using definitions (\[dumtit-1\]) and (\[dumtit-2\]) for ${A_{\bf a}({\lambda})}$ and ${B_{\bf b}({\lambda})}$ and making a repeated use of the well known bivector identity $${\bf L}({\bf a},\,\lambda)\,{\bf L}({\bf a'},\,\lambda)\,=\,-\,{\bf a}\cdot{\bf a'}\,-\,
{\bf L}({\bf a}\times{\bf a'},\,\lambda)\,,\label{bititi}$$ the above inequality can be further simplified to $$\begin{aligned}
|{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'})\,+\,
{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})|\,&\leqslant\sqrt{\!4-4\,({{\bf a}}\times{{\bf a}'})\cdot({{\bf b}'}\times{{\bf b}})-
4\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}{{\bf L}}({\bf z},\,\lambda^k)\right]} \notag \\
&\leqslant\sqrt{\!4-4\,({{\bf a}}\times{\bf a'})\cdot({\bf b'}\times{{\bf b}})-
4\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\lambda^k\right]{{\bf D}}({\bf z})} \notag \\
&\leqslant\,2\,\sqrt{\,1-({{\bf a}}\times{\bf a'})
\cdot({\bf b'}\times{{\bf b}})\,-\,0\,}\,,\label{before-opppo-666}\end{aligned}$$ where ${{\bf z}=({\bf a}\times{\bf a'})\times({\bf b'}\times{\bf b})}$. The last two steps follow from the relation (\[OJS\]) between ${{\bf L}({\bf z},\,\lambda)}$ and ${{\bf D}({\bf z})}$ and the fact that the orientation ${\lambda}$ of ${S^3}$ is evenly distributed between ${+1}$ and ${-1}$. Finally, by noticing that trigonometry dictates $$-1\leqslant\,({\bf a}\times{\bf a'})\cdot({\bf b'}\times{\bf b})\,\leqslant +1\,,$$ the above inequality can be reduced to the form $$\left|\,{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'})\,+\,
{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})\,\right|\,\leqslant\,2\sqrt{2}\,,
\label{My-CHSH}$$ exhibiting the correct upper bound on the strength of possible correlations. Thus the stronger bounds of ${-2}$ and ${+2}$ calculated naïvely by Gill in his paper are simply incorrect. More importantly, as correctly done in the paper [@IJTP], the correlation function for the bomb fragments respecting the SU(2) Lie algebra su(2) can indeed be calculated to yield $${\cal E}({\bf a},\,{\bf b})=\!\!\lim_{\,n\,\gg\,1}\!\left[\frac{1}{n}\!\sum_{k\,=\,1}^{n}
\{{sign}\,(+{\bf s}^k\cdot{\bf a})\}\,
\{{sign}\,(-{\bf s}^k\cdot{\bf b})\}\right]\!= -\,{\bf a}\cdot{\bf b}\,,\label{correlations}$$ where ${n}$ is the total number of trials performed. Therefore the remaining comments by Gill are anything but justified.
In fact there is considerable confusion in Gill’s attempted misrepresentation of the proposed experiment [@Gillprint]. Rather surprisingly, his comments fail to distinguish between the [*theoretical*]{} derivation of the correlation presented in Eq.${\,}$(110) of Ref.${\,}$[@IJTP] and the [*practical*]{} calculation of the correlation expressed in the above equation in terms of the raw scores. For example, he asserts that “correlations should only be computed the usual way using actual experimental outcomes..." This reveals that Gill has either not bothered to read the experimental proposal discussed in paper [@IJTP], or has failed to understand the crucial difference between the traditional Bell-type experiments and the one proposed in the paper.
The important question here is: What would be the “actual experimental outcomes" in the proposed experiment? From the above discussion and the discussion in the Section 4 of Ref.${\,}$[@IJTP] it is quite clear that what would be actually observed in the experiment are the bivectors ${{\bf L}({\bf a},\,\lambda)}$ and ${{\bf L}({\bf b},\,\lambda)}$ about the directions ${\bf a}$ and ${\bf b}$, respectively. The raw scores ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ may then be calculated, but by an algorithm [*extraneous*]{} to the actual experiment, and [*long after*]{} the experiment has actually been completed. Thus they would not be an integral dynamical part of the physical experiment itself. On the other hand, thanks to the precise geometrical relation between the raw scores ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ and the standard scores ${{\bf L}({\bf a},\,\lambda)}$ and ${{\bf L}({\bf b},\,\lambda)}$ given by the equations (\[88-oi\]) and (\[99-oi\]), the statistical association between the raw scores can nevertheless be inferred by calculating the [*covariance*]{} of the corresponding standardized variables ${A_{\bf a}(\lambda)}$ and ${B_{\bf b}(\lambda)}$ as demonstrated above, thus predicting the correlation (\[correlations\]).
Finally, Gill falsely and disingenuously claims that Section 5 of Ref.${\,}$[@IJTP] reproduces a sign error which is also present in my earlier work [@Christian]. There is in fact no such error, as explained already in Ref.${\,}$[@refute] and in several chapters of Ref.${\,}$[@Christian].
In conclusion, the criticism of the proposed experiment by Gill is a travesty. No physicist should be deceived by it.
Refutations of Gill’s Mistaken Claims by Explicit Numerical Simulations
=======================================================================
The above refutations of Gill’s fallacious claims have been independently verified by Albert Jan Wonnink [@Wonnink] in an explicit numerical simulation of Eq.${\,}$(\[stand-exppeu\]), by means of a specialized program for geometric algebra based computations. In other words, Gill’s mistaken claims have been refuted by Wonnink by numerically computing the expectation value $${\cal E}({\bf a},\,{\bf b})\,=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\bf L}({\bf a},\,{\lambda}^k)\;{\bf L}({\bf b},\,{\lambda}^k)\right]=-{\bf a}\cdot{\bf b}.\label{stand-exsss}$$ To understand this computation, recall that ${{\bf L}({\bf a},\,{\lambda}^k=+1)=+\,I\cdot{\bf a}}$ and ${{\bf L}({\bf b},\,{\lambda}^k=-1)=-\,I\cdot{\bf b}}$ represent the two spins of the bomb fragments, where ${I:=e_x\wedge e_y\wedge e_z}$ with ${+I}$ representing the right-handed orientation of ${S^3}$ and ${-I}$ representing the left-handed orientation of ${S^3}$. Consequently we may consider the following two geometric products, $$(\,+\,I\cdot{\bf a})(\,+\,I\cdot{\bf b})\,
=\,-\,{\bf a}\cdot{\bf b}\,-\,(\,+\,I\,)\cdot({\bf a}\times{\bf b}) \label{id-1}$$ and $$(\,-\,I\cdot{\bf a})(\,-\,I\cdot{\bf b})\,
=\,-\,{\bf a}\cdot{\bf b}\,-\,(\,-\,I\,)\cdot({\bf a}\times{\bf b}), \label{id-2}$$ as discussed in my earlier reply to Gill (cf. Eqs.${\,}$(15) and (16) of Ref.${}$[@refute]). The two possible orientations of ${S^3}$ may then be thought of as the random hidden variables ${\lambda=\pm\,1}$ (or the initial states ${\lambda=\pm\,1}$) of the two bomb fragments.
Now in traditional geometric algebra as well as in the GAViewer program [@GAV] employed by Albert Jan Wonnink the volume form of the physical space is fixed [*a priori*]{} to be ${+I}$, by convention. This convention is inconsistent with the physical process of spin detections delineated in the Eqs.${\,}$(\[88-oi\]) and (\[99-oi\]) above. Therefore a translation of the geometric product ${{\bf L}({\bf a},\,{\lambda}^k=-1)\;{\bf L}({\bf b},\,{\lambda}^k=-1)=(\,-\,I\cdot{\bf a})(\,-\,I\cdot{\bf b})}$ for the GAViewer built on the right-handed form ${+I}$ is necessary, which can be inferred from Eqs.${\,}$(\[id-1\]) and (\[id-2\]) by recalling that ${{\bf b}\times{\bf a}=-\,{\bf a}\times{\bf b}}$ and rewriting Eq.${\,}$(\[id-1\]) as $$(\,+\,I\cdot{\bf b})(\,+\,I\cdot{\bf a})\,
=\,-\,{\bf a}\cdot{\bf b}\,+\,(\,+\,I\,)\cdot({\bf a}\times{\bf b})\,=\,-\,{\bf a}\cdot{\bf b}\,-\,(\,-\,I\,)\cdot({\bf a}\times{\bf b}). \label{id-3}$$ Comparing the right-hand sides of Eqs.${\,}$(\[id-2\]) and (\[id-3\]) it is now quite easy to recognize that the desired translation is $$(\,-\,I\cdot{\bf a})(\,-\,I\cdot{\bf b})\,\longrightarrow\,(\,+\,I\cdot{\bf b})(\,+\,I\cdot{\bf a}). \label{traa}$$ In terms of this translation the geometric products appearing in the expectation function (\[stand-exsss\]) can be expressed as $$\begin{aligned}
{\bf L}({\bf a},\,{\lambda}^k=+1)\;{\bf L}({\bf b},\,{\lambda}^k=+1)\,&=\,(\,+\,I\cdot{\bf a})(\,+\,I\cdot{\bf b}) \\
\text{and}\;\;\;{\bf L}({\bf a},\,{\lambda}^k=-1)\;{\bf L}({\bf b},\,{\lambda}^k=-1)\,&=\,(\,+\,I\cdot{\bf b})(\,+\,I\cdot{\bf a}).\end{aligned}$$ In other words, when ${\lambda^k}$ happens to be equal to ${+1}$, ${{\bf L}({\bf a},\,{\lambda}^k)\;{\bf L}({\bf b},\,{\lambda}^k)=(\,+\,I\cdot{\bf a})(\,+\,I\cdot{\bf b})}$, and when ${\lambda^k}$ happens to be equal to ${-1}$, ${{\bf L}({\bf a},\,{\lambda}^k)\;{\bf L}({\bf b},\,{\lambda}^k)=(\,+\,I\cdot{\bf b})(\,+\,I\cdot{\bf a})}$. Consequently, the expectation value (\[stand-exsss\]) reduces at once to $${\cal E}({\bf a},\,{\bf b})\,=\,\frac{1}{2}(\,+\,I\cdot{\bf a})(\,+\,I\cdot{\bf b})\,+\,\frac{1}{2}(\,+\,I\cdot{\bf b})(\,+\,I\cdot{\bf a})\,
=\,-\,{\bf a}\cdot{\bf b}\,+\,0\,,\label{stand-nossss}$$ because the orientation ${\lambda}$ of ${S^3}$ is necessarily a fair coin. Here the last equality follows from the Eqs.${\,}$(\[id-1\]) and (\[id-3\]).
Given the translation (\[traa\]), it is now easy to understand the simulation code of Ref.${\,}$[@Wonnink], with its essential lines being $$\begin{aligned}
\text{if}\;\;\lambda\,&=\,+1,\;\;\text{then add }\;\;(\,+\,I\cdot{\bf a})(\,+\,I\cdot{\bf b}), \\
\text{but if}\;\;\lambda\,&=\,-1,\;\;\text{then add }\;\;(\,+\,I\cdot{\bf b})(\,+\,I\cdot{\bf a}).\end{aligned}$$ Not surprisingly, the expectation value computed in this event-by-event simulation prints out to be ${{\cal E}({\bf a},\,{\bf b})=-\,{\bf a}\cdot{\bf b}}$.
In complement to the above unambiguous demonstration, it is also possible to verify the correlation (\[stand-nossss\]) numerically in an event-by-event simulation within a [*non*]{}-Clifford algebraic representation of ${S^3}$, as done, for example, in Ref.${\,}$[@rpub].
Thus, once again, the criticism of the proposed experiment by Gill is exposed to be vacuous by explicit computations.
Correlations Among the Raw Scores is Equal to the Covariance Among the Standard Scores
======================================================================================
For the completeness of the arguments presented above, in this appendix let us prove the following equality explicitly: $${\cal E}({\bf a},\,{\bf b})\,=\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\!\sum_{k\,=\,1}^{n}
\{{sign}\,(+{\bf s}^k\cdot{\bf a})\}\,
\{{sign}\,(-{\bf s}^k\cdot{\bf b})\}\right]=\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\bf L}({\bf a},\,{\lambda}^k)\;{\bf L}({\bf b},\,{\lambda}^k)\right]=-{\bf a}\cdot{\bf b}.\label{nexs-bss}$$ We begin with the central equalities in the prescription of the remote measurement events given in Eqs.${\,}$(\[88-oi\]) and (\[99-oi\]): $$\begin{aligned}
{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})\,=\,{\mathscr A}({\bf a},\,{\lambda^k})\,&=\,\lim_{{\bf s}\,\rightarrow\,{\bf a}}\left\{\,-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\,\right\} \label{ppp-oi}\\
\text{and}\;\;{sign}\,(\,-\,{\bf s}^k\cdot{\bf b})\,=\,{\mathscr B}({\bf b},\,{\lambda^k})\,&=\,\lim_{{\bf s}\,\rightarrow\,{\bf b}}\left\{\,+\,{\bf D}({\bf b})\,{\bf L}({\bf s},\,\lambda^k)\,\right\}, \label{000-oi}\end{aligned}$$ where the [*same*]{} scalar number ${{\mathscr A}({\bf a},\,{\lambda^k})=\pm1}$ is expressed in two different ways — as a grade-0 number ${{sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ on the left hand, and as a product ${-\,{\bf D}({\bf a})\,{\bf L}({\bf a},\,\lambda^k)}$ of two grade-2 numbers, ${-\,{\bf D}({\bf a})}$ and ${{\bf L}({\bf a},\,\lambda^k)}$, on the right hand.
Now the correlation between ${{\mathscr A}({\bf a},\,{\lambda^k})}$ and ${{\mathscr B}({\bf b},\,{\lambda^k})}$ can be quantified by the product-moment correlation coefficient $${\cal E}({\bf a},\,{\bf b})\,=\,\frac{{\displaystyle\lim_{\,n\,\gg\,1}}\left[{\displaystyle\frac{1}{n}\sum_{k\,=\,1}^{n}}\,
\left\{{\mathscr A}({\bf a},\,{\lambda}^k)-\overline{{\mathscr A}({\bf a},\,{\lambda})}\right\}\;\left\{{\mathscr B}({\bf b},\,{\lambda}^k)-\overline{{\mathscr B}({\bf b},\,{\lambda})}\right\}\right]}{\sigma({\mathscr A})\;\sigma({\mathscr B})}\,,\label{exs-bss}$$ where ${\overline{{\mathscr A}({\bf a},\,{\lambda})}}$ and ${\overline{{\mathscr B}({\bf b},\,{\lambda})}}$ are the average values of ${\mathscr A}$ and ${\mathscr B}$ and ${\sigma({\mathscr A})}$ and ${\sigma({\mathscr B})}$ are their standard deviations: $$\sigma({\mathscr A})\,=\,\sqrt{\frac{1}{n}\sum_{k\,=\,1}^{n}\,\left|\left|\,{\mathscr A}({\bf a},\,{\lambda}^k)\,-\,
{\overline{{\mathscr A}({\bf a},\,{\lambda})}}\;\right|\right|^2\,}\, \;\;\;\;\;\text{and}\;\;\;\;\;
\sigma({\mathscr B})\,=\,\sqrt{\frac{1}{n}\sum_{k\,=\,1}^{n}\,\left|\left|\,{\mathscr B}({\bf b},\,{\lambda}^k)\,-\,{\overline{{\mathscr B}({\bf b},\,{\lambda})}}\;\right|\right|^2\,}.
\label{defstan}$$ Accordingly, let us first consider ${{\mathscr A}({\bf a},\,{\lambda^k})={sign}\,(\,+\,{\bf s}^k\cdot{\bf a})}$ and ${{\mathscr B}({\bf b},\,{\lambda^k})={sign}\,(\,-\,{\bf s}^k\cdot{\bf b})}$ from the left sides of the equalities (\[ppp-oi\]) and (\[000-oi\]). Written in this form, these variables cannot be factorized into products of a random number and a non-random number. We are therefore forced to treat them as irreducible random variables. Moreover, since ${\lambda}$ is a fair coin, ${\bf s}$ has equal chance of being parallel and anti-parallel to ${\bf a}$. Consequently, it is easy to work out from the above formulae that in the present case ${\overline{{\mathscr A}({\bf a},\,{\lambda})}=\overline{{\mathscr B}({\bf b},\,{\lambda})}=0\,}$ and ${\,\sigma({\mathscr A})=\sigma({\mathscr B})=1}$, which immediately gives us $${\cal E}({\bf a},\,{\bf b})\,=\!\lim_{\,n\,\gg\,1}\!\left[\frac{1}{n}\!\sum_{k\,=\,1}^{n}
\{{sign}\,(+{\bf s}^k\cdot{\bf a})\}\,
\{{sign}\,(-{\bf s}^k\cdot{\bf b})\}\right]\!.\label{pers-bss}$$ Thus, we recognize that the above expression of the observed correlations — which is traditionally employed by the experimentalists — is simply a special case of the Pearson’s product-moment correlation coefficient defined in Eq.${\,}$(\[exs-bss\]).
Next, consider ${{\mathscr A}({\bf a},\,{\lambda^k})=-\,{\bf D}({\bf a})\,{\bf L}({\bf a},\,\lambda^k)}$ and ${{\mathscr B}({\bf b},\,{\lambda^k})=+\,{\bf D}({\bf b})\,{\bf L}({\bf b},\,\lambda^k)}$ from the right sides of the equalities (\[ppp-oi\]) and (\[000-oi\]). Written in this form, these very same variables are now factorized into geometric products of a random number and a non-random number. Evidently, the variable ${{\mathscr A}({\bf a},\,{\lambda}^k)}$ is a product of a random spin bivector ${{\bf L}({\bf a},\,\lambda^k)}$ (which is a function of the random variable ${\lambda}$) and a non-random detector bivector ${-\,{\bf D}({\bf a})}$ (which is independent of the random variable ${\lambda}$). In other words, the randomness within ${{\mathscr A}({\bf a},\,{\lambda^k})}$ originates entirely from the randomness of the spin bivector ${{\bf L}({\bf a},\,\lambda^k)}$, with the detector bivector ${-\,{\bf D}({\bf a})}$ merely specifying the scale of dispersion within ${{\bf L}({\bf a},\,\lambda^k)}$. Consequently, as random variables, ${{\mathscr A}({\bf a},\,{\lambda}^k)}$ and ${{\mathscr B}({\bf b},\,{\lambda}^k)}$ are generated with [*different*]{} standard deviations, or [*different*]{} sizes of the typical error. ${{\mathscr A}({\bf a},\,{\lambda}^k)}$ is generated with a typical error ${-\,{\bf D}({\bf a})}$, whereas ${{\mathscr B}({\bf b},\,{\lambda}^k)}$ is generated with a typical error ${+\,{\bf D}({\bf b})}$. And since errors in linear relations propagate linearly, the standard deviation ${\sigma({\mathscr A}\,)}$ of ${{\mathscr A}({\bf a},\,{\lambda}^k)}$ is equal to ${-\,{\bf D}({\bf a})}$ times the standard deviation of ${{\bf L}({\bf a},\,\lambda^k)}$, and the standard deviation ${\sigma({\mathscr B}\,)}$ of ${{\mathscr B}({\bf b},\,{\lambda}^k)}$ is equal to ${+\,{\bf D}({\bf b})}$ times the standard deviation of ${{\bf L}({\bf b},\,\lambda^k)}$, which can be worked out using formulae similar to (\[defstan\]): $$\begin{aligned}
\sigma({\mathscr A}\,)\,=\,-\,{\bf D}({\bf a})\;\sigma\!\left\{{\bf L}({\bf a},\,\lambda^k)\right\}\,&=\,-\,{\bf D}({\bf a})\,\sqrt{\frac{1}{n}\sum_{k\,=\,1}^{n}\,\left|\left|\,{\bf L}({\bf a},\,{\lambda}^k)\,-\,
{\overline{{\bf L}({\bf a},\,{\lambda})}}\;\right|\right|^2\,}\,=\,-\,{\bf D}({\bf a})\\
\text{and}\;\,\;\sigma({\mathscr B}\,)\,=\,+\,{\bf D}({\bf b})\;\sigma\!\left\{{\bf L}({\bf b},\,\lambda^k)\right\}\,&=\,+\,{\bf D}({\bf b})\,\sqrt{\frac{1}{n}\sum_{k\,=\,1}^{n}\,\left|\left|\,{\bf L}({\bf b},\,{\lambda}^k)\,-\,
{\overline{{\bf L}({\bf b},\,{\lambda})}}\;\right|\right|^2\,}\,=\,+\,{\bf D}({\bf b}).\;\,\text{}\end{aligned}$$ Here I have used ${\left|\left|\,{\bf L}({\bf a},\,\lambda^k)\,\right|\right|=\left|\left|\,{\bf L}({\bf b},\,\lambda^k)\,\right|\right|=1}$ because all ${{\bf L}({\bf s},\,\lambda^k)}$ are [*unit*]{} bivectors, and ${{\overline{{\bf L}({\bf a},\,{\lambda})}}={\overline{{\bf L}({\bf b},\,{\lambda})}}=0}$ on the account of ${\lambda}$ being a fair coin. These standard deviations are derived much more rigorously in Refs.${\,}$[@IJTP] and [@Christian].
If we now substitute ${{\mathscr A}({\bf a},\,{\lambda^k})=-\,{\bf D}({\bf a})\,{\bf L}({\bf a},\,\lambda^k)}$ and ${{\mathscr B}({\bf b},\,{\lambda^k})=+\,{\bf D}({\bf b})\,{\bf L}({\bf b},\,\lambda^k)}$ into the same formula (\[exs-bss\]) used previously for the product-moment correlation coefficient [@refute], together with ${\overline{{\mathscr A}({\bf a},\,{\lambda})}=\overline{{\mathscr B}({\bf b},\,{\lambda})}=0}$, ${\sigma({\mathscr A}\,)=-\,{\bf D}({\bf a})}$, ${\sigma({\mathscr B}\,)=+\,{\bf D}({\bf b})}$, ${\{{\mathscr A}({\bf a},\,{\lambda^k})/{\sigma({\mathscr A}\,)}\}={\bf L}({\bf a},\,\lambda^k)}$, and ${\{{\mathscr B}({\bf b},\,{\lambda^k})/{\sigma({\mathscr B}\,)}\}={\bf L}({\bf b},\,\lambda^k)}$, then it immediately reduces to $${\cal E}({\bf a},\,{\bf b})\,=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\bf L}({\bf a},\,{\lambda}^k)\;{\bf L}({\bf b},\,{\lambda}^k)\right]\!.\label{tands-exsss}$$ Consequently, putting the results (\[pers-bss\]), (\[tands-exsss\]), and (\[stand-nossss\]) together, we finally arrive at the equality we set out to prove: $${\cal E}({\bf a},\,{\bf b})\,=\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\!\sum_{k\,=\,1}^{n}
\{{sign}\,(+{\bf s}^k\cdot{\bf a})\}\,
\{{sign}\,(-{\bf s}^k\cdot{\bf b})\}\right]=\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\bf L}({\bf a},\,{\lambda}^k)\;{\bf L}({\bf b},\,{\lambda}^k)\right]=-{\bf a}\cdot{\bf b}.\label{meexs-bss}$$
A Simplified Local-Realistic Derivation of the EPR-Bohm Correlation [[@disproof]]{}
===================================================================================
As in Eqs.${\,}$(\[88-oi\]) and (\[99-oi\]), let the spin bivectors ${\mp\,{\bf L}({\bf s},\,\lambda^k)}$ be detected by the detector bivectors ${{\bf D}({\bf a})}$ and ${{\bf D}({\bf b})}$, giving $$\begin{aligned}
S^3\ni\,{\mathscr A}({\bf a},\,{\lambda^k})\,:=\,\lim_{{\bf s}\,\rightarrow\,{\bf a}}\left\{-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\right\}&=\,
\begin{cases}
+\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,+\,1 \\
-\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,-\,1
\end{cases} \Bigg\}\,\;\text{with}\;\, \Bigl\langle\,{\mathscr A}({\bf a},\,\lambda^k)\,\Bigr\rangle\,=\,0\\
\text{and}\;\;\;\;S^3\ni\,{\mathscr B}({\bf b},\,{\lambda^k})\,:=\,\lim_{{\bf s}\,\rightarrow\,{\bf b}}\left\{+\,{\bf L}({\bf s},\,\lambda^k)\,{\bf D}({\bf b})\right\}&=\,
\begin{cases}
-\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,+\,1 \\
+\,1\;\;\;\;\;{\rm if} &\lambda^k\,=\,-\,1
\end{cases} \Bigg\}\,\;\text{with}\;\,\Bigl\langle\,{\mathscr B}({\bf b},\,\lambda^k)\,\Bigr\rangle\,=\,0\,,
\label{aaa99-oi}\end{aligned}$$ where the orientation ${\lambda}$ of ${S^3}$ is assumed to be a random variable with 50/50 chance of being ${+1}$ or ${-\,1}$ at the moment of the pair-creation, making the spinning bivector ${{\bf L}({\bf n},\,\lambda^k)}$ a random variable [*relative*]{} to the detector bivector ${{\bf D}({\bf n})}$: $${\bf L}({\bf n},\,\lambda^k)\,=\,\lambda^k\,{\bf D}({\bf n})\,\,\Longleftrightarrow\,\,{\bf D}({\bf n})\,=\,\lambda^k\,{\bf L}({\bf n},\,\lambda^k)\,. \label{aaaOJS}$$ The expectation value of simultaneous outcomes ${{\mathscr A}({\bf a},\,{\lambda^k})=\pm1}$ and ${{\mathscr B}({\bf b},\,{\lambda^k})=\pm1}$ in ${S^3}$ then works out as follows: $$\begin{aligned}
{\cal E}({\bf a},\,{\bf b})\,&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\mathscr A}({\bf a},\,{\lambda}^k)\;{\mathscr B}({\bf b},\,{\lambda}^k)\right]\,\text{within}\,\;S^3:=\,\text{the set of all unit (left-handed) quaternions \cite{IJTP}\cite{Christian}} \\
&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\bigg[\lim_{{\bf s}\,\rightarrow\,{\bf a}}\left\{\,-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\right\}\bigg]\left[\lim_{{\bf s}\,\rightarrow\,{\bf b}}\left\{\,+\,{\bf L}({\bf s},\,\lambda^k)\,{\bf D}({\bf b})\right\}\,\right]\right]\;\;\text{(conserving total spin = 0)} \\
&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\lim_{\substack{{\bf s}\,\rightarrow\,{\bf a} \\ {\bf s}\,\rightarrow\,{\bf b}}}\left\{\,-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\,\,{\bf L}({\bf s},\,\lambda^k)\,{\bf D}({\bf b})\,\equiv\,{\bf q}({\bf a},\,{\bf b};\,{\bf s},\,\lambda^k)\right\}\right]\;\;\,\text{(by a property of limits)} \\
&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\lim_{\substack{{\bf s}\,\rightarrow\,{\bf a} \\ {\bf s}\,\rightarrow\,{\bf b}}}\left\{\,-\,\lambda^k\,{\bf L}({\bf a},\,\lambda^k)\,\,{\bf L}({\bf s},\,\lambda^k)\,{\bf L}({\bf s},\,\lambda^k)\,\,\lambda^k\,{\bf L}({\bf b},\,\lambda^k)\right\}\right]\text{(all bivectors in the spin basis)}\\
&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\lim_{\substack{{\bf s}\,\rightarrow\,{\bf a} \\ {\bf s}\,\rightarrow\,{\bf b}}}\left\{\,-\,{\bf L}({\bf a},\,\lambda^k)\,\,{\bf L}({\bf s},\,\lambda^k)\,{\bf L}({\bf s},\,\lambda^k)\,\,{\bf L}({\bf b},\,\lambda^k)\right\}\right]\,\text{(scalars ${\lambda^k}$ commute with bivectors)}\\
&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\bf L}({\bf a},\,\lambda^k)\,{\bf L}({\bf b},\,\lambda^k)\,\right]\,\text{(follows from the conservation of zero spin angular momentum)} \\
&=\,-\,{\bf a}\cdot{\bf b}\,-\!\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\bf L}({\bf a}\times{\bf b},\,\lambda^k)\,\right]\;\;\,\text{(NB: there is no ``third" spin about the direction ${{\bf a}\times{\bf b}}$)} \\
&=\,-\,{\bf a}\cdot{\bf b}\,-\!\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\lambda^k\,\right]{\bf D}({\bf a}\times{\bf b})\;\;\text{(summing over counterfactual detections of ``third" spins)} \\
&=\,-\,{\bf a}\cdot{\bf b}\,+\,0\,\;\text{(because the scalar coefficient of the bivector ${{\bf D}({\bf a}\times{\bf b})}$ vanishes in the ${n\rightarrow\infty}$ limit)}
\label{exppeuuu}\end{aligned}$$ Here the integrand of (C6) is necessarily a unit quaternion ${{\bf q}({\bf a},\,{\bf b};\,{\bf s},\,\lambda^k)}$ since ${S^3}$ remains closed under multiplication; (C7) follows from using (C3); (C8) follows from using ${\lambda^2 = +1}$; (C9) follows from the fact that all unit bivectors suchas ${{\bf L}({\bf s},\,\lambda^k)}$ square to ${-1}$; (C10) follows from the geometric product (\[bititi\]); (C11) follows from using (C3); (C12) follows from the fact that initial orientation ${\lambda}$ of ${S^3}$ is a fair coin; and (C6) follows from (C5) as a special case of the identity $$\bigg[\lim_{{\bf s}\,\rightarrow\,{\bf a'}}\left\{\,-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\right\}\bigg]\left[\lim_{{\bf s}\,\rightarrow\,{\bf b'}}\left\{\,+\,{\bf L}({\bf s},\,\lambda^k)\,{\bf D}({\bf b})\right\}\,\right]
=\lim_{\substack{{\bf s}\,\rightarrow\,{\bf a'} \\ {\bf s}\,\rightarrow\,{\bf b'}}}\bigg\{-\,{\bf D}({\bf a})\,{\bf L}({\bf s},\,\lambda^k)\,{\bf L}({\bf s},\,\lambda^k)\,{\bf D}({\bf b})\,\bigg\},$$ which can be easily verified either by immediate inspection or by recalling the elementary properties of limits. Note thatapart from the assumption (C3) of initial state ${\lambda}$ the only other assumption needed in this derivation is the conservation of zero spin angular momentum. These two assumptions are necessary and sufficient to dictate the singlet correlation: $$\begin{aligned}
{\cal E}({\bf a},\,{\bf b})\,&=\lim_{\,n\,\rightarrow\,\infty}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,
{\mathscr A}({\bf a},\,{\lambda}^k)\;{\mathscr B}({\bf b},\,{\lambda}^k)\right]
=\,-\,{\bf a}\cdot{\bf b}\,.\end{aligned}$$ This demonstrates that EPR-Bohm correlations are correlations among the scalar points of a quaternionic 3-sphere.
I am grateful to Fred Diether, Michel Fodje, and Albert Jan Wonnink for their efforts to simulate the above model.
J. Christian, [*Macroscopic Observability of Spinorial Sign Changes under ${2\pi}$ Rotations*]{}, Int. J. Theor. Phys., DOI 10.1007/s10773-014-2412-2; See also the last two appendices of arXiv:1211.0784.
J. S. Bell, Physics [**1**]{}, 195 (1964).
A. Peres, [*Quantum Theory: Concepts and Methods*]{} (Kluwer, Dordrecht, 1993), p 161.
J. Christian, [*Can Bell’s Prescription for Physical Reality be Considered Complete?*]{}, arXiv:0806.3078.
J. Christian, [*Disproof of Bell’s Theorem: Illuminating the Illusion of Entanglement*]{}, Second Edition (BrownWalker Press, Boca Raton, Florida, 2014). For the latest results see also <http://libertesphilosophica.info/blog/>.
C. Doran and A. Lasenby, [*Geometric Algebra for Physicists*]{} (Cambridge University Press, Cambridge, 2003).
R. D. Gill, [*Macroscopic Unobservability of Spinorial Sign Changes*]{}, arXiv:1412.2677.
J. Christian, [*Refutation of Richard Gill’s Argument Against my Disproof of Bell’s Theorem*]{}, arXiv:1203.2529.
A-J. Wonnink, <http://challengingbell.blogspot.co.uk/2015/03/numerical-validation-of-vanishing-of_30.html>, & C. F. Diether III, <http://challengingbell.blogspot.co.uk/2015/05/further-numerical-validation-of-joy.html>.
L. Dorst, D. Fontijne, and S. Mann, <http://geometricalgebra.org/gaviewer_download.html>.
J. Christian, <http://rpubs.com/jjc/84238> \[See also <http://rpubs.com/jjc/99993> and <http://rpubs.com/jjc/105450>\].
J. Christian, [*Disproof of Bell’s Theorem*]{} (see especially the second version) arXiv:1103.1879v2 \[quant-ph\] 15 October 2015.
|
---
abstract: 'We initiate a mathematical analysis of hidden effects induced by binning spike trains of neurons. Assuming that the original spike train has been generated by a discrete Markov process, we show that binning generates a stochastic process which is *not Markov* any more, but is instead a Variable Length Markov Chain (VLMC) with unbounded memory. We also show that the law of the binned raster is a Gibbs measure in the DLR (Dobrushin-Lanford-Ruelle) sense coined in mathematical statistical mechanics. This allows the derivation of several important consequences on statistical properties of binned spike trains. In particular, we introduce the DLR framework as a natural setting to mathematically formalize anticipation, i.e. to tell “how good” our nervous system is at making predictions. In a probabilistic sense, this corresponds to condition a process by its future and we discuss how binning may affect our conclusions on this ability. We finally comment what could be the consequences of binning in the detection of spurious phase transitions or in the detection of wrong evidences of criticality.'
author:
- 'B. Cessac[^1], A. Le Ny[^2], E. Löcherbach[^3]'
bibliography:
- 'biblio.bib'
title: On the mathematical consequences of binning spike trains
---
[^1]: Biovision team INRIA, Sophia Antipolis, France. INRIA, 2004 Route des Lucioles, 06902 Sophia-Antipolis, France. email: [email protected]
[^2]: Laboratoire LAMA, UMR CNRS 8050, Créteil, France. Université Paris Est Créteil (UPEC), 91 Avenue du Général de Gaulle, 94010 Créteil cedex, France. email: [email protected]
[^3]: Laboratoire AGM, UMR CNRS 8088, Cergy-Pontoise, France. Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. email: [email protected]
|
---
abstract: 'We report muon spin relaxation ($\mu$SR) and magnetization measurements, together with synthesis and characterization, of the Li-intercalated layered superconductors Li$_{x}$HfNCl and Li$_{x}$ZrNCl with/without co-intercalation of THF (tetrahydrofuran) or PC (propylene carbonate). The 3-dimensional (3-d) superfluid density $n_{s}/m^{*}$ (superconducting carrier density / effective mass), as well as the two dimensional superfluid density $n_{s2d}/m^{*}_{ab}$ (2-dimensional (2-d) area density of superconducting carriers / ab-plane effective mass), have been derived from the $\mu$SR results of the magnetic-field penetration depth $\lambda_{ab}$ observed with external magnetic field applied perpendicular to the 2-d honeycomb layer of HfN / ZrN. In a plot of $T_{c}$ versus $n_{s2d}/m^{*}_{ab}$, most of the results lie close to the linear relationship found for underdoped high-$T_{c}$ cuprate (HTSC) and layered organic BEDT superconductors. In Li$_{x}$ZrNCl without THF intercalation, the superfluid density and $T_{c}$ for $x$ = 0.17 and 0.4 do not show much difference, reminiscent of $\mu$SR results for some overdoped HTSC systems. Together with the absence of dependence of $T_{c}$ on average interlayer distance among ZrN / HfN layers, these results suggest that the 2-d superfluid density $n_{s2d}/m^{*}_{ab}$ is a dominant determining factor for $T_{c}$ in the intercalated nitride-chloride systems. We also report $\mu$SR and magnetization results on depinning of flux vortices, and the magnetization results for the upper critical field $H_{c2}$ and the penetration depth $\lambda$. Reasonable agreement was obtained between $\mu$SR and magnetization estimates of $\lambda$. We discuss the two dimensional nature of superconductivity in the nitride-chloride systems based on these results.'
address:
- ' Department of Physics, Columbia University, 538W 120th St., New York, NY 10027 '
- ' Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027 '
- ' Department of Applied Chemistry, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan '
- ' Department of Physics and Astronomy, McMaster Univ., Hamilton, ON L8S 4M1, Canada '
- ' Department of Superconductivity, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan '
author:
- 'T. Ito[@byline], Y. Fudamoto, A. Fukaya, I.M. Gat-Malureanu, M.I. Larkin, P.L. Russo, A. Savici, and Y.J. Uemura[@byline2]'
- 'K. Groves, and R. Breslow'
- 'K. Hotehama and S. Yamanaka'
- 'P. Kyriakou, M. Rovers, and G.M. Luke'
- 'K.M. Kojima'
title: 'Two dimensional nature of superconductivity in intercalated layered systems Li$_{x}$HfNCl and Li$_{x}$ZrNCl: muon spin relaxation and magnetization measurements'
---
\#1[[$\backslash$\#1]{}]{}
INTRODUCTION {#sec:level1}
============
Layered superconductors, such as high-$T_{c}$ cuprates (HTSC) or organic BEDT systems, have been a subject of extensive research effort for decades. These systems show rich novel phenomena, including superconducting fluctuations, pancake vortex, complicated vortex phase diagrams, and interlayer Josephson effects. High-$T_{\rm c}$ cuprate superconductors have been investigated extensively as prototypical layered superconductors. The cuprates have a merit that their carrier concentration can be controlled by chemical substitutions and/or oxygen contents. On the other hand, it has not been easy to control the interlayer distance in cuprates. In general, superconductivity of these layered systems is deeply related to in-plane features as well as interlayer couplings. For overall understanding of superconductivity in these systems, it would be essential to elucidate interplay between in-plane and interplane properties. Despite extensive research effort, however, detailed roles of dimensionality are yet to be clarified in these systems.
Recently, superconductivity was discovered in ZrNCl and HfNCl intercalated with alkali atoms (Li, Na, K) [@discovery_Zr; @discovery_Hf]. Systems based on ZrNCl have superconducting transition temperatures $T_{c} \leq
15$ K, while those based on HfNCl have $T_{c} \leq 25.5$ K. These systems can be co-intercalated with organic molecules, such as THF (tetrahydrofuran) or PC (propyrene carbonate). The parent compounds ZrNCl and HfNCl are insulators which have a layered structure as shown in Fig. 1(a). Zr(Hf)-N honeycomb double layers are sandwiched by Cl layers, and a composite Cl-(ZrN)-(ZrN)-Cl layers form a stacking unit. Adjacent stacking units are bonded by weak van der Waals force. Alkali metal atoms and polar organic molecules such as THF or PC can be co-intercalated into the van der Waals gap of the parent compounds as shown schematically in Fig. 1(b). Intercalated alkali metal atoms are supposed to release electrons into Zr(Hf)-N double layers, which makes the system metallic and superconducting. On the other hand, intercalated organic molecules expand interlayer distance without changing Zr(Hf)-N honeycomb double layers. So we can control two separate parameters, carrier concentration and the stacking unit distance, in a single series of nitride chlorides with the common superconducting slab. This unique feature could allow studies of layered superconductors from a new angle.
In this paper, we will present synthesis and characterization of a series of intercalated ZrNCl and HfNCl samples, together with studies of their superconducting properties using muon spin relaxation ($\mu$SR), magnetization, and resistivity measurements. A part of this work was presented in a conference [@ssc20], where we reported $\mu$SR results in HfNCl-Li$_{0.5}$-THF$_{0.3}$, showed that $T_{c}$ and the 2-d superfluid density in this system follow the correlations found in cuprates and BEDT systems, and discussed that this feature likely comes from departure from BCS condensation, which can be understood in terms of crossover from Bose-Einstein to BCS condensation. Subsequently, Tou [*et al.*]{} [@touprl] reported an NMR Knight shift study which inferred a rather small density of states at the Fermi level in HfNCl-Li-THF, and discussed difficulty in explaining the high transition temperature $T_{c}$ in terms of the conventional BCS theory. Tou [*et al.*]{} [@touprb] also reported a rather high upper critical field $H_{c2}(T\rightarrow 0) \sim 100$ kG, for the field applied perpendicular to the conducting planes, from magnetization and NMR measurements.
Extensive $\mu$SR measurements of the magnetic field penetration depth $\lambda$ have been performed to date in various superconducting systems, such as high-$T_{c}$ cuprates (HTSC) [@ssc6; @ssc7; @ssc8; @ssc12; @ssc13; @ssc14; @ssc15; @ssc25], fullerides [@ssc17; @ssc18], and two-dimensional (2-d) organic BEDT systems [@ssc19]. Universal nearly linear correlations have been found between $T_{c}$ and the muon spin relaxation rate $\sigma(T \to 0) \propto 1/\lambda^{2} \propto n_{s}/m^{*}$ (superconducting carrier density / effective mass) in the underdoped region of many HTSC systems and in some other exotic superconductors [@ssc6; @ssc7] Such correlations are seen also in HTSC superconductors having extra perturbations, such as overdoping [@ssc8; @ssc15; @ssc23], (Cu,Zn) substitutions [@ssc9], or spontaneous formation of nano-scale regions with static stripe spin correlations [@ssc10; @ssc11]. In all these systems, $T_{c}$ follows the correlations with superfluid density found for less perturbed standard HTSC systems. These results indicate that the superfluid density is a determining factor for $T_{c}$ in the cuprates [@sscreview].
In general, a strong dependence of $T_{c}$ on the carrier density is not expected in conventional BCS theory[@ssc45] where $T_{c}$ is determined by the mediating boson (phonon) energy scale and the density of states of carriers at the Fermi level which govern the charge-boson (electron-phonon) coupling. For 2-d metals, the density of states does not depend on the carrier density in the simplest case of non-interacting fermion gas. Therefore, the BCS theory has a fundamental difficulty in explaining the observed correlations. In contrast, an explicit dependence of $T_{c}$ can be expected for the condensation temperature $T_{B}$ in Bose-Einstein (BE) condensation of a simple Bose gas, as well as for the Kosterlitz-Thouless transition temperature $T_{KT}$ for a 2-d superfluid [@ssc40]. In BE and KT transitions, the transition temperature is determined simply by the number density and the mass, since the condensation is decoupled from the formation of condensing bosons in these two cases. The universality of the $T_{c}$ vs $n_{s}/m^{*}$ relationship observed beyond the difference of systems, such as cuprates, fullerides, orangics, etc., may be related to this feature. Pictures proposed to explain the correlations between $T_{c}$ and the superfluid density in the cuprates include crossover from Bose-Einstein to BCS condensation [@ssc47; @ssc48; @ssc42; @ssc49; @ssc50] and phase fluctuations [@ssc51]. In the present work, we extend our study to intercalated nitride-chloride systems, seeking further insights into such phenomenology.
We have also determined the upper critical field $H_{c2}$ of nitride-chloride systems from magnetization and resistivity measurements. Although it is not easy to determine $H_{c2}$ in layered superconductors due to the strong superconducting fluctuations, Hao [*et al.*]{} [@hao] developed an approach to overcome such difficulty using a model for reversible diamagnetic magnetization of type-II superconductors which have high $\kappa$ values. Here, $\kappa$ is the Ginzburg-Landau parameter defined as the ratio of the penetration depth $\lambda$ to the coherence length $\xi$. In this model, one calculates the free energy including the supercurrent kinetic energy, the magnetic-field energy, as well as the kinetic-energy and condensation-energy terms arising from suppression of the order parameter in the vortex core. This method has been successfully applied to various high-$T_{c}$ cuprate superconductors. We will apply this model to superconducting Hf(Zr)NCl in the temperature and field region where the effect of superconducting fluctuations is negligible.
SYNTHESIS AND CHARACTERIZATION {#sec:level2}
==============================
The parent compounds HfNCl and ZrNCl were synthesized by the reaction of Zr or Hf powder with vaporized NH$_{4}$Cl at 600 $^{\circ}$C for 30 minutes in N$_{2}$ flow [@prereaction]. The resulting powder was sealed in a quartz ampule and was purified by a chemical vapor transport method with temperature gradient [@chemical_transport]. For the purification, an end of the ampule with pre-reacted powder was kept at 800 $^{\circ}$C and the other end, where purified powder is collected, at 900 $^{\circ}$C for three days. No impurity phase was detected in the purified powder from x-ray diffraction.
Since a Li-intercalated sample is sensitive to air, intercalation was performed in a glove box. We used three intercalation methods [@intercalation] to prepare a variety of samples: (i) Li-intercalated ZrNCl samples were prepared by soaking parent ZrNCl powder in three kinds of butyllithium solution. We used 2.0 $M$ ($mol/l$) $n$-butyllithium solution in cyclohexane, 1.3 $M$ $sec$-butyllithium solution in cyclohexane, and 1.7$M$ $tert$-butyllithium solution in pentane for this purpose. In this order, reducing ability becomes stronger and therefore higher concentration of Li atoms can be intercalated into the parent compound. We used solution which contained butyllithium corresponding to more than 2, 5, and 5 equivalents of ZrNCl for $n$-, $sec$-, and $tert$-butyllithium, to avoid the dilution of butyllithium in the intercalation process. (ii) We performed co-intercalation of Li and organic molecules into ZrNCl by soaking Li intercalated ZrNCl powder (prepared by the method (i) using $n$-Butyllithium solution) in enough THF or PC. (iii) Li- and THF-co-intercalated HfNCl samples were prepared by soaking HfNCl powder in various concentrations (2.5-100 $mM$) of lithium naphthalene (Li-naph) solution in THF. Since phase separation was often observed in the samples prepared by the method (iii), we selected single phase samples for the measurements in this study via characterization from $T_{c}$ as well as c-axis lattice constant. The Li- and THF-co-intercalated HfNCl sample with the largest c-axis lattice constant according to the method (iii) was prepared at Hiroshima University, while all the other samples at Columbia University. Table 1 shows a list of these samples.
The chemical composition was determined by inductively coupled plasma atomic emission spectroscopy (ICP-AEM) and CHN elemental analysis. The powder samples were pressed into pellets under uniaxial stress and sealed in cells made of Kapton film and epoxy glue for x-ray measurements. The x-ray rocking curve of $(00l)$ peaks for cleaved surface (inside of a pellet) shows a peak with half width at half maximum of $\sim$ 8 degrees, which indicates that the bulk of the samples have well-aligned preferred orientation and are suitable for studies of anisotropic properties. Essentially similar rocking curves were also observed for as-prepared surfaces of pellets.
The superconducting transition temperature of the samples was determined from measurements of magnetic susceptibility $\chi$. In Fig. 2, we show the results of $\chi$ in intercalated ZrNCl specimens which were examined by $\mu$SR measurements. The values of $\chi = M/H$ are of the order for ideal perfect diamagnetism $-3/8\pi$ for spherical samples and $-1/4\pi$ for long cylindrical samples with the field parallel to the cylinder axis. The determination of the absolute values of $\chi$, however, can be affected by such factors as non-spherical sample shape, sample morphology and residual field in a SQUID magnetometer. These factors might have caused deviation of some of the zero-field cooled shielding values of $\chi$ from $-1/4\pi$.
In Table 1, we summarize the composition, synthesis method, distance between adjacent stacking units (1/3 of the c-axis lattice constant $c_{0}$, see Fig. 1), and the superconducting transition temperature $T_{c}$ for these samples. The stacking unit distance is 9.4 Å for the Li-intercalated samples prepared by the method (i) without co-intercalation of organic molecules. This value is almost the same as that for unintercalated parent compound (9.3 Å) without Li. With increasing Li concentration $x$ from 0.17 to 0.6 (in the samples without co-intercalation of organic molecules), $T_{c}$ decreases from 14.2 K to 11.7 K. Figure 3(a) shows the $x$ dependence of $T_{c}$ in Li$_{x}$ZrNCl and Li$_{x}$HfNCl samples, with/without co-intercalation, obtained in the present study. In Fig. 3 and Table 1, we find that: (1) $T_{c}$ shows a slow reduction with increasing $x$; (2) for close $x$ values, $T_{c}$ does not depend much on the stacking-unit distance $c_{o}$/3 (see Fig. 3(b)). These results are qualitatively consistent with the reported results in the Li, K, and Na doped ZrNCl samples with/without co-intercalation of organic molecules [@kawaji]. Decrease of $T_{c}$ with increasing charge doping is reminiscent of the case of overdoped high-$T_{c}$ cuprate superconductors.
To the best of our knowledge, this is the first report of success in Li-THF intercalation into ZrNCl by $sec$- and $tert$-butyllithium. The stacking unit thickness is 13.3 or 18.7 Å for the methods (ii) and (iii). As has been reported, $T_{c}$ is almost unaffected by the expansion of the stacking-unit distance from $\sim$ 9.4 Å to $\sim$ 13.3 Å for the samples with Li content $x \sim 0.17$ (see Fig. 2 (b)). Systems based on HfNCl has $T_{c}$ = 25.5 K, nearly a factor of two higher than that for intercalated ZrNCl.
$\mu$SR: EXPERIMENTAL {#sec:level3}
=====================
Our $\mu$SR experiments were performed at TRIUMF, the Canadian National Accelerator Laboratory located in Vancouver, Canada, which provided a high intensity and polarized beam of positive muons. Each pressed pellet sample, with the c-axis aligned, was sealed in a sample cell which has a Kapton window and mounted in a He gas-flow cryostat with the c-axis parallel to the direction of muon beam. Transverse external field (TF) was applied parallel to the beam direction, while muons are injected with their initial spin polarization perpendicular to the field/beam direction. Low-momuntum (surface) muons with the incident momentum of 29.8 MeV/c were implanted in the pellet specimens. The average stopping depth, 100-200 mg/cm$^{2}$, assured that the majority of muons are stopped within the specimen, after going through Kapton windows of the cryostat and the sample cell.
Plastic sintillation counters were used to detect the arrival of a positive muon and its decay into a positron, and the decay-event histogram was obtained, as a function of muon residence time $t$ which corresponds to the time difference of the muon arrival and positron decay signals. The time evolution of muon spin direction/polarization was obtained from the angular asymmetry of positron histograms, after correction for the exponential decay $\exp(-t/\tau_{\mu})$, where $\tau_{\mu}$ = 2.2 $\mu$s is the mean lifetime of a positive muon. Details of $\mu$SR technique can be found, for example, in refs [@ssc1; @ssc2; @ssc3].
The asymmetry time spectra $A(t)$ were fit to a functional form; $$A(t)=A(0)\exp(-\sigma^{2}t^{2}/2)\times cos(\omega t + \phi),$$ where $A(0)$ is the initial decay asymmetry at $t$ = 0. The muon spin precesses at the frequencies $\omega = \gamma_{\mu}H_{ext}$, where $\gamma_{\mu}$ is the gyromagnetic ratio of a muon ($2\pi \times 13.554$ MHz/kG) and $H_{ext}$ denotes the transverse external magnetic field. As shown in Fig. 4, for an example of Li$_{0.17}$ZrNCl, this oscillation exhibits faster damping in the superconducting state due to inhomogeneous distribution of internal magnetic fields in the flux vortex structure. In pressed pellet samples of random or oriented powder, this relaxation can usually be approximated by a Gaussian decay which defines the muon spin relaxation rate $\sigma$.
For systems except for ZrNCl-Li-THF, the $\mu$SR results were analysed by assuming a single-component signal, which shows a reasonble agreement to the data as in Fig. 4(b). In ZrNCl co-intercalated with Li and THF, the relative value of the shielding susceptibility was significantly lower than those of other samples, as shown in Fig. 2. Although it is not clear, this reduced susceptibility could possibly imply a finite fraction of superconducting volume. As a precautionary measure, by fitting selected low-temperature signals in field-cooled and zero-field cooled procedures to an asymmetry function having two-component signals, we estimated an upper-limit of the relaxation rate for the superconducting fraction. This upper-limit is shown (in Figs. 6 and 7) by the error-bar placed to the right side of the main point for $\sigma$ which was obtained for a single-component asymmetry function.
$\mu$SR: SPECTRA AND RELAXATION RATE {#sec:level4}
====================================
Figure 4 shows time spectra of muon decay asymmetry for a representative sample above and below $T_{c}$. In the normal state above $T_{c}$, the oscillation shows a small relaxation due to nuclear dipole fields. We denote this relaxation rate as $\sigma_{n}$. Below $T_{c}$, the relaxation becomes faster due to additionaal field distribution from the flux vortex lattice. For each specimen, the zero-field $\mu$SR spectra obtained above and well below $T_{\rm c}$ did not show any difference. This assures that the temperature dependence of the relaxation rate observed in TF is due to superconductivity alone, and also implies that there is no detectable effect of time-reversal symmetry breaking, contrary to the case of UPt$_3$ [@UPt3] and Sr$_2$RuO$_4$ [@Sr2RuO4].
The effect of the superconducting vortex lattice can be obtained by subtracting this normal-state background $\sigma_{n}$ from the observed relaxation rate $\sigma_{ob}$. Since the nuclear dipolar broadening and superconducting broadening of the internal fields do not add coherently, here we adopt quadratic subtraction to obtain the relaxation rate $\sigma$ due to superconductivity as: $$\sigma = \sqrt{\sigma_{ob}^2-\sigma_{n}^2}\ \ \
for\ \ \ (\sigma_{ob} \ge \sigma_{n})$$ and $$\sigma = -\sqrt{\sigma_{n}^2-\sigma_{ob}^2}\ \ \
for\ \ \ (\sigma_{ob} < \sigma_{n}).$$ Note that this procedure makes the error bar rather large around $\sigma$ = 0.
Figure 5(a) shows the temperature dependence of the relaxation rate $\sigma$ for the samples co-intercalated with organic molecules having expanded interlayer distance. With decreasing temperature, the relaxation rate begins to increase below the superconducting transition temperature $T_{c}$. At the flux-pinning temperature $T_{p}$, the zero-field-cooling (ZFC) curve begins to deviate from the field-cooling (FC) curve. In the ZFC procedures, flux vortices are required to enter the specimen from its edge and move a large distance before reaching their equilibrium position. Below the pinning temperature $T_{p}$, this long-distance flux motion could be prevented by the flux pinning, resulting in an highly inequilibrium flux lattice and more inhomogeneous field distribution at muon sites. This behavior has been observed in earlier $\mu$SR studies of HTSC [@wuprbbi2212], BEDT [@ssc19] and some other systems. In both systems shown in Fig. 5(a), we find that $T_{p}$ is much lower than $T_{c}$, which is a characteristic feature for highly 2-d superconductors. The $\mu$SR results of $T_{p}$ for the present systems agree well with those from magnetization measurements discussed in Section VII.
We performed FC measurements using a wide range of external transverse magnetic fields $H_{ext}$, and found no significant dependence of $\sigma$ on $H_{ext}$ from 40 G to 1000 G, as shown in Fig. 5. In TF-$\mu$SR measurements in highly 2-d superconductors, such as Bi2212 or (BEDT-TTF)$_{2}$Cu(NCS)$_{2}$, application of a high external magnetic field transforms 3-d flux vortex structure into 2-d pancake vortices, since higher field implies stronger coupling of flux vortices within a given plane and higher chance for the vortex location in each plane to be determined by random defect position on each plane [@3Dvortex]. The absence of field dependence in our measurements implies that corrections for the 2-d vortex effect is not necessary in the present study. This situation is expected for our c-axis aligned powder specimens. The relaxation rate $\sigma(T)$ shows a tendency of saturation at low temepratures in all of the measured samples of nitride-chloride systems in the present study. This behavior is characteristic for s-wave superconductors. However, experiments using high-quality single crystals are required for a conclusive determination of the superconducting pairing symmetry. In the case of HTSC cuprates, d-wave pairing was established only after $\mu$SR results on high-quality crystals of YBa$_{2}$Cu$_{3}$O$_{y}$ became available [@ssc4].
Figure 5(b) shows the temperature dependence of $\sigma$ for the samples without organic co-intercalant. The dependence of $\sigma$ on temperature $T$ and field $H_{ext}$ of the field-cooling results was essentially similar to that for specimens with co-intercalation in Fig. 5(a). In these systems, we determined the pinning temperature $T_{p}$ by magnetization measurements instead of by $\mu$SR, and show the results in section VII.
$\mu$SR: COMPARISON WITH OTHER SYSTEMS AND SUPERFLUID ENERGY SCALES {#sec:level5}
===================================================================
The $\mu$SR relaxation rate due to the penetration depth is related to the superconducting carrier density $n_s$, effective mass $m^*$, the coherence length $\xi$ and the mean free path $l$ as $$\sigma \propto \lambda^{-2} =
\frac{4\pi e^2}{c^2} \times \frac{n_{s}}{m^{*}}
\times \frac{1}{1+\xi/l}.$$ The proportionality to $n_{s}/m^{*}$ comes from the fact that this effect is caused by the superconducting screening current, and consequently reflecting the current density in a similar way to the normal state conductivity of a metal which is proportional to the carrier density divided by the effective mass. As will be shown later, $\xi$ is estimated to be 80-90 Å in the present nitride-chloride systems. The mean free path cannot be determined at the moment, since a high-quality single crystal is not yet available. In this situation, we proceed with the following arguments by assuming that the system falls in the clean limit ($\xi << l$). Clean limit has been confirmed in many other strongly type-II superconductors, such as the cuprates and BEDT systems.
In highly anisotropic 2-d superconductors, the penetration depth measured with the external field parallel and perpenducular to the conducting plane could be very different. For the geomentry with $H_{ext}$ perpendicular to the conducting plane, related to the in-plane penetration depth $\lambda_{ab}$ as in the present study, the superconducting screening current flows within the plane, resulting in the more effective partial screening of $H_{ext}$ and the shorter $\lambda$ compared to the case with $H_{ext}$ parallel to the planes. In the present work, our specimen has a highly-oriented c-axis direction within +/- 8 degrees, and we regard our specimen as equivalent to single crystal specimens in terms of anisotropy. A theory/simulation work [@ssc5] shows that for un-oriented ceramic specimens of highly 2-d superconductors, value of $\sigma$ should be reduced by a factor of 1/1.4 from the value for single crystals observed with $H_{ext}$ applied perpenducular to the conducting planes.
In Fig. 6, in a plot of $T_{c}$ versus the low-temperature relaxation rate $\sigma(T \rightarrow 0) \propto n_{s}/m^{*}$, we compare the results of the present nitride-chloride systems with those from cuprate and organic BEDT superconductors. The point for the BEDT system was obtained in $\mu$SR measurements using single crystal specimens [@ssc19]. The shaded area denoted as cuprates represents the universal linear correlations found for un-oriented ceramic specimens of underdoped YBCO systems: we multiplied the relaxation rate in these YBCO by a factor of 1.4 to account for the difference between single crystal and un-oriented ceramic specimens. The data points lie in possibly two different groups having different slopes in the $T_{c}$/$\sigma$ relation. The first group with a higher slope includes the present nitride-chloride systems with co-intercalation of organic molecules and the BEDT system, all of which having highly 2-d character as demonstrated by their depinning temperature $T_{p}$ being nearly a 1/3 to 1/2 of $T_{c}$. The second group includes YBCO cuprates and nitride-chloride systems without organic co-intercalation, all of which have more 3 dimensional (3-d) character in the flux pinning property with $T_{p}$ closer to $T_{c}$. The irreversibility and depinning behavior in Li$_{x}$ZrNCl without organic co-intercalation was studied not by $\mu$SR but by magnetization measurements as described in section VII.
The relaxation rate observed by $\mu$SR is determined by the 3-d superfluid density $n_{s}/m^{*}$, as this is a phenomenon caused by the screening supercurrent density in bulk specimens. With the knowledge of interlayer spacing $c_{int}$, one can convert 3-d density $n_{s}/m^{*}$ into 2-d density on each conducting plane as $n_{s2d}/m^{*} = n_{s}/m^{*} \times c_{int}$. For systems having double-layer conducting planes, such as the present nitride chlorides or some family of the cuprates, the average interlayer spacing depends on whether or not the double layer is regarded as a single conducting unit or two. In our previous reports for cuprates [@ssc42], we treated the double layer as two single layers. We shall follow this approach here, and define the $c_{int}$ to be a half of the stacking unit distance as $c_{int} = c_{0}/6$.
In Fig. 7, we show a plot of $T_{c}$ versus the 2-d superefluid density $n_{s2d}/m^{*}$ represented by the value $\sigma(T\rightarrow 0) \times c_{int}$. We include a point obtained in c-axis oriented ceramic specimen of YBa$_{2}$Cu$_{3}$O$_{7}$ [@YBC_oriented] (without multiplying a factor 1.4 to $\sigma$ since this specimen had an almost perfect alignment of c-axis direction). We find that most of the data points share a unique slope in Fig. 7. This result suggests two features: (1) within the nitride-chloride systems, 2-d superfluid density $n_{s2d}/m^{*}$ is a determining factor for $T_{c}$; and (2) the 2-d superfluid density may even be a fundamental determining factor for $T_{c}$ among different superconducting systems. However, the second conclusion (2) must be taken with caution, because, this analysis depends on our treatment regarding single versus double layers, and also because recent data on Tl2201 [@ssc8]. and Bi2201 [@tbp] cuprates, having very large interlayer distance $c_{int} > $12 Å, show universal behavior with the results from YBCO ($c_{int} \sim$ 6 Å) only in a 3-d plot like Fig. 6 but not in a 2-d plot like Fig. 7 [@ssc42]. In contrast, the conclusion (1) is more robust, since all the nitride-chloride systems have double conducting layers, and since the predominant 2-d character is consistent with the absence of dependence of $T_{c}$ on interlayer spacing in nitride-chlorides shown in Fig. 3(b).
Since the Fermi energy of a 2-d metal is proportional to the 2-d carrier density $n_{2d}$ divided by the in-plane effective mass $m^{*}$, as $T_{F} = (\hbar^{2}\pi)n_{2d}/m^{*}$, the horizontal axis of Fig. 7 can be converted into an energy scale representing superconducting condensate. This conversion from penetration depth to the superfluid energy scale was first attempted by Uemura [*et al.*]{} [@ssc7] in 1991 and later followed by other researchers, including Emery and Kivelson [@ssc51].
In order to do such a conversion, one needs to obtain absolute values of the penetration depth $\lambda$ from the relaxation rate $\sigma$. The numerical factor in this $\sigma$ to $\lambda$ conversion in $\sigma \propto \lambda^{-2}$ depends on models used for analyses of relaxation function line shapes, fitting range of data analyses, treatment of single crystal versus ceramic samples, and some other factors. The Gaussian decay, which fits most of the data from ceramic samples quite well, is significantly different from the ideal field distribution $P(H)$ expected for a perfect Abrikosov vortex lattice in triangular lattice. So, using a theoretical second moment for $P(H)$ in Abrikosov lattice is not necessarily appropriate for data analyses in real experiments. After various simulations and consistency checks, we decided to adopt a factor which gives $\lambda$ = 2,700 Å for $\sigma$ = 1 $\mu$s$^{-1}$ for a triangular lattice. Note that this conversion is for a standard triangular lattice, contrary to the statements of Tou and collabrators [@touprl; @touprb] who have erroneously cited that we calculated $\lambda$ for a square vortex lattice. Then we can derive $n_{s2d}/m^{*}$ from the observed values of $\sigma$ and known values of $c_{int}$. In the horizontal axis of Fig. 7, we attach the 2-d Fermi temperature $T_{F2d}$ corresponding to the 2-d superfluid density obtained in the above-mentioned procedure.
A 2-d superfluid of bose gas, such as thin films of liquid He, undergoes superfluid to normal transition via a thermal excitation of unbound flux vortices, as shown by Kosterlitz and Thouless (KT) [@ssc40]. For paired fermion systems composed of $n$ fermions with mass $m$, forming a superfluid with boson density $n/2$ and mass $2m$, the Kosterlitz-Thouless transition temprature $T_{KT}$ becomes 1/8 of the 2-d Fermi temperature $T_{F2d}$ of the corresponding fermion system. In the KT theory, the 2-d superfluid density at the transition temperture $T_{KT}$ should follow system-independent universal behavior: namely, $(\hbar^{2}\pi)\times n_{s2d}/m^{*}$ at $T = T_{KT}$ equals $T_{F2d}/8$. This universal relation was first confirmed by an experiment on He thin films [@KTjump]. In systems close to ideal Bose-gas, the 2-d superfluid density shows almost no reduction between $T=0$ and $T=T_{KT}$ [@KTjump]. Thus, in such a case, we would expect the points in Fig. 7 (based on $n_{s2d}/m^{*}(T=0)$) to lie on the $T_{KT}$ line. In thin films of BCS superconductors, the superfluid density shows much reduction from the value of $T=0$ to $T= T_{KT}$, and the “KT jump of superfluid density” becomes invisible. This corresponds to the situation where the points in Fig. 7 lie far in the right side of the $T_{KT}$ line. In Fig. 7, most of the points lie about a factor of 2 away from the $T_{KT}$ line. The linear relation between $T_{c}$ and $n_{s2d}/m^{*}(T=0)$ suggests relevance to the KT transition, as pointed out by Emery and Kivelson[@ssc51]. However, the deviation from the $T_{KT}$ line implies serious difference from the ideal KT situation.
MAGNETIZATION MEASUREMENTS: EXPERIMENTAL {#sec:level6}
========================================
Magnetization measurements were performed using a SQUID magnetometer (Quantum-Design) at Columbia. Aligned pressed samples were sealed in quartz ampules. The raw response curve was corrected by the subtraction of the quartz background curve measured in advance. In the normal state of the superconducting Hf(Zr)NCl samples, as well as the parent compounds, weak-ferromagnetic behavior is observed up to room temperature. This weak-ferromagnetic behavior changes by the intercalation. Therefore, we subtracted the weak-ferromagnetic contribution. We estimated this by extrapolating temperature dependence, assuming the Curie-Weiss law ($M = C / (T - \theta)$) and fitting the normal state magnetization in the temperature range of $2.5T_{c} \leq T \leq 5T_{c}$. In this temperature range, superconducting fluctuations can be neglected and the temperature dependence is slightly concave. We only used the data with the extrapolated weak-ferromagnetic contribution smaller than 10 % of the diamagnetic magnetization, to avoid an error from the assumption of the Curie-Weiss temperature dependence. The model developed by Hao [*et al.*]{} [@hao] was applied to the analysis of the reversible region. In this model, reduced (dimensionless) magnetization $M' = M/\sqrt{2} H_{c}(T)$ and field $H' = H/\sqrt{2} H_{c}(T)$ scales as a single function that contains the Ginzburg-Landau parameter $\kappa$ as a unique parameter. In our analysis, we optimized $H_{c}(T)$, in addition to $\kappa$ as a parameter independent of temperature. Resistivity measurements were performed using a well-aligned pressed sample with four electrodes that is sealed in a cell made of Kapton film and epoxy glue.
MAGNETIZATION MEASUREMENTS: SUPERCONDUCTING PROPERTIES {#sec:level7}
======================================================
Magnetization measurements were performed in Li$_{0.17}$ZrNCl and Li$_{0.15}$THF$_{0.08}$ZrNCl with magnetic fields applied parallel to the c-axis. Figure 8 shows the results obtained after the corrections for the quartz ample background and for the weak-ferromagnetic contribution. We note that a crossing point exists in the $M(T)$ curve under various magnetic fields for each system, which is characteristic of quasi-two-dimensional superconductors [@kesprl]. There are reversible temperature regions where ZFC and FC magnetization curves overlap each other. Below a certain temperature (the pinning temperature $T_{p}$), ZFC and FC magnetization curves deviate. We notice that the reversible region is wider for Li$_{0.15}$THF$_{0.08}$ZrNCl. This result is consistent with a picture that, by the expansion of the interlayer distance, the interlayer coupling become weaker and the pinning of the vortices becomes less effective.
In the data analyses in the reversible region of Li$_{0.17}$ZrNCl and Li$_{0.15}$THF$_{0.08}$ZrNCl, we confined to the temperature region apart from $T_{\rm c}(H)$ in order to avoid ambiguity due to the superconducting fluctuations. As shown in Fig. 9, the data scale quite well to Hao’s model in the whole reversible temperature range below $T_{c}$ for the both systems. For all the data, $M' << H'$ and hence the demagnetization factor can be ignored. This analysis yielded values of $\kappa$ ranging between 50 and 80 (see Table 1), which indicates that these compounds are extreme type-II superconductors. In Fig. 10, we show the values of the upper critical field $H_{c2,//c}(T)$ obtained down to $T=2$ K in this process using Hao’s model. The temperature dependence of $H_{c2,//c}(T)$ fits well to an empirical formula $H_{c2,//c}(0) [1-(T/T_{c})^2]$, as shown by the dashed lines in Fig. 10. We emphasize that the low temperature limit value $H_{c2,//c}(T\rightarrow 0)$ can be obtained almost without any extrapolation using this formula: the resulting values are shown in Table 1. The $H_{c2//c}$ value of $\sim$ 4-5 T in ZrNCl-Li-THF system is about a factor of 2 smaller than $H_{c2} \sim$ 10 T in HfNCl-Li-THF system reported by Tou [*et al*]{} [@touprb]. These results might indicate that $H_{c2}$ roughly scales with $T_{c}$. A similar nearly linear relation between $H_{c2}$ and $T_{c}$ can be found in the $H_{c2}$ values for high-$T_{c}$ cuprate superconductors in the optimum doping region.
The critical temperature $T_{c}$ obtained using Hao’s model is 14.9 K for both samples, which agrees with the estimate from the onset of diamagnetism due to superconductivity. We notice that at $H=55$ kG above $H_{c2,//c}(0)$, a diamagnetic behavior due to superconducting fluctuation was observed in magnetization as shown in Fig. 8. Similar results due to critical fluctuations have been reported in HTSC [@fl1; @fl2; @fl3] and BEDT [@fl4] systems. We obtained the in-plane coherence length $\xi_{ab}(0)$ and the in-plane penetration depth $\lambda_{ab,M}(0)$ using expressions $H_{c2}(0) = \phi_0/2\pi\xi(0)^2$ and $\kappa = \lambda/\xi$. These results are also summarized in Table 1. We note that $H_{c2,//c}(0)$, and $\xi_{ab}(0)$ are almost unaffected by the interlayer distance. This agrees with the view that the essence of the superconductivity in Hf(Zr)NCl is dominated in Hf(Zr)-N honeycomb double layers. The values of the penetration depth determined both from $\mu$SR and magnetization show reasonable agreement, although the former is $\sim20$ % smaller than the latter for both compounds.
In magentization measurements (see Fig. 8) and $\mu$SR measurements (see Fig. 5(a)), the results become history dependent below a pinning temperature $T_{p}$ for a given external field $H_{ext}$. This feature can be expressed by defining the irreversibility field $H_{irr}$ for a given temperature $T$ as $H_{ext}(T=T_{p}) = H_{irr}$. Figure 10 also includes $H_{irr}$ as a function of temperature, determined from magnetization and $\mu$SR measurements. The results obtained from the two different techniques exhibit excellent agreement. ZrNCl superconductors have a quite large reversible region in the $H$-$T$ plane. The temperature dependence of the irreversibility field fits well to a functional form $H_{irr}(T) = H_{irr}(0) (T_{c}/T-1)^n$, obtained for three-dimensionally fluctuating vortices[@crossover], with $n=1.5$ at low fields below $H \sim 0.4$ Tesla. This provides support to our assumption of 3-d vortex lines which we adopted in our analyses of TF-$\mu$SR spectra taken below $H=0.1$ Tesla. At higher fields, the fitting becomes worse, similarly to Ref. [@crossover]. This may be related to a dimensional crossover from 3- to 2-d vortex fluctuations. More careful measurements are necessary to conclude this point.
In order to provide a cross-check for the results of $H_{c2}(T)$, we performed magnetoresistance measurements on Li$_{0.17}$ZrNCl. The temperature dependence of resistivity for two sets of field and current configurations is shown in Fig. 11. High resistivity of the order of 100 $m\Omega cm$ and negative slope of the resistivity in the normal state at low temperatures could be due to grain boundaries and may not be intrinsic. The observed superconducting transition is broadened by superconducting fluctuations, weakly superconducting regions such as grain boundaries, and by vortex motion due to Lorenz force. We notice that the resistive broadening is slightly larger for $H // {c}$, which is a natural consequence of significant superconducting fluctuations only for $H // {c}$. We defined $T_{c}(H)$ where resisitivity shows 50 % drop of the maximum value. Figure 12 shows $H_{c2}(T)$ obtained in this procedure. These absolute values of $H_{c2}(0)$ for $H//c$ agree reasonably well with those from magnetization measurements, in spite of the unreliable definition due to the broad resistive transition. The difference between the temperature dependences of the magnetization (Fig. 10) and resistive (Fig. 12) $H_{c2}$ data may be due the above-mentioned limitations of the resistive measurements. The anisotropy ratio of the upper critical field $H_{c2,\perp c}/H_{c2,// c}$ is roughly 3 as shown in Fig. 12. Although we do not have data for the system with co-intercalation of THF or PC, the anisotropy ratio would presumably increase in more 2-d systems with larger stacking unit distance.
DISCUSSIONS AND CONCLUSIONS {#sec:level8}
===========================
The quasi-two-dimensional nature of the superconducting state appears in various superconducting properties of intercalated Hf(Zr)NCl. $T_{c}$ correlates with a 2-d superconducting carrier density $n_{s2d}$ devided by effective mass $m^{*}$ rather than the 3-d counterpart. Diamagnetic magnetization due to superconducting fluctuation for $H // {c}$ is observed at high temperatures and high fields. The crossing point exists in $M(T)$ curves measured at various fields. The reversible region of magnetization becomes larger with the increase of interlayer distance, suggesting weaker interlayer coupling.
In addition to these results, we note that $T_{c}$, $n_{s2d}/m^{*}$, and $\xi_{ab}$ ($H_{c2,//c}$), all show moderte dependence on chemical doping level which presumably represents the in-plane normal-state carrier concentration, while remaining almost independent of the stacking unit distance. These parameters are closely related to the superconductivity mechanism of this layered superconductor. Since the coherence length is a measure for the pair size, independence of $\xi_{ab}$ on interlayer distance implies that interlayer coupling does not affect the pair formation. It is then possible to consider a picture in which fluctuating superconductivity exists within a given layer, while the layers are coupled weakly by Josephson coupling to achieve 3-d bulk superconductivity.
On the other hand, $\lambda_{ab}$ and $T_{p}$ are strongly affected by the interlayer distance. The reduction of $\lambda_{ab}$ with increasing stacking-unit distance can be understood as a simple reduction of the supercurrent density caused by lower density of the planes. The strong dependence of $T_{p}$ on $c_{int}$ is not surprising: this behavior has been seen in many HTSC cuprates.
Our results show that Hf(Zr)NCl with variable interlayer distance as well as carrier concentration is suitable for systematic studies of layered superconductors. In addition, low $H_{c2}$ value of this compound makes it easier to cover the whole superconducting region in the $H$-$T$ plane, and helps our study of vortex phase diagram.
In Fig. 6 and Fig. 7, we have compared the results from the nitride-chlorides with other layered superconductors. All the arguments in the previous paragraphs, as well as Fig. 7, give an impression that 2-d properties are predominantly important factor of all of these layered superconductors. However, comparison among different cuprate systems having different interlayer spacing $c_{int}$ indicates that the 3-d interlayer coupling plays a very important role in determing $T_{c}$ in the cuprates. Furthermore, the observed results in Fig. 7 show about a factor 2 deviation from the $T_{KT}$ line. These results indicate that a simple theory for KT transition, whose $T_{c} = T_{KT}$ is unrelated to the interlayer coupling, is not applicable either to the cuprates nor to the nitride-chloride systems. Further experimental and theoretical studies are required to determine the origin of this deviation. Studies of crossover from Bose Einstein to BCS condensation, in the case of 2-d systems, might provide a clue for understanding this feature.
The absolute values of the penetration depth $\lambda$, obtained from the $\mu$SR and magnetization measurements, show a reasonable agreement. We notice, however, about 20-30 % difference in the values from the two different methods (see Table 1). $\mu$SR and magnetization estimates of $\lambda$ often exhibit some disagreement of this magnitude, as can be found also in the cases of HTSC and organic systems. Ambiguity of $\lambda$ with 20-30 % would, however, correspond to $\sim$ 50 % ambiguity in the estimate of the superfluid density. In this situation, it would be ideal if there were a method to cross-check the superfluid density derived from $\mu$SR results in a completely different perspective.
In Bi2212 cuprate systems, Corson [*et al.*]{} [@ssc53] measured the frequency dependent superfluid response, and found temperature $T_{KT}$ above which the superfluid density depends on the measuring frequency. The superfluid density observed at $T=T_{KT}$ was consistent with the value expected in the universal argument of KT. This provides an excellent system-independent calibration to the superfluid density. The difference between the superfluid density at $T_{KT}$ and at $T\rightarrow 0$ should correspond to the distance (in the horizontal direction) of the corresponding data point in Fig. 7 from the $T_{KT}$ line. The $\mu$SR Bi2212 data point in a plot like Fig. 7 lie about a factor 2-3 away from the $T_{KT}$ line. This factor agrees reasonably well with the reduction of the superfluid density from the $T=0$ value to the $T=T_{KT}$ value observed by Corson [*et al.*]{} [@ssc53] in Bi2212 system in a similar doping region. This satiafactory cross-check for the Bi2212 system indicates that our choice of the conversion factor between $\sigma$ and $\lambda$ was reasonable, and the superfluid density derived by $\mu$SR is very reliable. Of course, comparisons among $\mu$SR data for different systems in a relative scale can be performed without being affected by an ambiguity of their absolute values of the superfluid density.
We performed $\mu$SR measurements on three different specimens based on ZrNCl with Li concentraitons 0.15, 0.17 and 0.4, and found that the results of 2-d superfluid density $n_{s2d}/m^{*}$ for these systems do not show much difference among one another. This phenomenon could be explained by two different possibilities: (a) not all the Li atoms donnate carriers on the ZrN planes, and the Li concentration $x$ does not serve as an indicator of normal-state carrier concentraion; or (b) all the Li atoms donnate electrons to the ZrN planes, but only a finite fraction of those normal-state carriers participate in the superfluid. The situation (b) is similar to the case of overdoped HTSC cuprates [@sscreview; @uemuraodssc], where an energy-balance in the condensation process seems to determine the superfluid density. Further experiments on normal-state transport properties are required to distinguish between (a) and (b) in the nitride-chloride systems.
In conclusion, we have synthesized and characterized several different specimens of intercalated nitride-chloride superconductors, and performed $\mu$SR and magnetization measurements. The superconducting transition temperature $T_{c}$ and the upper critical field $H_{c2,//}$ exhibit a nearly linear relation with the 2-d superfluid density $n_{s2d}/m^{*}$, while showing almost no dependence on the stacking unit distance. These features suggest a highly two-dimensional nature of superconductivity in the nitride-chloride system.
ACKNOWLEDGEMENT {#sec:level9}
===============
We are grateful to A.R. Moodenbaugh for x-ray rocking curve measurement, Y. Mawatari for discussion about reversible magnetization, and H. Stormer for help in resistance measurements. This work was supported primarily by the National Science and Engineering Initiative of the National Science Foundation under NSF Award CHE-01-17752. The work at Columbia was also supported by NSF-DMR-01-02752 and NSF-INT-03-14058. The work at Hiroshima Univ. was supported by the Grant-in-Aid for Scientific Research (B) (No.14350461) and the COE Research (No. 13E2002)) of the Ministry of Education, Science, Sports, and Culture of Japan, and by CREST, Japan Society for Science and Technology (JST). Research at McMaster is supported by NSERC and CIAR (Quantum Materials Program).
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------------------------------ ----------------------------- ------------- ----------------- ------------------------------------------------- -------------- --------------------- ----------------------- ------------------------------
[Sample]{} [Synthesis method]{} [$c_0/3$]{} [$T_{\rm c}$]{} [$\lambda_{{\rm ab,}\mu{\rm SR}} (T \to 0)$]{} [$\kappa$]{} [$H_{c2,//c}(0)$]{} [$\xi_{\rm ab}(0)$]{} [$\lambda_{\rm ab,M} (0)$]{}
[Å]{} [K]{} [Å]{} [T]{} [Å]{} [Å]{}
Li$_{0.17}$ZrNCl [(i) $n$-Butyllithium]{} 9.4 14.2 3700 56 4.7 83 4700
Li$_{0.4}$ZrNCl [(i) $sec$-Butyllithium]{} 9.4 12.5 3900
Li$_{0.6}$ZrNCl [(i) $tert$-Butyllithium]{} 9.4 11.7
Li$_{0.15}$THF$_{0.08}$ZrNCl [(ii) THF]{} 13.3 14.4 5200 76 4.2 88 6700
Li$_{0.18}$PC$_{0.15}$ZrNCl [(ii) PC]{} 13.3 14.6
Li$_{0.24}$THF$_{0.14}$HfNCl [(iii) 8 $mM$ Li-Naph]{} 13.3 25.5
Li$_{0.5}$THF$_{0.3}$HfNCl [(iii) 100 $mM$ Li-Naph]{} 18.7 25.5 3900
------------------------------ ----------------------------- ------------- ----------------- ------------------------------------------------- -------------- --------------------- ----------------------- ------------------------------
: Synthesis method (see the text for details), stacking unit distance ($c_{0}/3$), superconducting transition temperature $T_{c}$ estimated from magnetization, magnetic field penetration depth $\lambda_{ab,\mu{SR}} (T \to 0)$ at zero temperature limit estimated from TF-$\mu$SR, Ginzburg-Landau parameter $\kappa$, upper critical field at zero temperature $H_{c2,//c}(0)$, coherence length at zero temperature $\xi_{ab}(0)$, and magnetic penetration depth at zero temperature $\lambda_{ab,M}(0)$ estimated from reversible magnetization for intercalated HfNCl and ZrNCl systems reported in the present work. TF-$\mu$SR and magnetization measurements were performed under the magnetic field parallel to the $c$-axis.
|
---
abstract: 'We propose a multilingual model to recognize Big Five Personality traits from text data in four different languages: English, Spanish, Dutch and Italian. Our analysis shows that words having a similar semantic meaning in different languages do not necessarily correspond to the same personality traits. Therefore, we propose a personality alignment method, GlobalTrait, which has a mapping for each trait from the source language to the target language (English), such that words that correlate positively to each trait are close together in the multilingual vector space. Using these aligned embeddings for training, we can transfer personality related training features from high-resource languages such as English to other low-resource languages, and get better multilingual results, when compared to using simple monolingual and unaligned multilingual embeddings. We achieve an average F-score increase (across all three languages except English) from 65 to 73.4 (+8.4), when comparing our monolingual model to multilingual using CNN with personality aligned embeddings. We also show relatively good performance in the regression tasks, and better classification results when evaluating our model on a separate Chinese dataset.'
author:
- |
Farhad Bin Siddique $^{12}$, Dario Bertero $^{12}$, Pascale Fung $^{123}$\
$^1$ Electronic and Computer Engineering Department\
$^2$ Center for Artificial Intelligence Research (CAiRE)\
$^3$ EMOS Technologies Inc.\
The Hong Kong University of Science and Technology\
Clear Water Bay, Hong Kong\
title: 'GlobalTrait: Personality Alignment of Multilingual Word Embeddings'
---
Introduction
============
According to [@allport1937personality], personality refers to the characteristic pattern in a person’s thinking, feeling, and decision making. It is a quality of a person across a relatively long period and is different from emotions, which can be perceived in the moment. We can think of it as, personality is to emotion what climate is to weather. The Big Five model of personality [@goldberg:93] is a common way of quantifying a person’s personality, and is recognized by most psychologists around the world. It tries to represent the traits as scores across five dimensions:
- **Extraversion vs Introverted** (*Extr*) - sociable, assertive, playful vs aloof, reserved, shy;
- **Conscientiousness vs Unconscientious** (*Cons*) - self-disciplined, organised vs inefficient, careless;
- **Agreeableness vs Disagreeable** (*Agr*) - friendly, cooperative vs antagonistic, faultfinding;
- **Neuroticism vs Emotionally Stable** (*Emot*) - insecure, anxious vs calm, unemotional;
- **Openness to Experience vs Cautious** (*Openn*) - intellectual, insightful vs shallow, unimaginative.
Personality traits affect the usage of language in people [@mairesse2007using], and it is an integral part of human-human interaction [@long2000personality; @berry2000affect]. As we develop smarter dialogue systems, future virtual agents need to detect and adapt to different user personalities in order to express empathy [@fung2016towards]. Although traditionally the user personality can be identified by having the user fill out a self-assessment form such as the NEO Personality Inventory [@costa:2008], this method is not feasible for many applications where we may wish to identify user personality, such as dialogue systems. Therefore, work on automatic personality assessment has become increasingly important recently with the rise in popularity of applications such as Human Resources (HR) screening, personalized marketing, and other social media related user-profiling.
[.5]{} 
[.5]{} 
Currently, personality labeled data is scarce, especially in the multilingual setting, which makes it essential to use the relatively larger size of data in English to help recognize traits in other languages. Previous work on multilingual personality recognition have tried to use the word-level or character-level similarity across languages to develop a multilingual model [@liu2016language; @siddique2017bilingual]. However, our experiments show that words used by people with different personality traits differ among languages or cultures, which is not captured by distributional semantics alone, making it necessary to learn a personality-based mapping of words to express each personality trait. Therefore, we propose GlobalTrait, which trains personality trait-based alignment of multilingual embeddings from the source language(s) to the target language, such that words that correlate positively to each trait are closer together in the multilingual (global) vector space. We show that taking such mapping or alignment of embeddings as input to our model gives us better multilingual results in the task of personality recognition.
Related Work
============
Automatic personality recognition has been done since as early as 2006 [@oberlander2006whose], where the personality of blog authors were identified using Naive Bayes algorithm and n-gram features. [@mairesse2007using] used two sources of lexical features, Linguistic Inquiry and Word Count (LIWC) [@pennebaker2001linguistic] and Medical Research Council (MRC) Psycholinguistics Database [@coltheart1981mrc] features, to identify personality from written and spoken transcripts. More recently, for tasks such as the Workshop on Computational Personality Recognition [@celli2013workshop], people have worked on identifying personality from social media texts (Facebook status updates and Youtube vlog transcriptions). [@verhoeven2013ensemble] used 2000 frequent trigrams as features and trained a SVM classifier, and [@farnadi2013recognising] used LIWC features to train SVM, Naive Bayes and K Nearest Neighbor (KNN) algorithms.
Deep Learning models such as Convolutional Neural Networks (CNNs) have gained popularity in the task of text classification [@kalchbrenner2014convolutional; @kim2014convolutional]. This is because CNNs are good at capturing text features via its convolution operation, which can be applied on the text by taking the distributed representation of the words, called word embeddings, as input. Learning such distributed representation comes from the hypothesis that words that appear in similar contexts have similar meaning [@harris1954distributional]. Different works have been carried out in the past to learn such representations of words, such as [@mikolov2013distributed; @pennington2014glove], and more recently [@bojanowski2016enriching]. Cross-lingual or multilingual word embeddings try to capture such semantic information of words across two or more languages, such that the words that have similar meaning in different languages are close together in the vector space [@faruqui2014improving; @upadhyay2016cross]. For our task we use a more recent approach [@conneau2017word], which does not require parallel data and learns a mapping from the source language embedding space to the target language in an unsupervised fashion.
Methodology
===========
We propose GlobalTrait, a text-based model that uses multilingual embeddings across languages to train personality alignment per trait, such that words in the languages that correspond positively to one trait are closer together in the multilingual vector space: which are then fed to a CNN model for binary classification of each trait. We first train multilingual embeddings in the given languages, followed by identifying the most significant words in each language corresponding positively to each trait, which are used to learn a mapping from each source to the target language. The target language is the language in which we have the most labeled data available, in our case English. The mapping is essentially a personality trait based alignment that tries to bring the distributional representation of words correlating positively to each trait closer together. The initial multilingual embeddings along with the GlobalTrait aligned embeddings are then fed into a two channel CNN model to extract the relevant features for classification via a fully connected layer followed by softmax.
Multilingual Embeddings Training
--------------------------------
Learning distributed representation of words (word embeddings) comes from the hypothesis that words that appear in similar contexts have similar meaning [@harris1954distributional]. Different works have been done to learn such representations [@mikolov2013distributed; @pennington2014glove; @bojanowski2016enriching]. Cross-lingual or multilingual word embeddings try to capture such semantic information of words across two or more languages [@faruqui2014improving; @upadhyay2016cross]. We use the methodology of Multilingual Unsupervised and Supervised Embeddings (MUSE) [@conneau2017word] to first train multilingual embeddings across the four languages. MUSE tries to learn a mapping $W$ of dimension $d \times d$, where $d=300$ is the embedding dimension we use, such that:
$$W^* = \operatorname*{argmin}_{W \in O_d(\mathbb{R})} || WX - Y ||_F$$
where, $O_d(\mathbb{R})$ suggests that $W$ is an orthogonal matrix consisting of real numbers, $X$ and $Y$ are the $d \times n$ matrices representing the word embeddings of $n$ words in the source and target languages respectively. It is important that the mapping matrix $W$ is orthogonal, so that we are performing a rotational mapping on the embedding space, which does not disrupt the monolingual semantic information of the original embeddings. Also, being orthogonal gives us the following Procrustes solution to equation (1):
$$W^* = UV^T, \text{with } U\Sigma V^T = SVD(YX^T).$$
$W$ mapping matrix is trained via a Generative Adversarial Network (GAN) training approach [@goodfellow2014generative; @ganin2016domain], where a generator and a discriminator network are both trained in parallel. It is a two player game where the discriminator tries to differentiate between a source embedding and a mapped embedding, and the generator tries to fool the discriminator by making $WX$ as similar to $Y$ as possible. For our monolingual embeddings used for each of the individual languages, we use the pre-trained word embeddings via fastText[^1]. The issue with multilingual embeddings is that it only captures the semantic information of words across languages, so words that share similar context will appear close together in the multilingual vector space. As we can see in Figure \[extr\] (explained further in the *Experiments* section), words that correspond to the same trait do not always share a similar semantic meaning across languages. This motivates the need for a mapping of the embeddings from the source to the target language, hence the notion of personality trait-based alignment.
![Simplified architecture of our two channel CNN model, one channel taking the unaligned multilingual embeddings as input, while the other taking the GlobalTrait aligned embeddings, fed into a one layer CNN, and the extracted features are concatenated, followed by a max-pooling layer and a fully connected layer to softmax for binary classification.[]{data-label="cnn"}](cnn.png)
GlobalTrait - One Alignment Per Trait
-------------------------------------
We propose GlobalTrait, which does a personality trait-based alignment of the multilingual embeddings from the source to the target language. From the dataset, we use Term Frequency - Inverse Document Frequency (*tf-idf*) features to obtain the *n* most significant words that correspond positively to each trait per language, and get the multilingual embeddings corresponding to the words. Using these embeddings, we learn a second mapping from each source language to our target language, English, with the idea that the mapped or aligned embeddings will represent the trait, by being closer together in the vector space. There has to be one alignment per trait per language - a rotational mapping from the source to the target language space. We end up training 5 different mapping matrices for each source language, one for each of the Big Five traits.
### Training Procedure
We use the same training approach as MUSE to train our personality mapping. For each trait, we take the words having the highest significance in the source languages, and train a second mapping of their corresponding multilingual embeddings to the target language space. We can call such mapping of the Agreeableness trait, for example, as $W_{a}$, and therefore, we try to achieve the following equation, $W_{a}A = B$, where $A$ and $B$ are the trained multilingual embeddings of words in the source and target language respectively, which correlate positively to the Agreeableness trait. Let $X=\{x_1,...,x_n\}$ be the n multilingual embeddings in the source language of the words corresponding most positively to the Agreeableness trait. Likewise, $Y=\{y_1,...,y_m\}$ are the m multilingual embeddings of words in the target language, again for the Agreeableness trait. We try to train a discriminator such that it can differentiate between random samples taken from $W_{a}X = \{W_{a}x_1,...,W_{a}x_n\}$ and $Y$. We train $W_a$ matrix such that the discriminator is unable to differentiate between the two, therefore making the mapping from source as close to the target as possible, bringing the notion of personality based alignment.
- **Discriminator**: Let $\theta_D$ be the parameters of the discriminator, and $P_{\theta_D} (source = 1 | z)$ is the probability that a vector $z$ is the element of a source embedding according to the discriminator, and not the target (mapped) embedding. Therefore, for the Agreeableness trait, the discriminator loss function is as follows: $$\begin{split}
L_D (\theta_D|W_{a}) = -\frac{1}{n} \sum_{i=1}^n \log P_{\theta_D} (source = 1 | W_{a}x_i) \\ - \frac{1}{m} \sum_{i=1}^m \log P_{\theta_D} (source = 0 | y_i)
\end{split}$$
- **Mapping Matrix**: We try to train $W_a$ such that the discriminator is unable to differentiate between the source and the target (mapped) embeddings: $$\begin{split}
L_W (W_{a}|\theta_D) = -\frac{1}{n} \sum_{i=1}^n \log P_{\theta_D} (source = 0 | W_{a}x_i) \\ - \frac{1}{m} \sum_{i=1}^m \log P_{\theta_D} (source = 1 | y_i)
\end{split}$$
Therefore, the discriminator and the weight matrix objective functions work alternative to each other, and they finally converge in a *min-max* solution. Similarly, we train a mapping matrix for each of the other four traits, which gives us the GlobalTrait personality aligned embeddings, where the embeddings closer in the multilingual vector space reflect the same trait.
Convolutional Neural Network
----------------------------
Deep learning models such as Convolutional Neural networks (CNNs) have gained popularity in the task of text classification [@kalchbrenner2014convolutional; @kim2014convolutional]. Our CNN model is a two channel mode, where one channel takes the multilingual embeddings, while the other takes the GlobalTrait personality aligned embeddings as input. The first channel with multilingual embeddings is kept trainable, or we can call it a dynamic channel, which means the embeddings are also taken as training parameters, and can change as the training goes on. The other channel, where the personality alignment has been trained already, is kept static. For both channels, we choose window sizes to be 3, 4 and 5, which essentially extracts 3, 4 and 5-gram features from the text, and we have a max pooling operation that keeps the maximum features per window from both the channels. The total features are concatenated and passed to a fully connected layer with a single hidden layer and non-linear activation (tanh), ultimately mapping the features to a binary classification of each trait via softmax.
Experiments
===========
Lang Model Extr Agr Cons Emot Openn *Average*
------ ----------------- ---------- ---------- ---------- ---------- ---------- -----------
en Lgr-mono 64.0 **53.3** 64.5 72.5 60.5 63.0
CNN-mono **74.4** 48.2 **72.8** **74.9** **67.7** **67.6**
Lgr-mono 73.2 72.1 70.0 56.9 69.0 68.2
Lgr-multi 73.2 70.1 72.6 57.0 **69.5** 68.5
es Lgr-GlobalTrait 75.9 74.7 82.4 59.0 69.0 72.2
CNN-mono 74.7 74.7 70.0 **69.0** 67.2 71.1
CNN-GlobalTrait **79.4** **76.0** **83.3** 67.3 67.0 **74.6**
Lgr-mono 53.3 60.3 52.5 60.4 63.3 58.0
Lgr-multi 64.2 71.2 51.5 62.1 **64.2** 62.6
it Lgr-GlobalTrait 66.2 75.2 49.7 65.7 63.5 64.1
CNN-mono **67.2** 74.3 **60.3** 75.4 63.1 68.1
CNN-GlobalTrait 64.0 **77.5** 58.3 **78.0** 63.2 **68.2**
Lgr-mono 76.0 **67.4** 67.9 65.0 67.7 68.8
Lgr-multi 74.2 58.2 66.8 66.2 67.0 66.5
nl Lgr-GlobalTrait 76.9 52.9 62.0 68.2 66.4 65.3
CNN-mono 76.4 60.6 61.8 78.6 64.6 68.4
CNN-GlobalTrait **85.3** 58.4 **83.3** **85.8** **74.5** **77.5**
Dataset
-------
We used the 2015 Author Profiling challenge dataset (PAN 2015) [@rangel2015overview], which includes user tweets in four languages - English (en), Spanish (es), Italian (it) and Dutch (nl), where the personality labels were obtained via self-assessment using the BFI-10 item personality questionnaire [@rammstedt2007measuring]. Only the training set was released to us from the PAN2015 website [^2]. Their test data was not available to us as we did not take part in the Author Profiling competition of 2015. The dataset consists of 152 English (14,166 tweets), 110 Spanish (9,879 tweets), 38 Italian (3,687 tweets), and 34 Dutch (3,350 tweets) users in total. Our task is to identify personality in user-level, so we concatenated all the tweets made by a single user to create one single training/test data point. As preprocessing, we tokenized each tweet using Twokenizer [@owoputi2013improved], and replaced all usernames and URL mentions with generic words (*@username* and *@url*), so the model is not affected by mentions in the tweet that are not influenced by the user’s personality. Since we are interested in the binary classification of each big five trait, we carried out a median split of the scores, to obtain positive and negative samples (users) for each of the five traits. For our results shown, we carried out a stratified k-fold cross validation, by making k=5 splits of the training set into training/validation, and then show the average result across the 5 different validation sets.
Experimental Setup
------------------
For our MUSE training, we used a discriminator with 2 hidden layers, each having a dimension of 2048, and we ran our training for 5 epochs with 100,000 iterations in each epoch. When training the personality alignment, we took the top 3000 significant words corresponding positively to each trait per language. For our evaluation, we built a source to target language dictionary and used mean cosine distance as the validation metric. For our CNN model, we used 64 filters per filter size, and for our fully connected layer, we set the hidden layer dimension to 100, and we ran our training for 100 epochs for each model with batch size = 10. We used binary cross entropy as our loss function, and used Adam optimizer with learning rate of $1e^{-4}$.
Text Personality Analysis - Visualization of Embeddings
-------------------------------------------------------
To check if words having similar semantic information across languages contribute to similar Big Five traits, we carried out some text-based analysis of our data. For each of the Big Five traits, we first obtained the most significant words in each language using term frequency-inverse document frequency (*tf-idf*) features. We then took the top 750 words for each language having the highest *tf-idf* and plot their trained multilingual embeddings (trained using MUSE from monolingual embeddings in each language) on a 2-D space by performing t-distributed Stochastic Neighbor Embedding (*t-SNE*) on the 300-dimensional vectors. As an example, plots for the Extraversion and Conscientiousness traits are shown in Figure \[extr\].
As we can see in the figure, for both traits there is very little overlap in the embedding space between the four languages, and most of the words are clustered per language. This shows that words corresponding to each trait might not have the same semantic meaning across the languages. Similar to the Extraversion and Conscientiousness trait, in all five traits, we see a similar trend, some overlap between English and Spanish, some overlap between Spanish and Italian, but very little or no overlap between Dutch and any of the languages. Therefore, this gives rise to the need for a certain mapping from each language to our target language, English. Some examples of words corresponding positively to the traits per language is shown in table \[words\].
Binary Classification
---------------------
Lang Model Extr Agr Cons Emot Openn *Average*
------ ------------------------------------------------ ----------- ----------- ----------- ----------- ----------- -----------
Char Bi-RNN [@liu2016language] 0.148 **0.143** 0.157 0.177 **0.136** 0.152
es tf-idf linear regression [@sulea2015automatic] 0.152 0.148 **0.114** 0.181 0.142 **0.147**
*CNN-GlobalTrait* **0.142** 0.150 0.135 **0.169** 0.151 0.149
Char Bi-RNN [@liu2016language] 0.124 0.130 **0.095** **0.144** 0.131 0.125
it tf-idf linear regression [@sulea2015automatic] 0.119 **0.122** 0.101 0.150 **0.130** **0.124**
*CNN-GlobalTrait* **0.107** 0.128 0.120 0.147 0.134 0.127
\[regression\]
We first carried out binary classification of the users based on the median split of scores in each trait. Classification is of more importance to us as dialogue systems and other similar applications require us to classify each person into positive/negative for each trait, which can then be used to make decisions such as adapting to the given personality.
### Baseline
We implemented a simple logistic regression classifier, which takes in the average embeddings of the words as input features, in order to compare our aligned embeddings result with the monolingual counterpart. We present the results of the following experiments for comparison:
- **Lgr-mono**: logistic regression using monolingual embeddings
- **Lgr-multi**: logistic regression using unaligned multilingual embeddings
- **Lgr-GlobalTrait**: logistic regression using our personality aligned multilingual embeddings
- **CNN-mono**: CNN using monolingual embeddings as input
- **CNN-GlobalTrait**: two channel CNN using multilingual embeddings plus the GlobalTrait aligned embeddings
In both ‘-multi’ and ‘-GlobalTrait’, models, the training set includes both the English and the source language’s training data, and is tested on the source language’s validation sets, while ‘-mono’ is just the monolingual model for the respective source language. Results are reported in table \[results\].
### Results
We achieve an average F-score of 74.6 in Spanish, 68.2 in Italian, and 77.5 in Dutch when using our multilingual CNN model with the GlobalTrait aligned embeddings, which are the highest performance achieved in each of the three languages. As we can see in table \[results\], CNN-GlobalTrait performs the best except for two traits in Spanish and three traits in Italian. The discrepancies can be due to the imbalanced nature of the dataset, and the logistic regression being a simple classifier, can converge better than CNN, especially for smaller datasets. For logistic regression, using multilingual and then GlobalTrait aligned embeddings improves on the monolingual results. In general, our multilingual results perform better than monolingual, except for one trait in Spanish, two traits in Italian, and one trait in Dutch. This shows that we can use the features retrieved from English to help us recognize personality in the other languages, and our personality alignment makes it easier for such kind of transfer learning.
Regression
----------
We also carried out regression experiments to compare our model with a recent paper [@liu2016language] that tries to perform multilingual personality recognition on the same dataset. They use a character to word to sentence for personality traits (C2W2S4PT) model, which uses a two layer bi-directional RNN model with Gated Recurrent Units (GRU) followed by a fully connected layer to achieve the results.
For our regression task, we used the scores given in the dataset, and did not carry out the median split anymore. We kept our same training procedure as our classification task to train the multilingual and the personality aligned embeddings using MUSE. Our CNN model was also the same except for our last fully connected layer, where we did not have the softmax layer and instead of cross entropy, we used mean-squared error as our objective (loss) function:
$$L(\theta) = \frac{1}{n} \sum_{i=1}^n (y_{t_i} - \hat{y}_{t_i})^2$$
where $y_{t_i}$ is the ground truth personality score of the $t_i$ tweet, and $\hat{y}_{t_i}$ is the predicted score, $\theta$ being the collection of all parameters being trained. As our evaluation metric, we use Root Mean Squared Error (RMSE), which tries to measure the performance via the average error of the model across all users:
Lang Model Extr Agr Cons Emot Openn *Average*
------ ----------------- ---------- ---------- ---------- ---------- ---------- -----------
Lgr-mono 58.2 59.0 57.4 56.9 56.5 57.6
Lgr-multi 62.1 60.0 61.4 60.5 58.2 60.4
ch Lgr-GlobalTrait 64.1 **62.1** 61.3 62.9 59.1 61.9
CNN-mono 60.6 58.4 59.3 58.2 57.5 58.8
CNN-GlobalTrait **64.2** 61.9 **63.0** **62.5** **60.1** **62.3**
\[chinese\]
$$RMSE_{user} = \sqrt{\frac{1}{n} \sum_{i=1}^U (y_{user_i} - \hat{y}_{user_i})^2}$$
where $y_{user_i}$ and $\hat{y}_{user_i}$ are the true and predicted personality trait score of the $i^{th}$ user, and $U$ is the total number of users. Table \[regression\] shows our results compared to the [@liu2016language] model (we only show our results for Spanish and Italian, since they did not use the Dutch data in their paper). We also compare our results to the PAN 2015 participants [@sulea2015automatic] who used character n-gram based *tf-idf* features to train a regression model, and achieved one of the highest results in the competition.
As we can see from our results in table \[regression\], our model gets the best performance in Extraversion trait for both languages, and it performs comparably in the other traits. This could mean that our GlobalTrait personality alignment works better for Extraversion, such that the words in different languages that correspond positively to Extraversion are indeed closer together in the multilingual vector space. However, it does not perform as well in the Openness trait, for example. It is important to take into consideration that [@sulea2015automatic] uses a monolinugual model for each language, and therefore is not expandable to multiple languages. Also, unlike [@liu2016language] our model does not use character based RNN, which enables us to train on languages that do not share the same characters as English. To show this, we carried out separate classification experiments on a Chinese dataset.
Personality Classification on Chinese
-------------------------------------
Personality labeled data is currently rare in languages such as Chinese, which necessitates a model like GlobalTrait, enabling us to use English as additional training data to help us recognize personality in the Chinese test set. We use a Chinese personality labeled dataset called the BIT Speaker Personality Corpus [@zhang2017social], collected and released to us by the Beijing Institute of Technology. It consists of 498 Chinese speech clips, each around 9-13 seconds and labeled with Big Five Personality scores given by five judges. We take a mean of the five scores for each clip, and carry out a median split for our binary classification task on each trait. We use an automatic speech recognition (ASR) system to get the speech transcriptions, and use Jieba segmenter [^3] to tokenize the Chinese text into words, since words in Chinese are not separated by a blank space.
We get pre-trained monolingual embeddings of Chinese from fastText and use MUSE to train multilingual embeddings in English and Chinese. We then use our GlobalTrait alignment method to map the positively correlated Chinese words in each trait to our target language space of English. The average of our 5 fold cross-validation experimental results are shown in table \[chinese\] and we train the same five models that were defined earlier. The results show us that, using our GlobalTrait aligned embeddings undoubtedly improves performance on the Chinese evaluation, which indicates a connection between the English and Chinese data captured via the personality alignment.
Final Discussion and Future Work
================================
We have seen from our results that using the larger data available in English, we are able to improve our multilingual results, when applied to other languages such as Spanish, Italian, Dutch and Chinese. The personality alignment is particularly interesting, as it shows us how the words used to express different personality traits compare and contrast between multiple languages. Since we train a mapping for each trait per language, one word can have five different embeddings, based on the five different trait mappings. For example, our analysis show that the mapped embedding of the word ‘mundo’ (world) in Spanish is closest to the words ‘travel, flights, fresh’, etc. in English, for the Openness trait, while the same word ‘mundo’ for the Extraversion trait gets closest to ‘parties, love, life’, etc.
We plan to explore more in the future to get more insight into such mappings for our GlobalTrait alignment, and also apply our model to other datasets in different languages. It will also be interesting to apply our GlobalTrait aligned embeddings to other models such as the Bi-directional RNN model we saw implemented by [@liu2016language], and other hierarchical attention networks, where our sequential data will be the tweets of a single user, based on the chronological order of the user tweets. Another work would be to include a much larger English personality labeled database, possibly the Facebook data released by myPersonality project [^4], where they collected data of around 154,000 users, with a total of 22 million status updates. Such a large database will help us find a better relation from other languages to English, thereby giving us a more meaningful personality alignment.
Conclusion
==========
We propose the use of personality aligned embeddings, GlobalTrait, which maps the embeddding space from the source language to our high-resource target language (English), thereby enabling us to get better multilingual results. We have shown in our paper that conventional methods that try to use monolingual or even multilingual word similarity for personality recognition may not always give better results, as words corresponding to personality traits might not have similar semantic meaning across multiple languages. Such a method like GlobalTrait can give us a better understanding of how people express personality across different cultures and languages, and therefore enable us to train better language-independent models for multilingual personality recognition.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially funded by CERG \#16214415 of the Hong Kong Research Grants Council and RDC \#1718050-0 of EMOS.AI.
[^1]: https://github.com/facebookresearch/\
fastText
[^2]: https://pan.webis.de/clef15/pan15-web/\
author-profiling.html
[^3]: https://github.com/fxsjy/jieba
[^4]: http://mypersonality.org/
|
---
abstract: |
This paper develops a deterministic model of quantum mechanics as an accumulation-and-threshold process. The model arises from an analogy with signal processing in wireless communications. Complex wavefunctions are interpreted as expressing the amplitude and phase information of a modulated carrier wave. Particle transmission events are modeled as the outcome of a process of signal accumulation that occurs in an extra (non-spacetime) dimension.
Besides giving a natural interpretation of the wavefunction and the Born rule, the model accommodates the collapse of the wave packet and other quantum paradoxes such as EPR and the Ahanorov-Bohm effect. The model also gives a new perspective on the ‘relational’ nature of quantum mechanics: that is, whether the wave function of a physical system is “real" or simply reflects the observer’s partial knowledge of the system. We simulate the model for a 2-slit experiment, and indicate possible deviations of the model’s predictions from conventional quantum mechanics. We also indicate how the theory may be extended to a field theory.
author:
- Chris Thron
- Johnny Watts
date: 'Received: date / Accepted: date'
title: A Signal Processing Model of Quantum Mechanics
---
Wavefunction analogy in wireless communications {#sec:wireless}
===============================================
Several physical systems are characterized by a process of accumulation (of energy, charge, etc.), which leads to an activation event once the accumulation attains a certain threshold. Examples of such systems include lightning and nerve impulse transmission. In many cases the accumulation process is described in terms of a continuous field, while attaining the threshold triggers a discrete event. This simultaneous presence of discrete and continuous aspects is reminiscent of quantum mechanics.
Signal acquisition in wireless digital communications also follows this same general pattern. Consider a mobile receiver moving randomly within a region in which a modulated carrier wave is broadcast. The carrier wave is modulated both in amplitude and phase. In order to detect the broadcasted signal, the receiver accumulates its received signal until a detection threshold is reached. We shall construct a mathematical model of a system, in which that the location where detection occurs obeys a probability distribution reminiscent of the quantum wavefunction. In our model, the wireless signal has the following characteristics:
- The carrier frequency is $\omega$, so that the signal has the general mathematical form $A(\mathbf{r},t)\sin(\omega t+ \phi(\mathbf{r},t))$. Such a signal is commonly represented by its “complex amplitude" $A(\mathbf{r},t) e^{i\phi(\mathbf{r},t)}$.
- The transmitted signal (at the transmitter) has constant complex amplitude over time intervals of length $\delta$, where $\delta >> 2\pi / \omega$ ($\delta$ is called the “chip width” in digital communications ([@Proakis])). The probability distribution of complex amplitudes is Gaussian, so that real and imaginary parts are independent, identically distributed (i.i.d) standard normal random variables with mean 0 and variance 1.
- The ratio of field amplitude to transmitted signal amplitude (denoted by $\psi(\mathbf{r}))$ depends on the field location $\mathbf{r}$, but is independent of time. For mathematical simplicity, we assume that the ratio assumes one of a finite set of complex values $\lbrace \psi_1,\ldots \psi_K \rbrace$; and that within the (finite) region of interest, the sets $\lbrace \mathbf{r} | \psi(\mathbf{r}) = \psi_k~ (k=1,\ldots K) \rbrace$ all have equal area (see Figure 1).
![Wireless communication model[]{data-label="fig:1"}](QSfigure1.png)
The receiver has the following characteristics:
- The receiver consists of an oscillating circuit with natural frequency $\omega$, which is driven by the signal field at the receiver’s current location.
- The receiver moves slowly enough so that its field amplitude does not change significantly over time intervals of length $M\delta$, where $M$ is an integer $>> 1$.
- The receiver moves in such a way that its position uniformly samples the entire region of interest (for instance, by random walk).
- Our mathematical proof (see Appendix) requires that the receiver’s fields over the time intervals $(m_1,m_1+1)M\delta$ and $(m_2,m_2+1)M\delta$ are statistically independent whenever $ m_1 \neq m_2$. Strictly speaking, a receiver moving under random walk will not satisfy this condition: instead, the receiver would have to make uniformly-distributed random jumps at times $M\delta, 2M\delta, \ldots $ A rigorous treatment with random-walk motion would require a more careful analysis.
- The receiver detects the signal when the receiver’s amplitude exceeds a fixed threshold $\Theta$.
Note this simple model does not include any effects from polarization, propagation delay or Doppler phase shifting. With the above assumptions, the field at receiver position r can be expressed (with the aid of complex amplitudes) as follows: $$A(\mathbf{r},t) = \Re[ \psi(\mathbf{r})\cdot \nu_{\lceil t/ \delta \rceil} \cdot e^{i\omega t}] \label{eq:1.1}$$ where
: $\psi(\mathbf{r})$ = (field amplitude at $\mathbf{r}$) / (signal amplitude at transmitter)
: $\nu_{\lceil t/\delta \rceil}$ may be written as $\nu_n=\alpha_n+i\beta_n$, where $\alpha_n,\beta_n$ are i.i.d. standard normal random variables.
: $\lceil x \rceil$ denotes the “ceiling" function, i.e. next largest integer greater than $x$.
We now suppose that the trajectory of the receiver is given by the function $\mathbf{r}(t)$. It follows that the equation for the amplitude $x(t)$ of the driven oscillator is: $$x''+ \omega^2x = A(\mathbf{r}(t),t) \label{eq:1.2}$$ This equation may be expressed as the real part of the complex equation $$z'' + \omega^2z = \psi(\mathbf{r}(t))\cdot \nu_{\lceil t/ \delta \rceil} \cdot e^{i\omega t} \label{eq:1.3}$$ The solution of (\[eq:1.3\]) which satisfies $z(0) = z'(0) = 0$ is $$z(t)= \frac{-i}{2\omega} \int_0^t \psi(\mathbf{r}(u))\cdot \nu_{\lceil u/\delta \rceil} du \cdot e^{i\omega t} + \frac{i}{2 \omega} \int_0^t \psi(\mathbf{r}(u)) \cdot \nu_{\lceil u/ \delta \rceil} \cdot e^{2i \omega u} du \cdot e^{-i\omega t} \label{eq:1.4}$$ According to our assumptions, the factor $e^{2i \omega u}$ in the second integrand oscillates rapidly compared to the rest of the integrand, which causes the second integral to be negligible compared to the first. The model assumptions imply that $\psi(\mathbf{r}(u))$ can be treated as constant over time intervals of length $M\delta$. Using the notation $\Psi_{\lceil u/(M\delta) \rceil} \equiv \psi(\mathbf{r}(u))$, we have: $$z(t) \approx \frac{-i\delta}{2\omega} \sum_{n=1}^{\lceil t/\delta \rceil} \Psi_{\lceil n/M \rceil} \cdot \nu_n \cdot e^{i \omega t} \label{eq:1.5}$$ The oscillation at time $N \delta$ has complex amplitude $(-i\delta/2\omega) \cdot S(N)$, where $$S(N) \equiv \sum_{n=1}^N \Psi_{\lceil n/M \rceil} \cdot \nu_n \label{eq:1.6}$$ According to the model assumptions, each $\Psi_j$ is one of the values $\lbrace\psi_1, \ldots \psi_K \rbrace$. Define the random variable $\kappa(m)$ to be the index $k$ corresponding to random variable $\Psi_m$: that is $$\kappa(m) \equiv \lbrace k | \Psi_m = \psi_k \rbrace \label{eq:1.7}$$ Define $N_\Theta$ as the time index at which $|S(N)|$ first passes a given threshold $\Theta$: $$N_\Theta \equiv \min_N \lbrace N| \sim |S(N)| \ge \Theta \rbrace \label{eq:1.8}$$ Our goal is to evaluate the probability distribution of $\kappa (\ldots )$ corresponding to the first passing of the threshold: $$\Pr\left[\kappa \left( \left \lceil N_{\Theta} / M \right \rceil \right)=k\right] \qquad k =1,\ldots ,K, \label{eq:1.9}$$ This corresponds to the probability distribution of the value of field $\psi$ at the location of detection. It is a well-known fact in signal processing that the rate of accumulation of a random signal is proportional to the signal power, which is in turn proportional to the squared signal amplitude ([@Proakis]). It stands to reason that given a signal that assumes different power levels at different times but with equal probabilities, the chance of the accumulated signal passing a fixed threshold while at a certain power level should be proportional to that power level. This is in fact the case; in the Appendix we prove that $$\Pr\left[\kappa \left( \left \lceil N_{\Theta} /M \right \rceil \right)=k\right] \propto |\psi_k |^2, \qquad k =1,\ldots ,K, \label{eq:1.10}$$ which are exactly the Born probabilities for the spatial wavefunction $\psi$.
Single quantum detection event model {#sec:single}
====================================
In this section, we present a model (based on the model in the previous section) that explains quantum detection probabilities. The model is discretized for conceptual clarity and computational tractability; it is fairly straightforward to see how the model could be taken to a continuous limit.
We emphasize that the probability distribution in the previous model arose from the outcome of a process. The process involved sampling the entire region of potential detection before the actual detection was made. This representative sampling was necessary in order for the field strengths to translate into relative probabilities. We want similar characteristics for the quantum process.
In our previous model the process variable is time; this was appropriate because we were only concerned about the spatial position of the receiver at the moment of detection. However, in quantum mechanics, we are concerned about the location of detection events within space-time. It is impossible to have a process that unfolds in time that at the same time samples all space-time locations before determining the detection location. For this reason, it is necessary to introduce a new process variable, so that the process of signal accumulation takes place in a non-observable dimension which we will call the $a$-dimension.
Our wireless communications model had a physical receiver which moves within the state space of possible detection locations. Quantum detection (say of a particle on a screen) does not appear to have any corresponding receiver. We therefore introduce the notion of a *detectron*, which plays the same role as the receiver in our previous model. The accumulation takes place as the detectron jumps around and uniformly samples the set of all potential detection locations. This “jumping around” takes place in the $a$-dimension; for fixed $a$, the detectron’s space-time location is fixed. We emphasize that the detectron is a mathematical construct, and should not be considered as a physical particle; we will say more about the physical nature of detectrons in Section \[sec:extension\].
We also postulate a carrier wave that oscillates as a function of $a$ (*not* as a function of time) having the mathematical form $\sin\omega a$. The frequency $\omega$ is unknown, and does not correspond to any measurable quantity in space-time. The signal has the following characteristics:
- The signal has constant complex amplitude over $a$-intervals of length $\delta$, where $\delta >> 2\pi/\omega$. The distribution of complex amplitudes is mean-zero Gaussian, with i.i.d. standard normal real and imaginary parts;
- The signal is multiplied by a complex field amplitude $\psi(\mathbf{r},t)$ which is independent of $a$. For mathematical simplicity, we assume that the amplitude assumes one of a finite set of complex values $\lbrace \psi_1, \ldots \psi_K \rbrace$; and that within the space-time confines of the detector, the sets $\lbrace \mathbf{r},t | \psi(\mathbf{r},t) = \psi_k \rbrace~ (k=1,\ldots K)$ all have equal 4-volume.
The detectron has the following characteristics:
- Associated with the detectron is an oscillator (which varies sinusoidally with $a$) with natural frequency $\omega$, which is driven by the signal field at the detectron’s current space-time location;
- The detectron moves in space-time (as a function of $a$) slowly enough so that its field amplitude does not change significantly over a-intervals of length $M\delta$, where $M$ is an integer $>> 1$;
- The detectron moves in such a way that it uniformly samples the space-time extent of the detector.
- The detectron becomes a detection when its oscillator’s amplitude exceeds a fixed threshold $\Theta$.
We can apply this model to the two-slit setup shown in Figure 2. The detectron moves within the space-time confines of the detection screen. The complex field amplitude $\psi(\mathbf{r},t)$ corresponds to the conventional Schrödinger wavefunction at the screen, which in the ray approximation is given by: $$\psi(0,L,z,t) \propto d_1^{-1} e^{i(k'd_1-\omega' t)}+ d_2^{-1} e^{i(k'd_2-\omega' t)} \label{eq:1.11}$$ where $k'$, $\omega'$ are the (observable) wave number and frequency, and $d_1 =(L_1^2+h^2)^{1/2} + (L_2^2+(z-h)^2)^{1/2}$, $d_2 =(L_1^2+h^2)^{1/2} + (L_2^2+(z+h)^2)^{1/2}$.
![Notation for quantum two-slit experiment[]{data-label="fig:2"}](QSfigure2.png){width="3in"}
We simulated this system using MATLAB, with the parameters shown in Table 1 . We only considered a single time slice, and restricted to the $x=0$ portion of the screen. The $z$ locations were discretized into 100 bins; the detectron jumped uniformly randomly from bin to bin every $M=400$ iteration steps. At each iteration, the signal was incremented by $\nu_n \cdot \psi(0,L,z,0)$, where $\nu_n$ are i.i.d. complex random variables with standard normal real and imaginary parts. Each time the detection threshold $\Theta=500$ was reached, a detection was logged and the simulation was restarted. Altogether 100,000 detections were logged. Figure 3 shows the detection probability distribution obtained in the simulation. Agreement is very close with the theoretical result $|\psi(z)|^2$, with $\psi$ given by (\[eq:1.11\]). Note that in the simulation, distribution peaks are slightly lower than theoretical values. If quantum probabilities are indeed the result of such an accumulation process, it is possible that measured probability values may be lower than the conventional quantum prediction. Unfortunately, it is not possible to predict the extent of the lowering from our model, because it depends on details of the accumulation process that are not accessible to measurement.
![Simulation and theory for double-slit experiment with the ray approximation[]{data-label="fig:3"}](QSfigure3.png){width="4in"}
[lll]{} Parameter symbol& Parameter signifigance (distances in wavelengths)&Value\
$h$ & 1/2 the distance between slits & 5\
$L_1$ & Distance from source to slit screen & $1 \times 10^4$\
$L_2$ & Distance from slit screen to detection screen & $1 \times 10^6$\
$M$ & Number from slit screen to detection screen & 400\
$N_{\textrm{detect}}$ & Number of detections & $1 \times 10^5$\
$Z$ & Screen half-width & $1 \times 10^6$\
$z_n$ & Number of bins (discretization) & 100\
$\Theta$ & Detection threshold & 500\
Wavefunction formation via accumulation {#sec:accumulation}
=======================================
The preceding section describes how to obtain quantum-like detection probabilities given that a certain “broadcast field" is present. However, it provides no mechanism for the creation of the broadcast field itself. In this section, we show how the broadcast field can be modeled as the result of a process of accumulation that parallels the signal accumulation described above. The well-known path-integral expression for the propagator $K(\mathbf{r}_1,t_1;\mathbf{r}_2,t_2)$ is given by ([@Kleinert]): $$K(\mathbf{r}_1,t_1;\mathbf{r}_2,t_2)=\int Dq(t) e^{iS[q(t)]/\hbar} \label{eq:1.12}$$ Here $S[q(t)]$ is the action, and the notation $\int Dq(t)$ denotes an equally-weighted summation over all possible paths from $(\mathbf{r}_1,t_1)$ to $(\mathbf{r}_2,t_2)$. This integral may be seen as the outcome of an accumulation process. We may envision a succession of carrier-wave *blips*, where each blip corresponds to a single path $q(t)$ and makes a differential contribution to the field $\psi(\mathbf{r}_2,t_2)$ which is proportional to $e^{iS[q(t)]/\hbar}\psi(\mathbf{r}_1,t_1)$ (as shown in Figure 4). Recall that $\psi(\mathbf{r},t)$ corresponds to the amplitude and phase of a modulated carrier wave; thus $e^{iS[q(t)]/\hbar}$ expresses the influence of a source at $(\mathbf{r}_1,t_1)$ on the amplitude and phase of the wave at $(\mathbf{r}_2,t_2)$ when a blip passes between them.
![Single “blip" with field perturbation at $(\mathbf{r}_2,t_2)$[]{data-label="fig:4"}](QSfigure4.png){width="4in"}
These blips may be associated with detectrons as follows. If $(\mathbf{r}_2,t_2)$ is a possible event detection location, then a path $q(t)$ that passes through $(\mathbf{r}_2,t_2)$ can be identified with a detectron location of $(\mathbf{r}_2,t_2)$. We postulate that simultaneously with causing an incremental change in the field $\psi(\mathbf{r}_2,t_2)$, the blip also increments the overall complex detection signal amplitude by $\psi(\mathbf{r}_2,t_2) \cdot \sum_n \nu_n$, as described in the communication model in the previous section. In this way, the blips perform a dual mathematical function: they both build up the field, and furnish the uniform random sampling of possible detection sites that is required to obtain Born-rule probabilities (as was shown in the previous section). The process of detection signal buildup and detection is shown in Figure \[fig:5\].
![Detection signal buildup and detection[]{data-label="fig:5"}](QSfigure5.png){width="4in"}
To see how this works in practice, we focus specifically on the two-slit experiment shown in Figure 2 in two space dimensions, using non-relativistic electrons of fixed energy $\hbar \omega'$ as particles. We consider the spatial distribution of detections at time $t_2=0$; due to invariance, this distribution will be independent of $t_2$. The detection screen corresponds to spatial locations $(L_2,z)$. We assume the source is configured so that $\psi(s_{+},t)=\psi(s_{-},t)= Ae^{-i\omega' t}$, where $s_{+}$ and $s_{-}$ denote the spatial locations of the two slits ( $(0,h)$ and $(0,-h)$, respectively). We also assume that $A>0$ is large enough so that the signal accumulation process has negligible effect on the size of $A$. We replace $t$ with negative $t$ (since only negative times contribute to detection at $t_2=0$) and obtain an expression the Schrödinger kernel for paths that exit through the upper slit $s_+$: $$K(s_+,-t;\mathbf{r}_2,0) \equiv \kappa(z-h,t) \equiv (B/t) \cdot \exp\left(\frac{i(L_2^2+(z-h)^2))}{2 \hbar t }\right) \label{eq:1.13}$$ where $B$ is a constant of proportionality. Similarly, the kernel for paths that exit through the lower slit is $K(s_-,-t;\mathbf{r}_2,0)=\kappa(z+h,t)$. The theoretical expression for the wavefunction $\psi(z,0)$ is $$\psi(z,0) \propto \int_0^{\infty} \kappa(z-h,t) e^{i\omega't} dt+\int_0^{\infty} \kappa(z+h,t) e^{i \omega't} dt \label{eq:1.14}$$ which evaluates to $$\psi(z,0) \propto K_0 \left(-2i \sqrt{\beta_{-} \omega'}\right) + K_0 \left(-2i \sqrt{\beta_{+} \omega'}\right) \label{eq:1.15}$$ where $$\beta_{\pm}=\frac{L_2^2+(z \pm h)^2)}{2\hbar/m}. \label{eq:1.16}$$ It was computationally intractable for us to simulate the path integrals that give rise to the Schröedinger kernels themselves. Instead, we assumed the kernels and simulated the cumulative effect of different source points. We use the following algorithm to accumulate both the fields at different screen locations and the overall detection signal:
`Initialize:` $\psi(z)=0$ for all detection locations $z; n=0$\
`While` $|\textrm{Signal}|<\Theta$\
$n = n+1$\
`If ` $n$ divides $m$\
Change current detectron location $z_n$;\
Choose random time $t$ (uniformly distributed);\
`End If`\
$\psi(z_n)=\psi(z_n)+ \kappa(z_n \pm h,t)\cdot e^{i\omega't}$;\
$\nu= \textrm{Normal}(0,1)+ i\cdot \textrm{Normal}(0,1)$;\
\
`End While`\
Record detection at location $z_n$\
[lll]{} Parameter symbol& Description &Value (in mks units)\
$h$ & 1/2 the distance between slits & $7.26 \times 10^{-10}$ m\
$L_2$ & Distance from slits to detection screen & $7.26 \times 10^{-9}$ m\
$M$ & Number of iterations between detectron jumps & 25\
$N_{\textrm{detect}}$ & Number of detections & 3600\
$T_{\textrm{min}}$ & Minimum value for random time $t$ & $3.37 \times 10^{-15}$ m sec\
$T_{\textrm{max}}$ & Maximum value for random time $t$ & $3.02 \times 10^{-14}$ sec\
$\omega'$ & Electron frequency & $5 \times 10^{15}$ Hz\
$Z$ & Screen half-width & $1.45 \times 10^{-8}$ m\
$z_n$ & Number of bins (discretization) & 43\
$\Theta$ & Detection threshold & $1\times 10^5$\
Table 2 shows the parameters of the simulation. Figure \[fig:6\] shows the results for 3600 detections. The figure shows counts from 1/2 of the detection screen ($z<0$), which was discretized into 43 bins (with the edge of bin 43 at the center of the screen). The simulation relative counts per bin and the accumulated field $|\psi(z)|^2$ are plotted, as well as theoretical detection probabilities from (\[eq:1.15\]). In general, the theoretical curve lies within error bars of the simulation counts; the simulation counts are consistently slightly higher than theoretical probabilities near the interference pattern nulls.
![Accumulated field and detection frequencies for 2-slit experiment[]{data-label="fig:6"}](QSfigure6.png){width="5in"}
Multiple quantum events {#sec:multiple}
=======================
In the preceding sections we have described how accumulation of the signal associated with the moving detectron eventually leads to detection when the accumulated signal passes a threshold. This description treats a single quantum event in isolation without considering interactions between quantum events. Accordingly, we now postulate that every space-time event corresponds to a detectron, and all detectron signals at a given $a$-instant are *multiplied* before accumulation. The situation thus remains as shown in Figure \[fig:5\], except that the jumps in detection signal magnitude are caused by products of detectron signals. In fact, the effects of this multiplication are already included in the model in Section \[sec:single\] via the random Gaussian factors ${\nu_n}$ which multiply the field $\psi(\mathbf{r},t)$ at the detectron: these fluctuations in the signal amplitude correspond to the variation of other detectron signals.[^1]
Along with the change in detection signal, at each $a$-instant the complex amplitude of the field at each detectron location is incremented as follows. Let $(\mathbf{r}_1,t_1), \ldots (\mathbf{r}_M,t_M)$ be the (time-ordered) sequence detectron space-time locations for the pre-universe at $a$. Then the complex wavefunction amplitude at $(\mathbf{r}_m,t_m)$ is incremented by $\prod_{j<m} e^{i S[q_{j,j+1}(t)]/\hbar}$, where $q_{j,j+1}(t)$ is a path from $(\mathbf{r}_j,t_j)$ to $(\mathbf{r}_{j+1},t_{j+1})$. Given that the paths $q_{j_1,j_1+1}(t)$ and $q_{j_2,j_2+1}(t)$ are statistically independent for $j_1 \neq j_2$, this gives rise to the expression (\[eq:1.12\]) for the propagator.
In summary, each pre-universe shown in Figure \[fig:5\] gives a single contribution to the overall detection signal, that accumulates as the pre-universes unfold in the $a$-dimension. Once the detection signal attains a threshold for a particular value of $a$, the space-time universe that we experience is actualized as a “snapshot” at that $a$-instant.
Extension to a field theory {#sec:extension}
===========================
In the previous sections, we have presented detectrons as moving placeholders for possible detection sites. This picture presumes a division of the universe between fields and detectors. This division of course is not realistic, for the detectors themselves are represented by fields in their own right. We may eliminate this dichotomy by identifying detectrons as field configurations (including the transmitted particle field and the field representing the atoms of the detection screen) that produce contributions to the detection signal. The transition to a field theory may then be accomplished by replacing paths with field configurations. In other words, the pre-universe at each $a$-instant has a particular field configuration that produces an overall contribution to the detection signal. As $a$ increases, the field configurations vary and their contributions to the detection signal are accumulated. At the $a$-instant where the threshold is attained, we arrive at the field configuration corresponding to the observable universe.
Explanations of quantum paradoxes
=================================
*Collapse of the wave packet*: In our model, the wave function is seen as an “actual" field that develops via a process of accumulation. The field is not merely a representation of the observer’s partial knowledge; but on the other hand, the field is not directly observable via physical events in space-time. The apparent “collapse” of the wave packet is due to the fact that the physical universe is only a single “$a$-slice” of the entire process.
*EPR and Bell’s inequality*: Non-local effects pose no problem for this model, for the model itself is inherently non-local. An EPR experiment where the two spin detectors are aligned parallel will always detect anti-aligned particles because both detections correspond to the same blip and are thus perfectly correlated. If the detectors are not parallel, the detection probabilities are still determined according to the quantum expression appropriate for that configuration.
*Aharanov-Bohm Effect*: The Aharanov-Bohm effect shows that fields that are localized in a region where a particle can never be detected can still have an effect on the motion of the particle. This poses no difficulty to our model, because in our model the so-called particle is not an object that travels through space-time but rather a correlated series of detection events.
*Identical particle statistics*: More work is needed to introduce spin into the model. However, in light of our framework it is not surprising that “particles" obey special statistics, because “particles" are not separate objects at all. What we call a “particle" is simply a series of correlated events.
Comparison With Other Interpretations of Quantum Mechanics
==========================================================
We briefly compare our interpretation with other alternative interpretations of quantum mechanics.
*Everett’s “Many-worlds" interpretation*[@DeWitt] requires exponential plethorization of space-times. Our model, which embeds space-time within one additional dimension, possesses a much simpler state space.
*Bohm’s quantum mechanics*[@Bohm] posits that particles such as electrons are able to track along with pilot waves. This appears to imply that these particles have some sort of inner structure. In our model particles are not “objects" at all, so no such complications appear.
*Cramer’s transactional quantum mechanics*[@Cramer] interprets $\psi^*$ as a wave traveling backwards in time, but gives no explanation why $\psi\psi^*$ should be interpreted as a probability. Furthermore, transactional quantum mechanics is not very clear about the order in which “transactions" are determined. In our model, all transactions are determined “simultaneously" (at $a=a_\Theta)$, and a single accumulation process is used to determine all interaction events.
We also remark that none of these alternative models explains why the wavefunction is complex, nor why the squared amplitude is interpreted as a probability.
Possible experimental verification
==================================
If true, then our model indicates that the usual formula for a quantum wavefunction is a statistical approximation, and small deviations from the probabilities predicted by the wave quation should be expected. In particular, in our simulations we consistently found that detection rates near theoretical wavefunction nulls were higher that the conventional quantum predictions. Unfortunately, the size of these effects would depend on aspects of the process that cannot be directly measured.
Conclusions
===========
Our model presents a radically different picture of reality. Traveling “particles" are replaced with series of detection events; as a visual analogy, imagine a series of fireflies in a line that flash successively, giving the impression that a single firefly is moving along the line. Our model replaces temporal causality with atemporal causality; past, present, future are actualized together as the result of a process that occurs in a different dimension. Apparent temporal causality is due to correlation and not causation. The wavefunction is given a physical interpretation as a dynamical field; and the Born rule based on the wavefunction is derived as the natural result of a thresholding process involving this field. The model includes possible differences from conventional quantum mechanics. A lowering of peak probabilities in quantum interference patterns compared to the conventional quantum-mechanical prediction is a possible result of the model.
Appendix: Mathematical derivation of the Born rule
==================================================
In this section, we prove the Born probability rule, $$\Pr\left[\kappa \left( \left \lceil N_{\Theta} / M \right \rceil \right)=k\right] \propto |\psi_k |^2, \qquad k =1,\ldots ,K, \label {eq:1.17}$$ for the wireless communication scenario described in Section \[sec:wireless\]. We will use the notation and definitions of that section.
In order to investigate the dependence of $Pr\left[\kappa \left( \left \lceil N_{\Theta} / M \right \rceil \right)=k\right]$ on $\psi_k$, for each fixed $m'>0$ we will investigate the event $$E_{m',k} \equiv\left[ m'=\left \lceil N_{\Theta} / M \right \rceil \text{~and~} \kappa(m' )=k\right]$$ conditioned on fixed sequences of $(m'-1)$ initial $\psi'$s, corresponding to the $K^{m'-1}$ events $$\begin{aligned}
&F_{m'}(\lbrace k'_{1},\ldots k'_{m'-1}\rbrace) \equiv \lbrace \kappa(m) = k'_m , 1\le m<m',\notag\\
&\qquad{} \textrm{where} ~ 1\le k'_m \le K ~ \textrm{are~ fixed}\rbrace \label{eq:1.18}\end{aligned}$$ We shall show that $$\Pr[E_{m',k} | F_m'(\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)]= C(m',\lbrace k'_{1'},\ldots k'_{m'-1}\rbrace) \cdot|\psi_k |^2, \label{eq:1.19}$$ where $C(\ldots )$ is independent of $k$. The events $ \lbrace F_m'(\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)\rbrace $ for fixed $m'$ partition the sample space and $\Pr[F_{m'}(\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)] = K^{1-m'}$. Furthermore, the events $\left\{E_{m',k} \right\}_{m'=1,2,\ldots }$ partition the event $\left\{\left[\kappa \left( \left \lceil N_{\Theta} / M \right \rceil \right)=k\right]\right\}$, so we obtain $$\begin{aligned}
&\Pr\left[\kappa \left( \left \lceil N_{\Theta} / M \right \rceil \right)=k\right] \notag\\
&= \sum_{m'}\sum_{k'_{1},\ldots ,k'_{m'}}\Pr[E_{m',k} | F_{m'} (\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)]\Pr[F_{m'} (\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)] \label{eq:1.20}\\
&= |\psi_k |^2 \sum_{m'}\sum_{k'_{1},\ldots ,k'_{m'}}C(m',k'_1,\ldots k'_{m'-1}) \cdot K^{1-m'} \notag\\
&\propto |\psi_k |^2 \notag\end{aligned}$$ which is the desired result.
We prove (\[eq:1.19\]) as follows. Conditioned on event $F_m'(\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)$ , we have $$S(N)= \sum_{n=1}^N(\psi_{k'_{\lceil n/M \rceil}} \cdot \nu_n),\qquad (N \le m'M) \label{eq:1.22}$$ were the $\lbrace \nu_n \rbrace$ have i.i.d. standard normal real and imaginary parts (we write this as: $\nu_n \tilde N(0,1) + iN(0,1)$). It follows that $S(N)$ is a random walk in the complex plane with independent (but not identically distributed) steps. We also have $$E[|S(N)|^2 ]=2 \sum_{n=1}^N \left|\psi_{k'_{\lceil n/M \rceil }} \right|^2 \label{eq:1.23}$$ We shall assume that $\Theta >> \max_k|\psi_k |$, so $\Theta$ is much greater than any individual term in $S(N)$. It follows that the distribution of $\lbrace \Theta^{-1}S(N)\rbrace$ for all sample paths $S$ can be approximated as a standard Brownian motion $B(\tau)$, where the time variable $\tau$ is given by $$\tau(N) \equiv E \left[|\Theta^{-1} S(N)|^2 \right] \approx 2\Theta^{-2} \sum_{n=1}^N \left|\psi_{k'_{\lceil n/M \rceil}} \right|^2 . \label{eq:1.24}$$ The sample paths comprised in the event $ \left \lceil N_{\Theta} / M \right \rceil \ge m' $ correspond in the Brownian motion picture to sample paths for which $|B(t)| <1$ for all $t \le \tau((m'-1)M)$. For these sample paths, the distribution of $B(\tau((m'-1)M))$ corresponds to the position probability density for a standard Brownian motion with absorbing barrier at $|z|=1$.
Now there is a close connection between Brownian motion and the heat equation as follows. Let $\beta(z,T)$ be the probability density at time $T$ of a Brownian motion with absorbing barrier at $|z|=1$. Then $\beta(z,T)$ can be found by solving the heat equation with corresponding boundary and initial conditions, which in this case are:
: Boundary conditions: $\beta(z,T) = 0$ for $|z|=1$;
: Initial conditions: $\beta(z,0)=\delta(z)$,
where $\delta$(…) is the Dirac delta function. We do not need the complete solution for $\beta(z,T)$ (which can be expressed in terms of the Bessel functions $\lbrace J_0(\alpha_n r/\Theta)\rbrace n=1,2,\ldots )$, but we will make use of the following properties:
a)
: $\beta$ is radial, so we may write $\beta(z,T)$ as $\beta(r,T)$
b)
: $\beta(r,T)$ is $C^{\infty}$ for $r \le \Theta$ and $T > 0$;
c)
: $\beta_r(1,T) < 0$ for all $T > 0$;
These properties can be mathematically proven, but are also intuitive consequences of the physical interpretation of $\beta(r,T)$ as an evolving temperature distribution within a disk where the boundary is held at zero temperature. In light of property c), at time $T \beta(r,T)$ can be approximated near the boundary $|z| = 1$ as $$\beta(r,T) = (1-r)|\beta_r(1,T)| + O[(1-r)^2]. \label{eq:1.25}$$ It follows from our identification of $\lbrace S(N)/\Theta \rbrace$ with $B(\tau(N))$ that $$dP \left[|S((m'-1) \cdot M)| = r\Theta ~ \textrm{and} ~ \left \lceil N_{\Theta} / M \right \rceil \ge m' \right]= A(1-r) + O(1-r)^2. \label{eq:1.26}$$ Since $\Psi_m=\psi_k$,it follows that the terms $\Psi_{\lceil n/M \rceil} \cdot \nu_n \sim |\psi_k | \cdot [N(0,1)+i \cdot N(0,1)]$ for $n=(m'-1)M+1\ldots m'M$. By rotating in the complex plane we have that $$\begin{aligned}
&\Pr \left[|S(n)|<\Theta,n=(m'-1)M+1 \ldots m'M \right. \notag \\
&\qquad \qquad \left| \Psi_{m'}=\psi_k ~\textrm{and}~ |S((m'-1)M)|=r\Theta \right] \label{eq:1.27}\\
&= \Pr \left[ \left|r+|\psi_k/\Theta| \sum_{j=1 \ldots J} \nu'_j )\right|<1,J=1 \ldots M \right], \notag\end{aligned}$$ where $\nu'_j \sim N(0,1)+i \cdot N(0,1)$.
In the case where $\Theta>> \max_k |\psi_k|$ and $r\approx 1$, we have $$\left| r+|\psi_k/\Theta| \sum_{j=1 \ldots J} \nu'_j \right| = \Re\left[r+|\psi_k/\Theta| \sum_{j=1 \ldots J} \nu'_j\right] + O([|\max_k |\psi_k| / \Theta]^2). \label{eq:1.28}$$ Thus the condition $\left| r+|\psi_k/\Theta| \sum_{j=1 \ldots J} \nu'_j \right|<1$ can be replaced to a very close approximation by the condition $\Re \left[ r+|\psi_k/\Theta| \sum_{j=1 \ldots J} \nu'_j \right]< 1$ (see also Figure 7) and $$\begin{aligned}
&\Pr[|S(n)|<\Theta,n=(m'-1)M+1 \ldots m'M \notag\\
&\qquad{} | \Psi_{m'}=\psi_k \textrm{and}~ |S((m'-1)M)|=r\Theta ] \label{eq:1.29}\\
&\approx \Pr \left[\Re\left[r+|\psi_k/\Theta| \sum_{j=1 \ldots J} \nu'_j \right]<1,J=1 \ldots M\right] \notag\\
&= \Pr\left[\Re\left[\sum_{j=1 \ldots J} \nu'_j\right]<\Theta(1-r)/|\psi_k |,J=1\ldots M\right] \notag\\
&\equiv \phi \left(\frac{\Theta(1-r)}{|\psi_k |} \right). \notag\end{aligned}$$
![Depiction of random sum[]{data-label="fig:7"}](QSfigure7.png){width="2.5in"}
Note that $\phi(x)=1$ for $x\ge J$ since $\Re[|\nu'_j|] \le 1$. Now for sample paths with $\left \lceil N_{\Theta}/M \right \rceil \ge m',$ the event $\left \lceil N_{\Theta}/M \right \rceil = m',$ is the complement of the event $\lbrace |S(n)|< \Theta,n=(m'-1)M+1 \ldots m'M \rbrace$. It follows in summary that $$\begin{aligned}
&\Pr[E_{m',k} | Fm'(\lbrace k'_1,\ldots k'_{m'-1}\rbrace)] \notag\\
&=\Pr\left[ \lceil N_{\Theta} / M \rceil = m' ~\text{and}~ \Psi_m=\psi_k \right. \left| F_{m'} (\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)\right] \notag\\
&= \Pr\left[ \left( \lceil N_{\Theta} / M \rceil = m' | \Psi_m=\psi_k \right) \right. \left| F_{m'} (\lbrace k'_{1},\ldots k'_{m'-1}\rbrace)\right]\cdot \Pr[\Psi_m=\psi_k] \notag\\
&\equiv \int_0^1 \left(1-\phi \left(\frac{\Theta(1-r)}{|\psi_k|}\right)\right) \notag\\
&\qquad \qquad \cdot dP \left[|S((m'-1)\cdot M)|= r\Theta \textrm{~and}~ \lceil N_{\Theta}/M \rceil \ge m' \right] \cdot K^{-1}. \label{eq:1.30}\end{aligned}$$
Note that $(1-\phi(\Theta(1-r)/|\psi_k |))=0$ unless $0< (1-r) < J \cdot |\psi_k |/\Theta$; and since $\Theta >> J \cdot |\psi_k|$, the approximation (\[eq:1.26\]) holds on this range. Our integral becomes $$\approx \int_{1-J|\psi_k |/\Theta}^1 \left(1-\phi\left(\frac{\Theta(1-r)}{|\psi_k |}\right)\right) \cdot K^{-1} A(1-r)\cdot dr. \label{eq:1.31}$$ Changing variable to $x \equiv(1-r) / |\psi_k |$, we have
$$\begin{aligned}
&\approx \int_0^{J/\Theta} (1-\phi(x \Theta)) \cdot |\psi_k |^2 \cdot K^{-1} Ax \cdot dx \label{eq:1.32}\\
&\propto |\psi_k |^2. \notag\end{aligned}$$
Thanks to Walter Wilcox for many helpful suggestions.
J. Proakis, *Digital Communications* $4^{th}$ Edition, McGraw Hill (2000). H. Kleinert, *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets* $4^{th}$ Edition, World Scientific, Singapore (2004). B. S. DeWitt, R. N Graham, eds, *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press (1973). D. Bohm and B. J. Hiley, *The Undivided Universe: An Ontological Interpretation of Quantum Theory*, Routledge & Kegan Paul, London (1993). J.G. Cramer, “An Overview of the Transactional Interpretation of Quantum Mechanics", *International Journal of Theoretical Physics* 27, 227 (1988).
[^1]: The actual distribution arising from a product of random signals will be lognormal rather than normal: we used normal random variables for computational simplicity. The results are not sensitive to the particular shape of the distribution.
|
---
abstract: 'We propose a new mechanism for the prompt emission of gamma-ray burst. In our model electrons are continuously accelerated in the post shock region via plasma turbulence. Using the Monte Carlo technique, we mimic the second-order Fermi acceleration due to plasma turbulence and obtain photon spectra. Since the acceleration balances with the synchrotron cooling, the observed low-energy spectral index is naturally explained. The resultant spectra can be consistent with observed spectra at least below $\sim 1$ MeV. The model also predicts delayed GeV-TeV emission due to inverse Compton and broad pulse profile of optical emission in some cases. Although nontrivial assumptions are required to reproduce MeV-GeV power-law spectra, the model implies the possibility to explain various kinds of luminosity correlations.'
author:
- '<span style="font-variant:small-caps;">Katsuaki Asano and Toshio Terasawa</span>'
title: |
Slow Heating Model of Gamma-Ray Burst:\
Photon Spectrum and Delayed Emission
---
Introduction {#sec:intro}
============
In the widely discussed internal shock scenario [see, e.g., reviews by @pir05; @mes06], the prompt emission of gamma-ray bursts (GRBs) is due to collisions among inhomogeneities within ultrarelativistic outflows, which lead to formation of shocks. The nonthermal photons, whose typical energy $\varepsilon_{\rm p}
\sim$ a few hundred keV, are emitted from shock-accelerated electrons in highly magnetized plasma. However, several open problems for the internal shock model have been pointed out such as the radiative efficiency, various kinds of luminosity correlations, and so on. In this paper, we focus on the two open problems: the energy transfer problem and the low-energy spectral index problem. The standard model postulates that a large fraction of the kinetic energy carried by protons should be efficiently converted into that of relativistic electrons. However, it is apparent that the Coulomb interaction cannot transport the internal energy of heated protons into electrons to achieve energy equipartition, because the timescale of the Coulomb interaction is much longer than the dynamical timescale. While the simple first-order Fermi acceleration at the shock front is assumed to transfer the energy into electrons in the standard model, some unknown plasma processes may play an important role in the energy transfer.
The second problem is in the spectral shape of the prompt emission. The observed spectra of GRBs are well fitted with the conventional Band function [@ban93]; the photon number spectrum $\propto \varepsilon^{\alpha} \exp{[-(2+\alpha)\varepsilon/\varepsilon_{\rm p}]}$ below $(\alpha-\beta)\varepsilon_{\rm p}/(2+\alpha)$, and $\propto \varepsilon^{\beta}$ above it. The typical fitted value of the low energy spectral index is $\alpha=-1.0$ [@pre00], while the standard model predicts that photons from cooled electrons dominate the low energy region below $\varepsilon_{\rm p}$, which leads to $\alpha=-1.5$. To resolve this problem several alternative models, such as the thermal emission from photosphere [@mes00; @iok07 and references therein], have been considered. The Klein-Nishina effect on synchrotron self-Compton (SSC) process, which can affect the low energy synchrotron spectrum, has been discussed frequently [@der01; @bos09; @nak09; @wan09]. Recently, @pee06b have suggested that the decay of magnetic fields (§\[sec:dec\]) may resolve the problem in the low energy spectral index. However, in that model, we have not yet found a reason why the decay timescale should always be comparable to the cooling timescale (§\[sec:model\]). In a particle-in-cell (PIC) simulation for electron-positron plasma [@cha08], the decay timescale is close to the requirement of @pee06b, but long-term evolution of magnetic fields is still controversial, partially because the effect of accelerated particles is not quantitatively unveiled yet [@kes09].
Motivated by these problems, we propose an alternative model for the prompt emission of GRBs (§\[sec:simple\]). In our model electron-heating (second-order Fermi acceleration) due to plasma turbulence continues during photon emission in shocked plasmas [@ghi99], so that the resultant spectral index of low energy photons can be consistent with the observations at least below $\sim 1$ MeV. Since the assumed heating timescale is longer than that in the standard scenario, we call this model the “slow heating model”. After the free energy for the plasma instabilities is dissipated, the magnetic fields may decay and the synchrotron emission will cease. The further possibility to reproduce MeV-GeV power-law spectra within the framework of this model is discussed in §\[sec:var\]. In addition, our model naturally predicts delayed GeV-TeV emission due to inverse Compton (IC) and broad pulse profile of optical emission under certain conditions (§\[sec:del\]). Finally, we summarize the results of our model, and compare them with the observed luminosity correlations in GRBs (§\[sec:sum\]). We should note that our synchrotron model is different from the IC models using the same terminology “slow heating” in @pee06 and @gia08, in which the electrons are heated slowly in a timescale comparable to the shell-expansion timescale [@ghi99; @ste04; @vur09].
Magnetic Field: Generation and Decay {#sec:dec}
====================================
In the standard scenario, the magnetic fields are assumed to be generated/amplified in the region around shocks via plasma instabilities, such as the Weibel instability [e.g., @kaz98; @med99; @sil03; @nis05; @kat07] or the two-stream instability due to high-energy particles accelerated at the shock [e.g., @bel04]. The difference in temperatures of electrons and protons may also arouse some plasma instabilities. Generated magnetic fields efficiently interact with particles, whose Larmor radii are comparable to the typical scale of the turbulence. This situation is definitely different from the ideal magnetohydrodynamic (MHD) approximation. For example, where the electron Larmor radii are finite, the off-diagonal terms in the electron pressure tensor could appear and catalyze magnetic reconnection [e.g., @moz09]. The energy of magnetosonic perturbations can be transferred to resonant particles via the transit-time damping process [e.g., @sch98]. It is natural, therefore, to consider that the generated magnetic fields may decay via interaction with particles after the free energy for instability excitation (anisotropy, inhomogeneity, different temperatures of electrons and ions etc.) is dissipated with a decay timescale $t_{\rm dec}$. Actually, recent PIC simulations of electron-positron plasmas show that magnetic turbulences induced by the Weibel instability decay [e.g., @cha08; @kes09]. For $t >t_{\rm dec}$, electrons stop emitting photons via synchrotron radiation. Therefore, the decay of magnetic fields may suppress the photon emission from cooled electrons, which resolves the problem in the index $\alpha$.
Numerical Model: Standard Case {#sec:model}
==============================
First let us revisit the effects of decay of magnetic fields with the standard manner of the electron injection. In Figure \[fig:test\], changing the decay timescale, we show GRB spectra obtained by numerical calculations with the same code in @asa07 [details will be explained in the following section]. Throughout this paper, all spectra are shown in terms of the observed fluence versus photon energy, assuming a GRB redshift of $z=0.1$. The vertical axes denote $\varepsilon f(\varepsilon)$, so that photon spectra with a spectral index $\alpha$ are plotted as $ \propto \varepsilon^{\alpha+2}$. The model parameters are estimated as follows. The emitting region for a pulse is a homogeneous shell expanding with the Lorentz factor $\Gamma$ at radius $R$ from the central engine. We adopt $l=R/\Gamma$ for the comoving width of the shell, so that the pulse timescale in the observer frame is $\Delta t=R/\Gamma^2 c$ [@sar97]. Here, we choose parameters, $\Gamma=300$, $\Delta t=0.1$ s, which implies $R=2.7 \times 10^{14}$ cm. The energy density of accelerated electrons in the shell $U_{\rm e}=\epsilon_{\rm e} U$ ($U$ is the total energy density of the shocked plasma) is a parameter that can be directly related to the isotropic-equivalent energy of photons from a single pulse $E_{\rm sh}$ (here we adopt $10^{51}$ erg) as $E_{\rm sh} = U_{\rm e}{\cal V}$, where ${\cal V} \equiv 4 \pi R^3/\Gamma$ is the comoving volume.
In the standard scenario, relativistic electrons are injected at the shock front with a power-law energy distribution $\dot{N}(\gamma_{\rm e}) \propto \gamma_{\rm e}^{-p}$ for $\gamma_{\rm e} \ge \gamma_{\rm e,m}$, where $\gamma_{\rm e}$ is the electron Lorentz factor in the plasma rest frame. This scenario requires a sharp low-energy cutoff for the electron injection spectrum; the minimum Lorentz factor $\gamma_{\rm e,m}$ is evaluated in the literature by giving the energy density of electrons $U_{\rm e}=\epsilon_{\rm e} U$ together with the total number density of electrons. Therefore, in the standard scenario, $\gamma_{\rm e,m}$ has been conventionally described by the phenomenological parameter $\epsilon_{\rm e}$, though the energy scale corresponding to $\gamma_{\rm e,m}$ should be derived from physics in relativistic plasmas. Here, instead of $\epsilon_{\rm e}$, we take $\gamma_{\rm e,m}$ to be a parameter, because we do not concern the non-observable parameter $U$.
The photon energy $\varepsilon_{\rm p}$ corresponding to $\gamma_{\rm e,m}$ is given by $$\begin{aligned}
\varepsilon_{\rm p} \simeq
\frac{\hbar e B \gamma_{\rm e,m}^2}{m_{\rm e} c} \Gamma.\end{aligned}$$ The cooling timescale for electrons of $\gamma_{\rm e,m}$ is written as $$\begin{aligned}
t_{\rm c}(\gamma_{\rm e,m})=\frac{6 \pi m_{\rm e} c}{\sigma_{\rm T} B^2 \gamma_{\rm e,m}},
\label{tc}\end{aligned}$$ where $\sigma_{\rm T}$ is the Thomson cross section. With a non-dimensional parameter $\epsilon_B$, the magnetic energy density $U_B \equiv B^2/8 \pi$ is given as $\epsilon_B U= (\epsilon_B/\epsilon_{\rm e}) U_e$. In Figure \[fig:test\], we set $B=$ 3200 G and $\gamma_{\rm e,m}=3900$, which correspond to $\epsilon_B/\epsilon_{\rm e}=0.1$ and $\varepsilon_{\rm p} \sim$ a few hundred keV, respectively.
We numerically follow electron cooling via synchrotron and IC emissions, adopting the Klein-Nishina cross section, and artificially stop the calculation after $t_{\rm dec}$ to mimic the decay of magnetic fields. The effects of $\gamma \gamma$ pair production and synchrotron self-absorption are also taken into account. Since $\gamma_{\rm e,m} \varepsilon_{\rm p}/\Gamma > m_{\rm e} c^2$ in our choice, the Klein-Nishina effect cannot be neglected for IC emission. The dynamical timescale $t_{\rm dyn}=l/c=30$ s is much longer than the cooling time $t_{\rm c}=0.02$ s.
The blue curve with $t_{\rm dec} = 0.01 t_{\rm c}\ll t_{\rm c}$ corresponds to the slow cooling case [@sar98]. In this case, $\varepsilon_{\rm p}$ is determined by the lowest energy of electrons that can cool within the timescale $t_{\rm dec}$. As is well known, the index in the slow cooling case is $\alpha=-(p+1)/2$, which is softer than the typical observed $\alpha$ for our choice of $p=2.5$. If we adopt a very hard injection index $p \simeq 1$, $\alpha$ can be $\sim -1$, but the high-energy index $\beta = -(p+2)/2 \sim -1.5$ contradicts the typical value $\beta < -2$. On the other hand, for the two cases, $t_{\rm dec} = 0.1 t_{\rm c}$ and $ t_{\rm c}$ (green and black curves), the index becomes $\alpha \simeq -1$ below $\varepsilon_{\rm p}$. These cases are what @pee06b suggested to solve the problem of the low-energy spectral index.
It is further seen in Figure \[fig:test\] that for the case of $t_{\rm dec} = 10 t_{\rm c}$ (red curve) the spectrum shows $\alpha \simeq -1.5$. This is the prediction by the standard model as referred in the introduction: electrons injected with $\gamma_{\rm e}=\gamma_{\rm e,m}$ are cooled after $t=t_{\rm c}$, and the low-energy spectrum becomes soft owing to emissions from such cooled electrons. It is noted that this case shows a spectral bump in the GeV band due to IC emission, whose contribution is boosted up by enhancement of low-energy seed photons. We can also see cutoffs above 10 GeV and below 30 eV. They are $\gamma \gamma$-absorption and synchrotron self-absorption, respectively.
The above results indicate that only with the case, $t_{\rm dec} \sim t_{\rm c}(\gamma_{\rm e,m})$, the decaying magnetic field can explain the low-energy index $\alpha$. However, there is no definite physical reason to expect such a matching between $t_{\rm dec} $ and $ t_{\rm c}(\gamma_{\rm e,m})$. The jitter radiation [@med00; @fle06] instead of the synchrotron radiation is worthwhile to consider, because the typical scale of turbulence excited by plasma instabilities can be much shorter than the Larmor radii of radiating electrons. While the typical photon energy in the jitter radiation, which is determined by the coherence scale of the disturbed magnetic field, differs from in the usual synchrotron radiation, the introduction of the jitter radiation does not significantly change the low-energy spectral shape: $\alpha$ remains $\simeq -1.5$ as long as $t_{\rm dec} \gg t_{\rm c}$ (fast cooling).
Slow Heating Model: start {#sec:simple}
=========================
As we mentioned in §\[sec:dec\], turbulent magnetic fields may be generated in the plasmas around shocks. Such turbulent waves may play a role in energy transfer from protons to electrons until the magnetic fields decay. In this section we present our new model, the slow heating model, to resolve the index problem. While the standard picture postulates a prompt acceleration of electrons, whose timescale is much shorter than $t_{\rm c}(\gamma_{\rm e,m})$, our model assumes slower energy transfer from the background plasma to electrons via some unknown plasma instabilities (see Figure \[fig:sch\]).
In order to mimic the energy transfer we consider the second-order Fermi acceleration, even though there may exist not only Alfvén waves but also other types of acceleration mechanisms such as electric fields around ion current channels [@hed04] or coherent wave-particle interactions resulting from parametric instabilities [@mat09], which can also contribute to particle acceleration. When a particle is scattered by a wave or magnetized cloud preserving its energy in the wave (cloud) frame, twice Lorentz transformations give us energy gain due to this collision $\xi \equiv \Delta E/E$ as $$\begin{aligned}
\xi =\gamma_0^2 \left(
1-\beta_0 \mu_1+\beta_0 \mu'_2-\beta_0^2 \mu_1 \mu'_2
\right)-1,\end{aligned}$$ where $\gamma_0=1/\sqrt{1-\beta_0^2}$, $\mu_1$, and $\mu'_2$ are the Lorentz factor of the wave (cloud), cosines of incident angle in the reference frame, and scattering angle in the wave (cloud) frame, respectively. If the wave velocity is non-relativistic ($\beta_0 \ll 1$), the mean energy gain $\overline{\xi} \sim \beta_0^2$ under the assumption of isotropic wave distribution $\overline{\mu_1}=-\beta_0/2$ and isotropic scattering $\overline{\mu'_2}=0$. So the second-order Fermi acceleration is a slower acceleration process than the first-order one ($\overline{\xi} \sim \beta_0$) in non-relativistic cases. However, GRB internal shock is relativistic so that we can expect turbulent magnetic fields with $\beta_0 \sim 1$.
The Fokker-Planck equation for ultrarelativistic particles can be written as $$\begin{aligned}
\frac{\partial N}{\partial t}=\frac{\partial}{\partial E}
D_{EE} \frac{\partial N}{\partial E}-\frac{\partial}{\partial E}
\left[ \left(2 \frac{D_{EE}}{E}-\dot{E}_{\rm cool} \right) N \right],
\label{FP}\end{aligned}$$ where $D_{EE}$ is the energy diffusion coefficient [see, e.g., @liu06]. Defining the mean free time of particles $t_{\rm coll}$, we can write $$\begin{aligned}
D_{EE}=\frac{\overline{\xi} E^2}{2 t_{\rm coll}},\end{aligned}$$ and the acceleration timescale $t_{\rm acc}=t_{\rm coll}/\bar{\xi}$. At present we have no reliable model of relativistic turbulence in GRBs. For reference, let us look in stochastic acceleration in non-relativistic plasma. When we express the diffusion coefficient as $D_{EE} \propto E^n$ ($t_{\rm acc} \propto E^{2-n}$), we obtain $n=m$ for isotropic Alfvén turbulence of spectral energy density per unit wavenumber $W_k \propto k^{-m}$ [see, e.g., @mil89]. The model with $n=2$ is often adopted for small scale MHD turbulences [see, e.g., @liu06]. Another value $n=5/3$ is frequently used for the Kolmogorov turbulence. The strong turbulence limit (Bohm limit), where the mean free path becomes comparable to Larmor radii, corresponds to the case of $n=1$, where the dependence on $m$ disappears. Relativistic shocks in electron-positron plasmas in PIC simulations [@cha08] generate magnetic turbulence with $m=0$ for small $k$ ($kc \ll$ plasma frequency), but $m \simeq 2$ for large $k$. Although we have no definite shape of $D_{EE}$ for GRBs yet, future long-term PIC simulations may reveal the property and evolution of magnetic turbulence.
From eq. (\[FP\]) we may write $\overline{\Delta E^2}=(\overline{\xi}-\overline{\xi}^2) E^2$, so that we assume the probability function of $\xi$ per collision as a Gaussian form, $$\begin{aligned}
P(\xi) =\frac{1}{\sqrt{2 \pi}\sigma} \exp{\left[
-\frac{(\xi-\overline{\xi})^2}{2 \sigma^2}\right]},
\quad \sigma=\sqrt{\overline{\xi}-\overline{\xi}^2}.
\label{prob}\end{aligned}$$ Hereafter, we assume a constant value of $\overline{\xi}=0.1$. Considering synchrotron and IC emissions, the energy loss rate due to radiative cooling is expressed as $$\begin{aligned}
\dot{E}_{\rm cool}=\frac{4}{3} \sigma_{\rm T} c \gamma_{\rm e}^2
U_B \left( 1+K(\gamma_{\rm e}) \frac{U_{\rm ph}}{U_B} \right),
\label{cool}\end{aligned}$$ where $U_{\rm ph}$ and $K(\gamma_{\rm e})$ are the photon energy density and the correction coefficient due to the Klein-Nishina effect, respectively. In the Thomson limit, $K(\gamma_{\rm e})=1$.
We employ the Monte Carlo numerical code of @asa07 to follow the radiative cooling and stochastic energy gain/loss processes according to eqs. (\[cool\]) and (\[prob\]) with a time step, $$\begin{aligned}
\delta t=\min(t_{\rm coll}/30,E/\dot{E}_{\rm cool}/30,t_{\rm dec}/30),\end{aligned}$$ and at $t=t_{\rm dec}$, we artificially halt the calculations to mimic the decay of magnetic fields. For each time step, we judge the occurrence of collision and estimate energy loss due to radiation using random numbers. If a collision occurs, the energy gain/loss due to the collision is counted with evaluated $\xi$.
Since highly disturbed magnetic fields are assumed, it is meaningful to consider the jitter radiation [@med00; @fle06]. But, for simplicity, we consider usual synchrotron radiation, using the synchrotron function [@ryb79]. As for IC emission we numerically estimate the spectral photon emission rate and $K(\gamma_{\rm e})$ by integrating the photon energy distribution given in advance with the Klein-Nishina cross section $\sigma_{\rm KN}$ [@ryb79]. We assume a uniform and isotropic photon field within a shell with width $l=R/\Gamma$ in the shell frame. To obtain photon spectra, the energy distributions of photons and particles are simulated iteratively until the resultant spectrum and presupposed spectrum are identical.
In addition we take into account $\gamma \gamma$ pair production and synchrotron self-absorption. However, these processes are not so important in this paper, so that we omit the explanation of the method to include these effects [see @asa07].
The studies for plasma turbulences in the post shock region by many authors are ongoing now. Although remarkable development is seen in recent PIC simulations and MHD simulations [e.g., @wzha09], a definite picture of shocked plasma is not understood yet. Here, we consider a simple toy model assuming that $D_{EE} \propto E^2$, which means that $t_{\rm coll}$ does not depend on the energy of electrons. Although we take into account IC emission, it is not a main subject to discuss in this paper. In order to concentrate on synchrotron photon spectra, we adopt a stronger magnetic field $B=10^4$ G. The other parameters are the same as those in §\[sec:model\] except for the electron injection. The typical photon energy is expected to be emitted from electrons, whose energy loss rate is balanced with the second-order Fermi acceleration. Therefore, we adjust $t_{\rm coll}$ to make $t_{\rm acc}=t_{\rm c}$ at $\gamma_{\rm e}=\gamma_{\rm typ}=3100$ ($t_{\rm c}(\gamma_{\rm typ}) \equiv t_{\rm c,typ} \sim 2 \times 10^{-3}$ s) that implies the typical photon energy $\varepsilon_{\rm p}(\gamma_{\rm typ}) \sim$ a few hundred keV. The Klein-Nishina effect is important for electrons of $\gamma_{\rm e}=\gamma_{\rm typ}$ even in this case. The number of electrons is roughly adjusted to make $E_{\rm sh}=10^{50}$-$10^{51}$ erg considering the heating rate and $t_{\rm dec}$ (we may not exactly forecast the final photon energy in advance). Below (above) $\gamma_{\rm typ}$ the acceleration timescale is shorter (longer) than the synchrotron cooling timescale. The heating due to turbulence reduces the effective number of electrons below $\gamma_{\rm typ}$ so that the low-energy photon spectrum is expected to be harder than the standard one (the red line in Figure \[fig:test\]) even for $t_{\rm dec} \gg t_{\rm c,typ}$. At $t=0$ electrons are injected with monochromatic energy of $\gamma_{\rm e}=\gamma_{\rm inj}=\gamma_{\rm typ}/10$. We have confirmed that a run with 5000 particle histories is enough to converge. In order to verify that the low-energy photon spectra become hard enough ($\alpha \sim -1$) even for a longer decay timescale than the cooling timescale, we adopt $t_{\rm dec}=30 t_{\rm c,typ}$. The result is shown in Figure \[fig:2nd\].
Photons at the spectral peak are emitted from electrons of $\gamma_{\rm e} \sim \gamma_{\rm typ}$ as anticipated in advance. Since electrons are accelerated immediately, the injection parameter $\gamma_{\rm inj}$ does not affect the resultant spectrum very much. The low-energy spectral index is well approximated as $\sim -1$. The spectral bump at $\sim$ GeV is due to IC emission, whose contribution is small because of high $B$ ($\epsilon_B/\epsilon_{\rm e} \sim 3$) and Klein-Nishina effect. Compared to Figure \[fig:test\], the lower density of target photons ($\geq 2$ MeV) weakens the $\gamma \gamma$ absorption effect on IC emission. The overall shape of the spectrum is different from the Band function. We magnify the spectrum for 10 keV–1 MeV range in Figure \[fig:2nd\](b). In this energy range, the spectrum with artificial errors (10%) does not contradict the Band function very much, even though the model is quite simple.
Slow Heating Model: modification {#sec:var}
================================
While the model spectra in §\[sec:simple\] may be fitted with the Band function below $\sim$ MeV, some GRBs show power-law spectra in the MeV-GeV range with $\beta \sim -2$ [see @abd09 as one of the recent examples]. As numerous simulations for the broadband prompt emission spectrum have shown [e.g., @pee06; @gup07; @asa07; @bos09], the usual simple power-law injection for electrons can easily reproduce the MeV-GeV power-law spectra. In the slow heating model, one of the simplest interpretation to overcome this difficulty is that such power-law spectra are superpositions of multiple components with different $\varepsilon_{\rm p}$. This explanation requires fine adjust of the amplitude of multiple components. Although this interpretation remains viable so far, we search for alternative ideas within the slow heating model in this section.
In §\[sec:simple\], the index $n$ of energy dependence of $D_{EE} (\propto E^n)$ was taken 2. To make the resultant energy spectrum harder, we first test the cases with $n>2$ (note that $t_{\rm acc}<t_{\rm c}$ for $n>3$ in higher energy range). Our simulations show that the spectral shape becomes close to a power-law function as $n$ increases. However, even for an extreme choice of $n=3$ ($t_{\rm acc} \propto E^{-1}$), the resultant index $\beta=-3.2$ is still not hard enough.
Since we consider the decay of the magnetic fields, it is natural to let $D_{EE} = 2 \bar{\xi} E^2/ 2 t_{\rm coll}$ also time (or equivalently distance from the shock front) dependent. Our toy model assumes a power-law shape as $t_{\rm coll} \propto \gamma_{\rm e}^0 t^{\chi}$ with upper and lower limits. Namely, $$\begin{aligned}
t_{\rm coll}=\min\left[\bar{\xi} t_{\rm c,typ},
\max\left\{ t_{\rm min} \left(
\frac{t}{t_{\rm min}} \right)^{\chi},
t_{\rm min} \right\}
\right],\end{aligned}$$ where $t_{\rm min} \equiv \bar{\xi} t_{\rm c,typ}/100$ in our model. In this case, the initial short timescale of acceleration makes $\varepsilon_{\rm p}$ higher, and as the acceleration timescale elongates with time, $\varepsilon_{\rm p}$ will be settled around a few hundred keV. Here, to harden spectra, we adopt $\gamma_{\rm inj}=10 \gamma_{\rm typ}$. Since the acceleration time from $\gamma_{\rm inj}$ to $100 \gamma_{\rm typ}$ is $\sim 10 t_{\rm min} \gg t_{\rm min}$, the highest photon energy may be $\sim 100$ MeV emitted from electrons of $\gamma_{\rm e}=\gamma_{\rm inj}$ (10 GeV photons from electrons of $\gamma_{\rm e}=100
\gamma_{\rm typ}$ may not be produced so much). Assuming $t_{\rm dec}=300 t_{\rm c,typ}$ (other parameters are the same as before), we calculate spectra (see Figure \[fig:Tdep\]).
We can see that the spectra above $\varepsilon_{\rm p}$ for $\chi=0.3$ and $0.4$ are well approximated by power-law functions. Even for $\chi=0.5$, the spectrum from $\sim 500$ keV to $\sim 3$ MeV can be accepted as a power-law function. The peak energy $\varepsilon_{\rm p}$ for $\chi=0.3$ becomes above MeV, because $t_{\rm acc}$ at $t=t_{\rm dec}$ is still shorter than the final acceleration timescale assumed in advance, $\overline{\xi} t_{\rm c,typ}$. For $\chi=0.5$, after $t=10 t_{\rm c,typ} \ll t_{\rm dec}=300 t_{\rm c,typ}$, the timescale $t_{\rm coll}$ attains the upper limit $\overline{\xi} t_{\rm c,typ}$, which makes $\gamma_{\rm e} \sim \gamma_{\rm typ}$. Compared to the timescale of stay around $\gamma_{\rm inj}$, the longer stayover around $\gamma_{\rm typ}$ yields the spectrum bump around a few hundred keV as set in advance. So one may easily understand that the spectra in this model depend on the timescale $t_{\rm dec}$. Figure \[fig:Tdep2\] shows that the spectral bump around $\varepsilon_{\rm p}$ grows as $t_{\rm dec}$ extends. It is apparent that the upper limit for $t_{\rm coll}$, whom we set up to adjust $\varepsilon_{\rm p}$, causes the bumps. If $t_{\rm coll}$ is elongated monotonically, $\varepsilon_{\rm p}$ is determined by $t_{\rm coll}$ at $t=t_{\rm dec}$ as the case of $\chi=0.3$ in Figure \[fig:Tdep\].
The models of $D_{EE}$ in this section are toy models to demonstrate the capability of MeV-GeV power-law spectrum in the slow heating model. In addition, we have assumed the monochromatic injection of electrons. The actual plasma turbulence and electron injection mechanism may be more complicated than the models we tested. The strength of magnetic fields may evolve with $D_{\rm EE}$, or the first-order Fermi acceleration or surfing/drift/wake-field acceleration [see, e.g., @ama07; @hos08] may work as the injection mechanism at shock front. Although we need the nontrivial shape of $D_{EE}$ to reproduce the high-energy power-law spectra, the actual GRB plasmas may provide favorable conditions for MeV-GeV emissions. PIC simulations can be strong tools to verify this scenario. For example, @cha08 show that the energy density of magnetic turbulence in electron-positron plasma evolves as $\propto t^{-2/3}$ initially, then steepens to $\propto t^{-1}$ later. Such results encourage the model we discussed in this section.
Delayed Emission {#sec:del}
================
In the simulations discussed in the previous sections, we have artificially halted the calculations at $t=t_{\rm dec}$ with constant magnetic fields. However, the actual magnetic fields may not disappear suddenly, and we may expect residual magnetic fields at $t>t_{\rm dec}$. In this section, we consider emissions after the decay of magnetic fields. Here, we assume a simple exponential decay and residual magnetic fields as $$\begin{aligned}
B=\max\left( B_0 e^{-t/t_{\rm dec}},B_{\rm min} \right),\end{aligned}$$ where $B_0$ and $B_{\rm min}$ are constants. As for the acceleration timescale, to get rid of the heating effect smoothly, we assume the rapid evolution of $t_{\rm coll}$ as $$\begin{aligned}
t_{\rm coll}=\min\left(\bar{\xi} t_{\rm c,typ} e^{(t/t_{\rm dec})^2},t_{\rm dyn}\right).\end{aligned}$$ We numerically follow electron cooling/heating and photon emission during a period of $t_{\rm sim}=t_{\rm dyn}=l/c \gg t_{\rm dec}$ with $\gamma_{\rm inj}=\gamma_{\rm typ}/10$ and $t_{\rm dec}=30 t_{\rm c,typ}$ (see Figure \[fig:Gdel\]). Two parameter sets are adopted; one is the same as that in §\[sec:simple\] ($\Gamma=300$, $\gamma_{\rm typ}=3100$, $R=2.7 \times 10^{14}$ cm, $B_0=10^4$ G), and another parameter set describes a higher $\Gamma$ case with the same $\varepsilon_{\rm p}$ and $\Delta t$: $\Gamma=800$, $\gamma_{\rm typ}=7800$, $R=1.9 \times 10^{15}$ cm, and $B_0=530$ G. For the final magnetic fields, $B_{\rm min}=1$ G is adopted in both the two cases, though there is no clue to the residual magnetic fields at present. For $t \ll t_{\rm dec}$, the photon emission mechanism is the same as those in §\[sec:simple\]. However, after $t=t_{\rm dec}$, the main cooling mechanism is switched from synchrotron to IC, because the photon density is assumed to be constant within the shell of width $l$.
Comparing the thin dotted line with solid lines in Figure \[fig:Gdel\], it is clearly shown that the residual energy of electrons at $t=t_{\rm dec}$ is emitted via IC emission in GeV-TeV ranges. The typical photon energy due to IC largely depends on $\Gamma$. As shown by the long dashed line in Figure \[fig:Gdel\], $\gamma \gamma$ absorption affects the final spectrum for $\Gamma=300$ above 10 GeV (close to the cases of Figure \[fig:test\]), while it is negligible for $\Gamma=800$ owing to the lower photon density. Therefore, this case indicates a delayed onset of GeV-TeV photons compared to MeV photons with a timescale of $\sim t_{\rm dec}/\Gamma$. On the other hand, the spectral shape in the low-energy region is not altered in this model as seen in Figure \[fig:Gdel\].
Recent GRBs detected by [*Fermi*]{}-LAT tend to show such a delayed onset of high-energy ($>100$ MeV) emission [GRB 080916C, GRB 080825C, etc.; @abd09; @abd09b]. However, it is noted that the broadband spectral shape for GRB 080916C is not well reproduced by our simple toy model. Future observations with [*Fermi*]{} or Cerenkov telescopes will testify the model prospects and yield a clue to the refinement of the model.
Next, let us consider photon emission in longer timescales than the dynamical timescale $t_{\rm dyn}$. In the standard scenario, emission after the dynamical timescale may be negligible because of the fast cooling of electrons. However, in our scenario, the residual electron energy may be released via synchrotron radiation, which can contribute to optical emissions as seen in some GRBs [@ves05; @bla05]. Our results for such cases are shown in Figure \[fig:Odel\], where the parameters are the same as those for $\Gamma=300$ in Figure \[fig:Gdel\], but $t_{\rm sim}=10 t_{\rm dyn}$, $B_{\rm min}=1$, 10 and 30 G, and rapid decay of the photon density $\propto \exp{[-(t/t_{\rm dec})^2]}$. For simplicity, we neglect the effects of adiabatic cooling due to shell expansion.
As seen in Figure \[fig:Odel\], optical synchrotron emissions become more luminous than the fluence of the power-law extrapolation from the X-ray spectra. Most of the optical photons are emitted after $t=t_{\rm dyn}$ in the weak magnetic fields so that broader pulse profile is expected for optical than $\gamma$/X-ray bands. Such longer variability timescales in prompt optical emission is seen in GRB 080319B [@rac08], though the flux is brighter than the extrapolation from the X-ray spectra by 3-4 orders of magnitude (1-2 orders in our results). More luminous optical emissions as compared to X-ray may be possible, if we change $t_{\rm dec}$. Since our objective in this paper is not to reproduce spectra of specific GRBs, we do not further discuss the fraction of optical flux here.
Summary and Discussion {#sec:sum}
======================
Motivated by the energy transfer and low-energy spectrum problems, we propose a new model to reproduce GRB prompt emissions. In this model, electrons are continuously heated via plasma turbulences within a timescale longer than the cooling timescale. The acceleration timescale is assumed to be much longer than that in the standard picture. Emissions from cooled electrons are suppressed so that the low-energy spectral index $\alpha$ is close to the observed value $-1$. At least below MeV, the model spectrum does not contradict the Band function very much. Considering that most of the GRB spectra were obtained in energy ranges below MeV, the model spectra may be consistent with a large fraction of GRBs. In order to explain power-law spectra in MeV-GeV range observed in some GRBs, we need a superposition of multiple components with different $\varepsilon_{\rm p}$, or nontrivial shape and evolution of the diffusion coefficient. Our model, under certain conditions, predicts delayed GeV-TeV emission via IC or delayed optical emission with broad pulse profile via synchrotron. We expect that the accumulation of many GRB observations will verify the characteristics predicted by our model in near future.
Roughly speaking, the energy release in this model is estimated as $\Gamma N (t_{\rm dec}/t_{\rm c,typ})
\gamma_{\rm typ} m_{\rm e} c^2$, where $N$ is the total number of accelerated electrons. Given the total isotropic energy $E_{\rm iso}=4 \pi R^3 U$, the required number of electrons depends on $t_{\rm dec}$. Thus, the fraction of accelerated electrons can be much less than unity in this model, while many authors have frequently assumed that all electrons are accelerated [see @eic05].
One interesting point in our model is that the physical explanation for the spectral peak energy $\varepsilon_{\rm p}$ is clear. The balance between synchrotron cooling and heating provides us the typical electron energy $\gamma_{\rm typ} m_{\rm e} c^2$, from which we can estimate $\varepsilon_{\rm p}$. The spectral peak and low-energy index are reproduced by this mechanism unless $t_{\rm acc} \gg t_{\rm c}$. One remaining problem is why shocked plasma in GRBs always adjust $t_{\rm coll}$ to make $\varepsilon_{\rm p}$ observed range of $\sim 1$ MeV. To explain this we need another assumption; for example, $D_{\rm EE} \propto U_B$, which implies $t_{\rm acc} \propto B^{-2}$. The balance between cooling time $t_{\rm c}
\propto B^{-2}$ and $t_{\rm acc}$ gives us $\gamma_{\rm typ} \propto B^0$. Since we may write the luminosity as $L_{\rm iso}=E_{\rm iso}/(R/c \Gamma^2)= 4 \pi (\epsilon_{\rm e}/\epsilon_B)
U_B R^2 \Gamma^2$, we obtain $$\begin{aligned}
\varepsilon_{\rm p} \propto \Gamma B \gamma_{\rm typ}^2 \propto
(\epsilon_{\rm e}/\epsilon_B)^{-1/2} L_{\rm iso}^{1/2} R^{-1},\end{aligned}$$ where $\Gamma$-dependence disappears. These results are consistent with the Yonetoku relation [$\varepsilon_{\rm p} \propto L_{\rm iso}^{0.5}$; @yon04], except for the factor $R^{-1}$. Of course, the above assumption is not trivial. When we assume $D_{EE} \propto U_B^x$, the correlation becomes $\varepsilon_{\rm p}
\propto %B^{3-2x} \Gamma \propto
L_{\rm iso}^{(3-2x)/2} \Gamma^{2x-2} R^{2x-3}$. If the jitter radiation is applicable, in which the typical photon energy depends on the coherent scale of the field, the Yonetoku relation implies some correlations between the typical scale of turbulence and luminosity. In any case, fundamental studies of the long-term evolution of relativistic plasmas, based on PIC simulations, etc., are indispensable to verify the model and luminosity correlations.
For the [*Fermi*]{}-LAT GRB 080916C, by a process of elimination, @zha09 conclude that this GRB is emitted from a magnetically dominated outflow. In such a case, the dissipation of the bulk kinetic energy may not be due to internal shocks, because the Alfvén velocity is very close to the light speed. In order to produce non-thermal particles, the dissipation of magnetic fields such as magnetic reconnection, etc. should occur in the outflow. The dissipation processes of magnetic fields may generate magnetic turbulences so that we can expect the second-order Fermi acceleration in this case too.
We appreciate the anonymous referee for the useful advice. This work is partially supported by the Grant-in-Aid for Scientific Research, No. 21540259 from the MEXT of Japan.
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---
abstract: '*Shield synthesis* is an approach to enforce a set of safety-critical properties of a reactive system at runtime. A shield monitors the system and corrects any erroneous output values instantaneously. The shield deviates from the given outputs as little as it can and recovers to hand back control to the system as soon as possible. This paper takes its inspiration from a case study on mission planning for unmanned aerial vehicles (UAVs) in which *$k$-stabilizing* shields, which guarantee recovery in a finite time, could not be constructed. We introduce the notion of *admissible* shields, which improves *$k$-stabilizing* shields in two ways: (1) whereas $k$-stabilizing shields take an adversarial view on the system, admissible shields take a collaborative view. That is, if there is no shield that guarantees recovery within $k$ steps regardless of system behavior, the admissible shield will attempt to work with the system to recover as soon as possible. (2) Admissible shields can handle system failures during the recovery phase. In our experimental results we show that for UAVs, we can generate admissible shields, even when $k$-stabilizing shields do not exist.'
author:
- 'Laura Humphrey, Bettina Könighofer, Robert Könighofer, Ufuk Topcu'
bibliography:
- 'main.bib'
title: 'Synthesis of Admissible Shields[^1]'
---
Introduction
============
Technological advances enable the development of increasingly sophisticated systems. Smaller and faster microprocessors, wireless networking, and new theoretical results in areas such as machine learning and intelligent control are paving the way for transformative technologies across a variety of domains – self-driving cars that have the potential to reduce accidents, traffic, energy consumption, and pollution; and unmanned systems that can safely and efficiently operate on land, under water, in the air, and in space. However, in each of these domains, concerns about safety are being raised [@4772749],[@dalamagkidis2011integrating]. Specifically, there is a concern that due to the complexity of such systems, traditional test and evaluation approaches will not be sufficient for finding errors, and alternative approaches such as those provided by formal methods are needed [@Lygeros96].
Formal methods are often used to verify systems at design time, but this is not always realistic. Some systems are simply too large to be fully verified. Others, especially systems that operate in rich dynamic environments or those that continuously adapt their behavior through methods such as machine learning, cannot be fully modeled at design time. Still others may incorporate components that have not been previously verified and cannot be modeled, e.g., proprietary components or pre-compiled code libraries.
Also, even systems that have been fully verified at design time may be subject to external faults such as those introduced by unexpected hardware failures or human inputs. One way to address this issue is to model nondeterministic behaviours (such as faults) as disturbances, and verify the system with respect to this disturbance model [@Mancini14]. However, it is impossible to model all potential unexpected behavior at design time.
An alternative in such cases is to perform *runtime verification* to detect violations of a set of specified properties while a system is executing [@leucker2009]. An extension of this idea is to perform *runtime enforcement* of specified properties, in which violations are not only detected but also overwritten in a way that specified properties are maintained.
A general approach for runtime enforcement of specified properties is *shield synthesis*, in which a shield monitors the system and instantaneously overwrites incorrect outputs. A shield must ensure both *correctness*, i.e., it corrects system outputs such that all properties are always satisfied, as well as *minimum deviation*, i.e., it deviates from system outputs only if necessary and as rarely as possible. The latter requirement is important because the system may satisfy additional noncritical properties that are not considered by the shield but should be retained as much as possible.
Bloem et al. [@BloemKKW15] proposed the notion of $k$-stabilizing shields. Since we are given a safety specification, we can identify wrong outputs, that is, outputs after which the specification is violated (more precisely: after which the environment can force the specification to be violated). A wrong trace is then a trace that ends in a wrong output. The idea of shields is that they may modify the outputs so that the specification always holds, but that such deviations last for at most $k$ consecutive steps after a wrong output. If a second violation happens during the $k$-step recovery phase, the shield enters a mode where it only enforces correctness, but no longer minimizes the deviation. This proposed approach has two limitations with significant impact in practice. (1) The $k$-stabilizing shield synthesis problem is unrealizable for many safety-critical systems, because a finite number of deviations cannot be guaranteed. (2) $k$-stabilizing shields make the assumption that there are no further system errors during the recovery phase.
In this paper, we introduce *admissible* shields, which overcome the two issues of $k$-stabilizing shields. To address shortcoming (1), we guarantee the following: (a) Admissible shields are subgame optimal. That is, for any wrong trace, if there is a finite number $k$ of steps within which the recovery phase can be guaranteed to end, the shield will always achieve this. (b) The shield is *admissible*, that is, if there is no such number $k$, it always picks a deviation that is optimal in that it ends the recovery phase as soon as possible for some possible future inputs. (This is defined in more detail below.) As a result, admissible shields work well in settings in which finite recovery can not be guaranteed, because they guarantee correctness and may well end the recovery period if the system does not pick adversarial outputs. To address shortcoming (2), admissible shields allow arbitrary failure frequencies and in particular failures that arrive during recovery, without losing the ability to recover.
As a second contribution, we demonstrate the use of admissible shields through a case study involving mission planning for an unmanned aerial vehicle (UAV). Manually creating and executing mission plans that meet mission objectives while addressing all possible contingencies is a complex and error-prone task. Therefore, having a shield that changes the mission only if absolutely necessary to enforce certain safety properties has the potential to lower the burden on human operators, and ensures safety during mission execution. We show that admissible shields are applicable in this setting, whereas $k$-stabilizing shields are not.
**Related Work:** Our work builds on synthesis of reactive systems [@Pnueli1989], [@BloemJPPS12] and reactive mission plans [@EhlersKB15] from formal specifications, and our method is related to synthesis of robust [@BloemCGHHJKK14] and error-resilient [@EhlersT14] systems. However, our approach differs in that we do not synthesize an entire system, but rather a shield that considers only a small set of properties and corrects the output of the system at runtime. Li et al. [@LiSSS14] focused on the problem of synthesizing a semi-autonomous controller that expects occasional human intervention for correct operation. A human-in-the-loop controller monitors past and current information about the system and its environment. The controller invokes the human operator only when it is necessary, but as soon as a specification is violated ahead of time, such that the human operator has sufficient time to respond. Similarly, our shields monitor the behavior of systems at run time, and interfere as little as possible. Our work relates to more general work on runtime enforcement of properties [@FalconeFM12], but shield synthesis [@BloemKKW15] is the first appropriative work for reactive systems, since shields act on erroneous system outputs immediately without delay. While [@BloemKKW15] focuses on shield synthesis for systems assumed to make no more than one error every $k$ steps, this work assumes only that systems generally have cooperative behavior with respect to the shield, i.e., the shield ensures a finite number of deviations if the system chooses certain outputs. This is similar in concept to cooperative synthesis as considered in [@BloemEK15], in which a synthesized system has to satisfy a set of properties (called guarantees) only if certain environment assumptions hold. The authors present a synthesis procedure that maximizes the cooperation between system and environment for satisfying both guarantees and assumptions as far as possible. Our construction of admissible shields is related to the construction of fine automata [@KupfermanL06]. A fine automaton is a nondeterministic automaton on finite words that must accept at least one bad prefix of every infinite computation that does not satisfy a safety specification. The key idea behind the construction is based on a subset construction: even though it is impossible to bound the length of a bad prefix, it is possible to bound the number of bad “events” in a run on it.
**Outline:** In what follows, we begin in Section \[sec:ex\] by motivating the need for admissible shields through a case study involving mission planning for a UAV. In Sections \[sec:prelim\], \[sec:def\], \[sec:sol\], we define preliminary concepts, review the general shield synthesis framework, and describe our approach for synthesizing admissible shields. Section \[sec:exp\] provides experimental results, and Section \[sec:conc\] concludes.
Motivating Example {#sec:ex}
==================
In this section, we apply shields on a scenario in which a UAV must maintain certain properties while performing a surveillance mission in a dynamic environment. We show how a shield can be used to enforce the desired properties, where a human operator in conjunction with a lower-level autonomous planner is considered as the reactive system that sends commands to the UAV’s autopilot. We discuss how we would intuitively want a shield to behave in such a situation. We show that the *admissible* shields provide the desired behaviors and address the limitations of $k$-stabilizing shields.
To begin, note that a common UAV control architecture consists of a ground control station that communicates with an autopilot onboard the UAV [@chao_2010]. The ground control station receives and displays updates from the autopilot on the UAV’s state, including position, heading, airspeed, battery level, and sensor imagery. It can also send commands to the UAV’s autopilot, such as waypoints to fly to. A human operator can then use the ground control station to plan waypoint-based routes for the UAV, possibly making modifications during mission execution to respond to events observed through the UAV’s sensors. However, mission planning and execution can be very workload intensive, especially when operators are expected to control multiple UAVs simultaneously [@donmez_2010a]. To address this issue, methods for UAV command and control have been explored in which operators issue high-level commands, and automation carries out low-level execution details. Several errors can occur in this type of human-automation paradigm [@chen_2012]. For instance, in issuing high-level commands to the low-level planner, a human operator might neglect required safety properties due to high workload, fatigue, or an incomplete understanding of exactly how the autonomous planner might execute the command. The planner might also neglect these safety properties either because of software errors or by design. Waypoint commands issued by the operator or planner could also be corrupted by software that translates waypoint messages between ground station and autopilot specific formats or during transmission over the communication link.
As the mission unfolds, waypoint commands will be sent periodically to the autopilot. If a waypoint that violates the properties is received, a shield that monitors the system inputs and can overwrite the waypoint outputs to the autopilot would be able to make corrections to ensure the satisfaction of the desired properties.
Consider the mission map in Fig. \[fig:map\] [@Feng16], which contains three tall buildings (illustrated as blue blocks), over which a UAV should not attempt to fly. It also includes two unattended ground sensors (UGS) that provide data on possible nearby targets, one at location $loc_1$ and one at $loc_x$, as well as two locations of interest, $loc_y$ and $loc_z$. The UAV can monitor $loc_x$, $loc_y$, and $loc_z$ from several nearby vantage points. The map also contains a restricted operating zone (ROZ), illustrated with a red box, in which flight might be dangerous, and the path of a possible adversary that should be avoided (the pink dashed line). Inside the communication relay region (large green area), communication links are highly reliable. Outside this region, communication relies on relay points with lower reliability.
![A map for UAV mission planning.[]{data-label="fig:map"}](figs/map){width="3in"}
Given this scenario, properties of interest include:
1. **Connected waypoints.** \[connected\] The UAV is only allowed to fly to directly connected waypoints.
2. **No communication.** The UAV is not allowed to stay in a location with reduced communication reliability.
3. **Restricted operating zones.** \[ROZ\] The UAV has to leave a ROZ within 2 time steps.
4. **Detected by an adversary.** Locations on the adversary’s path cannot be visited more than once over any window of 3 time steps.\[adversary\]
5. **UGS.** If a UGS reports a possible nearby target, the UAV should visit a respective waypoint within 7 steps (for $UGS_1$ visit $loc_1$, for $UGS_2$ visit $loc_5$, $loc_6$, $loc_7$, or $loc_8$).\[ugs\]
6. **Go home.** Once the UAV’s battery is low, it should return to a designated landing site at $loc_{14}$ within 10 time steps.\[home\]
The task of the shield is to ensure these properties during operation. In this setting, the operator in conjunction with a lower-level planner acts as a reactive system that responds to mission-relevant inputs; in this case data from the UGSs and a signal indicating whether the battery is low. In each step, the next waypoint is sent to the autopilot, which is encoded in a bit representation via outputs $l_4$, $l_3$, $l_2$, and $l_1$. We attach the shield as shown in Fig. \[fig:attach\_shield\]. The shield monitors mission inputs and waypoint outputs, correcting outputs immediately if a violation of the safety properties becomes unavoidable.
We represent each of the properties by a safety automaton, the product of which serves as the shield specification. Fig. \[fig:model\_map\] models the “connected waypoints” property, where each state represents a waypoint with the same number. Edges are labeled by the values of the variables $l_4\dots l_1$. For example, the edge leading from state $s_5$ to state $s_6$ is labeled by $\neg l_4 l_3 l_2 \neg l_1$. For clarity, we drop the labels of edges in Fig. \[fig:model\_map\]. The automaton also includes an error state, which is not shown. Missing edges lead to this error state, denoting forbidden situations.
How should a shield behave in this scenario? If the human operator wants to monitor a location in a ROZ, he or she would like to simply command the UAV to “monitor the location in the ROZ and stay there”, with the planner handling the execution details. If the planner cannot do this while meeting all the safety properties, it is appropriate for the shield to revise its outputs. Yet, the operator would still expect his or her commands to be followed to the maximum extent possible, leaving the ROZ when necessary and returning whenever possible. Thus, the shield should minimize deviations from the operator’s directives as executed by the planner.
![Safety automaton of Property \[connected\] over the map in Fig. \[fig:map\].[]{data-label="fig:model_map"}](figs/attach_shield){width="2in"}
![Safety automaton of Property \[connected\] over the map in Fig. \[fig:map\].[]{data-label="fig:model_map"}](figs/model_map){width="2.3in"}
**Using a $k$-stabilizing shield.** As a concrete example, assume the UAV is currently at $loc_3$, and the operator commands it to monitor $loc_{12}$. The planner then sends commands to fly to $loc_{11}$ then $loc_{12}$, which are accepted by the shield. The planner then sends a command to loiter at $loc_{12}$, but the shield must overwrite it to maintain Property \[ROZ\], which requires the UAV to leave the ROZ within two time steps. The shield instead commands the UAV to go to $loc_{15}$. Suppose the operator then commands the UAV to fly to $loc_{13}$, while the planner is still issuing commands as if the UAV is at $loc_{12}$. The planner then commands the UAV to fly to $loc_{13}$, but since the actual UAV cannot fly from $loc_{15}$ to $loc_{13}$ directly, the shield directs the UAV to $loc_{14}$ on its way to $loc_{15}$. The operator might then respond to a change in the mission and command the UAV fly back to $loc_{12}$, and the shield again deviates from the route assumed by the planner, and directs the UAV back to $loc_{15}$, and so on. Therefore, a single specification violation can lead to an infinitely long deviation between the UAV’s actual position and the UAV’s assumed position. A $k$-stabilizing shield is allowed to deviate from the planner’s commands for at most $k$ consecutive time steps. Hence, no $k$-stabilizing shield exists.
**Using an admissible shield.** Recall the situation in which the shield caused the actual position of the UAV to “fall behind” the position assumed by the planner, so that the next waypoint the planner issues is two or more steps away from the UAV’s current waypoint position. The shield should then implement a best-effort strategy to “synchronize” the UAV’s actual position with that assumed by the planner. Though this cannot be guaranteed, the operator and planner are not adversarial towards the shield, so it will likely be possible to achieve this re-synchronization, for instance when the UAV goes back to a previous waypoint or remains at the current waypoint for several steps. This possibility motivates the concept of an *admissible* shield. Assume that the actual position of the UAV is $loc_{14}$ and the its assumed position is $loc_{13}$. If the operator commands the UAV to loiter at $loc_{13}$, the shield will be able to catch up with the state assumed by the planner and to end the deviation by the next specification violation.
Preliminaries {#sec:prelim}
=============
We denote the Boolean domain by $\B=\{\true,\false\}$, the set of natural numbers by $\N$, and abbreviate $\N\cup\{\infty\}$ by $\N^\infty$. We consider a reactive system with a finite set $\din=\{i_1,\ldots,i_m\}$ of Boolean inputs and a finite set $\dout=\{o_1,\ldots,o_n\}$ of Boolean outputs. The input alphabet is $\dinalph=2^\din$, the output alphabet is $\doutalph=2^O$, and $\dalph=\dinalph \times \doutalph$. The set of finite (infinite) words over $\dalph$ is denoted by $\dalph^*$ ($\dalph^\omega$), and $\dalph^{\infty} = \dalph^* \cup \dalph^\omega$. We will also refer to words as *(execution) traces*. We write $|\dtrace|$ for the length of a trace $\dtrace\in \dalph^*$. For $\dintrace = x_0
x_1 \ldots \in \dinalph^\infty$ and $\douttrace = y_0 y_1 \ldots \in
\doutalph^\infty$, we write $\dintrace || \douttrace$ for the composition $(x_0,y_0) (x_1,y_1) \ldots \in \dalph^\infty$. A set $\lang
\subseteq \dalph^\infty$ of words is called a *language*. We denote the set of all languages as $\langset = 2^{\dalph^\infty}$.
**Reactive Systems.** A *Mealy machine* (reactive system, design) is a 6-tuple $\design = (\states, \init, \\\dinalph, \doutalph, \delta, \lambda)$, where $\states$ is a finite set of states, $\init\in \states$ is the initial state, $\delta:
\states \times \dinalph \rightarrow \states$ is a complete transition function, and $\lambda: \states \times \dinalph \rightarrow \doutalph$ is a complete output function. Given the input trace $\dintrace = x_0
x_1 \ldots \in \dinalph^\infty$, the system $\design$ produces the output trace $\douttrace = \design(\dintrace) = \lambda(q_0, x_0)
\lambda(q_1, x_1) \ldots \in \doutalph^\infty$, where $q_{i+1} =
\delta(q_i, x_i)$ for all $i \ge 0$. The set of words produced by $\design$ is denoted $\lang(\design) = \{\dintrace || \douttrace \in
\dalph^\infty \mid \design(\dintrace) = \douttrace\}$. Let $\design = (\states, \init, \dinalph, \doutalph, \delta, \lambda)$ and $\design' = (\states', \init', \dalph, \doutalph, \delta',
\lambda')$ be reactive systems. A serial composition of $\design$ and $\design'$ is realized if the input and output of $\design$ are fed to $\design'$. We denote such composition as $\design \comp \design'=(\hat{\states}, \hat{\init}, \dinalph, \doutalph, \hat{\delta},
\hat{\lambda})$, where $\hat{\states} = \states \times \states'$, $\hat{\init} = (\init, \init')$, $\hat{\delta}((q,q'),\dinletter) = (\delta(q,\dinletter),
\delta'(q',(\dinletter,\lambda(q,\dinletter))))$, and $\hat{\lambda}((q,q'),\dinletter) =
\lambda'(q',(\dinletter,\lambda(q,\dinletter)))$.
**Specifications.** A *specification* $\spec$ is a set $\lang(\spec) \subseteq
\dalph^\infty$ of allowed traces. $\design$ *realizes* $\spec$, denoted by $\design \models \spec$, iff $\lang(\design) \subseteq \lang(\spec)$. A specification $\spec$ is *realizable* if there exists a design $\design$ that realizes it. A *safety* specification $\spec^s$ is represented by an automaton $\spec^s = (\states, \init, \dalph, \delta, F)$, where $\dalph =
\dinalph\cup\doutalph$, $\delta : \states \times \dalph \rightarrow
\states$, and $F\subseteq \states$ is a set of safe states. The *run* induced by trace $\dtrace = \dletter_0 \dletter_1 \ldots
\in \dalph^\infty$ is the state sequence $\overline{q} = q_0 q_1
\ldots $ such that $q_{i+1} = \delta(q_i, \dletter_i)$; the run is *accepting* if $\forall i\geq 0 \scope q_i \in F$. Trace $\dtrace$ (of a design $\design$) *satisfies* $\spec^s$ if the induced run is accepting. The *language* $\lang(\spec^s)$ is the set of all traces satisfying $\spec^s$.
**Games.** A (2-player, alternating) *game* is a tuple $\game = (\gstates,
\ginit, \dinalph, \doutalph, \delta, \win)$, where $\gstates$ is a finite set of game states, $\ginit \in \gstates$ is the initial state, $\delta: \gstates \times \dinalph \times \doutalph \rightarrow \gstates$ is a complete transition function, and $\win: \gstates^\omega
\rightarrow \B$ is a winning condition. The game is played by two players: the system and the environment. In every state $g\in \gstates$ (starting with $\ginit$), the environment first chooses an input letter $\dinletter \in \dinalph$, and then the system chooses some output letter $\doutletter \in \doutalph$. This defines the next state $g' =
\delta(g,\dinletter, \doutletter)$, and so on. Thus, a (finite or infinite) word over $\Sigma$ results in a (finite or infinite) *play*, a sequence $\overline{g} = g_0 g_1 \ldots$ of game states. A play is *won* by the system iff $\win(\overline{g})$ is $\true$. A *safety game* defines $\win$ via a set $F^s\subseteq \gstates$ of safe states: $\win(g_0 g_1 \ldots)$ is $\true$ iff $\forall i \geq 0
\scope g_i \in F^s$, i.e., if only safe states are visited. Let $\inf(\overline{g})$ denote the states that occur infinitely often in $\overline{g}$. A *game* defines $\win$ via a set $F^b\subseteq \gstates$ of accepting states: $\win(\overline{g})$ is $\true$ iff $\inf(\overline{g}) \cap F^b \neq \emptyset$.
It is easy to transform a safety specification into a safety game such that a trace satisfies the specification iff the corresponding play is won. Given a safety specification $\spec^s$. A finite trace $\dtrace
\in \dalph^*$ is *wrong*, if the corresponding play is not won, i.e., if there is no way for the system to guarantee that any extension of the trace satisfies the specification. An *output* is called *wrong*, if it makes a trace wrong; i.e., given $\spec^s$, a trace $\dtrace \in \dalph^*$ an input $\dinletter \in
\dinalph$, and an output $\doutletter \in \doutalph$, $\doutletter$ is wrong iff $\dtrace$ is not wrong, but $\dtrace \cdot
(\dinletter,\doutletter)$ is.
A deterministic (memoryless) *strategy* for the environment is a function $\rho_e: \gstates \rightarrow \dinalph$. A deterministic (memoryless) *strategy* for the system is a function $\rho_s:
\gstates \times \dinalph \rightarrow \doutalph$. A strategy $\rho_s$ is *winning* for the system, if *for all* strategies $\rho_e$ of the environment the play $\overline{g}$ that is constructed when defining the outputs using $\rho_e$ and $\rho_s$ satisfies $\win(\overline{g})$. The *winning region* $W$ is the set of states from which a winning strategy exists. A strategy is *cooperatively winning* if there *exists* a strategy $\rho_e$ and $\rho_s$, such that the play $\overline{g}$ constructed by $\rho_e$ and $\rho_s$ satisfies $\win(\overline{g})$.
For a game $\mathcal{G}$ with accepting states $F^b$, consider a strategy $\rho_e$ of the environment, a strategy $\rho_s$ of the system, and a state $g\in G$. We set the distance $dist(g, \rho_e, \rho_s)=k$, if the play $\overline{g}$ defined by $\rho_e$ and $\rho_s$ reaches from $g$ an accepting state that occurs infinitely often in $\overline{g}$ in $k$ steps. If no such state is visited, we set $dist(g, \rho_e, \rho_s)=\infty$. Given two strategies $\rho_s$ and $\rho_s'$ of the system, we say that $\rho_s'$ *dominates* $\rho_s$ if: (i) for all $\rho_e$ and all $g\in G$, $dist(g,\rho_e,\rho_s')\leq dist(g,\rho_e,\rho_s)$ , and (ii) there exists $\rho_e$ and $g\in G$ such that $dist(g,\rho_e,\rho_s')< dist(g,\rho_e,\rho_s)$.
A strategy is *admissible* if there is no strategy that dominates it.
Admissible Shields {#sec:def}
==================
Bloem et al. [@BloemKKW15] presented the general framework for shield synthesis. A shield has two main properties: (i) For any design, a shield ensures *correctness* with respect to a specification. (ii) A shield ensures *minimal deviation*. We revisit these properties in Sec. \[sec:def\_shields\]. The definition of minimum deviation is designed to be flexible and different notions of minimum deviation can be realized. $k$-stabilizing shields represent one concrete realization. In Sec. \[sec:def\_admissible\_shields\], we present a new realization of the minimum deviation property resulting in admissible shields.
Definition of Shields {#sec:def_shields}
---------------------
A shield reads the input and output of a design as shown in Fig. \[fig:attach\_shield\]. We then address the two properties, correctness and minimum deviation, to be ensured by a shield.
**The Correctness Property.** With correctness we refer to the property that the shield corrects any design’s output such that a given safety specification is satisfied. Formally, let $\spec$ be a safety specification and $\shield = (\states', \init', \dalph,
\doutalph, \delta', \lambda')$ be a Mealy machine. We say that $\shield$ *ensures correctness* if for any design $\design =
(\states, \init, \dinalph, \doutalph, \delta, \lambda)$, it holds that $(\design \comp \shield) \models \spec$.
Since a shield must work for any design, the synthesis procedure does not need to consider the design’s implementation. This property is crucial because the design may be unknown or too complex to analyze. On the other hand, the design may satisfy additional (noncritical) specifications that are not specified in $\spec$ but should be retained as much as possible.
**The Minimum Deviation Property.** Minimum deviation requires a shield to deviate only if necessary, and as infrequently as possible. To ensure minimum deviation, a shield can only deviate from the design if a property violation becomes unavoidable. Given a safety specification $\spec$, a Mealy machine $\shield$ *does not deviate unnecessarily* if for any design $\design$ and any trace $\dintrace||\douttrace$ that is not wrong, we have that $\shield(\dintrace||\douttrace) = \douttrace$. In other words, if $\design$ does not violate $\spec$, $\shield$ keeps the output of $\design$ intact.
A Mealy machine $\shield$ is a *shield* if $\shield$ ensures correctness and does not deviate unnecessarily.
Ideally, shields end phases of deviations as soon as possible, recovering quickly. This property leaves room for interpretation. Different types of shields differentiate on how this property is realized.
Defining Admissible Shields {#sec:def_admissible_shields}
---------------------------
In this section we define admissible shields using their speed of recovery. We distinguish between two situations. In states of the design in which a finite number $k$ of deviations can be guaranteed, an admissible shield takes an adversarial view on the design: it guarantees recovery within $k$ steps regardless of system behavior, for the smallest $k$ possible. In these states, the strategy of an admissible shield conforms to the strategy of $k$-stabilizing shield. In all other states, admissible shields take a collaborative view: the admissible shield will attempt to work with the design to recover as soon as possible. In particular, an admissible shield plays an admissible strategy, that is, a strategy that cannot be beaten in recovery speed if the design acts cooperatively.
We will now define admissible shields. For failures of the system that are corrected by the shield, we consider four phases:
1. The *innocent phase* consisting of inputs $\dintrace$ and outputs $\douttrace$, in which no failure occurs; i.e., $(\dintrace||\douttrace) \models \spec$.
2. The *misstep phase* consisting of a input $\dinletter$ and a wrong output $\doutletter^f$; i.e., $(\dintrace||\douttrace) \cdot (\dinletter, \doutletter^f) \not\models \spec$.
3. The *deviation phase* consisting of inputs $\dintrace'$ and outputs $\douttrace'$ in which the shield is allowed to deviate, and for a correct output $\doutletter^c$ we have $(\dintrace||\douttrace) \cdot (\dinletter,\doutletter^c)\cdot (\dintrace'||\douttrace') \models \spec$.
4. The *final phase* consisting $\dintrace''$ and $\douttrace''$ in which the shield does not deviate, and $(\dintrace||\douttrace) \cdot(\dinletter,\doutletter^c)\cdot (\dintrace'||\douttrace') \cdot
(\dintrace''||\douttrace'') \models \spec$.
Adversely $k$-stabilizing shields have a deviation phase of length at most $k$.
A shield $\shield$ adversely $k$-stabilizes a trace $\dtrace =
\dintrace || \douttrace \in \dalph^*$, if for any input $\dinletter\in\dinalph$ and any wrong output $\doutletter^f\in\doutalph$, ***for any*** correct output $\doutletter^c\in\doutalph$ and ***for any*** correct trace $\dintrace' || \douttrace' \in \dalph^k$ there exists a trace $\doutletter^\# \douttrace^\# \in\doutalph^{k+1}$ such that for any trace $\dintrace'' || \douttrace'' \in \dalph^{\omega}$ such that $(\dintrace||\douttrace) \cdot(\dinletter,\doutletter^c)\cdot (\dintrace'||\douttrace') \cdot
(\dintrace''||\douttrace'') \models \spec$, we have $$\shield(\dtrace \cdot (\dinletter,\doutletter^f)\cdot (\dintrace' || \douttrace') \cdot
(\dintrace'' || \douttrace'')) = \douttrace \cdot \doutletter^\# \cdot \douttrace^\# \cdot
\douttrace''$$ and $$(\dintrace||\douttrace) \cdot(\dinletter,\doutletter^\#) \cdot (\dintrace'||\douttrace^\#) \cdot
(\dintrace''||\douttrace'') \models \spec.$$
Note that it is not always possible to adversely $k$-stabilize a shield for a given $k$ or even for any $k$.
\[Adversely $k$-Stabilizing Shields [@BloemKKW15]\] A shield $\shield$ is adversely $k$-stabilizing if it adversely $k$-stabilies any finite trace.
An adversely $k$-stabilizing shield guarantees to end deviations after at most $k$ steps and produces a correct trace under the assumption that the failure of the design consists of a transmission error in the sense that the wrong letter is substituted for a correct one. We use the term *adversely* to emphasize that finitely long deviations are guaranteed for *any* future inputs and outputs of the design.
A shield $\shield$ is *adversely subgame optimal* if for any trace $\dtrace \in \dalph^*$, $\shield$ adversely $k-$stabilizes $\dtrace$ and there exists no shield that adversely $l$-stabilizes $\dtrace$ for any $l<k$.
An adversely subgame optimal shield $\shield$ guarantees to deviate in response to an error for at most $k$ time steps, for the smallest $k$ possible.
A shield $\shield$ collaboratively $k$-stabilizes a trace $\dtrace =
\dintrace || \douttrace \in \dalph^*$, if for any input $\dinletter\in\dinalph$ and any wrong output $\doutletter^f\in\doutalph$, ***there exists*** a correct output $\doutletter^c\in\doutalph$, a correct trace $\dintrace' || \douttrace' \in \dalph^k$, and a trace $\doutletter^\# \douttrace^\# \in\doutalph^{k+1}$ such that for any trace $\dintrace'' || \douttrace'' \in \dalph^{\omega}$ such that $(\dintrace||\douttrace) \cdot(\dinletter,\doutletter^c)\cdot (\dintrace'||\douttrace') \cdot
(\dintrace''||\douttrace'') \models \spec$, we have $$\shield(\dtrace \cdot (\dinletter,\doutletter^f)\cdot (\dintrace' || \douttrace') \cdot
(\dintrace'' || \douttrace'')) = \douttrace \cdot \doutletter^\# \cdot \douttrace^\# \cdot
\douttrace''$$ and $$(\dintrace||\douttrace) \cdot(\dinletter,\doutletter^\#) \cdot (\dintrace'||\douttrace^\#) \cdot
(\dintrace''||\douttrace'') \models \spec.$$
A shield $\shield$ collaboratively $k$-stabilizes a trace $\dtrace =
\dintrace || \douttrace \in \dalph^*$, if for any traces $\dintrace' \in \dinalph^k$ and $\douttrace' \in \doutalph^k$ such that the first letter of $\douttrace'$ is wrong, for a trace $\douttrace^{\dagger} \in \doutalph^k$ that differs from $\douttrace'$ in only the first letter, and ***there exists*** a trace $\dintrace'' \in \dinalph^{\omega}$ and a trace $\douttrace'' \in
\doutalph^{\omega}$ such that $(\dintrace||\douttrace) \cdot (\dintrace'||\douttrace^{\dagger}) \cdot
(\dintrace''||\douttrace'') \models \spec$, we have $$\shield(\dtrace \cdot (\dintrace' || \douttrace') \cdot
(\dintrace'' || \douttrace'')) = \douttrace \cdot \douttrace^\# \cdot
\douttrace''$$ and $$(\dintrace||\douttrace) \cdot (\dintrace'||\douttrace^\#) \cdot
(\dintrace''||\douttrace'') \models \spec.$$
\[Collaborative $k$-Stabilizing Shield\] A shield $\shield$ is collaboratively $k$-stabilizing if it collaboratively $k$-stabilizes any finite trace.
A collaborative $k$-stabilizing shield requires that it must be possible to end deviations after $k$ steps, for some future input and output of $\design$. It is not necessary that this is possible for all future behavior of $\design$ allowing infinitely long deviations.
A shield $\shield$ is *collaborative subgame optimal* if for any trace $\dtrace \in \dalph^*$, $\shield$ collaboratively $k-$stabilizes $\dtrace$ and there exists no shield that adversely $l$-stabilizes $\dtrace$ for any $l<k$.
A shield $\shield$ is admissible if for any trace $\dtrace$, whenever there exists a $k$ and a shield $\shield'$ such that $\shield'$ adversely $k$-stabilizes $\dtrace$, then $\shield$ adversely $k$-stabilizes $\dtrace$. If such a $k$ does not exist for trace $\dtrace$, then $\shield$ collaboratively $k$-stabilizes $\dtrace$ for a minimal $k$.
An admissible shield ends deviations whenever possible. In all states of the design $\design$ where a finite number of deviations can be guaranteed, an admissible shield deviates for each violation for at most $k$ steps, for the smallest $k$ possible. In all other states, the shield corrects the output in such a way that there exists design’s inputs and outputs such that deviations end after $l$ steps, for the smallest $l$ possible.
Defining Admissible Shields {#sec:def_admissible_shields}
---------------------------
In this section we define admissible shields using their speed of recovery. We distinguish between two situations. In states of the design in which a finite number $k$ of deviations can be guaranteed, an admissible shield takes an adversarial view on the design, and guarantees recovery within $k$ steps regardless of system behavior. In these states, the strategy of an admissible shield conforms to the strategy of $k$-stabilizing shield. In all other states, admissible shields take a collaborative view: the admissible shield will attempt to work with the design to recover as soon as possible. In particular, an admissible shield plays an admissible strategy, that is, a strategy that cannot be beaten in recovery speed if the design acts cooperatively.
We will now define admissible shields. For failures of the system that are corrected by the shield, we consider three phases: the *innocent phase* consisting of inputs $\dintrace$ and outputs $\douttrace$, in which no failure occurs; the *deviation phase* consisting of $\dintrace'$ and $\douttrace'$ such that the first letter of $\douttrace'$ is wrong, and the *final phase* consisting $\dintrace''$ and $\douttrace''$ in which the shield does not deviate. Adversely $k$-stabilizing shields have a deviation phase of length at most $k$.
A shield $\shield$ adversely $k$-stabilizes a trace $\dtrace =
\dintrace || \douttrace \in \dalph^*$ if for any traces $\dintrace'
\in \dinalph^k$ and $\douttrace' \in \doutalph^k$ such that the first letter of $\douttrace'$ is wrong, there exists a trace $\douttrace^\#
\in \doutalph^k$ such that $$\shield(\dtrace \cdot (\dintrace' || \douttrace')) = \douttrace
\cdot \douttrace^\#$$ and *for any* $\dintrace'' \in \dinalph^{\omega}$ and $\douttrace'' \in
\doutalph^{\omega}$ we have $$\shield(\dtrace \cdot (\dintrace' || \douttrace') \cdot
(\dintrace'' || \douttrace'')) = \douttrace \cdot \douttrace^\# \cdot
\douttrace''$$ and if there is a $\douttrace^{\dagger} \in \doutalph^k$ that differs from $\douttrace'$ in only the first letter and $(\dintrace||\douttrace) \cdot (\dintrace'||\douttrace^{\dagger}) \cdot
(\dintrace''||\douttrace'') \models \spec$, then $(\dintrace||\douttrace) \cdot (\dintrace'||\douttrace^\#) \cdot
(\dintrace''||\douttrace'') \models \spec$.
Note that it is not always possible to adversely $k$-stabilize a shield for a given $k$ or even for any $k$.
\[Adversely $k$-Stabilizing Shields [@BloemKKW15]\] A shield $\shield$ is adversely $k$-stabilizing if it adversely $k$-stabilies any finite trace.
An adversely $k$-stabilizing shield guarantees to end deviations after at most $k$ steps and produces a correct trace under the assumption that the failure of the design consists of a transmission error in the sense that the wrong letter is substituted for a correct one. We use the term *adversely* to emphasize that finitely long deviations are guaranteed for *any* future inputs and outputs of the system.
A shield $\shield$ is *adversely subgame optimal* if for any trace $\dtrace \in \dalph^*$, $\shield$ adversely $k-$stabilizes $\dtrace$ and there exists no shield that adversely $l$-stabilizes $\dtrace$ for any $l<k$.
An adversely subgame optimal shield $\shield$ guarantees to deviate in response to an error for at most k time steps, for the smallest $k$ possible.
A shield $\shield$ is collaboratively $k$-stabilizes a trace $\dtrace
= \dintrace || \douttrace \in \dalph^*$ if for any traces $\dintrace'
\in \dinalph^k$ and $\douttrace' \in \doutalph^k$ such that the first letter of $\douttrace'$ is wrong, there exists a trace $\douttrace^\#
\in \doutalph^k$ such that $$\shield(\dtrace \cdot (\dintrace' || \douttrace')) = \douttrace
\cdot \douttrace^\#$$ and *there exist* $\dintrace'' \in \dinalph^{\omega}$ and $\douttrace'' \in
\doutalph^{\omega}$ such that $$\shield(\dtrace \cdot (\dintrace' || \douttrace') \cdot
(\dintrace'' || \douttrace'')) = \douttrace \cdot \douttrace^\# \cdot
\douttrace''$$ and if there is a $\douttrace^{\dagger} \in \doutalph^k$ that differs from $\douttrace'$ in only the first letter and $(\dintrace||\douttrace) \cdot (\dintrace'||\douttrace^{\dagger}) \cdot
(\dintrace''||\douttrace'') \models \spec$, then $(\dintrace||\douttrace) \cdot (\dintrace'||\douttrace^\#) \cdot
(\dintrace''||\douttrace'') \models \spec$.
\[Collaborative $k$-Stabilizing Shield\] A shield $\shield$ is collaboratively $k$-stabilizing if it collaboratively $k$-stabilizes any finite trace.
A collaborative $k$-stabilizing shield requires that it must be possible to end deviations after $k$ steps, for some future input and output of $\design$. It is not necessary that this is possible for all future behavior of $\design$ allowing infinitely long deviations.
A shield $\shield$ is *collaborative subgame optimal* if for any trace $\dtrace \in \dalph^*$, $\shield$ collaboratively $k-$stabilizes $\dtrace$ and there exists no shield that adversely $l$-stabilizes $\dtrace$ for any $l<k$.
A shield $\shield$ is admissible if for any trace $\dtrace$, whenever there exists a $k$ and a shield $\shield'$ such that $\shield'$ adversely $k$-stabilizes $\dtrace$, then $\shield$ adversely $k$-stabilizes $\dtrace$. If such a $k$ does not exist for trace $\dtrace$, then $\shield$ collaboratively $k$-stabilizes $\dtrace$ for a minimal $k$.
An admissible shield ends deviations whenever possible. In all states of the design $\design$ where a finite number of deviations can be guaranteed, an admissible shield deviates for each violation for at most $k$ steps, for the smallest $k$ possible. In all other states, the shield corrects the output in such a way that there exists design’s inputs and outputs such that deviations end after $l$ steps, for the smallest $l$ possible.
Synthesizing Admissible Shields {#sec:sol}
===============================
The flow of the synthesis procedure is illustrated in Fig. \[fig:c-stab\]. Starting from a safety specification $\spec = (Q, q_{0}, \dalph, \delta,F)$ with $\dalph=\dinalph\times\doutalph$, the admissible shield synthesis procedure consists of five steps.
![Outline of our admissible shield synthesis procedure.[]{data-label="fig:c-stab"}](figs/c_stab_overview){width="85.00000%"}
### Step 1. Constructing the Violation Monitor $\mathcal{U}$. {#sec:violation_monitor}
From $\spec$ we build the automaton $\mathcal{U} = (U, u_0, \dalph, \delta^u)$ to monitor property violations by the design. The goal is to identify the latest point in time from which a specification violation can still be corrected with a deviation by the shield. This constitutes the start of the *recovery* period, in which the shield is allowed to deviate from the design. In this phase the shield monitors the design from all states that the design could reach under the current input and a correct output. A second violation occurs only if the next design’s output is inconsistent with all states that are currently monitored. In case of a second violation, the shield monitors the set of all input-enabled states that are reachable from the current set of monitored states.
The first phase of the construction of the violation monitor $\mathcal{U}$ considers $\spec = (Q, q_{0}, \\ \dalph, \delta,F)$ as a *safety game* and computes its winning region $W\subseteq F$ so that every reactive system $\design\models\spec$ must produce outputs such that the next state of $\spec$ stays in $W$. Only in cases in which the next state of $\spec$ is outside of $W$ the shield is allowed to interfere.
The second phase expands the state space $Q$ to $2^{Q}$ via a subset construction, with the following rationale. If the design makes a mistake (i.e., picks outputs such that $\spec$ enters a state $q\not \in W$), we have to “guess” what the design actually meant to do and we consider all output letters that would have avoided leaving $W$ and continue monitoring the design from all the corresponding successor states in parallel. Thus, $\mathcal{U}$ is essentially a subset construction of $\spec$, where a state $u\in U$ of $\mathcal{U}$ represents a set of states in $\spec$.
The third phase expands the state space of $\mathcal{U}$ by adding a counter $d\in\{0,1,2\}$ and a output variable $z$. Initially $d$ is 0. Whenever a property is violated$d$ is set to 2. If $d>0$, the shield is in the recovery phase and can deviate. If $d=1$ and there is no other violation, $d$ is decremented to 0. In order to decide when to decrement $d$ from 2 to 1, we add an output $z$ to the shield. If this output is set to $\true$ and $d = 2$, then $d$ is set to 1.
The final violation monitor is $\mathcal{U} = (U, u_0, \dalph^u,
\delta^u)$, with the set of states $U = (2^{Q} \times \{0,1,2\})$, the initial state $u_0 = (\{q_0\}, 0)$, the input/output alphabet $\dalph^u = \dinalph \times \doutalph^u$ with $\doutalph^u = \doutalph \cup z$, and the next-state function $\delta^u$ , which obeys the following rules:
1. $\delta^u((u,d), (\dinletter, \doutletter)) =
\bigl(\{q' \kin W \mid \exists q\in u, \doutletter' \in \doutalph^u
\scope \delta(q,(\dinletter,\doutletter')) = q'\}, 2\bigr)$\
if $\forall q \in u \scope
\delta(q,(\dinletter, \doutletter)) \not\in W$, and \[eq:subset\_m\]
2. $\delta^u((u,d), \dletter) \!=\!
\bigl(\{q'\kin W \mid \exists q\kin u \scope\delta(q,\dletter) = q'\},
\textsf{dec}(d)\bigr)$ if $\exists q \kin u \scope \delta(q,\dletter) \kin W$, and $\textsf{dec}(0) = \textsf{dec}(1) = 0$, and if $z$ is $\true$ then $\textsf{dec}(2) = 1$, else $\textsf{dec}(2) = 2$. \[eq:subset\_n\]
Our construction sets $d=2$ whenever the design leaves the winning region, and not when it enters an unsafe state. Hence, the shield $\shield$ can take a remedial action as soon as “the crime is committed”, before the damage is detected, which would have been too late to correct the erroneous outputs of the design.
(s0) at (0,0) [$F$]{}; (s1) at (1.7,0) [$S$]{}; (s2) at (3.4,0) [$T$]{}; (s0) edge\[->\] node\[above\] [$o_2o_1$]{} (s1); (s1) edge\[->\] node\[above\] [$o_2o_1$]{} (s2); (s2) edge\[->, bend left=25\] node\[below\] [$o_2o_1$]{} (s0);
(s0) edge\[loop above\] node\[above\] [$\neg o_2 \neg o_1$]{} (s0);
(s0) at (0,0) [$t_0$]{}; (s1) at (2.7,0) [$t_1$]{}; (s0) edge\[->, bend right=15\] node\[below\] [$\sigma_o\neq \sigma_o'$]{} (s1); (s1) edge\[->, bend right=15\] node\[above\] [$\sigma_o=\sigma_o'$]{} (s0);
(s0) edge\[loop above\] node\[above\] [$\sigma_o=\sigma_o'$]{} (s0); (s1) edge\[loop above\] node\[above\] [$\sigma_o\neq\sigma_o'$]{} (s1);
\[ex:monitor\_U\] We illustrate the construction of $\mathcal{U}$ using the specification $\spec$ from Fig. \[fig:ex\_spec\] over the outputs $o_1$ and $o_2$. (Fig. \[fig:ex\_spec\] represents a safety automaton if we make all missing edges point to an (additional) unsafe state.) The winning region consists of all safe states, i.e., $W = \{F,S,T\}$. The resulting violation monitor is $\mathcal{U}=
(\{\text{F},\text{S},\text{T},\text{FS},\text{ST},\text{FT},\\\text{FST}\}
\times\{0,1,2\}, (\text{F},0), \dalph^u, \delta^u)$. The transition relation $\delta^u$ is illustrated in Table \[fig:ex1\_table\] and lists the next states for all possible present states and outputs. Lightning bolts denote specification violations. The update of counter $d$, which is not included in Table \[fig:ex1\_table\], is as follows: Whenever the design commits a violation $d$ is set to $2$. If no violation exists, $d$ is decremented in the following way: if $d=1$ or $d=0$, $d$ is set to 0. If $d=2$ and $z$ is $\true$, $d$ is set to 1, else $d$ remains 2. In this example, $z$ is set to $\true$, whenever we are positive about the current state of the design (i.e. in $(\{F\},d)$, $(\{S\},d)$, and $(\{T\},d)$).
Let us take a closer look at some entries of Table \[fig:ex1\_table\]. If the current state is $(\{F\},0)$ and we observe the output $\neg o_2 o_1$, a specification violation occurs. We assume that $\design$ meant to give an allowed output, either $o_2 o_1$ or $\neg o_2 \neg o_1$. The shield continues to monitor both $F$ and $S$; thus, $\mathcal{U}$ enters the state $(\{F,S\},2)$. If the next observation is $o_2 o_1$, which is allowed from both possible current states, the possible next states are $S$ and $T$, therefore $\mathcal{U}$ traverses to state $(\{S,T\},2)$. However, if the next observation is again $\neg o_2 o_1$, which is neither allowed in $F$ nor in $S$, we know that a second violation occurs. Therefore, the shield monitors the design from all three states and $\mathcal{U}$ enters the state $(\{F,S,T\},2)$.
$\neg o_1 \neg o_2$ $\neg o_1 o_2$ or $o_1\neg o_2$ $o_1o_2$
--------- --------------------- --------------------------------- ----------
{F} {F} {F,S} {S}
{S} {T} {T} {T}
{T} {F} {F} {F}
{F,S} {F} {F,S,T} {S,T}
{S,T} {F,T} {F,T} {F,T}
{F,T} {F} {F,S,T} {F,S}
{F,S,T} {F} {F,S,T} {F,S,T}
: $\delta^u$ of $\mathcal{U}$ of Example \[ex:monitor\_U\].[]{data-label="fig:ex1_table"}
### Step 2. Constructing the Deviation Monitor $\mathcal{T}$. {#sec:deviation_monitor}
We build $\mathcal{T} = (T, t_0, \doutalph \times \doutalph, \delta^t)$ to monitor deviations between the shield and design outputs. Here, $T = \{t_0, t_1\}$ and $\delta^t(t, (\doutletter,
\doutletter')) = t_0$ iff $\doutletter = \doutletter'$. That is, if there is a deviation in the current time step, then $\mathcal{T}$ will be in $t_1$ in the next time step. Otherwise, it will be in $t_0$. This deviation monitor is shown in Fig. \[fig:dev\_monitor\].
### Step 3. Constructing and Solving the Safety Game $\mathcal{G}^s$.
Given the automata $\mathcal{U}$ and $\mathcal{T}$ and the safety automaton $\spec$, we construct a safety game $\mathcal{G}^s = (G^s, g_0^s, \dinalph^s, \doutalph^s$ $\delta^s, F^s)$, which is the synchronous product of $\mathcal{U}$, $\mathcal{T}$, and $\spec$, such that $G^s= U \times
T \times Q$ is the state space, $g_0^s = (u_0, t_0, q_0)$ is the initial state, $\dinalph^s=\dinalph\times\doutalph$ is the input of the shield, $\doutalph^s=\doutalph\cup \{z\}$ is the output of the shield, $\delta^s$ is the next-state function, and $F^s$ is the set of safe states such that $\delta^s\bigl((u, t, q), (\dinletter, \doutletter),
(\doutletter',z)\bigr) = $ $$\bigl( \delta^u[u,(\dinletter, (\doutletter,z))],
\delta^t[t,(\doutletter, \doutletter')],
\delta[q, (\dinletter, \doutletter')]
\bigr),$$
and $F^s = \{(u, t, q)\in G^s \mid
q \in F \wedge u=(w,0) \rightarrow t=t_0\}$.
We require $q \in F$, which ensures that the output of the shield satisfies $\spec$, and that the shield can only deviate in the recovery period (i.e., if $d=0$, no deviation is allowed). We use standard algorithms for safety games (cf. [@Faella09]) to compute the winning region $W^s$ and the most permissive non-deterministic winning strategy $\rho_s: \gstates \times \dinalph \rightarrow 2^{\doutalph}$ that is not only winning for the system, but also contains all deterministic winning strategies.
### Step 4. Constructing the Game $\mathcal{G}^b$.
Implementing the safety game ensures correctness ($\design \comp \shield \models \spec$) and that the shield $\shield$ keeps the output of the design $\design$ intact, if $\design$ does not violate $\spec$. The shield still has to keep the number of deviations per violation to a minimum. Therefore, we would like the recovery period to be over infinitely often. This can be formalized as a winning condition. We construct the game $\mathcal{G}^b$ by applying the non-deterministic safety strategy $\rho^s$ to the game graph $\mathcal{G}^s$.
Given the safety game $\mathcal{G}^s=(G^s, g_0^s, \dinalph^s, \doutalph^s, \delta^s, F^s)$ with the non-deterministic winning strategy $\rho^s$ and the winning region $W^s$, we construct a game $\mathcal{G}^b=(G^b, g_0^b, \dinalph^s, \\\doutalph^s, \delta^b, F^b)$ such that $G^b=W^s$ is the state space, the initial state $g_0^b=g_0^s$ and the input/output alphabet $\dinalph^b=\dinalph^s$ and $\doutalph^b=\doutalph^s$ remain unchanged, $\delta^b=\delta^s\cap\rho^s$ is the transition function, and $F^b = \{(u, t, q)\in W^s \mid
(u=(w,0) \vee u=(w,1))\}$ is the set of accepting states. A play is winning if $d\leq1$ infinitely often.
### Step 5. Solving the Game $\mathcal{G}^b$. {#sec:solving_buchi}
Most likely, the game $\mathcal{G}^b$ contains reachable states, for which $d\leq1$ cannot be enforced infinitely often. We implement an admissible strategy that enforces to visit $d\leq1$ infinitely often whenever possible. This criterion essentially asks for a strategy that is winning with the help of the design.
The admissible strategy $\rho^b$ for a game $\mathcal{G}^b=(G^b, g_0^b, \dinalph^b, \doutalph^b, \delta^b, F^b)$ can be computed as follows[@Faella09]:
1. Compute the winning region $W^b$ and a winning strategy $\rho_w^b$ for $\mathcal{G}^b$ (cf. [@Mazala01]).
2. Remove all transitions that start in $W^b$ and do not belong to $\rho_w^b$ from $\mathcal{G}^b$. This results in a new game $\mathcal{G}_1^b=(G^b, g_0^b, \dinalph^b, \doutalph^b, \delta_1^b, F^b)$ with $(g,(\dinletter,\doutletter),g')\in\delta_1^b$ if $(g,\dinletter, \doutletter)\in\rho_w^b$ or if $\forall \doutletter' \in \doutalph^b \scope (g,\dinletter, \doutletter')\notin\rho_w^b \wedge (g,(\dinletter,\doutletter),g')\in\delta^b$.
3. In the resulting game $\mathcal{G}_1^b$, compute a cooperatively winning strategy $\rho^b$. In order to compute $\rho^b$, one first has to transform all input variables to output variables. This results in the game $\mathcal{G}_2^b=(G^b, g_0^b, \emptyset, \dinalph^b\times\doutalph^b, \delta_1^b, F^b)$. Afterwards, $\rho^b$ can be computed with the standard algorithm for the winning strategy on $\mathcal{G}_2^b$.
The strategy $\rho^b$ is an admissible strategy of the game $\mathcal{G}^b$, since it is winning and cooperatively winning [@Faella09]. Whenever the game $\mathcal{G}^b$ starts in a state of the winning region $W^b$, any play created by $\rho_w^b$ is winning. Since $\rho^b$ coincides with $\rho_w^b$ in all states of the winning region $W^b$, $\rho^b$ is winning. We know that $\rho^b$ is cooperatively winning in the game $\mathcal{G}_1^b$. A proof that $\rho^b$ is also cooperatively winning in the original game $\mathcal{G}^b$ can be found in [@Faella09].
A shield that implements the admissible strategy $\rho^b$ in the game $\mathcal{G}^b=(G^b, g_0^b, \dinalph^b, \doutalph^b, \delta^b, F^b)$ in a new reactive system $\shield = (G^b, g^b_0, \dinalph^b, \doutalph^b,
\delta', \rho^b)$ with $\delta'(g,\dinletter) = \empty
\delta^b(g,\dinletter,\rho^b(g,\dinletter))$ is an admissible shield.
First, the admissible strategy $\rho^b$ is winning for all winning states of the game $\mathcal{G}^b$. Since winning strategies for games are subgame optimal, a shield that implements $\rho^b$ ends deviations after the smallest number of steps possible, for all states of the design in which a finite number of deviations can be guaranteed. Second, $\rho^b$ is cooperatively winning in the game $\mathcal{G}^b$. Therefore, in all states in which a finite number of deviation cannot be guaranteed, a shield that implements the strategy $\rho^b$ recovers with the help of the design as soon as possible.
The standard algorithm for solving games contains the computation of attractors; the $i$-th attractor for the system contains all states from which the system can “force” a visit of an accepting state in $i$ steps. For all states $g\in G^b$ of the game $\mathcal{G}^b$, the attractor number of $g$ corresponds to the smallest number of steps within which the recovery phase can be guaranteed to end, or can end with the help of the design if a finite number of deviation cannot be guaranteed.
Let $\spec=\{Q, q_{0}, \dalph, \delta, F\}$ be a safety specification and $|Q|$ be the cardinality of the state space of $\spec$. An admissible shield with respect to $\spec$ can be synthesized in $\mathcal{O}(2^{|Q|})$ time, if it exists.
Our safety game $\mathcal{G}^s$ and our game $\mathcal{G}^b$ have at most $m=(2 \cdot 2^{|Q|}+|Q|)\cdot 2 \cdot |Q|$ states and at most $n=m^2$ edges. Safety games can be solved in $\mathcal{O}(m+n)$ time and games in $\mathcal{O}(m\cdot n)$ time [@Mazala01].
Experimental Results {#sec:exp}
====================
We implemented our admissible shield synthesis procedure in Python, which takes a set of safety automata defined in a textual representation as input. The first step in our synthesis procedure is to build the product of all safety automata and construct the violation monitor \[sec:violation\_monitor\]. This step is performed on an explicit representation. For the remaining steps we use Binary Decision Diagrams (BDDs) for symbolic representation. The synthesized shields are encoded in Verilog format. To evaluate the performance of our tool, we constructed three sets of experiments, the basis of which is the safety specification of Fig. \[fig:map\]. This example represents a map with 15 waypoints and the six safety properties \[connected\]-\[home\]. First, we reduced the complexity of the example by only considering 8 out of 15 waypoints. This new example, called *Map$_8$*, consists of the waypoints $loc_1$ to $loc_8$ with their corresponding properties. The second series of experiments, called *Map$_{15}$*, considers the original specification of Fig. \[fig:map\] over all 15 waypoints. The synthesized shields behave as described in Section \[sec:ex\]. The third series of experiments, called *Map$_{31}$*, considers a map with $31$ waypoints, essentially adding a duplicate of the map in Fig. \[fig:map\]. All results are summarized in Table \[tab:res1\] and in Table \[tab:res2\]. For both tables, the first columns list the set of specification automata and the number of states, inputs, and outputs of their product automata. The next column lists the smallest number of steps $l$ under which the shield is able to recover with the help of the design. The last column lists the synthesis time in seconds. All computation times are for a computer with a 2.6 GHz Intel i5-3320M CPU with 8 GB RAM running an 64-bit distribution of Linux. Source code, input files, and instructions to reproduce our experiments are available for download[^2].
Example Property $|Q|$ $|I|$ $|O|$ $l$ Time \[sec\]
------------ ---------- ------- ------- ------- ----- --------------
Map$_8$ 1 9 0 3 3 0.52
1+4 12 0 3 3 1.2
1+5a 46 1 3 4 6.2
1+5b 32 1 3 3 7
1+4+5a 55 1 3 4 17
1+4+5b 36 1 3 3 12
Map$_{31}$ 1 32 0 5 6 122
Map$_{31}$ 1+2 32 0 5 6 143
Map$_{31}$ 1+2+3 34 0 5 6 183
Map$_{31}$ 1+2+3+4 38 0 5 6 238
: Results of $map_8$ and $map_{31}$.[]{data-label="tab:res1"}
Example Property $|Q|$ $|I|$ $|O|$ $l$ Time \[sec\]
------------ ---------- ------- ------- ------- ----- --------------
Map$_{15}$ 1 16 0 4 5 12
1+2 16 0 4 5 14
1+2+3 19 0 4 5 19
1+2+3+4 23 0 4 5 28
1+5a 84 1 4 6 173
1+5a+2 84 1 4 6 205
1+5a+2+3 100 1 4 6 307
1+5b 64 1 4 6 169
1+5b+2 64 1 4 6 195
1+6 115 1 4 7 690
: Results of $map_{15}$.[]{data-label="tab:res2"}
Conclusion {#sec:conc}
==========
We have proposed a new shield synthesis procedure to synthesize *admissible shields*. We have shown that admissible shields overcome the limitations of previously developed $k$-stabilizing shields. We believe our approach and first experimental results over our case study involving UAV mission planning open several directions for future research. At the moment, shields only attend to safety properties and disregard liveness properties. Integrating liveness is therefore a preferable next step. Furthermore, we plan to further develop our prototype tool and apply shields in other domains such as in the distributed settings or for Safe Reinforcement Learning, in which safety constraints must be enforced during the learning processes. We plan to investigate how a shield might be most beneficial in such settings.
[^1]: This work was supported in part by the Austrian Science Fund (FWF) through the research network RiSE (S11406-N23), and by the European Commission through the project IMMORTAL (644905).
[^2]: [
http://www.iaik.tugraz.at/content/research/design\_verification/others/](
http://www.iaik.tugraz.at/content/research/design_verification/others/)
|
---
abstract: 'We will introduce a quantity which measures the singularity of a plurisubharmonic function $\varphi$ relative to another plurisubharmonic function $\psi$, at a point $a$. We denote this quantity by $ \nu_{a,\psi}(\varphi)$. It can be seen as a generalization of the classical Lelong number in a natural way: if $\psi=(n-1)\log| \cdot - a|$ where $n$ is the dimension of the set where $\varphi$ is defined, then $\nu_{a,\psi}(\varphi)$ coincides with the classical Lelong number of $\varphi$ at the point $a$. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form $ \{z: \nu_{z,\psi}(\varphi) \geq c \}$ where $c>0$, are in fact analytic sets, provided that the *weight* $\psi$ satisfies some additional conditions.'
address: |
A Lagerberg :Department of Mathematics\
Chalmers University of Technology and the University of Göteborg\
S-412 96 GÖTEBORG\
SWEDEN,\
author:
- Aron Lagerberg
title: 'A new generalization of the Lelong number.'
---
Introduction
============
In what follows, we let $\Omega$ denote an open subset of $\C^n$, $ \varphi$ a plurisubharmonic function in $\Omega$, and $\psi$ a plurisubharmonic function in $\Cn$. When we are dealing with constants, we often let the same symbol denote different values when the explicit value does not concern us. The object of this paper is to introduce a generalization of the classical Lelong number: The quantity we will consider depends on two plurisubharmonic functions $\varphi, \psi$ and it will be a measurement of the singularity of $\varphi$ relative to $\psi$. Moreover, if we let $\psi(z) = (n-1)\log|z-a|$ we get back the classical Lelong number of $\varphi$ at the point $a$. The main theorem of this paper (Theorem \[analyticity\_thm\]) tells us that this generalized Lelong number satisfies a semi-continuity property of the same type as the classical Lelong number does, namely, its super level-sets define analytic varieties. Also, we investigate what further properties this quantity satisfies. The paper is organized as follows: in this introduction we define the generalized Lelong number and discuss the motivation behind it. In section (\[properties\_and\_\_examples\]) we explore some basic properties and examples of the generalized Lelong number, obtaining as corollaries classical results concerning the classical Lelong number. Section \[analyticity\_section\] concerns the theorem stating that the upper level-sets of the generalized Lelong number defines an analytic set. In section \[approx\_section\] we prove a theorem due to Demailly, which states that one can approximate plurisubharmonic functions well with Bergman functions with respect to a certain weight. Using a different weight we obtain a slightly better estimate. In section \[kiselmans\_section\] we relate our generalized Lelong number to another generalization due to Kisleman (cf. [@Kiselman]).\
\
Let us begin by recalling some relevant definitions. For $r>0$ define \[leldef\] ( ,a,r) := . The function in the nominator can actually be seen to be a convex function of $\log r$ (cf. [@Lelong]). Furthermore, the fraction is increasing in $r$, and so the limit as $r$ tend to 0 exists:
The (classical) Lelong number of $\varphi$ at $a\in\Omega$ is defined as \[lel\_def\_2\] (, a) = \_[r 0]{} ( ,a,r).
As can be seen from the definition, the Lelong number compares the behaviour of $\varphi$ to that of $\log(|z-a|)$, as $z \rightarrow a$. In fact (cf. [@Kiselman]), the following is true: if $\nu (\varphi, a)=\tau$ then, near the point $a$, \[liminfcond\] (z) |z-a| + O(1), and $\tau$ is the best constant possible. Two other ways to represent the classical Lelong number are given by the following equalities: $$\nu (\varphi, a) =
\liminf_{z \rightarrow a} \frac{ \varphi(z)}{\log|z-a|}$$ and $$\nu (\varphi, a)
= \lim_{r \rightarrow 0} \int_{B(a,r)} (d d^c \varphi(z)) \wedge (dd^c \log|z-a|^2)^{n-1}.$$ The first of these equalities is a simple consequence of (\[leldef\]) (cf.[@Rashkovskii]), while the other follows from Stokes’ theorem (cf.[@Demailly]). Two generalizations of the Lelong number, due to Rashkovskii and Demailly respectively, come from exchanging $\log|z-a|$ for a different plurisubharmonic function $\psi$ in the charactarizations of the Lelong number above (cf [@Rashkovskii] and [@Demailly] respectively). To that effect, the relative type of $\varphi$ with respect to a function $\psi$ is given by \[relativetypedef\] \_a (, ) = \_[z a]{} , and Demailly’s generalized Lelong number of $\varphi$ with respect to $\psi$ is given by \_[Demailly]{} (, ) = \_[r 0]{} \_[0: e\^[ - - 2 (n-1) | - a | ]{} L\_[Loc]{}\^1(a) } ]{}.$$ In [@Berndtsson] the following generalization of the classical Lelong number is indicated:
The *generalized Lelong number of* $\varphi$ at $a\in \Omega$ with respect to a plurisubharmonic function $\psi$, is defined as $$\nu_{a,\psi} (\varphi) = \inf{ \{ s>0: \zeta \mapsto e^{ - \frac{2 \varphi (\zeta)}{s} - 2 \psi( \zeta - a ) } \in L_{Loc}^1(a) \} }.$$
Obviously, some condition regarding the integrability of $e^{-2\psi}$ is needed for the definition to provide us with something of interest; for our purpose, it is sufficient to assume that \[integrabilitycondition2\] e\^[-2(1+)]{} L\_[Loc]{}\^1(a), for some $\tau>0$. We single out the following special case of the generalized Lelong number:
For $t \in [0,n)$ we define $$\nu_{a,t} (\varphi) = \inf{ \{ s>0: \zeta \mapsto e^{ - \frac{2 \varphi (\zeta)}{s} - 2 t \log \left| \zeta - a \right| } \in L_{Loc}^1(a) \} },$$ that is, $\nu_{a,t} (\varphi):= \nu_{a,\psi} (\varphi)$, with $\psi=t \log \left| \zeta - a \right|$.
Theorem \[berndtssons\_sats\] shows that $\nu_{a,n-1}$ equals the classical Lelong number, which we sometimes will denote by just $\nu_{a} $. For $t=0$, $\nu_{a,t}=\nu_{a,0}$ equals another well known quantity, the so called *integrability index of $\varphi$ at $a$*. Thus $\nu_{a,t }$ can be regarded as a family of numbers which interpolate between the classical Lelong number and the integrability index of $\varphi$, as $t$ ranges between $0$ and $n-1$. One should put this in context with the following important inequality, due to Skoda (cf. [@Skoda]):
(Skoda’s inequality) For $\varphi \in PSH(\Omega)$, \[skoda\_ineq\] \_[z,0]{}() \_[z,n-1]{}() n \_[z,0]{}().
Later, we will prove the following generalization of Skoda’s inequality: $$\nu_{z,0}(\varphi) \leq \nu_{z,n-1}(\varphi) \leq (n-t)\nu_{z,t}(\varphi) \leq n \nu_{z,0}(\varphi).$$
Observe that when $n=1$ the Lelong number and integrability index of a function coincide. This follows from, for instance, Theorem \[berndtssons\_sats\].
A well known important result concerning classical Lelong numbers, due to Siu (cf. [@Siu]), is the following:
The sets $\{ z \in \Omega : \nu_z( \varphi ) \geq \tau \} $ are analytic subsets of $\Omega$, for $\tau>0$.
In fact both the relative type and Demailly’s generalized Lelong number defined above, enjoy similar analyticity properties, provided that $e^{2 \psi}$ is Hölder continuous. A natural question arises: does an equivalent statement to Siu’s analyticity theorem hold for Berndtsson’s generalized Lelong number? More precisely, are the sets $$\{ z \in \Omega : \nu_{z,\psi}( \varphi) \geq \tau \}$$ analytic for $\tau>0$? In the case where the weight $e^{2 \psi}$ is Hölder continuous the affirmative answer is the content of theorem $\ref{analyticity_thm}$. The main idea of the proof is due to Kiselman (cf.([@Kiselman])) and consists in his technique of “attenuating the singularities of $\varphi$”. However, this is here done in a different manner than in [@Kiselman], following results from [@Berndtsson]. Attenuating the singularities means that we construct a plurisubharmonic function $\Psi$ satisfying the following properties: if the generalized Lelong number of $\varphi$ is large then its classical Lelong number is positive, and if the generalized Lelong number of $\varphi$ is small then its classical Lelong number vanishes. Using this function we can then realize the set $\{ z \in \Omega : \nu_{z,\psi}( \varphi ) \geq \tau \}$ as an intersection of analytic sets, which by basic properties of analytic sets is analytic.\
**Acknowledgements:**[ I would like thank my advisor Bo Berndtsson for introducing me to the topic of this article, for his great knowledge and inspiration, and for his continuous support. ]{}
Properties and examples {#properties_and__examples}
=======================
We begin with listing some properties which the generalized Lelong number satisfies. Let $\varphi, \varphi' \in PSH( \Omega)$, and assume that $\psi$ satisfies (\[integrabilitycondition2\]) . Then
1. For $c>0$ , $\nu_{a,\psi}(c \varphi ) = c \nu_{a,\psi}(\varphi) $.
2. [If $\varphi \leq \varphi'$ on some neighbourhood of $a \in \Omega$, then $\nu_{a,\psi}( \varphi ) \geq \nu_{a,\psi}(\varphi' )$.]{}
3. [ $\nu_{a,\psi}(\varphi + \varphi') \leq \nu_{a,\psi}(\varphi) +\nu_{a,\psi}(\varphi') $.]{}
4. [ $\nu_{a,\psi}(\max(\varphi, \varphi')) \geq \min(\nu_{a,\psi}({\varphi}), \nu_{a,\psi}(\varphi') )$.]{}
5. [ Assume that $\nu_{a,0}( \varphi ) \leq \sigma_a(\varphi, \psi):=\sigma$, where $\sigma_a(,)$ denotes the relative type as defined by (\[relativetypedef\]). Then, $$\nu_{a,\psi}(\varphi) \leq \frac{\nu_{a,0}(\varphi)}{1-\frac{\nu_{a,0}(\varphi)}{\sigma}}.$$ If $\nu_{a,0}( \varphi ) > \sigma_a(\varphi, \psi)$ then $\nu_{a, \psi}(\varphi)=0.$ ]{}
The first properties, (1) and (2), are immediate consequences of the definition.\
(3): Let $s_0>\nu_{a,\psi}(\varphi)$ and $s'_0 > \nu_{a,\psi}(\varphi')$. Put$$p=\frac{s_0+s'_0}{s_0}, q = \frac{s_0+s'_0}{s'_0}$$ so that $ \frac{1}{p} + \frac{1}{q}=1$. Hölder’s inequality with respect to the finite measure $ e^{-2\psi} d \lambda $ on some neighbourhood $U$ of $a$, gives us: $$\int_{U} e^{ - 2 \frac{\varphi + \varphi' }{s_0 + s'_0} - 2 \psi ( \zeta - a ) } \leq
\left( \int_{U} e^{ - 2 \frac{\varphi}{s_0} - 2 \psi ( \zeta - a ) } \right)^{1/p}
\left( \int_{U} e^{ - 2 \frac{\varphi'}{s'_0} - 2 \psi ( \zeta - a ) } \right)^{1/q} < +\infty$$ due to our choice of $s_0$ and $s'_0$. Thus $\nu_{a,\psi}(\varphi + \varphi') \leq s_0 + s'_0$. Since we can choose $s_0$ and $s'_0$ arbitrarily close to $\nu_{a,\psi}(\varphi)$ and $\nu_{a,\psi}(\varphi')$ respectively, we are done.\
(4): Let $s_0<\nu_{a,\psi}(\varphi)$ and $s'_0 < \nu_{a,\psi}(\varphi')$, and let $s=\min(s_0,s'_0)$. Then $$\int_{U} e^{ - 2 \frac{\max(\varphi,\varphi') }{s} - 2 \psi ( \zeta - a ) } = \infty.$$ Thus $\nu_{a,\psi}(\max(\varphi, \varphi')) \geq s$. The statement follows.\
(5): In [@Rashkovskii] it is deduced that $$\varphi(z) \leq \sigma_a(\varphi, \psi) \psi(z) + O(1),$$ as $z \rightarrow a$ ( cf. (\[liminfcond\])). Thus, if we choose $r>0$ small enough, and $ \sigma \leq \nu_{a,0}( \varphi )$, $$\int_{B(a,r)} e^{ - 2 \frac{\varphi(\zeta)}{s} - 2 \psi ( \zeta - a ) }
\leq
C \int_{B(a,r)} e^{ - 2 \frac{\varphi(\zeta)}{s} - 2\frac{\varphi(\zeta)}{\sigma} }$$ which is finite if (remember that $\nu_{a,0}$ denotes the integrability index) $$\frac{1}{s} + \frac{1}{\sigma} < \frac{1}{\nu_{a,0}(\varphi)}
\Leftrightarrow
s > \frac{\nu_{a,0}(\varphi)}{1-\frac{\nu_{a,0}(\varphi)}{\sigma}} .$$ Thus we obtain: $$\nu(\varphi,\psi) \leq \frac{\nu_{a,0}(\varphi)}{1-\frac{\nu_{a,0}(\varphi)}{ \sigma}}.$$ On the other hand, it is evident that if $\sigma > \nu_{a,0}( \varphi )$ then the integral above will always be infinite, whence $\nu(\varphi,\psi)=0.$
We proceed by listing properties the special case $\nu_{z,t \psi}$ satisfies:
\[concavity\_lemma\] For $\varphi, \psi$ plurisubharmonic we have:
1. [ The function $$t \mapsto \frac{1}{\nu_{z,t \psi}(\varphi)}$$ is concave while $$t \mapsto {\nu_{z,t \psi}(\varphi)}$$ is convex. ]{}
2. [ If $\psi $ is such that $e^{-2n\psi} \notin L_{Loc}^1(0)$ then the function $$t \mapsto (n-t) \nu_{z,t \psi}(\varphi)$$ is decreasing. ]{}
3. [ The following inequalities hold: \[generalized\_skoda\_inequality\] \_[z,0]{}() \_[z,n-1]{}() (n-t) \_[z,t]{} () n \_[z,0]{}(). ]{}
(1): Assume $z=0$ and put $$f(t) = 1/ \nu_{0,t}( \varphi) = \sup\{s>0 : e^{-s 2\varphi -2t \psi} \in L_{Loc}^1(0) \}.$$ By the definition of concavity, we need to show that for every $a,b \in [0,n)$ and $\lambda \in (0,1)$ the inequality $$f(\lambda a + (1 - \lambda)b) \geq \lambda f(a) + (1 -\lambda)f(b)$$ holds. Applying Hölder’s inequality once again, with $p=1/\lambda, q=1/(1-\lambda)$, we see that $$\int_0 e^{-2 (\lambda f(a) + (1-\lambda) f(b)) \varphi -2(\lambda a + (1-\lambda) b) \psi} \leq
\big( \int_0 e^{-2 f(a) \varphi -2 a \psi} \big)^{\lambda}
\big(\int_0 e^{-2 f(b) \varphi -2 b \psi} \big)^{1-\lambda},$$ which implies that $ f(\lambda a + (1 - \lambda)b) \geq \lambda f(a) + (1-\lambda) f(b)$ . Thus $f$ is a concave function of $t$. The exact same calculations with $f(t) = \nu_{0,t \psi}( \varphi)$ give convexity of $t \mapsto {\nu_{z,t \psi}(\varphi)}$. Note however that this statement is weaker than saying that $t \mapsto \frac{1}{\nu_{z,t \psi}(\varphi)}$ is concave.\
(2): One can show that if $f(t) \geq 0$ is a concave function with $f(0)=0$, then $t \mapsto f/t$ is decreasing. Since $ t \mapsto 1/\nu_{0,(n-t)\psi}(\varphi)$ is concave by property ($1$) and is equal to $0$ for $t=0$ by the condition on $\psi$, we see that $$\frac{1}{t\nu_{0,(n-t)\psi}(\varphi)}$$ is decreasing. This implies that $t \mapsto (n-t) \nu_{0,t \psi}(\varphi) $ is a decreasing function.\
(3): If we accept Skoda’s inequality ($\ref{skoda_ineq}$), the only new information is the inequality $$\nu_{0,n-1}(\varphi) \leq (n-t)\nu_{0,t}(\varphi) \leq n \nu_{0,0}(\varphi)$$ which follows immediately from property ($2$) with $\psi=\log| \cdot |$, that is, the fact that $t \mapsto (n-t)\nu_{0,t}(\varphi)$ is decreasing in $t$.
The proof of property (1) in Lemma \[concavity\_lemma\] can easily be adapted to show that something stronger holds: the function $$\psi \mapsto \frac{1}{\nu_{z,\psi}(\varphi)}$$ is concave on the set of plurisubharmonic functions $\psi$.
We proceed by calculating two special cases of the generalized Lelong number, which will give us some insight to what it measures.
\[example1\] We calculate $ \nu_{0,t}(\varphi)$ where $ \varphi(z_1,..z_n)=\frac{1}{2}\log (z_1\bar{z_1}+...z_k \bar{z_k})$ where $k$ lies between 1 and $n$. Thus we want to decide for which $s>0$ the following integral goes from being finite into being infinite: $$\int_{\Delta} \frac{d \lambda}{|z_1\bar{z_1}+...z_k \bar{z_k}|^{1/s} |z|^{2t}},$$ where $\Delta$ is some arbitrarily small polydisc containing the origin. In this integral we put $z'' = (z_{k+1},..z_n)$ and introduce polar coordinates with respect to $z'=(z_1,..z_k)$ to obtain that it is equal to $$\label{specint1}
C \int_{\Delta ''} \int_0^1 \frac{R^{2k-1-2/s} }{|R^2 + |z''|^2 |^{t}}dR d \lambda(z''),$$ were $C$ is some contant depending only on the dimension. This integral is easily seen to be finite if and only if $ 2k -2/s> 0$ and $ 2k -2t-2/s >2k-2n$. In other words $$\nu_{0,t}(\log (z_1\bar{z_1}+...z_k \bar{z_k})) = \max(\frac{1}{k},\frac{1}{n-t}) .$$
This example shows that when we look at sets of the type $\{ z_1=...=z_k=0 \}$ in $\mathbb{C}^n$ the generalized Lelong number, as a function of $t$, thus senses the (co-)dimension of the set: it is constant, and equal to the integrability index of $\frac{1}{2}\log (z_1\bar{z_1}+...z_k \bar{z_k})$, when $t$ is so small so that $n-t$ is larger than $k$ - the co-dimension of the set - and then grows linearly to $1$, which is the Lelong number of $\frac{1}{2}\log (z_1\bar{z_1}+...z_k \bar{z_k})$.
\[example2\] Next we compute $\nu_{0,t}(\varphi)$ for $ \varphi(z_1,..z_n)=\log (z_1^{\alpha_1} \cdots z_k^{\alpha_1})$ for $1\leq k \leq n$ and $(\alpha_1,..,\alpha_k) \in \ \mathbb{N}^k$, $\alpha_i \neq 0$. Thus we want to study the following integrals behaviour with respect to $s$: $$\int_{\Delta} \frac{d \lambda}{|z_1^{\alpha_1} \cdots z_k^{\alpha_k}|^{2/s} |z|^{2t}}.$$ Using Fubini’s thorem and changing to polar coordinates in each of the variables $z_1$ to $z_k$ we obtain: $$\int_{ \Delta''} \int_{\mathbb{R}^k \cap U} \frac{r_1^{1-2 \alpha_1 /s} \cdots r_k^{1-2 \alpha_k /s } d r_1 \cdots d r_k }{ |r_1^2 + ... + r_k^2 + |z''|^2|^{t}}d \lambda(z'')$$ for some open set $U$ in $\mathbb{R}^k$. We put $ N = \sum \alpha_i$ and in the inner integral we change to polar coordinates in $\mathbb{R}^k$ and obtain an integral of the same magnitude: $$\int_{ \Delta''} \int_{0}^r \frac{R^{2k-1-2 N/s} }{|R^2 + |z''|^2|^{t}}dR d \lambda(z'') \cdot \int_{S^{k-1}} \omega_1^{1-2 \alpha_1 /s} \cdots \omega_k^{1-2 \alpha_k /s} d \sigma( \omega ),$$ for some $r>0$. In the previous example saw that the transition into being infinite for the first integral (with $s$ replaced by $s/N$) was obtained for $s=\max(\frac{N}{k},\frac{N}{n-t})$. The second integral can be computed using gamma functions, and in fact equals $$\frac{\prod_{i=1}^k \Gamma(\frac{ 2 - 2 \alpha_i /s}{2})}{\Gamma(\frac{ |\alpha| +k}{2})} ;$$ hence the integral diverges for $ 1-2 \alpha_i /s=-m$ for $m>0$ (if the other factors are non-zero). However, the condition on $s$ becomes $ s = \frac{\max{\alpha_i}}{1+m} \leq \frac{N}{k} $, which will not give any contribution to $s=\max(\frac{N}{k},\frac{N}{n-t})$. Putting it together we see that: $$\nu_{0,t}(\log (z_1^{\alpha_1} \cdots z_k^{\alpha_1})) = \max(\frac{\sum \alpha_i}{k},\frac{\sum \alpha_i}{n-t}).$$
When we are looking on sets of the form $\{ z_1 \cdots z_k = 0\}$, which is the union of the $k$ coordinate planes $\{ z_k=0 \}$ (corresponding to the function $\varphi=\log|z_1 \cdots z_k | $), the generalized Lelong number thus senses how many coordinate planes the union is taken over.
These two examples show us that the two leftmost inequalities in (\[generalized\_skoda\_inequality\]) are sharp. More precisely: using $\varphi$ from example \[example1\], we see that if $t=n-k$, $$\nu_{0,n-1}(\varphi) \leq (n-t) \nu_{0,t}(\varphi) \Leftrightarrow 1 \leq \frac{n-t}{k} \Leftrightarrow 1 \leq 1.$$ However, if $\varphi$ is as in example \[example2\], with $t = n-k$ we see that $$(n-t)\nu_{0,t}(\varphi) \leq n \nu_{0,0}(\varphi) \Leftrightarrow \sum_i \alpha_i \leq \sum_i \alpha_i.$$
We now use our generalized Lelong number to obtain a classical result, namely, that $\nu_{a,t}$ is invariant under biholomorphic coordinate changes.
\[biholotheorem\] If $f:\Omega \rightarrow \Omega$ is biholomorphic, $f(0)=0$ and $\det f'(0) \neq 0$, then $$\nu_{0,t} (\varphi \circ f ) = \nu_{0,t} (\varphi ).$$
By a change of coordinates we get $$\int_0 e^{- \frac{2}{s}\varphi \circ f(\zeta) - 2t\log |\zeta|} d \lambda(\zeta) =
\int_0 e^{- \frac{2}{s}\varphi (z) - 2t\log |f^{-1}(z)|} |\det f'(z)|^{-1} d \lambda(z).$$ Around the origin we have that $|f^{-1} (z) |$ is comparable to $|z|$ and thus the last integral is of the same magnitude as $$\int_0 e^{- \frac{2}{s}\varphi (z) - 2t\log |z|} \frac{1}{|\det f'(z)|} d \lambda(z).$$ Since $C > \frac{1}{|\det f'(\cdot)|} > c>0$ in some neighbourhood of the origin, the first integral is infinite iff $$\int_0 e^{- \frac{2}{s}\varphi (\zeta) - 2t\log |\zeta|} d \lambda(\zeta)=\infty.$$ In other words, $$\nu_{0,t} (\varphi \circ f ) = \nu_{0,t} (\varphi ).$$
Since for $t=n-1$ we get the classical Lelong number, we obtain as a corollary the theorem of Siu found in [@Siu]:
The classical Lelong number is invariant under biholomorphic changes of coordinates.
Let $V \subset \Omega$ be a variety and pick a point $x \in V$ where $V$ is smooth. We can then find a neigbourhood $U$ of $x$ and $f_1,...,f_k \in \mathcal{O}(U)$ such that $$V \cap U = \{ f_1=...=f_k=0 \}.$$ Since $V$ was smooth at $x$, we can change coordinates via a function $g:U \rightarrow U$ such that in these new coordinates $$V \cap U = \{ z_1=...z_l=0 \}$$ for some $l \leq k$. This means that $f_i \circ g = z_i $ for $1 \leq i \leq l$ and $f_i \circ g = 0$ for $i \geq l$. By proposition \[biholotheorem\] we have that $$\nu_{x,t} ( \sum_1^k |f_i|^2 ) = \nu_{x,t} ( \sum_1^l |z_i|^2 ),$$ and thus, by example \[example1\] we see that $\nu_{x,t} ( \sum_1^k |f_i|^2 )$ senses the co-dimension of $V$ at $x$.
When considering the generalized Lelong number, our next technical lemma shows that we can “move” parts of the singularity from the plurisubharmonic function to the weight, provided the singularity is sufficiently large:
\[singularitylemma\] Let $ \delta > 0 $, and $\psi$ be a plurisubharmonic function such that $e^{-2(1+\tau)\psi} \in L_{Loc}^1(0)$, for some $\tau>0$. If $ \nu_{a,\psi} (\varphi) = 1 + \delta$, then with $0<\epsilon<\tau \delta$ we have that $$\int_{a} e^{ - 2 \varphi (\zeta) - 2(1 - \epsilon ) \psi ( \zeta - a ) } = \infty.$$
The hypothesis implies that for every neighbourhood $U$ of $a$, $$\int_U e^{ - \frac{2 \varphi (\zeta)}{1 + \delta'} - 2 \psi ( \zeta - a ) } = \infty,$$ when $\delta'<\delta$. The function $\zeta \mapsto e^{-2(1-\epsilon)\psi (\zeta - a)}$ is locally integrable around $a$, and we apply Hölder’s inequality with respect to the measure $e^{-2(1-\epsilon)\psi (\zeta - a)} d\lambda (\zeta)$ on $U$, with $p = 1+\delta'$ and $q=\frac{1+\delta'}{\delta'}$, to obtain $$\begin{aligned}
\int_U e^{ - \frac{2 \varphi (\zeta)}{1 + \delta'} - 2\psi (\zeta - a)}d \lambda (\zeta)
=
\int_U e^{ - \frac{2 \varphi (\zeta)}{1 + \delta'}} e^{-2(1-\epsilon)\psi (\zeta - a)} e^{- 2 \epsilon \psi (\zeta - a)} d\lambda (\zeta)
\leq \\
\leq \Big{(}\int_U e^{ - 2 \varphi (\zeta)} e^{-2(1-\epsilon)\psi (\zeta - a)} d\lambda (\zeta)\Big{)}^{1/p}
\Big{(} \int_{U} e^{-2(\epsilon q + 1 - \epsilon)\psi (\zeta - a)} d\lambda (\zeta) \Big{)}^{1/q}.\end{aligned}$$ Since the left hand side is infinite by hypothesis, and the second integral on the right hand side converges (after possibly shrinking $U$, since $\epsilon q + 1 - \epsilon \leq 1 + \tau$, if $\epsilon < \delta' \tau$ ), we see that $$\Big{(}\int_U e^{ - 2 \varphi (\zeta) - 2(1-\epsilon) \psi (\zeta - a)} d\lambda (\zeta)\Big{)}^{1/p} = \infty.$$ This implies the desired conclusion, since $\delta'$ can be choosen arbitrarily close to $\delta$.
We will now give a proof of the Skoda inequality (\[skoda\_ineq\]), based on the Oshawa-Takegoshi extension theorem, learned in a private communication with Mattias Jonsson. We will also use the same technique to give a simple proof of Theorem \[berndtssons\_sats\].
We begin by recalling the statement of the **Oshawa-Takegoshi theorem** (see e.g. [@oh-t]): Assume $V$ is a smooth hypersurface in $\mathbb{C}^m$ which in local coordinates can be written as $ V = \{ z_n = 0 \}$, and let $U$ be a neighbourhood in $\mathbb{C}^m $ whose intersection with $V$ is non-empty. We also assume $\varphi$ is such that $\int_V e^{-2\varphi} < \infty$. Then, if $h_0 \in \mathcal{O}(V \cap U)$, there exists a $h \in \mathcal{O}( U)$ with $h=h_0$ on $V$ which satisfies the following estimate: \[ohtineq\] \_[U]{} C\_[ ]{} \_[U V]{} |h\_0|\^2 e\^[-2 ]{}, for $0 < \delta < 1$ and some constant $C_{\delta}$ depending only on $U$, $\delta$ and $\varphi$.\
The hard part of Skoda’s inequality, and the part we will show, is the implication $\nu_{z,n-1}(\varphi)<1 \Rightarrow \nu_{z,0}(\varphi)<1.$ We record the core of the argument as a lemma.
\[jonssonlma\] Let $\varphi \in PSH(\Omega)$ and let $x \in \Omega$. Assume there exists a complex line $L$ through $x$ for which $$\int_{L \cap \Omega} e^{-2 \varphi} < \infty,$$ then there exists a neighbourhood $\omega \subset \Omega$ of $x$, for which $$\int_{\omega} e^{-2 \varphi} < \infty.$$
That is, in order to prove that $e^{-2 \varphi}$ is locally integrable at a point, we need only to find a complex line where the statement holds.
It suffices, of course, to prove this for $x=0$. Assume $L$ is a complex line through the origin for which $\int_{L \cap \Omega} e^{-2 \varphi|_L} < \infty$. Applying the Ohsawa-Takegoshi extension theorem inductively, we can extend the function $1 \in L^2(L \cap \Omega, e^{- \varphi|_L}) \bigcap \mathcal{O} (L \cap \Omega) $ to a function $ h \in L^2(\Omega,e^{- \varphi}) \bigcap \mathcal{O} (\Omega) $, where $\Omega$ is a neighbourhood in $\mathbb{C}^n$, with a bound on the $L^2$ norm: $$\int_{\Omega} |h|^2 e^{-2 \varphi } \leq C \int_{\Omega \bigcap L} e^{-2 \varphi |_L} < + \infty$$ for some constant $C$. This inequality is obtained from (\[ohtineq\]) by just discarding the denominator figuring in the left-hand-side integral. Since $h$ is equal to 1 on $L$, the quantity $|h|^2$ is comparable to 1 in a neighbourhood $\omega$ of the origin. Thus we obtain: $$\int_{\omega} e^{-2 \varphi } < + \infty$$ which is what we aimed for.
*Proof of Skoda’s inequality:*
Remember, we want to prove the implication $\nu_{z,n-1}(\varphi)<1 \Rightarrow \nu_{z,0}(\varphi)<1$. To that effect, assume $ \nu_{z,n-1}(\varphi) < 1.$ It is well known that the Lelong number (at the origin) of a function $\varphi$ is equal to the Lelong number of the same function restricted to a generic complex line passing through the origin (we will prove this later, see lemma \[linelemma\]), which coincides with the integrability index on that line. Thus we can find a complex line $L$ for which $\int_{\Omega \bigcap L} e^{-2 \varphi |_L} < + \infty$ and so by lemma (\[jonssonlma\]) we see that $\nu_{z,0}(\varphi)<1 $.\
\
One might hope that knowledge of the dimension of the set where $\nu_{z,n-1}(\varphi) \geq c$ would enable us to sharpen the estimate of Skoda’s inequality. The following example shows that unfortunately this information is not sufficient to succeed.
Let $n=2$ and $\varphi(z_1,z_2)= \log( | z_1 |^2 + |z_2|^{2a})$. Then one calculates:
- $\nu_{0,n-1}( \varphi) = 2$,
- $\nu_{0,0}( \varphi) = \frac{2}{1+1/a}$,
- $\{z: \nu_{z,n-1}( \varphi) \geq 1 \}=\{0 \}$.
This is the best scenario possible: the dimension of the upper-level set of the Lelong number is 0 and *still* the lower bound of the Skoda inequality is sharp, which one realizes by letting $a \rightarrow \infty$.
Let us see how we can apply the full strength of the estimate (\[ohtineq\]) of the Ohsawa-Takegoshi theorem to obtain a proof of Theorem \[berndtssons\_sats\]. We recall the statement of Theorem \[berndtssons\_sats\]: $$\nu_{a} (\varphi) \geq 1 \Longleftrightarrow \int_a e^{-2 \varphi(\zeta) - 2(n-1)\log |\zeta - a|} d \lambda(\zeta) = + \infty.$$ Assume $a=0$, and let $\Omega$ be a small neighbourhood of the origin in $\mathbb{C}^n$, and let $\varphi \in PSH(\Omega)$ satisfy $\nu_0(\varphi)<1 $. Then we know that the restriction of $e^{-2\varphi}$ to a generic complex line is integrable. However, since a rotation of $\varphi$ will not effect $\varphi$’s integrability properties in $\Cn$, we may assume that the line is given by $\{z_2=...=z_n=0 \}$. In fact we can assume that $\varphi$ is integrable along every coordinate axis. Thus $\varphi$ satisfies \[gen\_cond\] \_[ {z\_2=...=z\_n=0 } ]{} e\^[-2 ]{}d \_1 < + . We want to prove that $$\int_0 e^{-2 \varphi(\zeta) - 2(n-1)\log |\zeta|} d \lambda(\zeta) < +\infty.$$ If we consider the constant function 1 as an element of $\mathcal{O} (\mathbb{C} \cap \Omega )$ then, by the argument above, we obtain a function $h \in \mathcal{O} (\mathbb{C}^2 \cap \Omega)$, comparable to 1 in $\Omega$, thus aquiring the following inequality: $$\int_{\mathbb{C}^2 \cap \Omega'} \frac{ e^{-2 \varphi}}{|z_1|^{2-2 \delta}} \leq C_{ \delta} \int_{\mathbb{C} \cap \Omega} e^{-2 \varphi}< + \infty,$$ with $\Omega' \subset \Omega$. Since $0 < \delta < 1$ the function $ \varphi + (1- \delta) \log |z_1| $ is plurisubharmonic in $\Omega$. Thus we can repeat the argument with $\varphi$ exchanged for $\varphi + (1- \delta) \log |z_1| $ to obtain: $$\int_{\mathbb{C}^3 \cap \Omega''} \frac{ e^{-2 \varphi}}{|z_1|^{2-2 \delta}|z_2|^{2-2 \delta}} \leq C_{ \delta} \int_{\mathbb{C}^2 \cap \Omega} \frac{ e^{-2 \varphi}}{|z_1|^{2-2 \delta}} < + \infty,$$ with $\Omega'' \subset \Omega'. $ Iterating this procedure it is easy to realise that, after possibly shrinking $\Omega$, we obtain the inequality \[skodaintegral\] \_[ ]{} C\_[ ]{} \_[\^[n-1]{} ]{} < + . Using the trivial estimate $$\int_{ \Omega} \frac{e^{-2 \varphi}}{|z|^{2(n-1)(1 -\delta)}} \leq \int_{ \Omega} \frac{ e^{-2 \varphi}}{|z_1|^{2-2 \delta}...|z_{n-1}|^{2-2 \delta}}$$ we see that $$\int_{ \Omega} e^{-2 \varphi -2(n-1)(1 -\delta)\log|z|} \leq C_{\delta}< + \infty, \, \, \, \, \, \, \forall \delta>0,$$ which, by lemma \[singularitylemma\] with $\psi(\zeta)=(n-1) \log|\zeta|$, implies that $$\int_{ \Omega} e^{-2 \frac{\varphi}{1+\delta(n-1)} -2(n-1)\log|z|} \leq C_{\delta}< + \infty, \, \, \, \, \, \, \forall \delta>0.$$ In this argument, since $\nu_0(\varphi)<1$, we can exchange $\varphi$ for $\frac{\varphi}{1-r} $, where $r>0$, and still have $\nu_0(\frac{\varphi}{1-r})<1$. Thus, the hypothesis implies that $\nu_{0,n-1} (\varphi) < 1 $.\
The other direction is simpler: by introducing polar coordinates we see that $$\int_a e^{-2 \varphi(\zeta) - 2(n-1)\log |\zeta - a|} d \lambda(\zeta) = C \int_{\omega \in S^{2n-2}} \int_{t \in \mathbb{C}, |t|<1} e^{-2 \varphi(a + t\omega)}.$$ If $ \nu_a(\varphi) \geq 1$ then the integral of $\varphi$ over almost every complex line through $a$ is infinite, and thus the above integral is infinite which is the same as saying $\nu_{0,n-1} (\varphi) \geq 1$. We have proved Theorem \[berndtssons\_sats\].\
\
We will now describe the relation between the generalized Lelong number $ \nu_{a,k}$ and restrictions to linear subspaces of dimension $k$. In order to do this, we will have to recall the natural measure on the Grassmannian induced by the Haar measure on $U(n)$, where $U(n)$ denotes the unitary group of $\mathbb{C}^n$. Also, let $\vartheta$ denote the unique, unit Haar measure on $U(n)$. Then we can define a measure $d\mu$ on the Grassmannian $G(k,n)$ - the set of $k-$dimensional subspaces of $\Cn$ - by setting for some fixed $V \in G(k,n)$, and $A \subset G(k,n)$ the mass of $A$ to be $\mu(A)= \vartheta ( M \in U(n) : MV \in A ).$ This means that if $ P : U(n) \rightarrow G(k,n)$ is the function $P(M) = MV$, then $$\mu = P_{*} (\vartheta) .$$ The measure $\mu$ is easily seen to be invariant under actions of $U(n)$, that is, $\mu(MA)= \mu(A)$ if $M \in U(n)$, and also to be independent of our choice of $V$. For $f$ a function defined on $G(k,n)$, we have $$\int_{T \in G(k,n)} f(T) d \mu = \int_{ M \in U(n)} f(MV) d \vartheta.$$ We deduce that, for $g \in C_c^{\infty}(\Cn)$, $$\int_{T \in G(k,n)} \int_{T} g(z) d \lambda_k d \mu =
\int_{ M \in U(n)} \int_{ MV } g(z) d \lambda_k d \vartheta$$ where $\lambda_k$ is the $k-$dimensional Lebesgue measure. After changing to polar coordinates the above integral becomes $$\int_0^{\infty} \int_{M \in U(n)} \int_{ S_{MV}^{2k-1} } \rho^{2k-1} g( \rho \omega) dS(\omega) d \vartheta d \rho ,$$ where $S_{MV}^{2k-1}$ denotes the $(2k-1)-$dimensional sphere in the $k-$dimensional subspace defined by $MV$. Consider the linear functional on $C(\rho S^{2n-1})$, defined by \[functionaldef\] I\_(g) := \^[2n-1]{}\_[M U(n)]{} \_[S\_[MV]{}\^[2k-1]{} ]{} g( ) dS() d . Notice that, even though this functional is defined for functions on $S^{2n-1}$ while the integration takes place on the sphere $S^{2k-1}$, it is invariant under rotations on the sphere $S^{2n-1}$. By the Riesz representation theorem, this functional is given by integration against a measure $d \gamma$ on $\rho S^{2n-1}$, i.e., \[functionaldef2\] I\_(g)= \_[S\^[2n-1]{}]{} g() d (), where, since $I_\rho$ is rotational invariant, the measure $d \gamma $ is rotational invariant as well. Thus $d \gamma$ is equal to the surface measure on $\rho S^{2n-1}$ multiplied with a constant $c(\rho)$. This constant is easily determined by evaluating $I_{\rho}(1)$ using the two expressions (\[functionaldef\]),(\[functionaldef2\]) above (remember that $\vartheta$ was normalized so that $\vartheta(U(n))=1$) : $$\rho^{2n-1} c(\rho) = I_{\rho}(1) = \rho^{2n-1} \int_{S_{MV}^{2k-1} } dS(\omega).$$ So $c(\rho)=c_k=\int_{S^{2k-1} } dS(\omega)$ and is therefore independent of $\rho$.
Thus we see that the integral $$\int_{T \in G(k,n)} \int_{ T} g(z) d \lambda_k d \mu
=
\int_0^{\infty} \rho^{2(k-n)} I_\rho(g) d \rho$$ is equal to $$c_k \int_0^{\infty} \rho^{2(k-n)} \int_{\rho S^{2n-1}} g(\omega) d S (\omega) d \rho
=
c_k \int_{\Cn} |z|^{2(k-n)} g(z) d \lambda_n$$
Exchanging $ g(z)$ for $ g(z) |z|^{2(n-k)} $ we have proven the following formula, which generalizes the formula for changing to polar coordinates in an integral:
\[grassmanmeasurelemma\] For $g$ an integrable function, $$\int_{\Cn} g d \lambda_n = c_k^{-1}\int_{T \in G(k,n)} \int_{ T} |z|^{2(n-k)} g(z) d \lambda_k d \mu .$$
We have proven the formula under the condition that $g \in C_c^{\infty}(\Cn)$. The general case follows by approximating an arbitrary integrable function $g$ by functions in $C_c^{\infty}(\Cn)$ .
Of course, a similar formula holds if we instead consider $k-$planes through some arbitrary point $a \in \Cn$, and in the above discussion assume the spheres to be centered around the point $a$. This remark applies to the following result as well:
Let $k$ be an integer between 0 and $n-1$. Then $\nu_{0,n-k} (\varphi)<1$ $\Longleftrightarrow $ $ \nu_{0,0}(\varphi|_T)<1$ for almost every $T \in G(k,n)$ $\Longleftrightarrow$ $ \nu_{0,0}(\varphi|_T)<1$ for some $T \in G(k,n)$.
The assumption $\nu_{0,n-k} (\varphi)<1$ means that $$\int_{B(0,r)} e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta} - 2 (n-k) \log( \left| \zeta \right| ) } d \lambda_n < +\infty$$ for some $r>0$, and $\delta>0$ small. Using lemma \[grassmanmeasurelemma\] this integral equals $$\int_{T \in G(k,n)} \int_{ B(0,r) \cap T} e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta}} d \lambda_k d \mu.$$ Thus $\int_{ B(0,r) \cap T} e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta}} d \lambda_k $ must be finite for almost every $T$ (since by the lemma, $d \mu$ is a multiple of the Lebesgue measure), which implies that $ \nu_{0,0}(\varphi|_T)<1$ for almost every $T \in G(k,n)$. This, of course, implies that $ \nu_{0,0}(\varphi|_T)<1$ for some $T \in G(k,n)$. On the other hand, if $\int_{ B(0,r') \cap T} e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta}} d \lambda_k < +\infty$ for some $T$ and $\delta>0$, then the exact same argument involved in proving the Skoda inequality (using the Ohsawa-Takegoshi theorem), shows that in fact $$\int_{ B(0,r) } e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta'} -2(n-k)\log|z|} d \lambda_n < +\infty,$$ for some small $\delta'>0$. This implies that $\nu_{0,n-k}(\varphi)<1.$
A similar argument gives us the following classical statement (cf. [@Siu]):
For a generic $V \in G(k,n)$ we have that $$\nu_{0,n-1}(\varphi) = \nu_{0,k-1}(\varphi |_V).$$
Assume $\nu_{0,n-1}(\varphi) < 1$. Then by Lemma \[grassmanmeasurelemma\], with $g(\zeta)=\exp(-2\varphi( \zeta) - 2(n-1)\log|\zeta| )$ we get that for some small $\delta>0$, $$+ \infty >
\int_{B(0,r)} e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta} - 2 (n-1) \log \left| \zeta \right| } d \lambda_n =
\int_{V \in G(k,n)} \int_{ B(0,r) \cap V} e^{ - 2 \frac{\varphi (\zeta)}{1 - \delta} - 2 (k-1) \log \left| \zeta \right| } d \lambda_k d \mu.$$ Thus $\nu_{0}(\varphi |_V) < 1$ for a generic $V \in G(k,n)$. The other direction is proved by using the same Ohsawa-Takegoshi argument as before.
Taking $k=1$ we obtain again the classical result:
\[linelemma\] For almost every complex line $L$ through a point $a$, $$\nu_{a}(\varphi) = \nu_{a}(\varphi|_L).$$
That is, the Lelong number coincides with what it generically is on complex lines. Moreover, $\nu_{n-k}$ coincides with the integrability index of $\varphi$ restricted a generic $k-$plane.\
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The analyticity property of the upper level sets of the generalized Lelong number {#analyticity_section}
=================================================================================
In this section we prove that the upper level sets of our generalized Lelong number are analytic, provided that the weight is “good enough”. This we accomplish by considering the Bergman function, whose definition we will soon recall. First, however, we begin with discussing what properties the weight need to satisfy in order to be “good enough”.\
\
$\bullet$ We say that a plurisubharmonic function $\psi$ has an isolated singularity at the origin if there exists an $M>0$ such that $$\psi(z) \geq M \log|z|,$$ for $z$ close to $0$. It might be worth mentioning that in the case of analytic singularities, i.e., if $\psi = \log|f|$ where $f=(f_1,..,f_n)$ is a tuple of holomorphic functions with common intersection locus at the a single point, the least of all $M$ for which $$\log|f(z)| \geq M \log|z|$$ is called the Lojasiewicz exponent of $f$.\
$\bullet$ We assume as before that $$e^{-2(1+\tau)\psi} \in L_{Loc}^1(0),$$ for some $\tau>0$.\
$\bullet$ We also assume that $e^{2 \psi}$ is Hölder continuous with exponent $\alpha$, at $0$.\
$\bullet$ Finally we assume that $\nu_0(\psi) = l>0,$ so that $\psi$ carries some singularity at the origin.
We say that a plurisubharmonic function $\psi$ is an admissible weight, and write $\psi \in W(\tau,l,M,\alpha)$ if it satisfies the four properties above.
Admissible weights satisfy the following property, which we will make use of in the proof of the analyticity:
\[annulilemma\] Assume that $e^{2\psi}$ is Hölder continuous at the origin, with exponent $\alpha$ and satisfies $\psi \geq M \log|z|$ near the origin. Then there exists a $R>0$ and a constant $C>0$, such that for every $k\in \mathbb{N}$: $$e^{ -2 \psi(\zeta - a')} \geq C e^{ -2 \psi(\zeta)}$$ where $ 2^{-(k-1)} \leq |\zeta | \leq 2^{-k} $ and $|a'|^=2^{-Rk}$.
Fix $k\in \mathbb{N}$. We want to show that $$e^{ -2 \psi(\zeta-a')} \geq Ce^{ -2 \psi(\zeta)}$$ for $ 2^{-(k-1)} \leq |\zeta | \leq 2^{-k} $ and $|a'|^=2^{-Rk}$. Since $|\zeta|^R \geq c |a'|$ the assumption gives us that $$e^{2 \psi(\zeta)} \geq |\zeta|^{2 M} \geq c |a'|^{ \alpha}$$ if $R \alpha \geq 2 M$ (which is a condition independent of $k$). Now, using the Hölder continuity, we obtain $$e^{ 2 \psi(\zeta - a')} \leq e^{ 2 \psi(\zeta)} + |a'|^{\alpha} \leq c e^{ 2 \psi(\zeta)}$$ which implies $$e^{ -2 \psi(\zeta - a')} \geq C e^{ -2 \psi(\zeta)},$$ for some contant $C>0$.
Examples of plurisubharmonic functions which are admissible weights are given by $$\psi = \log\left( \sum_{i=1}^n |f_i|^{\alpha_i} \right)$$ where $f=(f_1,...,f_n)$ is an $n-$tuple of holomorphic functions with common zero locus at the origin. Here we have to assume that $\alpha_i>0$ are as small as needed in order for a $\tau>0$ to exist for which $$e^{-2(1+\tau)\psi} \in L_{Loc}^1(0).$$ Then $e^\psi$ is Hölder continuous with Hölder exponent $\min(1,\alpha_i)$, and $\psi$ have Lelong number equal to $\min{\alpha_i \nu_0(\log|f_i|)}.$ Also, the Lojasiewicz exponent, which is the smallest $M$ for which $ \sum_{i=1}^n |f_i|^{\alpha_i} \geq |z|^M$ is easily seen to be finite.
We now define the Bergman kernel with respect to a weight.
Let $a \in \Omega$, $\varphi \in PSH(\Omega)$ and $\psi \in W(\tau,l,M, \alpha)$. We define $\mathcal{H}_a = \mathcal{O}(\Omega) \cap L^2 ( \Omega, e^{-2\varphi( \cdot) - 2 \psi( \cdot - a )} ),$ which is a separable Hilbert space. The Bergman kernel for a point $z \in \Omega$ is defined as the unique function $B^{\psi}_a(\zeta,z)$, holomorphic in $\zeta$, satisfying $$h(z) = \int_{\Omega} h( \zeta ) \overline{B^{\psi}_a(\zeta,z)} e^{-2\varphi( \zeta) - 2 \psi( \zeta - a )} d \lambda(\zeta),$$ for every $h \in \mathcal{H}_a$.
The existence of the Bergman kernel is a (rather easy) consequence of the Riesz representation theorem for Hilbert spaces. Closely related to the Bergman kernel is the Bergman function:
For $a\in \Omega$ the Bergman function at a point $\zeta \in \Omega$ is defined as $$B^{\psi}_a(\zeta) := B^{\psi}_a(\zeta,\zeta).$$
We define $$\Lambda(a) = \{ h \in \mathcal{O}(\Omega) , \int_{\Omega} \left| h( \zeta ) \right| ^2
e^{ - 2 \varphi (\zeta) - 2\psi( \zeta-a )} d \lambda ( \zeta) \leq 1 \}$$ that is, those functions in $\mathcal{H}_a$ of norm less than or equal to 1. Let us calculate the norm of $\zeta \mapsto B^{\psi}_a(\zeta,z)$: $$\left\| B^{\psi}_a(\cdot,z) \right\|^2 =
\int_{\Omega} B^{\psi}_a(\zeta,z) \overline{B^{\psi}_a(\zeta,z)} e^{-2\varphi( \zeta) - 2 \psi( \zeta - a )} d \lambda(\zeta)
= B^{\psi}_a(z,z),$$ which in particular implies that $ B^{\psi}_a(z,z)$ is given by a non-negative real number. Consequently $$s(\zeta) = \frac{B^{\psi}_a(\zeta,z)}{ \sqrt{B^{\psi}_a(z,z)}} \in \Lambda(z)$$ and so \[bmanrealization\] |s(z)|\^2 = B\^\_a(z).\
Also, we have that \_[h (a)]{} |h(z)|\^2 = \_[h (a)]{} |(h,B\^\_a(,z))|\^2 = B\^\_a(,z) \^2 = B\^\_a(z), where $(,)$ denotes the inner product in $\mathcal{H}_a$, which gives us the following extremely useful characterization of the Bergman function: \[bmanchar\] B\_a\^(z) := { |h(z)|\^2 : h () , \_ | h( ) | \^2 e\^[ - 2 () - 2( -a )]{} d ( ) 1 } , and (\[bmanrealization\]) means that this supremum is actually realized by the function $s$.\
Bergman functions enjoy several nice properties, and one, critical for our purposes, is provided by the following theorem of Berndtsson (cf. [@Berndtsson]),
If $\Omega$ is pseudoconvex, then the function $(a,z) \mapsto \log B_a^{\psi}(z)$, is plurisubharmonic in $(a,z)$.
Thus we can talk about the Lelong number of the function $ z \mapsto \log B_z^{\psi}(z)$ in $\Omega$, and the following proposition relates it to the generalized Lelong number of $\varphi$. More specifically, it says that if the generalized Lelong number of $\varphi$ with respect to $\psi$ is larger than 1, then the classical Lelong number of $ z \mapsto \log B_z^{\psi}(z)$ is larger than 0, and if the generalized Lelong number is smaller than 1 the classical number is 0. In the terminology of Kiselman, we say that $\log B_z^{\psi}(z)$ attenuates the singularities of $\varphi$.\
\
Recall that by lemma (\[annulilemma\]) we can find a $R>0$ for which $$e^{ -2 \psi(\zeta-a')} \geq C e^{ -2 \psi(\zeta)}$$ for $ 2^{-(k-1)} \leq |\zeta | \leq 2^{-k} $ and $|a'|^=2^{-Rk}$, for every $k \in \mathbb{N}$.\
\
Also, by lemma (\[singularitylemma\]) we can choose an $\epsilon < \delta \tau$ (arbitrarily close to $\delta \tau$) for which \[epsilonn\] \_[B(a,1/2\^N)]{} e\^[ - 2 () - 2(1 - ) ( -a) ]{}= , if we fix an $N>0$ large enough.
\[kernellemma\] Let $\delta>0$ be small and let $\Omega$ be an open and pseudoconvex set containing the point $a$, and let $\psi \in W(\tau,l,M,\alpha)$. Assume $$\nu_{a,\psi} (\varphi)=1+\delta.$$ Then, with $\cdelta = \delta \tau l $, the classical Lelong number of $\log B_{\cdot}^{\psi}(\cdot)$ at $a$ is larger than or equal to $\cdelta/R$ , that is: $$\nu_a(\log B_{\cdot}^{\psi}(\cdot)) \geq \frac{\cdelta}{R}.$$ On the other hand, if we assume that $$\nu_{a,\psi} (\varphi)<1,$$ then $$\nu_a(\log B_{\cdot}^{\psi}(\cdot)) = 0.$$
Without loss of generality, we can assume that $a=0$. Recalling the definition of the classical Lelong number, we see that we want to show that \[lelongdef\] \_[r 0]{} C\_ /R . The idea of the proof is the following. The assumption $\nu_{0,\psi} (\varphi)=1+\delta$ essentially means that $$\int_0 e^{-2 \frac{\varphi}{1+ \delta} - 2\psi} = \infty.$$ Thus, the weight $\varphi + \psi$ has a rather large singularity at the origin. If we move the singularity of $\psi$ by translating it to an arbitrary point $a'$, then if $a'$ is small enough, the weight $\varphi(\zeta) + \psi(\zeta - a')$ will have a rather large singularity at the point $a'$. As we will show, the singularity will actually be so large that if $h \in \Lambda(a')$, that is if $h$ is holomorphic and satisfies $$\int_{\Omega} \left| h( \zeta ) \right| ^2
e^{ - 2 \varphi (\zeta) - \psi( \zeta-a' )} d \lambda ( \zeta) \leq 1 ,$$ then $h$ is forced to be small at the point $a'$. In fact, \[smallness\] |h(a’)|\^2 |a’|\^[C\_/ R]{} . This would be enough to prove the proposition, but the following observations show that in fact it will be enough to show something weaker. By Cauchy estimates we will see that it suffices to find just some point $z_0$ near the origin for which $|h(z_0)|^2 \leq |z_0|^{C_\delta }$, if $h \in \Lambda(a')$. This simplifies things considerably. Also, since we, in (\[lelongdef\]), are dealing with a limit, it suffices to find a sequence $r_k$ tending to 0, for which the inequality (\[lelongdef\]) holds. This means that instead applying the above idea to arbitrary points $a'$ near the origin, we merely need to apply it to points of a sequence $a_k$ tending to 0. We now turn to the details.\
\
\[sequencelemma\] Assume $\nu_{0,\psi} (\varphi)=1+\delta$. Fix any sequence $a_k \rightarrow 0$ with $|a_k|=2^{-Rk}$ and for every $k$ choose a corresponding $h^k \in \Lambda(a_k)$. Then $\{ a_k \}$ contains asubsequence $\{ a_{k_j} \}$, for which there exists $b_{k_j} \in B(0,2^{-{k_j}})$ with, $$|h^{k_j}( b_{k_j} )| \leq \left|b_{k_j} \right| ^{C_{\delta} }.$$
The lemma is proven under the assumption $C_\delta = \epsilon l$. The general case with $C_\delta = \delta \tau l$ then follows, since $\epsilon$ can be choosen arbitrarily close to $\delta \tau$. We will prove the lemma by arguing via contradiction. The negation of the statement is the following: For every $k$ larger than some finite number, which we can assume to be the $N$ figuring in ($\ref{epsilonn}$), $$|h^k( \zeta )| > \left|\zeta \right| ^{C_{\delta} },$$ for every $\zeta \in B(0,2^{-k})$. Let us assume this negation. Then, for every $a_k$ we have, since $h^k \in \Lambda(a_k) $, $$1 \geq
\int_{B(0, 2^{-k})} \left| \zeta \right| ^{{C_{\delta}}}e^{ - 2 \varphi (\zeta) - 2 \psi( \zeta -a_k)}d \lambda ( \zeta) \geq
\int_{A(k)} \left| \zeta \right| ^{{C_{\delta}}}e^{ - 2 \varphi (\zeta) - 2 \psi( \zeta -a_k)}d \lambda ( \zeta),$$ where $A(k)$ denotes the annulus $B(0,2^{-k}) \setminus B(0,2^{-k-1})$. Since for $\zeta \in A(k)$ $$e^{ -2 \psi(\zeta-a_k)} \geq C e^{ -2 \psi(\zeta)}$$ by lemma \[annulilemma\], we deduce that \[annulusintegral\] C \_[A(k)]{} | | \^[[C\_]{}]{}e\^[ - 2 () - 2 ( )]{}d ( ). Now, remember that $\epsilon$ was choosen so that $$\infty = \int_{B(0,1/2^N)} e^{ - 2 \varphi (\zeta) - 2(1 - \epsilon ) \psi( \zeta) }d \lambda( \zeta).$$ Thus, by covering the ball $B(0,1/2^N)$ by annuli $B(0,2^{-k}) \setminus B(0,2^{-k-1})$, we get $$\begin{aligned}
\infty = \int_{B(0,1/{2^N})} e^{ - 2 \varphi (\zeta) - 2(1 - \epsilon ) \psi( \zeta) }d \lambda( \zeta)
\leq
C \sum_{k}
\int_{A(k)} \left| \zeta \right|^{2 \epsilon l} e^{ - 2 \varphi (\zeta) - 2 \psi( \zeta) } d \lambda ( \zeta) \\
\leq
C \sum_k 2^{-k \epsilon l}
\int_{A(k)} \left| \zeta \right| ^{\epsilon l} e^{ - 2 \varphi (\zeta) - 2 \psi( \zeta )}d \lambda ( \zeta)
\leq (\ref{annulusintegral}) \leq
C \sum_k 2^{-k \epsilon l} < + \infty,
\end{aligned}$$ where we in the first inequality use the fact that $\nu_0(\psi)<l \Rightarrow e^{2 \epsilon \psi( \zeta)} \leq C | \zeta|^{2 \epsilon l} $ for $|\zeta|\leq2^{-N}$, if $N$ is large enough. This establishes the desired contradiction.
*(of Proposition (\[kernellemma\]))* Fix a point $a \in \Omega$ with $|a|=2^{-Rk}$ for some $k\in \mathbb{N}$, and choose an $h \in \Lambda(a),$ for which $|h( b )| \leq \left| b \right| ^{C_{\delta}}$ for some $b \in B(0,2^{-{k}})$. We claim that in fact such an $h$ satisfies the following estimate: \[contprop\] |h(a)| D |a|\^[C\_/R]{}, for some constant $D\geq0$ which does not depend on $h$ nor $k$: Since $ \varphi$ and $\psi$ are locally bounded from above, every $h\in\Lambda(a)$ satisfies: $$\int_{ \Omega } \left| h( \zeta ) \right| ^2 d \lambda ( \zeta) \leq C .$$ Thus, for $\Omega' \subset \subset \Omega$, by applying Cauchy estimates on $h$, we see that $$\left| h'( \zeta ) \right| ^2 \leq C ,$$ in $ \Omega'$, where $C$ is some constant independent of ${k}$. By using a first order Taylor expansion of $h$, we conclude that $$|h(a)| \leq |h(b)|+C|a-b| \leq |b|^{C_\delta} + C|b| \leq D \left|a \right|^{C_{\delta}/R},$$ for some constant $D\geq0$ independent of $h$ and $k$, as promised.\
Let us complete the proof: Assume that $\nu_{0, \psi}(\varphi) = 1 + \delta.$ Choose $\rho>0$ and fix a sequence $\{ a_k \} \subset \Omega$ with $|a_k|=2^{-Rk}$ satisfying the following: $$\sup_{|z|=2^{-Rk}} \log B_{z}^{\psi}(z) \leq \log B_{a_k}^{\psi}(a_k) + \rho,$$ for every $k$. In view of (\[bmanrealization\]), which says that we can actually find a holomorphic function in $ \Lambda(z) $ realising the Bergman function at $z$, we can find for each $k$, a funcion $h^k \in \Lambda(a_k)$ for which \[specreal\] h\^k(a\_k) = B\_[a\_k]{}\^(a\_k) . By lemma (\[sequencelemma\]) we can extract a subsequence $\{a_{k_j} \}$ with a corresponding sequence $\{ b_{k_j} \} $, where $|b_{k_j}| =2^{-k_j} $, for which $$|h^{k_j}(b_{k_j})| \leq |b_{k_j}|^{C_{\delta}} .$$ The estimate (\[contprop\]) implies that $$\log B_{a_{k_j}}^{\psi}(a_{k_j}) \leq \log| a_{k_j}|^{\cdelta / R} + D.$$
Thus we obtain (observe that the denominator is *negative*), $$\lim_{r \rightarrow 0} \frac{ \sup_{|z|=r} \log B_{z}^{\psi}(z) }{\log r}
=
\lim_{j \rightarrow \infty } \frac{ \sup_{|z|=2^{-R k_j}} \log B_{z}^{\psi}(z) }{\log 2^{-R k_j}}
\geq
\lim_{j \rightarrow \infty } \frac{ \log B_{a_{k_j}}^{\psi}(a_{k_j}) + \rho}{\log 2^{-R k_j}}
\geq
\cdelta / R,$$ and we are done in this case.\
\
If $\nu_a(\varphi,\psi)<1$, then an application of Hörmanders $L^2$-methods (cf. [@Hormander]) provides us with an holomorphic function $h$, satisfying $|h(a)|^2 > 0$, and the integral over $\Omega$ of $h$ with respect to the weight $ e^{ - 2 \varphi (\cdot) - 2 \psi( \cdot - a )} $ is less than $1$. In view of (\[bmanchar\]) this implies that $B_a^{\psi}(a) > 0$, hence $ \nu_a(\varphi, \psi)=0$.
We can now prove the analogue of Siu’s theorem for our generalized Lelong number, using an argument due to Kiselman.
\[analyticity\_thm\] Let $\Omega \in \mathbb{C}^n$ be open and pseudoconvex and $\varphi$ be a plurisubharmonic function in $\Omega$. Then if $\rho > 0$ , $$\{ z \in \Omega : \nu_{z,\psi}( \varphi) \geq \rho \}$$ is an analytic subset of $\Omega$.
We first note that if $\psi=0$ then $\nu_{a,0}( \varphi)$ is the same as the integrability index of $\varphi$ for which the conclusion of the theorem holds (see e.g. [@Kiselman]). Using the notation of proposition \[kernellemma\] we define $$\Psi(z) = 3 n \frac{\log B_z^{\psi} (z)}{\cdelta / R}, z \in \Omega.$$ The core of the proof is to show that $$\{z \in \Omega: \nu_{z,\psi}( \varphi) \geq 1 + \delta\}
\subset
\{z \in \Omega: e^{-2\Psi} \notin L_{Loc}^1(z) \}
\subset
\{z \in \Omega: \nu_{z,\psi}( \varphi) \geq 1\}.$$ This we can accomplish, as follows:\
\
If for $a \in \Omega$ we have that $\nu_{a,\psi}( \varphi) \geq 1 + \delta $ then due to Proposition \[kernellemma\], the classical Lelong number of $\Psi$ at $a$ is greater than $3n$ since $$\nu_a( \Psi) \geq \frac{3n \cdot \cdelta/R}{\cdelta /R} = 3 n.$$ By Skoda’s inequality ($\ref{skoda_ineq}$) we have that $\nu_a( \Psi) \leq n \nu_{a,0} (\Psi)$ which shows that the integrabilty index of $\Psi$ at $a$ is larger than or equal to $3$. In particular, this implies that $e^{-2 \Psi( \cdot ) }$ is not locally integrable at $a$ and proves the first of the inclusions.\
For the second one, assume that $$\nu_{a,\psi}(\varphi) < 1.$$ This implies that $ e^{ -2 \varphi (\zeta) - 2\psi ( \zeta - a )} $ is locally integrable at $a$. As noted above, an application of Hörmanders $L^2$-methods gives us a holomorphic function $h$ in $\Omega$ such that $|h(a)|^2 > 0$, and the integral of $h$ with respect to the weight $ e^{ - 2 \varphi (\cdot) - 2 \psi( \cdot - a )} $ is less than $1$. Thus the function $ z \mapsto B_z^{\psi} (z)$, being defined as that supremum of the modulus square of all holomorphic functions whose integral with respect to our weight is less than or equal to 1, is strictly positive at $a$, which implies $$\Psi(a)>-\infty.$$ But (see e.g.[@Hormander]) every plurisubharmonic function $u$ satisfies that $e^{-2u}$ is locally integrable around the points where it is finite, and thus we see that $$e^{-2 \Psi} \in L_{Loc}^1(a),$$ which proves the second inclusion.\
\
As noted above, we know that set $\{z \in \Omega: e^{-2\Psi} \notin L_{Loc}^1(z) \}$ is analytic in $\Omega$. Thus, by rescaling we obtain analytic sets $Z_{\delta,\rho}$ such that $$\{z \in \Omega : \nu_{z,\psi} ( \varphi) \geq \rho \}
\subset
Z_{\delta,\rho}
\subset
\{z \in \Omega: \nu_{z,\psi}( \varphi) \geq \frac{\rho}{1+\delta}\},$$ which implies $$\{z \in \Omega: \nu_{z,\psi}( \varphi) \geq \rho \} = \bigcap_{\delta>0} Z_{\delta,\rho}.$$ Since the intersection of any number of analytic sets is analytic, we are done.
As a conequence of this theorem we can define the following concept, introduced in the classical case by Siu([@Siu]):
For $Z$ an analytic set in $\Omega$, we define the *generic generalized Lelong number* of $\varphi$ by $$m_Z^{\psi}( \varphi) = \inf \{\nu_{z,{\psi}}( \varphi); z \in Z \}$$
We have the following lemma by precisely the same argument as in the classical case.
$\nu_{z,{\psi}} (\varphi) = m_Z^{\psi}( \varphi)$ for $z \in Z \setminus Z'$ where $Z'$ is a union of countably many proper analytic subsets of $Z$.
Put $Z' = \bigcup_{c>m_Z^{\psi}, c \in \mathbb{Q}} Z \cap E_c^{\psi}$, where $E_c^{\psi} = \{ z \in Z : \nu_{z,{\psi}} (\varphi) \geq c \}. $ Then each $Z \cap E_c^{\psi}$ is an analytic proper subset of $Z$ and $\nu_{z,{\psi}} (\varphi) = m_Z^{\psi}( \varphi)$ on $Z \setminus Z'$ by construction.
Approximation of plurisubharmonic functions by Bergman kernels {#approx_section}
==============================================================
A well known result due to Demailly (see for instance [@DemaillyKollar]) makes it possible to approximate a plurisubharmonic function $\varphi$, by the logarithm of the Bergman function ($\Psi^m$) with respect to the weight $e^{-2m\varphi}$, as $m$ tends to infinity. Furthermore, the approximation is continuous with respect to the (classical) Lelong number: $$\nu_{z,n-1}(\Psi^m) \rightarrow \nu_{z,n-1}(\varphi),$$ as $m \rightarrow \infty$. We will now show that the same holds true using the Bergman function with respect to the weight $e^{-2m\varphi-2\psi(\cdot - x)} $ where $x$ is the point at which we evaluate the Bergman function. The argument mimics closely that of Demailly’s (cf. [@DemaillyKollar]), with some minor changes to fit our case. To begin with, we modify the construction of $\mathcal{H}_a$ slightly:
For each $m\in \mathbb{N}$ and $a \in \Omega$ we let $$\mathcal{H}_a^m = \mathcal{O}(\Omega) \cap L^2 ( \Omega, e^{-2 m \varphi( \cdot) - 2 \psi( \cdot - a )} ).$$
Denote by $B_{a}^m$ (for notational convinience we supress the dependence on $\psi$) the Bergman function for $\mathcal{H}_a^m$, and put $$\Psi_a^m(z) = \frac{1}{2 m} \log{B_{a}^m}(z),$$ for $z\in \Omega$. Fix $a \in \Omega$. If $h \in \H_a^m$ and has norm bounded by 1, the mean value property for holomorphic functions shows that for $r=r(a)>0$ small enough, $$\begin{aligned}
|h(a)|^2
&\leq&
\frac{n!}{\pi^n r^{2n}} \int_{|a-\zeta|<r} |h(\zeta)|^2 d \lambda ( \zeta) \leq \nonumber \\
&\leq& \frac{n!}{\pi^n r^{2n}} e^{\sup_{|a-\zeta|<r} \{ 2 m \varphi( \zeta) + 2 \psi(\zeta-a) \} }
\int_{|a-\zeta|<r} |h(\zeta)|^2 e^{- 2 m \varphi( \zeta) - 2 \psi(\zeta-a)} d \lambda ( \zeta). \nonumber \end{aligned}$$
Thus, if we assume that \[psihypotes2\] ( -a ) l |-a|, we have that $$\Psi_a^m(a)
\leq
\sup_{|a-\zeta|<r} \{ \varphi( \zeta) + \frac{1}{2m} 2 \psi(\zeta-a) \} - \frac{1}{2m} \log r^{2n} + C
\leq
\sup_{|a-\zeta|<r} \{ \varphi( \zeta) \} +\frac{1}{m}(l-n) \log r + C/m.$$
Now, assume that \[psihypotes\] () (n-)||, for some small, fixed $\delta > 0$. Fix a point $a$ for which $\varphi(a)>- \infty$. By considering the 0-dimensional variety $\{ a \}$, we obtain, by the Ohsawa-Takegoshi theorem (see section \[properties\_and\_\_examples\]), the following: for every $\xi \in \C$, there exists an $h \in \O(\Omega)$, satisfying $h(a)=\xi$, and $C_{\delta}>0$ a constant, depending only on the dimension and $\delta$, such that $$\int_{ \Omega} |h(\zeta)|^2 e^{- 2 m \varphi( \zeta) - 2 (n-\delta)\log| \zeta-a|} d \lambda ( \zeta)
\leq
C_{\delta} e^{-2m \varphi(a) } |\xi|^2.$$ By the assumption (\[psihypotes\]) this implies that $$\int_{ \Omega} |h(\zeta)|^2 e^{- 2 m \varphi( \zeta) - 2 \psi(\zeta-a)} d \lambda ( \zeta)
\leq
C_{\delta} e^{-2m \varphi(a) } |\xi|^2.$$ Since this holds for every $\xi$ we can choose one such that the right-hand-side is equal to 1, i.e. $C_{\delta} e^{-2m \varphi(a) } |\xi|^2=1$. Then $h$ satisfies $$\int_{ \Omega} |h(\zeta)|^2 e^{- 2 m \varphi( \zeta) - 2 \psi(\zeta-a)} d \lambda ( \zeta)
\leq
1$$ and $$\log|h(a)|^2 = \log|\xi|^2 = -\log C_{\delta} + 2 m \varphi(a).$$ Thus, $$\Psi_a^m(a) \geq \varphi(a) - \frac{\log C_{\delta}}{2m}.$$ If $a$ is such that $\varphi(a)=- \infty$ this inequality is trivial. Thus, for every $m$ and $z \in \Omega$ we have that \[lelongapprox\] (z) - C\_ \_z\^m(z) \_[|a-|<r]{} { ( ) } +(l-n) r + C. We now want to show that this approximation behaves well with respect to generalized Lelong number with weight $\psi$. To this end, fix $a\in \Omega$ and let $\lambda > \nu_{a,\psi}( \Psi_{ a}^m( \cdot) )$, and put $$p = 1 +m \lambda, q=1+\frac{1}{m \lambda}.$$ Then $1/p +1/q=1$ and we apply Hölder’s inequality to the following integral, with $r(a)$ so small that $ \{ |a-\zeta|<r(a) \} \subset \subset \Omega$, $$\int_{|\zeta - a|<r} e^{-\frac{2m}{p} \varphi - 2 \psi(\zeta-a) } d \lambda(\zeta)
=
\int_{|\zeta - a|<r} e^{-\frac{2m}{p} \varphi - 2 \psi(\zeta-a)} (B_{a}^m(\zeta))^{1/p}(B_{a}^m(\zeta))^{-1/p} d \lambda(\zeta),$$ to obtain, since $-q/p = -1/(m \lambda)$, that it is dominated by $$\Big( \int_{|\zeta - a|<r} (B_{a}^m(\zeta)) e^{-2m \varphi} e^{- 2 \psi(\zeta-a)} d \lambda(\zeta) \Big)^{1/p}
\Big(\int_{|\zeta - a|<r} (B_{a}^m(\zeta))^{-1/m \lambda} e^{ - 2 \psi(\zeta-a)}d \lambda(\zeta)\Big)^{1/q}.$$ The first integral can be seen to be finite (cf. [@DemaillyKollar],p.21), as well as the second integral, since it equals $$\int_{|\zeta - a|<r} e^{- 2 \log B_a^m(\zeta)/{(2 m \lambda)} - 2 \psi(\zeta-a) } d \lambda(\zeta),$$ which is finite due to the assumption on $\lambda$. Since $\frac{1}{m/p} = \frac{1}{m} + \lambda$ this implies the inequality: \[psiestimate\] \_[a,]{}(()) \_[a,]{}(\_[a]{}\^m()) + 1/m.We need a lemma:
For every $m \in \mathbb{N}$ and $a \in \Omega$ we have that $$B_{a}^m(\zeta) \geq C |\zeta-a|^{(\frac{n-\delta}{l})(n+2)} \cdot B_{\zeta}^m(\zeta)$$ for every $\zeta$ in a small neighbourhood of $a$, where $C>0$ is an constant not depending on $\zeta$ or $a$.
For a fixed point $a$ in $\Omega$, choose $\zeta \neq a$ with distance less than $\min \{1,dist(a,\partial \Omega) \}$ to each other , and let $$M=\frac{n-\delta}{l}.$$ First, we claim that for $$z \in B(\zeta,2^{-M} |\zeta-a|^{M}),$$ the following inequality holds: $$\label{weightestimate}
e^{-2\psi(a-z)}\leq e^{-2\psi(\zeta-z)}.$$ Indeed, thanks to the assumptions (\[psihypotes2\]) and (\[psihypotes\]), for such $z$, $$e^{\psi(\zeta-z)}\leq |\zeta-z|^l \leq |a-z|^{n-\delta} \leq e^{\psi(a-z)},$$ since $$|\zeta-z|\leq 2^{(-M)} |\zeta-a|^{M} \leq |z-a|^{M},$$ where in the second inequality we used that $|\zeta-a|\leq 2|z-a| $. Now, let $$h\in\Lambda^m(\zeta):=\{h\in\mathcal{O}(\Omega),\int_{\Omega}\left|h(z)\right|^{2}e^{-2m\varphi(z)-2\psi(z-\zeta)}d\lambda(z)\leq1\}$$ be such that $h(\zeta)=B^m_{\zeta}(\zeta)$. To simplify notation we assume $m=1$, but the following calculations remains valid for any $m$. Take a smooth function $\theta$ with support in the ball $B(\zeta,2^{-M}|\zeta-a|^{M})$ satisfying $\theta=1$ in a neighbourhood of $B(\zeta,2^{-(1+M)}|\zeta-a|^{M})$, and $$|\bar{\partial}\theta(z)|\leq\frac{{1}}{|a-\zeta|^{2M}}.\label{eq:differentialineq}$$ Thus for every point $z\in supp\theta$ the inequality (\[weightestimate\]) holds. Moreover, we have that $$e^{-2(n+1)\log|z-\zeta|}=\frac{{1}}{|z-\zeta|^{2(n+1)}}\leq \frac{2^{(n+1)(M+1)}}{|a-\zeta|^{2M(n+1)}}$$ for $z\in supp\bar{\partial}\theta$ . Putting this information together we obtain the following estimate:
$$\label{integralestimate}
\int_{\Omega}|\bar{\partial}\theta h|^{2}e^{-2\varphi(z)-2\psi(a-z)-2(n+1)\log|z-\zeta|}d\lambda(z)
\leq$$
$$\leq\frac{2^{(n+1)(M+1)}}{|a-\zeta|^{2M(n+1)}}
\int_{\Omega}|\bar{\partial}\theta h|^{2}e^{-2\varphi(z)-2\psi(\zeta-z)}d\lambda(z)
\leq
\frac{2^{(n+1)(M+1)}}{|a-\zeta|^{2M(n+2)}},$$ since $h\in\Lambda(\zeta).$ Thus, by standard $L^2$-estimates, we can solve the equation $$\bar{\partial}u=\bar{\partial}(\theta h)=\bar{\partial}\theta\cdot h\label{eq:dbar}$$ with respect to the weight $e^{-2\varphi(z)-2\psi(a-z)-2(n+1)\log|z-\zeta|}.$ The singularity of the weight forces $u$ to vanish at $\zeta$, so if we define $F=\theta h-u,$ then $F(\zeta)=h(\zeta),$ and $F$ is holomorphic in $\Omega.$ Moreover, by the triangle inequality, $$\left(\int_{\Omega}|F|^{2}e^{-2\varphi(z)-2\psi(a-z)}d\lambda(z)\right)^{1/2}\leq\left(\int_{\Omega}|\theta h|^{2}e^{-2\varphi(z)-2\psi(a-z)}d\lambda(z)\right)^{1/2}+$$ $$+\left(\int_{\Omega}|u|^{2}e^{-2\varphi(z)-2\psi(a-z)}d\lambda(z)\right)^{1/2}.$$ Using (\[weightestimate\]) we have that $$\int_{\Omega}|\theta h|^{2}e^{-2\varphi(z)-2\psi(a-z)}d\lambda(z)\leq\int_{\Omega}|\theta h|^{2}e^{-2\varphi(z)-2\psi(\zeta-z)}d\lambda(z)\leq1,$$ and also we see that $$\int_{\Omega}|u|^{2}e^{-2\varphi(z)-2\psi(a-z)}d\lambda(z)\leq \int_{\Omega}|u|^{2}e^{-2\varphi(z)-2\psi(a-z)-2(n+1)\log|z-\zeta|}d\lambda(z)\leq$$ $$\leq C' \int_{\Omega}|\bar{\partial}\theta h|^{2}e^{-2\varphi(z)-2\psi(a-z)-2(n+1)\log|z-\zeta|}d\lambda(z),$$ where the first inequality comes from the assumption that $|z-\zeta|\leq|a-\zeta|\leq 1$, and the last inequality, as well as the constant $C'$ (which only depends on $\Omega$), comes from the $L^{2}-$estimate obtained from solving . Using (\[integralestimate\]) we arrive at $$\int_{\Omega}|F|^{2}e^{-2\varphi(z)-2\psi(a-z)}d\lambda(z)\leq1+C'\frac{2^{(n+1)(M+1)}}{|a-\zeta|^{2M(n+2)}}\leq\frac{C_1}{|a-\zeta|^{2M(n+2)}},$$ where $C_1$ is a constant independent of $\zeta$ and $a.$ Thus, if we define the function $$\tilde{F}(z)=\frac{|a-\zeta|^{M(n+2)}}{\sqrt{C_1}}F(z)$$ then $\tilde{F}$ belongs to $\Lambda(a)$ and satifies $$|\tilde{F(\zeta)|}=C_1^{-1/2}|B^{m}_{\zeta}(\zeta)|\cdot|a-\zeta|^{M(n+2)}.$$ This shows that for every $a$ and $\zeta$ (the inequality is trivial if $\zeta=a$), $$|B^{m}_{a}(\zeta)|\geq C |B^m_{\zeta}(\zeta)|\cdot|a-\zeta|^{M(n+2)},$$ where $C=C_1^{-1/2}$ does not depend on $\zeta$ or $a$.
Using the lemma we see that for each $a \in \Omega$ , $$\frac{{1}}{2m}\log|B^{m}_{a}(\zeta)|\geq\frac{{1}}{2m}\log|B^{m}_{\zeta}(\zeta)|+\frac{{M(n+2)}}{2m}\log|a-\zeta|+C/m,$$ for $\zeta$ close to $a$, which implies that$$\nu_{a,\psi}(\Psi_a^m(\cdot))\leq\nu_{a,\psi}(\Psi_{\cdot}^m(\cdot))+\frac{{C}}{2m}.$$ Combining with (\[psiestimate\]) we obtain that $$\nu_{a,\psi}(\varphi(\cdot)) \leq \nu_{a,\psi}(\Psi_{\cdot}^m(\cdot)) + C/m,$$ for every $a \in \Omega$.
On the other hand, the left-hand estimate of (\[lelongapprox\]) implies that $$\nu_{a,\psi}(\varphi(\cdot)) \geq \nu_{a,\psi}(\Psi_{\cdot}^m(\cdot)).$$ Thus we have proved:
Assume $\psi$ satisfies l|z| (z) (n-)|z|, for some small, fixed $\delta > 0$. Then for $\varphi \in PSH(\Omega)$, $m \in \mathbb{N}$, $z,a \in \Omega$ and every $r<d(z, \ds \Omega)$ we have that \[lel\_appr\] (z) - C\_ \_z\^m(z) \_[|z-|<r]{} { ( ) } +(l-n) r + C/m and \[lel\_appr\_label\] \_[a,]{}(()) - \_[a,]{}(\_\^m ()) \_[a,]{}(()), where $C$ is a constant depending on $\Omega,l,\delta$. In particular, $\Psi_{z}^m( z )$ converges to $\varphi(z)$ as $m \rightarrow + \infty$ both pointwise and in $L_{Loc}^1$.
If $\psi=0$, as in the original theorem of Demailly, then $\Psi_a^m$ are plurisubharmonic functions with analytic singularities, and thus we approximate $\varphi$ with plurisubharmonic functions with analytic singluarities. For instance, this gives a very simple proof of Siu’s analyticity theorem for the classical Lelong number. In our setting however, it is unclear, and an interesting question, if the presence of $\psi$ allows for $\Psi_{a}^m $ to have analytic singularities.
One can show that when comparing the approximations $\Psi_z^m$ to the classical Lelong number we can obtain, instead of (\[lel\_appr\_label\]), the following inequalities: $$\nu_{a,n-1}(\varphi(\cdot)) - \frac{n-l}{m} \leq \nu_{a,n-1}(\Psi_{\cdot}^m (\cdot)) \leq \nu_{a,n-1}(\varphi(\cdot)).$$ The approximation of Demailly, that is with $\psi=0$, satisfied these inequalities with $l=0$.
Kiselman’s directed Lelong number {#kiselmans_section}
=================================
In this section we relate the classical Lelong number to yet another integral. Using this relation, we can interpret Kiselman’s directional Lelong number (defined below) as our generalized Lelong number, in a generic sense. Equation (\[skodaintegral\]) shows us that if $ \nu_0 (\varphi)<1$ and the integral of $e^{-2 \varphi}$ restricted to the coordinate axes is finite, then $$\int_0 e^{ \frac{2\varphi}{1-\epsilon} -2(1-\delta) \sum_{i \neq j} \log|z_i| } d \lambda(z) < +\infty,$$ for every $ \delta >0$ and $j=1...n$, and for some small $\epsilon>0$. Applying the inequality between geometric and arithmetic mean (i.e. $ \frac{a_1+...+a_n}{n} \geq (a_1 \cdots a_n)^{\frac{1}{n}}$ for $ a_i \geq 0 $) we see that $$\frac{1}{n}\sum_{j=1}^n e^{-2 \sum_{i \neq j} \log|z_i| } = \frac{1}{n}\sum_{j=1}^n \frac{1}{\prod_{i \neq j} |z_i|^2} \geq \prod_{j=1}^n \frac{1}{\prod_{i \neq j} |z_i|^{\frac{2}{n}}} = \prod_{i=1}^n \frac{1}{|z_i|^{\frac{2(n-1)}{n}}}.$$ This last expression is equal to $$e^{-2 \frac{n-1}{n}\sum_{i = 1}^n \log|z_i| }$$ which tells us that $$\int_0 e^{ - \frac{2\varphi}{1-\epsilon} -2(1-\delta)\frac{n-1}{n}\sum_{i = 1}^n \log|z_i|} d \lambda(z) < + \infty.$$ Using a similar Hölder argument as in Lemma \[singularitylemma\], we get $$\int_0 e^{ - \frac{2\varphi}{1-\epsilon+\delta'} -2\frac{n-1}{n}\sum_{i = 1}^n \log|z_i|} d \lambda(z) < + \infty,$$ for $\delta'>0$ sufficiently small. Furthermore, if $\delta'$ is so small that $\delta' < \epsilon$, then $$\int_0 e^{ - 2\varphi -2\frac{n-1}{n}\sum_{i = 1}^n \log|z_i|} d \lambda(z) < + \infty.$$ On the other hand, if $$\int_0 e^{ - 2\varphi -2\frac{n-1}{n}\sum_{i = 1}^n \log|z_i|} d \lambda(z) < + \infty$$ then, using the pointwise estimate $ \log|z_i| \leq \log|z|$, we see that $$\int_0 e^{ - 2\varphi -2(n-1)\log|z|} d \lambda(z) < + \infty$$ which is equivalent to $ \nu_{0,n-1} (\varphi) <1.$ Thus we have proved the following lemma:
\[dir\_lel\_lemma\] For a plurisubharmonic function $\varphi$, which satisfies that the restriction of $e^{-2 \varphi}$ to the coordinate axes is integrable, the condition $\nu_{0,n-1} (\varphi) <1$ is equivalent to $$\int_0 e^{ - 2\varphi -2\frac{n-1}{n}\sum_{i = 1}^n \log|z_i|} d \lambda(z) < + \infty.$$
Instead of saying that the lemma holds under the condition that the integral of the restriction of $e^{-2 \varphi}$ to the coordinate axes is finite, we can say that it holds for some generic rotation of $e^{-2 \varphi}$. This is so since if the Lelong number of $\varphi$ is smaller than 1, then a generic rotation of $e^{-2 \varphi}$ is integrable along the coordinate axes.
We now define the directed Lelong number, due to Kiselman (cf. [@Kiselman]).
For $a_j \geq 0$ the directed Lelong number at a point $w$ is defined as $$\nu_{w} (\varphi,(a_1,...a_n)) = \limsup_{r \rightarrow 0} \frac{ \sup_{|z_i-w_i|=r^{a_i} } \varphi(z_1,..,z_n) }{\log r}.$$
It is proved in [@Kiselman] that the function which the limsup is taken over is increasing, and so we can exchange the limsup for a limit. Also, for $a_i=1$ for every $i$, we obtain the classical Lelong number.
For $a_i = \frac{p_i}{q} \in \mathbb{Q}_+$ the directional Lelong number satisfies $$\nu_{w} (\varphi,(a_1,...a_n)) = q^{-1}\nu_{w} (\varphi(z_1^{p_1},...,z_n^{p_n})).$$
Since the result is local we can assume that $w=0$. By homogenity of the directional Lelong number it is enough to consider the case $q=1$. We have, with $z_i=r_i e^{i \theta_i}$, that \[dir\_lel\_expr\] \_[|z\_i|=r\^[p\_i]{} ]{} (z\_1,..,z\_n) = \_[r\_i=r\^[p\_i]{} ]{} (r\_1 e\^[i \_1]{},..,r\_n e\^[i \_n]{}) = \_[r\_i=r ]{} (r\_1\^[p\_i]{} e\^[i \_1]{},..,r\_n\^[p\_n]{} e\^[i \_n]{}). Since $p_j \in \mathbb{N}_+$ this last expression is equal to $$\sup_{r_i=r } \varphi(r_1^{p_i} e^{i \theta_1 p_1},..,r_n^{p_n} e^{i \theta_n p_n}) =
\sup_{|z_i|=r } \varphi(z_1^{p_1} ,..,z_n^{p_n}),$$ and we are done.
\[thm\_gen\_dir\_lelong\_numbers\] Let $\varphi$ be plurisubharmonic in $\Omega$. Then, for a generic rotation of $\varphi$, $$\nu_{w} (\varphi,(a_1,...a_n))<1$$ iff $$\int_{w} e^{-\frac{2\varphi}{q} - \sum_{i=1}^n (1-\frac{1}{n p_i})\log|z_i-w_i| } d \lambda(z) < +\infty,$$ for $a_i = \frac{p_i}{q} \in \mathbb{Q}_+$.
Assume $w=0$. By the above lemma the hypotesis implies, $$\nu_{w} (q^{-1} \varphi(z_1^{p_1},...,z_n^{p_n})) < 1$$ which by Lemma \[dir\_lel\_lemma\] implies that \[sumintergal\] \_0 e\^[ - 2q\^[-1]{} ( z\_1\^[p\_1]{},...,z\_n\^[p\_n]{} ) -2 \_[j = 1]{}\^n |z\_j|]{} d (z) < + . By applying the change of coordinates $z_j^{p_j} = y_j$, we see that the integral in (\[sumintergal\]) is equal to $$\int_0 e^{ - 2 q^{-1} \varphi(y_1,...,y_n) -2 \sum_{j = 1}^n \log|y_j|({1- \frac{1}{p_j} + \frac{n-1}{n} \frac{1}{p_j}})} d \lambda(y)$$ Since $1- \frac{1}{p_j} + \frac{n-1}{n} \frac{1}{p_j} = 1 - \frac{1}{np_j}$, this equals $$\int_0 e^{ - 2q^{-1}\varphi(y_1,...,y_n) -2 \sum_{j = 1}^n \log|y_j|({1- \frac{1}{n p_j}})} d \lambda(y),$$ which consequently is finite. This proves the “only if” part. However, the exact same argument used in the opposite direction proves the “if” part.
It is easy to see that $\varphi$ must satisfy some condition in terms of integrability. Take for instance $\varphi = \log{z_1}$ for which $\int_{ \{ z_2=...=z_n=0\}} e^{-2 \varphi} d \lambda(z) = + \infty$. Then, it is easily verified that $\nu_{w} (\varphi,(1,...1)) = 1$ but of course, $$\int_{w} e^{-2\varphi - \sum_{i=1}^n (1-\frac{1}{n})\log|z_i-w_i| } d \lambda(z) = +\infty.$$
One knows that for the weight $ \psi = \log \max_i {|z_i|^{1/a_i} }$ the relative type, $\sigma(\varphi, \psi)$ and Demailly’s generalized Lelong number $\nu_{Demailly}(\varphi, \psi)$ (defined in the introduction) both coincide with Kiselman’s directed Lelong number. However, it is unkown if there exists a weight $\psi$ for which $\nu_{z,\psi}$ coincides with the directional Lelong number in more than the generic sense found above. \#1\#2\#3[[\#1]{}: [*\#2*]{}, \#3.]{}
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abstract: 'Recently, Chen *et al. \[Phys. Rev. A [**84,**]{} 033835 (2011)\] reported observation of anticorrelated photon coincidences in a Mach-Zehnder interferometer whose input light came from a mode-locked Ti:sapphire laser that had been rendered spatially incoherent by passage through a rotating ground-glass diffuser. They provided a quantum-mechanical explanation of their results, which ascribes the anticorrelation to two-photon interference. They also developed a classical-light treatment of the experiment, and showed that it was incapable of explaining the anticorrelation behavior. Here we show that semiclassical photodetection theory—i.e., classical electromagnetic fields plus photodetector shot noise—does indeed explain the anticorrelation found by Chen *et al. The key to our analysis is proper accounting for the disparate time scales associated with the laser’s pulse duration, the speckle-correlation time, the interferometer’s differential delay, and the duration of the photon-coincidence gate. Our result is consistent with the long-accepted dictum that laser light which has undergone linear-optics transformations is classical-state light, so that the quantum and semiclassical theories of photodetection yield quantitatively identical results for its measurement statistics. The interpretation provided by Chen *et al. for their observations implicitly contradicts that dictum.***'
author:
- 'Jeffrey H. Shapiro'
- Eric Lantz
date: 'January 6, 2012'
title: 'Comment on “Observation of anticorrelation in incoherent thermal light fields”'
---
The recent paper by Chen *et al. [@Chen] reports the following experiment. A continuous-wave mode-locked Ti:sapphire laser operating at $\lambda = 780$nm wavelength with 78MHz pulse-repetition frequency, and a $\tau_p\sim$150fs pulse duration illuminated an interference filter, to somewhat increase the pulse duration, followed by a rotating ground-glass diffuser, to render the light spatially incoherent. The diameter $D=4.5$mm output beam from the diffuser was divided by a 50-50 beam splitter, with the resulting beams propagating $d\approx 200$mm (from the diffuser) to collection planes, each of which contained the tip of a single-mode optical fiber. These fibers routed the light they collected to another 50-50 beam splitter whose outputs illuminated single-photon detectors. By sufficiently offsetting, in their respective planes, the transverse coordinates of the fiber tips that collected light from the diffuser, Chen *et al. ensured that there was no first-order interference in the Mach-Zehnder interferometer formed by the two 50-50 beam splitters and the intervening fibers. They measured photon coincidences between the two detectors, with a $T\sim 1$ns coincidence gate, as one of the collection fibers was moved longitudinally to create a $-$2ps $\le \delta t\le 2$ps differential delay. What they observed was a pronounced dip—a photon anticorrelation—in the coincidence rate, despite the absence of any delay-dependence in the singles rates. See [@Chen] Fig. 3 for a diagram of the Chen *et al. experiment, and [@Chen] Fig. 4 for their observations of anticorrelation.***
Chen *et al. provided a quantum-mechanical explanation for the anticorrelation seen in their experiment, which shows that it is due to two-photon interference. Because light is quantum mechanical, and photodetection is a quantum measurement, there must be a quantum explanation for the results in [@Chen]. But the authors of [@Chen] do more than provide a quantum explanation for their observations. They presents a classical-field analysis that, they claim, proves that *only a quantum treatment can account for the anticorrelation they found. Were these authors correct, their work would present a very significant conundrum for quantum optics. Laser light, except for any excess noise it may carry, is coherent-state light. Passage through a ground-glass diffuser, free-space propagation, beam splitting, and fiber propagation are all linear optical effects, with the first best modeled as a random process while the rest can be taken to be deterministic. Taken together, the preceding two sentences imply that the joint quantum state of the fields illuminating the two detectors in Fig. 3 of Chen *et al. is *classical, viz., it is a random mixture of coherent states. It has long been known that the quantum and semiclassical [@footnote1] theories of photodetection yield quantitatively *identical predictions for classical-state illumination, see [@Shapiro] for a detailed review of this topic.*****
So, in view of the preceding discussion, we can say that one of three things must be true: (1) despite what is argued in [@Chen], there *is a classical explanation for the anticorrelation reported therein; or (2) laser light that has undergone linear transformation is *not in a coherent state or a random mixture of coherent states; or (3) the quantum and semiclassical theories of photodetection *can make different quantitative predictions for the measurement statistics of classical-state illumination. To assert the truth of items (2) and/or (3), as Chen *et al. implicitly do, would constitute a major upheaval in quantum optics. We shall show that item (1) holds. The key to doing so is proper accounting for the disparate time scales associated with the laser’s pulse duration, the speckle-correlation time, the interferometer’s differential delay, and the duration of the photon-coincidence gate.****
With the interference filter in place, the duration of the laser pulse that illuminated the ground-glass diffuser in Fig. 3 of [@Chen] was increased to either $\tau_p \sim 345\,$fs or $\tau_p \sim 541$fs, depending on which of two interference filters was employed. The linear velocity of the rotating ground-glass where it was illuminated was $\sim$0.8m/s [@Chen], so that for either interference filter it is fair to assume that the ground glass was completely stationary while a single laser pulse propagated through it. In other words, the speckle correlation time greatly exceeded $\tau_p$. The differential delay over which Chen *et al. traced out coincidence rates was $|\delta t|\le 2$ps. Thus the duration of the photon-coincidence gate in Fig. 3 of [@Chen] obeyed $T \gg |\tau_p \pm \delta t|$.*
Chen *et al. used single-mode fibers to collect spatial samples of the two light beams that had propagated $d\approx 200$mm from the diffuser and been separated by the initial 50-50 beam splitter in their Fig. 3. Coherence theory [@MandelWolf] shows that the fields at that distance from the diffuser have $\ell_c\sim\lambda d/D \approx 35\,\mu$m transverse coherence lengths [@footnote2]. The data in Fig. 4 of [@Chen] was collected with more than $40 \ell_c$ transverse separation, in their respective collection planes, between the tips of the single-mode fibers, whose core diameters we shall assume to be much smaller than $\ell_c$. Hence the field injected in each fiber comes from a unique coherence cell, in time as well in space. This ensures that every fiber-collected femtosecond pulse is coherent, although with a random phase and amplitude. Moreover, the pulses in each fiber arise from different coherence cells, and so their random behaviors are statistically independent. Nevertheless, it is incorrect to assert (cf. Sec. 4 of [@Chen]) that the light beams emerging from the two fibers do not interfere. Rather, they produce fringes that are random between pulses separated by more than the decorrelation time of the pseudothermal source. More importantly, energy conservation implies there will be anticorrelation at output ports 1 and 2 in the Chen *et al. experiment, viz., a bright fringe in port 1 is always accompanied by a dark fringe in port 2. As noted in [@Chen], this anticorrelation would not depend on the interferometer’s differential delay for continuous-wave (statistically stationary) pseudothermal light. Chen *et al., however, used femtosecond pulses, for which the anticorrelation disappears when the pulses do not overlap in time at the second 50-50 beam splitter, and this loss of anticorrelation occurs even though the necessary differential delay is much shorter than the photodetectors’ nanosecond coincidence window.***
The argument presented in the preceding paragraph constitutes a complete explanation of the Chen *et al. anticorrelation in terms of classical interference behavior. We will now expand upon that classical-field explanation to provide a full quantitative treatment. We define $E_+(t)$ and $E_-(t)$ to be the $\sqrt{\mbox{photons/s}}$-units positive-frequency *classical fields entering the single-mode fibers from a single pulse occurring at time $t=0$ [@footnote3]. Given that the speckle is frozen over a single laser pulse, and that the fibers have core diameters which are much smaller than $\ell_c$, it is fair to write these fields as follows: $$E_{\pm}(t) = v_\pm f(t\pm\delta t/2)e^{-i\omega_0t},
\label{Epm}$$ where $v_+$ and $v_-$ are independent, identically distributed, zero-mean, isotropic, complex-valued Gaussian random variables with common mean-squared strength $$\langle |v_+|^2\rangle = \langle |v_-|^2\rangle = N,$$ and $$f(t) \equiv \frac{e^{-t^2/\tau_p^2}}{(\pi\tau_p^2/2)^{1/4}},
\label{pulse}$$ is a transform-limited Gaussian pulse normalized to satisfy $$\int\!dt\,|f(t)|^2 = 1.$$ Physically, $v_+$ and $v_-$ are the constant-in-time speckle values for the given laser pulse, whose independence is guaranteed by the large transverse separation of the fibers in their respective collection planes. Our $f(t)$ normalization then implies that $N\hbar \omega_0$, with $\omega_0 = 2\pi c/\lambda$, is the average energy entering each of the fibers from the given laser pulse. Thus $N$ measures the average energy of these classical fields in photon units and, because the measurements reported in [@Chen] were made in the photon-counting regime, we will assume $N \ll 1$. The fields that illuminate the photodetectors, which will denote $ E_1(t)$ and $ E_2(t)$, as was done in [@Chen], are then given by $$E_1(t) \equiv \frac{E_+(t) + E_-(t)}{\sqrt{2}}
\label{EaDefn}$$ and $$E_2(t) \equiv \frac{E_+(t) -E_-(t)}{\sqrt{2}}
\label{EbDefn}.$$ Furthermore, because $N\ll 1$, we can say that the average singles rates (counts/gate) and coincidence rate (coincidences/gate) obey [@Shapiro] $$S_K = \eta\int_{-T/2}^{T/2}\!dt\,\langle |E_K(t)|^2\rangle\quad\mbox{for $K= 1,2$,}$$ and $$C_{12} = \eta^2\int_{-T/2}^{T/2}\!dt\int_{-T/2}^{T/2}\!du\,\langle | E_1(t)|^2| E_2(u)|^2\rangle,$$ where $\eta$ is the photodetectors’ quantum efficiency. All that remains is to evaluate these rates.**
Using the statistical independence of $v_+$ and $v_-$ and their common mean-squared value, we immediately find that $$\begin{aligned}
\lefteqn{S_1 = S_2 =} \nonumber \\[.1in]
&& \frac{\eta N}{2} \int_{-T/2}^{T/2}\!dt\,(|f(t+\delta t/2)|^2 + |f(t-\delta t/2)|^2)\\[.1in] &\approx& \eta N,\end{aligned}$$ where the approximation follows from $|\tau_p \pm \delta t/2| \ll T$ and Eq. (\[pulse\]). Similarly, for the coincidence rate, the statistical independence of $v_+$ and $v_-$ leads to [@footnote4] $$\begin{aligned}
C_{12} &=& \frac{\eta^2}{4}\int_{-T/2}^{T/2}\!dt\,\int_{-T/2}^{T/2}\!du\,[\langle |v_+|^4\rangle |f(t_+)|^2|f(u_+)|^2 \nonumber\\[.1in]
&+& \langle |v_+|^2\rangle \langle |v_-|^2\rangle |f(t_+)|^2|f(u_-)|^2 \nonumber \\[.1in]
&+& \langle |v_+|^2\rangle \langle |v_-|^2\rangle |f(u_+)|^2|f(t_-)|^2 \nonumber\\[.1in]
&-&2\langle |v_+|^2\rangle \langle |v_-|^2\rangle{\rm Re}[f^*(t_+)f^*(u_-)f(t_-)f(u_+)] \nonumber \\[.1in]
&+& \langle |v_-|^4\rangle |f(t_-)|^2|f(u_-)|^2],
\label{C12dev}\end{aligned}$$ where $t_\pm \equiv t \pm \delta t/2$ and $u_\pm \equiv u \pm \delta t/2$. Now, using the Gaussian moment-factoring theorem [@WJ], $|\tau_p\pm\delta t/2| \ll T$, and Eq. (\[pulse\]), we can reduce the preceding expression to $$C_{12} \approx \frac{\eta^2N^2}{2}\left(3-\left|\int_{-T/2}^{T/2}\!d\tau\,f^*(\tau+\delta t/2)f(\tau-\delta t/2)\right|^2\right).$$ Using $|\tau_p\pm \delta t| \ll T$ and Eq. (\[pulse\]) then gives us our final result, $$C_{12} \approx \frac{\eta^2N^2}{2}\left(3-e^{-\delta t^2/\tau_p^2}\right).
\label{Cab}$$
Equation (\[C12dev\]) can be obtained in a slightly different way to emphasize the presence of anticorrelated fringes at the output ports. The intensities at these ports can be obtained from Eqs. (\[EaDefn\]) and (\[EbDefn\]) as: $$\begin{aligned}
| E_1(t)|^2&=& \frac{1}{2}(|v_+|^2 |f(t_+)|^2+|v_-|^2 |f(t_-)|^2\nonumber \\[.1in]
&&+2|v_+||v_-|\,{\rm Re}[f^*(t_+) f(t_-)e^{i\Delta\varphi}]\label{fringes1} \\[.1in]
| E_2(u)|^2&=& \frac{1}{2}(|v_+|^2 |f(u_+)|^2+|v_-|^2 |f(u_-)|^2\nonumber \\[.1in]
&& -2|v_+||v_-|\,{\rm Re}[f^*(u_+) f(u_-)e^{i\Delta\varphi}]
\label{fringes2}\end{aligned}$$ where $\Delta\varphi \equiv \varphi_--\varphi_+$ in terms of the phases, $\varphi_+$ and $\varphi_-$, associated with $v_+$ and $v_-$. Equation (\[C12dev\]) can be retrieved from Eqs. (\[fringes1\]) and (\[fringes2\]) by noting that the amplitude and phase of $v_\pm$ are statistically independent, with $\varphi_\pm$ being uniformly distributed on $0\le \varphi_\pm \le 2\pi$, so that $\langle e^{i\varphi_\pm}\rangle = \langle e^{i2\varphi_\pm}\rangle = 0$. Equations (\[fringes1\]) and (\[fringes2\]) also show that the interference—and hence the anticorrelation—disappears when the pulses no longer overlap in time at the second 50-50 beam splitter, because $$|\delta t| >> \tau_p \Rightarrow f^*(t_+) f(t_-)=0,\,\, \forall\, t.$$
At this point we have accomplished our objective. Our simple classical-field theory predicts singles rates that are independent of the differential delay, and a coincidence rate that exhibits a pronounced dip (anticorrelation) within a (post interference-filter) laser pulse duration, in agreement with the experimental results from [@Chen]. We shall close by delving a little deeper into how our work stacks up against those experiments. We have assumed $N\ll 1$, i.e., that the average photon number coupled into each fiber from a single laser pulse is much smaller than one. Our theory gives $$\max(C_{12})/S_1 = 3\eta N/2.$$ From Fig. 4 of [@Chen] we then get $3\eta N/2 \approx 0.004$ that, for reasonable values of $\eta$ (say, $\eta \sim 0.1$), is consistent with $N$ being much smaller than one [@footnote5].
Our theory predicts that the anticorrelation dip has visibility $$\mathcal{V} \equiv \frac{\max(C_{12})-\min(C_{12})}{ \max(C_{12}) + \min(C_{12})} = 1/5,$$ which is in reasonable agreement with the experimental results from Fig. 4 of [@Chen]. If we eliminate accidental coincidences (the terms that Chen *et al. refer to as “self-intensity correlations”) from our theory—by subtracting from $C_{12}$ in Eq. (\[Cab\]) the coincidence rate when $E_+(t)=0$ and the coincidence rate when $E_-(t) = 0$—we get $$C_{12} \approx \frac{\eta^2N^2}{2}\left(1-e^{-\delta t^2/\tau_p^2}\right),$$ which implies the anticorrelation dip has perfect, $\mathcal{V} = 1$, visibility. When Chen *et al. do the like correction to their anticorrelation data, they find near-unity visibility, in agreement with our theory.**
In conclusion, we have provided a classical explanation for the anticorrelation experimental results reported in [@Chen]. Thus those experimental results do *not require us to abandon the well-accepted precepts that laser light through linear-optics transformations can be modeled as a coherent-state or a classical mixture of coherent states, and that the photodetection measurement statistics for such states can be computed from semiclassical theory, in which the light is treated classically.*
The work of J.H.S. was supported by the DARPA Information in a Photon Program under U.S. Army Research Office Grant No. W911NF-10-1-0404.
[2]{}
H. Chen, T. Peng, S. Karmakar, Z. Xie, and Y. Shih, Phys. Rev. A [**84,**]{} 033835 (2011). In the semiclassical theory of photodetection, light is taken to be a classical electromagnetic wave, and the discreteness of the electron charge leads to shot noise as the fundamental noise in photodetection. Because the shot noises from physically independent photodetectors are statistically independent, the explanation we will present below for the anticorrelation seen in [@Chen] will derive solely from randomness in classical electromagnetic fields. J. H. Shapiro, IEEE J. Sel. Top. Quantum Electron. [**15,**]{}1547 (2009). L. Mandel and E. Wolf, *Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995). This distance is also the characteristic size of the speckles cast by the diffuser, see J. W. Goodman, *Speckle Phenomena in Optics: Theory and Applications(Roberts & Co., Englewood Colo. 2007). Throughout our analysis we will neglect all propagation delays with the exception of the interferometer’s differential delay, $\delta t$. This same result can be obtained from the Gaussian moment-factoring theorem, which implies that $\langle | E_1(t)|^2| E_2(u)|^2\rangle$ = $\langle | E_1(t)|^2\rangle\langle | E_2(u)|^2\rangle$ + $|\langle E^*_1(t) E_2(u)\rangle|^2$ for the case at hand. It is here where the classical-field analysis in Chen *et al. goes astray. In particular, our Eqs. (\[Epm\]), (\[EaDefn\]), and (\[EbDefn\]) yield $\langle E^*_1(t) E_2(u)\rangle = N[f^*(t_+)f(u_+) - f^*(t_-)f(u_-)]/2$, which is dependent on the interferometer’s differential delay $\delta t$. In contrast, Chen *et al. incorrectly assert, below their Eqs. (28) and (29), that no such $\delta t$ dependence exists in the corresponding terms from their moment-factoring analysis. \[The reader should note, in this regard, that our $E_+(t)$ and $E_-(t)$ correspond, respectively, to $ E_A(t)$ and $ E_B(t)$ from Chen *et al.\] J. M. Wozencraft and I. M. Jacobs, *Principles of Communication Engineering (Wiley, New York 1965). Strictly speaking, our classical-field theory only needs $\eta N\ll 1$, rather than $N\ll 1$, so that no supposition about $\eta$ is actually necessary.******
|
---
abstract: 'In this paper we introduce the *robust random number generation* problem where the goal is to design an abstract tile assembly system (aTAM system) whose terminal assemblies can be split into $n$ partitions such that a resulting assembly of the system lies within each partition with probability 1/$n$, regardless of the relative concentration assignment of the tile types in the system. First, we show this is possible for $n=2$ (a *robust fair coin flip*) within the aTAM, and that such systems guarantee a worst case $\BO(1)$ space usage. We accompany our primary construction with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility, and also prove some negative results. As an application, we combine our coin-flip system with a result of Chandran, Gopalkrishnan, and Reif to show that for any positive integer $n$, there exists a $\BO(\log n)$ tile system that assembles a constant-width linear assembly of expected length $n$ for any concentration assignment. We then extend our robust fair coin flip result to solve the problem of robust random number generation in the aTAM for all $n$. Two variants of robust random bit generation solutions are presented: an unbounded space solution and a bounded space solution which incurs a small bias. Further, we consider the harder scenario where tile concentrations change arbitrarily at each assembly step and show that while this is not possible in the aTAM, the problem can be solved by exotic tile assembly models from the literature.'
author:
- Cameron Chalk
- Bin Fu
- Eric Martinez
- Robert Schweller
- Tim Wylie
bibliography:
- 'tam.bib'
title: 'Concentration Independent Random Number Generation in Tile Self-Assembly[^1]'
---
Department of Computer Science\
The University of Texas - Rio Grande Valley\
Edinburg, TX, 78539-2999\
[{cameron.chalk01, bin.fu, eric.m.martinez02, robert.schweller, timothy.wylie}@utrgv.edu]{}
Introduction {#sec:introduction}
============
Definitions and Model: Tiles, Assemblies, and Tile Systems {#sec:definitions}
==========================================================
Robust Fair Coin Flipping in the aTAM {#sec:bounded}
=====================================
Robust Simulation of Randomized Linear Assemblies {#sec:linear}
=================================================
Robust Fair Coin Flipping at Temperature 1 {#sec:boundedt1}
==========================================
Robust Random Number Generation in the aTAM {#sec:rng}
===========================================
1-Extensible Robust Fair Coin Flipping in the aTAM {#sec:1ext}
==================================================
Robust Fair Coins with Unstable Concentrations {#sec:unstable}
==============================================
Other Self-Assembly Models {#sec:alternative}
==========================
Conclusions and Future Work {#sec:conclusion}
===========================
[^1]: This research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1555626.
|
---
abstract: 'Increased demands in the field of scientific computation require a more efficient implementation of algorithms. Maintaining correctness, in addition to efficiency, is a challenge software engineers in the field have to face. In this report, we share our first impressions and experiences with the applicability of formal methods to such design challenges arising in the development of scientific computation software, in the field of material science. We investigated two different algorithms, one for load distribution and one for the computation of convex hulls, and demonstrate how formal methods have been used to discover counterexamples to the correctness of the existing implementations, and to prove the correctness of a revised algorithm. The techniques employed include SMT solvers, and automatic and interactive verification tools.'
author:
- Bernhard Beckert
- Britta Nestler
- Moritz Kiefer
- |
\
Michael Selzer
- Mattias Ulbrich
bibliography:
- 'paper.bib'
title: |
Experience Report:\
Formal Methods in Material Science
---
Introduction
============
The demands on precision and the extent of simulations, and other scientific computation applications increase continuously. In contrast to these requirements, Moore’s law for technological advances of processors faces a foreseeable stagnation, which will make a future development of more efficient software unavoidable and necessary in this area. With the requirement to design scientific software with more sophisticated algorithms, using distributed computing, efficient memory technologies, *etc.*, the number of pitfalls grows, by which flaws could accidentally be introduced into the code.
Bugs in scientific software are hard to discover by testing, since they may only occur at inputs arising after a longer (simulation) runtime. This challenge in software development thus appears to be an opportunity for static formal analyses which analyse the code symbolically, and thus cover all possible runtime situations – regardless of the time required to reach the value during an actual run.
In order to verify the hypothesis “Formal methods can be employed successfully in the design process for scientific software”, we tested it against two case studies from real scientific applications, for which the location of problematic code had previously been identified. In one case, the challenge was to find a correct implementation which meets the intention of the original code, without repeating its flaws. In the other case, the flaw had already been corrected. The challenge was to verify that the correction solves the existing problem.
The paper continues with a presentation of the two case studies, which is followed by a summary of the observations made during the analysis process, and it concludes with an outlook into future work.
Case Study : Load Distribution {#sec:load-distr}
==============================
For an efficient use of resources, provided by large computing clusters, load distribution is a crucial challenge. In this case study, we analyse an algorithm to rebalance the load within a cluster, in case the number of tasks to be computed changes. An original version of the algorithm in question has been in use in a scientific computation software, and its correctness could not be established by manual code inspection.
This document reports how formal analyses helped us to formulate a correct load balancing algorithm by
1. finding a subtle flaw in original implementation, using a custom fuzzer, and by
2. proving the correctness of an improved implementation, using [*Why3*]{} [@why3].
#### Problem Statement {#sec:prob-stm-i}
The starting point for this case study was an algorithm taken from a scientific computation library which was actively in use. The algorithm computes the distribution of $\mathbf{s}$ tasks onto $\mathbf{n}$ cluster nodes from a given distribution for $\mathbf{t}$ tasks. This routine is called within a simulation framework when the number of simulated entities is refined (usually increased).
More formally, the problem can be stated as follows: Given a set $S=\{s_{1},\ldots,s_{n}\}$ with $s_{i}\in \mathbb{N}_{+}$ and $\mathbf{s} = \sum_{i=1}^{n} s_{i}$ and a natural number $\mathbf{t}\in\mathbb{N}_{+}$, produce a set $T=\{t_{1},\ldots,t_{n}\}$ such that $\mathbf{t} = \sum_{i=1}^{n} t_{i}$. The respective values $s_i$ and $t_i$ are the number of tasks to be run on the $i$-th node, before and after load balancing. The objective of the algorithm is to obtain a new load distribution that is “close” to the old distribution, i.e., $\frac{t_{i}}{s_{i}}$ should be close to $\frac{\mathbf{t}}{\mathbf{s}}$.
#### Starting Point
The original algorithm shown in Listing. \[lst:original\] used floating-point numbers to calculate $s_{i}\frac{\mathbf{t}}{\mathbf{s}}$ for each $i$. The integral part of this value then was assigned to $t_{i}$, while the fractional parts were accumulated until they made up $1$ full node, which was then assigned to the current node. This algorithm computes a correct distribution among the nodes, if executed on precise rational numbers. However, when using floating-point values to approximate rationals, the imprecisions can result in tasks getting lost in the balancing process, i.e., it can be that $\sum_{i=1}^{n} t_{i} < \mathbf{t}$.
#### Automatic discovery of counterexamples
$s_{1}$ $s_{2}$ $\mathbf{t}$ rest
---------- ----------- --------------- -------------------
1048627 524206 1099511627744 0.9998779296875
32779 536870892 1099511627779 0.999881774187088
67108824 33554439 1099511627792 0.9998779296875
Providing concrete counterexamples first requires a definition of the `isNearlyEqual` method. In the original implementation, this function was implemented as an absolute $\epsilon$-comparison using `FLT_EPSILON` as the value for $\epsilon$. `FLT_EPSILON` is defined as the difference between 1 and the smallest floating-point number of type `float` that is greater than 1 [@glibc-fp]. Under the assumption that `float` refers to the 32bit floating-point type defined in IEEE754, as is usually the case, this value is $2^{-23}$.
A careful analysis leads to the conclusion that it should be possible to find a counterexample with $n=2$. However, the search space consisting of the three `long` values $s_{1}$, $s_{2}$ and $\mathbb{t}$ is still too large to be explored exhaustively. Thus, we first reduced the search space further, and then used random fuzzing to discover counterexamples. To reduce the search space, we drew the values $s_{1},s_{2},\mathbf{t}$ randomly from $\Set{ 2^{e} + \delta | e \in \Set{0,\ldots,40}, \delta \in
\Set{-100,\ldots,100}}$. The rationale for this choice is that small offsets from large integer powers use a large range of the precision, and thereby are likely to lead to imprecisions during calculations.
The fuzzer finds numerous counterexamples within a matter of seconds, a few of which can be seen in Fig. \[fig:counterexamples\], including the final value of `rest` which is below $0.9999$, and thereby is well below `1 - FLT_EPSILON`.
#### Improved Algorithm
To remedy the problems in the original algorithm, we eliminated all uses of floating-point numbers in the original algorithm in favour of integer operations. Then we verified three different properties of this algorithm using [*Why3*]{}:
1. No tasks are lost, i.e., $\sum_{i=1}^{n} t_{i} = \mathbf{t}$.
2. The resulting values $t_{i}$ are close to $s_{i} \frac{\mathbf{t}}{\mathbf{s}}$. In particular, the following holds:\
$\lfloor \frac{\mathbf{t}}{\mathbf{s}} \rfloor \leq
\frac{t_{i}}{s_{i}} \leq \lceil \frac{\mathbf{t}}{\mathbf{s}}
\rceil$.
3. The integer algorithm is equivalent to the original algorithm, if rationals are used instead of floating-point values (i.e., if all computations are exact, no rounding effects).
Property 3 is particularly interesting, since proving the functional equivalence of the original algorithm to the new integer algorithm ensures that the intentions of the original author, which can include domain knowledge not available to the verification engineer, are preserved.
While the verification of property 1 was possible automatically using [*Why3*]{} after adding lemmas to assist the proof search, property 2 and 3 required reasoning about properties, involving floating-points, which is a weakness of most automatic theorem provers. We thus had to resort to time-intensive interactive proofs using the [*Coq*]{} theorem prover [@coq].
double rest = 0.0;
for (int i = 0; i < num_tasks; ++i) {
double share =
(double)tasks[i]/(double)total_tasks;
double real_size =
share * (double)new_total_tasks;
double floor_size = floor(real_size);
rest += real_size - floor_size;
new_tasks[i] = (int)floor_size;
if (isNearlyEqual(rest, 1.0)) {
new_tasks[i] += 1;
rest -= 1;
}
}
int rest = 0;
for (int i = 0; i < num_tasks; ++i) {
int scaled = new_total_tasks * tasks[i];
int floor_size = scaled / total_tasks;
rest += scaled % total_tasks;
new_tasks[i] = floor_size;
if (rest >= total_tasks) {
new_tasks[i] += 1;
rest -= total_tasks;
}
}
module Resize
lemma small_rest : forall n:int, a:int, b:int, c:int.
n>=0 /\ a>=0 /\ b>=0 /\ c>=0 /\ c<n /\ n*a=n*b+c -> c=0
lemma floor_ceil : forall r:real.
from_int (floor r) < r -> floor r + 1 = ceil r
lemma floor_rest : forall i:int, r:real.
i>=0 /\ 0.0<=r /\ r<1.0 -> floor (from_int i + r) = i
lemma floor_div : forall a:int, b:int.
a>=0 /\ b>0 -> div a b = floor (from_int a / from_int b)
lemma floor_le_ceil : forall r:real. floor r <= ceil r
let resize (tasks : array int) (total_tasks: int) (new_total_tasks: int)
requires { total_tasks = sum tasks 0 (length tasks) }
requires { forall i:int. 0 <= i < length tasks -> tasks[i] >= 0 }
requires { 0 < total_tasks /\ 0 <= new_total_tasks }
ensures { new_total_tasks = sum result 0 (length result) }
ensures {
forall i:int. 0 <= i < length result ->
let exact = from_int tasks[i] * from_int new_total_tasks /
from_int total_tasks
in floor exact <= result[i] <= ceil exact
}
= let new_tasks = make (length tasks) 0 in
let rest = ref 0 in
for i = 0 to length tasks - 1 do
invariant { sum tasks 0 i * new_total_tasks
= sum new_tasks 0 i * total_tasks + !rest }
invariant { sum new_tasks 0 i >= 0 }
invariant { 0 <= !rest < total_tasks }
invariant { forall j:int. 0 <= j < i ->
let exact = from_int tasks[j] * from_int new_total_tasks /
from_int total_tasks
in floor exact <= new_tasks[j] <= ceil exact
}
let floor_size = div (new_total_tasks * tasks[i]) total_tasks in
rest := !rest + mod (new_total_tasks * tasks[i]) total_tasks;
new_tasks[i] <- floor_size;
if !rest >= total_tasks then (
new_tasks[i] <- new_tasks[i] + 1;
rest := !rest - total_tasks
);
done;
assert { total_tasks * new_total_tasks =
total_tasks * sum new_tasks 0 (length tasks) + !rest };
assert { !rest = 0 };
new_tasks
end
Case Study : Convex Hull
========================
Calculating the convex hull of a set of points is a problem which is commonly found in scientific computation applications. However, while the algorithms for the computation of the hull are mathematically simple and straightforward, numerical errors caused by using floating-point values in implementations instead of real numbers, can lead to a wrong result.
In the two-dimensional case, the implications are not too severe: It might be that a point close to an edge of the convex hull is wrongly included in (or wrongly excluded from) the hull polygon. But the result is always a valid polygon which is close to the desired result.
The situation is different if three-dimensional data is taken as input. The additional dimension requires keeping a record of the facets making up the convex polyhedron. This situation is computationally considerably more sphisticated than the two-dimensional case as it requires the calculation of the side of a facet (front or back) that is faced by a point. For points close to the facet, such a calculation may come up with the wrong result due to floating-point imprecision. The situation becomes bad if this computation for two facets errs for one point in different directions: Then the convex hull, which relies on these computations, wrongly includes or excludes facets from the hull, with the catastrophic outcome that the result is not only imprecise, but not a (closed) polyhedron at all – which is a far more severe problem.
This problem, which was reported by Barber et. al. [@quickhull], when presenting Quickhull, a widely employed algorithm for convex hull computation, has been known for a long time.
The existing implementation given here was known to suffer from such errors, and it was also known that errors had occurred in practice. As workarounds were implemented into the code to mitigate the problem, the number of observed errors decreased, but the effectiveness of the solution for the general case was unclear. The workarounds are focused on a particular method which given a plane spanned by three points and a fourth point decides whether that point is above, below or coplanar to the plane.
As mentioned above, due to floating-point imprecision effects, the wrong decision might be made with fatal effects. In an attempt to avoid faulty results due to rounding errors, the original method has been modified to compute the output three times. Each time, a different point is chosen from the three spanning points as the base point of the plane. As final result, the result computed by the majority, is returned.
In this case study, we did not apply formal methods to prove a correctness hypothesis, like in Section \[sec:load-distr\], but we succeeded in *disproving* that the introduced workaround works in all cases. We used the state-of-the art SMT solver *Z3* [@z3] to find inputs were the majority vote computes a different result from the exact result (on real numbers, not floats). Since this verification task heavily depended on the semantics of the floating-point arithmetic, which is very difficult to handle in a deductive fashion, we chose to model it using the SMT solver. This technology (in almost all cases) reduces problems on floating-point values to problems on corresponding bitvectors according to the standard IEEE 754. These are then resolved into an instance of the propositional SAT problem which can then be solved using a Boolean satisfiability solver.
While the search was successful, it required a careful analysis of the problem to reduce the search space. Floating-point problems are known to have the tendency to lead to SAT instances that are difficult to solve. A thorough manual analysis of the situation allowed us to narrow the search space for the floating-point values considerably. Even with the reduced search space, *Z3* required approx. 100h[^1] to find a counterexample. On the other hand, was virtually impossible to manually come up with concrete numbers that constitute a counterexample, because the dependencies of individual bits in floating-point arithmetic are difficult to follow.
Observations
============
In this report, we have looked at how formal methods can be applied to two orthogonal problems. The formal techniques employed in this study can be separated into finding counterexamples to disprove the fact that existing algorithms satisfy certain properties, and into verifying the correctness of algorithms. The latter can again be separated into using formal verification to prove the correctness, as given by an abstract specification, and into relational techniques which prove the equivalence to an existing algorithm.
While it is still too early to draw definitive conclusions from these early investigations, they give rise to some interesting observations of the potential applicability of formal methods in the field.
One interesting observation is that it is often possible to isolate the problems faced in large projects to relatively small and isolated pieces of code. While the formal analysis of the original project might not be feasible due to size and complexity, analysing these small pieces of code is significantly easier. However, extracting such a piece of code is a process that often requires domain knowledge. The provision of tools which help or automate this extraction, such that it can be done by the domain experts themselves, could reduce the time needed to produce code samples suitable for formal analysis.
While we have been able to find counterexamples automatically using SMT solvers, this relies heavily on an upfront reduction of the search space, which requires experience with formal methods, and cannot be applied by scientists themselves. Automating this process such that it could be applied by the developers of the original algorithm would be highly desirable, but requires further research. When comparing the use of fuzzers and SMT solvers to find counterexamples, fuzzers are significantly better at providing fast feedback which is suitable for interactive use. For intricate problems, random fuzzing is not likely to discover edge cases, on the other hand. A combination of both SMT solvers and fuzzers, while only requiring a single specification, could help to combine the respective benefits.
Overall, our experience shows that formal techniques can be employed successfully to assist scientists. However, this relies on the ability to isolate problems to smaller code samples, which is a tedious process. Furthermore, while the large effort required for a formal verification can be justified for algorithms used for longer periods of time, immediate feedback provided for scientists during development would be highly desirable, but is not easily achievable using existing techniques.
[^1]: on a virtualised Intel Xeon E3 with 2.6GHz
|
---
abstract: |
In this article the neutrino bremsstrahlung process is considered in presence of strong magnetic field, though the calculations for this process in absence of magnetic field are also carried out simultaneously. The electrons involved in this process are supposed to be highly degenerate and relativistic. The scattering cross sections and energy loss rates for both cases, in presence and absence of magnetic field, are calculated in the extreme-relativistic limit. Two results are compared in the range of temperature $5.9\times 10^{9}$ K $< T\leq 10^{11}$ K and magnetic field $10^{14} - 10^{16}$ G at a fixed density $\sim
10^{15}$ $gm/cc$, a typical environment during the cooling of magnetized neutron star. The interpretation of our result is briefly discussed and the importance of this process during the stellar evolution is speculated.
author:
- |
Indranath Bhattacharyya\
Department of Applied Mathematics\
University of Calcutta, Kolkata-700 009, INDIA\
E-mail : $i_{-}[email protected]$\
title: Neutrino Bremsstrahlung Process in highly degenerate magnetized electron gas
---
.3in
Introduction:
=============
The neutrino emission process plays an important role in the late stages of the stellar evolution. It is known that the radiation of neutrino could be dominated over the ordinary electromagnetic radiation in case of highly dense and hot stellar structures, such as, white dwarves or neutron stars. Unlike the other mechanisms the neutrinos are produced directly from their point of origin and do not require the transport of energy to the surface of the stellar object before getting radiated. As a consequence the energy outflow is given directly by the rate at which the neutrinos are produced. Photo-neutrino process $(e^{-}+\gamma\longrightarrow \nu+\overline{\nu})$, Pair-annihilation process $(e^{-}+e^{+}\longrightarrow
\nu+\overline{\nu})$ and Plasma neutrino process $(\Gamma\longrightarrow \nu+\overline{\nu})$ are the sole mechanisms that carry away the energy from star during its evolution period, although there are few more processes which might play more important role under some special environments. Chiu and his collaborators [@Chiu1960; @Chiu1961; @Chiu] calculated some important neutrino emission processes and pointed out their important role in astrophysics. In 1972 Dicus [@Dicus1972] reconsidered a few such processes in the framework of electro-weak interaction theory and calculated the energy-loss rate at the various stages of the stellar evolution. According to the Standard Model the neutrino has the minimal properties such as zero mass, zero charge etc. Introduction of neutrino mass compatible with the experimental data is a bare minimum extension of the Standard Model. In the frame work of Standard Model with this little extension some calculations [@Dicus2000; @Dodelson] related to the neutrino process have already been carried out. Itoh et al. [@Itoh] considered some neutrino emission processes to calculate them numerically.\
Pontecorvo [@Pontecorvo] gave an idea that the neutrino may be emitted by the interaction of electron with the nucleus, called bremsstrahlung process. Gandel’man and Pinaev [@Gandel'man] carried out the detailed calculations for this process in the non-relativistic electron gas. After that Festa and Ruderman [@Festa] extended this calculations for relativistic limit with considering the screening effect that becomes important in the high density. Dicus et al. [@Dicus] considered the process according to the Standard Model and compare the result with different screening effect. Saha [@Saha] calculated the bremsstrahlung process according to photon-neutrino weak coupling theory which is very much satisfactory in explaining the neutrino-synchrotron process [@Raychaudhuri1970]. In this article we have considered the bremsstrahlung process in presence of strong magnetic field and this is the first attempt to do so. It has been shown by Festa and Ruderman [@Festa] and then by Dicus et al. [@Dicus] that the bremsstrahlung process has maximum effect in the relativistic degenerate region; thus in the stellar object such as newly born born neutron star the process is supposed to be very much effective. Here we are to verify whether the presence of strong magnetic field, which may be generated in the rotating neutron stars, will have any effect on the bremsstrahlung process. We have calculated the scattering cross section and then obtained the energy-loss rate for the bremsstrahlung both in presence and absence of strong magnetic field. We are going to study the influence of high magnetic field on the bremsstrahlung process and the region where its presence may take a crucial role in the neutrino emission. The ordinary neutrino bremsstrahlung process is very much significant for the neutrino energy generation process in the relativistic highly degenerate region; therefore a comparative study is required for this process in presence and also in absence of strong magnetic field.
Calculation of scattering cross section:
========================================
In the bremsstrahlung process the electron interacts with the nucleus having the coulomb potential $$\Phi(\overrightarrow{r})=\frac{Ze}{\mid\overrightarrow{r}\mid}
e^{-\mid\overrightarrow{r}\mid /\lambda_{d}}\eqno{(2.1)}$$ i.e. a single charge with an exponential screening cloud (Yukawa like charge distribution), where $\lambda_{d}$ is the Debye screening length given by $$\frac{1}{\lambda_{d}^{2}}=\frac{4e^{2}}{\pi}E_{F}P_{F}\eqno{(2.2)}$$ Here $P_{F}$ and $E_{F}$ represent the Fermi momentum and energy respectively. We consider the potential, given by (2.1), because the screening effect is important in the high density region. There will be four different Feynman Diagrams shown in the Figure-1 (Z-exchange diagram) and Figure-2 (W-exchange diagram). In our calculations the presence of magnetic field plays a crucial role, so it is to be handled with care. Without any loss of generality we can take the direction of magnetic field along $z$-axis. In presence of magnetic field the energy momentum relation of the electron becomes $$(p_{n}^{0})^{2}=m_{e}^{2}+ p_{z}^{2}+2n\frac{H}{H_{c}}m_{e}^{2}\eqno{(2.3)}$$ where $H_{c}=4\cdot 414$ G stands for critical magnetic field. Here $n$ represents the Landau level for the electron in the magnetic field. \[The value of $s$ is taken as $\pm 1$ as per the spin of the electron is directed towards or opposite to the direction of magnetic field (along z-axis) respectively.\]\
The component of electron momentum along the direction of magnetic field remains unaffected. It is clear that the effect of magnetic field on the electron quantizes its energy to the direction perpendicular to $H$ and thus transverse components would get replaced by $p_{x}^{2}+p_{y}^{2}\longrightarrow2nm_{e}^{2}\frac{H}{H_{c}}$, whereas the longitudinal component $p_{z}$ would be directed along the magnetic field. The Feynman diagrams for the bremsstrahlung process are same for both in presence and absence of magnetic field; only we have to keep in our mind that four momenta of the electronic lines, present in the diagrams, should be modified. First we shall calculate the ordinary bremsstrahlung process i.e. the process in absence of magnetic field. The matrix element can be constructed as $$M_{fi}=-ie\frac{G_{F}}{\sqrt{2}}A_{0}(\overrightarrow{k})J_{\mu}\mathcal{M}^{\mu}\eqno{(2.4)}$$ where, $$\mathcal{M}^{\mu}=\overline{u}(p')[\gamma^{\mu}(C_{V}-C_{A}\gamma_{5})
\frac{(p^{'\tau}\gamma_{\tau}+q^{\tau}\gamma_{\tau}+m_{e})}
{(p'+q)^{2}-m_{e}^{2}+i\epsilon}\gamma^{0}+\gamma^{0}
\frac{(p^{\tau}\gamma_{\tau}-q^{\tau}\gamma_{\tau}+m_{e})}
{(p-q)^{2}-m_{e}^{2}+i\epsilon}\gamma^{\mu}(C_{V}-C_{A}\gamma_{5})]u(p)\eqno{(2.5)}$$ $$J_{\mu}=\overline{u}_{\nu}(q_{1})\gamma_{\mu}(1-\gamma_{5})v_{\nu}(q_{2})\eqno{(2.6)}$$ $$A_{0}(\overrightarrow{k})=\int\Phi(\overrightarrow{r})
e^{-(\overrightarrow{k}.\overrightarrow{r})}d^{3}r=-\frac{4\pi
Ze}{\mid k^{2}+q_{sc}^{2}\mid}\eqno{(2.7)}$$ and $$q=q_{1}+q_{2}$$ The energy momentum conservation leads to $$k+p=p'+q$$ where $k$ is purely space like as in the case of photo-coulomb neutrino process [@Rosenberg; @Bhattacharyya1]. The term $q_{sc}$ present in the equation (2.7) arises due to the screening effect and can be expressed as $$q_{sc}=\frac{1}{\lambda_{d}}\eqno{(2.8)}$$ Using some simplifications we can write the term $J_{\mu}\mathcal{M}^{\mu}$ as follows: $$J_{\mu}\mathcal{M}^{\mu}=[\frac{(p'J)}{q^{2}/2+(p'q)}+\frac{(pJ)}{q^{2}/2-(pq)}]
\overline{u}(p')\gamma^{0}(C_{V}-C_{A}\gamma_{5})u(p)\eqno{(2.9)}$$ We can put this expression to the equation (2.4) to get the expression for the scattering matrix. Now the squared sum of the scattering matrix over the final spins is to be integrated over the final momenta. The squared sum of the expression $[\overline{u}(p')\gamma^{0}(C_{V}-C_{A}\gamma_{5})u(p)]$ gives $$\sum\mid\overline{u}(p')\gamma^{0}(C_{V}-C_{A}\gamma_{5})u(p)\mid^{2}=(C_{V}^{2}-C_{A}^{2})
+(C_{V}^{2}+C_{A}^{2})\frac{(p'_{0}p_{0}-\overrightarrow{p}'.\overrightarrow{p})}{m_{e}^2}\eqno{(2.10)}$$ Note that in the equation (2.10) no neutrino momentum is present, so this part will not be taken into account during the integration over the final momenta of neutrinos. Let us now evaluate the squared spin sum of the expression $\mid\frac{(p'J)}{q^{2}/2+(p'q)}+\frac{(pJ)}{q^{2}/2-(pq)}\mid$ and then integrating over final momenta of the neutrinos we can obtain \[See Appendix-A\] $$\int\sum\mid\frac{(p'J)}{q^{2}/2+(p'q)}+\frac{(pJ)}{q^{2}/2-(pq)}\mid^{2}
\frac{d^{3}q_{1}d^{3}q_{2}}{(2\pi)^{3}2q^{0}_{1}(2\pi)^{3}2q^{0}_{2}}(2\pi)
\delta(q^{0}-q^{0}_{1}-q^{0}_{2})$$ $$\approx\frac{1}{18(2\pi)^{3}m_{\nu}^{2}}(p^{0}-p^{'0})^{3}
[\frac{\mid\overrightarrow{p}-\overrightarrow{p}'\mid}{p^{0}+p^{'0}}]^{2}\eqno{(2.11)}$$ Up to this step the calculations for the neutrino bremsstrahlung process would be same in both situations i.e. in presence as well as absence of magnetic field. It is assumed that in presence of magnetic field quantized transverse components of the momentum do not participate directly during the interaction between nucleus and electron. Only the z-component of the electron momentum takes part in this process. Thus in this case we can obtain an expression almost similar to the equation (2.11) with replacing $\mid\overrightarrow{p}'\mid$ and $\mid\overrightarrow{p}\mid$ by $p'_{z}$ and $p_{z}$ respectively. Now to integrate the squared sum of the matrix element over all final momenta we shall utilize the result obtained in the equation (2.10) and (2.11), but we have to take care when the magnetic field is present. In this case the phase space factor $d^{3}p'$ takes the form \[See Appendix-B\] $$\int d^{3}p'=\pi \frac{H}{H_{c}}m_{e}^{2}\int dp_{z}'\eqno{(2.12)}$$ whereas in the ordinary bremsstrahlung process the integration over the final momentum of electron is done in the usual manner. In the center of mass frame and assuming the electron momentum is much high relative to its rest mass we can carry out the calculations. In this extreme relativistic limit we can evaluate the integral over the final momentum of electron and obtain the following expression. $$\int\sum\mid M\mid^{2}\frac{d^{3}q_{1}d^{3}q_{2}d^{3}p'}
{(2\pi)^{3}2q^{0}_{1}(2\pi)^{3}2q^{0}_{2}(2\pi)^{3}2p^{'0}}(2\pi)\delta(q^{0}-q^{0}_{1}-q^{0}_{2})
=\frac{8G_{F}^{2}\alpha^{2}Z^{2}}{9(2\pi)^{3}(1+r^{2})^{2}}(C_{V}^{2}+C_{A}^{2})\frac{p_{0}^{3}}
{m_{e}^{2}m_{\nu}^{2}}\eqno{(2.13)}$$ This expression is obtained for the ordinary bremsstrahlung process when there will be no magnetic field. In presence of magnetic field this expression becomes $$\frac{2G_{F}^{2}\alpha^{2}Z^{2}}{9(2\pi)^{3}}(C_{V}^{2}+C_{A}^{2})\frac{p_{0}}
{m_{\nu}^{2}}(\frac{H}{H_{c}})\eqno{(2.14)}$$ Note that the term $r$ arises due to the weak screening effect. It is given by $$r\approx \frac{q_{sc}}{p^{0}}$$ The expression for $C_{V}$ and $C_{A}$ for the electron type of neutrino emission will differ from those in case of muon and tau neutrino emission, since W-boson exchange diagrams are present only when the electron neutrino anti-neutrino pair is emitted. Inserting the above terms into the expression of the scattering cross section for both of those cases and returning to the C.G.S. unit we can finally obtain $$\sigma\approx1\cdot76\times10^{-50}(\frac{E}{m_{e}c^{2}})^{2}\frac{1}{(1+r^{2})^{2}}
\hspace{0.5cm}cm^{2}\eqno{(2.15)}$$ in absence of magnetic field, whereas $$\sigma_{mag}\approx4\cdot41\times10^{-51}(\frac{H}{H_{c}})
\hspace{0.5cm}cm^{2}\eqno{(2.16)}$$ in presence of magnetic field.\
It is worth noting that all three type of neutrinos are taken into account in our calculations. Our result (equation 2.16) shows that the scattering cross section for the bremsstrahlung process in presence of magnetic field will not depend on the energy of the incoming electron, but on the intensity of the magnetic field present in the surroundings.
Calculation of energy loss rate:
================================
In the extreme relativistic case the energy loss rate in erg per nucleus per second for the neutrino bremsstrahlung process is calculated by the formula $$\mathcal{E}_{\nu}^{Z}=\frac{2}{(2\pi)^{3}\hbar^{3}}
\int\frac{d^{3}p}{[e^{\frac{E-E_{F}}{\kappa T}}+1]}c\sigma E
e^{\frac{E-E_{F}}{\kappa T}}\eqno{(3.1)}$$ where $E_{F}$ stands for Fermi energy of the electron. We are considering the case in which the electrons are highly degenerate. It is well known that in the degenerate region the energy of the electron remains below the Fermi energy level. To obtain the energy-loss rate in erg per gram per second $\mathcal{E}_{\nu}^{Z}$ is divided by $Am_{p}$ and it is obtained as $$\mathcal{E_{\nu}}=\frac{Z^{2}}{A}\times5\cdot 26\times10^{-3}\times
\frac{x_{F}^{6}e^{1-x_{F}}} {(1+r^{2})^{2}}T_{10}^{6}\hspace{0.5
cm}erg/gm-sec\eqno{(3.2)}$$ where, $T_{10}=T\times 10^{-10}$\
The $x_{F}$ represents the ratio of the Fermi temperature to the maximum temperature of the degenerate electron gas. The degeneracy will be attained only when the following condition will be satisfied [@Chandrasekhar]. $$x_{F}^{2}>2\pi^{2}\eqno{(3.3)}$$ It can be obtained $x_{F}\approx 6$, considering the fact that temperature and density of the electron gas present in the core of a newly born neutron star would be approximately $10^{12}$ K and $10^{15}$ $gm/cc$ respectively. The term $r$ arises due to the weak screening, related to $x_{F}$ by $r\approx 0\cdot 096 \times
x_{F}$. Thus we can calculate the term $r$, present in the equation (3.2). Finally the expression for the energy-loss rate becomes $$\mathcal{E_{\nu}}=\frac{Z^{2}}{A}\times0\cdot93\times10^{12}\times
T_{10}^{6} \hspace{0.5 cm}erg/gm-sec\eqno{(3.4)}$$ In the same manner we can obtain the energy loss rate in presence of magnetic field. In that case the the phase space factor will be replaced according to the rule defined in (B6). In the same manner energy-loss rate can be calculated as $$\mathcal{E_{\nu}}^{mag}=\frac{Z^{2}}{A}\times0\cdot51\times10^{6}
\times H_{13}^{2}\times
T_{10}^{2}\hspace{0.5cm}erg/gm-sec\eqno{(3.5)}$$ where, $H_{13}=H\times 10^{-13}$\
We have computed (Table-1) the logarithmic value of the energy loss rate in the temperature range $0\cdot 8\times 10^{10} - 10^{11}$ K and the magnetic field $10^{14}- 10^{16}$ G at a fixed density $\rho=10^{15}$ $gm/cc$.
discussion:
===========
The neutrino bremsstrahlung process is an important energy generation mechanism during the stellar evolution and very much effective in the highly degenerate region, for examples, in the cores of low mass red giants, white dwarves etc. In addition to this degenerate nature if the electron gas is highly relativistic the energy-loss rate through the bremsstrahlung process would be significantly high. It has already been calculated that the neutrino luminosity in the crust of neutron star is high enough [@Maxwell], but it is yet to be verified what would be the effect of neutrino emission by the bremsstrahlung process in the core region, particularly when the core is strongly magnetized. The discovery of radio pulsars showed that the collapse of normal stars results not only in supernova explosions, but may also generate strong magnetic field. In some stellar objects like neutron stars and magnetars the magnetic field may reach to $10^{16}$ G and influences the neutrino emission process. It is known that the neutron star is born as a result of type II supernova explosion. In the newly born neutron star the core temperature becomes $10^{12}$ K which drops down to $10^{11}$ K within few seconds of its birth and then slowly cools down until the temperature reaches to $2\times10^{8}$ K, after which the electromagnetic emission dominates over the neutrino emission [@Raffelt]. It is worth noting that the electron gas, still left in the stellar core, is highly degenerate and relativistic.\
We are going to verify the role of the magnetic field on the neutrino bremsstrahlung process. In the neutrino synchrotron radiation [@Landstreet; @Raychaudhuri1970; @Canuto; @Bhattacharyya2] neutrino anti-neutrino pair emission takes place since the electron changes its Landau levels, but in the bremsstrahlung the Landau levels are assumed to be unchanged throughout the process. The process occurs through the change of magnitude of the component of electron linear momentum directed along the magnetic field. It is found from Table-1 that in the temperature range $10^{10}$ K $\leq T\leq 10^{11}$ K and at the density $10^{15}$ $gm/cc$ the energy loss rate for the ordinary bremsstrahlung process is greater than that in presence of strong magnetic field ($10^{14} - 10^{16}$ G). We can interpret that in the early stage of neutron star cooling, when the temperature remains above $10^{10}$ K, the effect of the bremsstrahlung process get lowered due to the presence of magnetic field, although the energy loss rate is still very high. If the strength of the magnetic field goes below the critical value, the process would become free from the influence of magnetic field, and a greater amount of neutrino energy is produced. It is evident from our work that though, in general, the magnetic field makes the bremsstrahlung process a bit less effective, but there exists a particular region ($5\cdot 9\times 10^{9}$ K $<T< 10^{10}$ K, $H\sim 10^{16}$ G and $\rho\sim 10^{15}$ $gm/cc$) during neutron star cooling, where the bremsstrahlung process contributes a greater amount of neutrino energy loss by the influence of magnetic field compared to the situation when there would be no magnetic field at all. Therefore, it can be predicted that if the temperature falls below $10^{10}$ K, the process would give maximum effect due to the presence of super strong magnetic field having intensity $10^{16}$ G. Our study reveals that the neutrino bremsstrahlung process is an important energy generation mechanism in the late stages of the stellar evolution, even in presence of magnetic field.
Acknowledgement:
================
I am very much thankful to Prof. [**Probhas Raychaudhuri**]{} of The Department of Applied Mathematics, University of Calcutta, for his continuous help, suggestions and guidance during preparation of this manuscript. I like to thank [**CSIR**]{}, India for funding this research work. My special thank goes to [**ICTP**]{}, Trieste, Italy for giving me an opportunity for two months visit and providing some excellent facilities and information, which has facilitated to carry out this work.
Appendix-A:
===========
We have chosen a frame in which $$\overrightarrow{q}=\overrightarrow{q}_{1}+\overrightarrow{q}_{2}\eqno{(A1)}$$ In this frame we obtain $$\frac{q^{2}}{2}+(p'q)=\frac{(p^{'0}-p^{0})(p^{'0}+p^{0})}{2}\eqno{(A2)}$$ $$\frac{q^{2}}{2}-(pq)=-\frac{(p^{'0}-p^{0})(p^{'0}+p^{0})}{2}\eqno{(A3)}$$ Thus, $$\sum\mid\frac{(p'J)}{q^{2}/2+(p'q)}+\frac{(pJ)}{q^{2}/2-(pq)}\mid^{2}
=\frac{4}{(p^{'0}-p^{0})^{2}(p^{'0}+p^{0})^{2}}\sum\mid(p'-p)J\mid^{2}\eqno{(A4)}$$ Let us assume $$P=p-p'=q-k\eqno{(A5)}$$ we have, $$\sum\mid(PJ)\mid^{2}=\sum\mid\overline{u}_{\nu}(q_{1})P^{\mu}\gamma_{\mu}(1-\gamma_{5})
v(q_{2})\mid^{2}$$ $$\hspace{1.5cm}=\frac{2}{m_{\nu}^{2}}[2(q_{1}P)(q_{2}P)-(q_{1}q_{2})P^{2}]$$ $$\hspace{6cm}=\frac{2}{m_{\nu}^{2}}[(m_{\nu}P^{0})^{2}+\{(1-2cos^{2}\alpha)
\mid\overrightarrow{q}_{1}\mid^{2}+(q_{1}^{0})^{2}\}\mid
\overrightarrow{P}\mid^{2}]\eqno{(A6)}$$ Now using $$\int d^{3}q_{2}=\frac{4\pi}{3}\mid\overrightarrow{q}_{2}\mid^{3}
=\frac{4\pi}{3}\mid\overrightarrow{q}_{1}\mid^{3}\eqno{(A7)}$$ and $$d^{3}q_{1}=\mid\overrightarrow{q}_{1}\mid^{2}
d\mid\overrightarrow{q}_{1}\mid sin\alpha d\alpha
d\phi\eqno{(A8)}$$ we can obtain $$\int\sum\mid (PJ)\mid^{2}\frac{d^{3}q_{1}}{2q_{1}^{0}}\frac{d^{3}q_{2}}{2q_{2}^{0}}
\delta(2q_{1}^{0}-q^{0})$$ $$=\frac{2\pi^{2}}{3m_{\nu}^{2}}\int\int_{\alpha=0}^{\pi}
\frac{\mid\overrightarrow{q}_{2}\mid^{4}}{q_{1}^{0}}[(m_{\nu}P^{0})^{2}+\{(1-2cos^{2}\alpha)
\mid\overrightarrow{q}_{1}\mid^{2}+(q_{1}^{0})^{2}\}\mid
\overrightarrow{P}\mid^{2}]\delta(q_{1}^{0}-\frac{q_{0}}{2})dq_{1}^{0}sin\alpha
d\alpha d\phi$$ $$=\frac{4\pi^{2}}{3m_{\nu}^{2}}\int\frac{\mid\overrightarrow{q}_{2}\mid^{4}}{q_{1}^{0}}
[(m_{\nu}P^{0})^{2}+\{\frac{\mid\overrightarrow{q}_{1}\mid^{2}}{3}+(q_{1}^{0})^{2}\}\mid
\overrightarrow{P}\mid^{2}]\delta(q_{1}^{0}-\frac{q_{0}}{2})dq_{1}^{0}$$ $$\approx \frac{\pi^{2}}{18m_{\nu}^{2}}(p^{0}-p^{'0})^{5}\mid
\overrightarrow{p}-\overrightarrow{p}'\mid^{2}\eqno{(A9)}$$ We have assumed $m_{\nu}\ll q_{1}^{0}<q_{0}$ and used the following criteria $$m_{\nu}\longrightarrow 0$$ $$P^{0}=q^{0}=p^{0}-p^{'0}$$ $$\overrightarrow{P}=\overrightarrow{k}=\overrightarrow{p}-\overrightarrow{p}'$$ since $k$ is space like, whereas $q$ is time like in our chosen frame.\
Now introducing normalized factors and also using $(A4)$ we obtain $$\int\sum\mid\frac{(p'J)}{q^{2}/2+(p'q)}+\frac{(pJ)}{q^{2}/2-(pq)}\mid^{2}
\frac{d^{3}q_{1}d^{3}q_{2}}{(2\pi)^{3}2q^{0}_{1}(2\pi)^{3}2q^{0}_{2}}(2\pi)
\delta(q^{0}-q^{0}_{1}-q^{0}_{2})$$ $$\approx\frac{1}{18(2\pi)^{3}m_{\nu}^{2}}(p^{0}-p^{'0})^{3}[\frac{\mid\overrightarrow{p}
-\overrightarrow{p}'\mid}{p^{0}+p^{'0}}]^{2}\eqno{(A10)}$$ This is same as the equation (2.11).
Appendix-B
==========
In presence of magnetic field the phase space factor is replaced by the following relation [@Roulet] $$\frac{2}{(2\pi)^{3}}\int d^{3}p=\frac{1}{(2\pi)^{2}}\sum_{n=0}^{n_{max}}g_{n}
\int dp_{z}\eqno{(B1)}$$ where $g_{n}$ represents degeneracy factor of the Landau levels i.e. $$g_{0}=1,\hspace{4cm} g_{n}=2\hspace{2cm}(n\geq 1)\eqno{(B2)}$$ The maximum Landau level $n_{max}$ can be obtained from the following relation $$n_{max}=\frac{1}{2m_{e}^{2}}(\frac{H}{H_{c}})[(p^{0}_{n_{max}})^{2}-(p^{0})^{2}]\eqno{(B3)}$$ where, $$(p^{0})^{2}=p_{z}^{2}+m_{e}^{2}\eqno{(B4)}$$ For $n_{max}<1$ we have, $$H > \frac{1}{2m_{e}^{2}}[(p^{0}_{n_{max}})^{2}-(p^{0})^{2}]H_{c}\eqno{(B5)}$$ It shows that for a very high magnetic field only $n=0$ Landau level would contribute in the phase space. In this article we consider the environment is highly magnetized, which gives $$(p^{0}_{n_{max}})^{2}-(p^{0})^{2}> 2m_{e}^{2}$$ and therefore $$H > H_{c}$$ In that case the phase space factor will take the form $$\int d^{3}p=\pi \frac{H}{H_{c}}m_{e}^{2}\int dp_{z}\eqno{(B6)}$$ It is same as the equation (2.12).\
If the magnetic field is comparatively lower the higher Landau levels contribute in the phase space as per the condition (B3).
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[**Figure Caption :**]{}\
[**Figure-1:**]{} Feynman diagram for the neutrino bremsstrahlung process in presence of magnetic field with Z boson exchange.\
[**Figure-2:**]{} Feynman diagram for the neutrino bremsstrahlung process in presence of magnetic field with W boson exchange.
$T_{10}$ $log(\frac{A}{Z^{2}}\mathcal{E_{\nu}})$
---------- ------------ ------------- ----------------------------------------- ---------------------------
Presence of magnetic field Absence of magnetic field
$10^{14}$ $10^{15}$ $10^{16}$
$0.8$ $7\cdot51$ $9\cdot51$ ${\bf 11\cdot51}$ $11\cdot39$
$0.9$ $7\cdot62$ $9\cdot61$ $11\cdot62$ $11\cdot69$
$1$ $7\cdot71$ $9\cdot71$ $11\cdot71$ $11\cdot97$
$2$ $8\cdot31$ $10\cdot31$ $12\cdot31$ $13\cdot77$
$3$ $8\cdot66$ $10\cdot67$ $12\cdot67$ $14\cdot83$
$4$ $8\cdot91$ $10\cdot91$ $ 12\cdot91$ $15\cdot58$
$5$ $9\cdot10$ $11\cdot10$ $13\cdot10$ $16\cdot16$
$6$ $9\cdot26$ $11\cdot26$ $13\cdot26$ $16\cdot64$
$7$ $9\cdot40$ $11\cdot40$ $ 13\cdot40$ $17\cdot04$
$8$ $9\cdot51$ $11\cdot51$ $13\cdot51$ $17\cdot39$
$9$ $9\cdot62$ $11\cdot62$ $13\cdot62$ $17\cdot69$
$10$ $9\cdot71$ $11\cdot71$ $13\cdot71$ $17\cdot97$
: Logarithmic expression for energy loss rate at $\rho=
10^{15}$ $gm/cm^{3}$, and magnetic field $H=10^{16}$, $10^{15}$, $10^{14}$ G due to the neutrino bremsstrahlung process in presence and absence of magnetic field respectively in the temperature range $0.8\times 10^{10} - 10^{11}$ K. The bold number indicates that the former process dominates over the later.
|
---
abstract: 'We compute the homotopy type of the space of possibly empty proper $d$-dimensional submanifolds of $\bR^n$ with a topology coming from a Hausdorff distance. Our methods give also a different proof of the Galatius–Randal-Williams theorem on the homotopy type of their space of submanifolds.'
address: '[Mathematisches Institut, Universit[ä]{}t M[ü]{}nster, Einsteinstra[ß]{}e 62, 48149 M[ü]{}nster, Germany]{}'
author:
- Federico Cantero
bibliography:
- 'biblio-article.bib'
title: 'The space of merging submanifolds in $\bR^n$'
---
Introduction
============
In [@GMTW], the classifying space of the $d$-dimensional cobordism category was found to be homotopy equivalent to a delooping of the infinite loop space associated to the Thom spectrum $\mathbf{MTO}(d)$, whose $n$-th space is the Thom space of the affine Grassmannian $\gamma_{d,n}^\perp$ of $d$-planes in $\bR^n$, seen as a vector bundle over the linear Grassmannian $\Gr_d(\bR^n)$. This was proven again with different methods by [@GR-W], who introduced the space $\P_d(\bR^n)$ of submanifolds of $\bR^n$. A crucial step in their proof was the following
The inclusion $\Th(\gamma_{d,n}^\perp)\hookrightarrow \P_d(\bR^n)$ is a homotopy equivalence.
$\P_d(\bR^n)$ is a topological space whose points are possibly empty proper $d$-submanifolds of $\bR^n$ (in the sense that the intersection of such submanifold with a compact subset of $\bR^n$ is compact). Before describing its topology, let us discuss what natural topologies one can put in its underlying set $\psi_d(\bR^n)$:
If $W$ is a proper submanifold of $\bR^n$, and we identify $\bR^n\cup \{\infty\}\cong S^n$, then $W\cup\{\infty\}$ is a compact subset of $S^n$, so one has an inclusion $$\psi_d(\bR^n)\lra \Power(S^n)$$ into the space of non-empty compact subsets of $S^n$ with the Hausdorff distance. The induced topology on $\psi_d(\bR^n)$ is coarser than the one defined by Galatius and Randal-Williams, in part because it does not take into account the smoothness of the submanifolds. Instead one can include $$\label{eq:009}
\psi_d(\bR^n)\lra \Power(S^n\times\Gr_d(\bR^n) / \{\infty\}\times \Gr_d(\bR^n))$$ using the affine Gauss map, that sends a submanifold to the collection of pointed affine tangent planes to it together with the point at infinity. This takes into account the $C^1$-information of the submanifolds. One could also improve the target of this inclusion to account for the complete $C^\infty$-information of the submanifold. We denote by $\pp$ the set $\psi_d(\bR^n)$ endowed with the topology induced by the Hausdorff distance on the right hand-side, and we refer to it as the *space of merging submanifolds* of $\bR^n$.
This topology is still coarser than the topology in $\P_d(\bR^n)$, but it is very close to it: A sequence of compact submanifolds in $\Pr_d(\bR^n)$ converging to a compact submanifold $W$ eventually takes values in covering spaces of $W$ (for non-compact submanifolds there is an analogous condition). If we refine the topology $\Pr_d(\bR^n)$ imposing these covering spaces to be single-sheeted (i.e., diffeomorphisms), then we arrive to the topology $\P_d(\bR^n)$ defined by Galatius and Randal-Williams. All the topologies described so far give rise to topological sheaves over $\bR^n$.
The purpose of this note is to find the homotopy type of the space $\pp$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This paper has benefited from many conversations with Abdó Roig, as well as from comments from Oscar Randal-Williams and Martin Palmer. The paper by Karcher in the references was kindly pointed out to me by Igor Belegradek at mathoverflow.
Spaces of submanifolds
======================
We begin by recalling the definition of the topology in $\P_d(U)$, when $U$ is an open subset of $\bR^n$ ([@GR-W §2]). For the sake of clearness, we will give a $C^1$-version of this topology.
Recall, for a submanifold $W$ of $U$, the partially defined function $$\exp_W\colon NW\dasharrow U.$$ defined in a neighbourhood $B$ of the zero section $z$, where $NW$ is the normal bundle of $W$.
The space $\P_d(U)$ has as underlying set the collection of all proper $d$-dimensional submanifolds of $U$, together with the empty submanifold. Its topology is given by the following neighbourhood basis of any proper submanifold $W$:
- *If $W\neq \emptyset$*, then every compact subset $K\subset U$ and every $\epsilon>0$ define a basic neighbourhood $(K,\epsilon)^\P$ of $W$; a submanifold $W'$ belongs to $(K,\epsilon)^\P$ if there is a section $f$ of the normal bundle $NW\to W$ such that
1. $\exp_W(f(W))\cap K = W'\cap K$ and
2. \[bla\] $\|f(x)\| + \|D(f-z)(x)\| < \epsilon$ for all $x\in W$ such that $\exp_W\circ f(x)\in W'\cap K$.
- *If $W=\emptyset$*, then every compact subset $K\subset \bR^n$ defines a basic neighbourhood $(K)^\P$ of $\emptyset$; and a submanifold $W'$ belongs to $(K)^\P$ if $W'\cap K=\emptyset$.
We now define the topology in $\pp$, the only difference being that instead of requiring $W'\cap K$ to be the image of a global section of $NW$, we only ask it to be the union of images of local sections of $NW$ whose domains cover $W$. The fact that this topology agrees with the one coming from the Hausdorff distance in is shown in [@FC-thesis §4.3].
The space $\Pr_d(U)$ has the same underlying set as $\P_d(U)$, with neighbourhood basis of a proper submanifold $W$:
- *If $W\neq \emptyset$*, then every compact subset $K\subset U$ and every $\epsilon>0$ define a basic neighbourhood $(K,\epsilon)^{\Pr}$ of $W$; a submanifold $W'$ belongs to $(K,\epsilon)^{\Pr}$ if there is a subset $Q\subset NW$ such that the composite $Q\subset NW\to W$ is a covering map and
1. \[qui2\] $\exp_W(Q)\cap K = W'\cap K$,
2. \[bla2\] $\|f(x)\| + \|D(f-z)(x)\| < \epsilon$ for each local section $f$ of the covering map.
- *If $W=\emptyset$*, then every compact subset $K\subset \bR^n$ defines a basic neighbourhood $(K)^{\Pr}$ of $\emptyset$; and a submanifold $W'$ belongs to $(K)^{\Pr}$ if $W'\cap K=\emptyset$.
1. One can also define the topology in $\Pr(U)$ following the definition of the topology in $\Psi(U)$, but replacing the normal bundle of $W$ by the fibrewise Ran space on the normal bundle of $W$ and imposing the cardinality of the image of the section to be locally constant.
2. Let $\Gr_d(\bR^n)$ be endowed with its natural metric. Then from the definitions above it follows that if $W'\in (K,\epsilon)^\Psi$ (respectively, $W'\in (K,\epsilon)^{\Pr}$) and $f$ is a global (resp. local) section defining $W'$, then $$\label{eq:5}
d(x,f(x)) + d(T_xW,T_{f(x)}W')<\epsilon$$ for all $x\in W$ such that $\exp_W\circ f(x)\in W'\cap K$.
3. Condition (\[bla\]) in both definitions says that $f$ is $\epsilon$-close to the zero section in the $C^1$-topology. One can instead impose that condition in the $C^\infty$-topology. This would give, in the case of $\Psi_d(U)$, the actual definition given by Galatius and Randal-Williams. Both definitions give homotopy equivalent spaces.
4. When $d=0$, the subspace of $\Psi_0(\bR^n)$ consisting of $0$-submanifolds contained in the unit ball, is the unordered configuration space on the unit ball, whereas that subspace in $\widetilde{\Psi}_0(\bR^n)$ is the Ran space of the unit ball.
Let $\ll$ be the subspace of $\pp$, consisting only on unions of parallel affine planes, together with the empty set. The inclusion $i\colon \ll\hookrightarrow \pp$ is a homotopy equivalence and the space $\ll$ is weakly contractible.
The strategy to construct a homotopy inverse to $i$ will be the following: We continuously change $W$ through a homotopy $H_t$ that ends in a submanifold $V = H_0(W)$ whose tangent planes are close to each other, seen as points in $\Gr_d(\bR^n)$. To the submanifold $V$ we can continuously assign a plane $\mu(V)$ obtained by averaging all tangent planes of $V$. The plane $\mu(V)$ will have another property: every tangent plane to $V$ is transverse to $\mu(V)^\perp$. Finally, we stretch out $V$ in the direction $\mu(V)$, obtaining several parallel copies of $\mu(V)$, one for each point in $\mu(V)^\perp\cap V$.
1. The argument here applies as well to give a different proof of the Galatius–Randal-Williams theorem, as the analogous subspace $\cL_d(\bR^n)\subset \P_d(\bR^n)$ is easily seen to be homotopy equivalent to $\Th(\gamma_{d,n}^\perp)$.
2. The arguments in [@GR-W §3] are enough to show that if one topologises the cobordism category in that paper using the topology $\pp$ instead of $\P_d(\bR^n)$, then its classifying space is contractible.
3. The assignment $\Pr$ defines a sheaf on the site of manifolds and open embeddings, and in Section \[s:4\] we show that this sheaf is not microflexible, so the methods of [@R-WEmbedded §3-6] do not generalize to this situation.
Proof
=====
The action of a embedding space on the space of merging submanifolds
--------------------------------------------------------------------
The next lemma can be proven following [@GR-W §2.2] and making the appropriate modifications. Instead, we will take advantage of knowing that the topology of $\Pr_d(U)$ is induced by a Hausdorff metric.
Let $U,V$ be open subsets of $\bR^n$. Then the map $$\label{lemma:cont}\Emb(U,V)\times \Pr_d(V)\lra \Pr_d(U)$$ given by sending a pair $f,W$ to $f^{-1}(W)$ is continuous.
Here we take the $C^1$ compact-open topology on $\Emb(U,V)$.
If we let $\overline{X}$ denote the one-point compactification of a locally compact space $X$, then there is a continuous map $$\mathrm{O}\Emb(X,Y)\lra \map(\overline{Y},\overline{X})$$ from the space of open embeddings of $X$ into $Y$ to the mapping space between $\overline{Y}$ and $\overline{X}$, both endowed with thecompact-open topology (this is an adaptation of the second part of the proof of Theorem 4 in [@Arens], c.f. [@Cantero:collapse-mathoverflow]). It is given by sending an embedding $e$ to the map that sends a point $y$ to $e^{-1}(y)$ if the latter exists and to $\infty$ otherwise. Denote by $\Power(\overline{X})$ the set of non-empty compact subsets of $\overline{X}$ with the Hausdorff metric. There is a continuous map [@Cantero:hyperspace] $$\map(\overline{Y},\overline{X})\lra \map(\Power(\overline{Y}),\Power(\overline{X}))$$ to the mapping space between $\Power(\overline{Y})$ and $\Power(\overline{X})$ with the compact-open topology. Composing these two maps with the evaluation map, we obtain a continuous map $$\Emb(X,Y)\times \Power(\overline{Y})\lra \Power(\overline{X}).$$ Finally, if $B\subset \Power(\overline{Y})$ is a subspace and the latter map restricted to this subspace takes values in a subspace $A\subset \Power(\overline{X})$, then $$\Emb(X,Y)\times B\lra A$$ is continuous as well. We now take $X=U\times \Gr_d(\bR^n)$, $Y=V\times\Gr_d(\bR^n)$, $A=\Pr_d(U)$ and $B=\Pr_d(V)$, obtaining a continuous map $$\Emb(U\times \Gr_d(\bR^n),V\times \Gr_d(\bR^n))\times \Pr_d(V)\lra \Pr_d(U)$$ and the lemma follows by precomposing with the map $$\Emb(U,V)\lra \Emb(U\times \Gr_d(\bR^n),V\times \Gr_d(\bR^n))$$ that sends an embedding $e$ to the embedding given by $(x,L)\mapsto (e(x),De(L))$.
A more convenient assumption
----------------------------
Let $B^n$ be the open unit ball in $\bR^n$. By the previous lemma, the diffeomorphism $f\colon B^n\to \bR^n$ given by $f(x) = \frac{x}{1-\|x\|}$ induces a homeomorphism $\qq\to \pp$. Therefore our theorem will be proven by showing that the inclusion $\widetilde{\mathcal{L}}_d(B^n)\to \qq$ is a homotopy equivalence, where $\widetilde{\mathcal{L}}_d(B^n)\cong f^{-1}(\widetilde{\mathcal{L}}_d(\bR^n))$ consists on intersections of affine planes with the unit ball. The reason for this change is that it will be convenient to make use of conformal maps between $B^n$ and balls of smaller radius, and this is not possible if we take $\bR^n$ instead.
Averaging the tangent planes of a submanifold of $B^n$
------------------------------------------------------
Recall that, for a submanifold $W$ of $B^n$, the Gauss map $W\to \Gr_d(\bR^n)$ sends a point to its tangent plane.
Let us fix once and for all a Riemannian metric on $\Gr_d(\bR^n)$ and write $\diam$ for diameter. A submanifold $V\in \qq$ is *compactifiable* if it is of the form $e^{-1}(W)$ for some relatively compact embedding $e\colon B^n\to B^n$ and some $W\in \qq$. A compactifiable submanifold has finite volume, so it makes sense to integrate over it.
For each non-empty compactifiable $W\in \qq$ and each plane $L\in \Gr_d(\bR^n)$, define $$\lambda(W,L) = \frac{1}{2\int_{W} (1-\|x\|)dx}\int_{W} (1-\|x\|)d(L,T_xW)^2dx.$$ By [@Karcher Theorem 1.2], there exists a $\delta>0$ (that depends only on the metric of $\Gr_d(\bR^n)$) such that if $$\label{eq:101}
\Gauss(W)\subset B_\delta\quad \text{(for some ball $B_\delta$ of radius $\delta$),}$$ then the above function is convex in $B_\delta$. In this case, we define $\mu(W)$ to be the $L\in B_\delta$ that minimizes $\lambda(W,-)$, and observe that $$\begin{aligned}
\label{d-mu}
d(\mu(W),T_xW) &\leq \diam\circ\Gauss(W) \text{ for all $x\in W$.}\end{aligned}$$
The assigment $\mu$ defines a continuous map from the subspace $\qq^\delta\subset \qq$ of compactifiable submanifolds satisfying to the space $\Gr_d(\bR^n)$.
We will prove that the function $\lambda(L,-)$ is continuous in the second variable. Since in our case the functions $\lambda(-,W)$ have a unique minimum, it follows that $\min_L(\lambda(L,-))$ is also continuous in the second variable.
Let $W'\in (K_t,\epsilon)^{\Pr}$, a neighbourhood of $W$, where $K_t$ is a disc of radius $t$. Then there is a covering map $q\colon Q\subset NW\to W$ such that
1. $\exp_W(Q)\cap K \subset W'\cap K$,
2. $\|f(x)\| + \|D(f-\Id)(x)\| < \epsilon$ for each local section $f$ of the covering map.
Let $W_t:= W\cap K_t$ and let $W'_t = q^{-1}(W_{t-\epsilon})\subset W'\cap K_t$. Therefore we have that, if $\nabla_W:= \int_W(1-\|x\|)dx$, then $$\begin{aligned}
\frac{1}{2\nabla_{W'}}\int_{W'} (1-\|x\|)d(L,T_xW')^2dx =\\
\frac{1}{2\nabla_{W'}}\left(\int_{W'_t} (1-\|x\|)d(L,T_xW')^2dx + \int_{W'\setminus W'_t} (1-\|x\|)d(L,T_xW')^2dx \right) \\
\frac{1}{2\nabla_{W'}}\left(\int_{W'_t} (1-\|x\|)(d(L,T_{q(x)}W) + \alpha(x))^2dx + \beta(x) \right)\\
\frac{\#\text{sheets of $q$}}{2\nabla_{W'}}\left(\int_{W_t} (1-\|q(x)\|+\nu(x))(d(L,T_{x}W) + \alpha(x))^2\gamma(\epsilon)dx + \beta(x) \right)\end{aligned}$$ and $|\alpha(x)|<d(T_xW',T_{q(x)}W)<\epsilon$ and $0\leq\beta(x)\leq \nabla_{W'\setminus W'_t}(1-t)\delta^2$, $\gamma(\epsilon) = \det(Dq)$ and $|\nu(x)|<\epsilon$. Therefore, when $t\to 1$ and $\epsilon\to 0$, the above integral converges to $$\begin{aligned}
\frac{\#\mathrm{sheets}}{2\nabla_{W'}}\int_{W} (1-\|x\|)d(L,T_{x}W)^2dx\end{aligned}$$ and similarly $\nabla_{W'}$ tends to $\#\mathrm{sheets}\cdot \nabla_W$ as $t\to 1$ and $\epsilon\to 0$.
Let $\alpha$ be smaller than the $\delta$ provided by Karcher’s theorem, and also smaller than half the distance between any two planes $P,Q\in \Gr_d(\bR^n)$ such that $P\cap Q^\perp\neq 0$.
Controlling the value $\diam\circ\Gauss(W\cap B(\epsilon))$ {#controlling-the-value-diamcircgausswcap-bepsilon .unnumbered}
-----------------------------------------------------------
Let $F_\geq([0,1))$ be the set of non-decreasing functions from $[0,1)$ to $[0,\infty)$ that preserve $0$, and let $C_>([0,1))$ be the subset of continuous increasing functions. We topologize this space as follows: For each pair of positive real numbers $(t,\epsilon)$ with $\epsilon<t$, there is a basic neighbourhood $(t,\epsilon)^F$ of a function $f$. Another function $g$ is in the $(t,\epsilon)$-neighbourhood of $f$ if for all $s < 1-t$, $f(s-\epsilon)-\epsilon \leq g(s) \leq f(s+\epsilon)+\epsilon$. The subspace $C_>([0,1))$ inherits the compact-open topology.
Let us define $$\vartheta\colon \qq\lra F_\geq([0,1))$$ as the adjoint of the map $\qq\times [0,1)\to [0,\infty)$ that sends a pair $(W,\epsilon)$ to $\diam\circ\Gauss(W_\epsilon)$, where $W_\epsilon = W\cap B(\epsilon)$ and $B(\epsilon)$ is the ball of radius $\epsilon$ centered at the origin.
The map $\vartheta$ is continuous.
Let $(t,\epsilon)^F$ be a neighbourhood of $\vartheta(W)$. Let $K(r)\subset \bR^n$ be the closed disc with radius $r = 1-t$, and let $(K(r),\epsilon)$ be a neighbourhood of $W$. If $s<r$ and $W'\in (D(s),\epsilon)^{\widetilde{\Psi}}$, then, by condition (\[qui2\]) and by in the definition of the topology of $\pp$ the followig holds: for each $y\in W'_{s}$, there is a $x\in W$ at distance at most $\epsilon$ (so $x\in W_{s+\epsilon}$), and such that $T_yW'$ and $T_xW$ are also at distance at most $\epsilon$. Therefore for all $s<r$: $$\diam\circ \Gauss(W'_s)\leq \diam\circ \Gauss(W_{s+\epsilon}) + \epsilon.$$ Similarly, using , $$\diam\circ \Gauss(W'_s)\geq \diam\circ\Gauss(W_{s-\epsilon}) - \epsilon.$$ As a consequence, if we write $f=\vartheta(W)$ and $g=\vartheta(W')$, $$f(s-\epsilon)-\epsilon \leq g(s)\leq f(s+\epsilon) + \epsilon.$$ Hence $\vartheta((D(r),\epsilon)^{\Pr})\subset (t,\epsilon)^F$.
The following lemma and its use was suggested to me by Abdó Roig.
There is a continuous map $a\colon F_\geq(\bR^+) \lra C_>(\bR^+)$ such that $a(f)\geq f$, $a(f)(x)\geq \alpha x$ and $a(0)(x) = \alpha x$
The function $b(f)(x) = \frac{1}{x}\int_x^{2x}{f(y)dy}$ is continuous and non-decreasing. In addition, being $f$ non-decreasing, $b(f)(x) \geq \frac{1}{x}(f(x)(2x-x)) = f(x)$. To make it strictly increasing, first define $g(x) = \alpha x$ and replace $f$ by $f + g$, which is always positive and bigger than $f$. Therefore $a(f) = b(f+g)$ gives the desired function.
Let $\varphi\colon \qq\to (0,\infty)$ the continuous function obtained applying $\vartheta$, then $a$, then taking the inverse of the resulting function, and finally evaluating that function on $\alpha$. It has the following properties: $$\begin{aligned}
\diam\circ\Gauss(W_{\varphi(W)})&<\alpha & \text{for all $W$} \label{diam-gauss-varphi}\\
\varphi(W)&=1 &\text{if $W$ is a union of parallel planes}.\label{varphi-linear} \\
\varphi(W)&<1 &\text{if $W$ is not a union of parallel planes}.\label{varphi-nonlinear}\end{aligned}$$
Proof of the first part {#proof-of-the-first-part .unnumbered}
-----------------------
Let $f_{r,t}\in \Emb(B^n,B^n)$ be the isotopy of embeddings $f_{r,t}(x) = (t+(1-t)r)x $. Let $H_t\colon \qq\to \qq$ be the induced homotopy that sends a submanifold $W$ to $f_{\varphi(W),t}^{-1}(W)$. This is a continuous map by . Now, $f_{r,t}$ is conformal, and this implies that $\Gauss(W_{\varphi(W)}) = \Gauss(H_0(W))$, and it follows from that $H_0(W)$ satisfies . In addition, $H_0(W)$ is compactifiable as a consequence of (observe that unions of affine planes are always compactifiable). Hence $H_0(W)\in \qq^\delta$.
If $P\in \Gr_d(\bR^n)$, and $\pi,\pi^\perp$ are the projections onto $P$ and its orthogonal complement, then define an isotopy of embeddings $g_{t,P}\colon B^n\to B^n$ by $g_t(x) = t\cdot \pi(x) + \pi^\perp(x)$. Let $G_t\colon \qq^\delta \to \qq$ be the induced homotopy that sends a submanifold $W$ to $g_{t,\mu(W)}^{-1}(W)$. Because $\mu$ is continuous in $\qq^\delta$ and because of , it follows that $G_t\circ H_0$ is continuous as well except possibly at $t=0$.
The submanifold $g_{t,\mu(W)}^{-1}(W)$ is well-defined for $t=0$ if and only if $W$ intersects $\mu(W)^\perp$ transversally, in which case $g_{0,\mu(W)}^{-1}(W)$ is a union of affine planes parallel to $P$ whose origins (their closest points to the origin of $B^n$) are precisely $\mu(W)^\perp\cap W$. On the other hand, the tangent planes of the submanifold $V:=H_0(W)$ are all very close because its diameter is bounded by $\alpha$, and, by definition, close to $\mu(V)$, and therefore they meet $\mu(V)^\perp$ transversally: for any $x\in W$ and any $L$ with $L\cap \mu(V)^\perp\neq 0$, using and $$\begin{aligned}
d(T_xV,L)&\geq d(\mu(V),L) - d(\mu(V),T_xV) \\
&\geq 2\alpha - \diam\circ\Gauss(V) \\
&= 2\alpha - \diam\circ\Gauss(W_{\varphi(W)}) \\
&> 2\alpha - \alpha = \alpha>0.\end{aligned}$$ Since $V$ meets $\mu(V)^\perp$ transversally, $g_t^{\mu(V)}(V)$ is well-defined, and so is $H_0\circ G_0$.
Therefore $h:= G_0\circ H_0$ is well-defined and lands in $\widetilde{{\mathcal L}}_d(B^n)$, and $h\circ i$ is the identity (see ), whereas performing $H_t$ and then $G_t$ defines a homotopy between $i\circ h$ and the identity.$\square$
Proof of the second part {#proof-of-the-second-part .unnumbered}
------------------------
Let $C_d(\bR^n)\subset \ll$ be the subspace of those non-empty unions of affine planes, all of whose origins are at distance at most $1$ from the origin of $\bR^n$. This is a closed subset. Let $U\subset \ll$ be the subspace of those (possibly empty) unions of planes that do not contain the origin. Then, there is a pushout square $$\xymatrix{
U\cap C_d(\bR^n) \ar[d]\ar[rr] && U\ar[d] \\
C_d(\bR^n) \ar[rr] && \ll
}$$ which is also a homotopy pushout square because the upper horizontal arrow is a cofibration. Now, a point in $C_d(\bR^n)$ is a collection of parallel planes, and remembering the underlying linear plane defines a map $C_d(\bR^n)\to \Gr_d(\bR^n)$ that is also a fibre bundle. Its fibre over a plane $P$ is the space $C_0(P^\perp)$, which is the Ran space of $P^\perp$ and is well-known to be weakly contractible [@Gaitsgory Appendix]. Therefore $C_d(\bR^n)\simeq \Gr_d(\bR^n)$. The same argument proves that $U\cap C_d(\bR^n)\simeq \Gr_d(\bR^n)$, and since the left vertical map is a map over $\Gr_d(\bR^n)$, it is a weak homotopy equivalence. On the other hand, $U$ is contractible as the homotopy $(W,t)\mapsto \frac{1}{1-t}W$ defines a contraction of $U$ to the empty submanifold. As a consequence, the homotopy pushout $\ll$ is weakly contractible as well.$\square$
Microflexibility {#s:4}
================
The spaces $\P_d(U)$ and $\Pr_d(U)$ glue together to form sheaves $\P_d$ and $\Pr_d$ on $\bR^n$: If $U\subset U'$ is a pair of open subsets then the restriction map $\Pr_d(U')\to \Pr_d(U)$ sends a submanifold $W$ to the intersection $W\subset W'$. A sheaf of topological spaces in $\bR^n$ extends canonically to a sheaf of topological spaces in the site of manifolds and open embeddings [@R-WEmbedded Theorem 3.3].
At this point one is tempted to use the methods in the latter article to extend our theorem to the space of merging submanifolds in an arbitrary open manifold: In that article, it was proven that the sheaf $\Psi$ on a manifold $M$ is $\Diff(M)$-equivariant and that it is *microflexible*. By a theorem of Gromov [@Gromov], this automatically implies that for connected non-compact manifolds $M$ a certain map $$\Psi(M)\lra \Gamma(\Psi^\fib(TM)\to M)$$ is a homotopy equivalence. The space on the right is the space of sections of the fibrewise space of submanifolds of the tangent bundle of $M$. By the Galatius–Randal-Williams theorem, the fibre over each point is homotopy equivalent to the Thom space $\Th\gamma_{d,n}^\perp$.
The sheaf $\Pr$ is in fact $\Diff(M)$-equivariant, and if it were also microflexible, then one could deduce that a certain map $$\Pr(M)\lra \Gamma(\Pr^\fib(TM)\to M)$$ is a homotopy equivalence. Since the space of sections on the right has weakly contractible fibres by Theorem \[bla\], one would have that $\Pr(M)$ would be weakly contractible when $M$ is connected and non-compact. But this is a castle in the sky:
The sheaf $\widetilde{\Psi}$ is not microflexible.
Let us recall first the definition of microflexibility.
A sheaf $\Phi$ on a manifold $M$ is microflexible if for each pair $C'\subset C$ of compact subspaces of $M$, and each pair $C'\subset U',C\subset U$ of open subsets of $M$ such that $U'\subset U$, and for each diagram $$\xymatrix{
P\times \{0\}\ar[r]^f\ar[d] & \Phi(U)\ar[d]^r \\
P\times [0,1]\ar[r]^h & \Phi(U')}$$ there exists an $\epsilon>0$ and a pair of open subsets $C'\subset V'\subset U'$ and $C\subset V\subset U$ such that $V'\subset V$, and a dashed arrow $$\xymatrix{
P\times \{0\}\ar[r] \ar[d]& \Phi(U)\ar[r] & \Phi(V) \ar[d] \\
P\times [0,\epsilon)\ar[r]\ar@{-->}[urr] & \Phi(U')\ar[r] & \Phi(V')
}$$
Let $W'$ be a connected compact submanifold of $\bR^n$, and let $C'=U'$ be a tubular neighbourhood of $W'$ (which we implicitely identify with $NW$ from now on).
Let $W''\subset W\subset W'$ be codimension $0$ submanifolds such that $W''$ is closed as a subset and $W$ is open, and the first inclusion is a homotopy equivalences.
Let $U$ and $C$ be the restrictions of $U'$ to $W$ and $W''$ respectively.
Let $C_k(NW)$ be the fibrewise configuration space of $k$ unordered points in the normal bundle of $W$. A section $f$ of this bundle defines
1. a $k$-sheeted covering of $W$ and
2. an element in $\Pr_d(NW)\subset \Pr_d(U)$.
Since the fibre of $NW$ is a vector space, we can multiply any subset of it by a real number. Then we can define a path $$[0,1]\lra \Pr_d(NW)$$ by sending $t>0$ to $t\cdot f(W)$ and $t=0$ to $W$. If we have a family of sections of $C_k(NW)$ indexed by $P$, we obtain a map $$g\colon P\times [0,1]\lra \Pr_d(NW)$$ whose restriction to $P\times \{0\}$ is constant. Suppose now that we have a microflexible solution for the diagram $$\xymatrix{
P\times \{0\}\ar[r]^c\ar[d] & \Pr(U')\ar[d]^r \\
P\times [0,1]\ar[r]^g & \Pr(U)}$$ where $c$ is the constant map with value $W'$. This means that we can find an $\epsilon>0$ and an open subset $C\subset V\subset U$ such that $$\xymatrix{
P\times \{0\}\ar[r]^c \ar[d]& \Phi(U)\ar[r] & \Phi(V) \ar[d] \\
P\times [0,\epsilon)\ar[r]^g\ar[urr]^h & \Phi(U')\ar[r] & \Phi(U').
}$$ Then, for small values of $\delta\in [0,\epsilon)$, the map $h$ takes values in the space of sections of $C_k(NW)$ (the $k$ is determined because $h$ is extending the section $g$ that takes values in $C_k(NW)$, and because the inclusion $W\subset W'$ induces an epimorphism in components), and therefore defines for each $p\in P$ a $k$-sheeted covering of $W$. As a consequence, the above solution gives also a lift to the following diagram (where $\mathrm{Cov}(W)$ denotes the space of finite sheeted coverings of $W$): $$\xymatrix{
& \mathrm{Cov}(W') \ar[d] \\
P\ar[r]^g\ar[ur]^h & \mathrm{Cov}(W).
}$$ But this would mean that any family of coverings of $W$ can be extended to a family of coverings of $W'$ which is false (for instance, if $W' = S^2$ and $W$ is a equatorial band in $S^2$).
|
---
abstract: 'We consider a model of the early universe which consists of two scalar fields: the inflaton, and a second field which drives the stabilisation of the Planck mass (or gravitational constant). We show that the non-minimal coupling of this second field to the Ricci scalar sources a non-adiabatic pressure perturbation. By performing a fully numerical calculation we find, in turn, that this boosts the amplitude of the primordial power spectrum after inflation.'
author:
- Carsten van de Bruck
- Adam Christopherson
- Mathew Robinson
title: Stabilising the Planck mass shortly after inflation
---
Introduction
============
Cosmological observations put strong constraints on processes which could have occurred in the early universe. For example, models of inflation are tested with the properties of the cosmic microwave background radiation (CMB), such as the CMB anisotropies, CMB polarisation, non–Gaussianity and spectral distortions to the black–body spectrum. The [Planck]{} satellite [@Ade:2013uln; @Planck:2015xua] provides the most recent observational data of the CMB. Non–Gaussian statistics originating from inflationary physics can furthermore be probed with studies of the large scale structures (LSS) in the universe [@Desjacques:2010jw]. This is one of the goals of state-of-the-art current and future experiments, such as the Dark Energy Survey [@Abbott:2005bi], the Large Synoptic Survey Telescope (LSST) [@Ivezic:2008fe] and the [Euclid]{} satellite [@Amendola:2012ys].
The number of different inflationary models is vast. The simplest models consist of a single scalar field minimally coupled to gravity. However, the phenomenology of even these simple models is rich, with hundreds of different choices for the inflationary potential [@Martin:2014vha]. More complex models arise from including more than one scalar field, which can lead to qualitative differences. These differences occur due to fluctuations not just in one field direction, but now in more than one direction (i.e. isocurvature or non-adiabatic pressure perturbations, to which we will return later). Then, one could consider a single field with a non-standard kinetic term, such as k-inflation [@Garriga:1999vw]; these such models are often motivated by theories with extra dimensions, e.g., DBI inflation from string theory [@Alishahiha:2004eh]. For more complexity, the scalar field could have a non-minimal coupling to gravity, such as in the newest version of the Higgs inflation model [@Bezrukov:2007ep] (however, for the majority of cases, this can be treated as a field with a minimal coupling by moving to the Einstein frame and modifying the potential). Finally, the most complex inflationary models contain multiple scalar fields non-minimally coupled to gravity, and with non-standard kinetic terms [@Kaiser10a; @Kaiser:2013sna].
By comparing each model’s predictions with observational data, we can rule out regions of model space, with the ultimate goal to obtain a single inflationary model which best fits the data. Recent data provides bounds on the gravitational wave signature for which the simplest single field inflationary model with an $m^2\phi^2$ potential is disfavoured [@Ade:2015tva]. Therefore, it is particularly important to continue to investigate the dynamics and observational predictions of inflationary models beyond the simplest single scalar field model. One interesting model not belonging to the single field class is the curvaton model [@Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct]. This model consists of a second field, the curvaton, in addition to the inflaton. The curvaton is dynamically unimportant during inflation, but its fluctuations source the curvature perturbation.
In this paper, we address the question of whether a possible stabilisation of the Planck mass (or gravitational constant) just after inflation can have a sizeable effect on the primordial power spectrum of the curvature perturbation. In theories in which the four–dimensional Planck mass are not constant, its dynamics is usually driven by one or several moduli fields. These describe for example the size of the extra–dimensional space. Since the time evolution of the gravitational constant is strongly constrained by experiments (see e.g. [@Will:2014xja] for a recent update on experimental tests of General Relativity), any stabilisation of the moduli field(s) must have happened in the early universe[^1]. The stabilisation could have happened well before inflation ended, affecting scales well outside the visible horizon. If the stabilisation happened during the last 60 e–folds of inflation, possible signatures in the curvature perturbation power–spectra can be produced [@Ashoorioon:2014yua]. If the stabilisation happened later, in the radiation dominated epoch, the rapid oscillations of the scalar field(s) can affect the expansion history [@Steinhardt:1994vs; @Perivolaropoulos:2002pn]. In the scenario discussed in this paper, Newton’s constant stabilised a few e–folds after inflation. We take into account the possibility that the moduli field driving the evolution of Newton’s constant can decay into other particles as well. Our setup is therefore related to the standard curvaton scenario. In the absence of a non-adiabatic pressure perturbation, the curvature perturbation $\zeta$ on uniform density hypersurfaces is known to be conserved on superhorizon scales [@Wands2]. However, this is not necessarily the case if several fields are dynamically important. Even if inflation has ended, the decay of fields at a later stage can significantly influence the evolution of $\zeta$ (see e.g. [@Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct; @Ashcroft:2004rs; @Bassett:1998wg; @Bassett:1999mt; @Bassett:1999ta]). In the case of a scalar field driving the evolution of the Planck mass, we find that the non–minimal coupling to the Ricci scalar can boost the amplitude of the curvature perturbation by several orders of magnitude, even if the Planck mass varies only by a very small amount.
The paper is structured as follows: in the next section, we present the model, and the governing equations for the background and perturbations of the model. Then, in Section \[sec:numerics\] we describe our numerical procedure, before presenting results in Section \[sec:results\]. Finally, we conclude in Section \[sec:conclusion\].
Theory and analytical results {#sec:theory}
=============================
The model we consider consists of two scalar fields, namely of the inflaton $\phi$ and the field $\sigma$, which describes the evolution of the Planck mass. In the Jordan frame, the action is given by $$\begin{aligned}
\label{eq:action}
\mathcal{S} = \int \mathrm{d}^4 x \sqrt{-g} \left[ \frac{1}{2}f(\sigma)R - \frac{1}{2}g^{\mu\nu}\left(\partial_{\mu}\phi\partial_{\nu}\phi + \partial_{\mu}\sigma\partial_{\nu}\sigma\right) - V(\phi, \sigma) + \mathcal{L}_{\rm int} \right]\,,\end{aligned}$$ where $g_{\mu\nu}$ is the metric tensor, $V(\phi, \sigma)$ the potential and $\mathcal{L}_{\rm int}$ is the interaction Lagrangian, describing the perturbative decay [@Huston13] of both the $\phi$ and $\sigma$ into radiation. By working in the Jordan frame, the decay rates can be calculated in the standard way. We denote them by $\Gamma^\phi$ and $\Gamma^\sigma$ respectively. Since we are interested in the effect of stabilising the Planck mass, we expand $f(\sigma)$ around its minimum, keeping only the leading term: $$\begin{aligned}
f(\sigma) = 1 + \frac{\alpha}{2}(\sigma - \sigma_{\rm min})^2\,.\end{aligned}$$ We denote the masses of the fields by $m_\phi$ and $m_\sigma$ and assume that the fields are not directly interacting. Note that we are working in units with reduced Planck mass $M_{\rm PL}=1$.Therefore, the potential is given by $$V(\phi,\sigma) = \frac{1}{2}m_\phi^2 \phi^2 + \frac{1}{2}m_\sigma^2 \sigma^2.$$
To consider the evolution of cosmological perturbations, we work in the longitudinal gauge, in which the line element takes the form [@Bardeen80; @Kodama84; @Mukhanov:1990me] $$\begin{aligned}
ds^2 = -(1+2\Phi)dt^2 + a^2(t)\left(1-2\Psi\right)\delta_{ij}dx^i dx^j~.\end{aligned}$$ Here, $a(t)$ is the scale factor, $\Phi$ and $\Psi$ are independent metric perturbations, which depend on all coordinates. In the Jordan frame, $\Phi$ and $\Psi$ are not equal even in the absence of anisotropic stress (see Eq. (\[PhiPsi\]) below). The equations of motion for the system can be obtained by varying the action in Eq. (\[eq:action\]). In the background, we have evolution equations for the two scalar fields $$\begin{aligned}
\ddot{\phi} &=& - V_\phi - 3H\dot{\phi} -\Gamma^\phi\dot{\phi}\,,\\
\ddot{\sigma} &=& - V_\sigma - 3H\dot{\sigma} + R f_\sigma/2 - \Gamma^\sigma\dot{\sigma} \label{curvaton} \,, \end{aligned}$$ in addition to an energy conservation equation for the radiation fluid $$\dot{\rho_\gamma} = -4H\rho_\gamma + \Gamma^\phi\dot{\phi}^2+ \Gamma^\sigma\dot{\sigma}^2\,,$$ and the Friedmann equation $$H^2 = \frac{1}{3f}\left[ \frac{\dot{\phi}^2}{2} + \frac{\dot{\sigma}^2}{2} + V + \rho_\gamma \right] - \frac{f_\sigma \dot{\sigma}H}{f}$$ In these equations, a subscript $\phi$ or $\sigma$ denotes a partial derivative with respect to the field, and we have written the derivatives with respect to cosmic time, $t$. We shall also use the slow roll parameter defined by [@Liddle:2000cg], $$\epsilon\equiv-\frac{\dot{H}}{H^2}\,,$$ in order to more simply write the Ricci scalar, which can be expressed as $R = 6H^2(2-\epsilon)$ along with its perturbation $$\delta R = -6\ddot{\Psi} - 6H(\dot{\Phi} + 4\dot{\Psi}) - 2R\Phi + 2\frac{k^2}{a^2}(\Phi - 2\Psi)\,.$$
Considering now the linear perturbations, we obtain a Klein-Gordon equation for each field [@Kaiser10a] $$\begin{aligned}
\ddot{\delta\phi} &= -3H\dot{\delta\phi} - \left( \frac{k^2}{a^2} + V_{\phi\phi}\right)\delta\phi - V_{\phi\sigma}\delta\sigma + \dot{\phi}(\dot{\Phi} + 3\dot{\Psi}) - 2V_\phi\Phi \,,\\
%
\label{dsigdashdash}
\ddot{\delta\sigma} &= -3H\dot{\delta\sigma} - \left( \frac{k^2}{a^2} + V_{\sigma\sigma} - \frac{f_{\sigma\sigma}R}{2}\right)\delta\sigma - V_{\sigma\phi}\delta\phi + \dot{\sigma}(\dot{\Phi} + 3\dot{\Psi}) - 2V_\sigma\Phi + \frac{f_\sigma}{2}(2R\Phi + \delta R) \,,\end{aligned}$$ along with a conservation equation for the radiation fluid $$\dot{\delta\rho_\gamma} = -4H\delta\rho_\gamma + 4\rho_\gamma\dot{\Phi}- 2\frac{k^2}{a^2}(\dot{\Psi} + H\Phi) + 2\Gamma_\phi(\dot{\phi}\dot{\delta\phi} - \frac{\dot{\phi}^2}{2}\Phi) + 2\Gamma_\sigma(\dot{\sigma}\dot{\delta\sigma} - \frac{\dot{\sigma}^2}{2}\Phi)\,.$$ The metric perturbation $\Psi$ satisfies the following evolution equation $$\begin{aligned}
\ddot{\Psi} &= - 3H\dot{\Psi} - H\dot{\Phi} - H^2(3-2\epsilon)\Phi \nonumber\\
& + \frac{1}{2f} \Bigg[\dot{\phi}\dot{\delta\phi} + \dot{\sigma}\dot{\delta\sigma} - (\dot{\phi}^2 + \dot{\sigma}^2)\Phi - V_\phi\delta\phi - V_\sigma\delta\sigma - 2\ddot{f}\Phi - \dot{f}(\dot{\Psi} + 2H\Phi) \Bigg. \nonumber\\
&\left. \qquad \quad- \frac{\delta f}{f}\left( \frac{\dot{\phi}^2}{2} + \frac{\dot{\sigma}^2}{2} - V + \ddot{f} + 2H\dot{f} \right) + \ddot{\delta f} + 2H\dot{\delta f} + \frac{k^2}{a^2}\delta f \right]
\label{dPsidashdash}\end{aligned}$$ along with the constraint $$\label{PhiPsi}
\Phi = \Psi - \frac{\delta f}{f}\,.$$ The predictions from inflationary models are mapped onto observations (such as the temperature fluctuations of the CMB) in a simple way by introducing a curvature perturbation. The curvature perturbation on uniform density hypersurfaces, $\zeta$, is defined as $$\zeta=-\Psi-\frac{H}{\dot{\rho}}\delta\rho\,,$$ where here, $\rho$ and $\delta\rho$ are the energy density and perturbation for the entire matter content of the universe. We can obtain an evolution equation for $\zeta$ which, in the large-scale limit, takes the form $$\dot{\zeta}=-\frac{H}{(\rho+P)}\delta P_{\rm nad}\,,$$ where the non-adiabatic pressure perturbation, $\delta P_{\rm nad}$, is defined as $$\delta P_{\rm nad}\equiv\delta P-\frac{\dot{P}}{\dot{\rho}}\delta\rho\,.$$ For a minimally coupled single field model of inflation (or, equivalently, for a universe containing a single fluid), the curvature perturbation, $\zeta$, is conserved for both canonical and non-canonical models of inflation, independent of the theory of gravity [@Wands2; @Rigopoulos:2003ak; @Christopherson:2008ry]. This allows us to compare inflationary predictions directly to observations by mapping the field fluctuations onto the curvature perturbation. Since it is conserved, we do not need to worry about the mechanism by which inflation ends and the universe reheats.
However, moving beyond these simple models, the non-adiabatic pressure (or entropy) perturbation is non-zero, and therefore the curvature perturbation can continue to evolve and be enhanced on super-horizon scales. This feature has been exploited in numerous scenarios containing multiple minimally coupled scalar fields (see, e.g., Refs. [@GarciaBellido:1995qq; @Bassett:2005xm; @Bassett:1998wg; @Bassett:1999mt; @Bassett:1999ta; @Huston:2011fr; @Huston13]). In these models, we must take into account the reheating phase in order to make reliable predictions. Models with non-minimally coupled scalar fields, on the other hand, will produce a distinct source of entropy perturbations, arising due to the coupling of the scalars. In the model we present above, it is expected that, after inflation has ended and during the reheating phase, these entropy perturbations can become sizeable due to the fact that $\dot{f}$ and $\ddot{f}$ no longer need to remain small [@Kaiser10a]. It is these non-adiabatic pressure perturbations, and the resulting amplification of the power spectrum, that we will investigate in the remainder of the paper.
$\sigma_{\rm{ini}}$ The initial value of $\sigma$
--------------------- ----------------------------------------------------------------------------------------
$\sigma_{\rm{end}}$ The value $\sigma$ reaches at the end of inflation, before rolling down to its minimum
$\sigma_{\rm{min}}$ The minimum in the expansion of $f(\sigma)$
: \[tab:notation\]A table clarifying our notation for the subscripts denoting various stages in the evolution of $\sigma$.
We will solve the system of equations derived above. It will be necessary to follow the evolution of the secondary field, $\sigma$, throughout the inflationary phase, the decay of the inflationary field, $\phi$, and the radiation epoch right up until $\sigma$ itself has decayed and no longer contributes to the overall energy density of the universe. This is important so as not to restrict ourselves to the case where the auxiliary field starts at its minimum, $\sigma_{\rm min}$ (see Table \[tab:notation\] for subscript notation), and is pushed away, but to also include cases where the field evolves towards $\sigma_{\rm min}$ during inflation. It is often the case that when studying subdominant, curvaton-like fields, the calculation begins in a post-inflationary radiation-dominated phase and proceeds from there. For the usual, minimally coupled fields this is sufficient, since $\sigma$ does not evolve until late on, after the end of inflation [@Lyth:2001nq]. However, as this no longer holds in our case, we must track it throughout. It is still important for this coupling to remain small so as not allow $\sigma$ to contribute too much and impact upon the dynamics of inflation itself.
This feature, which distinguishes our setup from the standard curvaton scenario, arises due to the explicit coupling to the Ricci scalar causes the field to obtain an effective mass, which might not be small compared to $H^2$. In the slow–roll approximation we find that $\sigma$ evolves according to $$\sigma \propto e^{3\alpha\frac{2-\epsilon}{3-\epsilon} N},$$ where $\epsilon$ is assumed to be roughly constant and $N$ is the e–fold number. This equation follows directly from Eq. ([\[curvaton\]]{}), writing this equation in terms of e–fold number $N$ and neglecting the bare mass of the field. Therefore, it is often the case that the field–value of $\sigma$ at horizon crossing is different from the value of $\sigma$ at the end of inflation.
Numerical method {#sec:numerics}
================
We will solve the governing equations derived in the previous section using a code written in Python. This starts at the beginning of inflation and runs right through to the end of the decay of the second field, when the power spectrum has reached its final value. This is the full numerical simulation including field perturbations, their gravitational counterparts and those in the radiation fluid created in the final stages. We begin by running through the background in order to ascertain the values needed to set up the initial conditions for each mode, $k$, such that $k_* = 50a_{\rm i} H_{\rm i}$. We then set the initial conditions of the perturbations as those of the Bunch-Davies vacuum [@Bunch:1978yq] at this point and begin the full perturbation code. The perturbation equations are then each integrated twice, independently, by first setting the initial value of $\delta\sigma$ to zero whilst leaving $\delta\phi$ to take its Bunch-Davies vacuum form:
$$\begin{aligned}
\delta\phi, \delta\sigma &=& \frac{e^{-ik\tau}}{\sqrt{2ka_{\rm i}}}\,,\\
\delta\phi', \delta\sigma' &=& \frac{-ik e^{-ik\tau}}{\sqrt{2ka_{\rm i}^3 H_{\rm i}^2}}\,,\end{aligned}$$
and then vice versa. We normalize the number of e-foldings to be $N = 0$ at horizon exit, and so plot our results around this value.
The code is split into four sections, each solved successively with the end values to each one used as the initial conditions in the next:
1. [**Inflation**]{}: [This covers the period from $N=0$ through to when the inflaton crosses the minimum of the potential, at which point we switch the decay, $\Gamma_\phi$ on.]{.nodecor}
2. [**Inflaton decay**]{}: [Covering the period through the first part of reheating, but before the secondary field has begun to decay.]{.nodecor}
3. [**Overlap decay**]{}: [The inflaton still contributes a significant amount to the overall energy density but the secondary field, $\sigma$ too has started to decay, so we switch on $\Gamma_\sigma$.]{.nodecor}
4. [**Secondary field decay**]{}: [Finally, we switch off the evolution of the $\phi$ field altogether as it is so difficult and time consuming to follow the vastly different scales involved in both this and much smaller secondary field, $\sigma$. This then continues until the power spectra settles on a specific value and all the energy density of the universe is held within the radiation.]{.nodecor}
We take the decay parameters to be $\Gamma^\phi=10^{-8}M_{\rm PL}$ and $\Gamma^\sigma=10^{-14}M_{\rm PL}$, and the masses of the scalar fields to be $m_\phi=10^{-7}M_{\rm PL}$ and $m_\sigma=10^{-10}M_{\rm PL}$. These values are chosen to be close to those in Ref. [@Huston13], and compatible with the limits in Eq. (11) of Ref. [@Bartolo:2002vf]
While the first two sections can be integrated in a matter of minutes, the latter sections can take some considerable time to track the oscillatory phases throughout decay of the secondary field. This is due to the small scales involved in comparison to the inflationary phases. Since the sudden decay approximation does not hold in this case [@Lyth:2002my], the later sections are crucial in our numerical procedure in order to obtain an accurate result. Even in the standard case, with $\alpha = 0$, we find that the ratio of $\sigma$ to the other components in the universe, $r_{\rm{dec}}$, evaluated when $H = \Gamma^\sigma$ is not the true point at which $r_{\rm{dec}}$ reaches its maximum value. An improved (and increased) value can be attained slightly before this at $H \sim 3 - 5\Gamma^\sigma$ or, more accurately still, read off from its numerical maximum. The need for following the decay in full becomes even more apparent when we look at the results, in the next section. We find that oscillations in the non-adiabatic pressure perturbation towards the end of curvaton decay play an important role too.
Results {#sec:results}
=======
The case: $\sigma_{\rm{min}} = \sigma_{\rm{ini}}$ {#sec:results1}
-------------------------------------------------
For this section we set $\sigma_{\rm{min}} = \sigma_{\rm{ini}}$ in order to exclude any evolution of the secondary field during inflation (see Figure \[om\_-0.005\]). By keeping $\sigma_{\rm{min}} = \sigma_{\rm{ini}}= 0.1$ and comparing to a standard curvaton scenario, for which $\alpha = 0$ and $r_{\rm{dec}} \simeq 0.18$ (Figure \[om\_-0.005\]), we see a significant change in the amplitude of the final power spectrum for a given $k$. In Figure \[Pdnad1\] we plot the final twelve efolds as the inflaton decays, followed by the radiation dominated and secondary field dominated phases in terms of both $P_{\mathcal{\zeta}}$ (the dimensionless power spectrum) and $\delta P_{\rm{nad}}$. In this and later plots in the paper, we take $k=0.05 \,{\rm Mpc}^{-1}$. This shows the influence of the non-adiabatic pressure on the final power spectra; the $\delta P_{\rm{nad}}$ survives for around an efold longer and has a maximum amplitude of up to roughly 100 times that of the standard case. Figure \[om\_-0.005\] shows that this increase in amplitude is not due to a more dominant secondary field, as the value of $\Omega_\sigma$ at the time of decay remains roughly constant (the change is of the order of $0.1\%$).
![[*Top*]{}: The evolution of the relative energy density in each species, $\Omega_i$ for both the $\alpha = 0$ and $\alpha = -0.005$ cases, which overlap throughout. [*Bottom left*]{}: The background evolution of $\sigma$ for both $\alpha$ values and for $\sigma_{\rm{min}} = \sigma_{\rm{ini}} = 0.1$. [*Bottom right*]{}: The evolution of the effective Planck mass for $\alpha = -0.005$ when $\sigma_{\rm{min}} = \sigma_{\rm{ini}} = 0.1$.[]{data-label="om_-0.005"}](omegas.png){width="\linewidth"}
![The power spectrum of the curvature perturbation ([*left*]{}) for both $\alpha = 0$ and $\alpha = -0.005$ cases and the associated $\delta P_{\rm nad}$ ([*right*]{}).[]{data-label="Pdnad1"}](alphaneg005.png){width="\linewidth"}
For the case with $\sigma_{\rm{ini}} = 0.1$, as above, we see in Figure \[Pvasym\] that the dependence on $\alpha$ is independent of sign. This might be expected due to the boost in power spectrum coming as the secondary field oscillates and decays. Terms such as $\dot{f}$ and $\ddot{f}$, which both contribute to $\delta P_{\rm{nad}}$, effectively average out as their sign changes back and forth. For this reason, in the case of $\sigma_{\rm{min}} = \sigma_{\rm{ini}}$ we shall now only look at the effect of $\lvert \alpha\rvert$. For the cases when $\sigma_{\rm{min}} \neq \sigma_{\rm{ini}}$, which we will consider in the next section, this may no longer remain true as the sign of $\alpha$ will introduce a scaling of the final field value, $\sigma_{\rm{end}}$, which can in turn affect the final power spectrum. We also observe a slight dip on either side of $\alpha = 0$, for which the amplitude decreases before increasing again. We have checked that this is not a numerical artefact. We do not have a physical explanation for this dip and the complexity of the governing equations makes it difficult to address this question analytically.
![The amplitude of the power spectrum as a function of $\alpha$ normalised to the $\alpha=0$ power spectrum: ${\mathcal{P}}_{\mathcal{\zeta}}(\alpha)/{\mathcal{P}}_{\mathcal{\zeta}}(0)$.[]{data-label="Pvasym"}](Pvasym.png){width="\linewidth"}
Finally, in these simple cases, it is useful to compare how $\sigma_{\rm {end}}$ affects the outcome for various values of $\lvert \alpha\rvert$. We will specifically focus on $\sigma_{\rm{end}} = 0.1,0.2,0.3$ which results in $r_{\rm{dec}}$ values of $0.17, 0.45$ and $0.62$ respectively. For these final values of $\sigma$ we would expect a varying increase in the amplitude of the power spectrum arising simply from the standard curvaton results. At the end of inflation we find $P_{\mathcal{\zeta}}= 3.01\times 10^{-13}$ and this value is boosted by factors of $2.51, 11.13$ and $13.85$ respectively by the end of curvaton decay with $\alpha = 0$. In Figure \[alphas1\] this small boost is apparent in the values at $\alpha = 0$ but is insignificant in comparison to the subsequent amplitude increases as we increase $\alpha$ from $0$. The results for each $\sigma_{\rm{end}}$ diverge for increasing $\alpha$ due to the fact that for each $\sigma_{\rm{end}}$ we also have $\sigma_{\rm{min}} = \sigma_{\rm{end}}$, so that the difference between the true minimum ($ \sigma \simeq 0$) and the local minimum associated with $f(\sigma_{\rm min})$ increases as the values for $\sigma_{\rm end}$ increase.
![Amplitude of the power spectrum ${\mathcal{P}}_{\mathcal{\zeta}}$ as a function of $\alpha$ for three different values of $\sigma_{\rm end} = 0.1,0.2,0.3$[]{data-label="alphas1"}](alphas0.png){width="\linewidth"}
The case: $\sigma_{\rm min} \neq \sigma_{\rm ini}$
--------------------------------------------------
In the more general case we have two possibilities, namely $\alpha > 0$ and $\alpha < 0$. This choice plays a role in the evolution of $\sigma$ during inflation which can in turn affect $\sigma_{\rm end}$ and $r_{\rm dec}$. By choosing $\alpha$ to be negative, we can pull $\sigma$ towards its local minimum before decay; a positive $\alpha$ has the opposite effect and pushes it away. This second case soon becomes unworkable for values of $\alpha$ approaching $0.05$ or greater due to the exponential increase apparent in the value of $\sigma$. Due to the symmetry of $\alpha$ shown in Figure \[Pvasym\] this need not be too concerning, however, as once $\sigma$ reaches its final value at the end of inflation we can still study the subsequent effects purely by using negative values. The only benefit of using $\alpha > 0$ comes in the ability to further vary the trajectory of $\sigma$ to test that our results are independent of it. This can be done by setting $\sigma$ to a number of different initial values and using $\alpha$ to control its final value in order to compare results.
In each case we find that the power spectrum is dominated by the value of $\sigma_{\rm min}$ with a lesser but still noticeable dependence on $\sigma_{\rm end}$. This is most simply demonstrated by Figure \[min01\] in which we show two cases, both with $\sigma_{\rm min} = 0.1$ but with $\sigma_{\rm ini} = 0.1$ and $0.3$ respectively. We let $\alpha$ run over the same values used previously which, for $\sigma_{\rm ini} = 0.3$, gives various values of $\sigma_{\rm end}$: $0 > \alpha > -0.03$ results in $0.3 > \sigma_{\rm end} > 0.1$, while $\alpha < -0.03$ gives $\sigma_{\rm end} = 0.1$ (see the right hand side of Figure \[min01\]). It is clear from the left hand side of Figure \[min01\] that while $\alpha$ remains small, the final amplitude of the power spectrum differs from that of the $\sigma_{\rm ini} = \sigma_{\rm min}$ case. This can be explained by the observation that in these cases the field has not had enough time to reach $\sigma_{\rm end} = 0.1$ due to the smallness of $\alpha$. For larger $\alpha$, however, the two cases converge because $\sigma_{\rm end}$ now equals $0.1$ in each of these examples.
![[*Left:*]{} The power spectra for varying $\alpha$ with $\sigma_{\rm ini} = 0.3$ and $\sigma_{\rm min} = 0.1$ (red) in comparison to the case of $\sigma_{\rm ini} = \sigma_{\rm min} = 0.1$ (blue). [*Right*]{}: The background trajectories for $\sigma$ for each of these cases. []{data-label="min01"}](min01.png){width="\linewidth"}
The case: $\sigma_{\rm min} = 0$
--------------------------------
Finally, we give an example which demonstrates both that the evolution of $\sigma$ throughout inflation has little to no impact (other than the dependence on $\sigma_{\rm end}$) on the final power spectra amplitudes and that we get no noticeable boost when $\sigma_{\rm min} = 0$. Here we take $\sigma_{\rm ini} = 0.05$ and $\alpha = \{0.0056, 0.011, 0.0145\}$ which gives $\sigma_{\rm end} = \{0.1, 0.2, 0.3\}$, respectively. From these we find that the amplitude of the final power spectrum is multiplied by factors of $1.03, 1.29$ and $1.51$. However, in comparison to the factors involved in the standard curvaton-like case given in Section \[sec:results1\], for these values of $\sigma_{\rm end}$ we see that the changes represent additional increases of only $1-3\%$. These are insignificant when taking into account the usual approximations inherent in the curvaton model and those increases found earlier in the paper for $\sigma_{\rm min} \neq 0$.
Conclusion {#sec:conclusion}
==========
In this paper we have studied a model of the early universe consisting of two scalar fields: the inflaton and a second field which controls the stabilisation of the Planck mass. We work in the Jordan frame, for which the second field is non-minimally coupled to gravity; this choice allows us to deal with the decay of the fields in the usual way. We have investigated numerically the effects of this coupling on the power spectrum of primordial fluctuations. It has previously been shown that a non-minimally coupled scalar field can induce changes in the curvature perturbation on super-horizon scales via the introduction of terms proportional to $f, \dot{f}$ and $\ddot{f}$ in the non-adiabatic pressure perturbation, $\delta P_{\rm nad}$ [@Kaiser10a]. Here, we have quantified this effect. We have shown that it can play an important role on the amplitude of the power spectrum in a non-minimally coupled curvaton-like case, in which the secondary field decays only after inflation is complete. Allowing the effective Planck mass to evolve in such a way, even by the smallest of amounts, leads to dramatic changes in the amplitude of the final power spectrum in comparison to the standard curvaton scenario.
The effect of this amplitude boost can also be linked to the spectral index, $n_s$ and tensor-scalar ratio, $r_{TS}$ by parameterising the power spectrum as [@Enqvist:2013paa] $$\begin{aligned}
\mathcal{P}_\zeta = \mathcal{P}_\zeta^{(\phi)}+ \mathcal{P}_\zeta^{(\sigma)} = (1+R)\mathcal{P}_\zeta^{(\phi)}\,,\end{aligned}$$ where $$\begin{aligned}
R = \frac{\mathcal{P}_\zeta^{(\sigma)}}{\mathcal{P}_\zeta^{(\phi)}}\,.\end{aligned}$$ This gives $$\begin{aligned}
n_s - 1 = -2\epsilon + 2\eta_\sigma - \frac{4\epsilon-2\eta_\phi}{1+R} \qquad\text{and}\qquad r_{TS} = \frac{16\epsilon}{1+R} \,,\end{aligned}$$ using the usual definitions of the slow roll parameters, evaluated at horizon crossing. The important point to note here is that $n_s$ and $r_{TS}$ depend only on the ratio, $R$, not the mechanism by which the curvaton, or curvaton-like field, sources the final curvature perturbation. This places tight constraints on the values that $\alpha, \sigma_{\rm min}$ and hence $f(\sigma)$ can take according to the latest [Planck]{} data [@Byrnes:2014xua]. We soon find ourselves with a spectral index approaching 0.98 – as in the pure curvaton limit – as $R$ becomes large with relatively small changes in $f$. This is also largely independent of any evolution in $\sigma$ during inflation because the inflaton dominates the universe at horizon crossing, when both the slow roll parameters are evaluated and the tensor perturbations freeze in.
It remains to be seen whether a similar scenario to the one discussed will arise from fundamental theories of particle physics. If so, it will have an impact on inflationary model building in such theories.
CvdB is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant ST/L000520/1. AJC is supported by the U.S. Department of Energy under Grant No. DE-FG02-97ER41029 and, during the early stages of this work, was supported by the Sir Norman Lockyer Fellowship of the Royal Astronomical Society.
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[^1]: Alternatively, the post-inflation evolution of the field(s) could be very slow.
|
---
abstract: 'We prove that the diameter of the commuting graph of the full matrix ring over the real numbers is at most five. This answers, in the affirmative, a conjecture proposed by Akbari-Mohammadian-radjavi-Raja, for the special case of the field of real numbers.'
---
\[section\]
\[Theorem\][Definition]{}
\[Theorem\][Corollary]{}
\[Theorem\][Lemma]{}
**[C. Miguel]{}**
Instituto de Telecomunucações,
Pólo de Covilhã,
[email protected]
[**Subject Classification:**]{} 15A21; 15A27; 05C50\
[**Keywords:**]{} Jordan form; full matrix ring; commuting graph; diameter
Introduction
=============
For a ring $R$, we denote the center of $R$ by $Z(R)$, that is, $Z(R)=\{x\in R\;:\;xr=rx\;\forall r\in R\}$. The [*commuting graph*]{} of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertices are all non-central elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $ab=ba$. In particular, the set of neighbors of a vertex $a$ is the set of all non-central elements of the centralizer of $a$ in $R$, that is, of $C_R(a)=\{x\in R : ax=xa\}$. The commuting graph has been studied extensively in recent years by several authors, e.g.[@ak; @ak1; @ar; @dol; @do; @do1; @gi; @ma; @om]. Additional information about algebraic properties of the elements can be obtained by studying the properties of a commuting graph. For example, if $R$ is a ring with identity such that $\Gamma(R)\cong\Gamma(M_2(\mathds F))$, for a finite field $\mathds F$, then $R\cong M_2(\mathds F)$, see [@ma]. It is conjectured that this is also true for the full matrix ring $M_n(\mathds F)$, where $\mathds F$ is a finite field and $n>2$.
We next recall some concepts from graph theory. In a graph $G$, a [*path*]{} is an ordered sequence $v_1-v_2-\ldots -v_l$ of distinct vertices of $G$ in which every two consecutive vertices are adjacent. The graph $G$ is called [*connected*]{} if there is at least one path between every pair of its vertices. The distance between two distinct vertices $u$ and $v$, denoted by $d(u , v)$, is the length of the shortest path connecting them (if such a path does not exists, then $d(u , v)=\infty$). The [*diameter*]{} of the graph is the longest distance between any two vertices of the graph $G$ and will be denoted by $diam(G)$.
Much research has concerned the diameter of commuting graphs of certain classes of rings, e.g.[@ak2; @dol; @do; @gi]. For the matrix ring $M_n(\mathds F)$, over an algebraically closed field $\mathds F$, the commuting graph is connected and the diameter is always equal to four, provided $n\geq 3$ [@ak2]. Note that for $n=2$ the commuting graph is always disconnected [@ak4 remark 8]. If the field $\mathds F$ is not algebraically closed, the commuting graph $\Gamma(M_n(\mathds F))$ may be disconnected for an arbitrarily large integer $n$ [@ak3]. However, in $\cite{ak2}$ it is proved that for any field $\mathds F$ and $n\geq 3$, if $\Gamma(M_n(\mathds F))$ is connected, then the diameter is between four and six [@ak2]. Also in [@ak2] the authors advanced the following conjecture.
\[Akbari, Mohammadian, Radjavi, Raja\] Let $\mathds{F}$ be a field. If $\Gamma(M_n(\mathds F))$ is a connected graph, then its diameter is at most $5$.
The aim of the present paper is to prove this conjecture for the special case of the field $\mathds R$ of real numbers. It was shown in [@ak3 remark 8] that $\Gamma(M_n(\mathds R))$ is a connected graph, for any $n\geq 3$. It is well-know that the central elements in the matrix ring $M_n(\mathds R)$ are the scalar matrices. Hence, the vertices in the graph $\Gamma(M_n(\mathds R))$ are the non-scalar matrices. Our main result is the following theorem.
\[t1\]Let $\mathds R$ be the field of real numbers and $n\geq 3$. Then, $diam(\Gamma(M_n(\mathds R)))=4$ for $n\neq 4$ and $diam(\Gamma(M_4(\mathds R)))\leq 5$.
We will show that Theorem \[t1\] follows quickly from various results in the literature and from the real Jordan canonical form.
We note that the Theorem \[t1\] above generalizes easily to fields which have an algebraic closure which is a finite extension. This follows from the Artin-Schreier Theorem [@ja p.316] which asserts that such fields are precisely the real closed fields, which roughly speaking are the fields behaving like $\mathds R$ , and whose algebraic closures have degree $2$ and are given by adjoining a square root of $-1$. An example is the field of real algebraic numbers.
Some Lemmas
============
In this section we assemble the tools that we require to prove Theorem\[t1\]. One of the key tools is the real Jordan canonical form for a matrix over the field $\mathds R$ of real numbers. For the sake of completeness we recall it very briefly. Let $\mathds F$ be an algebraically closed field and $A\in M_n(\mathds F)$. The well-known Jordan canonical form theorem states that there is a regular matrix $S\in M_n(\mathds F)$ such that $$S^{-1}AS=diag(J_{n_1}(\lambda_1), \ldots , J_{n_k}(\lambda_k)).$$ Each $J_{n_i}(\lambda_i)$, $i=1, \ldots ,k$, is called a [*Jordan block*]{} of order $n_i$ and is of the form $$J_{n_i}(\lambda_i)=\lambda_iI+N_i,$$ where $I$ is the identity matrix and each $N_i$, $i=1, \ldots , k$, is a square matrix whose only non-zero entries are $1's$ on the super-diagonal (i.e. just above the diagonal). The matrix $S^{-1}AS=diag(J_{n_1}(\lambda_1), \ldots , J_{n_k}(\lambda_k))$ is called the [*Jordan canonical form*]{} of the matrix $A$. This canonical form was described by C. Jordan in 1870. If the field $\mathds F$ is not algebraically closed, the Jordan canonical form is no longer available for all matrices in $M_n(\mathds F)$. However, for matrices over the field of real numbers the complex eigenvalues come in complex conjugate pairs, and this can be used to give a real Jordan canonical form for real matrices. Let $\lambda=a+ib$, where $b\neq 0$, be a complex eigenvalue of a matrix $A\in M_n(\mathds R)$. Denote by $C(a , b)$ the $2$ by $2$ matrix $$C(a, b)=\left[\begin{array}{cc}a&b\\-b&a \end{array}\right]$$ and by $C_k(a, b)$ the matrix of order $2k$ of the form $$C_k(a, b)=\left[\begin{array}{ccccc}C(a, b) &I & 0 &
\ldots &0
\\ 0 & C(a, b)& I & \ddots & \vdots \\
0 & 0 & \ddots & \ddots & 0 \\
\vdots & \ddots & \ddots & C(a, b) & I \\
0 & \ldots & 0 & 0& C(a, b)
\end{array}\right]$$ where $I$ is the identity matrix of order $2$. Then, there is a regular matrix $S\in M_n(\mathds R)$ such that $$S^{-1}AS=diag(C_{n_1}(a_1, b_1), \ldots , C_{n_t}(a_t, b_t), J_{m_1}(\lambda_1), \ldots , J_{m_s}(\lambda_s))$$ where $a_j+ib_j$ are the nonreal eigenvalues of $A$, for $j=1, \ldots ,t$, and $\lambda_q$ are the real eigenvalues of $A$, for $q=1, \ldots ,s$. This is called the [*real Jordan canonical form*]{} of $A$. For more details about the real Jordan canonical form, including proofs, the reader is referred to section 6.7 of [@la].
It was show in [@dol] that for an algebraically closed field $\mathds F$, every matrix $A\in M_n(\mathds F)$ commutes with a rank one matrix. In the following result we show that for an arbitrary field $\mathds F$, a matrix $A\in M_n(\mathds F)$ commutes with a rank one matrix if it has an eigenvalue in $\mathds F$. The proof is almost identical to the proof presented in [@dol] for algebraically closed fields.
\[l1\] Let $\mathds F$ be a field and $A\in M_n(\mathds F)$. If $A$ has an eigenvalue in $\mathds F$, then there exists a rank one matrix $X\in M_n(\mathds F)$ such that $d(A, X)\leq 1$.
[**Proof.**]{} Let $x$ and $y$ be eigenvectors of $A$ and $A^T$, respectively, corresponding to the same real eigenvalue $\lambda$. Then $X=xy^T$ is a rank one matrix with $AX=(Ax)y^T=\lambda xy^T=x(A^Ty)^T=XA$. $\Box$
Many of the results about the commuting graph of matrix rings over an algebraically closed field are obtained with the help of rank one matrices, see e.g. [@dol]. For a matrix $A\in M_n(\mathds R)$, if $A$ has no real eigenvalues, it is no longer true that $A$ commutes with a rank one matrix. Therefore, we can not apply to real matrices the same techniques that we apply in the case of matrices over an algebraically closed field. However, in the next lemma we show that every matrix $A\in M_n(\mathds R)$ commutes with a rank two matrix. This allows us to adapt some techniques used with matrices over an algebraic closed field to matrices over the field of real numbers.
\[l2\] Let $A\in M_n(\mathds R)$ be a matrix without real eigenvalues. Then, there exists a rank two matrix $X\in M_n(\mathds R)$ such that $d(A, X)\leq 1$.
[**Proof.**]{} Since the matrix $A$ has no real eigenvalues, it follows that $n$ is even and the real Jordan canonical form of $A$ is of the form $$S^{-1}AS=diag(C_{n_1}(a_1, b_1), \ldots , C_{n_t}(a_t, b_t)).$$ Note that the field $\mathds C$ of complex numbers can be embedded in the matrix ring $M_2(\mathds R)$ by the matrix-valued function $M:\mathds C\rightarrow M_2(\mathds R)$ defined by $$\label{e2}M(a+ib)=\left[\begin{array}{cc}a&b\\-b&a\end{array}\right].$$ This embedding can be extended in the obvious way to an embedding function $\varphi : M_{n}(\mathds C)\rightarrow M_{2n}(\mathds R)$.
Now, observe that the real Jordan canonical form of $A$, that is, the matrix $S^{-1}AS$, belongs to the image of the embedding $\phi : M_{n/2}(\mathds C)\rightarrow M_{n}(\mathds R)$. Let $E\in M_{n/2}(\mathds C)$ be such that $\varphi(E)=S^{-1}AS$. By Lemma \[l1\] there exists a rank one matrix $R\in M_{n/2}(\mathds C)$ such that $d(E , R)\leq 1$. It follows that $$A(S\varphi(R)S^{-1})=(S\varphi(E)S^{-1})(S\varphi(R)S^{-1})= S\varphi(E R)S^{-1}=(S\varphi(R)S^{-1})(S\varphi(E)S^{-1})=$$ $$=(S\varphi(R)S^{-1})A.$$ To complete the proof we take $X=S\varphi(R)S^{-1}$.$\Box$
We conclude this section with three lemmas that have been proved in [@ak2] for the more general case of a division ring.
\[l3\] Let $\mathds F$ be a field and $n\geq 2$. Suppose $A, B\in M_n(\mathds F)$ are two matrices such that $ker(A)\cap ker(B)\neq \{0\}$. Then, $C_{M_n(\mathds F)}(\{A, B\})$ contains at least one matrix with rank one.
\[l4\] Let $\mathds F$ be a field and $n\geq 3$. If $N, M\in M_n(\mathds F)$ are two non-zero matrices such that $M^2=N^2=0$, then $d(M , N)\leq 2$ in $\Gamma(M_n(\mathds F))$.
\[l5\] Let $\mathds F$ be a field and $n\geq 3$. If $A, B\in M_n(\mathds F)$ are two non-scalar idempotent matrices, then $d(A , B)\leq 2$ in $\Gamma(M_n(\mathds F))$.
Proof of the Main Theorem
=========================
In this section we prove the main theorem. Throughout this section $n\geq 3$ is a natural number.
[**Proof of Theorem \[t1\].**]{} Suppose first $n\neq 4$. Let $A, B\in M_n(\mathds R)$ be matrices with no real eigenvalues. By Lemma \[l2\] there exist matrices $X, Y\in M_n(\mathds R)$, with rank $2$ and such that $d(A, X)\leq 1$ and $d(B, Y)\leq 1$. Since $dim(Ker(A))=dim(Ker(B))=n-2$, it follows that $dim(Ker(X)\cap Ker(Y))\geq n-4$. Hence, $Ker(X)\cap Ker(Y)\neq \{0\}$. By Lemma \[l3\] there is a matrix $Z\in C_{M_n(\mathds F)}(\{X, Y\})$, which implies that $A-X-Z-Y-B$ is a path in $\Gamma(M_n(\mathds R))$.
Let now $A, B\in M_n(\mathds R)$ be matrices such that both have a real eigenvalue. By Lemma \[l1\] there exist rank one matrices $X, Y\in M_n(\mathds R)$ such that $d(A , X)\leq 1$ and $d(B , Y)\leq 1$. Now we have $dim(Ker(X)\cap Ker(Y))\geq n-2$. Again by Lemma \[l3\] there is a matrix $Z\in C_{M_n(\mathds R)}(\{X, Y\})$. Hence, $A-X-Z-Y-B$ is a path in $\Gamma(M_n(\mathds R))$.
Finally, let us assume that $A, B\in M_n(\mathds R)$ are such that $A$ has a real eigenvalue and $B$ has no real eigenvalues. By Lemma \[l1\] there exists a rank one matrix $X\in M_n(\mathds R)$ such that $d(A , X)\leq 1$ and by Lemma \[l2\] there exists a rank two matrix $Y\in M_n(\mathds R)$ such that $d(B , Y)\leq 1$. In this case we have $dim(Ker(X)\cap Ker(Y))\geq n-3$. Since $A$ has no real eigenvalues, it follows that $n\geq 4$. Hence, $Ker(X)\cap Ker(Y)\neq\{ 0\}$. Again, there exists $Z\in M_n(\mathds R)$ such that $A-X-Z-Y-A$ is a path in $\Gamma(M_n(\mathds R))$. So, we have proved that for $n\neq 4$ $diam(\Gamma(M_n(\mathds R)))=4$.
Suppose now that $n=4$ and let $A, B\in M_4(\mathds R)$. Assume first that both $A$ and $B$ have a real eigenvalue. As in the previous case, there exist rank one matrices $X, Y\in M_4(\mathds R)$ such that $d(A , X)\leq 1$ and $d(B , Y)\leq 1$. Since $dim(Ker(X)\cap Ker(Y))\geq 2$, it follows that there exists $Z\in M_4(\mathds R)$ such that $A-X-Z-Y-B$ is a path $\Gamma(M_4(\mathds R))$. If $A$ has a real eigenvalue and $B$ has no real eigenvalue, then there exist a rank one matrix $X\in M_4(\mathds R)$ and a rank two matrix $Y\in M_4(\mathds R)$ such that $d(A , X)\leq 1$ and $d(B , Y)\leq 1$. Since in this case we have $dim(Ker(A)\cap Ker(B))\geq 1$, it follows that there exists $Z\in M_4(\mathds R)$ such that $A-X-Z-Y-B$ is a path in $\Gamma(M_4(\mathds R))$.
Finally, suppose that both $A$ and $B$ have no real eigenvalues. There are three possible cases for the real Jordan form of a matrix in $ M_4(\mathds R)$, namely
$$\left[\begin{array}{cc}R_1&0\\0&R_1\end{array}\right],\;\;\;\;\left[\begin{array}{cc}R_1&0\\0&R_2\end{array}\right]
\;\;or\;\;\left[\begin{array}{cc}R_1&I\\0&R_1\end{array}\right],$$
where $I$ is the identity matrix of order $2$ and $R_i$, for $i=1, 2$, is a square matrix of order 2 of the type $C(a, b)$. Therefore, for two matrices $A, B\in M_4(\mathds R)$ we have six possible cases. Let us study each case separately.
[*Case $1$.*]{}
$$A=S\left[\begin{array}{cc}A_1&0\\0&A_1\end{array}\right]S^{-1}\;\;\;and\;\;\;
B=L\left[\begin{array}{cc}B_1&0\\0&B_1\end{array}\right]L^{-1}.$$
In this case the matrices $A$ and $B$ commute with the following idempotent matrices $$I_1=S\left[\begin{array}{cc}I&0\\0&0\end{array}\right]S^{-1}\;\;\;and\;\;\;
I_2=L\left[\begin{array}{cc}I&0\\0&0\end{array}\right]L^{-1},$$ respectively. Since by Lemma \[l5\] there is a non scalar matrix $X\in C_{M_4(\mathds R)}(\{I_1 , I_2\})$, it follows that $A-I_1-X-I_2-B$ is a path in $\Gamma(M_4(\mathds R))$.
[*Case $2$.*]{}
$$A=S\left[\begin{array}{cc}A_1&0\\0&A_1\end{array}\right]S^{-1}\;\;\;and\;\;\;
B=L\left[\begin{array}{cc}B_1&0\\0&B_2\end{array}\right]L^{-1}.$$
This case is similar to the case $1$.
[*Case $3$.*]{}
$$A=S\left[\begin{array}{cc}A_1&0\\0&A_1\end{array}\right]S^{-1}\;\;\;and\;\;\;
B=L\left[\begin{array}{cc}B_1&I\\0&B_1\end{array}\right]L^{-1}.$$
In this case the matrices $A$ and $B$ commute with the following nilpotent matrices $$N_1=S\left[\begin{array}{cc}0&I\\0&0\end{array}\right]S^{-1}\;\;\;and\;\;\;
N_2=L\left[\begin{array}{cc}0&I\\0&0\end{array}\right]L^{-1},$$ respectively. Since $N_1^2=N_2^2=0$, by Lemma \[l4\], there is a non scalar matrix $X\in C_{M_4(\mathds R)}(\{N_1 , N_2\})$. Hence, $A-N_1-X-N_2-B$ is a path in $\Gamma(M_4(\mathds R))$.
[*Case $4$.*]{}
$$A=S\left[\begin{array}{cc}A_1&0\\0&A_2\end{array}\right]S^{-1}\;\;\;and\;\;\;
B=L\left[\begin{array}{cc}B_1&0\\0&B_2\end{array}\right]L^{-1}.$$
This case is similar to the case $1$.
[*Case $5$.*]{}
$$A=S\left[\begin{array}{cc}A_1&0\\0&A_2\end{array}\right]S^{-1}\;\;\;and\;\;\;
B=L\left[\begin{array}{cc}B_1&I\\0&B_1\end{array}\right]L^{-1}.$$
In this case $A$ commutes with the following idempotent matrix $$I_1=S\left[\begin{array}{cc}0&0\\0&I\end{array}\right]S^{-1}.$$ The matrix $B$ commutes with the following matrix
$$B^\prime=L\left[\begin{array}{cc}B_1&0\\0&B_1\end{array}\right]L^{-1}.$$
Since $B^\prime$ commutes with the idempotent $$I_2=L\left[\begin{array}{cc}I&0\\0&0\end{array}\right]L^{-1},$$ it follows that there is a path $A-I_1-X-I_2-B^\prime-B$ in $\Gamma(M_4(\mathds R))$.
[*Case $6$.*]{}
$$A=S\left[\begin{array}{cc}A_1&I\\0&A_1\end{array}\right]S^{-1}\;\;\;and\;\;\;
B=L\left[\begin{array}{cc}B_1&I\\0&B_1\end{array}\right]L^{-1}.$$
This case is similar to the case $3$. The proof is completed.$\Box$
We were unable to prove that the distance five, between two vertices in the graph $\Gamma(M_4(\mathds R))$, is attained. Therefore, the question $diam(\Gamma(M_4(\mathds R)))=5$ or $diam(\Gamma(M_4(\mathds R)))=4$ remains open.
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abstract: 'We have developed an analytical formulation to calculate the plasmon dispersion relation for a two-dimensional layer which is encapsulated within a narrow spatial gap between two bulk half-space plasmas. This is based on a solution of the inverse dielectric function integral equation within the random-phase approximation (RPA). We take into account the nonlocality of the plasmon dispersion relation for both gapped and gapless graphene as the sandwiched two-dimensional (2D) semiconductor plasma. The associated nonlocal graphene plasmon spectrum coupled to the “sandwich" system is exhibited in density plots, which show a linear mode and a pair of depolarization modes shifted from the bulk plasma frequency.'
author:
- 'Godfrey Gumbs$^{1,2}$, N. J. M. Horing$^3$, Andrii Iurov$^{ 4}$, and Dipendra Dahal$^{1}$'
title: Plasmon Excitations for Encapsulated Graphene
---
0.2in
Introduction {#sec1}
============
The properties of high-quality graphene encapsulated between two films such as hexagonal boron-nitride are just beginning to be explored and have now become an active area of research due to recent advances in device fabrication techniques [@encaps1; @encaps2; @encaps3; @encaps4; @encaps5; @encaps6; @encaps7; @encaps8; @encaps9; @encaps10]. Interest in the optical properties of these heterostructures has been focused on their unusual plasmonic behavior including their spatial dispersion and damping. This hybrid system may be employed for tailoring novel metamaterials. Additionally, this fabrication technique provides a clean environment for graphene. This brand new area of nanoscience not only poses challenges for experimentalists, but also for theoreticians seeking to formulate a theory for a model system. Although there already exists a copious literature on graphene on a single substrate [@GG; @ONB; @NJMH; @Pol1; @Pol2; @Pol3; @Pol4; @Pol5], this study shows that encapsulated graphene is vastly different in many ways.
The model we use in this paper consists of two identical semi-infinite metallic plasmas with planar boundaries at $z = \pm a/2$. Within the spatial separation between the two bulk conducting plasmas ($|z|<a/2$) is inserted a 2D monolayer graphene sheet at $z=0$, shown schematically in Fig. \[FIG:1\]. The natural first step in our calculations of the plasmon excitation spectrum is to set up and solve the random-phase approximation (RPA) integral equation for the inverse dielectric screening function of this hybrid system. We have solved this equation analytically in position representation for a narrow spatial gap between the bulk half-space plasmas, obtaining a closed-form formula for the inverse dielectric function in terms of the nonlocal polarizability for graphene and the bulk metallic polarizability, for which the latter is well approximated by the hydrodynamical model. Based on this newly derived formula, we have calculated the nonlocal plasmon dispersion relation numerically, considering both gapped and gapless graphene as the two-dimensional (2D) semiconductor plasma. The resulting nonlocal graphene plasmon spectra coupled to the “sandwich" system are exhibited in density plots, which show a linear mode and a pair of depolarization modes shifted from the bulk plasma frequency.
Hexagonal boron nitride has been the main substrate material that facilitates graphene based devices to exhibit micrometer-scale ballistic transport. The recent work of Kretinin, et al. [@encaps10] has shown that other atomically flat crystals may also be employed as substrates for making high-quality graphene heterostructures. Alternative substrates for encapsulating graphene include molybdenum and tungsten disulfides which have been found to exhibit consistently high carrier mobilities of about $6\times 10^4$ cm$^2$ V$^{-1}$ s$^{-1}$. On the other hand, when graphene is encapsulated with atomically flat layered oxides such as mica, bismuth strontium calcium copper oxide, or vanadium pentoxide, the result is remarkably low quality graphene with mobilities of about $ 10^3$ cm$^2$ V$^{-1}$ s$^{-1}$. This difference is due mainly to self-cleansing which occurs at interfaces between graphene, hexagonal boron nitride, and transition metal dichalcogenides. In our model calculations, we allow for the possibility that the substrate may affect the energy band structure of graphene by opening a gap in the energy band. We compare the resulting calculated plasmon spectra for encapsulated gapless and gapped graphene.
The work of Chuang, et al. [@DD1] is a study of the graphene-like high mobility for p- and n-doped WSe$_2$. Electron transport in the junction between two 2D materials $MoS_2$ and graphene have also been reported recently, showing how spatial confinement can influence physical properties [@DD2]. Mobility, transconductance and carrier inhomogeniety experiments have also been reported in [@DD3; @DD4; @DD5; @DD6] for monolayer and bilayer graphene fabricated on hexagonal BN as well as mica based substrates.
The outline of the rest of our paper is as follows. In Sec. \[sec2\], we give details of our calculation of the inverse dielectric function for a 2D layer sandwiched between two conducting substrates whose separation is very small. We explicitly derive the plasma dispersion equation when the thick substrate layers may be treated in the hydrodynamical model. The 2D RPA ring diagram polarization function for graphene at arbitrary temperature is employed in the dispersion equation. A careful determination of the plasmon spectra for a range of energy gap and carrier doping values is reported in Sec. \[sec3\]. We conclude with a discussion of the highlights of our calculations in Sec. \[sec4\].
![(Color online) Schematic illustration of a pair of thick conducting plasmas (taken to be semi-infinite in our formulation of the problem) encapsulating a 2D monolayer graphene sheet in a “sandwich" array.[]{data-label="FIG:1"}](fig1){width="35.00000%"}
Theoretical Formulation {#sec2}
========================
We consider two identical semi-infinite conductors on either side of a 2D semiconductor layer (Fig. \[FIG:1\]). One of the conductors extends from $z=-a/2$ to $z=-\infty$ while the other conductor has its surface at $z= a/2$ and extends to $z=\infty$. The 2D layer lies in a plane mid-way between the two conductors at $z=0$ in the gap region $|z|<a/2$. The inverse dielectric function satisfies
$$\begin{aligned}
K(z_1,z_2)&=& K_\infty(z_1,z_2)
\nonumber\\
&-& \int_{-\infty}^\infty dz^{\prime} \int_{-\infty}^\infty dz^{\prime\prime} \
K_\infty (z-z^\prime) \left[\alpha_{2D}(z^\prime,z^{\prime\prime})- \alpha_{gap}(z^\prime,z^{\prime\prime})
\right] K(z^{\prime\prime},z_2)\ ,
\label{eq:1}\end{aligned}$$
where $K_\infty(z_1,z_2)=K_\infty(z_1-z_2)$ is the bulk infinite space symmetric conducting medium inverse dielectric function and
$$\alpha_{2D}(z^\prime,z^{\prime\prime})=\frac{2\pi e^2}{\epsilon_s q_\parallel}
\Pi^{(0)}_{2D} (q_\parallel,\omega) e^{-q_\parallel |z^\prime|} \delta(z^{\prime\prime})
\equiv \tilde{\alpha}_{2D}(q_\parallel,\omega) \ e^{-q_\parallel |z^\prime|} \delta(z^{\prime\prime}) \ ,
\label{eq:2}$$
which defines $\tilde{\alpha}_{2D}(q_\parallel,\omega) $ in terms of the 2D ring diagram $\Pi^{(0)}_{2D} (q_\parallel,\omega)$. For a narrow gap between the two identical half-space slabs, i.e., $q_\parallel a \ll 1$, the gap polarizability is given by
$$\alpha_{gap}(z^\prime-z^{\prime\prime})= a\delta(z^{\prime\prime}) \alpha_{\infty}(z^\prime-z^{\prime\prime})
= a\delta(z^{\prime\prime}) \int_{-\infty}^\infty \frac{dp_z}{2\pi}\alpha_\infty (p_z,q_\parallel)
e^{ip_z ( z^\prime-z^{\prime\prime} ) } \ .
\label{eq:3}$$
Employing Eqs. (\[eq:2\]) and (\[eq:3\]) in Eq. (\[eq:1\]), we have $K(z_1,z_2)= K_\infty(z_1-z_2)- {\cal F}(z_1;q_\parallel,\omega) K(0,z_2)$, where ${\cal F}(z_1;q_\parallel,\omega)= {\cal F}_{2D}(z_1;q_\parallel,\omega)
+ {\cal F}_{gap}(z_1;q_\parallel,\omega) $ and
$$\begin{aligned}
{\cal F}_{2D}(z_1;q_\parallel,\omega)&\equiv & \frac{2\pi e^2}{\epsilon_s q_\parallel}
\Pi^{(0)}_{2D} (q_\parallel,\omega)
\int_{-\infty}^\infty dz^\prime\ K_\infty (z_1-z^\prime)
e^{-q_\parallel |z^\prime|} \ ,
\nonumber\\
{\cal F}_{gap}(z_1;q_\parallel,\omega)&\equiv &
-a\ \int_{-\infty}^\infty dz^\prime\ K_\infty (z_1-z^\prime) \alpha_\infty(z^\prime,0) \ .
\label{eq:5-2}\end{aligned}$$
Setting $z_1=0$ in the equation relating $K(0,z_2)$ to $K(z_1,z_2)$ above, we may solve for $K(0,z_2)$ and then obtain
$$K(z_1,z_2)= K_\infty(z_1-z_2)- \frac{{\cal F}(z_1;q_\parallel,\omega)}
{1+{\cal F}(z_1=0;q_\parallel,\omega)} K_\infty(0,z_2)\ ,
\label{eq:6}$$
so that the plasma excitation frequencies are determined by solving for the zeros of
$$\begin{aligned}
{\cal D}(q_\parallel,\omega) &\equiv & 1+{\cal F}(z_1=0;q_\parallel,\omega)
\nonumber\\
&=& 1+ \int_{-\infty}^\infty dz^\prime\ K_\infty (0-z^\prime))
\left[ \frac{2\pi e^2}{\epsilon_s q_\parallel} \Pi^{(0)}_{2D} (q_\parallel,\omega) e^{-q_\parallel |z^\prime|}
-a\ \alpha_\infty(z^\prime,0)\right] \ .
\label{eq:7}\end{aligned}$$
Expressing Eq. (\[eq:5-2\]) in terms of the Fourier transform of $K_\infty (z_1-z^\prime)$, we obtain
$$\begin{aligned}
{\cal F}_{2D}(z_1;q_\parallel,\omega)&= & \frac{2 e^2}{\epsilon_s}
\Pi^{(0)}_{2D} (q_\parallel,\omega)
\int_{-\infty}^\infty dp_z\ e^{ip_zz_1} \frac{K_\infty (p_z )}{p_z^2+q_\parallel^2} \ ,
\nonumber\\
{\cal F}_{gap}(z_1;q_\parallel,\omega)&= &
-a\ \int_{-\infty}^\infty \frac{dp_z}{2\pi}\ e^{ip_zz_1} K_\infty (p_z) \alpha_\infty(p_z) \ .
\label{eq:5-3}\end{aligned}$$
Using the hydrodynamical model for nonlocality of the conducting substrate [@EGUI; @Kamen], we have $\epsilon(q,\omega)=1-\omega_p^2/(\omega^2-\beta^2q^2)$, where $\omega_p$ is the local bulk metallic plasma frequency, $\beta$ is an adjustable parameter which mabe adjusted to give the correct dispersive shift of the bulk plasma frequency, i.e., $\beta^2=3v_F^{(3D)\ 2}/5$ (where $v_F^{(3D) }$ is the bulk Fermi velocity of the conducting substrate). This approximation leads to the dispersion equation which, after performing the $p_z$ integrations, we obtain
$$\begin{aligned}
{\cal D}(q_\parallel,\omega)&=&1+ \frac{2 \pi e^2}{\epsilon_s q_\parallel}
\Pi^{(0)}_{2D} (q_\parallel,\omega) \frac{1}{|A(q_\parallel,\omega)|}
\left( \frac{\omega_p^2 q_\parallel -\omega^2 |A(q_\parallel,\omega)|}{\omega_p^2-\omega^2} \right)
\nonumber\\
&-& \frac{ a \omega_p^2}{2\beta^2 |A(q_\parallel,\omega)|}\ ,
\label{xxx}\end{aligned}$$
where $A(q_\parallel,\omega) \equiv \beta ^{-1} \left( \omega_p^2+\beta^2 q_\parallel^2
-\omega^2\right)^{1/2} $.
The 2D RPA ring diagram polarization function for graphene with a gap $\Delta$ may be expressed as
$$\begin{aligned}
\label{A1}
\Pi^{(0)}_{2D}(q,\omega)
&& = \frac{g}{4 \pi^2} \int d^2 {\bf k} \sum\limits_{s,s' = \pm} \left( 1 + s s'
\frac{{\bf k} \cdot ({\bf k}+{\bf q}_{\parallel}) + \Delta^2}{\epsilon_k \,\, \epsilon_{\vert {\bf k}+{ \bf q}_{\parallel} \vert }} \right) \nonumber \\ && \times \frac{f(s \, \epsilon_{{\bf k}}) - f(s' \epsilon_{{\bf k}+{\bf q}_{\parallel}})}{s \, \epsilon_{{\bf k}_{\parallel}} - s' \epsilon_{{\bf k}+{\bf q}_{\parallel}} - \hbar \omega - i \eta^+ } \ ,\end{aligned}$$
where $s, s^\prime$ are band indices, $g=4$ accounts for both spin and valley degeneracies and $\epsilon_{{\bf k}}$ is the band energy. Since we limit our considerations to zero temperature, $T=0$, the Fermi-Dirac distribution function is reduced to the Heaviside step function $f(\epsilon, \mu; T \rightarrow 0) = \eta_{+}(\mu - \epsilon)$, and we use the results of Refs. \[\].
Numerical Results and Discussion {#sec3}
================================
In the figures, we present calculated results for the nonlocal plasmon excitations of encapsulated gapless and gapped graphene. As we demonstrated in previous work [@GG; @ONB], the hybrid plasmon modes and their damping are mainly determined by the doping concentration, i.e., the chemical potential $\mu$, along with the energy bandgap $\Delta$. Consequently, we have paid particular attention in our numerical investigations to the various regimes of the ratio $\Delta/\mu$. In Fig. \[FIG:2\], we exhibit results for gapless graphene. Figures \[FIG:3\] and \[FIG:4\] illustrate the case in which the graphene layer has an intermediate or large energy gap. The regions of strong damping arise from inter-band transitions which are forbidden by Pauli blocking (i.e., when the final transition states are filled, so that further transitions cannot occur). In the case of a [*single*]{} substrate and a 2D layer, the lowest acoustic branch vanishes due to damping in the long wavelength limit for a range of separations between the 2D layer and the surface. However, this is not the present case under consideration involving encapsulation of the graphene sheet within a small gap between the two surfaces. Within the “small gap" framework, $a<1/q_\parallel$, the lower branch is never damped in the long wavelength limit. This result is quite different from that obtained for a [*single*]{} substrate, in which strong damping of the acoustic branch appears in the long wavelength limit for the distance $a$ below a specific critical value. In regard to the two upper branches, attributed to bulk three-dimensional plasmons, their undamped parts appear as upper and lower arcs of a single loop on the left sides of the figures in the cases of low doping or low value of the chemical potential, as clearly demonstrated in Fig.\[FIG:2\]. It is noteworthy that both bulk plasmon modes start from $\omega_p$, [*not*]{} from $\omega_p/\sqrt{2}$ as we previously observed in the case of a single conducting substrate.
The presence of a bandgap in the graphene energy spectrum leads to interesting features. First, a finite energy gap modifies the location of each single-particle excitation region. It is especially unusual to observe the extension of the upper plasmon branches, which are understood to arise from a semi-infinite substrate. It is also apparent that the acoustic mode is broken into two separate undamped parts located between the two distinct single-particle excitation regions, as has been previously reported for free-standing gapped graphene [@pavlo] (see Figs. \[FIG:3\] (c) and (d)). The figures show that specifically acoustic plasmon mode behavior persists in our present case of encapsulated graphene. Within the framework of our assumption that the spacing between the conducting half-spaces is narrow, $a\ll 1/q_\parallel$, the results are relatively insensitive to the gap separation, and we consider only small values of $a$. Moreover, we examined modification of the plasmon spectra for various values of the Fermi energy, $\mu$, as depicted in Figs. \[FIG:2\] through \[FIG:4\].
In the matter of experimental realization of our results, it is necessary to achieve a situation in which the surface plasmons are not Landau damped, i.e., the corresponding branches are located outside of the upper inter-band part of the single-particle excitation spectrum. Accordingly, the frequency of the surface plasmon mode should be comparable with the Fermi energy in graphene, as well as with the corresponding $q^{1/2}$-2D-graphene plasmon energy. Only if this condition is satisfied will it be possible to observe the strong plasmon coupling we have reported here. Therefore, one must ensure that the bulk plasmon frequency is in the range of $\sim .1 \, eV$ and below [@NPo1]. Ref \[\] is an experimental paper on $Bi_2Se_3$ which is a heavily doped topological insulator, where the surface plasmon energy was found to be around $0.1 \, eV$. Evidence of mutual interaction between the surface plasmon and the Dirac plasmon of $Bi_2Se_3$ has been provided by using high-resolution electron energy loss spectroscopy. Additionally, at a graphene/$Bi_2Se_3$ interface, which was recently experimentally realized by Kepaptsoglou, et al. [@NPo2], the surface plasmon of $Bi_2Se_3$ is hybridized with acoustic plasmons in graphene. In this vein, the Fermi energy of free-standing graphene corresponding to an electron density $n=10^{16}\, m^{-2}$ is $ E _F = \hbar v_F \sqrt{\pi n } = 0.12\, eV$, so the two quantities are of the same order of magnitude.
![(Color online) Particle-hole modes and plasmon dispersion for doped, gapless graphene sandwiched between two semi-infinite conducting plasmas with separation $a=0.5\ k_F^{-1}$. Panels $(a)$ and $(b)$ present the regions where the plasmons are damped for $\mu=\hbar\omega_p$ and $\mu=1.5\hbar\omega_p$. respectively. Panels $(c)$ and $(d)$ show density plots for the plasmon excitation spectra corresponding to the parameters for zero energy gap and the same bulk plasma separation chosen in $(a)$ and $(b)$. Panels $(e)$ and $(f)$ show the plasmon excitations, both damped (dashed lines) and undamped (solid lines), obtained by solving $ \mbox{Re} \, D (q_\parallel,\omega) = 0 $ for the chosen distance $a=0.5k_F^{-1}$ between the bulk surfaces with the 2D layer at $z=0$, the same as in $(a)$ through $(d)$. The value of the parameter in the hydrodynamical model is $\beta=c_0 v_F$ in terms of the Fermi velocity $v_F$ for graphene and $c_0=(3/5)^{1/2}(v_F^{3D}/v_F)$, where $v_F^{3D}$ is the Fermi velocity of the conducting substrate (but we take $c_0=1$ here).[]{data-label="FIG:2"}](fig2){width="55.00000%"}
![(Color online) Particle-hole modes and plasmon dispersion for doped graphene encapsulated between two semi-infinite conducting plasmas with separation $a=0.5\ k_F^{-1}$. The graphene layer possesses an intermediate value for the energy bandgap $\Delta=0.2\ \mu$ determining the energy dispersion. Panels $(a)$ and $(b)$ present density plots for $ \mbox{Im}\, D^{-1} (q_\parallel,\omega)$, showing the regions where the plasmons are damped when $\mu=\hbar\omega_p$ and $\mu=1.5\hbar\omega_p$, respectively. Panels $(c)$ and $(d)$ exhibit density plots for the plasmon excitation spectra corresponding to the parameters for the energy gap $\Delta=0.2\mu$ and bulk plasma separation chosen in $(a)$ and $(b)$. Panels $(e)$ and $(f)$ show the plasmon excitations, both damped (dashed lines) and undamped (solid lines), obtained by solving $ \mbox{Re} \, D (q_\parallel,\omega) = 0 $ for the chosen distance between the bulk surfaces with the 2D layer at $z=0$, the same as $(a)$ through $(d)$. The value of the parameter used in the hydrodynamical model is $\beta=v_F$ in terms of the Fermi velocity for graphene, as in Fig. \[FIG:2\].[]{data-label="FIG:3"}](fig3){width="55.00000%"}
![(Color online) Particle-hole modes and the plasmons for graphene encapsulated between two semi-infinite conducting substrates. Here, we present a set of graphs, similar to those in Fig. \[FIG:3\], but for the case of a large energy gap $\Delta=0.6\ \mu$.[]{data-label="FIG:4"}](fig4){width="55.00000%"}
Concluding Remarks {#sec4}
==================
In this paper, we have investigated the properties of the plasmon spectra for a heterostructure consisting of a pair of identical semi-infinite conductors and a 2D graphene layer sandwiched between them. Our formulation is suitable when the separation between the two semi-infinite bulk conducting materials is small compared to the inverse Fermi wave number in graphene, and the whole system is symmetric in the $z-$direction perpendicular to the 2D layer.
We have obtained the plasmon dispersion relations for this encapsulated graphene system for both zero and finite energy bandgap and for various values of the chemical potential. In each case, we clearly obtain three hybridized plasmon modes, one of which (the acoustic branch) starts from the origin and is attributed primarily to the graphene layer and the other two modes originating at $\omega=\omega_p$ are considered as optical plasmons. This situation is novel and has not been encountered previously in a system involving a 2D layer with a single semi-infinite conductor. Each of the branches exhibits a specific behavior depending on the chemical potential and consists of various undamped parts which are determined by the energy bandgap and its ratio to the doping parameter, as discussed above.
The low-frequency branch has a linear dispersion at long wavelengths and becomes damped by the intra-band and inter-band particle-hole modes as wave vector is increased. There are two bulk plasmon branches which are depolarization shifted by the Coulomb interaction. In this regard, the results demonstrate that there is almost no dependence on the distance $a$ between the two substrates as long as the condition $a \ll k_F^{-1}$ is satisfied. Moreover, there is no evidence of critical damping of the acoustic plasmon branch in the long wavelength limit ($q_\parallel \rightarrow 0$), as was found in the previous comparative study of a monolayer of graphene interacting with a single conducting substrate [@GG]. Furthermore, some crucial properties of the plasmons in free-standing graphene with an energy bandgap, such as extension of the undamped branch and its separation into the two parts for intermediate energy gap $\Delta \backsimeq 0.22 \mu$, are also present in our results.
In summary, we have developed a new analytical model and obtained a complete set of numerical results for a two-dimensional layer (graphene) surrounded by two identical thick conducting substrates. While our previous work [@GG] confirmed and offered an adequate theoretical explanation for recent experimental findings [@Pol1; @Pol2; @Pol3; @Pol4], this paper is expected to predict the correct plasmon behavior of the totally realistic and novel situation of encapsulated graphene, which is now being very actively studied experimentally [@Gong; @Kamat]. Hybridized plasmon modes of a graphene-based nanoscale system are at the focus of significant interest in the current fields of technology and applications. The authors of [@encaps5] report evidence of the formation of electron-hole puddles for encapsulated graphene by hexagonal BN. This is an interesting effect which was not taken into account in our model calculations. These localization effects will be investigated in future work by including a concentration of impurities and defects in interaction with the 2D layer and two substrates. The role of impurities is expected to be modest at low concentration. An increase in the concentration of the impurities will lead to a decrease in the electron correlations. The Green’s function and consequently will, in this case, be expressible by vertex corrections by the usual rules of field theory when impurities are present.
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---
abstract: 'Let $f \colon X \rightarrow Y$ be a resolvable-measurable mapping of a metrizable space $X$ to a regular space $Y$. Then $f$ is piecewise continuous. Additionally, for a metrizable completely Baire space $X$, it is proved that $f$ is resolvable-measurable if and only if it is piecewise continuous.'
address: |
Department of Mathematical and Functional Analysis\
South Ural State University\
pr. Lenina, 76, Chelyabinsk, 454080 Russia
author:
- Sergey Medvedev
title: On piecewise continuous mappings of metrizable spaces
---
In an old question Lusin asked if any Borel function is necessarily countably continuous. This question was answered negatively by Keldiš [@K34], and an example of a Baire class 1 function which is not decomposable into countably many continuous functions was later found by Adyan and Novikov [@AN]; see also the paper of van Mill and Pol [@vMP].
The first affirmative result was obtained by Jayne and Rogers [@JR Theorem 1].
If $X$ is an absolute Souslin-$\mathcal{F}$ set and $Y$ is a metric space, then $f \colon X \rightarrow Y$ is $\mathbf{\Delta}^0_2$-measurable if and only if it is piecewise continuous.
Later Solecki [@S98 Theorem 3.1] proved the first dichotomy theorem for Baire class 1 functions. This theorem shows how piecewise continuous functions can be found among $\mathbf{\Sigma}^0_2$-measurable ones.
Let $f \colon X \rightarrow Y$ be a $\mathbf{\Sigma}^0_2$-measurable function from an analytic set $X$ to a separable metric space $Y$. Then precisely one of the following holds:
1. $f$ is piecewise continuous,
2. one of $L$, $L_1$ is contained in $f$, where $L$ and $L_1$ are two so-called Lebesgue’s functions.
Kačena, Motto Ros, and Semmes [@KMS Theorem 1] showed that Theorem JR holds for a regular space $Y$. They also got [@KMS Theorem 8] a strengthening of Solecki’s theorem from an analytic set $X$ to an absolute Souslin-$\mathcal{F}$ set $X$.
On the other hand, Banakh and Bokalo [@BB Theorem 8.1] proved among other things that a mapping $f \colon X \rightarrow Y$ from a metrizable completely Baire space $X$ to a regular space $Y$ is piecewise continuous if and only if it is $\mathbf{\Pi}^0_2$-measurable. Under some set-theoretical assumptions, examples of $\mathbf{\Pi}^0_2$-measurable mappings which are not piecewise continuous were constructed in the work [@BB].
Recently, Ostrovsky [@Ost] proved that every resolvable-measurable function $f \colon X \rightarrow Y$ is countably continuous for any separable zero-dimensional metrizable spaces $X$ and $Y$.
The main result of the paper (see Theorem \[t:4\]) states that every resolvable-measurable mapping $f \colon X \rightarrow Y$ of a metrizable space $X$ to a regular space $Y$ is piecewise continuous. Comparison of our result and the Banakh and Bokalo theorem shows that the condition on $X$ is weakened but $f$ is restricted to the class of resolvable-measurable mappings. Notice also that Theorem \[t:4\] generalizes and strengthens the Ostrovsky theorem.
In completely metrizable spaces, resolvable sets coincide with $\mathbf{\Delta}^0_2$-sets, see [@Kur1 p. 418]. Lemma \[L:5\] shows that every metrizable completely Baire space has such a property. This enables us to refine the above result of Banakh and Bokalo, see Theorem \[t:6\].
Theorem \[t:8\] states that in the study of $\mathbf{\Sigma}^0_2$-measurable mappings defined on metrizable completely Baire spaces it suffices to consider separable spaces. In a sense, Theorem \[t:8\] is similar to the non-separable version of Solecki’s Theorem S.
**Notation**. For all undefined terms, see [@Eng].
A subset $E$ of a space $X$ is *resolvable* if it can be represented as $$E= (F_1 \setminus F_2) \cup (F_3 \setminus F_4) \cup \ldots \cup (F_\xi \setminus F_{\xi+1}) \cup \ldots ,$$ where $\langle F_\xi \rangle$ forms a decreasing transfinite sequence of closed sets in $X$.
A metric space $X$ is said to be an *absolute Souslin*-$\mathcal{F}$ set if $X$ is a result of the $\mathcal{A}$-operation applied to a system of closed subsets of $\widehat{X}$, where $\widehat{X}$ is the completion of $X$ under its metric. Metrizable continuous images of the space of irrational numbers are called *analytic sets*.
A mapping $f \colon X \rightarrow Y$ is said to be
1. *resolvable-measurable* if $f^{-1}(U)$ is a resolvable subset of $X$ for every open set $U \subset Y$,
2. $\mathbf{\Delta}^0_2$-*measurable* if $f^{-1}(U) \in \mathbf{\Delta}^0_2(X)$ for every open set $U \subset Y$,
3. $\mathbf{\Sigma}^0_2$-*measurable* if $f^{-1}(U) \in \mathbf{\Sigma}^0_2(X)$ for every open set $U \subset Y$,
4. $\mathbf{\Pi}^0_2$-*measurable* if $f^{-1}(U) \in \mathbf{\Pi}^0_2(X)$ for every open set $U \subset Y$,
5. *countably continuous* if $X$ can be covered by a sequence $X_0, X_1, \ldots$ of sets such that the restriction $f \upharpoonright X_n$ is continuous for every $n \in \omega$,
6. *piecewise continuous* if $X$ can be covered by a sequence $X_0, X_1, \ldots$ of closed sets such that the restriction $f \upharpoonright X_n$ is continuous for every $n \in \omega$.
Obviously, every piecewise continuous mapping is countably continuous. Notice that every resolvable-measurable mapping of a metrizable space $X$ is $\mathbf{\Sigma}^0_2$-measurable because, by [@Kur1 p. 362], every resolvable subset of a metrizable space $X$ is a $\mathbf{\Delta}^0_2$-set, i.e., a set that is both $F_\sigma$ and $G_\delta$ in $X$. The following example shows that there exists a $\mathbf{\Delta}^0_2$-measurable mapping which is not resolvable-measurable.
Let $f \colon \mathbb{Q} \rightarrow D$ be a one-to-one mapping of the space $\mathbb{Q}$ of rational numbers onto the countable discrete space $D$. Clearly, $f$ is piecewise continuous and $\mathbf{\Delta}^0_2$-measurable. Gao and Kientenbeld [@GK Proposition 4] got a characterization of nonresolvable subsets of $\mathbb{Q}$. In particular, they showed that there exists a nonresolvable subset $A$ of $\mathbb{Q}$. Since $A = f^{-1}(f(A))$, the mapping $f$ is not resolvable-measurable.
The closure of a set $A \subset X$ is denoted by $\overline{A}$. Given a mapping $f \colon X \rightarrow Y$, let us denote by $\mathcal{I}_f$ the family of all subsets $A \subset X$ for which there is a set $S \in \mathbf{\Sigma}^0_2(X)$ such that $A \subset S$ and the restriction $f \upharpoonright S$ is piecewise continuous. In particular, $f$ is piecewise continuous if and only if $X \in \mathcal{I}_f$. From [@HZZ Proposition 3.5] it follows that the family $\mathcal{I}_f$ forms a $\sigma$-ideal which is $F_\sigma$ supported and is closed with respect to discrete unions, see also [@KMS].
To prove Theorem \[t:4\], we shall use the technique due to Kačena, Motto Ros, and Semmes [@KMS]. Therefore, the terminology from [@KMS] is applied. The sets $A, B \subset Y$ are *strongly disjoint* if $\overline{A} \cap \overline{B} = \emptyset$. Let $f \colon X \rightarrow Y$ be a mapping. Put $A^f = f^{-1}(Y \setminus \overline{A})$. As noted in [@KMS], if $A,B$ are strongly disjoint and $A^f , B^f \in \mathcal{I}_f$, then $X \in \mathcal{I}_f$.
Let $x \in X$, $X^\prime \subset X$, and $A \subset Y$. The pair $(x, X^\prime)$ is said to be $f$-*irreducible outside* $A$ if for every open neighborhood $V \subset X$ of $x$ we have $A^f \cap X^\prime \cap V \notin \mathcal{I}_f$. Otherwise we say that $(x, X^\prime)$ is $f$-*reducible outside* $A$, i.e., there exist a neighborhood $V$ of $x$ and a set $S \in \mathbf{\Sigma}^0_2(X)$ such that $A^f \cap X^\prime \cap V \subset S$ and $f \upharpoonright S$ is piecewise continuous. Clearly, $x \in \overline{A^f \cap X^\prime}$ if $(x, X^\prime)$ is $f$-irreducible outside $A$.
\[lem1\] Let $X$ be a metrizable space and $Y$ a regular space. Suppose $f \colon X \rightarrow Y$ is a $\mathbf{\Sigma}^0_2$-measurable mapping, $X^\prime$ is a subset of $X$, and $A \subset Y$ is an open set such that $X^\prime \subset A^f$. Then the following assertions are equivalent:
1. $X^\prime \notin \mathcal{I}_f$,
2. there exist a point $x \in \overline{X^\prime}$ and an open set $U \subset Y$ strongly disjoint from $A$ such that $f(x) \in U$ and the pair $(x, X^\prime)$ is $f$-irreducible outside $U$.
\[lem2\] Let $f \colon X \rightarrow Y$ be a mapping of a metrizable space $X$ to a regular space $Y$, $x \in X$, $X^\prime \subset X$, $A \subseteq Y$, and let $U_0, \ldots, U_k$ be a sequence of pairwise strongly disjoint open subsets of $Y$. If $(x; X^\prime)$ is $f$-irreducible outside $A$, then there is at most one $i \in \{ 0, \ldots, k \}$ such that $(x, X^\prime)$ is $f$-reducible outside $A \cup U_i$.
Recall that a set $A \subset Y$ is *relatively discrete* in $Y$ if for every point $a \in A$ there is an open set $U \subset Y$ such that $U \cap A = \{a \}$.
\[lem3\] Let $X$ be a metrizable space and $Y$ be a regular space. Suppose $f \colon X \rightarrow Y$ is a $\mathbf{\Sigma}^0_2$-measurable mapping which is not piecewise continuous. Then there exists a subset $Z \subset X$ such that:
1. $Z$ is homeomorphic to the space of rational numbers,
2. the restriction $f \upharpoonright Z$ is a bijection,
3. the set $f(Z)$ is relatively discrete in $Y$,
4. $\dim \overline{Z} = 0$.
Fix a metric $\rho$ on $X$. Denote by $2^{<\omega}$ the set of all binary sequences of finite length. The construction will be carried out by induction with respect to the order $\preceq$ on $2^{<\omega}$ defined by $$s \preceq t \, \Longleftrightarrow \, {\operatorname{length}}(s) < {\operatorname{length}}(t) \vee ({\operatorname{length}}(s) = {\operatorname{length}}(t) \wedge s \leq_\mathrm{lex} t),$$ where $\leq_\mathrm{lex}$ is the usual lexicographical order on $2^{{\operatorname{length}}(s)}$. We write $s \prec t$ if $s \preceq t$ and $s \neq t$.
We will construct a sequence $\langle x_s \colon s \in 2^{<\omega} \rangle$ of points of $X$, a sequence $\langle V_s \colon s \in 2^{<\omega} \rangle$ of subsets of $X$, and a sequence $\langle U_s \colon s \in 2^{<\omega} \rangle$ of open subsets of $Y$ such that for every $s \in 2^{<\omega}$:
1. if $t \subset s$ then $V_s \subset V_t$,
2. $V_s$ is an open ball in $X$ with the centre $ x_s$ and radius $\leq 2^{- {\operatorname{length}}(s)}$,
3. if $s = t^\wedge 0$ then $x_s = x_t,$
4. $f(x_s) \in U_s$,
5. $(x_t, V_t)$ is $f$-irreducible outside $A$ for every $t \preceq s$, where $A = \bigcup_{u \preceq s}U_u$,
6. the family $\{V_t \colon t \in 2^{n} \}$ is pairwise strongly disjoint for every $n \in \omega$,
7. the family $\{U_t \colon t \preceq s \}$ is pairwise strongly disjoint.
Since $f$ is not piecewise continuous, we can apply Lemma \[lem1\] with respect to $X^\prime = X$ and $A = \emptyset$ to obtain the point $x \in X$ and the open set $U \subset Y$. Then put $ x_\emptyset = x$ and $U_\emptyset = U$. Let $V_\emptyset= B(x_\emptyset, 1)$ be an open ball in $X$ with the centre $ x_\emptyset$ and radius 1.
Assume that $x_t$, $V_t$, and $U_t$ have been constructed for any $t \preceq s$. Put $x_{s^\wedge 0} = x_s$ and $U_{s^\wedge 0} = U_s$.
Let $A= \bigcup_{t \prec s^\wedge 1}U_t$ and $O= Y \setminus \overline{A}$. By the inductive hypothesis, the pair $(x_s, V_s)$ is $f$-irreducible outside $A$. Take a neighborhood $W$ of $x_s$ such that $\overline{W} \subset V_s$. Then $(x_s, W)$ is $f$-irreducible outside $A$ and $f^{-1}(O) \cap W = A^f \cap W \notin \mathcal{I}_f$. By Lemma \[lem1\] there exist a point $x^\prime \in \overline{f^{-1}(O) \cap W}$ and an open set $U_{x^\prime} \subset Y$ strongly disjoint from $A$ such that $f(x^\prime) \in U_{x^\prime}$ and the pair $(x^\prime, f^{-1}(O) \cap W)$ is $f$-irreducible outside $U_{x^\prime}$. Notice that $x^\prime \neq x_s$ because $f(x_s) \in A$ and $\overline{U_{x^\prime}} \cap \overline{A} = \emptyset$. If the pair $(x^\prime, f^{-1}(O) \cap W)$ is $f$-irreducible outside $A \cup U_{x^\prime}$, put $x^* = x^\prime$ and $U^* = U_{x^\prime}$.
Consider the case when the pair $(x^\prime, f^{-1}(O) \cap W)$ is $f$-reducible outside $A \cup U_{x^\prime}$. Take a neighborhood $W^\prime$ of $x^\prime$ such that $\overline{W^\prime} \subset V_s$. Let $$O^\prime = Y \setminus (\overline{A} \cup \overline{U_{x^\prime}}) \text{ and } X^\prime = f^{-1}(O^\prime) \cap W \cap W^\prime.$$ Then the pair $(x^\prime, X^\prime)$ is $f$-irreducible outside $U_{x^\prime}$ and $X^\prime \notin \mathcal{I}_f$. As above, by Lemma \[lem1\] there exist a point $x^{\prime \prime} \in \overline{X^\prime}$ and an open set $U_{x^{\prime \prime}} \subset Y$ strongly disjoint from $A \cup U_{x^\prime}$ such that $f(x^{\prime \prime}) \in U_{x^{\prime \prime}}$ and the pair $(x^{\prime \prime}, X^\prime)$ is $f$-irreducible outside $U_{x^{\prime \prime}}$. Notice that $x^{\prime \prime} \neq x_s$ and $x^{\prime \prime} \neq x^\prime$. From Lemma \[lem2\] it follows that the pair $(x^{\prime \prime}, X^\prime)$ is $f$-irreducible outside $A \cup U_{x^{\prime \prime}}$. Then put $x^* = x^{\prime \prime}$ and $U^* = U_{x^{\prime \prime}}$.
Let $k = | \{t \in 2^{< \omega} \colon t \prec s^\wedge 1 \} |$, $z_0 = x^*$, and $U_0 = U^*$. Repeating the above construction, for $j = 0, \ldots, k$ recursively construct $z_j \in V_s$ and $U_j$ such that $f(z_j) \in U_j$, $U_j$ is strongly disjoint from $A_j= A \cup \bigcup_{i <j}U_i$, and the pair $(z_j, V_s \cap (A_j)^f)$ is $f$-irreducible outside $A \cup U_j$. From Lemma \[lem2\] it follows that for each $t \prec s^\wedge 1$ there is at most one $j \in \{ 0, \ldots, k \}$ such that $(x_t, V_t)$ is $f$-reducible outside $A \cup U_j$. The pigeonhole principle implies that there exists $\ell \in \{ 0, \ldots, k \}$ such that the pair $(z_\ell, V_s \cap (A_\ell )^f)$ is $f$-irreducible outside $A \cup U_\ell$ and $(x_t, V_t)$ is $f$-irreducible outside $A \cup U_\ell$ for each $t \prec s^\wedge 1$. Finally, set $x_{s^\wedge 1} = z_\ell$ and $U_{s^\wedge 1} = U_\ell$.
Since $x_{s^\wedge 0}$ and $x_{s^\wedge 1}$ are two distinct points from $V_s$, we can choose their neighborhoods $V_{s^\wedge 0}$ and $V_{s^\wedge 1}$, respectively, according to (1),(2), and (6).
One readily verifies that conditions (1)–(7) are satisfied.
The set $Z = \bigcup \{x_s \colon s \in 2^{<\omega} \}$ is countable and has no isolated points by (1) and (2). According to the Sierpiński theorem (see [@Eng Exercise 6.2.A]), $Z$ is homeomorphic to the space of rational numbers. By construction, the set $\bigcup \{f(x_s) \colon s \in 2^{<\omega} \}$ consists of isolated points. From conditions (4) and (5) it follows that the restriction $f \upharpoonright Z$ is a bijection.
From conditions (1) and (2) it follows that the family $\mathcal{V}_n = \{V_t \colon t \in 2^n \}$ forms a cover of $Z$ by open sets of diameter $\leq 2^{1-n}$ for each $n \in \omega$. Then $$\overline{Z} \subset \bigcap \bigl\{\bigcup \{\overline{V_t} \colon t \in 2^n \} \colon n \in \omega \bigr\}.$$
Since the family $\mathcal{V}_n$ is finite and pairwise strongly discrete, we can find a pairwise strongly discrete open family $\mathcal{W}_n = \{W_t \colon t \in 2^{n} \}$ such that $\mathrm{diam}(W_t) < 2^{2-n}$ and $\overline{V_t} \subset W_t$ for each $t \in 2^{n}$. Without loss of generality, each $\mathcal{W}_{n+1}$ is a refinement of $\mathcal{W}_n$. Every family $\{W \cap \overline{Z} \colon W \in \mathcal{W}_n \}$, $n \in \omega$, forms a discrete open cover of $\overline{Z}$. From the Vopěnka theorem (see [@Eng Theorem 7.3.1]) it follows that $\dim \overline{Z} = 0$.
\[t:4\] Every resolvable-measurable mapping $f \colon X \rightarrow Y$ of a metrizable space $X$ to a regular space $Y$ is piecewise continuous.
Suppose towards a contradiction that there is a resolvable-measurable mapping $f \colon X \rightarrow Y$ which is not piecewise continuous. Using Lemma \[lem3\], we can find a subset $Z \subset X$ such that $Z$ is homeomorphic to the space of rational numbers, the restriction $f \upharpoonright Z$ is a bijection, and $f(Z)$ is relatively discrete. Since $f$ is a resolvable-measurable mapping, $f \upharpoonright Z$ is the same. On the other hand, $f \upharpoonright Z$ fails to be resolvable-measurable as shown in Example.
Let $f \colon X \rightarrow Y$ be a bijection between metrizable spaces $X$ and $Y$ such that $f$ and $f^{-1}$ are both resolvable-measurable mappings. Then $\dim X = \dim Y$.
Theorem \[t:4\] implies that $X = \bigcup_{n \in \omega}A_n $, where each $A_n$ is closed in $X$ and each restriction $f \upharpoonright A_n$ is continuous. Similarly, $Y = \bigcup_{k \in \omega}B_k $, where each $B_k$ is closed in $Y$ and each restriction $f^{-1} \upharpoonright B_k$ is continuous. The sequence $\langle A_n \cap f^{-1}(B_k) \colon n \in \omega, k \in \omega \rangle$ forms a cover of $X$ by closed sets. Similarly, the sequence $\langle f(A_n) \cap B_k \colon n \in \omega, k \in \omega \rangle$ forms a cover of $Y$ by closed sets. Since $f \upharpoonright (A_n \cap f^{-1}(B_k))$ is a homeomorphism, we have $$\dim (A_n \cap f^{-1}(B_k)) = \dim (f(A_n) \cap B_k).$$ The corollary follows from the countable sum theorem [@Eng Theorem 7.2.1].
A topological space $X$ is called a *Baire space* if the intersection of countably many dense open sets in $X$ is dense; or equivalently every nonempty open set in $X$ is not of the first category. A space $X$ is *completely Baire* if every closed subspace of $X$ is a Baire space. Recall that $F \subset X$ is a *boundary set* in $X$ if its complement is dense, i.e., if $\overline{X \setminus F} = X$.
\[L:5\] For a metrizable space $X$ the following conditions are equivalent:
1. no closed subspace of $X$ is homeomorphic to the space $\mathbb{Q}$ of rational numbers,
2. $X$ is a completely Baire space,
3. the $\mathbf{\Delta}^0_2(X)$-sets coincide with the resolvable sets in $X$.
(i)$\Rightarrow$(ii): Suppose towards a contradiction that $X$ is not a completely Baire space. Then there is a closed set $F \subset X$ which is not Baire. Hence we can find a nonempty open (in $F$) set $U \subset F$ of the first category in $F$. The closure $\overline{U}$ is of the first category on itself. According to [@M86 Corollary 1] (see also [@D87]) $\overline{U}$ contains a closed copy of $\mathbb{Q}$, a contradiction.
(ii)$\Rightarrow$(iii): By [@Kur1 p. 362], every resolvable set in a metrizable space is a $\mathbf{\Delta}^0_2$-set.
Conversely, let $E \in \mathbf{\Delta}^0_2(X)$ and $F$ be an arbitrary non-empty closed set. According to [@Kur1 p. 99], we have to show that that either $F \cap E$ or $F \setminus E$ is not a boundary set in $F$. Otherwise, the sets $F \cap E$ and $F \setminus E$ would be of the first category in $F$ (because every boundary $\mathcal{F}_\sigma$-set is of the first category), so their union $F = (F \cap E) \cup (F \setminus E)$ would be of the first category on $F$. This contradicts the fact that $F$ is a Baire space.
(iii)$\Rightarrow$(i): Striving for a contradiction, suppose that $X$ contains a closed set $F$ which is homeomorphic to $\mathbb{Q}$. As shown in Example, there is a nonresolvable set $A \in \mathbf{\Delta}^0_2(F)$. The set $A$ is the same in $X$ because $F$ is closed in $X$.
\[t:6\] Let $f \colon X \rightarrow Y$ be a mapping of a metrizable completely Baire space $X$ to a regular space $Y$. Then the following conditions are equivalent:
1. $f$ is resolvable-measurable,
2. $f$ is piecewise continuous,
3. $f$ is $\mathbf{\Pi}^0_2$-measurable.
The implication (i)$\Rightarrow$(ii) follows from Theorem \[t:4\].
(ii)$\Rightarrow$(i): By definition, there are closed sets $X_n \subset X$, $n \in \omega$, such that $\bigcup_{n \in \omega}X_n = X$ and each $f \upharpoonright X_n$ is continuous. Then $$f^{-1}(A) = \bigcup \{X_n \cap f^{-1}(A) \colon n \in \omega \}$$ is an $\mathcal{F}_\sigma$-set in $X$ for every open (or closed) set $A \subset Y$. Hence $f^{-1}(U) \in \mathbf{\Delta}^0_2(X)$ for every open $U \subset Y$. From Lemma \[L:5\] it follows that $f^{-1}(U)$ is a resolvable set in $X$.
Banakh and Bokalo [@BB Theorem 8.1] got (ii)$\: \Leftrightarrow$ (iii).
Let $X$ be a completely metrizable space and $Y$ a regular space. Then $f \colon X \rightarrow Y$ is resolvable-measurable if and only if $f$ is $\mathbf{\Pi}^0_2$-measurable.
According to [@KMS Corollary 6], for an absolute Souslin-$\mathcal{F}$ set $X$, if $f \colon X \rightarrow Y$ is $\mathbf{\Sigma}^0_2$-measurable and not piecewise continuous, then there is a copy $K \subset X$ of the Cantor space $2^\omega$ such that $f \upharpoonright K$ has the same properties. The following theorem shows that a similar statement is valid for metrizable completely Baire spaces. However, such a set $K$ from Theorem \[t:8\] need not be homeomorphic to the Cantor space. In fact, every Bernstein set is a metrizable completely Baire space but it contains no copy of the Cantor space.
\[t:8\] Let $X$ be a metrizable completely Baire space and $Y$ a regular space. If $f \colon X \rightarrow Y$ is $\mathbf{\Sigma}^0_2$-measurable and not piecewise continuous, then there is a zero-dimensional separable closed set $K \subset X$ such that the restriction $f \upharpoonright K$ is the same.
Let $K = \overline{Z}$, where the set $Z \subset X$ is obtained by Lemma \[lem3\]. Clearly, $f \upharpoonright K$ is $\mathbf{\Sigma}^0_2$-measurable.
Suppose towards a contradiction that $f \upharpoonright K$ is piecewise continuous. Then there are closed sets $K_n \subset X$, $n \in \omega$, such that $\bigcup_{n \in \omega}K_n = K$ and $f \upharpoonright K_n$ is continuous. Since $K$ is a Baire space, there exists a $K_j$ with the nonempty interior $V_j$ (in $K$). Clearly, $f \upharpoonright \overline{V_j \cap Z}$ is continuous. Take a point $q \in V_j \cap Z$. Fix a neighborhood $U_q \subset Y$ of $f(q)$ such that $U_q \cap f(Z) = f(q)$. From continuity of $f \upharpoonright \overline{V_j \cap Z}$ it follows that there is a neighborhood $V \subset V_j$ (in $K$) of $q$ such that $f(V) \subset U_q$. Then $V \cap Z = \{ q \}$, i.e., $q$ is an isolated point of $Z$. This contradicts the fact that the set $V_j \cap Z$ has no isolated points.
The last theorem yields
Let $f \colon X \rightarrow Y$ be an $F_\sigma$-measurable mapping of a metrizable completely Baire space $X$ to a regular space $Y$. If the restriction $f \upharpoonright Z$ is piecewise continuous for any zero-dimensional separable closed subset $Z$ of $X$, then $f$ is piecewise continuous.
[HD]{}
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abstract: 'We study hadronic molecular states in a coupled system of ${J/\psi}N - {\Lambda_c}{\bar{D}}^{(*)} - {\Sigma_c}^{(*)}{\bar{D}}^{(*)}$ in $I(J^P) = \frac{1}{2}(\frac{3}{2}^-)$ channel, using the complex scaling method combined with the Gaussian expansion method. We construct the potential including one pion exchange and one $D^{(*)}$ meson exchange with $S$-wave orbital angular momentum. We find that the both mass and width of the pentaquark $P_c(4380)$ can be reproduced within a reasonable parameter region, and that its main decay mode is ${\Lambda_c}{\bar{D}^*}$. We extend our analysis to a coupled system of ${\Lambda_c}D^{(*)} - {\Sigma_c}^{(*)}D^{(*)}$ in $I(J^P) = \frac{1}{2}(\frac{3}{2}^-)$ channel. We find that there exists a doubly charmed baryon of $ccqq\bar{q}$ type as a hadronic molecule, the mass and width of which are quite close to those of $P_c(4380)$.'
author:
- Yuki Shimizu
- Masayasu Harada
title: 'Hidden Charm Pentaquark $P_c(4380)$ and Doubly Charmed Baryon $\Xi_{cc}^*(4380)$ as Hadronic Molecule States'
---
Introduction {#sec:Intro}
============
In 2015, the LHCb experiment announced the observation of the hidden charm pentaquark $P_c(4380)$ and $P_c(4450)$ [@Aaij:2015tga; @Aaij:2016phn; @Aaij:2016ymb]. The mass and width of $P_c(4380)$ are $M=4380\pm8\pm29$MeV and $\Gamma=205\pm18\pm86$ MeV and those of $P_c(4450)$ are $M = 4449.8\pm1.7\pm2.5$ and $\Gamma = 39\pm5\pm19$MeV. Their spins and parities are not well determined; most likely $J^P=(3/2^- , 5/2^+)$.
Some theoretical works were done before the LHCb result in Refs. [@Wu:2010vk; @Yang:2011wz; @Wang:2011rga; @Wu:2012md; @Uchino:2015uha]. After the LHCb announcement, there are many theoretical analyses based on the hadronic molecule picture [@Chen:2015loa; @He:2015cea; @Chen:2015moa; @Huang:2015uda; @Roca:2015dva; @Meissner:2015mza; @Xiao:2015fia; @Burns:2015dwa; @Kahana:2015tkb; @Chen:2016heh; @Chen:2016otp; @Shimizu:2016rrd; @Yamaguchi:2016ote; @He:2016pfa; @Ortega:2016syt; @Azizi:2016dhy; @Geng:2017hxc], diquark-diquark-antiquark (diquark-triquark) picture [@Maiani:2015vwa; @Lebed:2015tna; @Anisovich:2015cia; @Li:2015gta; @Wang:2015epa; @Zhu:2015bba], compact pentaquark states [@Santopinto:2016pkp; @Takeuchi:2016ejt; @Wu:2017weo], and triangle singularities [@Guo:2015umn; @Liu:2015fea; @Mikhasenko:2015vca; @Liu:2016dli; @Guo:2016bkl; @Bayar:2016ftu]. The decay behaviors are studied in Refs.[@Wang:2015qlf; @Lu:2016nnt; @Shen:2016tzq; @Lin:2017mtz].
In Ref. [@Shimizu:2016rrd], effect of $\Sigma_c^*\bar{D}-\Sigma_c\bar{D}^*$ coupled channel is studied in the hadronic molecule picture for the hidden charm pentaquark with $I(J^P) = 1/2(3/2^-)$, by using the one-pion exchange potential with $S$-wave orbital angular momentum. It was shown that there exists a bound state with the binding energy of several MeV below $\Sigma_c^\ast\bar{D}$ threshold, which is mainly made from a $\Sigma_c^\ast$ and a $\bar{D}$. In Ref. [@Yamaguchi:2016ote], the coupled channel effect to $\Lambda_c\bar{D}^{(*)}$ was shown to be important to investigate the $P_c$ pentaquarks. In Ref. [@Lin:2017mtz], decay behaviors of hadronic molecule states of $\Sigma_c^*\bar{D}$ and $\Sigma_c\bar{D}^*$ to $J/\psi N$ are studied and it was shown that the contribution of $J/\psi N$ is small for the $P_c(4380)$ as the $\Sigma_c^*\bar{D}$ molecule. However, in our best knowledge, study of the effect of $J/\psi N$ in full coupled channel analysis, which reproduce both the mass and width of $P_c(4380)$, was not done so far.
In this paper, we make a coupled channel analysis including ${J/\psi}N$ in addition to $\Lambda_c\bar{D}^{(*)} - \Sigma_c^{(*)}\bar{D}^{(*)}$ with $S$-wave orbital angular momentum. Here we construct a relevant potential from exchange of one pion and $D^{(\ast)}$ mesons. Our result shows that both the mass and width of $P_c(4380)$ are within experimental errors for reasonable parameter region, and that the effect from $J/\psi N$ channel is very small. In other word, the observed mass and width of $P_c(4380)$ are well reproduced dominantly by one-pion exchange potential for $\Lambda_c\bar{D}^{(*)} - \Sigma_c^{(*)}\bar{D}^{(*)}$ coupled channel.
Since the one-pion exchange potential for $\Lambda_c{D}^{(*)} - \Sigma_c^{(*)}{D}^{(*)}$ coupled channel is same as the one for $\Lambda_c\bar{D}^{(*)} - \Sigma_c^{(*)}\bar{D}^{(*)}$ coupled channel, we expect the existence of a doubly charmed baryon with $I(J^P) = \frac{1}{2}\left(\frac{3}{2}^-\right)$ having the mass and width close to those of $P_c(4380)$, which we call $\Xi_{cc}^*(4380)$. In the latter half of this paper, we demonstrate that $\Xi_{cc}^*(4380)$ does exist in our model, which actually has the mass and width quite close to those of $P_c(4380)$.
This paper is organized as follows: In Sec. \[sec:pot\], we show the potentials which we use in our analysis. We study the pentaquark $P_c(4380)$ in Sec. \[sec:pc4380\], and the doubly charmed baryon $\Xi_{cc}^*(4380)$ in Sec. \[sec:xicc4380\]. Finally, we will give a brief summary and discussions in Sec. \[sec:summary\].
Potential {#sec:pot}
=========
In this section, we construct a potential for our coupled channel analysis based on the heavy quark symmetry and the chiral symmetry. We include one-pion exchange contribution for $\Lambda_c\bar{D}^{(*)} - \Sigma_c^{(*)}\bar{D}^{(*)}$ coupled channel and $D^{(*)}$ meson exchange for adding $J/\psi N$ channel.
For constructing effective interactions of $D$ and $D^\ast$ mesons, it is convenient to use the following heavy meson field $H$ defined as [@Falk:1991nq; @Wise:1992hn; @Cho:1992gg; @Yan:1992gz] $$\begin{aligned}
H &= \frac{1+{v\hspace{-.47em}/}}{2}\left[ D_{\mu}^{*}\gamma^{\mu} + iD\gamma_{5} \right]\ , \label{def H}\\
\bar{H} &= \gamma_0 H^{\dagger} \gamma_0~.\end{aligned}$$ where $D$ and $D^\ast$ are the pseudoscalar and vector meson fields, respectively, and $v$ denotes the velocity of the heavy mesons.
The pion field is introduced by the spontaneous chiral symmetry breaking $\textrm{SU}(2)_{\textrm{R}}\times\textrm{SU}(2)_{\textrm{L}} \to \textrm{SU}(2)_{\textrm{V}}$. The fundamental quantity is $$\begin{aligned}
A_{\mu} = \frac{i}{2}\left( \xi^{\dagger}{\partial}_{\mu}\xi - \xi{\partial}_{\mu}\xi^{\dagger} \right)~,\end{aligned}$$ where $\xi = \exp(i\hat{\pi}/\sqrt{2}{f_{\pi}})$. The pion decay constant is ${f_{\pi}}\sim 92.4$MeV and the pion field $\hat{\pi}$ is defined by a $2 \times 2$ matrix $$\begin{aligned}
\hat{\pi} = \left(
\begin{array}{cc}
\pi^0/\sqrt{2} & \pi^+ \\
\pi^- & -\pi^0/\sqrt{2}
\end{array}
\right)~.\end{aligned}$$
The interaction Lagrangian for the heavy meson and pions with least derivatives [@Wise:1992hn; @Yan:1992gz; @Cho:1992gg] is given by $$\begin{aligned}
\mathcal{L}_{HH\pi} &= g{\textrm{Tr}}\left[ \bar{H}H\gamma_{\mu}\gamma_5A^{\mu} \right]~,\end{aligned}$$ where $g$ is a dimensionless coupling constant. The explicit interaction terms can be written as $$\begin{aligned}
\mathcal{L}_{D^*D^*\pi} &= \frac{\sqrt{2}ig}{{f_{\pi}}}\epsilon^{\mu\nu\rho\sigma}\bar{D}_{\mu}^*D_{\nu}^*{\partial}_{\rho}\hat{\pi} v_{\sigma}~, \\
\mathcal{L}_{D^*D\pi} &= \frac{\sqrt{2}ig}{{f_{\pi}}} \left( \bar{D}_{\mu}^*D{\partial}^{\mu}\hat{\pi} - \bar{D}D_{\mu}^*{\partial}^{\mu}\hat{\pi} \right)~,\end{aligned}$$ by expanding the $A_{\mu}$ and $H$ fields. Note that the $DD\pi$ interaction term is prohibited by the parity invariance. The coupling constant $g$ is determined as $|g| = 0.59$ from the decay of $D^* \to D\pi$ [@Olive:2016xmw]. The sign of $g$ cannot be decided by the above decay, however we use $g = 0.59$ in the following analysis.
For introducing $\Sigma_c$ and $\Sigma_c^\ast$, we define the following superfield $S_{\mu}$ for $\Sigma_c$ and $\Sigma_c^*$ [@Liu:2011xc]: $$\begin{aligned}
S_{\mu} &= \Sigma_{c\mu}^* - \sqrt{\frac{1}{3}}\left( \gamma_{\mu} + v_{\mu} \right)\gamma_5\Sigma_c~,
\label{eq:superfield}\end{aligned}$$ where the single heavy baryon fields $\Lambda_c$ and $\Sigma_c$ are expressed by the $2 \times 2$ matrices as $$\begin{aligned}
\Lambda_c &= \left(
\begin{array}{cc}
0 & \Lambda_{c}^{+} \\
-\Lambda_{c}^{+} & 0
\end{array}
\right) , \quad \Sigma_c = \left(
\begin{array}{cc}
\Sigma_{c}^{++} & \frac{1}{\sqrt{2}}\Sigma_{c}^{+} \\
\frac{1}{\sqrt{2}}\Sigma_{c}^{+} & \Sigma_{c}^{0} \\
\end{array}
\right)~,\end{aligned}$$ and the matrix field of $\Sigma_c^*$ is defined similarly to the $\Sigma_c$. The interaction Lagrangian for the heavy baryon and pions is given by [@Falk:1991nq; @Yan:1992gz] $$\begin{aligned}
\mathcal{L}_{BB\pi} &= \frac{3ig_1}{2}v_{\sigma}\epsilon^{\mu\nu\rho\sigma}{\textrm{Tr}}\left[ \bar{S}_{\mu}A_{\nu}S_{\rho} \right] {\nonumber}\\
&\hspace{8mm} + g_4{\textrm{Tr}}\left[ \bar{S}^{\mu}A_{\mu}\Lambda_c \right] + H.c.~,\end{aligned}$$ where $g_1$ and $g_4$ are dimensionless coupling constants. We use $g_4=0.999$ determined from the $\Sigma_c^* \to \Lambda_c \pi $ decay. The value of $g_1$ cannot be determined by experimental decay, so we use $g_1=\frac{\sqrt{8}}{3}g_4 = 0.942$ estimated by the quark model in Ref. [@Liu:2011xc] as a reference value, and vary its value about 20%, $0.753$-$1.13$.
We include $J/\psi$ together with $\eta_c$ using a $\bar{c}c$ spin doublet field $\mathcal{J}$ as [@Jenkins:1992nb; @Wang:2015xsa] $$\begin{aligned}
\mathcal{J} &= \frac{1+{v\hspace{-.47em}/}}{2}\left( \left( J/\psi\right)^{\mu}\gamma_{\mu} - \eta_c\gamma_5 \right)\frac{1-{v\hspace{-.47em}/}}{2}~.\end{aligned}$$ In the following analysis, we use only $\left(J/\psi\right)^{\mu}$ field. The interaction of ${\mathcal J}$ to the heavy mesons $D^{(*)}$ and its anti-particles $\bar{D}^{(*)}$ is expressed as [@Wang:2015xsa] $$\begin{aligned}
\mathcal{L}_{\mathcal{J}HH} &= G_1{\textrm{Tr}}\left[ \mathcal{J}\bar{H}_A{\overleftrightarrow}{{\partial}}_{\mu}\gamma^{\mu}\bar{H} + H.c. \right]~,
\label{eq:JHH}\end{aligned}$$ where ${\overleftrightarrow}{{\partial}}_{\mu} = \overrightarrow{{\partial}}_{\mu} - \overleftarrow{{\partial}}_{\mu}$. The field $H$ is defined in Eq. (\[def H\]), and its anti-particle field $H_A$ is defined as $$\begin{aligned}
H_A &= \left[ \bar{D}_{\mu}^{*}\gamma^{\mu} + i\bar{D}\gamma_{5} \right] \frac{1-{v\hspace{-.47em}/}}{2} \ .\end{aligned}$$ We estimate the value of the coupling constant $G_1$ by comparing it with the $\phi KK$ coupling. Regarding the strange hadrons as heavy hadrons, we can write the effective Lagrangian for $\phi K \bar{K}$ in the same form as the one in Eq. (\[eq:JHH\]). Using the value of $\phi K \bar{K}$ coupling $G_{1(\phi KK)}$ determined from the $\phi \to K \bar{K}$ decay: $G_{1(\phi KK)}=4.48[\textrm{GeV}^{-3/2}]$, we estimate the value of $G_1$ as $$\begin{aligned}
G_1 = G_{1(\phi KK)}\sqrt{\frac{m_{\phi}m_K^2}{m_{{J/\psi}}m_D^2}} = 0.679[\textrm{GeV}^{-3/2}].\end{aligned}$$
The Lagrangian for the interactions among single heavy baryons, $D^{(*)}$ mesons and nucleons is given by $$\begin{aligned}
\mathcal{L}_{BHN} &=
G_2 \left(\tau_2\bar{S}_{\mu}\right)H\gamma_5\gamma^{\mu}N + H.c. {\nonumber}\\
&+ G_3 \left(\tau_2\bar{\Lambda}_{c}\right)HN + H.c. ~.\end{aligned}$$ We estimate the values of $G_2$ and $G_3$ using $g_{\Sigma_c DN}=2.69$ and $g_{\Lambda_c DN}=13.5$ [@Lu:2016nnt; @Lin:2017mtz; @Garzon:2015zva; @Liu:2001ce]. Considering the differences of the normalization of a heavy meson field by $\sqrt{m_D}$, we estimate them as $$\begin{aligned}
G_2 &= - \frac{g_{\Sigma_c DN}}{\sqrt{3m_D}} = -1.14[\textrm{GeV}^{-1/2}]~, \\
G_3 &= \frac{g_{\Lambda_c DN}}{\sqrt{m_D}} = 9.88[\textrm{GeV}^{-1/2}]~.\end{aligned}$$ Here the factor $-\frac{1}{\sqrt{3}}$ comes from the coefficient in Eq. (\[eq:superfield\]). The estimations of the values of $G_{1,2,3}$ are very rough. We will discuss the effects of ambiguities in the folllowing sections.
We constract the one-pion exchange potential and one $D^{(*)}$ meson exchange potential from the above interaction Lagrangians. We introduce the monopole-type form factor, $$\begin{aligned}
F(\vec{q}) = \frac{\Lambda^2-m_a^2}{\Lambda^2+|\vec{q}|^2}~,\end{aligned}$$ at each vertex, where $\Lambda$ is a cutoff parameter, $m_a$ and $\vec{q}$ are the mass and momentum of exchanging particle, respectively. Although the cutoff parameter $\Lambda$ may be different for pion and $D^{(*)}$ meson, we use the same value in the present analysis for simplicity. Including this form factor, the exchange potentials are written as $$\begin{aligned}
V_{ij}^{a}(r) &= G_{ij}C_a(r, \Lambda, m_a)~,
\label{pot part}\end{aligned}$$ where $G_{ij}$ denotes the coefficients, coupling constants, spin factors, and isospin factors for each $(i, j)$ channel. $C_a(r, \Lambda, m_a)$ is defined as $$\begin{aligned}
C_a(r, \Lambda, m_a) &= \frac{m_a^2}{4\pi}\left[ \frac{e^{-m_ar} - e^{-\Lambda r} }{r} - \frac{\Lambda^2-m_a^2}{2\Lambda}e^{-\Lambda r} \right]~.\end{aligned}$$ The explicit forms of potential are shown in the following sections.
Numerical result for pentaquark $P_c(4380)$ {#sec:pc4380}
===========================================
We consider the ${J/\psi}N - \Lambda_c\bar{D}^* - \Sigma_c^*\bar{D} - \Sigma_c\bar{D}^* - \Sigma_c^*\bar{D}^*$ coupled system with $S$-wave orbital angular momentum. We solve the coupled channel Schr[ö]{}dinger equation, using the potential $V(r)$ given by a $5\times5$ matrix expressed as
$$\begin{aligned}
V(r) &= \left(
\begin{array}{ccccc}
0 & G_1G_3(C_D + C_{D^*}) & -2\sqrt{6}G_1G_2C_{D^*} & \sqrt{2}G_1G_2(3C_D-C_{D^*}) & 2\sqrt{10}G_1G_2C_{D^*} \vspace{2pt} \\
G_1G_3(C_D + C_{D^*}) & 0 & -\frac{gg_4}{\sqrt{6}{f_{\pi}}^2}C_{\pi} & \frac{gg_4}{3\sqrt{2}{f_{\pi}}^2}C_{\pi} & -\frac{\sqrt{10}gg_4}{6{f_{\pi}}^2}C_{\pi} \vspace{2pt} \\
-2\sqrt{6}G_1G_2C_{D^*} & -\frac{gg_4}{\sqrt{6}{f_{\pi}}^2}C_{\pi} & 0 & \frac{gg_1}{2\sqrt{3}{f_{\pi}}^2}C_{\pi} & -\frac{\sqrt{15}gg_1}{9{f_{\pi}}^2}C_{\pi} \vspace{2pt} \\
\sqrt{2}G_1G_2(3C_D-C_{D^*}) & \frac{gg_4}{3\sqrt{2}{f_{\pi}}^2}C_{\pi} & \frac{gg_1}{2\sqrt{3}{f_{\pi}}^2}C_{\pi} & -\frac{gg_1}{3{f_{\pi}}^2}C_{\pi} & \frac{\sqrt{5}gg_1}{6{f_{\pi}}^2}C_{\pi} \vspace{2pt} \\
2\sqrt{10}G_1G_2C_{D^*} & -\frac{\sqrt{10}gg_4}{6{f_{\pi}}^2}C_{\pi} & -\frac{\sqrt{15}gg_1}{9{f_{\pi}}^2}C_{\pi} & \frac{\sqrt{5}gg_1}{6{f_{\pi}}^2}C_{\pi} & -\frac{2gg_1}{9{f_{\pi}}^2}C_{\pi}
\end{array}
\right)~,
\label{eq:pot5by5}\end{aligned}$$
where $C_a$ is defined in Eq. (\[pot part\]). The wave function has five components; $$\begin{aligned}
\Psi(r) = \left(
\begin{array}{c}
\psi_{{J/\psi}N} \\ \psi_{{\Lambda_c}{\bar{D}^*}} \\ \psi_{{\Sigma_c^*}{\bar{D}}} \\ \psi_{{\Sigma_c}{\bar{D}^*}} \\ \psi_{{\Sigma_c^*}{\bar{D}^*}}
\end{array}
\right)~.\end{aligned}$$ We use $m_{\pi}=137.2$, $m_{N}=938.9$, $m_{D}=1867.2$, $m_{D^*}=2008.6$, $m_{{\Lambda_c}}=2286.5$, $m_{{\Sigma_c}}=2453.5$, $m_{{\Sigma_c^*}}=2518.1$ and $m_{{J/\psi}}=3096.9$ MeV for the hadron masses [@Olive:2016xmw]. The thresholds for the hadronic molecules are shown in Table \[tab:pc4380\]. In this calculation, we vary the cutoff parameter $\Lambda$ from $1000$ to $1500$ MeV. For the coupling constant $g_1$, we use $g_1=0.942$ estimated in a quark model [@Liu:2001ce] as a reference value, and study the $g_1$ dependence of the results using $g_1=0.753$ and $1.13$. To obtain the bound and resonance solutions, we use the complex scaling method [@Aguilar:1971ve; @Balslev:1971vb; @Aoyama:2006csm] and Gaussian expansion method [@Hiyama:2003cu; @Hiyama:2012sma].
The resultant complex energies are shown in Table \[tab:pc4380\]. When the cutoff parameter $\Lambda$ becomes larger, the mass and width become smaller. In our ranges of $\Lambda$ and $g_1$, the bound state solution which has the real energy below the ${J/\psi}N$ threshold does not appear. The solutions of $\Lambda=1200$ and 1300 MeV for $g_1=0.942$ can reproduce the observed mass of $P_c(4380)$, $4380\pm8\pm29$MeV and width, $205\pm18\pm86$MeV. However, there exists another resonance state solution, the mass of which is $4283.1$MeV for $\Lambda = 1200$MeV and $4227.1$MeV for $\Lambda = 1300$MeV. These lower states are not observed in LHCb experiment, therefore we consider that these parameter sets are unlikely. On the other hand, for the $\Lambda=1000$MeV and $g_1=0.753$, we obtain only one resonance state which corresponds to $P_c(4380)$. Its mass, $4390.2$MeV, is slightly above the $\Sigma_c^*\bar{D}$ threshold, so this state is interpreted as a resonance state of $\Sigma_c^*\bar{D}$ molecule.
\[!htbp\]
$\Lambda$ \[MeV\]
------- ------------------ --------------------- ---------------------------------- --------------------------------- --------------------------------- -----------------------------------
$g_1$ $1000$ $1100$ $1200$ $1300$ $1400$ $1500$
threshold\[MeV\] ${J/\psi}N$(4035.8) ${\Lambda_c}{\bar{D}^*}$(4295.1) ${\Sigma_c^*}{\bar{D}}$(4385.3) ${\Sigma_c}{\bar{D}^*}$(4462.1) ${\Sigma_c^*}{\bar{D}^*}$(4526.7)
\[tab:pc4380\]
Doubly charmed baryon $\Xi_{cc}^*(4380)$ {#sec:xicc4380}
========================================
We study the doubly charmed baryon as a hadronic molecular state in this section. Replacing $\bar{D}^{(*)}$ with $D^{(*)}$ and excluding the ${J/\psi}N$ channel from the calculation in Sec. \[sec:pc4380\], we construct the $ccqq\bar{q}$ state which has the same flavor quantum number as the $ccq$ baryon has. The interactions of one-pion exchange is not changed by the replacement of $D^{(*)}$ meson. Therefore, the corresponding potential matrix is a bottom-right 4$\times$4 block of Eq. (\[eq:pot5by5\]): $$\begin{aligned}
V(r) &= \left(
\begin{array}{cccc}
0 & -\frac{gg_4}{\sqrt{6}{f_{\pi}}^2} & \frac{gg_4}{3\sqrt{2}{f_{\pi}}^2} & -\frac{\sqrt{10}gg_4}{6{f_{\pi}}^2} \vspace{2pt} \\
-\frac{gg_4}{\sqrt{6}{f_{\pi}}^2} & 0 & \frac{gg_1}{2\sqrt{3}{f_{\pi}}^2} & -\frac{\sqrt{15}gg_1}{9{f_{\pi}}^2} \vspace{2pt} \\
\frac{gg_4}{3\sqrt{2}{f_{\pi}}^2} & \frac{gg_1}{2\sqrt{3}{f_{\pi}}^2} & -\frac{gg_1}{3{f_{\pi}}^2} & \frac{\sqrt{5}gg_1}{6{f_{\pi}}^2} \vspace{2pt} \\
-\frac{\sqrt{10}gg_4}{6{f_{\pi}}^2} & -\frac{\sqrt{15}gg_1}{9{f_{\pi}}^2} & \frac{\sqrt{5}gg_1}{6{f_{\pi}}^2} & -\frac{2gg_1}{9{f_{\pi}}^2}
\end{array}
\right)C_{\pi}~.
\label{eq:pot4by4}\end{aligned}$$ The wave function has four components; $$\begin{aligned}
\Psi(r) = \left(
\begin{array}{c}
\psi_{{\Lambda_c}{\bar{D}^*}} \\ \psi_{{\Sigma_c^*}{\bar{D}}} \\ \psi_{{\Sigma_c}{\bar{D}^*}} \\ \psi_{{\Sigma_c^*}{\bar{D}^*}}
\end{array}
\right)~.\end{aligned}$$
We investigate the dependence on the cutoff $\Lambda$ and coupling constant $g_1$ in the same range as in Sec.\[sec:pc4380\], and show the numerical results in Table \[tab:Xi4380\].
\[!htbp\]
$\Lambda$ \[MeV\]
------- ------------------ -------------------------- ------------------------- ------------------------- --------------------------- --------
$g_1$ $1000$ $1100$ $1200$ $1300$ $1400$ $1500$
threshold\[MeV\] ${\Lambda_c}D^*$(4295.1) ${\Sigma_c^*}D$(4385.3) ${\Sigma_c}D^*$(4462.1) ${\Sigma_c^*}D^*$(4526.7)
\[tab:Xi4380\]
Comparing the results of Table \[tab:pc4380\] and Table \[tab:Xi4380\], they have close mass and decay width. For $\Lambda = 1200$-$1500$MeV, we obtain bound state solutions whose masses are below the threshold of $\Lambda_cD^*$. Since the mass and width of $P_c(4380)$ are not within experimental errors for $\Lambda \ge 1100$MeV, [^1] the bound state below $\Lambda_c D^\ast$ is unlikely to exist. On the other hand, when $\Lambda = 1000$MeV and $g_1 = 0.753$ are used, for which the mass and width of $P_c(4380)$ are within experimental errors, the mass and width of the doubly charmed baryon are $M= 4370.1$MeV and $\Gamma = 68.7$MeV, which are close to those of $P_c(4380)$. This means that, when the hidden charm pentaquark $P_c(4380)$ exist as a hadronic molecular state, a doubly charmed baryon with same spin and parity exists, and its mass and width are close to $P_c(4380)$, which we call this doubly charmed baryon $\Xi_{cc}^*(4380)$.
Summary and Discussions {#sec:summary}
=======================
We investigated the coupled channel of the ${J/\psi}N - \Lambda_c\bar{D}^* - \Sigma_c^*\bar{D} - \Sigma_c\bar{D}^* - \Sigma_c^*\bar{D}^*$ in $J^P=3/2^-$ with $S$-wave orbital angular momentum. We constructed the one-pion exchange and one-$D^{(*)}$ meson exchange potential and solved the complex scaled Schr[ö]{}dinger-type equation. We showed that, for $\Lambda= 1200$-$1300$MeV, there exists another state having mass and with smaller than $P_c(4380)$, while for $\Lambda = 1000$MeV and $g_1 = 0.753$, there exists only one molecular state having the mass and width within errors of experimental values. This shows that hidden charm pentaquark $P_c(4380)$ can be explained as a $S$-wave hadronic molecular state.
We studied the coupled channel of the $\Lambda_cD^* - \Sigma_c^*D - \Sigma_cD^* - \Sigma_c^*D^*$ in $J^P=3/2^-$ with $S$-wave orbital angular momentum. Since the one-pion interactions for $\bar{D}^{(*)}$ mesons are the same as the ones for $D^{(*)}$ mesons, we obtain a $\Xi_{cc}$ state with $J^P= \frac{3}{2}^-$ as a resonance state whose mass and width are very close to those of $P_c(4380)$, which we call $\Xi_{cc}^{\ast}(4380)$.
We think that the same mechanism applies for $P_c(4450)$: When $P_c(4450)$ is described as a hadronic molecular state, there exists a doubly charmed baryon which has a mass and a width quite close to $P_c(4450)$.
Although we do not evaluate the partial decay width for $J/\psi N$ in this paper, we can see that the partial width is much narrower than that for $\Lambda_c\bar{D}^*$ in the following way: When we omit the contribution from $J/\psi N$ channel to $P_c(4380)$, the relevant potential become the same as that for $\Xi_{cc}^*(4380)$ . This implies that the resultant mass and width without $J/\psi N$ channel is already close to the ones with $J/\psi N$ channel. This is consistent with the analysis of decay behaviors in Ref. [@Lin:2017mtz].
Comparing the results of $P_c(4380)$ and $\Xi_{cc}^*(4380)$, we can see that the contribution of the $J/\psi N$ channel to $P_c(4380)$ is small. This is consistent with the naive prospect of the supression of $D^{(*)}$ meson exchange potentials. Our evaluation of the coupling to the $J/\psi N$ was very rough, so that the values used in this analysis include some ambiguities. Furthermore, there may exist other contributions which couple the $J/\psi N$ channel to $ \Lambda_c\bar{D}^* - \Sigma_c^*\bar{D} - \Sigma_c\bar{D}^* - \Sigma_c^*\bar{D}^*$. We think that these ambiguities do not change our results, since the contribution from $J/\psi N$ channel is very small consistently with the result in Ref. [@Lin:2017mtz].
In the present analysis, we do not include the decay of $\Sigma_c^* \to \Lambda_c\pi$ for $\Sigma_c^*\bar{D}^{(*)}$ state. The width of this decay is about 15MeV [@Olive:2016xmw], so it makes the total width of $P_c(4380)$ broader [@Lin:2017mtz].
We used only one-pion exchange potential for $\Lambda_c{D}^{(*)} - \Sigma_c^{(*)}{D}^{(*)}$ coupled channel in the analysis of $\Xi_{cc}^*(4380)$, which is the same as the one for $\Lambda_c\bar{D}^{(*)} - \Sigma_c^{(*)}\bar{D}^{(*)}$ coupled channel in the analysis of $P_c(4380)$. Then, we obtained the mass and width of $\Xi_{cc}^*(4380)$ very close to those of $P_c(4380)$. When we include the effects of $\omega$ meson exchange, difference between $DD\omega$ and $D\bar{D}\omega$ will generate some differences of the mass and width [@Chen:2017vai].
There are some theoretical predictions of ordinary $ccq$-type baryons in $J^P=3/2^-$ [@Migura:2006ep; @Chiu:2005zc; @Wang:2010it; @Karliner:2014gca; @Padmanath:2015jea; @Wei:2015gsa; @Shah:2017liu]. In Ref. [@Shah:2017liu], the mass of $3P$-state spin-$\frac{3}{2}$ $\Xi_{cc}$ is predicted to be about $4.41$GeV. This state might mix with $\Xi_{cc}^*(4380)$ predicted in this analysis.
We expect that the precise properties of $P_c$ pentaquarks and the existence of excited $\Xi_{cc}$ baryons would be revealed in future experiments.
We would like to thank Yuji Kato for useful discussion. The work of Y.S. is supported in part by JSPS Grant-in-Aid for JSPS Research Fellow No. JP17J06300. The work of M.H. is supported in part by the JSPS Grant-in-Aid for Scientific Research (C) No. 16K05345.
[^1]: As we stated in the previous section, there are a few parameter choices for which the mass and width of $P_c(4380)$ are reproduced even for $\Lambda \ge 1100$MeV. However, there is another state lighter than $P_c(4380)$, so that these parameter choices are unlikely.
|
---
abstract: 'We study the co-evolution of supermassive black holes (SMBHs) with galaxies by means of semi-analytic model (SAM) of galaxy formation based on sub-halo merger trees built from Millennium and Millennium-II simulation. We utilize the simulation results from Guo 2013 and Henriques 2015 to study two aspects of the co-evolution, *i.e.* the stochastic gravitational wave (GW) background generated by SMBH merger and the SMBH/galaxy clustering. The characteristic strain amplitude of GW background predicted by Guo 2013 and Henriques 2015 models are $A_{yr^{-1}}=5.00\times10^{-16}$ and $A_{yr^{-1}}=9.42\times10^{-17}$, respectively. We find the GW amplitude is very sensitive to the galaxy merger rate. The difference in the galaxy merger rate between Guo 2013 and Henriques 2015, results in a factor $5$ deviation in the GW strain amplitude. For clusterings, we calculate the spatially isotropic two point auto- and cross-correlation functions (2PCFs) for both SMBHs and galaxies by using the mock catalogs generated from Guo 2013 model. We find that all 2PCFs have positive dependence on both SMBH and galaxy mass. And there exist a significant time evolution in 2PCFs, namely, the clustering effect is enhanced at lower redshifts. Interestingly, this result is not reported in the active galactic nuclei samples in SDSS. Our analysis also shows that, roughly, SMBHs and galaxies, with galaxy mass $10^2\sim10^3$ larger than SMBH mass, have similar pattern of clustering, which is a reflection of the co-evolution of SMBH and galaxy. Finally, we calculate the first ten multiples of the angular power spectrum of the energy density of GW background. We find the amplitude of angular power spectrum of the first ten multiples is about $10\%$ to $60\%$ of the monopole component in the whole frequency range.'
author:
- |
Qing Yang,$^{1}$ Bin Hu,$^{1}$[^1] Xiao-Dong Li,$^{2}$\
$^{1}$Department of Astronomy, Beijing Normal University, Beijing, 100875, China\
$^{2}$School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510297, P. R. China
bibliography:
- 'mnras\_SMB.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Co-evolution of supermassive black holes with galaxies from semi-analytic model: stochastic gravitational wave background and black hole clustering'
---
\[firstpage\]
galaxies: formation, (galaxies:) quasars: supermassive black holes
Introduction
============
Observational evidence shows that supermassive black holes (SMBHs) are located in the center of nearly all massive galaxies [@Soltan:1982vf; @Kormendy:1995er; @Magorrian:1997hw]. Although the evolution mechanism of SMBHs is not very well known yet, observational evidence shows that there are strong correlations between mass of SMBHs and observational properties of their host galaxies, such as velocity dispersion, star formation rate and bulge stellar mass [@Madau:1996aw; @Boyle:1997sm; @Magorrian:1997hw; @Ferrarese:2000se; @Ueda:2003yx; @Zheng:2009ac]. It is also expected that when galaxies merge, the SMBHs inside them should form SMBH binaries, emit gravitational waves during inspiral and merge eventually (we refer the readers to the latest review [@Sesana:2014wta]). Gravitational torques induced by galaxy-galaxy mergers drive inflows of cold gas toward the center of galaxies, triggering the central starbursts and also accretion on to SMBHs [@Hernquist:1989ew; @Barnes:1991zz; @Barnes:1996qt; @Mihos:1994wj; @Mihos:1995ri; @DiMatteo:2005ttp]. Galaxy/SMBH merger has been proposed to be a way by which central active galactic nuclei (AGN) could be triggered and SMBHs could grow [@Hopkins:2007hc; @Sanders:1988rz; @Treister:2012ag].
Gravitational waves (GWs) from inspiralling SMBHs are expected to form a GW background at frequency range of $10^{-9}\sim 10^{-6}\mathrm{Hz}$. The detection of this GW background would have fundamental and far-reaching importance in cosmology and galaxy evolution not accessible by any other means. Precision timing of an array of millisecond pulsars (PTA) is a unique way to detect low frequency GW signal [@Sazhin:1978gy; @Detweiler:1979wn; @Blandford:1984hf; @Foster:1990sl]; [@Blandford:1984hf]. Recently, European Pulsar Timing Array (EPTA) [@Ferdman:2010xq], Parkes Pulsar Timing Array (PPTA) [@Manchester:2012xd] and North American Nanohertz Observatory for Gravitational Waves (NANOGrav) [@Jenet:2009xf], joining together in the International Pulsar Timing Array (IPTA) [@Hobbs:2009yy], are constantly improving their sensitivity in this frequency range, thus provide an important opportunity to get the very first low-frequency GW background detection. As PTA are promoting their upper limits on the GW background from SMBH mergers, several works have also reported their predictions on this GW background based on phenomenological models or simulations [@Jaffe:2002rt; @Sesana:2012ak; @Sesana:2016yky; @Kelley:2016gse]. The predicted characteristic amplitude ($A_{yr^{-1}}$) is roughly in the range of $1\times10^{-16}$ to $5\times10^{-15}$.
The predictions of the characteristic amplitude focus on the isotropic property of the GW background. An isotropic GW background signal is an ideal case when the sources for the GW background has an infinite population, and is expected to produce the famous Hellings and Downs curve in the PTA observation [@Hellings:1983fr]. In realisty, the number of the SMBH binary is always finite, and the GW background signal they generated must has anisotropic components besides the dominated isotropic one. The anisotropic effect on PTA experiments and extension of the Hellings and Downs curve method to analyse anisotropies in the GW background has been developed in [@Cornish:2013aba], [@Mingarelli:2013dsa], [@Taylor:2013esa] and [@Cornish:2014rva]. At the same time, a first constraint on the anisotropy has been obtained with European Pulsar Timing Array data [@Taylor:2015udp]. On the other hand, the galaxy/SMBH clustering may also provide a way to study the SMBH growth and its co-evolution with galaxy. Recent large-scale surveys, such as the Sloan Digital Sky Survey (SDSS), provide observational sample over hundreds of thousands AGNs [@Schneider:2010]. The auto-correlation of AGN and cross-correlation between AGNs and galaxies are studied with large samples [@Shen:2006ti; @Shen:2008ez; @Ross:2009sn; @Coil:2009bi; @Mountrichas:2008jf; @Donoso:2009wd; @Krumpe:2011ra; @Komiya:2013vja; @Shirasaki:2015lu]. The resulted correlations showed a positive dependence on BH mass and radio loudness, while no clear dependence was found on redshift or colour.
In this paper, we are going to utilize the semi-analytic galaxy formation model (SAM) based on sub-halo merger trees built from Millennium simulation [@Springel:2005nw]. We will focus on the Munich model, namely, Guo 2013 [@Guo:2010ap; @Guo:2012fy] and Henriques 2015 [@Henriques:2014sga], to make predictions on the rates at which SMBH form binaries and evolve to coalescence, the distribution of SMBH merger event, as well as the resulted characteristic strain amplitude of GW background and its anisotropic properties. We will also investigate the clustering property of both SMBHs and galaxies, as well as the cross-correlation between them with the mock catalogs generated from Guo 2013 [@Guo:2012fy]. The dependence of resulted correlation on redshift and BH/galaxy mass will be shown.
The rest of the paper is organized as follows. In section \[sec:gw\], we will first give the formula that we will use for the GW background, then briefly introduce the semi-analytical galaxy formation model, and compare the merger event distribution and merger rate derived from Guo 2013 and Henriques 2015. We will give our predictions on the characteristic strain amplitude derived from these two galaxy formation models, and compare them with previous results and PTA upper limits. In section \[sec:correlation\], we will investigate clustering properties of both SMBHs and galaxies, and also the dependence of clustering amplitude on redshift and BH/galaxy mass. Motivated by the results of the clustering property derived in the last section, In section \[sec:anisotropy\], we will study the anisotropy property of the GW background besides the monopole component by calculating the angular power spectrum of the energy density function. Section \[sec:conclusion\] is devoted to summaries and discussions.
stochastic gravitational wave background {#sec:gw}
========================================
Gravitational wave strain
-------------------------
In this subsection we review the formulae for GW background generated by a superposition of GW signal from SMBH binary sources [@Phinney:2001di; @Sesana:2014wta]. Consider the inspiral phase of SMBH binaries, without making any restrictive assumption about their kinetics, such as their semi-major axis and eccentricity evolution, we can write the characteristic strain spectrum $h_c^2$ of the background GW signal generated by the overall population as: \[hcfirst\] h\^2\_c(f)&=&\_0\^dz\_0\^dM\_1\_0\^1dq\
&&h\^2(f\_[,r]{})\_[n=1]{}\^, where we have assumed the progenitor masses are $M_1$ and $M_2$ with $M_1>M_2$. Then $d^4N/dzdM_1dqdt_r$ is the differential cosmological coalescence rate of SMBH binaries per unit redshift ($z$), primary mass ($M_1$), mass ratio ($q=M_2/M_1<1$) and merge time ($t_r$). $dt_r/d\mathrm{ln}f_{\mathrm{K},r}$ is the time spent by the binary at each logarithmic frequency interval, where $f_{\mathrm{K},r}$ is the frequency of Keplerian motion measured in the rest frame of the binary. Together with $dt_r/d\mathrm{ln}f_{\mathrm{K},r}$, $d^4N/dzdM_1dqdt_r$ give the instantaneous population of orbiting binaries in a given logarithmic Keplerian frequency interval per unit redshift, primary mass and mass ratio. $h(f_{\mathrm{K},r})$ denotes the GW strain emitted by a circular binary at a Keplerian rest frame frequency $f_{\mathrm{K},r}$. The averaged strain over source orientations reads (see Thorne 1987 “in Three Hundred Years of Gravitation”, ed. S. W. Hawking, W. Israel, Cambridge University Press) \[hcircular\] h(f\_[,r]{})=(2f\_[,r]{})\^[2/3]{}, where $\mathcal{M}$ is the chirp mass, which is related to the progenitor masses by $\mathcal{M}=(M_1M_2)^{3/5}/(M_1+M_2)^{1/5}$, and $D$ is the luminosity distance to the source. Equation states that the background GW signal is a composition of the GW signal from each SMBH binary sources. Note that having averaged over all the radiation orientation of the source, eq. can be considered equivalently as an isotropic monopole radiation. The function $g(n,e)$ accounts for the fact that the binary radiates GW in the whole spectrum of harmonics $f_{r,n}=nf_{\mathrm{K},r}(n=1,2,...)$. In the circular case that we will consider throughout this paper, $g(n,e)=\delta_{n2}$, i.e. $g(n,e)=1$ for $n=2$, while is zero in other cases. The time duration spent on the logarithmic interval, $dt/d\mathrm{ln}f$, is given by the standard quadrupole formula as below [@Peters:1964zz] \[dtdf\] dt/df=\^[-5/3]{}f\_r\^[-8/3]{}. Besides these, we also have \[dn\] =, where $n$ is the comoving number density of coalescence, and $dV_c$ is the comoving volume shell lying between $z$ and $z+dz$. Plugging eq. , and into , we get the following background GW spectrum for the circular and quadrupole radiation \[hc\] h\_c\^2(f)=dzd\^[5/3]{}. We see that in this case, $h_c\propto f^{-2/3}$, it is therefore customary to write the characteristic strain amplitude in the form $h_c=A(f/f_0)^{-2/3}$, where $A$ is the amplitude of the signal at the reference frequency $f_0$. Observational limits on the GW background are usually given in terms of $A$. Hereafter we denote $A$ with $f_0=1yr^{-1}$ as $A_{yr^{-1}}$.
Semi-analytic model of galaxy formation
---------------------------------------
In this subsection we’ll first briefly introduce the general picture of semi-analytic model. And then, we will go into some details of the black hole self-regulated growth, in particular, the ‘quasar’ and ‘radio’ modes. Finally, we will compare the differences between the simulated results of Guo 2013 and Henriques 2015.
The semi-analytic model (SAM) of galaxy formation treats the baryonic evolution by post-processing cosmological N-body simulations in a way, that makes it possible to explore a wide model and parameter space in a reasonable amount of time. In this work, we utilize the Munich model/`L-Galaxies` code[^2], which is based on the sub-halo merger trees built from the Millennium [@Springel:2005nw]/Millennium-II simulations [@BoylanKolchin:2009nc] (MS/MS-II), and applied to WMAP [@Guo:2010ap; @Guo:2012fy], Planck cosmology [@Henriques:2014sga] . This model is developed based on a series of seminal works [@Springel:2005nw; @Croton:2005fe; @DeLucia:2006szx]. For readers who are interested in more details, we recommend to review papers [@Baugh:2006pf; @Benson:2010ei; @Benson:2010de]. In the following paragraphs, due to the restriction of this topic, we very briefly summarize the general model and highlight the black hole self-regulated growth and relevant feedback mechanisms.
It is commonly believed that, galaxies form at the centers of dark matter halos. They gain stars by formation from interstellar medium (ISM) and by accretion of satellite galaxies. Galactic disc is formed from the materials in ISM. And those materials are replenished both by diffuse infall from the surroundings and by gas from accreted satellite galaxies. There are two main channels for the diffuse infall. One is the direct infall of cold flow from intergalactic medium (IGM), the other is through cooling of the surrounded hot halos. Evolution of each galaxy is driven by the overall baryonic astrophysical complex network rather than a single process. This network includes not only the interactions among the processes mentioned above, but also the interactions of these processes with flows driven by SNe and by active galactic nuclei (AGN).
Due to the complexity of this system, our current understanding of most of these baryonic processes is mainly inspired by the simplified numerical simulations and by the phenomenology from observations. SAM may offer the best means to constrain them empirically using observational data. The baryonic content of galaxies contains five components: stellar bulge, stellar disc, gas disc, hot gas halo as well as ejecta reservoir. These components exchange materials through a variety of processes and gain mass via accreting IGM. The model parameters are estimated by using the observed abundance, structure and clustering of low-redshift galaxies as a function of stellar mass, luminosity and colour.
After reviewing the general picture, now we turn to the black hole self-regulated growth and feedback mechanisms. Following [@Croton:2005fe], we can separate black hole growth into ‘quasar’ mode and ‘radio’ mode. The detailed recipes vary among different versions of the codes, here we take [@Guo:2010ap] as an example.
The ‘quasar’ mode describes the black hole growth during gas-rich mergers. During this process, the major black hole grows both by absorbing the minor and by accreting cold gas. Hence, the final black hole mass can be expressed as $$\begin{aligned}
M_{{\rm bh},f} & = & M_{{\rm bh, maj}}+M_{{\rm bh,min}}+\Delta M_{{\rm bh},Q}\;,\\
\label{eq:bhQ}\Delta M_{{\rm bh},Q} & = & \frac{f_{\rm bh}(M_{\rm min}/M_{\rm maj})M_{\rm cold}}{1+280~{\rm km}~{\rm s}^{-1}/V_{vir}}\;,\end{aligned}$$ where $M_{{\rm bh, maj}}$, $M_{{\rm bh,min}}$, $M_{\rm cold}$, $V_{vir}$, $M_{\rm maj}$ and $M_{\rm min}$ are the black hole mass in the major and minor progenitors, the total cold gas in the two progenitors, virial velocity, the total baryon masses of the major and minor progenitors, respectively. $f_{\rm bh}$ is a free parameter, which is fixed to $0.03$ in order to reproduce the observed local $M_{\rm bh}-M_{\rm bulge}$ relation [@Croton:2005fe]. Both major mergers and gas rich minor mergers contribute significantly to this channel. The feedback in the ‘quasar’ mode is not explicitly written down in SAM. In some sense, we can treat the starburst as some indirect form of feedback in ‘quasar’ mode.
‘Radio’ mode growth is through hot gas accretion on to central black holes. The rate in this mode is modelled as [@Croton:2005fe]
$$\label{eq:Mbhdot}\dot{M}_{{\rm bh}} = \kappa\left(\frac{f_{{\rm hot}}}{0.1}\right) \left(\frac{V_{vir}}{200~{\rm km}~{\rm s}^{-1}}\right)^3 \left(\frac{M_{{\rm bh}}}{10^8 h^{-1}M_{\odot}}\right)M_{\odot}~{\rm yr}^{-1}$$
where $f_{\rm hot}$ is the ratio of hot gas mass to dark matter mass for the main subhalo case, and the ratio within some strip scales for a type-1 galaxy in a satellite subhalo case. The parameter $\kappa$ sets the efficiency of hot gas accretion. This hot gas accretion deposits energy in relativistic jets with $10\%$ efficiency. And this energy is transformed into heat in the atmosphere. In [@Guo:2010ap], the energy input rate is assumed as $$\begin{aligned}
\dot{E}_{{\rm radio}} & = & 0.1\dot{M}_{{\rm bh}}c^2\;.
\label{eq:feedback}\end{aligned}$$ Thus, the effective mass cooling rate is $$\begin{aligned}
\dot{M}_{{\rm cool,eff}} & = & {\rm max}\left[\dot{M}_{{\rm cool}}-\frac{2\dot{E}_{{\rm radio}}}{V_{200c}^2},0\right]\;.
\label{eq:cooling}\end{aligned}$$ It is the cooling onto the disk, the fuel of the star formation. Eq. (\[eq:Mbhdot\]) describes the growth/accretion of BH, and Eq. (\[eq:feedback\]) describes the energy of AGN feedback in reheating the gas. Here reheating gas means to heat up the gas to virial temperature, that is why there is $V_{vir}$ appearing in Eq. (\[eq:Mbhdot\]). Basically, $\dot{M}_{{\rm cool,eff}}$ describes the amount of gas that could have been cooled onto the disk if there were no AGN feedback. In the case of AGN feedback, $2\dot E_{{\rm radio}}/V^2_{200c}$ amount of the cooling gas are reheated, so that the cooling rate is reduced according to Eq. (\[eq:cooling\]).
In [@Henriques:2014sga] version, eq. (\[eq:bhQ\]) and (\[eq:Mbhdot\]) are replaced with $$\begin{aligned}
\Delta M_{{\rm bh},Q} & = & \frac{f_{\rm bh}(M_{\rm min}/M_{\rm maj})M_{\rm cold}}{1+(V_{\rm bh}/V_{200c})^2}\;,\\
\dot{M}_{{\rm bh}} & = & k_{{\rm AGN}}\left(\frac{M_{{\rm hot}}}{10^{11}M_{\odot}}\right)\left(\frac{M_{{\rm bh}}}{10^8M_{\odot}}\right)\;,\end{aligned}$$ and other equations keep the same.
After running SAM code, we get the whole galaxy merge history through the simulated time range. In the following, we will compare the simulation result of Guo 2013 and Henriques 2015 by mass-redshift distribution of SMBH merger events and galaxy merger rate. The SMBH merger history is related to the galaxy merger history assuming the two SMBHs will merge as soon as their host galaxies merged. Then both SMBH merger mass distribution and merger rate can be extracted from the simulated SMBH merger history.
![The 2D contour plot the distribution of $d\log N(z,\mathcal{M})/(dzd\log (\mathcal{M}/M_{\odot}))$, namely, the logarithmic number of SMBH merger events from the whole box volume $V=[500\mathrm{Mpc}/h]^3$ per unit redshift and logarithmic mass-interval. Top panel: result from Guo 2013. Bottom panel: result from Henriques 2015.[]{data-label="nummerger"}](nummergernewbin1log.png "fig:"){width="50.00000%"} ![The 2D contour plot the distribution of $d\log N(z,\mathcal{M})/(dzd\log (\mathcal{M}/M_{\odot}))$, namely, the logarithmic number of SMBH merger events from the whole box volume $V=[500\mathrm{Mpc}/h]^3$ per unit redshift and logarithmic mass-interval. Top panel: result from Guo 2013. Bottom panel: result from Henriques 2015.[]{data-label="nummerger"}](nummergernewbin2log.png "fig:"){width="50.00000%"}
In Fig. \[nummerger\] the logarithmic number of SMBH merger events from the whole box volume $V=[500\mathrm{Mpc}/h]^3$ per unit redshift and logarithmic mass-interval, i.e. $d\log N(z,\mathcal{M})/(d zd\log (\mathcal{M}/M_{\odot}))$, on redshift and logarithmic chirp mass plane are shown. We see that in both cases, SMBHs merge, on average, much more frequently in the low redshift region than in the high one. Its maximum locates around $z=0$, $\log\left(\mathcal{M}/M_{\odot}\right)=7$ (Guo 2013) and $\log\left(\mathcal{M}/M_{\odot}\right)=6$ (Henriques 2015), respectively. Then the differential event number gradually decrease to less than $10^{0.8}$ in the boundary of the plane, where either redshift is high or chirp mass is away from the maximum value. Furthermore, Fig. \[nummerger\] shows that SMBHs merger is more frequent in Guo 2013 than in Henriques 2015. The area of the maximum, i.e., the area where the differential number of merger events reaches about $10^6$, is larger in Guo 2013 than in Henriques 2015. Finally, there are more massive binaries, i.e. binaries with chirp mass larger than $10^9 M_{\odot}$ in Guo 2013 than in Henriques 2015, and the mass of the progenitor SMBHs is, on average, smaller in Henriques 2015. To sum up, systematically, the SMBHs merger is less massive and less frequent in Henriques 2015 than in Guo 2013.
In Fig. \[mergerate\], we plot the merger rate for galaxies as a function of redshift, for $q>1/4$ and descendant galaxies with stellar mass larger than $10^{10}M_{\odot}$ for these two models. Here $q$ refers to the stellar mass ratio between two progenitors as before. The merger rate is defined as merger rate=n\_[r]{}/(n\_[g]{}t), where $n_r$ is the number of merger remanent at certain redshift, $n_g$ is the total number of galaxies at the same redshift, and $\Delta t$ is the comoving time step between different redshift snapshots. We can see that the merger rate in Guo 2013 is larger than that in Henriques 2015 in the whole redshift range. The enhanced factor is roughly $1.1$ around $z=0$, and reaches $2.5$ at $z=4$. In the following, we will see that the differences in the merger rate in Guo 2013 and Henriques 2015 will result in different predictions on $h_c$ according to eq. .
We briefly mention here the possible reasons for the merger rate differences between the two models. Compared to Guo 2013 and previous models, Henriques 2015 changed the time scale of reincorporation of gas ejected by supernova-driven winds, to make the galaxy massfunction and redshift dependence consistent with observations at higher redshift. The change in this process resulted in a reduction of the number and stellar mass of galaxies, and thus a smaller cross section for galaxy-galaxy collisions. This maybe the main reason for the lower merger rate of Henriques 2015. But baryonic processes considered in SAM are complex and all coupled, many parameters in Henriques 2015 has been changed, a detailed description of Henriques 2015 and Guo 2013 and their behavior in galaxy growth can be found in [@Henriques:2014sga], [@Guo:2007wv] and [@Guo:2010ap]. A comparison between Henriques 2015 and De Lucia & Blaizot 2007 [@DeLucia:2006szx] can be found in [@Vulcani:2015fka]. The specific reasons why the merger rates of Henriques 2015 and Guo 2013 differ so significantly is still under discussion, and will be discussed in following papers.
![Merger rate, defined as $n_{r}/(n_{g}\Delta t)$, for descendant galaxies with stellar mass larger than $10^{10}M_{\odot}$ and mass ratio greater than $1/4$ extracted from Guo 2013 and Henriques 2015 as a function of redshift. Here $n_r$ is the number of merge remnant galaxies, and $n_g$ is the total number of galaxies at a certain redshift. $\Delta t$ is the elapsed comvoing time between two redshift snapshots.[]{data-label="mergerate"}](mergerate.png){width="50.00000%"}
Results for the gravitational wave background amplitude
-------------------------------------------------------
After inserting the results of $d^2n/dzd\mathcal{M}$ (can be derived from the events distribution shown in Fig. \[nummerger\]) into eq., we get the characteristic strain amplitude $A_{yr^{-1}}$ for the two galaxy formation models. Our results are $A_{yr^{-1}}=5.00\times10^{-16}$ (Guo 2013) and $A_{yr^{-1}}=9.42\times10^{-17}$ (Henriques 2015), respectively. In Fig. \[hcpic\], we plot our results together with several predictions made by previous papers, including the predictions derived in [@Jaffe:2002rt], which used phenomenological galaxy merger rate from CNOC2 and CFGRS redshift surveys and $M_{bh}-$spheroid mass relationship; [@Wyithe:2002ep], which used semi-analytic calculation of the merger rate history at all redshifts, and phenomenological $M_{bh}-$velocity dispersion relationship; [@Kelley:2016gse], which used coevolved populations of SMBH and galaxies from hydrodynamic, cosmological simulations; and [@Sesana:2016yky], which, as in [@Sesana:2012ak], utilised several observed galaxy mass functions and pair counts to phenomenological SMBH-host relations, and assuming merger timescale prescriptions derived by detailed hydrodynamical simulations of galaxy mergers, but selection bias is considered in SMBH-galaxy mass relationship. We summerize these predictions for characteristic strain amplitude in Table. \[table1\], with a brief summary of the methods they used.
Several recent pulsar timing array (PTA) upper limits (summarized in Table. \[table2\]), such as EPTA [@Lentati:2015qwp], NANOGrav [@Arzoumanian:2015liz], PPTA [@Shannon:2015ect], are also included for reference. First of all, as shown in Fig. \[hcpic\], both our results are still below the most stringent observational upper limits. Secondly, we see that the characteristic strain amplitude derived from Guo 2013 (upper red curve) is well consistent with most of previous results, while $h_c$ from Henriques 2015 (lower red curve) is a little bit lower than the result from [@Jaffe:2002rt] (black dashed curve), and is the lowest of all the predictions shown in the figure. The low prediction of GW strain amplitude in Henriques 2015 is a result of its low merger rate and low chirp mass distribution. And more importantly, our results reveal the fact that *difference in the galaxy merger rate between Guo 2013 and Henriques 2015 (shown in Fig. \[mergerate\]), results in a factor $5$ deviation in the GW strain amplitude (shown in Fig. \[hcpic\]).*
![The characteristic GW strain amplitude computed in this work. For comparison, we also include the existed results in the literatures. The upper red curve is obtained from Guo 2013 model, and the lower red curve is from Henriques 2015. The upper limits on the GW background from PTAs are also shown. The black vertical line highlights the reference frequency $f_0=1yr^{-1}$.[]{data-label="hcpic"}](strain2.png){width="50.00000%"}
$\log{A_{yr^{-1}}}$ Methods
------------------- --------------------- -----------------------------------
This work -15.30 SAM, Guo 2013
This work -16.05 SAM, Henriques 2015
[@Kelley:2016gse] -15.15 Cosmo-Hydro
[@Sesana:2016yky] $-15.4\pm0.4$ phenomenological, bias considered
[@Sesana:2012ak] $-15.1\pm0.3$ phenomenological
[@Wyithe:2002ep] -14.3 SAM+phenomenological
[@Jaffe:2002rt] -16 phenomenological
PTA $A_{yr^{-1}}$ $A_{f_{0}}$ $f_0[yr^{-1}]$
--------------------------------- --------------------- --------------------- ----------------
EPTA [@Lentati:2015qwp] $3.0\times10^{-15}$ $1.1\times10^{-14}$ 0.16
NANOGrav [@Arzoumanian:2015liz] $1.5\times10^{-15}$ $4.1\times10^{-15}$ 0.22
PPTA [@Shannon:2015ect] $1.0\times10^{-15}$ $2.9\times10^{-15}$ 0.2
IPTA [@Verbiest:2016] $1.5\times10^{-15}$ $ - $ $ - $
Black hole clustering {#sec:correlation}
=====================
![The auto-2PCFs of SMBHs for different redshift-mass bins. Top panel: 2PCFs at different redshifts for $5<\log[M_{bh}/M_{\odot}]<6$. Middle panel: 2PCFs at different redshifts for $6<\log[M_{bh}/M_{\odot}]<7$. Bottom panel: 2PCFs at different redshifts for $\log[M_{bh}/M_{\odot}]>7$. Jackknife errors are also shown in the figures.[]{data-label="correlationBH"}](correlationBH56r2.png "fig:"){width="50.00000%"} ![The auto-2PCFs of SMBHs for different redshift-mass bins. Top panel: 2PCFs at different redshifts for $5<\log[M_{bh}/M_{\odot}]<6$. Middle panel: 2PCFs at different redshifts for $6<\log[M_{bh}/M_{\odot}]<7$. Bottom panel: 2PCFs at different redshifts for $\log[M_{bh}/M_{\odot}]>7$. Jackknife errors are also shown in the figures.[]{data-label="correlationBH"}](correlationBH67r2.png "fig:"){width="50.00000%"} ![The auto-2PCFs of SMBHs for different redshift-mass bins. Top panel: 2PCFs at different redshifts for $5<\log[M_{bh}/M_{\odot}]<6$. Middle panel: 2PCFs at different redshifts for $6<\log[M_{bh}/M_{\odot}]<7$. Bottom panel: 2PCFs at different redshifts for $\log[M_{bh}/M_{\odot}]>7$. Jackknife errors are also shown in the figures.[]{data-label="correlationBH"}](correlationBH70r2.png "fig:"){width="50.00000%"}
![The auto-2PCFs of galaxies for different redshift-mass bins. Top panel: 2PCFs at different redshifts for $8<\log[M_{g}/M_{\odot}]<9$. Middle panel: 2PCFs at different redshifts for $9<\log[M_{g}/M_{\odot}]<10$. Bottom panel: 2PCFs at different redshifts for $\log[M_{g}/M_{\odot}]>10$.[]{data-label="correlationG"}](correlationG89r2.png "fig:"){width="50.00000%"} ![The auto-2PCFs of galaxies for different redshift-mass bins. Top panel: 2PCFs at different redshifts for $8<\log[M_{g}/M_{\odot}]<9$. Middle panel: 2PCFs at different redshifts for $9<\log[M_{g}/M_{\odot}]<10$. Bottom panel: 2PCFs at different redshifts for $\log[M_{g}/M_{\odot}]>10$.[]{data-label="correlationG"}](correlationG90r2.png "fig:"){width="50.00000%"} ![The auto-2PCFs of galaxies for different redshift-mass bins. Top panel: 2PCFs at different redshifts for $8<\log[M_{g}/M_{\odot}]<9$. Middle panel: 2PCFs at different redshifts for $9<\log[M_{g}/M_{\odot}]<10$. Bottom panel: 2PCFs at different redshifts for $\log[M_{g}/M_{\odot}]>10$.[]{data-label="correlationG"}](correlationG00r2.png "fig:"){width="50.00000%"}
![The cross-2PCFs between SMBHs and galaxies at $z=0$. Top panel: the cross-2PCFs of different galaxy mass bins with $5<\log[M_{bh}/M_{\odot}]<6$. Middle panel: the cross-2PCFs of different galaxy mass bins with $6<\log[M_{bh}/M_{\odot}]<7$. Bottom panel: the cross-2PCFs of different galaxy mass bins with $\log[M_{bh}/M_{\odot}]>7$. []{data-label="crosscorrelation"}](crosscorrelation1.png "fig:"){width="50.00000%"} ![The cross-2PCFs between SMBHs and galaxies at $z=0$. Top panel: the cross-2PCFs of different galaxy mass bins with $5<\log[M_{bh}/M_{\odot}]<6$. Middle panel: the cross-2PCFs of different galaxy mass bins with $6<\log[M_{bh}/M_{\odot}]<7$. Bottom panel: the cross-2PCFs of different galaxy mass bins with $\log[M_{bh}/M_{\odot}]>7$. []{data-label="crosscorrelation"}](crosscorrelation2.png "fig:"){width="50.00000%"} ![The cross-2PCFs between SMBHs and galaxies at $z=0$. Top panel: the cross-2PCFs of different galaxy mass bins with $5<\log[M_{bh}/M_{\odot}]<6$. Middle panel: the cross-2PCFs of different galaxy mass bins with $6<\log[M_{bh}/M_{\odot}]<7$. Bottom panel: the cross-2PCFs of different galaxy mass bins with $\log[M_{bh}/M_{\odot}]>7$. []{data-label="crosscorrelation"}](crosscorrelation3.png "fig:"){width="50.00000%"}
It is generally believed that gas accretion onto SMBHs in the galaxy center is the energy source of AGN. In order to understand the relationship between growth of SMBHs and their surrounding environment, it is important to investigate the clustering properties of both SMBHs and their host galaxies. The auto-correlation function of AGNs was studied by using the SDSS sample [@Shen:2006ti; @Shen:2008ez; @Ross:2009sn]. They found that while no significant evolution of SDSS quasar clustering amplitude can be seen for $z<2.5$, clustering strength does increase at higher redshift. [@Shen:2008ez] further studied the dependence of the two-point auto-correlation function of quasars on luminosity, BH mass, colour, and radio loudness, and found positive dependence on radio-loudness, weak or no dependence on virial BH mass for $z<2.5$. The clustering property of intermediate redshift quasars using the final SDSS III-BOSS sample was investigated in [@Eftekharzadeh:2015ywa]. The clustering of galaxies around AGNs in the areas of deep surveys was also investigated by [@Coil:2009bi] and [@Mountrichas:2008jf].
Recently, cross-correlation between AGNs and galaxies was studied using large samples [@Donoso:2009wd; @Krumpe:2011ra]. The results show that that radio-loud AGNs are clustered more strongly than radio-quiet ones, while no significant difference was found between X-ray selected and optically selected broad-line AGNs. [@Donoso:2009wd] found a positive dependence of the cross-correlation amplitude on stellar mass $M_*$, but within a narrow range of stellar mass ($10^{11}M_{\odot}<M_*<10^{12}M_{\odot}$). The mass dependence of cross-correlation between AGN and galaxies is further investigated in [@Komiya:2013vja] with $9394$ AGNs for $z = 0.1-1$ over a wide BH mass ($10^{6}M_{\odot}<M_*<10^{10}M_{\odot}$). There is an indication of an increasing trend of correlation with BH mass for $M_{bh}>10^8M_{\odot}$, while no BH mass dependence for $M_{bh}\lesssim10^8M_{\odot}$. [@Shirasaki:2015lu] studied the cross-correlation with updated UKIDSS catalog and reconfirmed the findings of [@Komiya:2013vja] that the clustering of galaxies around AGNs with the most massive SMBH is larger than those with less massive SMBH, and AGN bias was derived for each BH mass group.
On the simulation side, the clustering property of quasars or SMBH are also investigated in several works. [@Oogi:2015cxi] studied the clustering properties of quasars via quasar bias. [@DeGraf:2016bbt] also studied the clustering properties of SMBHs for Illustris simulation.
In order to justify/falsify the galaxy formation model, here we investigate the clustering properties of both SMBHs and galaxies, and corresponding cross-correlations between them. To do this, we use the mock catalog produced by Guo 2013, which allows us to study their dependence over a wide range of redshift and SMBH/galaxy mass with large samples. We selected four redshift snapshots which contains a total of 8668809 SMBHs and 51538704 galaxies lying in a box size of $[500\mathrm{Mpc}/h]^3$ created by applying the Guo 2013 SAM model to the Millennium simulation. The Millennium simulation was created in a cosmology of $(\Omega_b, \Omega_m, \sigma_8, h) = (0.0456, 0.273, 0.809, 0.704)$ using $2160^3$ particles. For the data completeness, the redshift and SMBH/galaxy mass range we are interested in are $0<z<2.07$, $M_{bh}>10^6M_{\odot}$ and $M_{g}>10^8M_{\odot}$, where $M_{bh}$ is the mass of SMBHs, and $M_g$ is the stellar mass of galaxies. We abandoned the data in the lower mass and higher redshift regime because of the limited resolution and small catalog size, respectively.
The galaxy and SMBH clustering is adequately described by the spatially isotropic two-point correlation function (2PCF), which is computed by the excess of data-data number counts relative to those of random pairs [@Landy:1993yu] \[correlation\] (r)=, where $DD(r)$ is the data-data number counts at different clustering scales, and $DR(r)$, $RR(r)$ being the data-random, random-random number counts, respectively.
To calculate the errors on 2PCFs, we adopt the ‘delete one jackknife’ method. We divide the full galaxy/SMBH sample into 125 sub-boxes, and calculate the 2PCF for $125$ sub-samples consisting all but the $k$-th sub-box. This allows us to construct a jackknife defined covariance matrix C(r\_i,r\_j)=\_[k=1]{}\^[125]{}\[(r\_i)-\_k(r\_i)\]\[(r\_j)-\_k(r\_j)\], where $\xi_k(r)$ refers to the value of the correlation obtained by omitting the $k^{\mathrm{th}}$ sub-box, $\overline{\xi}(r_i)$ is the average correlation value for all the subsamples, and $i$, $j$ refers to the $i^{\mathrm{th}}$ and $j^{\mathrm{th}}$ bins, respectively. The binned errors $\sigma_i$ can be obtained from the diagonal elements of the covariance metric as \^2\_i=C\_[i,i]{}.
To perform the calculation of correlation, we divide the SMBH and galaxy simulation sample into $12$ redshift-mass groups. For redshift, we choose $4$ snapshots (shown in eq. \[eq:zbin\]). SMBH and galaxy masses are divided into $3$ bins, shown in eq. (\[eq:bhmassbin\]) and (\[eq:gmassbin\]), respectively. z=0, 0.509, 1.079, 2.070.\[eq:zbin\]
5<<6, 6<<7, >7.\
\[eq:bhmassbin\]
8<<9, 9<<10, >10.\
\[eq:gmassbin\]
The auto-2PCFs are calculated in all redshift bins both for SMBHs and galaxies. The corresponding results are shown in Fig. \[correlationBH\] (SMBH) and Fig. \[correlationG\] (galaxy). The $z=2.07$ 2PCF in the first panel of Fig. \[correlationBH\] has larger error bars due to the limited SMBH samplings in the corresponding mass and redshift range. Comparing the 2PCFs in different plots with the same redshift and same population (SMBHs or galaxies), we see that 2PCFs have a positive dependence on mass for both SMBHs and galaxies, namely, SMBHs and galaxies are more correlated if they are more massive. This is consistent with the structure formation scenario, which states that more massive objects are formed in higher dense regions and thus have stronger clustering strength.
Another behavior is that the 2PCFs seems to increase faster with mass at higher redshifts than at lower redshifts for both SMBHs and galaxies. In particular, in the most massive cases, *i.e.* the last plot of Fig. \[correlationG\], we see that the correlation of the highest redshift bin has run over that of the lower redshift ones. The reason is that, at higher redshift galaxies/SMBHs as a whole population are less massive, by imposing a same mass cut at all redshifts, at high redshifts we are systematically selecting more biased objects, who reside in larger density contrast regions and thus have larger clustering strength.
In most panels of Fig. \[correlationBH\] and Fig. \[correlationG\], we see a clear redshift dependence of the 2PCF amplitude, namely, the clustering of both SMBHs and galaxies is enhanced at lower redshifts. This is due to the gravitational growth of structure with the evolution of time. Interestingly, this behavior is not reported in previous papers with SDSS samples [@Shen:2006ti; @Shen:2008ez; @Ross:2009sn], where they didn’t find significant evolution or just a weak positive redshift dependence of clustering for $z<2.5$ . This should result from the difference between the properties of the two samples. The SDSS sample analyzed in [@Ross:2009sn] is a flux limited sample, so the bias of objects rapidly increases at higher redshifts (where only very bright objects can be observed); the sparseness of the sample also weakens the power of statistics and makes the detection of redshift evolution relatively difficult. The samples considered in this analysis are denser and have constant mass cut at all redshifts. As a result, we can clearly detect the redshift evolution of clustering strength. In all, the combination of structure growth and selection effect determines how the amplitude of correlation evolves with redshift.
As for the shape of correlation functions, they are not evolving a lot at $z>1$, but starts to have larger slope at lower redshifts. This is expectable, since the non-linear growth of structure at later epoch results in enhancement of clustering in relatively small scales. Due to the same reason, correlation functions of more massive objects have larger slope than less massive objects results (more massive objects are more biased and they experienced more non-linear growth of structure). Comparing the 2PCFs between SMBHs and galaxies, since SMBHs reside in galaxies, the apparent clustering of SMBHs is actually a result of galaxy clustering, so it is not surprising that the amplitude and shape of SMBH correlation functions are similar to those of galaxies. This analysis shows that, very roughly, SMBHs and galaxies, with galaxy mass $10^2\sim10^3$ larger than SMBH mass, have similar pattern of clustering (strength and scale dependence).
The results for cross-correlation between different SMBH and galaxy mass bins at $z=0$ are shown in Fig. \[crosscorrelation\]. We calculate the cross-correlation between each of the three mass bins of SMBH and galaxy, so there are nine 2PCFs in total. And each panel in Fig. \[crosscorrelation\] is for a fixed SMBH mass. We see in each plot, the cross-correlations become larger for higher galaxy stellar mass. Comparing the 2PCFs in different plots with the same $M_g$, we also find that the correlation amplitude become larger for higher SMBH mass. That is, in summary, the cross-correlation between SMBH and galaxies has a positive dependence on both SMBH and galaxy mass, which is consistent with previous results from observed AGN and galaxy samples [@Donoso:2009wd; @Komiya:2013vja; @Shirasaki:2015lu].
Anisotropy of the gravitational wave background {#sec:anisotropy}
===============================================
The spatial inhomogeneities of the SMBH clustering should produce some amount of anisotropies in the GW background signal. In this section we will investigate the anisotropic property of the GW background generated by binary SMBH sources with a mock merger event catalog produced by Guo 2013. Anisotropy can be investigated by decomposing the energy density of the GW background [@Taylor:2013esa; @Kuroyanagi:2016ugi], which now is a function of the angular coordinate $\hat\Omega$ on the 2D sphere, in terms of the spherical harmonic functions as ()=\^\_[l=0]{}\^[l]{}\_[m=-l]{}c\_[lm]{}Y\_[lm]{}(). \[rhodecompose\] So the spherical harmonic coefficients $c_{lm}$ can be calculated by integrating the GW background signal over all directions c\_[lm]{}=d()Y\_[lm]{}(). \[clm\] The statistical isotropic angular power spectrum of GW background density reads C\_l=\_m|c\_[lm]{}|\^2/(2l+1). \[spectrum\] In the realistic case for the GW background generated by SMBH binaries, the number of sources is always discrete and finite, so eq. can be replaced by a discrete form: c\_[lm]{}=\^N\_[i=1]{}\_iY\_[lm]{}(\_i) \[clmdiscrete\] where $\rho_i$ is the energy density generated by the $i$th source, and is given by =\[f(1+z)\]\^[10/3]{}. \[rho\]
Our mock catalog from Guo 2013 contains 8426 galaxy merger events with binary SMBHs in total, each merger event carries information about their location and the mass of the progenitor SMBHs. To calculate the energy density from eq. , we still need to assign frequency to each of the SMBH binaries inside the merging galaxies. We assume that the seperation between the two black holes of the SMBH binaries lies between $10^{-2}\mathrm{pc}$ and their inner most stable orbit, which corresponds to minimum and maximum frequency, respectively. We assign frequency to each SMBH binary according to a normalized probability that is proportional to eq. , i.e., \[probability\] P(f)=C\^[-5/3]{}f\_r\^[-8/3]{}, where C is a normalization constant. This assignment has a meaning that the probability for a logarithmic frequency bin, that a SMBH binary may lie in, is proportional to the evolution time the binary spends in that frequency bin. Now with frequency, progenitor mass and location, we can calculate energy density for each SMBH binary. We plot a skymap of all the SMBH binary sources from the mock catalog in Fig. \[skymap\], the relative size of each source is a indication of the GW energy density. We also plot an energy flux distribution histogram for all the sources in Fig. \[histogram\], where the GW energy flux $\mathcal{F}$ is related to the energy density by $\mathcal{F}=c\rho$, and $c$ is the speed of light. As can be seen from the skymap and the histogram, there’re several bright sources shine upon the majority dim ones, and may give a major contribution to the result of the angular power spectrum.
![The skymap of all the SMBH binary sources from the mock merger event catalog of Guo 2013. 8426 SMBH binaries is contained in the catalog in total. Frequency is assigned to each binary according to $P(\mathrm{ln}f)\propto\mathcal{M}^{-5/3}f_r^{-8/3}$, and the relative size of each source is a indication of the GW energy density.[]{data-label="skymap"}](mapbig2.png){width="50.00000%"}
![The energy flux distribution histogram for all the GW sources from the mock merger event catalog of Guo 2013. Here $\mathcal{F}$ is the energy flux of the GWs from each source, and $n$ is the number of sources that fall into a corresponding logarithm flux bin.[]{data-label="histogram"}](fluxbin.png){width="50.00000%"}
![Our result for the anisotropic power spectrum normalized by the monopole component, i.e., $C_l/C_0$ up to $l=10$. The results are sorted into different frequency bins from $f=1.51\times 10^{-9}\mathrm{Hz}$ to $f=2.26\times 10^{-7}\mathrm{Hz}$.[]{data-label="powerspectrum"}](clplot2more.png){width="50.00000%"}
Now, we can calculate the angular power spectrum for the SMBH binaries that fall into the PTA band. We calculate the first ten multiples of the angular power spectrum for each frequency bin, and our result is shown in Fig. \[powerspectrum\]. From the plot we see that, in the whole frequency range, $C_l$ can have about 10% to 60% the power of the monopole component. This is a considerable level of anisotropy, and may come from the several bright sources which can be seen from Fig. \[skymap\]. For $l=1$, $C_1$ is in the range of 40% to 70% the amount of the monopole component. For higher multiples, $C_l$ drop quickly for in low frequency range $f<8.28\times10^{-8}\mathrm{Hz}$. Their amplitudes are of order $\mathcal{O}(10\%)$ w.r.t. the power of the monopole component. As of the high frequency modes, such as $f=1.36\times10^{-7}\mathrm{Hz}$ and $f=2.26\times10^{-7}\mathrm{Hz}$, the higher multiples are of order $\mathcal{O}(60\%)$. This is because the number of low frequency SMBH binaries are much more than those in high frequency band. Hence, the spatial distribution anisotropy is obviously larger for the small population. For example, in frequency bin above $2.26\times10^{-7}\mathrm{Hz}$, there’s only $18$ merger events.
Conclusions {#sec:conclusion}
===========
In this paper, we investigated the co-evolution of supermassive black holes (SMBHs) with galaxies by studying the stochastic gravitational wave background radiation generated by SMBH merger and the SMBH/galaxy clustering, namely, the two point auto- and cross-correlation functions, by using the mock catalogs generated by the semi-analytic model (SAM) of galaxy formation. For SAM, we utilize the Munich model, which is based on the sub-halo merger trees built from the Millennium/Millennium-II simulations, and applied to WMAP (Guo 2013), Planck cosmologies (Henriques 2015) .
For the stochastic gravitational wave background, we firstly compared the mass-redshift distribution of SMBH merger events and galaxy merger rate for Guo 2013 and Henriques 2015 models. We found that SMBHs merger is less massive and less efficient in Henriques 2015 than in Guo 2013 and galaxy merger rate for Guo 2013 is higher than that for Henriques 2015 in the whole simulated redshift range. Quantitatively, the maximum of differential event number of SMBH merger locates around $z=0.5$, $\log\left(\mathcal{M}/M_{\odot}\right)=7$ (Guo 2013) and $\log\left(\mathcal{M}/M_{\odot}\right)=5.5$ (Henriques 2015), respectively. And it reaches above $2100$ in Guo 2013, while it is below 2000 in Henriques 2015. As for the galaxy merger rate, with stellar mass ratio $q>1/4$ and stellar mass greater than $10^{10}M_{\odot}$, Guo 2013 model is systematically higher than Henriques 2015. The enhanced factor is roughly $1.1$ around $z=0$, and reaches $2.5$ at $z=4$.
We then predicted the characteristic strain amplitude of GW background for Guo 2013 and Henriques 2015 model to be $A_{yr^{-1}}=5.00\times10^{-16}$ and $A_{yr^{-1}}=9.42\times10^{-17}$, respectively. We shall emphasize that, the GW amplitude is very sensitive to the galaxy merger rate. The difference in galaxy merger rate between Guo 2013 and Henriques 2015 (shown in Fig. \[mergerate\]), results in a factor $5$ deviation in the GW strain amplitude (shown in Fig. \[hcpic\]). Furthermore, we compared our result with those in literatures with different methods. We found that $h_c$ from Guo 2013 is more closer to other studies while Henriques 2015 model gives the lowest prediction on the GW signal.
We now briefly discuss the simplifications we have used. In calculating the GW background amplitude, we have neglected environmental effects. We assumed that the orbits of the binary SMBHs are all circular and they emit GWs through quadrupole formula. Furthermore, we have assumed that binary SMBHs merge only through GW emission and with 100% efficiency. In the reality cases, merger processes are more complex than the situations we have considered. First of all, in realistic situations, the interactions between binary SMBHs and gas or stellar environment may increase the binaries’ eccentricity and the binary SMBHs emitting GWs in the PTA band may have non-negligible eccentric orbits [@Armitage:2005xq; @Matsubayashi:2005eg; @Berentzen:2008yw; @Sesana:2010qb; @Preto:2011gu; @Khan:2011gi], and the GW background spectrum will be changed and deviate from the simple $-2/3$ case here [@Enoki:2007sl; @Ravi:2014aha; @Sesana:2014wta]. On the other hand, binary SMBHs inside galaxies generally undergo dynamical friction, loss-cone stellar scattering, viscous drag to reach the GW domain regime. Among these processes, the efficiency of loss-cone scattering is still quite uncertain, as is pointed out by [@Merritt:2013vk], and generated the famous “last parsec problem" [@Milosavljevic:2002ht; @Merritt:2004gc]. We have assumed binary SMBHs coalesce simultaneously when host galaxies merge on kpc scale, for simplicity. Although our treatment is simpler than some of the studies in the literature, but our approach is fast and flexible, hence, we can update our result once the more reliable SAM model are presented.
For clusterings, we calculated the spatially isotropic two point auto- and cross-correlation functions (2PCFs) for both SMBHs and galaxies by using the mock catalogs generated from Guo 2013 model. We studied their dependence through a wide range of redshift as well as black hole and galaxy stellar mass. We showed that all 2PCFs have positive dependence on both SMBH and galaxy mass. And there exist a significant time evolution in both the SMBH and galaxy 2PCFs due to the gravitational growth of structure with the evolution of time, namely, the clustering effect is enhanced at lower redshift. Interestingly, this behavior is not reported in the previous AGN samples from SDSS survey, which should result from the increase of bias objects at higher redshifts and the sparseness of their sample. As for the shape of the 2PCFs, we found they always have larger slope at lower redshifts due to non-linear growth of structure and enhancement of clustering in relatively small scales at later epoch. We also showed that roughly, SMBHs and galaxies, with galaxy mass $10^2\sim 10^3$ larger than SMBH mass, have similar pattern of clustering.
For the GW background anisotropy, we calculated the angular power spectrum up to $l=10$ with a mock merger event catalog form Guo 2013. The catalog contains $8426$ SMBH binaries in total. We assign frequency to the SMBH binaries by assuming a probability proportional to their evolution time in the corresponding logarithmic frequency bin. We found a considerable amount of anisotropy. The corresponding angular power spectrum of the first ten multiples are about $10\%$ to $60\%$ w.r.t. the monopole component in the whole frequency range of $1.51\times10^{-9}\mathrm{Hz}$ to $2.26\times10^{-7}\mathrm{Hz}$. Several bright sources of the catalog offered the major contribution to this level of anisotropy.
Environmental effects may also affect the results of anisotropy. Recall that we have used a probability proportional to eq. , which is a purely quadrupole formula. According to this probability, a binary has much a greater chance to be found in the low frequency regime than in the high one. Environmental effects, if taken into consideration, would accelerate the merging process of the two SMBHs before the GW dominated era, and suppress the evolution time difference between frequency bands. Thus the opportunity to have a binary in the high frequency regime shall increase. The corresponding result for the angular power spectrum with high frequency should be suppressed due to a larger binary population.
In this paper, we studied the GW background signal (both isotropic and anisotropic components) and clustering status of SMBHs/galaxies, respectively. Since the clusterings of SMBHs and GW radiation are both the results of galaxy formations, these two aspects are tightly related with each other. Our paper is a tentative trial where these two aspects are investigated. In the future, we will consider the possible relationship between the GW background anisotropic signal and the galaxy/SMBH clustering status, or the use of the anisotropic component to distinguish the different scenario of production of GW background. We leave these for future studies.
As the PTA experiments are constantly improving their sensitivity, the first detection of background GW signal may not be far in the future. The recent large sky surveys are also aiming at broader dynamical ranges and larger sample sizes. All these experimental improvements are providing us new ways to study the SMBH growth and its co-evolution with galaxy. It is hopefully that our results may offer clue to the theoretical progress in relative fields. On the other hand, the comparison of our results with observations may help improve the galaxy formation model building.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Qi Guo for useful discussion. QY and BH are supported by the Beijing Normal University Grant under the reference No. 312232102 and by the National Natural Science Foundation of China Grants No. 210100088. BH is also partially supported by the Chinese National Youth Thousand Talents Program under the reference No. 110532102 and the Fundamental Research Funds for the Central Universities under the reference No. 310421107.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: <http://galformod.mpa-garching.mpg.de/public/LGalaxies/>
|
---
abstract: 'The satisfiability problem for SPARQL patterns is undecidable in general, since SPARQL 1.0 can express the relational algebra. The goal of this paper is to delineate the boundary of decidability of satisfiability in terms of the constraints allowed in filter conditions. The classes of constraints considered are bound-constraints, negated bound-constraints, equalities, nonequalities, constant-equalities, and constant-nonequalities. The main result of the paper can be summarized by saying that, as soon as inconsistent filter conditions can be formed, satisfiability is undecidable. The key insight in each case is to find a way to emulate the set difference operation. Undecidability can then be obtained from a known undecidability result for the algebra of binary relations with union, composition, and set difference. When no inconsistent filter conditions can be formed, satisfiability is decidable by syntactic checks on bound variables and on the use of literals. Although the problem is shown to be NP-complete, it is experimentally shown that the checks can be implemented efficiently in practice. The paper also points out that satisfiability for the so-called ‘well-designed’ patterns can be decided by a check on bound variables and a check for inconsistent filter conditions.'
author:
- |
Xiaowang Zhang\
School of Computer Science and Technology\
Tianjin University[^1]
- |
Jan Van den Bussche\
Universiteit Hasselt[^2]
- |
François Picalausa\
[[email protected]]([email protected])
bibliography:
- 'database.bib'
title: |
On the satisfiability problem\
for SPARQL patterns[^3]
---
Introduction
============
The Resource Description Framework [@RDFprimer] is a popular data model for information in the Web. RDF represents information in the form of directed, labeled graphs. The standard query language for RDF data is SPARQL [@sparql1.1]. The current version 1.1 of SPARQL extends SPARQL 1.0 [@sparql] with important features such as aggregation and regular path expressions [@chili_yotta]. Other features, such as negation and subqueries, have also been added, but mainly for efficiency reasons, as they were already expressible, in a more involved manner, in version 1.0. Hence, it is still relevant to study the fundamental properties of SPARQL 1.0. In this paper, we follow the elegant formalization of SPARQL 1.0 by Arenas, Gutierrez and Pérez [@perez_sparql_tods; @semanticsparql] which is eminently suited for theoretical investigations.
The fundamental problem that we investigate is that of *satisfiability* of SPARQL patterns. A pattern is called satisfiable if there exists an RDF graph under which the pattern evaluates to a nonempty set of mappings. For any query language, satisfiability is clearly one of the essential properties one needs to understand if one wants to do automated reasoning. Since SPARQL patterns can emulate relational algebra expressions [@ag_expsparql; @polleres_sparqldatalog; @chile_sparql_pods], and satisfiability for relational algebra is undecidable [@ahv_book], the general satisfiability problem for SPARQL is undecidable as well.
Whether or not a pattern is satisfiable depends mainly on the filter operations appearing in the pattern. The goal of this paper is to precisely delineate the decidability of SPARQL fragments that are defined in terms of the constraints that can be used as filter conditions. The six basic classes of constraints we consider are bound-constraints; equalities; constant-equalities; and their negations. In this way, fragments of SPARQL can be constructed by specifying which kinds of constraints are allowed as filter conditions. For example, in the fragment ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$, filter conditions can only be bound constraints, nonequalities, and constant-nonequalities.
Our main result states that the only fragments for which satisfiability is decidable are the two fragments ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ and ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ and their subfragments. Consequently, as soon as either negated bound-constraints, or constant-equalities, or combinations of equalities and nonequalities are allowed, the satisfiability problem becomes undecidable. Each undecidable case is established by showing how the set difference operation can be emulated. This was already known using negated bound-constraints [@ag_expsparql; @chile_sparql_pods]; so we show it is also possible using constant-equalities, and using combinations of equalities and nonequalities, but in no other way. Undecidability can then be obtained from a known undecidability result for the algebra of binary relations with union, composition, and set difference [@tony_da_arxiv].
In the decidable cases, satisfiability can be decided by syntactic checks on bound variables and the use of literals. Although the problem is shown to be NP-complete, it is experimentally shown that the checks can be implemented efficiently in practice.
At the end of the paper we look at a well-behaved class of patterns known as the ‘well-designed’ patterns [@perez_sparql_tods]. We observe that satisfiability of well-designed patterns can be decided by combining the check on bound variables with a check for inconsistent filter conditions.
This paper is further organized as follows. In the next section, we introduce syntax and semantics of SPARQL patterns and introduce the different fragments under consideration. Section \[secsat\] introduces the satisfiability problem and shows satisfiability checking for the fragments ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ and ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$. Section \[secund\] shows undecidability for the fragments ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$, ${{\text{SPARQL}(=_c)}}$, and ${{\text{SPARQL}(=,\allowbreak \neq)}}$. Section \[secwell\] considers well-designed patterns.
Section \[secexp\] reports on experiments that test our decision methods in practice. In Section \[sec1.1\] we briefly discuss how our results extend to the new operators that have been added to SPARQL 1.1. We conclude in Section \[seconcl\].
SPARQL and fragments
====================
In this section we recall the syntax and semantics of SPARQL patterns, closely following the core SPARQL formalization given by Arenas, Gutierrez and Pérez [@perez_sparql_tods; @semanticsparql; @chile_sparql_pods].[^4] The semantics we use is set-based, whereas the semantics of real SPARQL is bag-based. However, for satisfiability (the main topic of this paper), it makes no difference whether we use a set or bag semantics [@schmidt_sparqloptim Lemma 1].
In this section we will also define the language fragments defined in terms of allowed filter conditions, which will form the object of this paper.
RDF graphs
----------
Let ${I}$, ${B}$, and ${L}$ be infinite sets of *IRIs*, *blank nodes* and *literals*, respectively. These three sets are pairwise disjoint. We denote the union $I \cup B
\cup L$ by $U$, and elements of $I \cup L$ will be referred to as *constants*. Note that blank nodes are not constants.
A triple $(s, p, o) \in ({I}\cup {B}) \times {I} \times U$ is called an *RDF triple*. An *RDF graph* is a finite set of RDF triples.
Syntax of SPARQL patterns
-------------------------
Assume furthermore an infinite set $V$ of *variables*, disjoint from $U$. The convention in SPARQL is that variables are written beginning with a question mark, to distinguish them from constants. We will follow this convention in this paper.
SPARQL *patterns* are inductively defined as follows.
- Any triple from $({I}\cup {L} \cup {V}) \times ({I} \cup
{V}) \times ({I} \cup {L} \cup {V}$) is a pattern (called a *triple pattern*).
- If $P_{1}$ and $P_{2}$ are patterns, then so are the following:
- $P_{1} {\mathbin{\mathrm{UNION}}}P_{2}$;
- $P_{1} {\mathbin{\mathrm{AND}}}P_{2}$;
- $P_{1} {\mathbin{\mathrm{OPT}}}P_{2}$.
- If $P$ is a pattern and $C$ is a constraint (defined next), then $P {\mathbin{\mathrm{FILTER}}}C$ is a pattern; we call $C$ the *filter condition*.
Here, a *constraint* can have one of the six following forms:
1. *bound-constraint:* ${\mathrm{bound}}(?x)$
2. *negated bound-constraint:* $\neg {\mathrm{bound}}(?x)$
3. *equality:* $?x={?y}$
4. *nonequality:* $?x\neq {?y}$ with $?x$ and $?y$ distinct variables
5. *constant-equality:* $?x = c$ with $c$ a constant
6. *constant-nonequality:* $?x \neq c$
We do not need to consider conjunctions and disjunctions in filter conditions, since conjunctions can be expressed by repeated application of filter, and disjunctions can be expressed using UNION. Hence, by going to disjunctive normal form, any predicate built using negation, conjunction, and disjunction is indirectly supported by our language.
Moreover, real SPARQL also allows blank nodes in triple patterns. This feature has been omitted from the formalization [@perez_sparql_tods; @semanticsparql; @chile_sparql_pods], because blank nodes in triple patterns can be equivalently replaced by variables.
Semantics of SPARQL patterns
----------------------------
The semantics of patterns is defined in terms of sets of so-called *solution mappings*, hereinafter simply called *mappings*. A solution mapping is a total function $\mu : S \to U$ on some finite set $S$ of variables. We denote the domain $S$ of $\mu$ by ${\mathrm{dom}(\mu)}$.
We make use of the following convention.
\[covve\] For any mapping $\mu$ and any constant $c \in I \cup L$, we agree that $\mu(c)$ equals $c$ itself.
In other words, mappings are by default extended to constants according to the identity mapping.
Now given a graph $G$ and a pattern $P$, we define the semantics of $P$ on $G$, denoted by ${{\llbracket P \rrbracket_{G}}}$, as a set of mappings, in the following manner.
- If $P$ is a triple pattern $(u, v, w)$, then $${{\llbracket P \rrbracket_{G}}} :=
\{\mu : \{u, v, w\} \cap V \to U \mid (\mu(u),\mu(v),\mu(w)) \in G\}.$$ This definition relies on Convention \[covve\] formulated above.
- If $P$ is of the form $P_1 {\mathbin{\mathrm{UNION}}}P_2$, then $${{\llbracket P \rrbracket_{G}}} := {{\llbracket P_1 \rrbracket_{G}}} \cup {{\llbracket P_2 \rrbracket_{G}}}.$$
- If $P$ is of the form $P_1 {\mathbin{\mathrm{AND}}}P_2$, then $${{\llbracket P \rrbracket_{G}}} := {{\llbracket P_1 \rrbracket_{G}}} \Join {{\llbracket P_2 \rrbracket_{G}}} ,$$ where, for any two sets of mappings $\Omega_1$ and $\Omega_2$, we define $$\Omega_1 \Join \Omega_2 =
\{\mu_1 \cup \mu_2 \mid
\text{$\mu_1 \in \Omega_1$ and
$\mu_2 \in \Omega_2$ and $\mu_1 \sim \mu_2$} \}.$$ Here, two mappings $\mu_1$ and $\mu_2$ are called *compatible*, denoted by $\mu_1 \sim \mu_2$, if they agree on the intersection of their domains, i.e., if for every variable $?x \in {\mathrm{dom}(\mu_1)} \cap {\mathrm{dom}(\mu_2)}$, we have $\mu_1(?x) = \mu_2(?x)$. Note that when $\mu_1$ and $\mu_2$ are compatible, their union $\mu_1 \cup \mu_2$ is a well-defined mapping; this property is used in the formal definition above.
- If $P$ is of the form $P_1 {\mathbin{\mathrm{OPT}}}P_2$, then $${{\llbracket P \rrbracket_{G}}} := ({{\llbracket P_1 \rrbracket_{G}}} \Join {{\llbracket P_2 \rrbracket_{G}}} )
\cup ({{\llbracket P_1 \rrbracket_{G}}} \smallsetminus {{\llbracket P_2 \rrbracket_{G}}} ),$$ where, for any two sets of mappings $\Omega_1$ and $\Omega_2$, we define $$\Omega_1 \smallsetminus \Omega_2 =
\{ \mu_1 \in \Omega_1 \mid \neg \exists \mu_2 \in \Omega_2 :
\mu_1 \sim \mu_2\}.$$
- Finally, if $P$ is of the form $P_1 {\mathbin{\mathrm{FILTER}}}C$, then $${{\llbracket P \rrbracket_{G}}} := \{\mu \in {{\llbracket P_1 \rrbracket_{G}}} \mid \mu \models C\}$$ where the satisfaction of a constraint $C$ by a mapping $\mu$, denoted by $\mu \models C$, is defined as follows:
1. $\mu \models {\mathrm{bound}}(?x)$ if $?x \in {\mathrm{dom}(\mu)}$;
2. $\mu \models \neg {\mathrm{bound}}(?x)$ if $?x \notin {\mathrm{dom}(\mu)}$;
3. $\mu \models {?x={?y}}$ if $?x,?y \in {\mathrm{dom}(\mu)}$ and $\mu(?x)=\mu(?y)$;
4. $\mu \models {?x\neq {?y}}$ if $?x,?y \in {\mathrm{dom}(\mu)}$ and $\mu(?x)\neq\mu(?y)$;
5. $\mu \models {?x=c}$ if $?x \in {\mathrm{dom}(\mu)}$ and $\mu(?x)=c$;
6. $\mu \models {?x\neq c}$ if $?x \in {\mathrm{dom}(\mu)}$ and $\mu(?x)\neq c$.
Note that $\mu \models {?x \neq {?y}}$ is not the same as $\mu
\not \models {?x = {?y}}$, and similarly for $\mu \models {?x
\neq c}$. This is in line with the three-valued logic semantics for filter conditions used in the official semantics [@semanticsparql]. For example, if $?x \notin {\mathrm{dom}(\mu)}$, then in three-valued logic $?x=c$ evaluates to $\mathit{error}$ under $\mu$; consequently, also $\neg {?x=c}$ evaluates to $\mathit{error}$ under $\mu$. Accordingly, in the semantics above, we have both $\mu \not \models {?x=c}$ and $\mu \not
\models {?x\neq c}$.
SPARQL fragments
----------------
We can form fragments of SPARQL by specifying which of the six classes of constraints are allowed as filter conditions. We denote the class of bound-constraints by ‘bound’, negated bound-constraints by ‘$\neg{\mathrm{bound}}$’, equalities by ‘$=$’, nonequalities by ‘$\neq$’, constant-equalities by ‘$=_c$’, and constant-nonequalities by ‘$\neq_c$’. Then for any subset $F$ of $\{{\mathrm{bound}},\neg {\mathrm{bound}},=,\neq,=_c,\neq_c\}$ we can form the fragment ${\text{SPARQL}(F)}$. For example, in the fragment ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$, filter conditions can only be bound constraints, equalities, and constant-nonequalities.
Satisfiability: decidable fragments {#secsat}
===================================
A pattern $P$ is called *satisfiable* if there exists a graph $G$ such that ${{\llbracket P \rrbracket_{G}}} $ is nonempty. In general, checking satisfiability is a very complicated, indeed undecidable, problem. But for the two fragments ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ and ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$, it will turn out that there are essentially only two possible reasons for unsatisfiability.
The first possible reason is that the pattern specifies a literal value in the first position of some RDF triple, whereas RDF triples can only have literals in the third position. For example, using the literal 42, the triple pattern $(42,?x,?y)$ is unsatisfiable. Note that literals in the middle position of a triple pattern are already disallowed by the definition of triple pattern, so we only need to worry about the first position.
This discrepancy between triple patterns and RDF triples is easy to sidestep, however. In the Appendix we show how, without loss of generality, we may assume from now on that *patterns do not contain any triple pattern $(u,v,w)$ where $u$ is a literal.*
The second and main possible reason for unsatisfiability is that filter conditions require variables to be bound together in a way that cannot be satisfied by the subpattern to which the filter applies. For example, the pattern $$((?x,a,?y) {\mathbin{\mathrm{UNION}}}(?x,b,?z)) {\mathbin{\mathrm{FILTER}}}({\mathrm{bound}}(?y) \land {\mathrm{bound}}(?z))$$ is unsatisfiable. Note that bound constraints are not strictly necessary to illustrate this phenomenon: if in the above example we replace the filter condition by $?y={?z}$ the resulting pattern is still unsatisfiable.
We next prove formally that satisfiability for patterns in ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ and ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ is effectively decidable, by catching the reason for unsatisfiability described above. Note also that the two fragments can not be combined, since satisfiability for ${\text{SPARQL}(=,\neq)}$ is undecidable as we will see in the next Section.
Checking bound variables {#seccheck}
------------------------
To perform bound checks on variables, we associate to every pattern $P$ a set $\Gamma(P)$ of schemes, where a *scheme* is simply a set of variables, in the following way.[^5]
- If $P$ is a triple pattern $(u,v,w)$, then $\Gamma(P) :=
\{\{u,v,w\}\cap V\}$.
- $\Gamma(P_1 {\mathbin{\mathrm{UNION}}}P_2) := \Gamma(P_1) \cup \Gamma(P_2)$.
- $\Gamma(P_1 {\mathbin{\mathrm{AND}}}P_2) := \{S_1 \cup S_2 \mid S_1 \in \Gamma(P_1)$ and $S_2 \in \Gamma(P_2)\}$.
- $\Gamma(P_1 {\mathbin{\mathrm{OPT}}}P_2) := \Gamma(P_1 {\mathbin{\mathrm{AND}}}P_2) \cup \Gamma(P_1)$.
- $\Gamma(P_1 {\mathbin{\mathrm{FILTER}}}C) := \{S \in \Gamma(P_1) \mid S \vdash
C\}$, where $S \vdash C$ is defined as follows:
- If $C$ is of the form ${\mathrm{bound}}(?x)$ or $?x=c$ or $?x \neq c$, then $S \vdash C$ if $?x \in S$;
- If $C$ is of the form $?x={?y}$ or $?x\neq{?y}$, then $S \vdash C$ if $?x,?y \in S$;
- $S \vdash {\neg {\mathrm{bound}}(?x)}$ if $?x \notin S$.
\[exgammaunion\] Consider the pattern $$P = (?x,p,?y) {\mathbin{\mathrm{OPT}}}( (?x,q,?z) {\mathbin{\mathrm{UNION}}}(?x,r,?u)).$$ For the subpattern $P_1 =
(?x,q,?z) {\mathbin{\mathrm{UNION}}}(?x,r,?u)$ we have $\Gamma(P_1) =
\{\{?x,?z\},\allowbreak \{?x,?u\}\}$. Hence, $\Gamma((?x,p,?y) {\mathbin{\mathrm{AND}}}P_1) =
\{\{?x,?y,?z\},\{?x,?y,?u\}\}$. We conclude that $\Gamma(P) =
\{\{?x,?y\},\{?x,?y,?z\},\{?x,?y,?u\}\}$.
\[exgamma\] For another example, consider the pattern $$P = ((?x, p, ?y) {\mathbin{\mathrm{OPT}}}((?x, q, ?z)
{\mathbin{\mathrm{FILTER}}}{?y = ?z})) {\mathbin{\mathrm{FILTER}}}{?x \neq c}.$$ We have $\Gamma(?x,q,?z)
= \{\{?x,?z\}\}$. Note that $\{?x,?z\} \not \vdash {?y = ?z}$, because $?y \notin \{?x,?z\}$. Hence, for the subpattern $P_1 =
(?x, q, ?z) {\mathbin{\mathrm{FILTER}}}{?y = ?z}$ we have $\Gamma(P_1)=\emptyset$. For the subpattern $P_2 = (?x,p,?y) {\mathbin{\mathrm{OPT}}}P_1$ we then have $\Gamma(P_2)=\Gamma(?x,p,?y)=\{\{?x,?y\}\}$. Since $\{?x,?y\}
\vdash {?x \neq c}$, we conclude that $\Gamma(p) = \{\{?x,?y\}\}$.
We now establish the main result of this Section.
\[theordecidable\] Let $P$ be a ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ or ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ pattern. Then $P$ is satisfiable if and only if $\Gamma(P)$ is nonempty.
The only-if direction of Theorem \[theordecidable\] is the easy direction and is given by the following Lemma \[lemoif\]. Note that this lemma holds for general patterns; it can be straightforwardly proven by induction on the structure of $P$.
\[lemoif\] Let $P$ be a pattern. If $\mu \in {{\llbracket P \rrbracket_{G}}} $ then there exists $S
\in \Gamma(P)$ such that ${\mathrm{dom}(\mu)} = S$.
The if direction of Theorem \[theordecidable\] for ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ is given by the following Lemma \[lemifeq\].
In the following we use ${\mathrm{var}(P)}$ to denote the set of all variables occurring in a pattern $P$.[^6]
\[lemifeq\] Let $P$ be a pattern in ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$. Let $c \in I$ be a constant that does not appear in any constant-nonequality filter condition in $P$. With the constant mapping $\mu : {\mathrm{var}(P)} \to \{c\}$, let $G$ be the RDF graph consisting of all possible triples $(\mu(u),\mu(v),\mu(w))$ where $(u,v,w)$ is a triple pattern in $P$.
Then for every $S \in \Gamma(P)$ there exists $S' \supseteq S$ such that $\mu|_{S'}$ belongs to ${{\llbracket P \rrbracket_{G}}}$.
By induction on the structure of $P$. If $P$ is a triple pattern $(u,v,w)$ then $S = \{u,v,w\}\cap V$. Since $(\mu|_S(u),\mu|_S(v),\mu|_S(w))=(\mu(u),\mu(v),\mu(w)) \in G$, we have $\mu|_S \in {{\llbracket P \rrbracket_{G}}} $ and we can take $S'=S$.
If $P$ is of the form $P_1 {\mathbin{\mathrm{UNION}}}P_2$, then the claim follows readily by induction.
If $P$ is of the form $P_1 {\mathbin{\mathrm{AND}}}P_2$, then we have $S=S_1 \cup S_2$ with $S_i \in \Gamma(P_i)$ for $i=1,2$. By induction, there exists $S_i' \supseteq S_i$ such that $\mu|_{S_i'} \in {{\llbracket P_i \rrbracket_{G}}} $. Clearly $\mu|_{S_1'} \sim
\mu|_{S_2'}$ since they are restrictions of the same mapping. Hence $\mu|_{S_1'} \cup \mu|_{S_2'} = \mu_{S_1' \cup S_2'} \in
{{\llbracket P \rrbracket_{G}}}$ and we can take $S' = S_1' \cup S_2'$.
If $P$ is of the form $P_1 {\mathbin{\mathrm{OPT}}}P_2$, then there are two possibilities.
- If $S \in \Gamma(P_1 {\mathbin{\mathrm{AND}}}P_2)$ then we can reason as in the previous case.
- If $S \in \Gamma(P_1)$ then by induction there exists $S'_1
\supseteq S$ so that $\mu|_{S'_1} \in {{\llbracket P_1 \rrbracket_{G}}}$. Now there are two further possibilities:
- If $\Gamma(P_2)$ is nonempty then by induction there exists some $S_2'$ so that $\mu|_{S_2'} \in {{\llbracket P_2 \rrbracket_{G}}}$. We can now reason again as in the case $P_1 {\mathbin{\mathrm{AND}}}P_2$.
- Otherwise, by Lemma \[lemoif\] we know that ${{\llbracket P_2 \rrbracket_{G}}} $ is empty. But then ${{\llbracket P \rrbracket_{G}}} = {{\llbracket P_1 \rrbracket_{G}}} $ and we can take $S'
= S'_1$.
Finally, if $P$ is of the form $P_1 {\mathbin{\mathrm{FILTER}}}C$, then we know that $S
\in \Gamma(P_1)$ and $S \vdash C$. By induction, there exists $S' \supseteq S$ such that $\mu|_{S'} \in
{{\llbracket P_1 \rrbracket_{G}}} $. We show that $\mu|_{S'} \in
{{\llbracket P \rrbracket_{G}}}$ by showing that $\mu|_{S'} \models C$. There are three possibilities for $C$.
- If $C$ is of the form ${\mathrm{bound}}(?x)$, then we know by $S \vdash C$ that $?x \in S'$. Hence $\mu|_{S'} \models C$.
- If $C$ is of the form $?x = {?y}$, then we again know $?x,?y \in S'$, and certainly $\mu|_{S'} \models C$ since $\mu$ maps everything to $c$.
- If $C$ is of the form $?x \neq d$, then we have $d \neq c$ by the choice of $c$, so $\mu|_{S'} \models C$ since $\mu(?x)=c$.
To illustrate the above Lemma, consider the pattern $$P = ((?x,p,?y) {\mathbin{\mathrm{FILTER}}}{?x \neq a}) {\mathbin{\mathrm{OPT}}}( (?x,q,?z) {\mathbin{\mathrm{UNION}}}(?x,r,?u) )$$ which is a variant of the pattern from Example \[exgammaunion\]. As in that example, we have $\Gamma(P) = \{\{?x,?y\},\{?x,?y,?z\},\{?x,?y,?u\}\}$. In this case, the mapping $\mu$ from the Lemma maps $?x$, $?y$, $?z$ and $?u$ to $c$. The graph $G$ from the Lemma equals $\{(c,p,c),(c,q,c),(c,r,c)\}$, and ${{\llbracket P \rrbracket_{G}}} = \{\mu_1,\mu_2\}$ where $\mu_1 = \mu|_{\{?x,?y,?z\}}$ and $\mu_2 = \mu|_{\{?x,?y,?u\}}$. Now consider $S = \{?x,?y\} \in
\Gamma(P)$. Then for $S'=\{?x,?y,?z\}$ we indeed have $S'
\supseteq S$ and $\mu|_{S'} = \mu_1 \in {{\llbracket P \rrbracket_{G}}}$. Note that in this example we could also have chosen $\{?x,?y,?u\}$ for $S'$.
The counterpart to Lemma \[lemifeq\] for the fragment ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ is given by the following Lemma, thus settling Theorem \[theordecidable\] for that fragment.
\[lemifneq\] Let $P$ be a pattern in ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$. Let $W$ be the set of all constants appearing in a constant-nonequality filter condition in $P$. Let $Z \subseteq
I$ be a finite set of constants of the same cardinality as ${\mathrm{var}(P)}$, and disjoint from $W$. With $\mu : {\mathrm{var}(P)} \to Z$ an arbitrary but fixed injective mapping, let $G$ be the RDF graph consisting of all possible triples $(\mu(u),\mu(v),\mu(w))$ where $(u,v,w)$ is a triple pattern in $P$.
Then for every $S \in \Gamma(P)$ there exists $S' \supseteq S$ such that $\mu|_{S'}$ belongs to ${{\llbracket P \rrbracket_{G}}} $.
We prove for every subpattern $Q$ of $P$ that for every $S \in
\Gamma(Q)$ there exists $S' \supseteq S$ such that $\mu|_{S'}
\in {{\llbracket Q \rrbracket_{G}}} $. The proof is by induction on the height of $Q$. The reasoning is largely the same as in the proof of Lemma \[lemifeq\]. The only difference is in the case where $Q$ is of the form $Q_1 {\mathbin{\mathrm{FILTER}}}C$. In showing that $\mu_{S'} \models C$, we now argue as follows for the last two cases:
- If $C$ is of the form $?x \neq {?y}$, then $\mu|_{S'} \models C$ since $\mu$ is injective.
- If $C$ is of the form $?x \neq c$, then $\mu|_{S'} \models C$ since $Z$ and $W$ are disjoint.
Computational complexity
------------------------
In this section we show that satisfiability for the decidable fragments is NP-complete. Note that this does not immediately follow from the NP-completeness of SAT, since boolean formulas are not part of the syntax of the decidable fragments.
Theorem \[theordecidable\] implies the following complexity upper bound:
The satisfiability problem for ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ patterns, as well as for ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ patterns, belongs to the complexity class NP.
By Theorem \[theordecidable\], a ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ or ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ pattern $P$ is satisfiable if and only if there exists a scheme in $\Gamma(P)$. Following the definition of $\Gamma(P)$, it is clear that there is a polynomial-time nondeterministic algorithm such that, on input $P$, each accepting possible run computes a scheme in $\Gamma(P)$, and such that every scheme in $\Gamma(P)$ is computed by some accepting possible run.
Specifically, the algorithm works bottom-up on the syntax tree of $P$ and computes a scheme for every subpattern. At every leaf $Q$, corresponding to a triple pattern in $P$, we compute the unique scheme in $\Gamma(Q)$. At every UNION operator we nondeterministically choose between continuing with the scheme from the left or from right child. At every AND operator we continue with the union of the left and right child schemes. At every OPT operator, we nondeterministically choose between treating it as an AND, or simply continuing with the scheme from the left. At every FILTER operation with constraint $C$ we check for the child scheme $S$ whether $S \vdash C$. If the check succeeds, we continue with $S$; if the check fails, the run is rejected. When the computation has reached the root of the syntax tree and we can compute a scheme for the root, the run is accepting and the computed scheme is the output.
We next show that satisfiability is actually NP-hard, even for patterns not using any OPT operators and using only bound constraints in filter conditions.
\[prophard\] The satisfiability problem for OPT-free patterns in the fragment ${\text{SPARQL}({\mathrm{bound}})}$ is NP-hard.
We define the problem Nested Set Cover as follows:
Input:
: A finite set $T$ and a finite set $E$ of sets of subsets of $T$. (So, every element of $E$ is a set of subsets of $T$.)
Decide:
: Whether for each element $e$ of $E$ we can choose a subset $S_e$ in $e$, so that $\bigcup_{e \in E} S_e = T$.
Let us first describe how the above problem can be reduced in polynomial time to the satisfiability problem at hand. Consider an input $(T,E)$ for Nested Set Cover. Without loss of generality we may assume that $T$ is a set of variables $\{?x_1,?x_2,\dots,?x_n\}$. Fix some constant $c$. For any subset $S$ of $T$, we can make a pattern $P_S$ by taking the AND of all $(x,c,c)$ for $x \in S$. Now for a set $e$ of subsets of $T$, we can form the pattern $P_e$ by taking the UNION of all $P_S$ for $S \in e$. Finally, we form the pattern $P_E$ by taking the AND of all $P_e$ for $e \in E$.
Now consider the following pattern which we denote by $P_{(T,E)}$: $$P_E
{\mathbin{\mathrm{FILTER}}}{{\mathrm{bound}}(?x_1)}
{\mathbin{\mathrm{FILTER}}}{{\mathrm{bound}}(?x_2)}
\ldots
{\mathbin{\mathrm{FILTER}}}{{\mathrm{bound}}(?x_n)}$$
We claim that $P_{(T,E)}$ is satisfiable if and only if $(T,E)$ is a yes-instance for Nested Set Cover. To see the only-if direction, let $G$ be a graph such that ${{\llbracket P_{(T,E)} \rrbracket_{G}}}$ is nonempty, i.e., has as an element some solution mapping $\mu$. Then in particular $\mu \in {{\llbracket P_E \rrbracket_{G}}}$. Hence, for every $e \in E$ there exists $\mu_e \in {{\llbracket P_e \rrbracket_{G}}}$ such that $\mu = \bigcup_{e \in E} \mu_e$. Since $P_e$ is the UNION of all $P_S$ for $S \in e$, for each $e \in E$ there exists $S_e \in e$ such that $\mu_e \in {{\llbracket P_{S_e} \rrbracket_{G}}}$. Since $P_{S_e}$ is the AND of all $(x,c,c)$ for $x \in S_e$, it follows that ${\mathrm{dom}(\mu_e)}=S_e$. Hence, since ${\mathrm{dom}(\mu)} = \bigcup_{e\in E}
{\mathrm{dom}(\mu_e)}$, we have ${\mathrm{dom}(\mu)} = \bigcup_{e \in E} S_e$. However, by the bound constraints in the filters applied in $P_{(T,E)}$, we also have ${\mathrm{dom}(\mu)} = \{?x_1,\dots,?x_n\} = T$. We conclude that $T = \bigcup_{e \in E} S_e$ as desired.
For the if-direction, assume that for each $e \in E$ there exists $S_e \in e$ such that $T = \bigcup_{e \in E} S_e$. Consider the singleton graph $G = \{(c,c,c)\}$. For any subset $S$ of $T$, let $\mu_S : S \to \{c\}$ be the constant solution mapping with domain $S$. Clearly, $\mu_{S} \in {{\llbracket P_S \rrbracket_{G}}}$, so $\mu_{S_e}
\in {{\llbracket P_e \rrbracket_{G}}}$ for every $e \in E$. All the $\mu_S$ map to the same constant, so they are all compatible. Hence, for $\mu =
\bigcup_{e \in E} \mu_{S_e}$, we have $\mu \in {{\llbracket P_E \rrbracket_{G}}}$. Since ${\mathrm{dom}(\mu)} = \bigcup_{e\in E} {\mathrm{dom}(\mu_{S_e})} = \bigcup_{e \in E}
S_e = T = \{?x_1,\dots,?x_n\}$, the mapping $\mu$ satisfies every constraint ${\mathrm{bound}}(?x_i)$ for $i=1,\dots,n$. We conclude that $\mu \in {{\llbracket P_{(E,T)} \rrbracket_{G}}}$ as desired.
It remains to show that Nested Set Cover is NP-hard. Thereto we reduce the classical CNF-SAT problem. Assume given a boolean formula $\phi$ in CNF, so $\phi$ is a conjunction of clauses, where each clauses is a disjunction of literals (variables or negated variables). We construct an input $(T,E)$ for Nested Set Cover as follows. Denote the set of variables used in $\phi$ by $W$.
For $T$ we take the set of clauses of $\phi$. For any variable $x \in W$, consider the set ${\rm Pos}_x$ consisting of all clauses that contain a positive occurrence of $x$, and the set ${\rm Neg}_x$ consisting of all clauses that contain a negative occurrence of $x$. Then we define $e_x$ as the pair $\{{\rm
Pos}_x,{\rm Neg}_x\}$.
Now $E$ is defined as the set $\{e_x \mid x \in W\}$. It is clear that $\phi$ is satisfiable if and only if the constructed input is a yes-instance for Nested Set Cover. Indeed, truth assignments to the variables correspond to selecting either ${\rm Pos}_x$ or ${\rm Neg}_x$ from $e_x$ for each $x \in W$.
Undecidable fragments {#secund}
=====================
In this Section we show that the two decidable fragments ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ and ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ are, in a sense, maximal. Specifically, the three minimal fragments not subsumed by one of these two fragments are ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$, ${{\text{SPARQL}(=,\allowbreak \neq)}}$, and ${{\text{SPARQL}(=_c)}}$. The main result of this Section is:
Satisfiability is undecidable for ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$ patterns, for ${{\text{SPARQL}(=,\allowbreak \neq)}}$ patterns, and for ${{\text{SPARQL}(=_c)}}$ patterns.
We will first present the proof for ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$; after that we explain how the proof can be adapted for the other two fragments.
${{\text{SPARQL}(\neg{\mathrm{bound}})}}$
-----------------------------------------
Our approach is to reduce from the satisfiability problem for the algebra of finite binary relations with union, difference, and composition [@tony_da_arxiv]. This algebra is also called the Downward Algebra and denoted by DA. The expressions of DA are defined as follows. Let $R$ be an arbitrary fixed binary relation symbol.
- The symbol $R$ is a DA-expression.
- If $e_1$ and $e_2$ are DA-expressions, then so are $e_1 \cup
e_2$, $e_1 - e_2$, and $e_1 \circ e_2$.
Semantically, DA-expressions represent binary queries on binary relations, i.e., mappings from binary relations to binary relations. Let $J$ be a binary relation. For DA-expression $e$, we define the binary relation $e(J)$ inductively as follows:
- $R(J) = J$;
- $(e_1 \cup e_2)(J) = e_1(J) \cup e_2(J)$;
- $(e_1 - e_2)(J) = e_1(J) - e_2(J)$ (set difference);
- $(e_1 \circ e_2)(J) = \{(x,z) \mid \exists y : (x,y) \in e_1(J)$ and $(y,z) \in e_2(J)\}$.
A DA-expression is called *satisfiable* if there exists a finite binary relation $J$ such that $e(J)$ is nonempty.
An example of a DA-expression is $e=(R \circ R)-R$. If $J$ is the binary relation $\{(a,b),(b,c),(a,c),(c,d)\}$ then $e(J) =
\{(b,d),(a,d)\}$. An example of an unsatisfiable DA expression is $(R \circ R - R) \circ R - R \circ R \circ R$.
We recall the following result. It is actually well known [@andreka_memoir] that relational composition together with union and complementation leads to an undecidable algebra; the following result simplifies matters by showing that undecidability already holds for expressions over a single relation symbol and using set difference instead of complementation. The following result has been proven by reduction from the universality problem for context-free grammars.
The satisfiability problem for DA-expressions is undecidable.
We are now ready to formulate the reduction from the satisfiability problem for DA to the satisfiability problem for ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$.
\[reductionlemma\] Let $r \in I$ be an arbitrary fixed constant. For any binary relation $J$, let $G_J$ be the RDF graph $\{(c,r,d) \mid (c,d) \in J\}$. Then for every DA-expression $e$ there exists a ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$ pattern $P_e$ with the following properties:
1. there exist two distinct fixed variables $?x$ and $?y$ such that for every RDF graph $G$ and every $\mu \in {{\llbracket P_e \rrbracket_{G}}}$, $?x$ and $?y$ belong to ${\mathrm{dom}(\mu)}$;
2. for every binary relation $J$, we have $$e(J) =
\{(\mu(?x),\mu(?y)) \mid \mu \in {\llbracket P_e \rrbracket_{G_J}}\};$$
3. for every RDF graph $G$, we have ${{\llbracket P_e \rrbracket_{G}}} = {\llbracket P_e \rrbracket_{G^r}}$, where $G^r := \{(u,v,w) \in G \mid v=r\}$.
By induction on the structure of $e$. If $e$ is $R$ then $P_e$ is the triple pattern $(?x,r,?y)$.
If $e$ is of the form $e_1 \cup e_2$, then $P_e$ is $P_{e_1}
{\mathbin{\mathrm{UNION}}}P_{e_2}$.
If $e$ is of the form $e_1 \circ e_2$, then $P_e$ is $P'_{e_1}
{\mathbin{\mathrm{AND}}}P'_{e_2}$, where $P'_{e_1}$ and $P'_{e_2}$ are obtained as follows. First, by renaming variables, we may assume without loss of generality that $P_{e_1}$ and $P_{e_2}$ have no variables in common other than $?x$ and $?y$. Let $?z$ be a fresh variable. Now in $P_{e_1}$, rename $?y$ to $?z$, yielding $P'_{e_1}$, and in $P_{e_2}$, rename $?x$ to $?z$, yielding $P'_{e_2}$.
Finally, if $e$ is of the form $e_1 - e_2$, then we use a known idea [@chile_sparql_pods]. As before we may assume without loss of generality that $P_{e_1}$ and $P_{e_2}$ have no variables in common other than $?x$ and $?y$. Let $?u$ and $?w$ be two fresh variables. Then $P_e$ is equal to $$\bigl ( P_{e_1}
{\mathbin{\mathrm{OPT}}}(P_{e_2} {\mathbin{\mathrm{AND}}}(?u,r,?w)) \bigr ) {\mathbin{\mathrm{FILTER}}}{\neg {\mathrm{bound}}(?u)}.$$
The above lemma provides us with a reduction from satisfiability for DA to satisfiability for ${{\text{SPARQL}(\neg{\mathrm{bound}})}}$, thus showing undecidability of the latter problem. Indeed, if $e$ is satisfiable, then clearly $P_e$ is satisfiable as well, by property 2 of the lemma. Conversely, if $P_e$ is satisfiable by some RDF graph $G$, then, by property 3 of the lemma, ${\llbracket P_e \rrbracket_{G^r}}$ is nonempty. Now define the binary relation $J = \{(c,d) \mid (c,r,d) \in G\}$. Then $G_J = G^r$, so by property 2 of the lemma we obtain the nonemptiness of $e(J)$ as desired.
${{\text{SPARQL}(=,\allowbreak \neq)}}$ {#seceqneq}
---------------------------------------
We now consider a minor variant of satisfiability for DA-expressions where we restrict attention to binary relations over at least two elements. Formally, the *active domain* of a binary relation $J$ is the set of all entries in pairs belonging to $J$, so ${\mathrm{adom}(J)} := \{x \mid
\exists y : (x,y) \in J$ or $(y,x) \in J\}$. Then a DA-expression $e$ is called *two-satisfiable* if $e(J)$ is nonempty for some $J$ such that ${\mathrm{adom}(J)}$ has at least two distinct elements.
Clearly, two-satisfiability is undecidable as well, for if it were decidable, then satisfiability would be decidable too. Indeed, $e$ is satisfiable if and only if it is two-satisfiable, or satisfiable by a binary relation $J$ over a single element. Up to isomorphism there is only one such $J$ (the singleton $\{(x,x)\}$), and DA-expressions commute with isomorphisms.
Lemma \[reductionlemma\] can now be adapted as follows. Property 2 of the lemma is only claimed for every binary relations $J$ over at least two distinct elements. In the proof for the case where $e$ is $e_1 - e_2$, we use six fresh variables $?u$, $?u'$, $?v$, $?v'$, $?w$, and $?w'$. We use the abbreviation ${\mathit{adom}_{?u}}$ for $(?u,r,?w) {\mathbin{\mathrm{UNION}}}(?v,r,?u)$ and similarly for ${\mathit{adom}_{?u'}}$. We now use the following pattern for $P_e$: $$\begin{gathered}
\Bigl ( \bigl ( P_{e_1} {\mathbin{\mathrm{OPT}}}(
(P_{e_2} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u}} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u'}}) {\mathbin{\mathrm{FILTER}}}{?u \neq {?u'}}
) \bigr) \\
{} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u}} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u'}} \Bigr ) {\mathbin{\mathrm{FILTER}}}{?u={?u'}}. \end{gathered}$$
Let us verify that $P_e$ satisfies the three properties of Lemma \[reductionlemma\].
1. By induction, $P_{e_1}$ has the property that every returned solution mapping has $?x$ and $?y$ in its domain. Since $P_e$ is of the form $$(P_{e_1} {\mathbin{\mathrm{OPT}}}\ldots) {\mathbin{\mathrm{FILTER}}}\ldots$$ the same property holds for $P_e$.
2. Let $J$ be a binary relation on at least two distinct elements. To prove the equality $$e(J) = \{(\mu(?x),\mu(?y)) \mid \mu \in {\llbracket P_e \rrbracket_{G_J}}\}$$ we are going to consider both inclusions. For easy reference we name some subpatterns of $P_e$ as follows.
- $P_2$ denotes $(P_{e_2} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u}} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u'}}) {\mathbin{\mathrm{FILTER}}}{?u \neq {?u'}}$;
- $P_3$ denotes $P_{e_1} {\mathbin{\mathrm{OPT}}}P_2$.
- Thus, $P$ is $(P_3 {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u}} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u'}}) {\mathbin{\mathrm{FILTER}}}{?u = {?u'}}$.
To prove the inclusion from right to left, let $\mu \in {\llbracket P_e \rrbracket_{G_J}}$. Then $\mu = \mu_3 \cup \varepsilon$, where $\mu_3 \in
{\llbracket P_3 \rrbracket_{G_J}}$ and $\varepsilon$ is a mapping defined on $?u,$ and $?u'$ such that $\varepsilon(?u) = \varepsilon(?u')$. In particular, $\mu_3 \sim
\varepsilon$. Since $P_3 = P_{e_1} {\mathbin{\mathrm{OPT}}}P_2$, there are two possibilities for $\mu_3$:
- $\mu_3 \in {\llbracket P_{e_1} \rrbracket_{G_J}}$ and there is no $\mu_2 \in {\llbracket P_2 \rrbracket_{G_J}}$ such that $\mu_3 \sim \mu_2$. By induction, both $?x$ and $?y$ belong to ${\mathrm{dom}(\mu_3)}$, so $(\mu(?x),\mu(?y))$ equals $(\mu_3(?x),\mu_3(?y))$, which belongs to $e_1(J)$ again by induction. So it remains to show that $(\mu(?x),\mu(?y)) \notin
e_2(J)$. Assume the contrary. Then there exists $\mu'_2 \in {\llbracket P_{e_2} \rrbracket_{G_J}}$ such that $(\mu_3(?x),\mu_3(?y)) = (\mu'_2(?x),\mu'_2(?y))$. Since ${\mathrm{adom}(J)}$ has at least two distinct elements, $\mu_2'$ can be extended to a mapping $\mu_2 \in {\llbracket P_2 \rrbracket_{G_J}}$. Since $?x$ and $?y$ are the only variables common to ${\mathrm{var}(P_{e_1})}$ and ${\mathrm{var}(P_2)}$, we conclude $\mu_3 \sim \mu_2$ which is a contradiction.
- $\mu_3 = \mu_1 \cup \mu_2$ with $\mu_1 \in {\llbracket P_{e_1} \rrbracket_{G_J}}$ and $\mu_2 \in {\llbracket P_2 \rrbracket_{G_J}}$. In particular, $\mu_3$ is defined on $?u$ and $?u'$ and $\mu_3(?u) \neq \mu_3(?u')$. On the other hand, since $\mu_3 \sim \varepsilon$, and $\varepsilon(?u) =
\varepsilon(?u')$, also $\mu_3(?u) = \mu_3(?u')$. This is a contradiction, so the possibility under consideration cannot happen.
To prove the inclusion from left to right, let $(c,d) \in e(J)$. Since $(c,d) \in e_1(J)$, there exists $\mu_1 \in {\llbracket P_{e_1} \rrbracket_{G_J}}$ such that $(c,d) = (\mu_1(?x),\mu_1(?y))$. Assume, for the sake of argument, that there *would* exist $\mu_2
\in {\llbracket P_2 \rrbracket_{G_J}}$ such that $\mu_1 \sim \mu_2$. Mapping $\mu_2$ contains a mapping $\mu_2' \in {\llbracket P_{e_2} \rrbracket_{G_J}}$, by definition of $P_2$. Since $(\mu_2'(?x),\mu_2'(?y)) \in e_2(J)$ and $\mu_1 \sim \mu_2$, it follows that $(c,d) \in e_2(J)$ which is a contradiction.
So, we now know that there does *not* exist $\mu_2 \in {\llbracket P_2 \rrbracket_{G_J}}$ such that $\mu_1 \sim \mu_2$. Hence, $\mu_1 \in {\llbracket P_3 \rrbracket_{G_J}}$. Note that the six variables $?u$, $?u'$, $?v$, $?v'$, $?w$, and $?w'$ do not belong to ${\mathrm{dom}(\mu_1)}$. Since $J$ is nonempty, $\mu_1$ can thus be extended to a mapping $\mu \in
{\llbracket P \rrbracket_{G_J}}$. We conclude $(c,d) = (\mu_1(?x),\mu_1(?y)) =
(\mu(?x),\mu(?y))$ as desired.
3. The third property of Lemma \[reductionlemma\] holds because ${\llbracket {\mathit{adom}_{?u}} \rrbracket_{G}} = {\llbracket {\mathit{adom}_{?u}} \rrbracket_{G^r}}$ (and similarly for ${\mathit{adom}_{?u'}}$).
Using the adapted lemma, we can now reduce two-satisfiability for DA to satisfiability for ${{\text{SPARQL}(=,\allowbreak \neq)}}$. Indeed, a DA-expression $e$ is two-satisfiable if and only if the pattern $$P_e {\mathbin{\mathrm{AND}}}(({\mathit{adom}_{?u}} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u'}}) {\mathbin{\mathrm{FILTER}}}{?u \neq {?u'}})$$ is satisfiable, where all variables used in ${\mathit{adom}_{?u}}$ and ${\mathit{adom}_{?u'}}$ are distinct and disjoint from those used in $P_e$.
${{\text{SPARQL}(=_c)}}$ {#secfrageqc}
------------------------
We consider a further variant of two-satisfiability, called *$ab$-satisfiability*, for two arbitrary fixed constants $a,b \in I$ that are distinct from the constant $r$ already used for Lemma \[reductionlemma\]. A DA-expression is called $ab$-satisfiable if $e(J)$ is nonempty for some binary relation $J$ where $a,b \in {\mathrm{adom}(J)}$.
Since DA-expressions do not distinguish between isomorphic binary relations, $ab$-satisfiability is equivalent to two-satisfiability, and thus still undecidable.
We now again adapt Lemma \[reductionlemma\], as follows. Property 2 is only claimed for every binary relation $J$ such that $a,b \in
{\mathrm{adom}(J)}$. In the proof for the case $e=e_1- e_2$, we now use the following pattern for $P_e$: $$\Bigl ( \bigl ( P_{e_1} {\mathbin{\mathrm{OPT}}}((P_{e_2} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u}}) {\mathbin{\mathrm{FILTER}}}{?u = a})\bigr )
{\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u}} \Bigr ) {\mathbin{\mathrm{FILTER}}}{?u=b}.
$$
The proof correctness of this construction is analogous to the proof given in the previous Section \[seceqneq\]; instead of exploiting the inconsistency between $?u \neq ?u'$ and $?u = ?u'$ as done in that proof, we now exploit the inconsistency between $?u=a$ and $?u=b$.
We then obtain that $e$ is $ab$-satisfiable if and only if $$P_e
{\mathbin{\mathrm{AND}}}({\mathit{adom}_{?u}} {\mathbin{\mathrm{AND}}}{\mathit{adom}_{?u'}}) {\mathbin{\mathrm{FILTER}}}{?u=a} {\mathbin{\mathrm{FILTER}}}{?u'=b}$$ is satisfiable, establishing a reduction from $ab$-satisfiability for DA to satisfiability for ${{\text{SPARQL}(=_c)}}$.
Recall that literals cannot appear in first or second position in an RDF triple. Patterns using constant-equality predicates can be unsatisfiable because of that reason. For example, using the literal 42, the pattern $(?x,?y,?z) {\mathbin{\mathrm{FILTER}}}{?y=42}$ is unsatisfiable. However, we have seen here that the use of constant-equality predicates leads to undecidability of satisfiability for a much more fundamental reason, that has nothing to do with literals, namely, the ability to emulate set difference.
Satisfiability of well-designed patterns {#secwell}
========================================
The *well-designed* patterns [@perez_sparql_tods] have been identified as a well-behaved class of SPARQL patterns, with properties similar to the conjunctive queries for relational databases [@ahv_book]. Standard conjunctive queries are always satisfiable, and conjunctive queries extended with equality and nonequality constraints, possibly involving constants, can only be unsatisfiable if the constraints are inconsistent. An analogous behavior is present in what we call *AF-patterns*: patterns that only use the AND and FILTER operators. We will formalize this in Proposition \[propaf\]. We will then show in Theorem \[theorwell\] that a well-designed pattern is satisfiable if and only if its reduction to an AF-pattern is satisfiable. In other words, as far as satisfiability is concerned, well-designed patterns can be treated like AF-patterns.
Satisfiability of AF-patterns
-----------------------------
In Section \[seccheck\] we have associated a set of schemes $\Gamma(P)$ to every pattern $P$. When $\Gamma(P)$ is empty, $P$ is unsatisfiable (Lemma \[lemoif\]).
Now when $P$ is an AF-pattern and $\Gamma(P)$ is nonempty, the satisfiability of $P$ will turn out to depend solely on the equalities, nonequalities, constant-equalities, and constant-nonequalities occurring as filter conditions in $P$. We will denote the set of these constraints by $C(P)$.
Any set $\Sigma$ of constraints is called *consistent* if there exists a mapping that satisfies every constraint in $\Sigma$.
We establish:
\[propaf\] An AF-pattern $P$ is satisfiable if and only if $\Gamma(P)$ is non-empty and $C(P)$ is consistent.
The only-if direction of this proposition is given by Lemma \[lemoif\] together with the observation that if $\mu \in {{\llbracket P \rrbracket_{G}}}$, then $\mu$ satisfies every constraint in $C(P)$. Since $P$ is satisfiable, such $G$ and $\mu$ exist, so $C(P)$ is consistent.
For the if direction, since $P$ does not have the UNION and OPT operators, $\Gamma(P)$ is a singleton $\{S\}$. Since $C(P)$ is consistent, there exists a mapping $\mu : S \to U$ satisfying every constraint in $C(P)$. Let $G$ be the graph consisting of all triples $(\mu(u),\mu(v),\mu(w))$ where $(u,v,w)$ is a triple pattern in $P$. It is straightforward to show by induction on the height of $Q$ that for every subpattern $Q$ of $P$, we have $\mu|_{S'} \in {{\llbracket Q \rrbracket_{G}}}$, where $\Gamma(Q)=\{S'\}$. Hence $\mu \in {{\llbracket P \rrbracket_{G}}}$ and $P$ is satisfiable.
Note that $\Gamma(P)$ can “blow up” only because of possible UNION and OPT operators, which are missing in an AF-pattern. Hence, for an AF-pattern $P$, we can efficiently compute $\Gamma(P)$ by a single bottom-up pass over $P$. Morever, $C(P)$ is a conjunction of possibly negated equalities and constant equalities. It is well known that consistency of such conjunctions can be decided in polynomial time [@decisionprocedures]. Hence, we conclude:
\[afptime\] Satisfiability for AF-patterns can be checked in polynomial time.
AF-reduction of well-designed patterns
--------------------------------------
A well-designed pattern is defined as a union of union-free well-designed patterns. Since a union is satisfiable if and only if one of its terms is, we will focus on union-free patterns in what follows. Formally, a union-free pattern $P$ is called *well-designed* [@perez_sparql_tods] if
1. for every subpattern of $P$ of the form $Q \, {\mathbin{\mathrm{FILTER}}}\, C$, all variables mentioned in $C$ also occur in $Q$; and
2. for every subpattern $Q$ of $P$ of the form $Q_1 {\mathbin{\mathrm{OPT}}}Q_2$, and every $?x \in {\mathrm{var}(Q_2)}$, if $?x$ also occurs in $P$ outside of $Q$, then $?x \in {\mathrm{var}(Q_1)}$.
We associate to every union-free pattern $P$ an AF-pattern $\rho(P)$ obtained by removing all applications of OPT and their right operands; the left operand remains in place. Formally, we define the following:
- If $P$ is a triple pattern, then $\rho(P)$ equals $P$.
- If $P$ is of the form $P_1 {\mathbin{\mathrm{AND}}}P_2$, then $\rho(P) = \rho(P_1)
{\mathbin{\mathrm{AND}}}\rho(P_2)$.
- If $P$ is of the form $P_1 {\mathbin{\mathrm{FILTER}}}C$, then $\rho(P) = \rho(P_1)
{\mathbin{\mathrm{FILTER}}}C$.
- If $P$ is of the form $P_1 {\mathbin{\mathrm{OPT}}}P_2$, then $\rho(P) =
\rho(P_1)$.
For further use we note that $\Gamma(P)$ and $\Gamma(\rho(P))$ are related in the following way. The proof by induction is straightforward.
\[lemstuck\] Let $S \in \Gamma(P)$ and let $S' \in \Gamma(\rho(P))$. Then $S'
\subseteq S$.
The announced result is now given by the following theorem. The if direction of this theorem is already known from a result by Pérez et al. [@perez_sparql_tods Lemma 4.3].
\[theorwell\] Let $P$ be a union-free well-designed pattern. Then $P$ is satisfiable if and only if $\rho(P)$ is.
Since $\rho(P)$ can be efficiently computed from $P$, the above Theorem and Corollary \[afptime\] imply:
\[ptimecor\] Satisfiability of union-free well-designed patterns can be tested in polynomial time.
Proof
-----
We prove the only-if direction of Theorem \[theorwell\]. We begin by introducing two auxiliary notations.
1. For any pattern $P$ and subpattern $Q$ of $P$, we denote by ${\mathrm{var}^{P}(Q)}$ the set of variables from ${\mathrm{var}(Q)}$ that also occur in $P$ outside of $Q$.
2. When $P$ is an AF-pattern with nonempty $\Gamma(P)$, it is readily seen that $\Gamma(P)$ in that case consists of a single scheme. We denote the unique scheme in $\Gamma(P)$ by $S(P)$.
The following lemma connects the above two notations:
\[lemrho\] Let $P$ be a union-free well-designed pattern, and let $Q$ be a subpattern of $P$ such that $\Gamma(Q)$ is nonempty. Then $\Gamma(\rho(Q))$ is nonempty as well, and ${\mathrm{var}^{P}(Q)} \subseteq S(\rho(Q))$.
By induction on the height of $Q$. If $Q$ is a triple pattern $(u, v, w)$, then we have $Q = \rho(Q)$ and ${\mathrm{var}^{P}(Q)} \subseteq {\mathrm{var}(Q)} = \{u,v,w\}\cap V = S(Q) =
S(\rho(Q))$ as desired.
If $Q$ is of the form $Q_1 {\mathbin{\mathrm{AND}}}Q_2$, then the definition of $\Gamma(Q)$ immediately implies that $\Gamma(Q_1)$ and $\Gamma(Q_2)$ must both be nonempty. Since $\rho(Q)=\rho(Q_1)
{\mathbin{\mathrm{AND}}}\rho(Q_2)$ we then obtain $S(\rho(Q)) = S(\rho(Q_1)) \cup
S(\rho(Q_2))$. Any $?x \in {\mathrm{var}^{P}(Q)}$ belongs to ${\mathrm{var}^{P}(Q_1)}$ or ${\mathrm{var}^{P}(Q_2)}$; we assume the former case as the latter case is analogous. By induction, we then have $?x \in S(\rho(Q_1))
\subseteq S(\rho(Q))$ as desired.
If $Q$ is of the form $Q_1 {\mathbin{\mathrm{OPT}}}Q_2$, then $\rho(Q)=\rho(Q_1)$. Recall that $\Gamma(Q)=\Gamma(Q_1) \cup \Gamma(Q_1 {\mathbin{\mathrm{AND}}}Q_2)$. If $\Gamma(Q_1)$ is nonempty we obtain by induction that $\Gamma(\rho(Q_1))=\Gamma(\rho(Q))$ is nonempty; if $\Gamma(Q_1
{\mathbin{\mathrm{AND}}}Q_2)$ is nonempty we obtain $\Gamma(\rho(Q_1))$ nonempty as in the case for AND. So, $S(\rho(Q))$ exists and is equal to $S(\rho(Q_1))$. Now let $?x \in {\mathrm{var}^{P}(Q)}$. If $?x \in {\mathrm{var}^{P}(Q_1)}$ then $?x \in S(\rho(Q_1))$ by induction. But if $?x \in
{\mathrm{var}^{P}(Q_2)}$, then also $?x \in {\mathrm{var}^{P}(Q_1)}$ since $P$ is well-designed. Hence we are done with this case.
Finally, let $Q$ be of the form $Q_1 {\mathbin{\mathrm{FILTER}}}C$. Since $\Gamma(Q)$ is nonempty, $\Gamma(Q_1)$ is nonempty as well. To show that $\Gamma(\rho(Q))$ is nonempty we must show that $S(\rho(Q_1)) \models C$. Thereto, consider a variable $?x$ mentioned in $C$. Since $P$ is well-designed, $?x \in {\mathrm{var}(Q_1)}$ and thus $?x \in {\mathrm{var}^{P}(Q_1)}$. By induction we obtain $?x \in S(\rho(Q_1))$. By Lemma \[lemstuck\], then also $?x
\in S$ for every $S \in \Gamma(Q_1)$. In other words, $S \not
\models \neg {\mathrm{bound}}(?x)$ for every $S \in \Gamma(Q_1)$. This rules out the possibility that $C$ is a negated bound-constraint, since we are given that $\Gamma(Q)$ is nonempty. On the other hand, this argument also shows that $S(\rho(Q_1))
\models C$ in the other cases, where $C$ is a bound-constraint or an (constant) (non)equality, as desired.
It remains to show that ${\mathrm{var}^{P}(Q)} \subseteq S(\rho(Q)) =
S(\rho(Q_1))$. Let $?x \in {\mathrm{var}^{P}(Q)}$. If $?x \in {\mathrm{var}(Q_1)}$ the result follows by induction. If $?x$ occurs in $C$ then, because $P$ is well-designed, also $?x \in {\mathrm{var}(Q_1)}$ and thus we are done.
We mention in passing an interesting corollary of the reasoning in the above proof, to the effect that well-designedness rules out any nontrivial use of negated bound-constraints:
If $P$ is a union-free well-designed pattern and $Q$ is a subpattern of $P$ of the form $Q_1
{\mathbin{\mathrm{FILTER}}}{\neg {\mathrm{bound}}(?x)}$, then $\Gamma(Q)$ is empty, in particular, $Q$ is unsatisfiable.
We are now ready to make the final step in the proof of Theorem \[theorwell\]:
\[lemlast\] Let $P$ be a union-free well-designed pattern. If $\mu \in {{\llbracket P \rrbracket_{G}}}$ and $\Gamma(\rho(P))$ is nonempty, then $\mu|_{S(\rho(P))} \in {{\llbracket \rho(P) \rrbracket_{G}}}$.
By induction on the structure of $P$. If $P$ is a triple pattern, then the claim is trivial.
So let $P$ be of the form $P_1 {\mathbin{\mathrm{AND}}}P_2$. Since $\Gamma(\rho(P))$ is nonempty and $\rho(P)=\rho(P_1) {\mathbin{\mathrm{AND}}}\rho(P_2)$, also $\Gamma(\rho(P_i))$ is nonempty for $i=1,2$. Then by induction, $\mu|_{S(\rho(P_i))} \in {{\llbracket \rho(P_i) \rrbracket_{G}}}$. Since they are restrictions of the same mapping $\mu$, we also have $\mu|_{S(\rho(P_1))} \sim \mu|_{S(\rho(P_2))}$, so the mapping $\mu|_{S(\rho(P_1))} \cup \mu|_{S(\rho(P_2))}$ belongs to ${{\llbracket \rho(P) \rrbracket_{G}}}$. Since $S(\rho(P)) = S(\rho(P_1)) \cup S(\rho(P_2))$, we obtain $\mu|_{S(\rho(P))} \in {{\llbracket \rho(P) \rrbracket_{G}}}$ as desired.
If $P$ is of the form $P_1 {\mathbin{\mathrm{OPT}}}P_2$, then we have $\rho(P)=\rho(P_1)$, so we are given that $\Gamma(\rho(P_1))$ is nonempty. By induction, $\mu|_{S(\rho(P_1))} \in {{\llbracket \rho(P_1) \rrbracket_{G}}} = {{\llbracket \rho(P) \rrbracket_{G}}}$ as desired.
Finally, if $P$ is of the form $P_1 {\mathbin{\mathrm{FILTER}}}C$ then by the nonemptiness of $\Gamma(\rho(P))$ we know that $S(\rho(P_1)) \models C$ and $S(\rho(P)) = S(\rho(P_1))$. Hence, by induction, $\mu|_{S(\rho(P_1))} \in {{\llbracket \rho(P_1) \rrbracket_{G}}}$. It remains to show that $\mu|_{S(\rho(P_1))} \models C$, but this follows immediately because $\mu \models C$ and $S(\rho(P_1))
\models C$.
With the above lemmas in hand, the only-if direction of Theorem \[theorwell\] can now be argued as follows. Since $P$ is satisfiable, $\Gamma(P)$ is nonempty by Lemma \[lemoif\]. By Lemma \[lemrho\] applied to $Q=P$, also $\Gamma(\rho(P))$ is nonempty. Since $P$ is satisfiable, there exist $G$ and $\mu$ such that $\mu \in {{\llbracket P \rrbracket_{G}}}$. Now applying Lemma \[lemlast\] yields that ${{\llbracket \rho(P) \rrbracket_{G}}}$ is nonempty. We conclude that $\rho(P)$ is satisfiable.
Experimental evaluation {#secexp}
=======================
We want to evaluate experimentally the positive results presented so far:
1. Wrong literal reduction (Proposition \[propeasy\]);
2. Satisfiability checking for the two fragments ${{\text{SPARQL}({\mathrm{bound}},\allowbreak =,\allowbreak \neq_c)}}$ and ${{\text{SPARQL}({\mathrm{bound}},\allowbreak \neq,\allowbreak \neq_c)}}$ by computing $\Gamma(P)$ (Theorem \[theordecidable\]);
3. Satisifiability checking for well-designed patterns, by reduction to AF-patterns (Proposition \[propaf\] and Theorem \[theorwell\]).
Our experiments follow up on those reported earlier by the third author and Vansummeren [@pica_realsparql]. As test datasets of real-life SPARQL queries, we use logs of the SPARQL endpoint for DBpedia, available at <ftp://download.openlinksw.com/support/dbpedia/>. This data source contains the “query dumps” from the year 2012, divided into 14 logfiles. Out of these we chose the three logs 20120913, 20120929 and 20121031 to obtain a span of roughly three months; we then took a sample of $100\,000$ queries from each of them. A typical query in the log has size between 75 and 125 (size measured as number of nodes in the syntax tree). About 10% of the queries in each log is not usable because they have syntax errors or because they use features not covered by our analysis.
The implementation of the tests was done in Java 7 under Windows 7, on an Intel Core 2 Duo SU94000 processor (1.40GHz, 800MHz, 3MB) with 3GB of memory (SDRAM DDR3 at 1067MHz).
Our tests measure the time needed to perform the analyses of SPARQL queries presented above. The timings are averaged over all queries in a log, and each experiment is repeated five times to smooth out accidental quirks of the operating system. Although we give absolute timings, the main emphasis is on the percentage of the time needed to analyse a query, with respect to the time needed simply to read and parse that query. If this percentage is small this demonstrates efficient, linear time complexity in practice. It will turn out that this is indeed achieved by our experiments, as shown in Table \[tabwl\].
In the following subsections we discuss the results in more detail.
Wrong literal reduction
-----------------------
Testing for and removing triple patterns with wrong literals in a pattern $P$ is performed by the reduction $\lambda(P)$ defined in the Appendix. From the definition of $\lambda(P)$ it is clear that it can be computed by a single bottom-up traversal of $P$ and this is indeed borne out by our experiments. Table \[tabwl\] shows that on average, wrong-literal reduction takes between 3 and 5% of the time needed to read and parse the input.
logfile baseline WL $\Gamma(P)$ AF
---------- ----------- ----------- ------- ------------- ------- ----------- --------
20120913 $39\,422$ $41\,254$ $5\%$ $44\,395$ $8\%$ $48\,329$ $10\%$
20120929 $34\,281$ $35\,868$ $5\%$ $38\,102$ $7\%$ $41\,087$ $9\%$
20121031 $32\,286$ $33\,186$ $3\%$ $34\,419$ $4\%$ $36\,993$ $8\%$
: Timings of experiments (averaged over five repeats). Times are in ms. Baseline is time to read and parse $1000\,000$ queries; WL stands for baseline plus time for wrong-literal reduction. $\Gamma(P)$ stands for WL plus time for computing $\Gamma(P)$. AF stands for baseline, plus testing well-designedness, plus doing AF-reduction and testing satisfiability (Proposition \[propaf\]). The percentages show the increases relative to the baseline.[]{data-label="tabwl"}
Interestingly, some real-life queries with literals in the wrong position were indeed found; one example is the following:
SELECT DISTINCT *
WHERE { 49 dbpedia-owl:wikiPageRedirects ?redirectLink .}
Computing $\Gamma(P)$
---------------------
In Section \[secsat\] we have seen that satisfiability for the decidable fragments can be tested by computing $\Gamma(P)$, but that the problem is NP-complete. Intuitively, the problem is intractable because $\Gamma(P)$ may be of size exponential in the size of $P$. This actually occurs in real life; a common SPARQL query pattern is to use many nested OPTIONAL operators to gather additional information that is not strictly required by the query but may or may not be present. We found in our experiments queries with up to 50 nested OPT operators, which naively would lead to a $\Gamma(P)$ of size $2^{50}$. A shortened example of such a query is shown in Figure \[figexp\].
SELECT DISTINCT *
WHERE {
?s a <http://dbpedia.org/ontology/EducationalInstitution>,
<http://dbpedia.org/ontology/University> .
?s <http://dbpedia.org/ontology/country> <http://dbpedia.org/resource/Brazil> .
OPTIONAL {?s <http://dbpedia.org/ontology/affiliation> ?ontology_affiliation .}
OPTIONAL {?s <http://dbpedia.org/ontology/abstract> ?ontology_abstract .}
OPTIONAL {?s <http://dbpedia.org/ontology/campus> ?ontology_campus .}
OPTIONAL {?s <http://dbpedia.org/ontology/chairman> ?ontology_chairman .}
OPTIONAL {?s <http://dbpedia.org/ontology/city> ?ontology_city .}
OPTIONAL {?s <http://dbpedia.org/ontology/country> ?ontology_country .}
OPTIONAL {?s <http://dbpedia.org/ontology/dean> ?ontology_dean .}
OPTIONAL {?s <http://dbpedia.org/ontology/endowment> ?ontology_endowment .}
OPTIONAL {?s <http://dbpedia.org/ontology/facultySize> ?ontology_facultySize .}
OPTIONAL {?s <http://dbpedia.org/ontology/formerName> ?ontology_formerName .}
OPTIONAL {?s <http://dbpedia.org/ontology/head> ?ontology_head .}
OPTIONAL {?s <http://dbpedia.org/ontology/mascot> ?ontology_mascot .}
OPTIONAL {?s <http://dbpedia.org/ontology/motto> ?ontology_motto .}
OPTIONAL {?s <http://dbpedia.org/ontology/president> ?ontology_president .}
OPTIONAL {?s <http://dbpedia.org/ontology/principal> ?ontology_principal .}
OPTIONAL {?s <http://dbpedia.org/ontology/province> ?ontology_province .}
OPTIONAL {?s <http://dbpedia.org/ontology/rector> ?ontology_rector .}
OPTIONAL {?s <http://dbpedia.org/ontology/sport> ?ontology_sport .}
OPTIONAL {?s <http://dbpedia.org/ontology/state> ?ontology_state .}
OPTIONAL {?s <http://dbpedia.org/property/acronym> ?property_acronym .}
OPTIONAL {?s <http://dbpedia.org/property/address> ?property_address .}
OPTIONAL {?s <http://www.w3.org/2003/01/geo/wgs84_pos#lat> ?property_lat .}
OPTIONAL {?s <http://www.w3.org/2003/01/geo/wgs84_pos#long> ?property_long .}
OPTIONAL {?s <http://dbpedia.org/property/established> ?property_established .}
OPTIONAL {?s <http://dbpedia.org/ontology/logo> ?ontology_logo .}
OPTIONAL {?s <http://dbpedia.org/property/website> ?property_website .}
OPTIONAL {?s <http://dbpedia.org/property/location> ?property_location .}
FILTER ( langMatches(lang(?ontology_abstract), "es") ||
langMatches(lang(?ontology_abstract), "en") )
FILTER ( langMatches(lang(?ontology_motto), "es") ||
langMatches(lang(?ontology_motto), "en") )
}
In practice, however, the blowup of $\Gamma(P)$ can be avoided as follows. Recall that Theorem \[theordecidable\] states that $P$ is satisfiable if and only if $\Gamma(P)$ is nonempty. The elements of $\Gamma(P)$ are sets of variables. Looking at the definition of $\Gamma(P)$, a set may be removed from $\Gamma(P)$ only by the application of a FILTER. Hence, only variables that are mentioned in FILTER conditions can influence the emptiness of $\Gamma(P)$; other variables can be ignored. For example, in the query in Figure \[figexp\], only two variables appear in a filter, namely [`?ontology_abstract`]{} and [`?ontology_motto`]{}, so that the maximal size of $\Gamma(P)$ is reduced to $2^2$.
In our experiments, it turns out that typically few variables are involved in filter conditions. Hence, the above strategy works well in practice.
Another practical issue is that, in this paper, we have only considered filter conditions that are bound checks, equalities, and constant-equalities, possibly negated. In practice, filter conditions typically apply built-in SPARQL predicates such as the predicate `langMatches` in Figure \[figexp\]. For the experimental purpose of testing the practicality of computing $\Gamma(P)$, however, such predicates can simply be treated as bound checks. In this way we can apply our experiments to 70% of the queries in the testfiles.
With the above practical adaptations, our experiments show that computing $\Gamma(P)$ is efficient: Table \[tabwl\] shows that it requires, on average, between 4 and 8% of the time needed to read and parse the input, and these timings even include the wrong-literal reduction.
Satisfiability testing for well-designed patterns
-------------------------------------------------
In Section \[secwell\] we have seen that testing satisfiability of a well-designed pattern can be done by testing satisfiability of the AF-reduction (Theorem \[theorwell\]). The latter can be done by testing nonemptiness of $\Gamma(P)$ and testing consistency of the filter conditions (Proposition \[propaf\]).
Computing the AF-reduction can be done by a simple bottom-up traversal of the pattern. Moreover, for an AF-pattern $P$, computing $\Gamma(P)$ poses no problems since it is either empty or a singleton. As far as testing consistency of filter conditions is concerned, our experiments yield a rather baffling observation: almost all well-designed patterns in the test sets have no filters at all. We cannot explain this phenomenon, but it implies that we have not been able to test the performance of the consistency checks on real-life SPARQL queries.
Anyhow, Table \[tabwl\] shows that doing the entire analysis of wrong-literal reduction, testing well-designedness, AF-reduction, computing $\Gamma(P)$, and consistency checking (in the few cases where the latter was necessary), incurs at most a 10% increase relative to reading and parsing the input.
Scalability
-----------
The experiments described above were run on sets of $100\,000$ queries each. We also did a modest scaling experiment where we varied the number of queries from $5\,000$ to $200\,000$. Table \[tabscale\] shows that the performance scales linearly.
input size $200\,000$ $100\,000$ $50\,000$ $10\,000$ $5\,000$ Pearson coeficient
------------- ------------ ------------ ----------- ----------- ---------- --------------------
baseline $74\,168$ $39\,422$ $21\,315$ $3\,596$ $1\,851$ $0.999924005$
WL $77\,800$ $41\,253$ $21\,876$ $3\,762$ $1\,942$ $0.999989454$
$\Gamma(P)$ $81\,730$ $44\,395$ $23\,552$ $4\,016$ $2\,036$ $0.999900948$
AF $91\,470$ $48\,329$ $26\,023$ $4\,463$ $2\,254$ $0.999044542$
: Scalability experiment (times in ms). Timings clearly scale linearly for increasing input size.[]{data-label="tabscale"}
Extension to SPARQL 1.1 {#sec1.1}
=======================
As already mentioned in the Introduction, SPARQL 1.0 has been extended to SPARQL 1.1 with a number of new operators for building patterns. The main new features are property paths; grouping and aggregates; BIND; VALUES; MINUS; EXISTS and NOT EXISTS-subqueries; and SELECT. A complete analysis of SPARQL 1.1 goes beyond the scope of the present paper. Nevertheless, in this section, we briefly discuss how our results may be extended to this new setting.
Property paths provide a form of regular path querying over graphs. This aspect of graph querying has already been extensively investigated, including questions of satisfiability and other kinds of static analysis such as query containment [@vrgoc_containment; @krrv_sparqlpp]. Therefore we do not discuss property paths any further here.
The SPARQL 1.1 features that we discuss can be grouped in two categories: those that cause undecidability, and those that are harmless as far as satisfiability is concerned. We begin with the harmless category.
SELECT operator and EXISTS-subqueries
-------------------------------------
SPARQL 1.1 allows patterns of the form ${{\textstyle \mathop{\mathrm{SELECT}}}}_S P$, where $S$ is a finite set of variables and $P$ is a pattern. The semantics is that of projection: solution mappings are restricted to the variables listed in $S$. Formally, we define $${{\llbracket {{\textstyle \mathop{\mathrm{SELECT}}}}_S
P \rrbracket_{G}}} = \{\mu|_{S \cap {\mathrm{dom}(\mu)}} \mid \mu \in {{\llbracket P \rrbracket_{G}}}\}.$$
This feature in itself does not influence the satisfiability of patterns. Indeed, patterns extended with SELECT operators can be reduced to patterns without said operators. The reduction amounts simply to rename the variables that are projected out by fresh variables that are not used anywhere else in the pattern; then the SELECT operators themselves can be removed. The resulting, SELECT-free, pattern is equivalent to the original one if we omit the fresly introduced variables from the solution mappings in the final result. In particular, the two patterns are equisatisfiable.
Rather than giving the formal definition of SELECT-reduction and formally stating and proving the equivalence, we give an example. Consider the pattern $P$: $$(c,p,?x) {\mathbin{\mathrm{OPT}}}( (?x,p,?y) {\mathbin{\mathrm{AND}}}{{\textstyle \mathop{\mathrm{SELECT}}}}_{?y} (?y,q,?z) {\mathbin{\mathrm{AND}}}{{\textstyle \mathop{\mathrm{SELECT}}}}_{?y} (?y,r,?z)
)$$ Renaming projected-out variables by fresh variables and omitting the SELECT operators yields the following pattern $P'$: $$(c,p,?x) {\mathbin{\mathrm{OPT}}}( (?x,p,?y) {\mathbin{\mathrm{AND}}}(?y,q,?z_1) {\mathbin{\mathrm{AND}}}(?y,r,?z_2)
)$$ Pattern $P'$ is equivalent to $P$ in the sense that for any graph $G$, we have ${{\llbracket P \rrbracket_{G}}} = \{\hat \mu \mid \mu \in {{\llbracket P' \rrbracket_{G}}}\}$, where $\hat \mu$ denotes the mapping obtained from $\mu$ by omitting the values for $?z_1$ and $?z_2$ (if at all present in ${\mathrm{dom}(\mu)}$).
Now that we know how to handle SELECT operators, we can also handle EXISTS-subqueries. Indeed, a pattern $P \, {\mathbin{\mathrm{FILTER}}}\, {{\mathop{\mathrm{EXISTS}}}(Q)}$ (with the obvious SQL-like semantics) is equivalent to ${{\textstyle \mathop{\mathrm{SELECT}}}}_{{\mathrm{var}(P)}}(P {\mathbin{\mathrm{AND}}}Q)$.
Features leading to undecidability
----------------------------------
In Section \[secund\] we have seen that as soon as one can express the union, composition and difference of binary relations, the satisfiability problem becomes undecidable. Since union and composition are readily expressed in basic SPARQL (${\mathbin{\mathrm{UNION}}}$ and ${\mathbin{\mathrm{AND}}}$), the key lies in the expressibility of the difference operator. In this subsection we will see that various new features of SPARQL 1.1 indeed allow expressing difference.
#### MINUS operator and NOT EXISTS subqueries
Any of these two features can quite obviously be used to express difference, so we do not dwell on them any further.
#### Grouping and aggregates
A known trick for expressing difference using grouping and counting [@sqlforsmarties] can be emulated in the extension of SPARQL 1.0 with grouping. We illustrate the technique with an example.
Consider the query $(?x,p,?y) {\mathbin{\mathrm{MINUS}}}(?x,q,?y)$ asking for all pairs $(a,b)$ such that $(a,p,b)$ holds but $(a,q,b)$ does not. We can express this query (with the obvious SQL-like semantics) as follows:
${{\textstyle \mathop{\mathrm{SELECT}}}}_{?x,?y} \bigl (
(?x,p,?y) {\mathbin{\mathrm{OPT}}}((?x,q,?y) {\mathbin{\mathrm{AND}}}(?xx,p,?yy)) \bigr )$\
$\mathrm{{GROUP\ BY}}\ {?x,?y}$\
$\mathrm{HAVING} \ {\mathrm{count}(?xx) = 0}$
Note that this technique of looking for the $(?x,?y)$ groups with a zero count for $?xx$ is very similar to the technique used to express difference using a negated bound constraint (seen in the proof of Lemma \[reductionlemma\]).
#### BIND and VALUES
We have seen in Section \[secfrageqc\] that allowing constant equalities in filter constraints allows us to emulate the difference operator. Two mechanisms introduced in SPARQL 1.1, BIND and VALUES, allow the introduction of constants in solution mappings. Together with equality constraints this allows us to express constant equalities, and hence, difference.
Specifically, using VALUES, we can express $P \, {\mathbin{\mathrm{FILTER}}}\, {?x=c}$ as $${{\textstyle \mathop{\mathrm{SELECT}}}}_{{\mathrm{var}(P)}}(P {\mathbin{\mathrm{AND}}}{{\textstyle\mathop{\mathrm{VALUES}}}}_{?x}(c)).$$ Using BIND, it can be expressed as $${{\textstyle \mathop{\mathrm{SELECT}}}}_{{\mathrm{var}(P)}}((P {\mathbin{\mathrm{BIND}}}_{?x'}(c)) {\mathbin{\mathrm{FILTER}}}{?x = {?x'}})$$ where $?x'$ is a fresh variable. Note the use of SELECT, which, however, does not influence satisfiability as discussed above. We conclude that SPARQL($=$) extended with BIND, or SPARQL($=$) extended with VALUES, have an undecidable satisfiability problem.
Conclusion {#seconcl}
==========
The results of this paper may be summarized by saying that, as long as the kinds of constraints allowed in filter conditions cannot be combined to yield inconsistent sets of constraints, satisfiability for SPARQL patterns is decidable; otherwise, the problem is undecidable. Moreover, for well-designed patterns, satisfiability is decidable as well. All our positive results yield straightforward bottom-up syntactic checks that can be implemented efficiently in practice.
We thus have attempted to paint a rather complete picture of the satisfiability problem for SPARQL 1.0. Of course, satisfiability is only the most basic automated reasoning task. One may now move on to more complex tasks such as equivalence, implication, containment, or query answering over ontologies. Indeed, investigations along this line for limited fragments of SPARQL are already happening [@pp_sparqlcontainment; @fransen_sparqlcontain; @kg_sparqlontology; @ox_cgiortp] and we hope that our work may serve to provide some additional grounding to these investigations.
We also note that in query optimization it is standard to check for satisfiability of subexpressions, to avoid executing useless code. Some specific works on SPARQL query optimization [@sequeda_ultrawrap; @groppe_sparqlosers] do mention that inconsistent constraints can cause unsatisfiability, but they have not provided sound and complete characterizations of satisfiability, like we have offered in this paper. Thus, our results will be useful in this direction as well.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank the anonymous referees for their critical comments on a previous version of this paper, which encouraged us to significantly improve the paper.
Appendix {#appendix .unnumbered}
========
Literals in the wrong place in triple patterns are easily dealt with in the following manner. We define the *wrong-literal reduction* of a pattern $P$, denoted by $\lambda(P)$, as a set that is either empty or is a singleton containing a single pattern $P'$:
- If $P$ is a triple pattern $(u,v,w)$ and $u$ is a literal, then $\lambda(P):=\emptyset$; else $\lambda(P):=\{P\}$.
- $\lambda(P_1 {\mathbin{\mathrm{UNION}}}P_2) := \lambda(P_1) \cup \lambda(P_2)$ if $\lambda(P_1)$ or $\lambda(P_2)$ is empty;
- $\lambda(P_1 {\mathbin{\mathrm{UNION}}}P_2) := \{P_1' {\mathbin{\mathrm{UNION}}}P_2' \mid P_1' \in
\lambda(P_1)$ and $P_2' \in \lambda(P_2)\}$ otherwise.
- $\lambda(P_1 {\mathbin{\mathrm{AND}}}P_2) := \{P_1' {\mathbin{\mathrm{AND}}}P_2' \mid P_1' \in
\lambda(P_1)$ and $P_2' \in \lambda(P_2)\}$.
- $\lambda(P_1 {\mathbin{\mathrm{OPT}}}P_2) := \emptyset$ if $\lambda(P_1)$ is empty;
- $\lambda(P_1 {\mathbin{\mathrm{OPT}}}P_2) := \lambda(P_1)$ if $\lambda(P_2)$ is empty but $\lambda(P_1)$ is nonempty;
- $\lambda(P_1 {\mathbin{\mathrm{OPT}}}P_2) := \{P_1' {\mathbin{\mathrm{OPT}}}P_2' \mid P_1' \in
\lambda(P_1)$ and $P_2' \in \lambda(P_2)\}$ otherwise.
- $\lambda(P_1 {\mathbin{\mathrm{FILTER}}}C) := \{P_1' {\mathbin{\mathrm{FILTER}}}C \mid P_1' \in
\lambda(P_1)\}$.
Note that the wrong-literal reduction never has a literal in the subject position of a triple pattern. The next proposition shows that, as far as satisfiability checking is concerned, we may always perform the wrong-literal reduction.
\[propeasy\] Let $P$ be a pattern. If $\lambda(P)$ is empty then $P$ is unsatisfiable; if $\lambda(P)=\{P'\}$ then $P$ and $P'$ are equivalent, i.e., ${{\llbracket P \rrbracket_{G}}} = {{\llbracket P' \rrbracket_{G}}}$ for every RDF graph $G$. Moreover, if $\lambda(P)=\{P'\}$ then $P'$ does not contain any triple pattern $(u,v,w)$ where $u$ is a literal.
Assume $P$ is a triple pattern $(u,v,w)$ and $u$ is a literal, so that $\lambda(P)=\emptyset$. Since $u$ is a constant, $\mu(u)$ equals the literal $u$ for every solution mapping $\mu$. Since no triple in an RDF graph can have a literal in its first position, ${{\llbracket P \rrbracket_{G}}}$ is empty for every RDF graph $G$, i.e., $P$ is unsatisfiable. If $u$ is not a literal, $\lambda(P)=\{P\}$ and the claims of the Proposition are trivial.
If $P$ is of the form $P_1{\mathbin{\mathrm{UNION}}}P_2$, or $P_1 {\mathbin{\mathrm{AND}}}P_2$, or $P_1
{\mathbin{\mathrm{FILTER}}}C$, the claims of the Proposition follow straightforwardly by induction.
If $P$ is of the form $P_1\, {\mathbin{\mathrm{OPT}}}\, P_2$, there are three cases to consider.
- If $\lambda(P_1)$ is empty then so is $\lambda(P)$. In this case, by induction, $P_1$ is unsatisfiable, whence so is $P$.
- If $\lambda(P_1) = \{P_1'\}$ is nonempty but $\lambda(P_2)$ is empty, then $\lambda(P)=\{P_1'\}$. By induction, $P_2$ is unsatisfiable. Hence, $P$ is equivalent to $P_1$, which in turn is equivalent to $P_1'$ by induction. That $P_1'$ does not contain any triple pattern with a literal in first position again follows by induction.
- If $\lambda(P_1)=\{P_1'\}$ and $\lambda(P_2)=\{P_2'\}$ are both nonempty, then $\lambda(P)=P_1' {\mathbin{\mathrm{OPT}}}P_2'$. By induction, $P_1$ is equivalent to $P_1'$ and so is $P_2$ to $P_2'$. Hence, $P$ is equivalent to $P_1' {\mathbin{\mathrm{OPT}}}P_2'$ as desired. By induction, neither $P_1'$ nor $P_2'$ contain any triple pattern with a literal in first position, so neither does $P_1' {\mathbin{\mathrm{OPT}}}P_2'$.
[^1]: School of Computer Science and Technology, Tianjin University, No.92 Weijin Road, Nankai District, Tianjin 300072, P.R. China, [[email protected]]([email protected]); work performed while at Universiteit Hasselt.
[^2]: Databases and Theoretical Computer Science, Universiteit Hasselt, Martelarenlaan 42, 3500 Hasselt, Belgium, [[email protected]]([email protected])
[^3]: This work has been funded by grant G.0489.10 of the Research Foundation Flanders (FWO).
[^4]: The cited works are seminal works on the semantics and complexity of SPARQL patterns, but they do not investigate the satisfiability of SPARQL patterns which is the main topic of the present paper. The cited works also extensively discuss minor deviations between the formalization and real SPARQL, and why these differences are inessential for the purpose of formal investigation.
[^5]: We define $\Gamma(P)$ for general patterns, not only for those belonging to the fragments considered in this Section, because we will make another use of $\Gamma(P)$ in Section \[secwell\].
[^6]: We also use the following standard notion of restriction of a mapping. If $f : X \to Y$ is a total function and $Z \subseteq X$, then the restriction $f|_Z$ of $f$ to $Z$ is the total function from $Z$ to $Y$ defined by $f|_Z(z)=f(z)$ for every $z \in Z$. That is, $f|_Z$ is the same as $f$ but is only defined on the subdomain $Z$.
|
---
abstract: 'A scalar model of wet active matter in the presence of an imposed temperature gradient, or chemical potential gradient, is considered. It is shown that there is a convective instability driven by a (negative) activity parameter. In this non-equilibrium steady state the generic long-ranged correlations are computed and compared and contrasted with the analogous results in a passive fluid. In addition, the non-equilibrium Casimir pressure or force is computed. Singularities in various physical quantities as the instability is approached are determined. Finally, we give the generalized Lorenz equations characterizing the fluid behavior above the instability and contrast these equations to the Lorenz equations for the Rayleigh-Bernard instability in a passive fluid.'
author:
- 'T.R. Kirkpatrick$^{1}$ and J.K. Bhattacherjee$^{1,2}$'
title: 'Fluctuations, renormalizations, and a convective instability in driven wet active matter'
---
In recent years there has been an enormous amount of research on various aspects of active matter [@Ramaswamy_2010; @Marchetti_et_al_2013]. The hydrodynamic description of active matter rest on identifying the relevant variables, conservation laws, and slow processes, and using symmetry to determine the allowed terms in the equations in some sort of gradient expansion [@Brand_et_al_2014]. This is exactly the case for passive matter, but the crucial differences between the active and passive matter description is i) the conservation laws are in general different. For example, active matter, or swimmers, chemically generate their own energy, so that the hydrodynamic energy or temperature equation is not a conservation law [@Loi_et_al_2008; @Marconi_et_al_2017], ii) the coefficients in the hydrodynamic equations may be very large compared to their passive counterparts, and even have different signs.
Active matter can be wet, that is, coupled to a momentum conserving solvent, or dry, that is, coupled to momentum absorbing boundaries. Wet active matter is more similar to usual passive fluid hydrodynamics because they have more conservation laws in common. Physically, wet active matter includes bacterial swarms in a fluid, the cytoskeleton of living cells, and biomimetic cell extracts. Many of these objects are approximately spherical objects.
Much of the work on active matter has focused on active liquid crystals. These have either a polar or nematic order parameter that lead to new active terms in the hydrodynamic stress tensor. Depending on the size of the activity, there can be completely new physics such as giant number fluctuations, and spontaneous flow instabilities above an activity threshold [@Simha_Ramaswamy_2002; @Voituriez_et_al_2005; @Narayan_et_al_2007]. Experiments on bacteria swarms and microtubule-based cell extracts [@Dombrowski_et_al_2004; @Sanchez_et_al_2012] seem to be closely related to numerical simulations [@Fielding_et_al_2011; @Giomi_et_al_2013; @Thampi_et_al_2013] of the active liquid crystal hydrodynamic equations .
Here we will be interested in simpler scalar models of wet active matter, and we will use these models to study a different aspect of wet active matter. In particular we are interested in active matter in a spatially dependent non-equilibrium steady state (NESS). We compare and contrast the generic long-range correlations that exist in non-equilibrium (NE) passive [@Kirkpatrick_Cohen_Dorfman_1982A; @Dorfman_Kirkpatrick_Sengers_1994; @DeZarate_Sengers_2006] and active matter, and the NE Casimir forces [@Kirkpatrick_DeZarate_Sengers_2013; @Kirkpatrick_DeZarate_Sengers_2014; @Kirkpatrick_DeZarate_Sengers_2016a; @Kirkpatrick_DeZarate_Sengers_2016b]. We show that for sufficiently large driving force, or activity, there is a convective instability, which we characterize. To this end, we use a model for wet (momentum conserving) active matter that was introduced by Tiribocchi et.al. [@Tiribocchi_et_al_2015]. It is an active fluid version of Model H in the Halperin-Hohenberg classification scheme [@Hohenberg_Halperin_1977]. The model has been used to study phase separation in active matter [@Tiribocchi_et_al_2015] and to illustrate some general properties of active matter [@Nardini_et_al_2017].
The hydrodynamic variables in the active fluid version of Model H are a concentration field, $\phi(\bf r, t)$, proportional to the density of active particles, that is *swimmers*, coupled to a momentum conserving solvent. The fluid velocity is $\bf{u}(\bf r, t)$. The equations of motion are, $$\dot{\phi}+\mathbf{u}\cdot\nabla\phi=D\nabla^2\phi,$$ and $$\dot{\mathbf{u}}+\mathbf{u}\cdot\mathbf{\nabla}\mathbf{u}=-\mathbf{\nabla}p+\nu\nabla^2\mathbf{u}+\mathbf{\nabla}\cdot\mathbf(\Sigma+\mathbf{P})$$ Here $D$ is a diffusion coefficient, $p$ is a pressure, which in general is a function of $\phi$ and a temperature $T$, $\nu$ is the kinematic viscosity, and $\mathbf P$ is a Gaussian thermal white noise Langevin force that is specified by it’s second moment, $$\begin{split}
\langle P_{ij}(\mathbf r,t)P_{kl}(\mathbf r',t')\rangle=2k_BT\nu\delta(\mathbf r-\mathbf r')\delta(t-t')\\(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl})
\end{split}$$ There is also a noise term in the concentration equation, but it is not important in what follows. Also in this equation there are higher order gradient terms, as well as non-linearities that we similarly neglect. $\mathbf\Sigma$ in Eq.(2) is a activity contribution to the stress tensor that is given by, $$\Sigma_{ij}=-\zeta(\partial_i\phi\partial_j\phi-\frac{\delta_{ij}}{3}(\nabla\phi)^2).$$ Such a term is allowed by symmetry in both passive and active fluids. Indeed, historically it is called a non-linear Burnett term [@Wong_et_al_1978]. For example, in passive fluids such a term is crucial for understanding the singular behavior of the viscosity as the liquid-gas critical point is approached [@Das_Bhattacharjee_2003]. In active matter there is no obvious restriction on the sign or the magnitude of $\zeta$ [@Ramaswamy_2010; @Tiribocchi_et_al_2015]. For contractile swimmers ($\zeta<0$) [@Williams_et_al_2014; @Thung_et_al_2017] the fluid flow increases as the swimmer density gradient increases. This case is of particular interest.
We consider the active fluid in a parallel plate geometry in the $z$-direction with the distance between the plates of size $L$, and the transverse direction $L_{\perp}\gg L$. A spatially dependent non-equilibrium steady state (NESS) is set up by having the plates at a different temperature, or chemical potential. In the former case the imposed temperature gradient will induce a average concentration gradient $\nabla\phi_0$ so that the average pressure gradient is zero. We assume that these average gradients are basically constant, or that there is a linear temperature and concentration profile so that $\nabla\phi_0=\Delta\phi_0/L$, where $\Delta\phi_0$ is the concentration difference between the two plates. We further assume that we can ignore the dynamical temperature fluctuations since they decay on a relatively fast time scale.
Linearizing the Eqs.(1) and (2) about the NESS allows us to determine the stability of the solution. Assuming no-slip boundary conditions we use the Fourier representations, $$\nonumber
\begin{split}
(\delta\phi(\mathbf{r},t), u_z(\mathbf{r}, t))=\frac{2}{L}\sum_{n=1}\int\frac{d\omega}{2\pi}\int_{\mathbf{k}_{\perp}}\\e^{i\mathbf{k}_{\perp}\cdot\mathbf{r}_{\perp}-i\omega t}\sin(\frac{n\pi z}{L})(\delta\phi(\mathbf{k}, \omega), u_z(\mathbf{k}, \omega)).
\end{split}$$ Here $\mathbf{k}_{\perp}=(k_x, k_y)$, $\mathbf{k}=(\mathbf{k}_{\perp}, n\pi/L)$, and $\mathbf{r}_{\perp}=(x,y)$. The interesting eigenfrequencies are, $$\omega_{\pm}=\frac{-ik^2}{2}(D+\nu)\pm\frac{i}{2}\sqrt{k^4(D-\nu)^2-4\zeta(\partial_z\phi_0)^2k_{\perp}^2},$$ with $k^2=k_{\perp}^2+n^2\pi^2/L^2$. For $\zeta=0$ there is a shear mode and a diffusion mode. For $\zeta>0$ and large, the two modes change from being diffusive to propagating. More interestingly, for $\zeta<0$, and large magnitude, the $\omega_+$ mode becomes unstable. Structurally this is very similar to what happens at the Rayleigh-Bernard (RB) instability. It first occurs at $k_z=k_{\perp}=\pi/L$. The analog of the Rayleigh number is $N\equiv(\Delta\phi_0)^2|\zeta|/D\nu$. Note that physically this is similar to the Rayleigh number, diffusion and viscosity suppress the instability (make $N$ smaller), while the activity parameter takes the place of gravity in driving the instability. The critical $N$ for the instability is $N_c=4\pi^2$. Near the instability, $$\omega_+\approx -i\frac{2\nu D}{(\nu+D)}([k_{\perp}-\frac{\pi}{L}]^2+\frac{\pi^2}{L^2}\epsilon),$$ where $N=N_c(1-\epsilon)$, with $\epsilon\ll1$. Note the implied critical slowing down as the instability is approached.
The feedback mechanism that causes the instability is that a negative (positive) average concentration gradient causes a concentration fluctuation to increase with positive (negative) $u_z$, which in turn causes an increase in the magnitude of $u_z$, etc. Opposing this positive feedback are the viscosity and diffusion. For sufficiently large $|\zeta|(\partial_z\phi_0)^2$ the feedback wins and there is an instability. The linear mathematics of this instability are identical, for example, to the instability in the Richardson combat/arms race model discussed in [@Kibble_Berkshire_2004]. If we exclude complex roots, we see below that the non-linear mathematics of the instability are similar to the pitchfork bifurcation that occurs in the RB problem.
It is interesting to compute various equal time correlation functions that characterize the generic long-ranged correlations in the NESS, as well as their amplification as the instability is approached. The largest one for $\zeta<0$ is, in what follows we use units where $k_BT=1$, $$\langle|\delta\phi(\mathbf{k})|^2\rangle=\frac{(\partial_z\phi_0)^2k_{\perp}^2}{D(\nu+D)k^2}\frac{1}{[k^4-Nk_{\perp}^2/L^2]}$$ (for $\zeta>0$ change the sign of $N$). Note that is the absence of activity ($N=0$), this long-range correlation is analogous to the experimentally well verified [@Law_et_al_1990] one that appears in a simple fluid in a temperature gradient [@Kirkpatrick_Cohen_Dorfman_1982A]. In the presence of activity it is enhanced (suppressed) compared to the passive fluid result for $\zeta<0$ ($\zeta>0$). Near the instability the singular contribution is, $$\langle|\delta\phi(\mathbf{k})|^2\rangle_{\mathrm{sing}}\approx\frac{(\Delta\phi_0)^2}{8\pi^2D(\nu+D)}\frac{1}{[(k_{\perp}-\pi/L)^2+\pi^2\epsilon/L^2]}.$$
The NE Casimir pressure, $p_{NE}(L)$, or force is also of interest. For a passive fluid in a NESS it has been discussed in great detail elsewhere [@Kirkpatrick_DeZarate_Sengers_2013; @Kirkpatrick_DeZarate_Sengers_2014; @Kirkpatrick_DeZarate_Sengers_2016a; @Kirkpatrick_DeZarate_Sengers_2016b; @Aminov_et_al_2015]. Physically, non-linear long-range fluctuations renormalize the pressure [@Kardar_Golestanain_1999], $$p_{NE}(L)=\frac{1}{2}\Big(\frac{\partial^2p}{\partial\phi^2}\Big)_T\overline{\langle\delta\phi(\mathbf{r})^2\rangle},$$ where the over-line denotes a spatial average. For small $|N|$ the equal time correlation function in Eq.(9) is, $$\overline{\langle\delta\phi(\mathbf{r})^2\rangle}_{|N|\ll 1}\approx\frac{(\Delta\phi_0)^2}{48\pi LD(D+\nu)}$$ Near the instability there is a singular contribution given by, $$\overline{\langle\delta\phi(\mathbf{r})^2\rangle}_{N=N_c(1-\epsilon)}\approx\frac{(\Delta\phi_0)^2}{32\pi^2LD(\nu+D)\sqrt{\epsilon}}.$$ For $\zeta>0$ and $|N|\gg 1$ one obtains, $$\overline{\langle\delta\phi(\mathbf{r})^2\rangle}_{|N|\gg 1}\approx\frac{(\Delta\phi_0)^2}{16LD(\nu+D)\sqrt{|N|}}.$$ Again we see that positive activity suppresses fluctuations effects, while negative activity enhances fluctuations.
As the instability is approached, the transport coefficients themselves become singularly renormalized. For example, the mode-coupling renormalization of the diffusion coefficient, $\delta D$ is given by, $$\delta D=-\frac{\langle u_z(\mathbf{r})\delta\phi(\mathbf{r})\rangle}{(\partial_z\phi_0)}.$$ Near the instability the singular contribution is, $$\delta D\approx\frac{1}{16L(D+\nu)\sqrt{\epsilon}}.$$ Numerically this is a very small (it is a $1/L$ effect) perturbation on the bare $D$ unless one is extraordinarily close to the instability. A similar result is obtained for the thermal diffusion coefficient near the RB instability [@Kirkpatrick_Cohen_1983]. In practice this means that more sophisticated self-consistent or renormalization group-like treatments are not needed to describe these instabilities [@Swift_Hohenberg_1977].
Finally, we consider the active fluid average NE motion above the instability threshold by constructing a three-mode Lorenz-like model [@Lorenz_1993]. The spatial structure of the convection rolls that occur for $N>N_c$ are determined by the critical wavenumbers being $k_{\perp}=k_z=\pi$ (here we use units where $L=1$) and by taking the fluid to be incompressible ($\nabla\cdot\mathbf{u}=0$). Consistent with this we define, $u_z=A(t)\cos\pi x\sin\pi z$, $u_x=-A(t)\sin\pi x\cos\pi z$, and $\delta\phi=B(t)\cos\pi x\sin\pi z+C(t)\sin 2\pi z$. The equations of motion for a scaled form of $A(t)$, $B(t)$ and $C(t)$ are, $$\dot x=\sigma(-x+ry+ryz) \nonumber$$ $$\dot y=-xz+x-y$$ $$\dot z=-2z+xy \nonumber$$ With $r=N/N_c$ and $\sigma=\nu/D$, the Prandtl number for this system [^1]. The non-linear term in the $\dot x$ equation reflects the activity non-linearity and is not present in the Rayleigh-Bernard problem. The fixed points of these Lorenz equations are, $$x=y=z=0 \quad(\mathrm{stable\quad for\quad} r<1)$$ and for $r>1$, $$\frac{x^2}{2}=r-1\pm\sqrt{(r-1)^2+r-1}\quad (\mathrm{two\quad\ real\quad roots})\nonumber$$ $$y=\frac{x}{(1+\frac{x^2}{2})}$$
$$z=\frac{xy}{2}\nonumber.$$
Compared to the RB Lorenz equations [@Kibble_Berkshire_2004; @Ott_1993] there are at least three interesting features associated with these Lorenz equations that warrant further study, i) The physical fixed points close to and above the convective instability are $(x,y,z)=(\pm 2^{1/2}(r-1)^{1/4}, \pm 2^{1/2}(r-1)^{1/4}, (r-1)^{1/2})$. In the RB problem, $(r-1)^{1/4}$ is replaced with $(r-1)^{1/2}$. This implies that the convective transport, which is proportional to $z$, is non-analytic or singular in the control parameter, unlike in the RB problem. ii) The crucial nonlinearity in the $\dot x$ equation is proportional to $r$, and thus increases with driving causing the time independent solution given by Eqs.(17) to become unstable at a smaller $r$ than in the RB problem. This implies that the Lorenz equations for this system are a more realistic representation of the active fluid hydrodynamics than the RB Lorenz equations are for the passive fluid hydrodynamics. iii) There are two additional complex fixed points of the Eqs.(17), compared to the RB problem. This suggest the Hopf-bifurcation and the transition to turbulence in this system will be qualitatively different than in the RB problem.
Indeed, it can be shown that the steady roll fixed point given by the Eqs.(17) will become unstable to time-dependent flow via a Hopf-bifurcation at a critical $r$, $r_H$, if $\sigma>3$. Typically $\sigma$ will be very large and in this limit a good approximate value for the critical $r$ value is, $$r_H=1+\frac{\sigma^2+6\sigma+\sigma\sqrt{\sigma^2+16\sigma+24}}{8(\sigma-3)}$$ For large $\sigma$ the frequency at the critical point is $\omega\approx\pm\sqrt{8\sigma(r_H-1)}$. We note that at $\sigma=10$, an exact solution of the cubic equation for the Hopf-bifurcation gives $r_H\approx 7.19$ [^2]. In contrast, for the RB Lorenz equations there is a Hopf-bifurcation if $\sigma>3$ at $r_H=\sigma(\sigma+5)/(\sigma-3)$, so that at $\sigma=10$, this $r_H\approx 21.4$. These two very different values of $r_H$ are consistent with the notion that the Lorenz equations given by Eq.(15) are a better model for the active matter full hydrodynamic equations than the original Lorenz equations are for the RB problem. Physically this is plausible because both sets of Lorenz equations ignore the fluid velocity convective nonlinearity in the $\dot x$ equation, but in the active matter case, this nonlinearity will be sub-leading to the activity nonlinearity if the activity coefficient is large.
We conclude with a number of further remarks:
1. The presumed size of $\zeta$ here and in [@Tiribocchi_et_al_2015] is quite large. In simple passive fluids at liquid state densities we can estimate the scale of $\zeta$ as follows. If $\phi$ is dimensionless then $\zeta$ has the dimensions of $\ell^4/\tau^2=v^2\ell^2$. Here $\ell$ is a length, $\tau$ is a time, and $v$ is a velocity. $\ell$ is the larger of the molecular diameter, $\sigma$, and the mean-free-path, which for liquid state densities would be $\sigma$, and $v$ is the thermal velocity. For water at STP this would numerically give $|\zeta|\approx 2\cdot 10^{-6}\mathrm{cm^4/sec^2}$, which is about an order of magnitude larger than $D\nu$ in water. The value of $N_c$ and this suggest that the activity part of $|\zeta|$ plays a qualitatively new role when it is larger than the passive one by a factor of $100$ to $1000$.
2. In general in both passive and active fluids a gradient expansion breaks down after Naiver-Stokes order (in simple fluids in three-dimensions) and the generalized description must be non-local [@Ernst_Dorfman_1975; @Ernst_et_al_1978; @Dorfman_Kirkpatrick_Sengers_1994; @Belitz_Kirkpatrick_Vojta_2005]. Technically one finds divergences in the calculation of higher order transport coefficients such as $\zeta$. These singular renormalization will have a scale set by the small passive generalized transport coefficients and will presumably not be important. This deserves further study.
3. The transition to turbulence in the Lorenz equations for this system is quite interesting [^3]. For $\sigma$ not too large, the Hopf-bifurcation for these equations is backwards. For $r$ close to but below $r_H$ there is a slowly decaying limit cycle about the stable fixed point. For $r>r_H$ there is no stable feature and the fluid motion is turbulent. For larger $\sigma$, the Hopf-bifurcation is forward and there is a stable limit cycle above $r_H$.
In comparison to the RB problem, the smaller values of $(x, y, z)$ at the fixed point, Eq.(17), seem to make the active fluid more stable near $r_H$.
Discussions with Jan Sengers are gratefully acknowledged. JKB would like to thank the IPST at the University of Maryland for support during the initial stages of this work. In addition, this work was supported by the National Science Foundation under Grant No. DMR-1401449.
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[^1]: This Prandtl number will generally be very large compared to the Prandtl number in the RB problem because particle diffusion coefficients are typically very small compared to thermal diffusion coefficients.
[^2]: Eq. 18 gives $r_H\approx 6.87$ for $\sigma=10$. For $\sigma=50$, both the full cubic equation and Eq.(18) give $r_H\approx16.1$.
[^3]: J. K. Bhattacherjee and T. R. Kirkpatrick, unpublished
|
---
abstract: 'We construct non-geometric compactifications by using the F-theory dual of the heterotic string compactified on a two-torus, together with a close connection between Siegel modular forms of genus two and the equations of certain K3 surfaces. The modular group mixes together the Kähler, complex structure, and Wilson line moduli of the torus yielding weakly coupled heterotic string compactifications which have no large radius interpretation.'
address:
- 'Department of Mathematics and Statistics, Colby College, Waterville, ME 04901'
- 'Department of Mathematics, University of California, Santa Barbara, CA 93106'
author:
- 'Andreas Malmendier and David R. Morrison'
date:
-
-
title: 'K3 surfaces, modular forms, and non-geometric heterotic compactifications'
---
Introduction {#introduction .unnumbered}
============
The traditional approach to producing low-dimensional physical models out of high-dimensional theories such as the string theories and M-theory has been to use a specific geometric compactification of the “extra” dimensions and derive an effective description of the lower-dimensional theory from the choice of geometric compactification. However, it has long been recognized that there are other possibilities: for example, one can couple perturbative string theory to an arbitrary superconformal two-dimensional theory (geometric or not) to obtain an effective perturbative string compactification in lower dimensions.
One way of making an analogous construction in non-perturbative string theory is to exploit the nonperturbative duality transformations which relate various compactified string theories (and M-theory) to each other. This idea was the basis of the construction of F-theory [@Vafa:1996xn], and more recently was used in constructions involving the type II theories [@Hellerman:2002ax] and the heterotic theories [@nongeometries]. We pursue a further non-geometric construction of heterotic compactifications in this paper.
Our construction of such models relies on a very concrete relationship between modular forms on the moduli space of certain K3 surfaces and the equations of those K3 surfaces [@MR2427457; @arXiv:1004.3335; @arXiv:1004.3503]. The K3 surfaces in question have a large collection of algebraic curve classes on them, generating a lattice known as $\Lambda^{1,1}\oplus E_8(-1) \oplus E_7(-1)$. The presence of these classes restricts the form of moduli space,[^1] which turns out to be a space admitting [*modular forms.*]{} The modular forms in question are the Siegel modular forms of genus two, which have previously made some appearances in the study of string compactification.[^2]
The close connection between modular forms and equations allows us to mimic the basic F-theory construction, and build an interesting class of non-geometric heterotic compactifications which have duals described in terms of families of K3 surfaces. The starting point is the heterotic string compactified on a torus, and we exploit the non-perturbative duality symmetries which this theory possesses. We give the construction in considerable detail.
The paper begins with a review of F-theory in Section \[sec:F\] and then proceeds to give a construction of non-geometric heterotic compactifications in Section \[sec:nongeom\]. These compactifications require certain $5+1$-dimensional soliton solutions (serving as sources for scalar fields) whose heterotic construction is discussed in Section \[sec:fiveB\].
After a brief digression in Section \[sec:so32\] to interpret our construction in the context of the $\mathfrak{so}(32)$-heterotic string, we specialize in Section \[sec:sixD\] to the case of compactifications to six dimensions. There we find a surprise: although non-geometric techniques were used for the construction, the models we obtain are not new, but were already known (at least in dual form). We conclude the paper with a discussion of this surprise and its implications.
Review of F-theory {#sec:F}
==================
One of the fundamental interpretations of F-theory is in terms of the type IIB string, where it depends on three ingredients: an ${\operatorname{SL}_2(\mathbb{Z})}$ symmetry of the theory, a complex scalar field $\tau$ (the axio-dilaton) with positive imaginary part (in an appropriate normalization) on which ${\operatorname{SL}_2(\mathbb{Z})}$ acts by fractional linear transformations, and D7-branes, which serve as a source for the multi-valuedness of $\tau$ if $\tau$ is allowed to vary.
In a standard compactification of the type IIB string, $\tau$ is a constant and D7-branes are absent. Vafa’s idea in proposing F-theory [@Vafa:1996xn] was to simultaneously allow a variable $\tau$ and the D7-brane sources, arriving at a new class of models in which the string coupling is never weak.
Since we cannot use the axio-dilaton $\tau$ directly in these models, it would be natural to identify the physically relevant quantity with $\mathbb{H}/{\operatorname{SL}_2(\mathbb{Z})}$ (where $\mathbb{H}$ denotes the complex upper half-plane), but this turns out to be slightly too simplistic. To obtain the full range of F-theory models, we need instead to consider some functions of $\tau$ which are not invariant under ${\operatorname{SL}_2(\mathbb{Z})}$, but rather transform in a specific way. A function $f(\tau)$ which satisfies $$f\left(\frac{a\tau + b}{c\tau +d}\right) = (c\tau + d)^m f(\tau)$$ for $\bigl(\begin{smallmatrix}
a&b\\c&d
\end{smallmatrix} \bigr) \in {\operatorname{SL}_2(\mathbb{Z})}$ is a [*modular form of weight $m$ for ${\operatorname{SL}_2(\mathbb{Z})}$*]{} and such forms turn out to provide the flexibility we need for F-theory.
A simple way to write down some modular forms of even weight $m=2k$ for ${\operatorname{SL}_2(\mathbb{Z})}$ is to use what are called [*normalized Eisenstein series*]{}, defined as $$E_{2k}(\tau) = \frac1{2\zeta(2k)}
\sum_{(0,0)\ne(m,n)\in \mathbb{Z}^2}
\frac1{(m\tau+n)^{2k}},$$ where $\zeta(2k)=\sum_{n\ge1} n^{-2k}$ is Riemann’s zeta function. These normalized Eisenstein series have a Fourier expansion (in $q=e^{2\pi i\tau}$) of the form $$E_{2k}(\tau) = 1 + O(q),$$ which is the reason for including the normalization factor. It is known that $E_4(\tau)$ and $E_6(\tau)$ generate the entire ring of modular forms for ${\operatorname{SL}_2(\mathbb{Z})}$. The combination $$\begin{aligned} \Delta_{12}(\tau) &= 4\left(-\frac13\, E_4(\tau)\right)^3 +
27 \left(-\frac2{27}\, E_6(\tau)\right)^2\\
&= -\frac4{27}\, E_4(\tau)^3 + \frac4{27}\, E_6(\tau)^2
\end{aligned}$$ plays a special role in the theory.[^3] In particular, if we compactify the parameter space ${\operatorname{SL}_2(\mathbb{Z})}$ to $\overline{{\operatorname{SL}_2(\mathbb{Z})}}$, then $\Delta_{12}(\tau)$ extends to the compactification and vanishes on the boundary (which corresponds to the $q\to0$ limit).
An F-theory compactification (regarded as a compactification of the type IIB string with variable axio-dilaton) takes as its staring point a compact space $W$, a complex line bundle $\mathcal{L}$ on $W$ and sections $f(w)$ and $g(w)$ of the associated bundles $\mathcal{L}^{\otimes 4}$ and $\mathcal{L}^{\otimes 6}$. Then there is a (possibly non-supersymmetric) F-theory model with a variable $\tau$ function and ${\operatorname{SL}_2(\mathbb{Z})}$ symmetry, in which $f(w)$ is identified with $-\frac13E_4(\tau)$ and $g(w)$ is identified with $-\frac2{27}E_6(\tau)$. One must also insert seven-branes of various kinds along the zeros of $$\Delta(w):=4f(w)^3+27g(w)^2.$$ The geometry behind this construction is a beautiful story from 19th-century mathematics: the Weierstrass $\wp$-function. In order to define a doubly-periodic meromorphic function in the complex plane (with periods $1$ and $\tau$), Weierstrass introduced a function of $z\in\mathbb{C}$ and $\tau$: $$\wp(z,\tau) = \frac1{z^2} + \sum_{(m,n)\ne(0,0)} \left(
\frac1{(z-m\tau-n)^2} - \frac1{(m\tau+n)^2}\right).$$ This has a Laurent expansion (using the normalized Eisenstein series as well as the special values $\zeta(4)=\pi^4/90$ and $\zeta(6)=\pi^6/945$): $$\begin{aligned} \wp(z,\tau) &= z^{-2}+ 6\, \zeta(4) \, E_4(\tau) \, z^2 + 10\,
\zeta(6)\, E_6(\tau) \, z^4
+O(z^6),\\
&= z^{-2}+ \frac{\pi^4}{15} \, E_4(\tau)\, z^2 + \frac{2\pi^6}{189}\,
E_6(\tau)\, z^4
+O(z^6),
\end{aligned}$$ from which Laurent expansions for $(\wp(z,\tau)')^2$ and $\wp(z,\tau)^3$ can be calculated: $$\begin{aligned}
(\wp(z,\tau)')^2 &= 4z^{-6}-\frac{8\pi^4}{15} \, E_4(\tau) \, z^{-2}-\frac{32\pi^6}{189} \, E_6(\tau)+O(z)\;,\\
\wp(z,\tau)^3 &= z^{-6} + \frac{\pi^4}5\, E_4(\tau)\, z^{-1}+\frac{6\pi^6}{189} \, E_6(\tau)+O(z).\end{aligned}$$ It follows that $$(\wp(z,\tau)')^2 - 4\, \wp(z,\tau)^3 +\frac{4\pi^4}3E_4(\tau)\, \wp(z,\tau)=
-\frac{8\pi^6}{27}E_6(\tau)+O(z)=
-\frac{8\pi^6}{27}E_6(\tau).$$ (This is an exact expression since the left hand side is an entire holomorphic function which is bounded since it is doubly-periodic, and hence constant.) If we set $x:=\frac1{\pi^2}\wp(z,\tau)$ and $y:=\frac1{2\pi^3}\wp(z,\tau)'$, we find an equation for the elliptic curve with modular parameter $\tau$: $$y^2 = x^3 - \frac13\, E_4(\tau)\, x - \frac2{27}\, E_6(\tau).$$
Conversely, if we start from an elliptic curve with an equation of the form $$\label{eq:ell}
y^2=x^3+fx+g$$ which is nonsingular, then we can recover $\tau$ up to ${\operatorname{SL}_2(\mathbb{Z})}$ transformation as $$\tau = \frac{\int_{\gamma_2} \frac{dx}{\sqrt{x^3+fx+g}}}
{\int_{\gamma_1}\frac{dx}{\sqrt{x^3+fx+g}}} ,$$ where $(\gamma_1,\gamma_2)$ is an oriented basis of the first homology of the elliptic curve, such that $f = -\frac{\lambda^4}3 E_4(\tau)$, $g=-\frac{2\lambda^6}{27}E_6(\tau)$ for some nonvanishing scale factor $\lambda$. The condition for nonsingularity of is that the quantity $$\Delta:=4f^3+27g^3$$ does not vanish. It is this close connection between geometry and modular forms which allows the construction of families of elliptic curves with certain knowledge of the behavior of $\tau$ in such families.
To understand when the corresponding F-theory models are supersymmetric, we follow the duality between F-theory and M-theory. That duality is based on a key fact: Compactifying M-theory on a torus whose complex structure is labeled by $\tau$ and whose area is $A$ gives a model dual to the type IIB string compactified on a circle of radius[^4] $A^{-3/4}$ whose axio-dilaton has value $\tau$ [@Schwarz:1995dk; @Aspinwall:1995fw]. This then gives a connection between the F-theory construction and a dual geometric compactification of M-theory: if the F-theory model is further compactified on $S^1$ (which can be done without breaking any supersymmetry that might be present), a model will be obtained which is dual to M-theory compactified on the total space of the family $$y^2=x^3+f(w)\, x+g(w)$$ of elliptic curves over $W$. (The insertions of seven-branes in the F-theory model go over to singular elliptic fibers in the M-theory model which may require special treatment, but for generic sections $f$ and $g$ the total space of the family is nonsingular and the compactification makes sense as it stands.)
One can then ask whether the geometric M-theory model breaks or preserves supersymmetry, and the answer is known: supersymmetry is preserved exactly when the total space of the family is a Calabi–Yau manifold, which happens exactly when the line bundle $\mathcal{L}$ is the anti-canonical bundle of the base, i.e., $\mathcal{L}=\mathcal{O}_W(-K_W)$. In this way, we recover the familiar conditions for a supersymmetric F-theory compactification.
To complete the story, we need to know what types of seven-branes need to be inserted. The answer here comes from algebraic geometry, through work of Kodaira [@MR0184257] and Néron [@MR0179172] which classifies the possible singular limits in one-parameter families of elliptic curves and thereby gives a catalog of the different types of seven-branes which must be inserted. This catalog is by now well-known, but we will reproduce it in Table \[tab:kodaira\], in which labels for the types of seven-branes are given using Kodaira’s notation. The type of brane depends on the orders of vanishing of $f$, $g$, and $\Delta$ at the singular point $P$, and determines both the type of singularity which appears in the M-theory dual, and the transformation in ${\operatorname{SL}_2(\mathbb{Z})}$ which describes how $\tau$ changes when the singular point is encircled. The $I_n$ type corresponds to a stack of $n$ D7-branes, while the $I_n^*$ type corresponds to a stack of $n{+}4$ D7-branes on top of an orientifold plane. The last line of the table (labeled “non-minimal”) can be avoided by a suitable choice of line bundle for the Weierstrass model.
brane type $\operatorname{ord}_P(f)$ $\operatorname{ord}_P(g)$ $\operatorname{ord}_P(\Delta)$ singularity transformation
------------------ --------------------------- --------------------------- -------------------------------- --------------- ----------------------------------------------------------------------------------------
$I_0$ $\ge0$ $\ge0$ $0$ none $\begin{pmatrix}\hphantom{-}1&\hphantom{-}0\\\hphantom{-}0&\hphantom{-}1\end{pmatrix}$
$I_n$, $n\ge1$ $0$ $0$ $n$ $A_{n-1}$ $\begin{pmatrix}\hphantom{-}1&\hphantom{-}n\\\hphantom{-}0&\hphantom{-}1\end{pmatrix}$
$II$ $\ge1 $ $ 1 $ $ 2 $ none $\begin{pmatrix}\hphantom{-}1&\hphantom{-}1\\-1&\hphantom{-}0\end{pmatrix}$
$III$ $ 1 $ $ \ge2 $ $ 3 $ $ A_1$ $\begin{pmatrix}\hphantom{-}0&\hphantom{-}1\\-1&\hphantom{-}0\end{pmatrix}$
$IV$ $ \ge2 $ $ 2 $ $ 4 $ $ A_2$ $\begin{pmatrix}\hphantom{-}0&\hphantom{-}1\\-1&-1\end{pmatrix}$
$I_0^*$ $\ge2$ $\ge3$ $6$ $D_{4}$ $\begin{pmatrix}-1&\hphantom{-}0\\\hphantom{-}0&-1\end{pmatrix}$
$I_n^*$, $n\ge1$ $2$ $3$ $n+6$ $D_{n+4}$ $\begin{pmatrix}-1&-n\\\hphantom{-}0&-1\end{pmatrix}$
$IV^*$ $\ge3$ $ 4$ $ 8$ $ E_6$ $\begin{pmatrix}-1&-1\\\hphantom{-}1&\hphantom{-}0\end{pmatrix}$
$III^*$ $ 3 $ $ \ge5 $ $ 9 $ $ E_7$ $\begin{pmatrix}\hphantom{-}0&-1\\\hphantom{-}1&\hphantom{-}0\end{pmatrix}$
$II^*$ $ \ge4$ $ 5 $ $ 10 $ $ E_8$ $\begin{pmatrix}\hphantom{-}0&-1\\\hphantom{-}1&\hphantom{-}1\end{pmatrix}$
non-minimal $\ge4$ $\ge6$ $\ge12$ non-canonical –
: Kodaira–Néron classification of singular fibers and monodromy[]{data-label="tab:kodaira"}
Non-geometric heterotic models {#sec:nongeom}
==============================
By analogy, we now wish to study as our basic theory the heterotic string compactified on $T^2$ to produce an eight-dimensional effective theory. (This will be our analogue of the type IIB string in the previous section.) This effective theory has a complex scalar field which, after symmetries are taken into account, takes its values in the Narain space[^5] [@Narain:1985jj] $$\mathcal{D}_{2,18}/O(\Lambda^{2,18})$$ which is the quotient of the symmetric space for $O(2,18)$, $$\mathcal{D}_{2,18} := (O(2)\times O(18))\backslash O(2,18) ,$$ by the automorphism group $O(\Lambda^{2,18})$ of the unique integral even unimodular lattice $\Lambda^{2,18}$ of signature $(2,18)$. (This discrete group is sometimes called $O(2,18,\mathbb{Z})$.) In an appropriate limit, this space decomposes as a product of spaces parameterizing the Kähler and complex structures on $T^2$ as well as Wilson line expectation values around the two generators of $\pi_1(T^2)$ (see [@Narain:1986am]). However, that decomposition is only preserved by a parabolic subgroup $\Gamma\subset
O(\Lambda^{2,18})$, which is much smaller. Letting the moduli of the entire space vary arbitrarily (i.e., employing the full $O(\Lambda^{2,18})$ symmetry) will produce a compactification which has a right to be called non-geometric, because the Kähler and complex structures on $T^2$, and the Wilson line values, are not distinguished under the $O(\Lambda^{2,18})$-equivalences but instead are mingled together. With no Kähler class, we lose track of geometry.[^6]
The construction we will give of non-geometric heterotic compactifications actually uses an index $2$ subgroup $O^+(\Lambda^{2,18})
\subset O(\Lambda^{2,18})$ defined by $$O^+(\Lambda^{2,18}):= O^+(2,18) \cap O(\Lambda^{2,18}) ,$$ where $O^+(p,q)$ denotes the subgroup of $O(p,q)$ preserving the orientation on positive $p$-planes. The group $O^+(\Lambda^{2,18})$ is the maximum subgroup of $O(\Lambda^{2,18})$ whose action preserves the complex structure on the symmetric space, and thus is the maximal subgroup for which modular forms can be holomorphic. The corresponding quotient $$\label{k3moduli}
\mathcal{D}_{2,18}/ O^+(\Lambda^{2,18})$$ is a degree two cover of the Narain moduli space. The group $O^+(\Lambda^{2,18})$ is still large enough to thoroughly mix the Kähler, complex, and Wilson line moduli.
The quotient space is the parameter space for elliptically fibered K3 surfaces with a section: this is a statement of the duality between F-theory and the heterotic string in eight dimensions [@Vafa:1996xn], and the identification of the discrete group for this moduli problem as $O^+(\Lambda^{2,18})$ is well-known in the mathematics literature (see, for example, [@MR2336040]).[^7] To use these K3 surfaces in a similar way to the way that elliptic curves were used in constructing F-theory, we would need a close connection between $O^+(\Lambda^{2,18})$-modular forms and the equations of the corresponding elliptically fibered K3 surfaces; unfortunately such a connection is not known.
However, by making a simple and natural restriction on our heterotic theories, we can find and exploit such a connection. Namely, let us consider heterotic models with only a single nonzero Wilson line expectation value. For definiteness, we restrict to the $\mathfrak{e}_8\oplus\mathfrak{e}_8$ heterotic string, and note that asking for an unbroken gauge algebra of $\mathfrak{e}_8\oplus\mathfrak{e}_7$ will ensure that only a single Wilson line expectation value is nonzero. (There is a similar story for the $\mathfrak{so}(32)$ string which we will describe in Section \[sec:so32\].)
Let $L^{2,3}$ be the lattice of signature $(2,3)$ which is the orthogonal complement of $E_8(-1) \oplus E_7(-1)$ in $\Lambda^{2,18}$. By insisting that the Wilson lines values associated to the[^8] $E_8(-1)\oplus E_7(-1)$ sublattice be trivial (which leaves the algebra $\mathfrak{e}_8\oplus\mathfrak{e}_7$ unbroken), we restrict to those heterotic vacua parameterized by the space $$\mathcal{D}_{2,3}/O(L^{2,3}).$$ The corresponding degree two cover is $$\mathcal{D}_{2,3}/O^+(L^{2,3}),$$ and this space parameterizes elliptically fibered K3 surfaces with section which have one fiber of Kodaira type $III^*$ or worse and another fiber of Kodaira type precisely[^9] $II^*$. Such K3 surfaces contain the lattice $\Lambda^{1,1} \oplus E_8(-1)\oplus E_7(-1)$ inside their Néron–Severi lattice, and are often referred to as “lattice-polarized K3 surfaces”. The $\Lambda^{1,1}$ summand is generated by the classes of the fiber and the section of the elliptic fibration.[^10]
As we will describe below, the modular forms for $O^+(L^{2,3})$ have the desired property: there is a close geometric connection to the corresponding lattice-polarized K3 surfaces. (A similar picture was developed in earlier work in the case of no nontrivial Wilson line expectation values, using modular forms for $O^+(\Lambda^{2,2})$ [@nongeometries].)
Let $\mathbb{H}_g$ denote the Siegel upper half-space of genus $g$, on which the Siegel modular group ${\operatorname{Sp}_{2g}(\mathbb{Z})}$ acts. As explained in Appendix \[app:discrete\], for $g=2$ there is a homomorphism ${\operatorname{Sp}_4(\mathbb{R})}\to O^+(2,3)$ which induces an isomorphism $$\label{eq:h2iso}
\mathbb{H}_2 \cong \mathcal{D}_{2,3}.$$ By a result of Vinberg [@vinberg-siegel] reviewed in Appendix \[app:discrete\], the image of ${\operatorname{Sp}_4(\mathbb{Z})}\to O^+(L^{2,3})$ is a subgroup of index $2$, and the ring of $O^+(L^{2,3})$-modular forms turns out to correspond to the ring of Siegel modular forms with $g=2$ [*of even weight.*]{}[^11] Igusa [@MR0141643] showed that this latter ring is a polynomial ring in four free generators of degrees $4$, $6$, $10$ and $12$. We explicitly describe Igusa’s generators $\psi_4$, $\psi_6$, $\chi_{10}$ and $\chi_{12}$ in Section \[Siegel\_modular\_forms\]. (Igusa later showed [@MR0229643] that for the full ring of ${\operatorname{Sp}_4(\mathbb{Z})}$-modular forms, one needs an additional generator $\chi_{35}$, also described in Section \[Siegel\_modular\_forms\], which is algebraically dependent on the others. In fact, $\chi_{35}^2$ is an an explicit polynomial in $\psi_4$, $\psi_6$, $\chi_{10}$ and $\chi_{12}$ which is given in .)
The key geometric fact, due in different forms to Kumar [@MR2427457] and to Clingher–Doran [@arXiv:1004.3335; @arXiv:1004.3503], is the equation for an elliptically fibered K3 surface whose periods give the point $\underline{\tau}$ in the Siegel upper halfspace $\mathbb{H}_2$, with the coefficients in the equation being Siegel modular forms of even weight. (This is analogous to the Weierstrass equation for the elliptic curve $\mathbb{C}/\langle 1,\tau\rangle$ with coefficients being Eisenstein series in $\tau$). That equation is: $$\label{eq:imp}
y^2 = x^3 - t^3 \, \left( \frac{1}{48} \psi_4(\underline{\tau}) \, t +4\chi_{10}(\underline{\tau})
\right) \, x + t^5 \, \left( t^2 - \frac{1}{864} \psi_6(\underline{\tau}) \,
t+\chi_{12}(\underline{\tau})\right).$$ Just as in the elliptic curve case, the statement has two parts: starting from $\underline{\tau}$, we obtain the equation of a K3 surface . But conversely, if we start with a K3 surface $S$ with an equation of the form[^12] $$\label{eq:genform}
y^2 = x^3 + a \, t^4 \, x + b \, t^6 + c\, t^3 \, x + d\, t^5 + t^7,$$ and we determine a point in $\mathcal{D}_{2,3}$ by calculating the periods of the holomorphic $2$-form on $S$ over a basis of the orthogonal complement of $\Lambda^{1,1} \oplus E_8(-1)\oplus E_7(-1)$ in $H^2(S,\mathbb{Z})$ (which in turn determines $\underline{\tau}\in \mathbb{H}_2$ using the isomorphism ), then for some nonzero scale factor $\lambda$, $$a = -\frac{\lambda^4}{48}\, \psi_4(\underline{\tau}), \
b = -\frac{\lambda^6}{864}\, \psi_6(\underline{\tau}), \
c = -4\, \lambda^{10}\, \chi_{10}(\underline{\tau}), \
d = \lambda^{12}\, \chi_{12}(\underline{\tau}).$$ We verify in Appendix \[K3fibration\] that the K3 surface defined by agrees with the ones found by Kumar and by Clingher–Doran.
The strategy for constructing a non-geometric heterotic compactification is now clear. Start with compact manifold $Z$ and a line bundle $\Lambda$ on $Z$. Pick sections $a(z)$, $b(z)$, $c(z)$, and $d(z)$ of $\Lambda^{\otimes 4}$, $\Lambda^{\otimes 6}$, $\Lambda^{\otimes 10}$, and $\Lambda^{\otimes 12}$, respectively. Then there is a non-geometric heterotic compactification on $Z$ with variable $\underline{\tau}$ and $O^+(L^{2,3})$ symmetry for which $$\begin{aligned}
a(z) &= -\frac{1}{48}\, \psi_4(\underline{\tau}), \\
b(z) &= -\frac{1}{864}\, \psi_6(\underline{\tau}), \\[0.4em]
c(z) &= -4 \, \chi_{10}(\underline{\tau}), \\[0.6em]
d(z) &= \chi_{12}(\underline{\tau}).
\end{aligned}$$ (We can eliminate the scale factor $\lambda$, if any, by making a change of coordinates $(x,y,t)\mapsto (\lambda^{14}x,\lambda^{21}y,\lambda^6t)$.) Appropriate five-branes must be inserted on $Z$ as dictated by the geometry of the corresponding family of K3 surfaces $$\label{eq:family}
y^2 = x^3 + a(z)\,t^4x + b(z)\,t^6 + c(z)\,t^3x + d(z)\,t^5 + t^7.$$ We will explore these five-branes in the next section.
To understand when the non-geometric heterotic compactifications we have constructed are supersymmetric, we follow the duality between the heterotic string and F-theory. The heterotic compactification on $T^2$ with parameter $\underline{\tau} \in \mathbb{H}_2$ is dual to the F-theory compactification on the elliptically fibered K3 surface $S_{\underline{\tau}}$ defined by , where $t$ is a local coordinate on the base $\mathbb{P}^1$ of the elliptic fibration. Note that at $t=\infty$, $S_{\underline{\tau}}$ has a Kodaira fiber of type precisely $II^*$: it can be no worse because the coefficient of $t^7$ in $\eqref{eq:imp}$ is $1$. At $t=0$, there is a Kodaira fiber of type $III^*$ or worse.
When $a(z)$, $b(z)$, $c(z)$, and $d(z)$ are sections of line bundles over $Z$, we wish to determine whether F-theory compactified on the elliptically fibered manifold is supersymmetric, and this in turn depends on whether the total space defined by is itself a Calabi–Yau manifold. The base of the elliptic fibration on the total space is a $\mathbb{P}^1$-bundle $\pi:W \to Z$ which takes the form $W=\mathbb{P}(\mathcal{O}\oplus \mathcal{M})$ for some line bundle $\mathcal{M}$ that coincides with the normal bundle of $\Sigma_0:=\{t=0\}$ in $W$. Restricting the various terms in to $\Sigma_0$, we find relations $\Lambda^{\otimes 4} \otimes \mathcal{M}^{\otimes 4}
= \Lambda^{\otimes 10} \otimes \mathcal{M}^{\otimes 3}
= (\mathcal{L}|_{\Sigma_0})^{\otimes 4}$ and $
\mathcal{M}^{\otimes 7}
=\Lambda^{\otimes 6} \otimes \mathcal{M}^{\otimes 6}
=\Lambda^{\otimes 12} \otimes \mathcal{M}^{\otimes 5}
= (\mathcal{L}|_{\Sigma_0})^{\otimes 6}$. It follows that that $\mathcal{M}=\Lambda^{\otimes 6}$ and $\mathcal{L}|_{\Sigma_0}=\Lambda^{\otimes 7}$ (up to torsion). In other words, our $\mathbb{P}^1$-bundle must take the form $W=\mathbb{P}(\mathcal{O}\oplus \Lambda^{\otimes 6})$. This property can be traced back to the fact that the coefficient of $t^7$ in is $1$.
Now to check the condition for supersymmetry, note that $$-K_W = \Sigma_0 + \Sigma_\infty + \pi^{-1}(-K_Z),$$ where $\Sigma_\infty:=\{t=\infty\}$. Since $\Sigma_0$ and $\Sigma_\infty$ are disjoint, it follows that the condition for supersymmetry $\mathcal{L}=\mathcal{O}_W(-K_W)$ is equivalent to $\Lambda = \mathcal{O}_Z(-K_Z)$.
Let us briefly comment on the relationship of our construction with the appearance of Siegel modular forms in string compactifications involving the “STUV” model, as described in [@Curio:1997si] and the references therein. If we take the mirror of our family of lattice-polarized K3 surfaces [@stringK3; @MR1420220], we will obtain a family of K3 surfaces depending on $17$ complex parameters whose quantum Kähler moduli space is $\mathcal{D}_{2,3}/O^+(L^{2,3})$. The K3 surfaces in the new family all contain a lattice $L^{1,2}$ within $H^{1,1}$ with the property that $L^{2,3}\cong \Lambda^{1,1}\oplus L^{1,2}$. If $X$ is a Calabi–Yau threefold which has a one-parameter family of such K3 surfaces on it, then the $(1,1)$ classes on $X$ also include the lattice $L^{1,2}$. Type IIA string theory compactified on $X$ is the dual theory of the heterotic STUV model, as discussed in [@Curio:1997si] and elsewhere. It is natural that quantum corrections of this gravitational theory would respect the symmetry group $O^+(L^{2,3})$ and so would turn out to be related to Siegel modular forms as well.
Five-branes {#sec:fiveB}
===========
The base $W$ of an elliptic fibration maps naturally to the compactification $\overline{\mathbb{H}/{\operatorname{SL}_2(\mathbb{Z})}}$ of the parameter space $\mathbb{H}/{\operatorname{SL}_2(\mathbb{Z})}$, and if this map is nonconstant, there must be singular fibers (at which the $j$-invariant approaches $\infty$). In fact, for a generic elliptic fibration, all seven-branes will have $j\to\infty$, and those correspond to familiar seven-brane constructions in type IIB string theory (D7-branes, possibly combined with orientifold planes).
The situation for fibrations of lattice-polarized K3 surfaces is very different. There is a Satake-Baily-Borel compactification [@MR0118775; @BailyBorel] $\overline{\mathcal{D}_{2,3}/O^+(L^{2,3})}$ of the parameter space whose boundary has codimension two, and this implies that a one-parameter family of lattice-polarized K3 surfaces [*need not reach the boundary!*]{} That would suggest that it might be possible to have a family which never degenerates (i.e., with no brane insertions needed), but this is not the case: there are codimension one loci where some elements of $O^+(L^{2,3})$ have fixed points, and there must always be branes associated with these fixed loci.
To find group elements with fixed points, note that a reflection in a lattice element of square $-2$ has a fixed locus of codimension one, belongs to $O^+(L^{2,3})$, and does not belong to $SO^+(L^{2,3})\cong {\operatorname{Sp}_4(\mathbb{Z})}$. As a consequence, such reflections must act as $-1$ on the ${\operatorname{Sp}_4(\mathbb{Z})}$-modular forms of odd weight, and the fixed locus of any such reflection must be contained in the vanishing locus of any ${\operatorname{Sp}_4(\mathbb{Z})}$-modular form of odd weight. The modular forms of odd weight are generated by Igusa’s form $\chi_{35}$, so that form must vanish along the fixed loci of our reflections.
From the point of view of K3 geometry, if the periods are preserved by the reflection in $\delta$ with $\delta^2=-2$, then $\delta$ must belong to the Néron-Severi lattice of the K3 surface. That is, the lattice $\Lambda^{1,1}\oplus E_8(-1)\oplus E_7(-1)$ must be enlarged by adjoining $\delta$. It is not hard to show (using methods of [@ISBF], for example), that there are only two ways this enlargement can happen (if we have adjoined a single element only): either the lattice is extended to $\Lambda^{1,1}\oplus E_8(-1)\oplus E_8(-1)$ or it is extended to $\Lambda^{1,1}\oplus E_8(-1)\oplus E_7(-1) \oplus
\langle -2 \rangle$. In the former case, the fibers in the elliptic fibration become $II^*$, $II^*$ and 4 $I_1$, whereas in the latter case, the fibers become $II^*$, $III^*$, $I_2$ and 4 $I_1$.
If we start with an elliptically fibered K3 surface , then it is easy to see what the condition is for the first enhancement: we want the fiber at $t=0$ to go from type $III^*$ to type $II^*$, and this is achieved by setting $c=0$.
To see the second enhancement requires a computation. Starting with , we compute the discriminant of the elliptic fibration to be $$\Delta
=
t^9 \left( 4(at+c)^3+27t(t^2+bt+d)^2 \right)$$ The zeros of $\Delta/t^9$ represent the location of the $I_1$ fibers, so to find out when they coincide, we calculate the discriminant of that polynomial of degree $5$ in $t$ (which will vanish precisely when there are multiple roots). That discriminant turns out to take the form $$2^83^{12}\ell(a,b,c,d)^3q(a,b,c,d),$$ where, if we assign weights $4$, $6$, $10$, $12$ to $a$, $b$, $c$, $d$, respectively, then $\ell$ is the polynomial of weighted degree $20$ $$\label{ell}
\ell(a,b,c,d) := a^2 \, d-a\,b\,c+c^2 \;,$$ and $q$ is the polynomial of weighted degree $60$ $$\label{q-eqn}
\begin{split}
q(a,b,c,d) &:=
11664\,{d}^{5}
+864\,{a}^{3}{d}^{4}
-5832\,{b}^{2}{d}^{4}
+16\,{a}^{6}{d}^{3}
+216\,{a}^{3}{b}^{2}{d}^{3}
\\ &
-2592\,{a}^{2}bc{d}^{3}
+16200\,a{c}^{2}{d}^{3}
+729\,{b}^{4}{d}^{3}
+888\,{a}^{4}{c}^{2}{d}^{2}
-5670\,a{b}^{2}{c}^{2}{d}^{2}
\\ &
-13500\,b{c}^{3}{d}^{2}
+16\,{a}^{7}{c}^{2}d
+216\,{a}^{4}{b}^{2}{c}^{2}d
-3420\,{a}^{3}b{c}^{3}d
+4125\,{a}^{2}{c}^{4}d
\\ &
+729\,a{b}^{4}{c}^{2}d
+6075\,{b}^{3}{c}^{3}d
-16\,{a}^{6}b{c}^{3}
+16\,{a}^{5}{c}^{4}
-216\,{a}^{3}{b}^{3}{c}^{3}
\\ &
+2700\,{a}^{2}{b}^{2}{c}^{4}
-5625\,ab{c}^{5}
-729\,{b}^{5}{c}^{3}
+3125\,{c}^{6}
\end{split}$$ (which we computed directly using computer algebra).
The role of the polynomial $\ell$ is easy to see: it vanishes on precisely those K3 surfaces for which $f$ and $g$ have a common zero (at $t=-c/a$). Those are cases in which two $I_1$’s are replaced by a fiber of type $II$, but such a change does not affect the lattice or the gauge algebra and so these are not the K3 surfaces we are looking for.
Since the two lattice enhancements occur at $c=0$ and $q(a,b,c,d)=0$, we predict that $c\cdot q(a,b,c,d)$ should vanish along the locus where there is some degeneration. Indeed it turns out (as verified in Section \[Siegel\_modular\_forms\]) that $$q\left(-\frac1{48}\psi_4,-\frac1{864}\psi_6,-4 \, \chi_{10}, \, \chi_{12}\right)
= 2^{-8} \, \chi_{35}^2/\chi_{10},$$ confirming the prediction.
Thus, a generic non-geometric compactification constructed from these lattice-polarized K3 surfaces will have two types of five-branes, corresponding to[^13] $c=0$ and $q(a,b,c,d)=0$. From the heterotic side, these five-brane solitons are easy to see. When $q(a,b,c,d)=0$, we have an additional gauge symmetry enhancement to include $\mathfrak{su}(2)$, and the parameters of the theory include a Coulomb branch for that gauge theory on which the Weyl group $W_{\mathfrak{su}(2)}=\mathbb{Z}_2$ acts. There is thus a five-brane solution in which the field has a $\mathbb{Z}_2$ ambiguity encircling the location in the moduli space of enhanced gauge symmetry.
The other five-brane solution is similar: at $c=0$, there is an enhancement from $\mathfrak{e}_7$ to $\mathfrak{e}_8$ gauge symmetry, and a similar $\mathbb{Z}_2$ acts on the moduli space, leading to a solution with a $\mathbb{Z}_2$ ambiguity. These two brane solutions are the analogue of the simplest brane (a single D7-brane) in F-theory.
Finding a complete catalog of five-brane solutions for this theory is quite challenging. As we explain in Appendix \[app:modulispaces\], the parameter space $\mathcal{D}_{2,3}/O^+(L^{2,3})$ for our construction is closely related to some other moduli spaces: the moduli space of homogeneous sextics in two variables, the moduli space of Abelian surfaces, and the moduli space of curves of genus two. A version of Kodaira’s classification was given for curves of genus two by Namikawa and Ueno [@MR0369362] and this can in principle be used to give a classification of degenerations of these lattice-polarized K3 surfaces. We illustrate how this works in a number of interesting cases in Appendix \[degs\_and\_branes\].
The $\mathfrak{so}(32)$ heterotic string {#sec:so32}
========================================
It turns out that the total space of a lattice-polarized K3 surface of the form with lattice polarization by $\Lambda^{1,1} \oplus E_8(-1) \oplus E_7(-1)$ always admits a second elliptic fibration with a different polarization [@arXiv:1004.3503] (see also [@nongeometries]), which can be related to the $\mathfrak{so}(32)$ heterotic string. To see this, consider the birational transformation $$x=X^2T, \quad y=X^2Y, \quad t=X$$ applied to . (In applying the transformation, we make the substitution and then divide by the common factor of $X^4$.) The result is the equation $$\label{eq:so32}
\begin{aligned}
Y^2 &= X^2T^3 + aX^2T + bX^2 + cXT + dX + X^3 \\
&= X^3 + (T^3+aT + b)X^2 + (cT+d)X.
\end{aligned}$$ To more easily see the structure, we introduce homogeneous coordinates $[S,T]$ on the base $\mathbb{P}^1$ and write the equation as $$\label{eq:homog}
Y^2 = X^3 +S (T^3+aS^2T + bS^3)X^2 + S^7(cT+dS)X.$$ It is a straightforward exercise to complete the cube and calculate the discriminant, which is $$\Delta = -S^{16}(cT+dS)^2 (T^6+2aS^2T^4+2bS^3T^3+a^2S^4T^2+(2ab-4c)S^5T+(b^2-4d)S^6).$$ Since $S$ divides the coefficient of $X^2$ and $S^2$ divides the coefficient of $X$ in , we conclude that the fiber at $S=0$ is type $I_{10}^*$, so the gauge algebra is enhanced to $\mathfrak{so}(28)$. In addition, at the point $[S,T]=[-c,d]$, the coefficient of $X^2$ is [*not*]{} divisible by $(cT+dS)$, so the Kodaira type is $I_2$ and there is an additional enhancement of the gauge algebra to $\mathfrak{su}(2)$. (For generic coefficients, the other factor in the discriminant contributes six fibers of type $I_1$.)
Since the constant term in vanishes, the section $X=Y=0$ defines an element of order $2$ in the Mordell-Weil group. It follows as in [@MR1416960; @pioneG] that the actual gauge group of this model is $(\operatorname{Spin}(28)\times SU(2))/\mathbb{Z}_2$.
The intrinsic property of elliptically fibered K3 surfaces which leads to an equation of the form is the requirement that there be a $2$-torsion element in the Mordell–Weil group, and that one fiber in the fibration be of type $I_n^*$ for some $n\ge 10$. Under these assumptions, we can choose coordinates so that the specified fiber is at $T=\infty$. If we were to simply ask that the fiber at $T=\infty$ be of type $I_{10}^*$ or worse (as well as having a $2$-torsion element), then a slight modification of the argument in section 4 of [@Aspinwall:1996vc] or appendix A or [@nongeometries] would show that the equation takes the form $$Y^2 = X^3 + (\alpha T^3 + \beta T^2 +aT + b)X^2 + (cT+d)X.$$ However, if $\alpha=0$ then it turns out that the coefficient $f$ of the Weierstrass form vanishes to order at least $3$ and the coefficient $g$ vanishes to order at least $4$, which means that the fiber no longer has type $I_n^*$. Thus, our requirement of being type $I_n^*$ for some $n\ge10$ implies that $\alpha\ne0$ (or in a family, that $\alpha$ has no zeros). Then the coordinate change $$(X,Y,T) \mapsto (\alpha^{-2} X, \alpha^{-3} Y, \alpha^{-1}(T-\frac13\beta))$$ (followed by multiplying the equation by $\alpha^6$) yields an equation of the form , i.e., one in which the coefficient of $T^3X^2$ is $1$ and the coefficient of $T^2X^2$ is $0$.
The lattice enhancements which we have discussed can also be interpreted for these models. When $c=0$, the gauge group enhances to $\operatorname{Spin}(32)/\mathbb{Z}_2$, and when $q(a,b,c,d)=0$, there is an additional enhancement of the gauge algebra by a factor of $\mathfrak{su}(2)$.
Six-dimensional compactifications {#sec:sixD}
=================================
We now specialize to six-dimensional non-geometric heterotic compactifications. The base $Z$ is a Riemann surface with an effective anti-canonical divisor, so it must either be an elliptic curve or the Riemann sphere. In the first case $Z=T^2$, the line bundle $\Lambda$ is trivial, and the entire construction is just a $T^2$ compactification of the eight-dimensional theory, with no monodromy or brane insertions needed. In particular, the parameters of the eight-dimensional theory do not vary, and the compactification is geometric.
More interesting is the case $Z=\mathbb{P}^1$. In this case, as derived at the end of Section \[sec:nongeom\], we have $\Lambda=\mathcal{O}_{\mathbb{P}^1}(2)$, $\Lambda^{\otimes 6}=\mathcal{O}_{\mathbb{P}^1}(12)$, and the base $W=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}
\oplus \mathcal{O}_{\mathbb{P}^1}(12))$ of the non-geometric model coincides with the Hirzebruch surface $\mathbb{F}_{12}$, and is similar to models first studied in [@FCY2]. In particular, a Calabi-Yau three-fold $\mathbf{\bar{X}} \to \mathbb{F}_{12}$ can be defined by the Weierstrass equation $$\label{WE_MV2_2}
\begin{split}
0 = - y^2 \, z + x^3 +& \, s^4 \, t^3 \, \Big(a(u,v) \, t + c(u,v) \, s\Big) \, x \, z^2 \\
+ & \, s^5 \, t^5 \, \Big( t^2 + b(u,v) \, s \, t + d(u,v) \, s^2\Big) \, z^3 \;,
\end{split}$$ where $[u:v]$ denotes the homogeneous coordinates for the $\mathbb{P}^1$ that constitutes the base of the Hirzebruch surface $\mathbb{F}_{12}$, and $[s:t]$ denotes the homogeneous coordinates of the $\mathbb{P}^1$ that constitutes the fiber, and the coefficients $a(u,v)$, $b(u,v)$, $c(u,v)$, and $d(u,v)$ have degrees $8$, $12$, $20$, and $24$, respectively, as homogeneous polynomials on $\mathbb{P}^1$. The two $\mathbb{C}^*$-torus actions that define $\mathbb{F}_{12}$ are given by $$(s,t,u,v) \sim (\lambda^{-12} s, t, \lambda u , \lambda v) \;,\quad (s,t,u,v) \sim (\lambda s, \lambda t, u , v) \;,$$ for $\lambda \in \mathbb{C}^*$. The model has a fiber of type $II^*$ over the section $\sigma_\infty$ of self-intersection $-12$ given by $s=0$, and a fiber of type $III^*$ over a disjoint section $\sigma_0$ given by $t=0$ with $\sigma_0 = \sigma_\infty + 12\, F$ where $F$ is the fiber class. The divisor class of $\Delta =0$ is $[\Delta]= - 12 K_{\mathbb{F}_{12}}$ where $K_{\mathbb{F}_{12}}= - 2 \, \sigma_\infty- 14 \, F$. The two curves $s=0$ and $t=0$ will account for a large portion of the divisor class $[\Delta]$. The remaining part $[\Delta']$ of the divisor not contained in $\sigma_0$ and $\sigma_\infty$ is $[\Delta'] = [\Delta] - 10 \, \sigma_\infty - 9 \, \sigma_0$. It follows that what is left of the discriminant divisor $[\Delta']$ will not collide with $\sigma_\infty$ since $\sigma_\infty \cdot [\Delta']=0$. On the other hand, the divisor $[\Delta']$ will collide with $\sigma_0$ in a total number of $\sigma_0 \cdot [\Delta']=60$ points counted with multiplicity.
Because of the choice of $\mathbb{F}_{12}$, from the $\mathfrak{e}_8\oplus \mathfrak{e}_7$ heterotic perspective there are no instantons on the $\mathfrak{e}_8$ summand but the $\mathfrak{e}_7$ summand has instanton number $24$. Since instantons for $\mathfrak{e}_8$ must be pointlike instantons, this allows the heterotic model to make sense perturbatively,[^14] with instantons of finite size on $\mathfrak{e}_7$.
The enhancement from $\mathfrak{e}_7$ to $\mathfrak{e}_8$ thus occurs at the $20$ points $\{c(z)=0\}$. These points are regarded as responsible for the matter representation of $\mathfrak{e}_7$ [@geom-gauge], giving $20$ half-hypermultiplets in the $56$-dimensional representation.
The enhancement from $\mathfrak{e}_8\oplus \mathfrak{e}_7$ to $\mathfrak{e}_8\oplus \mathfrak{e}_7\oplus \mathfrak{su}(2)$ occurs along the locus $q(a,b,c,d)=0$, which consists of $120$ points on $\mathbb{P}^1$. At these points, the $\mathbb{P}^1$ fiber of $\mathbb{F}_{12}$ is tangent to the residual discriminant divisor $\Delta'$.
Similarly, our requirement of a fiber of type $I_n^*$, $n\ge10$, in the $\mathfrak{so}(32)$ heterotic string will not allow for the “hidden obstructor” of [@Aspinwall:1996vc] to occur. (Such “hidden obstructor” points occur when the coefficient of $T^3X^2$ in vanishes; as previously discussed, this vanishing is inconsistent with the fiber being of type $I_n^*$.) Avoiding these “hidden obstructors” allows for a perturbative description in the $\mathfrak{so}(32)$ case as well. The twenty zeros of $c(z)$ give rise to half-hypermultiplets in the tensor product of the vector representation of $\operatorname{Spin}(28)$ with the fundamental representation of $\operatorname{SU}(2)$ (which is again a $56$-dimensional quaternionic representation).
Discussion
==========
Something rather surprising has just happened: although we started with a construction for non-geometric heterotic compactifications, the resulting compactifications in six dimensions are actually the familiar F-theory duals of geometric compactifications of the heterotic string on K3 surfaces! How did this happen?
Recall the original description of the heterotic/F-theory duality in six dimensions, as described in [@FCY2] and further amplified in [@Friedman:1997yq]: the large radius limit of the heterotic string corresponds to a degeneration limit of the F-theory space, in which the F-theory base actually splits into two components. This limit involves tuning a holomorphic parameter, and tuning holomorphic parameters away from constant has an interesting property: it is not possible to keep the parameter controllably close to the limiting value. Instead, once the holomorphically varying quantity is non-constant, it samples [*all*]{} values.
The conclusion, then, is that taking a heterotic compactification even a “small distance” from the large radius limit destroys the traditional semiclassical interpretation and no longer allows us to discuss the compactification as being that of a manifold with a bundle. This is not unlike what happens in type II compactifications, where the analysis of $\Pi$-stability [@douglas; @Douglas:2001hw; @Aspinwall:2001dz] shows that going any distance away from large radius limit, no matter how small, necessarily changes the stability conditions on some D-brane classes and so destroys the semiclassical interpretation of the theory.
It would be interesting to check whether this phenomenon persists in compactifications to four dimensions. There, the presence of fluxes may alter the structure of the moduli space, which is here discussed using purely geometric considerations. It may be that when fluxes are involved, some truly new non-geometric models can be constructed. We leave this question for future work.
As pointed out to us by the referee, there may be interesting lessons from this work for double field theory (see, for example, [@Aldazabal:2013sca; @Berman:2013eva; @Hohm:2013bwa] and references therein) and its heterotic extensions. We leave this for future work as well.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Chuck Doran and Sav Sethi for helpful discussions. The first author acknowledges the generous support of the Kavli Institute for Theoretical Physics, and the second author is grateful to the Kavli Institute for the Physics and Mathematics of the Universe for hospitality during the early stages of this project. The work of the second author is supported by National Science Foundation grants DMS-1007414 and PHY-1307513 and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
Discrete groups and modular forms {#app:discrete}
=================================
Modular forms for $O^+(\Lambda^{2,2})$
--------------------------------------
Explicit generators for $O(\Lambda^{2,2})$ are given in [@MR2013800], together with their actions on $\mathbb{H}\times \mathbb{H}$. It implies that we can identify $O^+(\Lambda^{2,2})$ with $P({\operatorname{SL}_2(\mathbb{Z})}\times {\operatorname{SL}_2(\mathbb{Z})})\rtimes \mathbb{Z}_2 $, where the automorphism $\mathbb{Z}_2$ acts to exchange $\rho$ and $\tau$.
The modular forms of weight $d$ for this group must be functions of $\rho$ and $\tau$ of bidegree $(d,d)$ invariant under the exchange. We claim that this ring of modular forms is a free polynomial algebra on $E_4(\rho)E_4(\tau)$, $E_6(\rho)E_6(\tau)$, and $\Delta_{12}(\rho)\Delta_{12}(\tau)$.
To see why this is true, let $t_4=E_4(\rho)E_4(\tau)$, $t_6=E_6(\rho)E_6(\tau)$, and $t_{12}=\Delta_{12}(\rho)\Delta_{12}(\tau)$ be elements of bidegree $(4,4)$, $(6,6)$, and $(12,12)$, respectively. If they are not algebraically independent, then there exists a nonvanishing homogeneous polynomial $P(T)$ in the graded ring $\mathbb{C}[T_4,T_6,T_{12}]$ satisfying $P(t_4,t_6,t_{12})=0$. We take as $P(T)$ the polynomial of minimal degree and write it in the form $P_0(T_4,T_6,T_{12}) T_{12} + P_1(T_4,T_6)$. In the equation $$P_0(t_4,t_6,t_{12}) t_{12} + P_1(t_4,t_6) = 0$$ we take the limit $\rho \to i \infty$. Notice that under $\rho \to i \infty$ we have $$E_4(\rho) \to 1 \;, \quad E_6(\rho) \to 1\; \quad \eta(\rho) \to 0 \;.$$ Therefore, we get $P_1\big(E_4(\tau),E_6(\tau)\big)=0$. But $E_4(\tau)$ and $E_6(\tau)$ are algebraically independent. It follows that $P_1(T_4, T_6)=0$ and $P_0(T)$ is different from zero with $P_0(t_4,t_6,t_{12})=0$. Since $P_0(T)$ is of smaller degree than $P(T)$ we get a contradiction to the assumed minimality of $P(T)$.
Fix $k\ge 0$ even and let $n(k)$ denote the dimension of the space of modular forms $M_k$ of weight $k$ for ${\operatorname{SL}_2(\mathbb{Z})}$. It is well-known that $$\label{dimension}
n(k):= \dim M_k = \left\lbrace \begin{array}{ll} \lfloor k/12\rfloor & \text{for} \; k \equiv 2 (4), \\ \lfloor k/12 \rfloor+1 & \text{otherwise}. \end{array}\right.$$ Equivalently, the dimension $n(k)$ equals the number of nonnegative integer solutions to the equation $k=4p+6q$ as $E_4$ and $E_6$ generate the ring of modular forms. Then, the dimension of $M_k \otimes M_k$ is $n(k)^2$, and the dimension of the linear subspace of bi-degree $(k,k)$ defined by $f(\rho,\tau)=f(\tau,\rho)$ is $\frac{1}{2}n(k)\big(n(k)+1\big)$. Let $R$ denote the graded subring generated by the algebraically independent $t_4, t_6, t_{12}$. The dimension of the subspace $R_k$ of bi-degree $(k,k)$ equals the number of nonnegative integer solutions to the equation $k=4p+6q+12r$. If we fix $r$, then the number of such solutions equals the dimension of $M_{k-12r}$. From Equation (\[dimension\]) it follows that for the dimension we have $\dim M_k = \dim M_{k-12} + 1$. Let $r_0 =\lfloor k/12 \rfloor$ then summing over possible values for $r$ we obtain the dimension of $R_k$: $$\dim R_k = \sum_{r=0}^{r_0} \dim M_{k-12r_0+12r} = \left\lbrace \begin{array}{ll} \sum_{r=1}^{r_0} r & \text{for} \, k \equiv 2 (4) \\[0.4em] \sum_{r=0}^{r_0} r & \text{otherwise} \end{array} \right. \; = \frac{1}{2} \, n(k) \, \big(n(k)+1\big) \;,$$ which agrees with the dimension of $\operatorname{Sym}^2(M_k)$. It follows that the elements $t_4$, $t_6$, $t_{12}$ generate $\operatorname{Sym}^2(M_*)$.
The moduli spaces {#app:modulispaces}
-----------------
It is worthwhile to straighten out several moduli spaces of relevance here. The key observation, due to Vinberg [@vinberg-siegel], is that under the natural homomorphism ${\operatorname{Sp}_4(\mathbb{R})}\to O^+(2,3)$, the arithmetic group ${\operatorname{Sp}_4(\mathbb{Z})}$ (which is a maximal discrete subgroup of ${\operatorname{Sp}_4(\mathbb{R})}$) maps to an index two subgroup $SO^+(L^{2,3}) \subset O^+(L^{2,3})$ where $O^+(2,3)$ denotes the subgroup of index 2 of $O(2,3)$ consisting of the elements whose spinor norm is equal to the determinant. The isomorphism $$\mathbb{H}_2/{\operatorname{Sp}_4(\mathbb{Z})}\cong \mathcal{D}_{2,3}/ SO^+(L^{2,3})$$ gives rise to an isomorphism between the algebra of Siegel modular forms of genus 2 and the algebra of automorphic forms of $\mathcal{D}_{2,3}$ with respect to the group $SO^+(L^{2,3})$. But the algebra of automorphic forms of $\mathcal{D}_{2,3}$ with respect to the group $O^+(L^{2,3})$ is the *even* part of the algebra of automorphic forms with respect to $SO^+(L^{2,3})$ and, hence, is isomorphic to the algebra of *even* Siegel modular forms of genus 2.
This means that the moduli space of principally polarized abelian surfaces $\mathbb{H}_2/{\operatorname{Sp}_4(\mathbb{Z})}$ has a degree two map to the moduli space of K3 surfaces with lattice polarization $\mathcal{D}_{2,3}/O^+(L^{2,3})$. This can be understood at the level of modular forms as follows. The Siegel modular forms of even weight, generated by $\psi_4$, $\psi_6$, $\chi_{10}$ and $\chi_{12}$, are invariant under $O^+(L^{2,3})$ and give homogenous coordinates on the Baily-Borel compactification of that space. On the other hand, the full ring of modular forms is invariant under ${\operatorname{Sp}_4(\mathbb{Z})}$, and the equation $\chi_{35}^2=\mathcal{F}(\psi_4, \psi_6,\chi_{10},\chi_{12})$ (where $\mathcal{F}$ is given in Equation (\[chi35sqr\])) expresses $\mathbb{H}_2/{\operatorname{Sp}_4(\mathbb{Z})}$ as a double cover of $\mathcal{D}_{2,3}/O^+(L^{2,3})$.
This same phenomenon carries over to the moduli of genus two curves. We can express a genus two curve in the form $y^2=f(x)$ (cf. Appendix \[moduli\_curves\_genus2\]) and so there is a map $\mathcal{M}_2 \to \mathcal{U}_6$ from the moduli space of genus two curves to the moduli space of degree six polynomials or sextics. This also turns out to be a map of degree two. The coordinates on the moduli space of degree six polynomials were worked out by Clebsch: in Igusa’s notation, they are $I_2(f), I_4(f), I_6(f), I_{10}(f)$ and given in Equations (\[IgusaClebschInvariants\]). On the other hand, the moduli space of genus-two curves has one additional invariant $R$ defined in Appendix \[moduli\_curves\_genus2\]. The point is that under the operation $f(x) \mapsto \tilde{f}(x)=f(-x)$ the odd invariant $R(f)$ is mapped to $R(\tilde{f})=-R(f)$ whereas the even invariants $I_2(f), I_4(f), I_6(f), I_{10}(f)$ remain the same, i.e., $I_{2k}(f)=I_{2k}(\tilde{f})$ for $k=2, 4,6 ,10$. The subtle point is that mapping $f \mapsto \tilde{f}$ and, hence, the ramification points $\theta_i \mapsto - \theta_i$ defines equivalent sextics, but different genus-two curves. In fact, genus-two curves invariant under this action are the ones with bigger automorphism group with an extra involution of order two and $R(f)=0$. Equation (\[Rsqr\]) expresses $R^2=\mathcal{F'}(I_2, I_4, I_6, I_{10})$ as a polynomial in terms of the *even* Igusa-Clebsch invariants. Therefore, this expresses $\mathcal{M}_2$ as a double cover of $\mathcal{U}_6$. The diagram in Figure \[fig:moduli\] summarizes the discussion.
$$\begin{CD}
\mathbb{H}_2/{\operatorname{Sp}_4(\mathbb{Z})}@.
\mathcal{D}_{2,3}/O^+(L^{2,3})
\\
\cap @. \cap\\
\overline{\mathcal{A}_2} = \operatorname{Proj} \, \mathbb{C}\big\lbrack \psi_4, \psi_6, \chi_{10}, \chi_{12}, \chi_{35} \big\rbrack @>2:1>> \operatorname{Proj} \, \mathbb{C}\big\lbrack \psi_4, \psi_6, \chi_{10}, \chi_{12} \big\rbrack \\
\text{{\footnotesize (principally polarized abelian surfaces)}} @. \text{{\footnotesize ($N$-polarized K3 surfaces)}}\\
@AA\operatorname{Jac}A @AAA\\
\overline{\mathcal{M}_2} = \operatorname{Proj} \, \mathbb{C}\big\lbrack I_2, I_4, I_6, I_{10}, R \big\rbrack @>2:1>> \; \; \overline{\mathcal{U}_6}= \operatorname{Proj} \, \mathbb{C} \big\lbrack I_2, I_4, I_6, I_{10} \big\rbrack\\
\text{{\footnotesize (genus-two curves)}} @. \text{{\footnotesize (sextics)}}\\
\cup @. \cup\\
\Big \lbrace y^2 = f(x) \Big\rbrace / \text{isomorphisms} @. \Big \lbrace f(x) \Big\rbrace / \mathrm{GL}(2,\mathbb{C})
\end{CD}$$
The moduli space of principally polarized abelian surfaces {#app:B}
==========================================================
The Siegel modular three-fold {#SiegelThreefold}
-----------------------------
The Siegel three-fold is a quasi-projective variety of dimension $3$ obtained from the Siegel upper half-plane of degree two which by definition is the set of two-by-two symmetric matrices over $\mathbb{C}$ whose imaginary part is positive definite, i.e., $$\label{Siegel_tau}
\mathbb{H}_2 = \left. \left\lbrace \underline{\tau} = \left( \begin{array}{cc} \tau_1 & z \\ z & \tau_2\end{array} \right) \right|
\tau_1, \tau_2, z \in \mathbb{C}\,,\; {\textnormal{Im}\;\;\!\!\!}{(\tau_1)} \, {\textnormal{Im}\;\;\!\!\!}{(\tau_2}) > {\textnormal{Im}\;\;\!\!\!}{(z)}^2\,, \; {\textnormal{Im}\;\;\!\!\!}{(\tau_2)} > 0 \right\rbrace \;,$$ quotiented out by the action of the modular transformations $\Gamma_2:=
{\operatorname{Sp}_4(\mathbb{Z})}$, i.e., $$\mathcal{A}_2 = \mathbb{H}_2 / \Gamma_2 \;.$$ Each $\underline{\tau} \in \mathbb{H}_2$ determines a principally polarized complex abelian surface $\mathbf{A}_{\underline{\,\tau}} = \mathbb{C}^2 / \langle \mathbb{Z}^2 \oplus \underline{\tau} \,
\mathbb{Z}^2\rangle$ with period matrix $(\underline{\tau}, \mathbb{I}_2) \in \mathrm{Mat}(2, 4;\mathbb{C})$. Two abelian surfaces $\mathbf{A}_{\underline{\,\tau}}$ and $\mathbf{A}_{\underline{\,\tau}'}$ are isomorphic if and only if there is a symplectic matrix $$M= \left(\begin{array}{cc} A & B \\ C & D \end{array} \right) \in \Gamma_2$$ such that $\underline{\tau}' = M (\underline{\tau}):=(A\underline{\tau}+B)(C\underline{\tau}+D)^{-1}$. It follows that the Siegel three-fold $\mathcal{A}_2$ is also the set of isomorphism classes of principally polarized abelian surfaces. The sets of abelian surfaces that have the same endomorphism ring form subvarieties of $\mathcal{A}_2$. The endomorphism ring of principally polarized abelian surface tensored with $\mathbb{Q}$ is either a quartic CM field, an indefinite quaternion algebra, a real quadratic field or in the generic case $\mathbb{Q}$. Irreducible components of the corresponding subsets in $\mathcal{A}_2$ have dimensions $0, 1, 2$ and are known as CM points, Shimura curves and Humbert surfaces, respectively.
The Humbert surface $H_{\Delta}$ with invariant $\Delta$ is the space of principally polarized abelian surfaces admitting a symmetric endomorphism with discriminant $\Delta$. It turns out that $\Delta$ is a positive integer $\equiv 0, 1\mod 4$. In fact, $H_{\Delta}$ is the image inside $\mathcal{A}_2$ under the projection of the rational divisor associated to the equation $$a \, \tau_1 + b \, z + c \, \tau_3 + d\, (z^2 -\tau_1 \, \tau_2) + e = 0 \;,$$ with integers $a, b, c, d, e$ satisfying $\Delta=b^2-4\,a\,c-4\,d\,e$ and $\underline{\tau}
= \bigl(\begin{smallmatrix}
\tau_1&z\\ z&\tau_2
\end{smallmatrix} \bigr) \in \mathbb{H}_2$. For example, inside of $\mathcal{A}_2$ sit the Humbert surfaces $H_1$ and $H_4$ that are defined as the images under the projection of the rational divisor associated to $z=0$ and $\tau_1=\tau_2$, respectively. Equivalently, these points are invariant under the $\mathbb{Z}_2$-action generated by $ \bigl(\begin{smallmatrix}
A&0\\ 0&A
\end{smallmatrix} \bigr) \in \Gamma_2$ with $A=\bigl(\begin{smallmatrix}
0&1\\ 1&0
\end{smallmatrix} \bigr)$ and $A=\bigl(\begin{smallmatrix}
1&0\\ 0&-1
\end{smallmatrix} \bigr)$, respectively. In fact, the singular locus of $\mathcal{A}_2$ has $H_1$ and $H_4$ as its two connected components. As analytic spaces, the surfaces $H_1$ and $H_4$ are each isomorphic to the Hilbert modular surface $$\label{modular_product2}
\Big( ({\operatorname{SL}_2(\mathbb{Z})}\times {\operatorname{SL}_2(\mathbb{Z})}) \rtimes \mathbb{Z}_2 \Big) \backslash \Big( \mathbb{H} \times \mathbb{H} \Big) \;.$$ For a more detailed introduction to Siegel modular form, Humbert surfaces, and the Satake compactification of the Siegel modular threefold we refer to Freitag’s book [@MR871067].
Siegel modular forms {#Siegel_modular_forms}
--------------------
In general, we can define the Eisenstein series $\psi_{2k}$ of degree $g$ and weight $2k$ (where we assume $2k>g+1$ for convergence) by setting $$\psi_{2k}(\underline{\tau}) = \sum_{(C,D)} \det(C\cdot\underline{\tau}+D)^{-2k} \;,$$ where the sum runs over non-associated bottom rows $(C,D)$ of elements in ${\operatorname{Sp}_{2g}(\mathbb{Z})}$ where non-associated means with respect to the multiplication on the left by $\mathrm{GL}(g,{\mathbb{Z}})$. For $g=1$ and $k>1$, we have ${\operatorname{Sp}_2(\mathbb{Z})}=
{\operatorname{SL}_2(\mathbb{Z})}$, $\operatorname{GL}(1,{\mathbb{Z}})={\mathbb{Z}}_2$, and we obtain $\psi_{2k}(\tau)=E_{2k}(\tau)$ where $E_{2k}(\tau) = 1 + O(q)$ with $q=\exp{(2\pi i \tau)}$ are the standard normalized Eisenstein series. The reason is that the series $E_{2k}$ be written as $$E_{2k}(\tau) = \sum_{\substack{(c,d) = 1\\ (c,d)\equiv (-c,-d)} } \frac{1}{(c \tau+d)^{2k}} \;,$$ where the sum runs over all pairs of co-prime integers up to simultaneous ${\mathbb{Z}}_2=\mathrm{GL}(1,{\mathbb{Z}})$ action. The connection to the Eisenstein series $G_{2k}$ is given by $$G_{2k}(\tau) = 2 \, \zeta(2k) \, E_{2k}(\tau)
= \sum_{(m,n)\in \mathbb{Z}^2\backslash(0,0)} \frac{1}{(m\tau+n)^{2k}} \;.$$ In the following, we will always assume $g=2$ in the definition of $\psi_{2k}$. For $z\to 0$, we then have that $$\label{relation1}
\psi_{2k} \left( \begin{array}{cc} \tau_1 & z \\ z & \tau_2 \end{array}\right) = E_{2k}(\tau_1) \; E_{2k}(\tau_2) + O(z^2)\;.$$ Following Igusa [@MR0141643] we define a cusp form of weight $12$ by $$\label{definition2}
\begin{split}
\chi_{12}(\underline{\tau}) & = \frac{691}{2^{13}\, 3^8\, 5^3\, 7^2}\left(3^2\, 7^2 \, \psi_4^3(\underline{\tau})+2\cdot 5^3 \, \psi_6^2(\underline{\tau})-691\, \psi_{12}(\underline{\tau})\right) \;.
\end{split}$$ We find that for $z\to 0$ its asymptotic behavior is given by $$\label{relation2}
\begin{split}
\chi_{12} \left( \begin{array}{cc} \tau_1 & z \\ z & \tau_2 \end{array}\right) & = \eta^{24}(\tau_1) \; \eta^{24}(\tau_2) + O(z^2)
\end{split}$$ where $\eta(\tau)$ is the Dedekind $\eta$-function and we have used that $$\begin{split}
1728 \, \eta^{24}(\tau_j) &= E_4^3(\tau_j) - E_6^2(\tau_j) \;,\\
691 \, E_{12}(\tau_j) & = 441 \, E_{4}(\tau_j)^3 + 250 \, E_{6}(\tau_j)^2 \;.
\end{split}$$ Igusa’s ‘original’ definition [@MR0141643 Sec. 8, p. 195] for $\chi_{12}$ is $$\label{definition2b}
\begin{split}
\tilde{\chi}_{12}(\underline{\tau}) & = \frac{131\cdot 593}{2^{13}\, 3^7\, 5^3\, 7^2 \, 337}\left(3^2\, 7^2 \, \psi_4^3(\underline{\tau})+2\cdot 5^3 \, \psi_6^2(\underline{\tau})-691\, \psi_{12}(\underline{\tau})\right) \\
& = \frac{3 \cdot 131 \cdot 593}{337 \cdot 691} \, \chi_{12}(\underline{\tau}) = 1.00078\dots \; \chi_{12}(\underline{\tau})\;.
\end{split}$$ But all results in [@MR0141643] that connect $\chi_{12}$ to the Igusa-Clebsch coefficients are based on the asymptotic expansion in Equation (\[relation2\]). Hence, the definition in Equation (\[definition2\]) must be used. Using Igusa’s definition [@MR0141643 Sec. 8, p. 195] we also define a second cusp form of weight $10$ by $$\begin{split}
\chi_{10}(\underline{\tau}) & = - \frac{43867}{2^{12}\, 3^5\, 5^2\, 7 \cdot 53} \left(\psi_4(\underline{\tau}) \, \psi_6(\underline{\tau}) - \psi_{10}(\underline{\tau})\right) \;.
\end{split}$$ We see that for $z\to 0$ its asymptotic behavior is given by $$\label{relation3}
\begin{split}
\chi_{10} \left( \begin{array}{cc} \tau_1 & z \\ z & \tau_2 \end{array}\right) & = \eta^{24}(\tau_1) \; \eta^{24}(\tau_2) \, (\pi\,z)^2 + O(z^4) \;.
\end{split}$$ Hence, the vanishing divisor of the cusp form $\chi_{10}$ is the Humbert surface $H_1$ because a period point $\underline{\tau}$ is equivalent to a point with $z=0$ if and only if $\chi_{10}\big(\underline{\tau}\big)=0$.
Igusa proved [@MR0229643; @MR527830] that the ring of Siegel modular forms is generated by $\psi_4$, $\psi_6$, $\chi_{10}$, $\chi_{12}$ and by one more cusp form $\chi_{35}$ of odd weight $35$ whose square is the following polynomial [@MR0229643 p. 849] in the even generators $$\label{chi35sqr}
\begin{split}
\chi_{35}^2 & = \frac{1}{2^{12} \, 3^9} \; \chi_{10} \, \Big(
2^{24} \, 3^{15} \; \chi_{12}^5 - 2^{13} \, 3^9 \; \psi_4^3 \, \chi_{12}^4 - 2^{13} \, 3^9\; \psi_6^2 \, \chi_{12}^4 + 3^3 \; \psi_4^6 \, \chi_{12}^3 \\
& - 2\cdot 3^3 \; \psi_4^3 \, \psi_6^2 \, \chi_{12}^3 - 2^{14}\, 3^8 \; \psi_4^2 \, \psi_6 \, \chi_{10} \, \chi_{12}^3 -2^{23}\, 3^{12} \, 5^2\, \psi_4 \, \chi_{10}^2 \, \chi_{12}^3 + 3^3 \, \psi_6^4 \, \chi_{12}^3\\
& + 2^{11}\,3^6\,37\,\psi_4^4\,\chi_{10}^2\,\chi_{12}^2+2^{11}\,3^6\,5\cdot 7 \, \psi_4 \, \psi_6^2\, \chi_{10}^2 \, \chi_{12}^2 -2^{23}\, 3^9 \, 5^3 \, \psi_6\, \chi_{10}^3 \, \chi_{12}^2 \\
& - 3^2 \, \psi_4^7 \, \chi_{10}^2 \, \chi_{12} + 2 \cdot 3^2 \, \psi_4^4 \, \psi_6^2 \, \chi_{10}^2 \, \chi_{12} + 2^{11} \, 3^5 \, 5 \cdot 19 \, \psi_4^3 \, \psi_6 \, \chi_{10}^3 \, \chi_{12} \\
& + 2^{20} \, 3^8 \, 5^3 \, 11 \, \psi_4^2 \, \chi_{10}^4 \, \chi_{12} - 3^2 \, \psi_4 \, \psi_6^4 \, \chi_{10}^2 \, \chi_{12} + 2^{11} \, 3^5 \, 5^2 \, \psi_6^3 \, \chi_{10}^3 \, \chi_{12} - 2 \, \psi_4^6 \, \psi_6 \, \chi_{10}^3 \\
& - 2^{12} \, 3^4 \, \psi_4^5 \, \chi_{10}^4 + 2^2 \, \psi_4^3 \, \psi_6^3 \, \chi_{10}^3 + 2^{12} \, 3^4 \, 5^2 \, \psi_4^2 \, \psi_6^2 \, \chi_{10}^4 + 2^{21} \, 3^7 \, 5^4 \, \psi_4 \, \psi_6 \, \chi_{10}^5 \\
& - 2 \, \psi_6^5 \, \chi_{10}^3 + 2^{32} \, 3^9 \, 5^5 \, \chi_{10}^6 \Big) \;.
\end{split}$$ Hence, $Q:= 2^{12} \, 3^9 \, \chi_{35}^2 /\chi_{10}$ is a polynomial of degree $60$ in the even generators. One then checks that $$q\left(-\frac1{48}\psi_4,-\frac1{864}\psi_6,-4 \, \chi_{10}, \, \chi_{12}\right) = \frac{1}{2^{20} \, 3^9} \, Q(\psi_4, \psi_6, \chi_{10}, \chi_{12}) \;,$$ where $q$ was defined in Equation (\[q-eqn\]). It is known that the vanishing divisor of $Q$ is the Humbert surface $H_4$ [@MR1438983] because a period point $\underline{\tau}$ is equivalent to a point with $\tau_1=\tau_2$ if and only if $Q\big(\underline{\tau}\big)=0$. Accordingly, the vanishing divisor of $\chi_{35}$ is the formal sum $H_1 + H_4$ of Humbert surfaces, that constitutes the singular locus of $\mathcal{A}_2$.
In accordance with Igusa [@MR0141643 Theorem 3] we also introduce the following ratios of Siegel modular forms $$\label{ratio_of_Siegel_forms}
\mathbf{x}_1 = \dfrac{\psi_4 \, \chi_{10}^2}{\chi_{12}^2} ,\quad
\mathbf{x}_2 = \dfrac{\psi_6\, \chi_{10}^3}{\chi_{12}^3} ,\quad
\mathbf{x}_3 = \dfrac{\chi_{10}^6}{\chi_{12}^5} \;,$$ as well as $$\label{ratio_of_Siegel_forms2}
\mathbf{y}_1 = \dfrac{\mathbf{x}_1^3}{\mathbf{x}_3} = \dfrac{\psi_4^3}{\chi_{12}} ,\quad
\mathbf{y}_2 = \dfrac{\mathbf{x}_2^2}{\mathbf{x}_3} = \dfrac{\psi_6^2}{\chi_{12}} ,\quad
\mathbf{y}_3 = \dfrac{\mathbf{x}_1^2 \, \mathbf{x}_2}{\mathbf{x}_3} = \dfrac{\psi_4^2 \, \psi_6 \, \chi_{10}}{\chi_{12}} \;,$$ where we have suppressed the dependence of each Siegel modular form on $\underline{\tau}$. These ratios have the following asymptotic expansion as $z\to 0$ [@MR0141643 pp. 180–182] $$\label{asymptotics2}
\begin{split}
\mathbf{x}_1 & = E_4(\tau_1) \; E_4(\tau_2) \, (\pi z)^{4} + O(z^{5}) \;, \\
\mathbf{x}_2 & = E_6(\tau_1) \; E_6(\tau_2) \, (\pi z)^{6} + O(z^{7}) \;, \\
\mathbf{x}_3 & = \eta^{24}(\tau_1) \; \eta^{24}(\tau_2) \, (\pi z)^{12} + O(z^{13})\;,\\
\end{split}$$ and $$\label{asymptotics3}
\begin{split}
\mathbf{y}_1 & = j(\tau_1) \, j(\tau_2) + O(z^2) \;, \\
\mathbf{y}_2 & = \Big(1728 - j(\tau_1)\Big) \, \Big(1728 -j(\tau_2)\Big) + O(z^2) \;, \\
\mathbf{y}_3 & = \dfrac{E_4^2(\tau_1) \, E_4^2(\tau_2) \, E_6(\tau_1) \, E_6(\tau_2)}{\eta^{24}(\tau_1) \, \eta^{24}(\tau_2)} \, (\pi z)^2 + O(z^3)\;,
\end{split}$$ where we have set $$\label{j_invariant}
\begin{split}
j(\tau_j) & = \dfrac{1728 \, E_4^3(\tau_j)}{E_4^3(\tau_j)-E_6^2(\tau_j)} =\dfrac{E_4^3(\tau_j)}{\eta^{24}(\tau_j)}\;, \\
1728 - j(\tau_j) & = \dfrac{1728 \, E_6^2(\tau_j)}{E_4^3(\tau_j)-E_6^2(\tau_j)} =\dfrac{E_6^2(\tau_j)}{\eta^{24}(\tau_j)}\;.
\end{split}$$ Notice that for the asymptotic behavior in Equations (\[asymptotics2\]) and (\[asymptotics3\]) the right normalization of $\chi_{12}$ was essential.
Sextics and Igusa invariants {#moduli_curves_genus2}
----------------------------
We write the equation defining a genus-two curve $C$ by a degree-six polynomial or sextic in the form $$\label{genus_two_curve}
C: \; y^2 = f(x) = a_0 \, \prod_{i=1}^6 (x-\theta_i)
= \sum_{i=0}^6 a_i \, x^{6-i}\;.$$ The roots $(\theta_i)_{i=1}^6$ of the sextic are the six ramification points of the map $C \to \mathbb{P}^1$. Their pre-images on $C$ are the six Weierstrass points. The isomorphism class of $f$ consists of all equivalent sextics where two sextics are considered equivalent if there is a linear transformation in $\mathrm{GL}(2,\mathbb{C})$ which takes the set of roots to the roots of the other. The action of the linear transformations on the Weierstrass points defines a 7-dimensional irreducible linear representation of ${\operatorname{SL}_2(\mathbb{C})}$. The corresponding invariants are called the invariants of the sextic.
Clebsch defined such invariants $I_2, I_4, I_6, I_{10}$ of weights $2, 4, 6, 10$, respectively, now called the *Igusa-Clebsch invariants* of the sextic curve in (\[genus\_two\_curve\]), as follows $$\label{IgusaClebschInvariants}
\begin{split}
I_2(f) & = a_0^2 \, \sum_{i<j, k<l, m<n} D^2_{ij} \, D^2_{kl} \, D^2_{mn} \;, \\
I_4(f) & = a_0^4 \sum_{\substack{i<j<k, l < m <n}} D^2_{ij} \, D^2_{jk} \, D^2_{ki} \, D^2_{lm} \, D^2_{mn} \, D^2_{nl}\;,\\
I_6(f) & = a_0^6 \sum_{\substack{i<j<k, l<m<n\\i<l', j<m', k<n'\\l', m', n' \in \lbrace l, m, n \rbrace}} D^2_{ij} \, D^2_{jk} \, D^2_{ki} \, D^2_{lm} \, D^2_{mn} \, D^2_{nl} \, D^2_{il'} \, D^2_{jm'} \, D^2_{kn'} \;,\\
I_{10}(f) & = a_0^{10} \, \prod_{i<j} D^2_{ij} \;,
\end{split}$$ where $D_{ij} =\theta_i - \theta_j$ and all indices take values in $\lbrace 1, \dots, 6\rbrace$. In the following, we will often suppress the argument $f$. The invariants $(I_2 , I_4 , I_6 , I_{10})$ are the same invariants as $( A' , B' , C' , D')$ in [@MR1106431 p. 319] and also the same invariants as $(A, B, C, D)$ in [@MR0141643 p. 176]. It follows from the work of Mestre [@MR1106431] that the Igusa-Clebsch invariants are arithmetic invariants, i.e., polynomials in the coefficients $a_0, \dots, a_6$ with integer coefficients, or $I_k \in \mathbb{Z}[a_0,\dots,a_6]$ for $k\in \{2, 4, 6, 10\}$. Furthermore, a theorem by Bolza and Clebsch states that two sextics given by $f$ and $f'$ are isomorphic if and only if there is a $\rho \in \mathbb{C}^*$ such that $I_{2k}(f')=\rho^{-2k}\, I_{2k}(f)$ for $k=1,2,3,5$. Thus, the invariants of a sextic define a point in a weighted projective space $[I_2 : I_4 : I_6 : I_{10}] \in \mathbb{WP}^3_{(2,4,6,10)}$. It was shown in [@MR0141643] that points in the projective variety $\operatorname{Proj}\, \mathbb{C} [I_2, I_4, I_6, I_{10}]$ which are not on $I_{10}=0$ form the variety $\mathcal{U}_6 $ of moduli of sextics. Equivalently, points in this weighted projective space $\{[I_2 : I_4 : I_6 : I_{10}] \in \mathbb{WP}^3_{(2,4,6,10)}:
I_{10} \not = 0\}$ are in one-to-one correspondence with isomorphism classes of sextics.
Often the *Clebsch invariants* of a sextic are used as well. The Clebsch invariants $(A,B,C,D)$ are related to the Igusa-Clebsch invariants by the equations $$\label{Clebsch_invariants}
\begin{split}
I_2 = &\; -120 \, A\;, \\
I_4 = & \; -720 \, A^2 + 6750 \, B\;,\\
I_6 = & \; \phantom{X;} 8640 \, A^3 - 108000 \, A \, B + 202500 \, C\;,\\
I_{10} = & \; -62208 \, A^5 + 972000 \, A^3 \, B + 1620000 \, A^2 \, C \\
& \; - 3037500 \, A \, B^2 - 6075000 \, B \, C - 4556250 \, D \;.
\end{split}$$ Conversely, the invariants $(A,B,C,D)$ are polynomial expressions in the Igusa invariants $(I_2, I_4, I_6, I_{10})$ with rational coefficients. Mestre [@MR1106431] also defined the following polynomials in the Clebsch invariants $$\label{Clebsch_invariants2}
\begin{split}
A_{11} & = 2 \, C + \frac{1}{3} \, A\, B\;, \\
A_{22} & = A_{31} = \, D\;,\\
A_{33} & = \frac{1}{2} \, B \, D + \frac{2}{9} \, C \, (B^2+ A \, C) \;,\\
A_{23} & = \frac{1}{3} \, B \, (B^2+ A \, C) + \frac{1}{3} \, C \, (2\, C + \frac{1}{3} \, A \, B) \;,\\
A_{12} & = \frac{2}{3} \, (B^2+ A\, C) \;.
\end{split}$$ According to [@MR1106431] one can obtain from a sextic $f$ three binary quadrics of the form $$\mathsf{y}_i(x) := \alpha_i \, x^2 + \beta_i \, x + \gamma_i$$ with $i=1,2,3$ by an operation called ‘Überschiebung’ [@MR1106431 p. 317]. To fix the normalization and order of the quadrics we remark that in the notation of [@MR1106431] we have $I_{10}=(\mathsf{y}_3\mathsf{y}_1)_2$. The quadrics $\mathsf{y}_i$ for $i=1,2,3$ have the property that their coefficients are polynomial expressions in the coefficients of $f$ with rational coefficients. Moreover, under the operation $f(x) \mapsto \tilde{f}(x)=f(-x)$ the quadrics change according to $\mathsf{y}_i(x) \mapsto \tilde{\mathsf{y}}_i(x)= \mathsf{y}_i(-x)$ for $i=1,2,3$. Hence, they are not invariants of the sextic. In contrast, $I_2(f), I_4(f), I_6(f), I_{10}(f)$ remain unchanged under this operation, i.e., $I_{2k}(f)=I_{2k}(\tilde{f})$ for $k=2, 4,6 ,10$. This latter statement is easily checked since Equations (\[IgusaClebschInvariants\]) are invariant under $f \mapsto \tilde{f}$ or, equivalently, $D_{ij} \mapsto - D_{ij}$.
We define $R$ to be $1/4$ times the determinant of the three binary quadrics $ \mathsf{y}_i$ for $i=1,2,3$ with respect to the basis $x^2, x, 1$. It is obvious that under the operation $f(x) \mapsto \tilde{f}(x)=f(-x)$ the determinant $R$ changes its sign, i.e., $R(f) \mapsto R(\tilde{f})=-R(f)$. A calculation shows that $$\label{Rsqr}
R^2 = \frac{1}{2} \, \left| \begin{array}{ccc} A_{11} & A_{12} & A_{31} \\ A_{12} & A_{22} & A_{23} \\ A_{31} & A_{23} & A_{33} \end{array}\right| \;,$$ where $A_{ij}$ are the Clebsch invariants (\[Clebsch\_invariants2\]). Like $R(f)^2$, the coefficients $A_{ij}(f)$ are invariant under the operation $f(x) \mapsto \tilde{f}(x)=f(-x)$ as they are polynomials in $(I_2, I_4, I_6, I_{10})$. Bolza [@MR1505464] described the possible automorphism groups of genus-two curves defined by sextics. In particular, he provided effective criteria for the cases when the automorphism group of the sextic curve in (\[genus\_two\_curve\]) is nontrivial. The results are as follows:
1. The curve has an extra involution other than the exchange of sheets $(x,y) \to (x,-y)$ if and only if $R^2=0$. The sextic is then isomorphic to $f(x)=x^6 + c_1 \, x^4 + c_2 \, x^2 + 1$ for some $c_1, c_2 \in \mathbb{C}$ with the extra involution $(x,y) \to (-x,y)$.
2. The automorphism group contains an element of order $5$ if and only if $I_2=I_4=I_6=0,I_{10}\not = 0$. The sextic is then isomorphic to $f(x)=x(x^5+1)$ with the element of order $5$ being $(x,y) \to (\zeta_5^{2} \, x, \zeta_5 y)$ where $\zeta_5 = \exp{(2\pi i/5)}$.
The moduli space of genus-two curves {#moduli_space_chi12}
------------------------------------
Suppose that $C$ is an irreducible projective nonsingular curve. If the self-intersection is $C\cdot C=2$ then $C$ is a curve of genus two. For every curve $C$ of genus two there exists a unique pair $(\mathrm{Jac}(C),j_C)$ where $\mathrm{Jac}(C)$ is an abelian surface, called the Jacobian variety of the curve $C$, and $j_C: C \to \mathrm{Jac}(C)$ is an embedding. One can always regain $C$ from the pair $(\mathrm{Jac}(C),\mathcal{P})$ where $\mathcal{P}=[C]$ is the class of $C$ in the Néron-Severi group $\mathrm{NS}(\mathrm{Jac}(C))$. Thus, if $C$ is a genus-two curve, then $\mathrm{Jac}(C)$ is a principally polarized abelian surface with principal polarization $\mathcal{P}=[C]$, and the map sending a curve $C$ to its Jacobian variety $\mathrm{Jac}(C)$ is injective. In this way, the variety of moduli of curves of genus two is also the moduli space of their Jacobian varieties with canonical polarization. Since we have $\mathcal{P}^2=2$, the transcendental lattice is $\mathrm{T}(\mathrm{Jac}(C)) = \Lambda^{2,2} \oplus \langle -2 \rangle$. Furthermore, Torelli’s theorem states that the map sending a curve $C$ to its Jacobian variety $\mathrm{Jac}(C)$ induces a birational map from the moduli space $\mathcal{M}_2$ of genus-two curves to the complement of the Humbert surface $H_1$ in $\mathcal{A}_2$, i.e., $\mathcal{A}_2 - {\textnormal{supp}}{(\chi_{10})}_0$.
One can then ask what the Igusa-Clebsch invariants of a genus-two curve $C$ defined by a sextic curve $f$ are in terms of $\underline{\tau}$ such that $(\underline{\tau}, \mathbb{I}_2) \in \mathrm{Mat}(2, 4;\mathbb{C})$ is the period matrix of the principally polarized abelian surface $\mathbf{A}_{\underline{\tau}}=\mathrm{Jac}(C)$. Based on the asymptotic behavior in Equations (\[asymptotics2\]) and (\[asymptotics3\]), Igusa [@MR0141643] proved that the relations are as follows: $$\label{invariants}
\begin{split}
I_2(f) & = \dfrac{\chi_{12}(\underline{\tau})}{\chi_{10}(\underline{\tau})} \;, \\
I_4(f) & = \frac{1}{2^4 \, 3^2} \, \psi_4(\underline{\tau}) \;,\\
I_6(f) & = \frac{1}{2^6 \, 3^4} \, \psi_6(\underline{\tau}) + \frac{1}{2^4 \, 3^3} \, \dfrac{\psi_4(\underline{\tau}) \, \chi_{12}(\underline{\tau})}{\chi_{10}(\underline{\tau})} \;,\\
I_{10}(f) & = \frac{1}{2 \cdot 3^5} \, \chi_{10}(\underline{\tau}) \;.
\end{split}$$ Thus, we find that the point $[I_2 : I_4 : I_6 : I_{10}]$ in weighted projective space equals $$\label{IgusaClebschProjective}
\begin{split}
\Big[ 2^3 \, 3 \, (3r\chi_{12})\, : \, 2^2 3^2 \, \psi_4 \, (r\chi_{10})^2 \, : \, 2^3\, 3^2\, \Big(4 \psi_4 \, (3r\chi_{12})+ \psi_6 \,(r\chi_{10}) \Big)\, (r\chi_{10})^2: 2^2 \, (r\chi_{10})^6 \Big]
\end{split}$$ with $r=2^{12}\, 3^5$. Substituting (\[invariants\]) into Equations (\[Clebsch\_invariants\]), (\[Clebsch\_invariants2\]) it also follows that $$\begin{split}
R(f)^2 & = 2^{-41} \, 3^{-42} \, 5^{-20} \;
\dfrac{Q\Big(\psi_4(\underline{\tau}), \psi_6(\underline{\tau}), \chi_{10}(\underline{\tau}), \chi_{12}(\underline{\tau})\Big)}{\chi_{10}(\underline{\tau})^3} \\
& = 2^{-29} \, 3^{-33} \, 5^{-20}\; \dfrac{\chi_{35}(\underline{\tau})^2}{\chi_{10}(\underline{\tau})^4}\;,
\end{split}$$ where $Q$ and $R^2$ where defined in Equation (\[chi35sqr\]) and (\[Rsqr\]), respectively.
If $\underline{\tau}$ is equivalent to a point with $\tau_1=\tau_2$ or $[\underline{\tau}] \in H_4 \subset \mathcal{A}_2$ then the corresponding sextic curve has an extra automorphism with $R(f)^2=0$. The transcendental lattice degenerates to $\mathrm{T}(\mathbf{A}_{\underline{\tau}}) = \Lambda^{1,1} \oplus \langle 2 \rangle \oplus \langle -2 \rangle$. If $\underline{\tau}$ is equivalent to a point with $z=0$ or $[\underline{\tau}] \in H_1 \subset \mathcal{A}_2$, then the principally polarized abelian surface is a product of two elliptic curves $\mathbf{A}_{\underline{\tau}}=E_{\tau_1} \times E_{\tau_2}$ because of Equations (\[weighted\_projective\_coords\]) and (\[asymptotics2\]). The transcendental lattice degenerates to $\mathrm{T}(\mathbf{A}_{\underline{\tau}}) = \Lambda^{2,2}$.
For $I_2 \not =0$ we use the variables $\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3$ from Equations (\[ratio\_of\_Siegel\_forms\]) to write $$\label{weighted_projective_coords}
\begin{split}
\Big\lbrack I_2 : I_4 : I_6 : I_{10} \Big\rbrack = \left\lbrack 1 : \frac{1}{2^4 \,3^2} \, \mathbf{x}_1 : \frac{1}{2^6 \, 3^4} \, \mathbf{x}_2 + \frac{1}{2^4 \, 3^3} \, \mathbf{x}_1
: \frac{1}{2 \cdot 3^5} \, \mathbf{x}_3\right\rbrack \in \mathbb{WP}^3_{(2,4,6,10)} \;.
\end{split}$$ Since the invariants $I_4, I_6, I_{10}$ vanish simultaneously at sextics with triple roots all such abelian surfaces are mapped to $[1:0:0:0] \in \mathbb{WP}^3_{(2,4,6,10)}$ with uniformizing affine coordinates $\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3$ around it. Blowing up this point gives a variety that parameterizes genus-two curves with $I_2 \not = 0$ and their degenerations. In the blow-up space we have to introduce additional coordinates that are obtained as ratios of $\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3$ and have weight zero. Those are precisely the coordinates $\mathbf{y}_1, \mathbf{y}_2, \mathbf{y}_3$ already introduced in Equation (\[ratio\_of\_Siegel\_forms2\]). It turns out that the coordinate ring of the blown-up space is $\mathbb{C}[\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3, \mathbf{y}_1, \mathbf{y}_2, \mathbf{y}_3]$.
If a Jacobian variety corresponds to a product of elliptic curves then $\underline{\tau}$ is equivalent to a point with $z=0$, i.e., $\underline{\tau}$ is located on the Humbert surface $H_1$. We then have $\chi_{10}(\underline{\tau})=0, \chi_{12}(\underline{\tau})\not = 0$ and $[I_2:I_4:I_6:I_{10}]=[1:0:0:0]$. Equations (\[asymptotics2\]) and (\[asymptotics3\]) imply $\mathbf{x}_1 = \mathbf{x}_2 = \mathbf{x}_3 = \mathbf{y}_3 = 0$ and $\mathbf{y}_1 = j(\tau_1) \, j(\tau_2) $ and $ \mathbf{y}_2 = (1728 - j(\tau_1)) \, (1728 -j(\tau_2))$.
K3 fibrations {#K3fibration}
=============
The work of Clingher-Doran
--------------------------
Clingher and Doran introduced the following four-parameter quartic family in $\mathbb{P}^3$ [@arXiv:1004.3503 Eq. (3)] with canonical $\Lambda^{1,1} \oplus E_8(-1)\oplus E_7(-1)$ lattice polarization that generalizes a special two-parameter family of K3 surfaces introduced by Inose $$\label{Inose}
\mathbf{Y}^2\mathbf{ZW}-4\, \mathbf{X}^3\mathbf{Z}+3\, \alpha \, \mathbf{XZW}^2 + \beta \, \mathbf{ZW}^3 + \gamma \, \mathbf{XZ}^2 \mathbf{W} -\frac{1}{2}(\delta \,
\mathbf{Z}^2\mathbf{W}^2+\mathbf{W}^4)=0.$$ They also find the parameters $(\alpha,\beta,\gamma,\delta)$ in terms of Siegel modular forms $$(\alpha,\beta,\gamma,\delta) = \left(\psi_4, \psi_6, 2^{12}3^5 \, \mathcal{C}_{10},
2^{12}3^6 \, \mathcal{C}_{12}\right) \;.$$ (A similar picture was developed in earlier work for the case of a $H \oplus E_8 \oplus E_8$ lattice polarization [@math.AG/0602146].)
Clingher and Doran determine an alternate elliptic fibration on (\[Inose\]) that has two disjoint sections and a singular fiber of Kodaira-type $I_{10}^*$. Here, we use a normalization consistent with F-theory and set $$\mathbf{X} = \dfrac{T \, X^3}{2^9 \, 3^5} \;, \quad \mathbf{Y}=\dfrac{X^2 \, Y}{2^{15/2} \, 3^{9/2}} \;,\quad \mathbf{W}=\dfrac{X^3}{2^{10} \, 3^6} \;, \quad \mathbf{Z}= \dfrac{X^2}{2^{16} \, 3^9} \;,$$ and obtain from Equation (\[Inose\]) the Jacobian elliptic fibration $$\label{WEq.bak.alt}
Y^2 = X^3 + \left( T^3 - \frac{\psi_4}{48} \, T - \frac{\psi_6}{864} \right) \, X^2 - \Big( 4 \, \mathcal{C}_{10} \, T - \mathcal{C}_{12} \Big) \, X\;$$ with special fibers of Kodaira-types $I_{10}^*$, $I_2$, and $6 \, I_1$, and the second section $(Y,X)=(0,0)$. However, we are interested in the Jacobian elliptic fibration with two distinct special fibers of Kodaira-types $II^*$ and $III^*$, respectively. Therefore, we set $$\mathbf{X} = \dfrac{t \, x}{2^9 \, 3^5} \;, \quad \mathbf{Y}=\dfrac{y}{2^{15/2} \, 3^{9/2}} \;,\quad \mathbf{W}=\dfrac{t^3}{2^{10} \, 3^6} \;, \quad \mathbf{Z}= \dfrac{t^2}{2^{16} \, 3^9} \;,$$ and obtain from Equation (\[Inose\]) the Jacobian elliptic fibration $$\label{WEq.bak}
y^2 = x^3 - t^3 \, \left( \frac{\psi_4}{48} \, t + 4 \, \mathcal{C}_{10} \right) \, x + t^5 \, \left( t^2 - \frac{\psi_6}{864} \, t + \mathcal{C}_{12} \right) \;.$$ Clingher and Doran also state [@arXiv:1004.3503 Thm. 1.7] that $$\begin{split}
\left[ I_2 : I_4 : I_6 : I_{10} \right] = \left[ 2^3 \, 3 \, \delta : 2^2 3^2 \alpha \gamma^2, 2^3\, 3^2\, (4 \alpha \delta+ \beta\gamma)\gamma^2: 2^2 \gamma^6 \right]
\end{split}$$ which equals $$\label{IgusaClebschProjective.bak}
\begin{split}
\Big[ 2^3 \, 3 \, (3r\mathcal{C}_{12})\, : \, 2^2 3^2 \, \psi_4 \, (r\mathcal{C}_{10})^2 \, : \, 2^3\, 3^2\, \Big(4 \psi_4 \, (3r\mathcal{C}_{12})+ \psi_6 \,(r\mathcal{C}_{10}) \Big)\, (r\mathcal{C}_{10})^2: 2^2 \, (r\mathcal{C}_{10})^6 \Big]
\end{split}$$ with $r=2^{12} 3^5$. Equation (\[IgusaClebschProjective.bak\]) implies $\mathcal{C}_{10}=\chi_{10}$ and $\mathcal{C}_{12}=\chi_{12}$ by comparison with Equation (\[IgusaClebschProjective\]). This choice makes Equation (\[WEq.bak\]) also be in agreement with Equation (\[eq:imp\]).[^15]
The work of Kumar
-----------------
To relate this to Kumar’s work, we must consider Igusa–Clebsch invariants. Kumar worked with the moduli space of curves of genus $2$, which correspond to $\chi_{10}\ne0$. Kumar’s basic theorem [@MR2427457 Theorem 11] states that a Weierstrass model for a family of K3 surfaces with $\Lambda^{1,1} \oplus E_8(-1)\oplus E_7(-1)$ lattice polarization is given by the equation $$\label{WE_MV2}
y^2 = x^3 + t^3 \, (a \, t + c) \, x + t^5 \, (e \, t^2 + b \, t +d) \;,$$ where $t$ is an affine coordinate on the base $\mathbb{P}^1$, $x, y$ are the affine coordinates on the fiber, and the parameters $(a,b,c,d,e)$ are expressed in terms of the Igusa-Clebsch invariants from Section \[moduli\_curves\_genus2\] as follows: $$\begin{split}
a & = - \dfrac{I_4}{12} = - \dfrac{\psi_4(\underline{\tau})}{2^6 \, 3^3} \;,\\
b & = \dfrac{I_2 \, I_4 - 3 \, I_6}{108} = - \dfrac{\psi_6(\underline{\tau})}{2^8 \, 3^6}\;,\\
c & = -1 \;,\\
d & = \dfrac{I_2}{24} = \dfrac{\chi_{12}(\underline{\tau})}{2^3 \, 3 \, \chi_{10}(\underline{\tau})} \;,\\
e & = \dfrac{I_{10}}{4} = \dfrac{\chi_{10}(\underline{\tau})}{2^3 \, 3^5} \;.
\end{split}$$ Here, we used Equations (\[invariants\]) to express the parameters in terms of Siegel modular forms. The discriminant of the elliptic fiber in Equation (\[WE\_MV2\]) is $$\label{Delta_full}
\begin{split}
\Delta & = t^9 \;
\Big( 27 \, e^2 \, t^5+54 \, e \, b \, t^4+54 \, e \, d \, t^3+27 \, b^2 \, t^3 \\
& + 4 \, a^3 \, t^3+12 \, a^2 \, c \, t^2+54 \, b \, d \, t^2+12 \, a \, c^2 \, t+27 \, d^2 \,t+4 \, c^3 \Big) \;.
\end{split}$$ Generically, the fibration has a singular fiber of Kodaira-type $III^*$ at $t=0$ and a singular fiber of Kodaira-type $II^*$ at $t=\infty$. Moreover, there are five singular fibers of Kodaira-type $I_1$ at those $t$ where the degree-five part of the discriminant vanishes. The Mordell-Weil group is $\mathrm{MW}(\pi)=\lbrace \mathrm{id} \rbrace$, the Néron-Severi lattice has signature $(1,16)$ and discriminant $2$, and the transcendental lattice is $\Lambda^{2,2} \oplus \langle -2 \rangle$.
We know from Section \[Siegel\_modular\_forms\] that $\chi_{10}(\underline{\tau}) \to 0$ as $z \to 0$ in $\underline{\tau}$. For the Weierstrass equation to remain well-defined for $z \to0$, we rescale Equation (\[WE\_MV2\]) as follows $$x \mapsto \dfrac{x}{\mu^6 \, \chi^2_{10}(\underline{\tau})} \;, \quad y \mapsto \dfrac{y}{\mu^9 \, \chi^3_{10}(\underline{\tau})} \;, \quad
t \mapsto \dfrac{9 \, t}{\chi_{10}(\underline{\tau})}$$ with $\mu = 2^{1/6}/3^{1/2}$. We then obtain the following rescaled parameters in Equations (\[WE\_MV2\]) $$\label{duality_equation}
a = -\dfrac{\psi_4(\underline{\tau})}{48} \;, \quad b = - \dfrac{ \psi_6(\underline{\tau}) }{864}\;, \quad
c = - 4 \, \chi_{10}(\underline{\tau}) \;, \quad d=\chi_{12}(\underline{\tau}) \;, \quad e= 1\;.$$ With this choice for the coefficients Equation (\[WE\_MV2\]) remains well-defined in the limit $z\to 0$. In fact, setting $z=0$ we obtain $$a = - 3 \, \dfrac{E_4(\tau_1) \, E_4(\tau_2)}{2^4 \, 3^2} \;, \quad b = - 2 \, \dfrac{ E_6(\tau_1) \, E_6(\tau_2)}{2^6 \, 3^3}\;, \quad
c = 0 \;, \quad d=\eta(\tau_1)^{24} \, \eta(\tau_2)^{24} \;, \quad e= 1\;,$$ and after rescaling by $y\mapsto \lambda^{18} y$, $x\mapsto \lambda^{12} x$, $t\mapsto \lambda^6 \, t$ with $\lambda=\eta(\tau_1)^{2} \, \eta(\tau_2)^{2}$ the Weierstrass equation $$\label{MV}
y^2 = x^3 - 3 \, A \, t^4 \, x + t^5 \, \big( t^2 - 2 \, B \, t + 1\big) \;,$$ with $$A = \dfrac{E_4(\tau_1) \, E_4(\tau_2)}{2^4 \, 3^2 \, \eta(\tau_1)^{8} \, \eta(\tau_2)^{8}} \;, \quad B = \dfrac{E_6(\tau_1) \, E_6(\tau_2)}{2^6 \, 3^3 \, \eta(\tau_1)^{12} \, \eta(\tau_2)^{12}} \;.$$ Equation (\[MV\]) matches precisely the family presented in [@FCY2]. Therefore, this computation provides yet another independent check of the normalization of $\chi_{12}$ in Equation (\[relation2\]).
Degenerations and five-branes {#degs_and_branes}
=============================
In this section we consider certain degenerations of the multi-parameter family of K3 surfaces in Equation (\[WE\_MV2\]). As we have seen, the parameters $a, b, c, d$ can be interpreted as Siegel modular forms of even degree using Equation (\[duality\_equation\]) or, equivalently, as the Igusa-Clebsch invariants of a binary sextic using Equation (\[invariants\]). On the other hand, Namikawa and Ueno gave a geometrical classification of all (degenerate) fibers in pencils of curves of genus two in [@MR0369362]. Given a family of curves of genus two over the complex line with affine coordinate $u \in \mathbb{C}$ which is smooth over $\mathbb{C} \backslash \lbrace 0 \rbrace$, a multi-valued holomorphic map into the Siegel upper half plane of degree two, i.e., the period map, can be defined that determines the family uniquely. Moreover, there are three invariants called ‘monodromy’, ‘modulus point’, and ‘degree’ which determine the singular fiber at $u=0$ uniquely. To each singular fiber, which is labeled in a fashion similar to Kodaira’s classification of singular fibers of elliptic surfaces, Namikawa and Ueno give a one-parameter family of genus-two curves with a singular fiber of each given type over $u=0$.
We note that this work of Namikawa and Ueno provides an important class of examples of degenerations in our situation, but cannot be complete. This is because they studied degenerations of genus two curves (with modular group ${\operatorname{Sp}_4(\mathbb{Z})}$) rather than of binary sextics (with modular group $O^+(L^{2,3})$). Nevertheless, their work provides an interesting first start at studying degenerations and the associated five-branes.
From their list, we took all families of genus-two curves from [@MR0369362] that develop degenerations of type $III$, in particular, parabolic points of type $[3]$ with monodromy of infinite order over $u=0$. These families realize all singular fibers with modulus point $\bigl(\begin{smallmatrix}
\tau_1&z\\ z&\infty
\end{smallmatrix} \bigr)$ for $z\not=0$ or $z=0$ and with $\tau_1 \not = \infty$. The families are listed in Table \[degenerations\] along with the Namikawa-Ueno type of the singular fiber over $u=0$ and the modulus point. For the families in the table we computed the Igusa-Clebsch invariants as polynomials in $u$ and determined their asymptotic expansion as $u \to 0$. By means of Equation (\[WE\_MV2\]), each of the families of genus-two curves then determines a degenerating family of K3 surfaces as $u$ approaches zero. The degeneration consists of two elliptic surfaces meeting along a rational curve. In the last column of Table \[degenerations\] we list the Kodaira-types of the singular fibers of these two rational elliptic surfaces. Among these singular fibers, the stable models for the period points $\bigl(\begin{smallmatrix}
\tau_1&z\\ z&\infty
\end{smallmatrix} \bigr)$ or $\bigl(\begin{smallmatrix}
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$ are given by families with degenerations of Namikawa-Ueno type $[I_{n-0-0}]$ or $[I_n-I_0-m]$ with $m, n >0$.
To determine one of the rational components in the degeneration limit, let $a, b, c, d$ be polynomials in $u$. If $c(u), d(u) \to 0$ as $u \to 0$, we obtain a degeneration $$y^2 = x^3 + a(0) \,t^4 \, x + \big(t + b(0) \big)\, t^6 \;,$$ i.e., a $(4,6,12)$-point at $u=0$. Blowing up by setting $y=t^3y_1$, $x=t^2x_1$, $u=tu_1$, we obtain as the proper transform at $u_1=0$ the rational elliptic surface $$y_1^2 = x_1^3 + a(0) \, x_1 + \big(t + b(0) \big)$$ with singular fibers of Kodaira-types $II^* , 2 \, I_1$. This rational elliptic surface further degenerates to an isotrivial rational elliptic surface with singular fibers $II^*, II$ and $j=0$ if $a(0)=0$.
To determine the second rational component in the degeneration limit, we will have to consider different vanishing orders for the coefficients $a, b, c, d$ corresponding to different Namikawa-Ueno types for the singular fiber of the family of genus-two curves over $u=0$. As an example, we first consider the family of genus-two curves $$y^2 = u^\kappa \, \big(x^3+ \alpha x + 1\big) \, \big( (x-\beta)^2 + u^n \big)$$ that develops a singularity of Namikawa-Ueno type $[I_{n-0-0}]$ and $[I^*_{n-0-0}]$ with modulus point $\bigl(\begin{smallmatrix}
\tau_1&*\\ *&\infty
\end{smallmatrix} \bigr)$ for $\kappa=0$ and $\kappa=1$, respectively. One checks that the asymptotic behavior of the Igusa-Clebsch invariants for this family of genus-two curves is given by $$\begin{split}
a(u)&=a_0 \, u^{4\kappa} + a_1 \, u^{4\kappa+n} + O(u^{4\kappa+2n}) \;,\\
b(u)&=b_0 \, u^{6\kappa} + b_1 \, u^{6\kappa+n} + O(u^{6\kappa+2n}) \;,\\
c(u)&= c_0 \, u^{10\kappa+n} + c_1 \, u^{10\kappa+2n} + O(u^{10\kappa+3n}) \;,\\
d(u)&= d_0 \, u^{12\kappa+n} + d_1 \, u^{12\kappa+2n} + O(u^{12\kappa+3n}) \;,
\end{split}$$ with $n>0$, $\kappa \in \lbrace 0,1 \rbrace$, and $a_0, b_0, c_0, d_0 \not=0$ and generic. From our discussion above, it follows that the proper transforms in the coordinate chart $(u_1,t,x_1,y_1)$ at $u_1=0$ are rational elliptic surfaces with singular fibers $II^*, 2\, I_1$ and $II^*, II$ for $\kappa=0$ and $\kappa=1$, respectively. On the other hand, setting $y=u^{21\kappa+3n} y_2$, $x=u^{14\kappa+2n} x_2$, $t=u^{6\kappa+n} t_2$ we obtain the rational elliptic surface $$y_2^2 = x_2^3 + t_2^3 \, \big(a_0 t_2 + c_0 \big) \, x_2 + t_2^5 \, \big(b_0 \, t_2 + d_0 \big)$$ with singular fibers $III^* , 3 \, I_1$ as the proper transform at $u=0$.
For each example in Table \[degenerations\], we constructed the corresponding family of degenerating K3 surfaces and recovered the two rational elliptic surfaces in the degeneration limit whose singular fibers are listed in the last column of Table \[degenerations\]. We recorded the leading exponents $\mu(a), \mu(b), \mu(c), \mu(d)$ in the asymptotic expansions of $a, b, c, d$ in Table \[degenerations2\] where $\kappa=1$ if there is an additional star-fiber and $\kappa=0$ otherwise. In Table \[degenerations2\], we also recorded the exponents $\mu(y), \mu(x), \mu(u)$ used in the coordinate change $y=u^{\mu(y)} y_2$, $x=u^{\mu(x)} x_2$, $t=u^{\mu(t)} t_2$ that recovers the second rational component in the degeneration limit.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
type modulus family of genus-two curves rat. components
------------------------------ ---------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------- --
\[-0.9em\] $[I_{n-0-0}]$ $\bigl(\begin{smallmatrix} $y^2 = \big(x^3+ \alpha x + 1\big) \, \big( (x-\beta)^2 + u^n \big)$ $\begin{array}{c} II^*, 2 I_1\\ III^*, 3 I_1\end{array}$
\tau_1&*\\ *&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[I^*_{n-0-0}]$ $\bigl(\begin{smallmatrix} $y^2 = u \, \big(x^3+ \alpha x + 1\big) \, \big( (x-\beta)^2 + u^n \big)$ $\begin{array}{c} II^*, II\\ III^*, 3 I_1\end{array}$
\tau_1&*\\ *&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[II_{n-0}]$ $\bigl(\begin{smallmatrix} $y^2 = \big(x^4+ \alpha \, u \, x^2 + u^2\big) \, \big( (x-1)^2 + u^{n-1} \big)$ $\begin{array}{c} II^*, II\\ III^*, III\end{array}$
\tau_1&*\\ *&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[II^*_{n-0}]$ $\bigl(\begin{smallmatrix} $y^2 = u \, \big(x^4+ \alpha \, u \, x^2 + u^2\big) \, \big( (x-1)^2 + u^{n-1} \big)$ $\begin{array}{c} II^*, II\\III^*, III\end{array}$
\tau_1&*\\ *&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[I_n-I_0-m]$ $\bigl(\begin{smallmatrix} $y^2 = \big(x^3+ \alpha \, u^{4m} \, x + u^{6m}\big) \, \big( (x-1)^2 + u^n \big)$ $\begin{array}{c} II^*, II\\ II^*, 2 I_1 \end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[I_n-I^*_0-m]$ $\bigl(\begin{smallmatrix} $y^2 = \big(x^3+ \alpha \, u^{4m+2} \, x + u^{6m+3}\big) \, \big( (x-1)^2 + u^n \big)$ $\begin{array}{c} II^*, II\\ II^*, II \end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[I_n-I^*_0-0]$ $\bigl(\begin{smallmatrix} $y^2 = \big(x^3+ \alpha \, u^{2} \, x + u^{3}\big) \, \big( (x-1)^2 + u^n \big)$ $\begin{array}{c} II^*, II\\ III^*, II, I_1 \end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.4em\]
\[-0.9em\] $[I_0-I_n^*-m]$ $\bigl(\begin{smallmatrix} $\begin{array}{c}y^2 = \big(x+u\big) \, \big(x^2+u^{n+2}\big) \, \\[0.2em] \times \big((x-1)^3+ \alpha \, u^{4m} \, (x-1) + u^{6m}\big)\end{array}$ $\begin{array}{c} II^*, II\\II^*, II\end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.6em\]
\[-0.9em\] $[I_0-I_n^*-0]$ $\bigl(\begin{smallmatrix} $\begin{array}{c}y^2 = \big(x+u\big) \, \big(x^2+u^{n+2}\big) \,\\[0.2em] \times \big((x-1)^3+ \alpha \, (x-1) + 1\big)\end{array}$ $\begin{array}{c} II^*, II\\III^*, II, I_1\end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.6em\]
\[-0.9em\] $[I^*_0-I_n^*-m]$ $\bigl(\begin{smallmatrix} $\begin{array}{c}y^2 = \big(x+u\big) \, \big(x^2+u^{n+2}\big) \,\\[0.2em] \times \, \big((x-1)^3+ \alpha \, u^{4m+2} \, (x-1) + u^{6m+3}\big)\end{array}$ $\begin{array}{c} II^*, II\\II^*, II\end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.6em\]
\[-0.9em\] $[I^*_0-I_n^*-0]$ $\bigl(\begin{smallmatrix} $\begin{array}{c} y^2 = \big(x+u\big) \, \big(x^2+u^{n+2}\big) \,\\[0.2em] \times \big((x-1)^3+ \alpha \, u^{2} \, (x-1) + u^{3}\big)\end{array}$ $\begin{array}{c} II^*, II\\III^*, II, I_1\end{array}$
\tau_1&0\\ 0&\infty
\end{smallmatrix} \bigr)$
\[0.6em\]
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Families of genus-two curves with degeneration of type $III$[]{data-label="degenerations"}
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[^1]: We use the term “moduli space” here as mathematicians do, denoting the parameter space of the geometric objects.
[^2]: For an early appearance, see [@Curio:1997si] and references therein. More recently [@Martucci:2012jk; @arXiv:1308.0553], U-duality of type IIB compactifications on K3 surfaces and the connection to modular forms of genus two was used to construct non-geometric compactifications analogous to the ones studied in this paper.
[^3]: The convention in F-theory slightly differs from the one used in number theory, where the discriminant is identified with the 24th power of the Ramanujan tau function, defined to be $\eta^{24}(\tau) = q \prod_{n \ge 1} (1-q^n)^{24}$. With our conventions for $\Delta_{12}$, we have $\Delta_{12}(\tau) = - 2^8 \, \eta^{24}(\tau)$.
[^4]: The radius is measured in Einstein frame.
[^5]: To better match both the algebraic geometry and modular forms literature, we take the Narain lattices for compactifications of the heterotic string on a $d$-torus to have signature $(d,16+d)$, which is the opposite of the usual convention in string theory.
[^6]: Note that the heterotic dilaton is [*not*]{} affected by this group action, so weakly coupled models of these non-geometric heterotic strings will exist.
[^7]: The slight mismatch in duality groups between $O(\Lambda^{d,16+d})$ for the heterotic string compactified on $T^d$ and $O^+(\Lambda^{d,16+d})$ for the dual theory occurs for the type I’ dual (interpreted as a real K3 surface) when $d=1$ [@Cachazo:2000ey], the F-theory dual when $d=2$ as indicated here, the M-theory dual when $d=3$ (described in [@stringK3] and based on [@MR849050; @MR1102278; @MR1066174]), and the type IIA dual when $d=4$ [@Nahm:1999ps]. It would be interesting to have a more complete understanding of this mismatch.
[^8]: The sign change on $E_8$ and $E_7$ is due to our sign conventions about the Narain lattice.
[^9]: As we will see in Section \[sec:sixD\], assuming that this fiber is [*precisely*]{} type $II^*$ avoids “pointlike instantons” on the heterotic dual after compactification to dimension six or below, at least for general moduli.
[^10]: For the lattice embedding of $\Lambda^{1,1}$ to correspond to an elliptic fibration with section we also need to require that its image in the Néron–Severi lattice contains a pseudo-ample class.
[^11]: Note that because the groups are different, families of genus two curves (such as were used in [@Martucci:2012jk; @arXiv:1308.0553]) are not equivalent to the families of elliptic K3 surfaces needed for our construction.
[^12]: The reader may wonder why the coefficients of $x^3$, $y^2$, and $t^7$ have been set equal to $1$ in . The choice of coefficient $1$ for $x^3$ and $y^2$ is a familiar one, and derives from an analysis by Deligne [@MR0387292] of families of elliptic curves: assuming that all fibers are generalized elliptic curves of an appropriate kind, it follows that the coefficients of $x^3$ and $y^2$ never vanish, and then by a change of coordinates these coefficients can be set to $1$. The story for $t^7$ is similar: we are assuming that the Kodaira fiber at $t=\infty$ is exactly $II^*$, and this implies that the $t^7$ term in the equation must always be present. Thus, we could allow a coefficient $\alpha$ for $t^7$ but it would never be allowed to vanish; as a consequence, the change of coordinates $(x,y,t)\mapsto (x/\alpha^2,y/\alpha^3,t/\alpha)$ would be well-defined, and would map $x^3-y^2+\alpha t^7$ to $\alpha^{-6}(x^3-y^2+t^7)$. In other words, by making such a change of coordinates and then rescaling the entire equation by $\alpha^6$, we may assume that the coefficient of $t^7$ is $1$.
[^13]: The loci $\{c=0\}$ and $\{q(a,b,c,d)=0\}$ correspond to the well-studied Humbert surfaces $H_1$ and $H_4$ described in Appendix \[app:B\].
[^14]: Note that this is a difference from the case of unbroken gauge algebra $\mathfrak{e}_8 \oplus
\mathfrak{e}_8$ considered in [@nongeometries], where all instantons are pointlike no matter how the instanton numbers are distributed between the two summands.
[^15]: Clingher and Doran claim $\mathcal{C}_{12}=\tilde{\chi}_{12}$ instead of $\mathcal{C}_{12}=\chi_{12}$, but we believe that this is the same slip as the one discussed in Equation (\[definition2b\])
|
---
author:
- |
Zhen-Mu Hong[^1] Jun-Ming Xu[^2]\
\
[School of Mathematical Sciences]{}\
[University of Science and Technology of China]{}\
[Wentsun Wu Key Laboratory of CAS]{}\
[Hefei, Anhui, 230026, China]{}
title: 'Vulnerability of super edge-connected graphs[^3] '
---
[**Abstract**]{}
A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity $\la^{(h)}(G)$ of $G$ is defined as the minimum cardinality over all $h$-extra edge-cuts of $G$. A graph $G$, if $\la^{(h)}(G)$ exists, is super-$\la^{(h)}$ if every minimum $h$-extra edge-cut of $G$ isolates at least one connected subgraph of order $h+1$. The persistence $\rho^{(h)}(G)$ of a super-$\la^{(h)}$ graph $G$ is the maximum integer $m$ for which $G-F$ is still super-$\la^{(h)}$ for any set $F\subseteq E(G)$ with $|F|\leqslant m$. Hong [*et al.*]{} \[Discrete Appl. Math. 160 (2012), 579-587\] showed that $\min\{\la^{(1)}(G)-\delta(G)-1,\delta(G)-1\}\leqslant
\rho^{(0)}(G)\leqslant \delta(G)-1$, where $\delta(G)$ is the minimum vertex-degree of $G$. This paper shows that $\min\{\la^{(2)}(G)-\xi(G)-1,\delta(G)-1\}\leqslant \rho^{(1)}(G)\leqslant
\delta(G)-1$, where $\xi(G)$ is the minimum edge-degree of $G$. In particular, for a $k$-regular super-$\la'$ graph $G$, $\rho^{(1)}(G)=k-1$ if $\la^{(2)}(G)$ does not exist or $G$ is super-$\la^{(2)}$ and triangle-free, from which the exact values of $\rho^{(1)}(G)$ are determined for some well-known networks.
Connectivity, extra edge-connected, super connectivity, fault tolerance, networks
0.4cm [**AMS Subject Classification:** ]{} 05C4068M1568R10
0.6cm
Introduction
============
We follow [@x03] for graph-theoretical terminology and notation not defined here. Let $G=(V, E)$ be a simple connected graph, where $V=V(G)$ is the vertex-set of $G$ and $E=E(G)$ is the edge-set of $G$. It is well known that when the underlying topology of an interconnection network is modeled by a connected graph $G=(V,E)$, where $V$ is the set of processors and $E$ is the set of communication links in the network, the edge-connectivity $\la(G)$ of $G$ is an important measurement for reliability and fault tolerance of the network. In general, the larger $\la(G)$ is, the more reliable a network is. Because the connectivity has some shortcomings, Fàbrega and Fiol [@ff94; @ff96] generalized the concept of the edge-connectivity to the $h$-extra edge-connectivity for a graph.
[Let $h\geqslant 0$ be an integer. A subset $F\subseteq E(G)$ is an [*$h$-extra edge-cut*]{} if $G-F$ is disconnected and every component of $G-F$ has more than $h$ vertices. The [*$h$-extra edge-connectivity*]{} of $G$, denoted by $\la^{(h)}(G)$, is defined as the minimum cardinality of an $h$-extra edge-cut of $G$. ]{}
Clearly, $\la^{(0)}(G)=\la(G)$ and $\la^{(1)}(G)=\la'(G)$ for any graph $G$, the latter is called the restricted edge-connectivity proposed by Esfahanian and Hakimi [@eh88], who proved that for a connected graph $G$ of order at least $4$, $\la'(G)$ exists if and only if $G$ is not a star.
In general, $\la^{(h)}(G)$ does not always exist for $h\geqslant 1$. For example, let $G^*_{n,h}$ ($n\geqslant h$) be a graph obtained from $n$ copies of a complete graph $K_h$ of order $h$ by adding a new vertex $x$ and linking $x$ to every vertex in each of $n$ copies. Clearly, $G^*_{n,1}$ is a star $K_{1,n}$. It is easy to check that $\la^{(h)}(G^*_{n,h})$ does not exists for $h\geqslant
1$.
A graph $G$ is said a [*$\la^{(h)}$-graph*]{} or to be [*$\la^{(h)}$-connected*]{} if $\la^{(h)}(G)$ exists, and to be [*not $\la^{(h)}$-connected*]{} otherwise. For a $\la^{(h)}$-graph $G$, an $h$-extra edge-cut $F$ is a [*$\la^{(h)}$-cut*]{} if $|F|=\la^{(h)}(G)$. It is easy to verify that, for a $\lambda^{(h)}$-graph $G$, $$\label{e1.1}
\lambda^{(0)}(G)\leqslant \lambda^{(1)}(G)\leqslant \lambda^{(2)}(G)\leqslant
\cdots \leqslant \lambda^{(h-1)}(G)\leqslant \lambda^{(h)}(G).$$
For two disjoint subsets $X$ and $Y$ in $V(G)$, use $[X,Y]$ to denote the set of edges between $X$ and $Y$ in $G$. In particular, $E_G(X)=[X,\overline X]$ and let $d_G(X)=|E_G(X)|$, where $\overline
X=V(G)\setminus X$. For a $\la^{(h)}$-graph $G$, there is certainly a subset $X\subset V(G)$ with $|X|\geqslant h+1$ such that $E_G(X)$ is a $\la^{(h)}$-cut and, both $G[X]$ and $G[\over{X}]$ are connected. Such an $X$ is called a [*$\la^{(h)}$-fragment*]{} of $G$.
For a subset $X\subset V(G)$, use $G[X]$ to denote the subgraph of $G$ induced by $X$. Let $$\xi_h(G)=\min\{d_G(X):\ X\subset V(G),\
|X|=h+1\ {\rm and}\ G[X] \ {\rm is\ connected}\}.$$ Clearly, $\xi_0(G)=\delta(G)$, the minimum vertex-degree of $G$, and $\xi_1(G)=\xi(G)$, the minimum edge-degree of $G$ defined as $\min\{d_G(x)+d_G(y)-2: xy\in E(G)\}$. For a $\lambda^{(h)}$-graph $G$, Whitney’s inequality shows $\lambda^{(0)}(G)\leqslant\xi_0(G)$; Esfahanian and Hakimi [@eh88] showed $\lambda^{(1)}(G)\leqslant\xi_1(G)$; Bonsma [*et al.*]{} [@buv02], Meng and Ji [@mj02] showed $\lambda^{(2)}(G)\leqslant\xi_2(G)$. For $h\geqslant 3$, Bonsma [*et al.*]{} [@buv02] found that the inequality $\lambda^{(h)}(G)\leqslant\xi_h(G)$ is no longer true in general. The following theorem shows existence of $\la^{(h)}(G)$ for any graph $G$ with $\delta(G)\geqslant h$ except for $G^*_{n,h}$.
[(Zhang and Yuan [@zy05])]{}\[thm1.2\] Let $G$ be a connected graph with order at least $2(\delta+1)$, where $\delta=\delta(G)$. If $G$ is not isomorphic to $G^*_{n,\delta}$, then $\la^{(h)}(G)$ exists and $$\la^{(h)}(G)\leqslant \xi_h(G)\ \ {\rm for\ any}\ h\ {\rm with}\
0\leqslant h\leqslant \delta.$$
A graph $G$ is said to be [*$\la^{(h)}$-optimal*]{} if $\la^{(h)}(G)=\xi_h(G)$. In view of practice in networks, it seems that the larger $\la^{(h)}(G)$ is, the more reliable the network is. Thus, investigating $\la^{(h)}$-optimal property of networks has attracted considerable research interest (see Xu [@x01]). A stronger concept than $\la^{(h)}$-optimal is super-$\la^{(h)}$.
[A $\la^{(h)}$-optimal graph $G$ is [*super $k$-extra edge-connected*]{} ([*super-$\la^{(h)}$*]{} for short), if every $\la^{(h)}$-cut of $G$ isolates at least one connected subgraph of order $h+1$. ]{}
By definition, a super-$\la^{(h)}$ graph is certainly $\la^{(h)}$-optimal, but the converse is not true. For example, a cycle of length $n\,(n\geqslant 2h+4)$ is a $\la^{(h)}$-optimal graph and not super-$\la^{(h)}$. The following necessary and sufficient condition for a graph to be super-$\la^{(h)}$ is simple but very useful.
\[lem1.4\] A $\la^{(h)}$-graph $G$ is super-$\la^{(h)}$ if and only if either $G$ is not $\la^{(h+1)}$-connected or $\la^{(h+1)}(G)>\xi_h(G)$ for any $h\geqslant 0$.
Faults of some communication lines in a large-scale system are inevitable. However, the presence of faults certainly affects the super connectedness. The following concept is proposed naturally.
[The [*persistence*]{} of a super-$\la^{(h)}$ graph $G$, denoted by $\rho^{(h)}(G)$, is the maximum integer $m$ for which $G-F$ is still super-$\la^{(h)}$ for any subset $F\subseteq E(G)$ with $|F|\leqslant
m$.]{}
It is clear that the persistence $\rho^{(h)}(G)$ is a measurement for vulnerability of super-$\la^{(h)}$ graphs. We can easily obtain an upper bound on $\rho^{(h)}(G)$ as follows.
\[thm1.6\] $\rho^{(h)}(G)\leqslant \delta(G)-1$ for any super-$\la^{(h)}$ graph $G$.
Let $G$ be a super-$\la^{(h)}$ graph and $F$ a set of edges incident with some vertex of degree $\delta(G)$. Since $G-F$ is disconnected, $G-F$ is not super-$\la^{(h)}$. By the definition of $\rho^{(h)}(G)$, we have $\rho^{(h)}(G)\leqslant \delta(G)-1$.
By Theorem \[thm1.6\], we can assume $\delta(G)\geqslant 2$ when we consider $\rho^{(h)}(G)$ for a super-$\la^{(h)}$ graph $G$. In this paper, we only focus on the lower bound on $\rho^{(1)}(G)$ for a super-$\la^{(1)}$ graph $G$. For convenience, we write $\la^{(0)}$, $\la^{(1)}$, $\la^{(2)}$, $\rho^{(0)}$ and $\rho^{(1)}$ for $\la$, $\la'$, $\la''$, $\rho$ and $\rho'$, respectively.
Very recently, Hong, Meng and Zhang [@hmz12] have showed $\rho(G)\geqslant \min\{\la'(G)-\delta(G)-1, \delta(G)-1\}$ for any super-$\la$ and $\la'$-graph $G$. In this paper, we establish $\rho'(G)\geqslant\min\{\la''(G)-\xi(G)-1,\delta(G)-1\}$, particularly, for a $k$-regular super-$\la'$ graph $G$, $\rho'(G)=k-1$ if $G$ is not $\la''$-connected or super-$\la''$ and triangle-free. As applications, we determine the exact values of $\rho'$ for some well-known networks.
The left of this paper is organized as follows. In Section 2, we establish the lower bounds on $\rho'$ for general super-$\la'$ graphs. In Section 3, we focus on regular graphs and give some sufficient conditions under which $\rho'$ reaches its upper bound or the difference between upper and lower bounds is at most one. In Section 4, we determine exact values of $\rho'$ for two well-known families of networks.
Lower bounds on $\bm{\rho'}$ for general graphs
===============================================
In this section, we will establish some lower bounds on $\rho'$ for a general super-$\la'$ graph. The following lemma is useful for the proofs of our results.
[(Hellwig and Volkmann [@hv05])]{}\[lem2.1\] If $G$ is a $\la'$-optimal graph, then $\la(G) = \delta(G)$.
\[lem2.2\] Let $G$ be a $\la'$-graph and $F$ be any subset of $E(G)$.
[(i)]{} If $G$ is $\la'$-optimal and $|F|\leqslant
\delta(G)-1$, then $G-F$ is $\la'$-connected.
[(ii)]{} If $G-F$ is $\la''$-connected, then $G$ is also $\la''$-connected. Moreover, $$\label{e2.1}
\la''(G-F)\geqslant \la''(G)-|F|.$$
Let $G$ be a $\la'$-graph of order $n$ and $F$ be any subset of $E(G)$. Clearly, $n\geqslant 4$.
\(i) Assume that $G$ is $\la'$-optimal and $|F|\leqslant \delta(G)-1$. It is trivial for $\delta(G)=1$. Assume $\delta(G)\geqslant 2$ below. Since $G$ is $\la'$-optimal, $\la(G)=\delta(G)$ by Lemma \[lem2.1\]. By $|F|\leqslant \delta(G)-1$, $G-F$ is connected. If $G-F$ is a star $K_{1,n-1}$, then $G$ has a vertex $x$ with degree $n-1$. Let $H=G-x$. Then $F=E(H)$ and $\delta(H)\geqslant
\delta(G)-1$. Thus, $$\delta(G)-1\geqslant |F|=|E(H)|\geqslant \frac12(n-1)(\delta(G)-1),$$ which implies $n\leqslant 3$, a contradiction. Thus, $G-F$ is not a star $K_{1,n-1}$, and so is $\la'$-connected.
\(ii) Assume that $G-F$ is $\la''$-connected, and let $X$ be a $\la''$-fragment of $G-F$. Clearly, $E_G(X)$ is a $2$-extra edge-cut of $G$, and so $G$ is $\la''$-connected and $d_G(X)\geqslant
\la''(G)$. Thus, $\la''(G-F)=d_{G-F}(X) \geqslant
d_G(X)-|F|\geqslant \la''(G)-|F|.$
By Lemma \[lem2.2\], we obtain the following result immediately.
\[thm2.3\] Let $G$ be a super-$\la'$ graph. If $G$ is not $\la''$-connected, then $\rho'(G)=\delta(G)-1$.
Since $G$ is super-$\la'$, $G$ is $\la'$-optimal. Let $F$ be any subset of $E(G)$ with $|F|\leqslant \delta(G)-1$. By Lemma \[lem2.2\] (i), $G-F$ is $\la'$-connected. If $G$ is not $\la''$-connected, then $G-F$ is also not $\la''$-connected by Lemma \[lem2.2\] (ii). By Lemma \[lem1.4\] $G-F$ is super-$\la'$, which implies $\rho'(G)\geqslant \delta(G)-1$. Combining this with Theorem \[thm1.6\], we obtain the conclusion.
By Theorem \[thm2.3\], we only need to consider $\rho'(G)$ for a $\la''$-connected super-$\la'$ graph $G$. A graph $G$ is said to be [*edge-regular*]{} if $d_G(\{x,y\})=\xi(G)$ for every $xy\in E(G)$, where $d_G(\{x,y\})$ is called the edge-degree of the edge $xy$ in $G$. Denote by $\eta(G)$ the number of edges with edge-degree $\xi(G)$ in $G$. For simplicity, we write $\la''=\la''(G)$, $\la'=\la'(G)$, $\rho'=\rho'(G)$, $\xi=\xi(G)$ and $\delta=\delta(G)$ when just one graph $G$ is under discussion.
\[thm2.4\] Let $G$ be a $\la''$-connected super-$\la'$ graph. Then
[(i)]{} $\rho'(G)\geqslant \min\{\la''-\xi-1,\delta-1\}$ if $\eta(G)\geqslant \delta$, or
[(ii)]{} $\rho'(G)\geqslant \min\{\la''-\xi,\delta-1\}$ if $G$ is edge-regular.
Since $G$ is $\la''$-connected and super-$\la'$, $\la''>\xi$ by Lemma \[lem1.4\]. If $\delta=1$, then $\rho'=0$. Assume $\delta\geqslant 2$ below. Let $$\label{e2.2}
\begin{array}{ll}
& m_1 =\min\{\la''-\xi-1,\delta-1\}, \\
& m_2 =\min\{\la''-\xi,\delta-1\} \ \ {\rm and}\ \ m=m_1\ \ {\rm or}\ \ m_2.
\end{array}$$ Since $\la''>\xi$ and $\delta\geqslant 2$, $0\leqslant m_1\leqslant \la''-\xi-1$, $1\leqslant m_2\leqslant \la''-\xi$ and $m\leqslant\delta-1$.
Let $F$ be any subset of $E(G)$ with $|F|= m$ and let $G'=G-F$. Since $G$ is $\la'$-optimal and $|F|\leqslant \delta-1$, $G'$ is $\la'$-connected by Lemma \[lem2.2\] (i). To show that $\rho'\geqslant m$, we only need to prove that $G'$ is super-$\la'$. If $G'$ is not $\la''$-connected, then $G'$ is super-$\la'$ by Lemma \[lem1.4\]. Assume now that $G'$ is $\la''$-connected. It follows from (\[e2.1\]) and (\[e2.2\]) that $$\label{e2.3}
\la''(G')\geqslant\la''(G)-|F|= \la''-m \geqslant
\left\{\begin{array}{ll}
\xi+1 & {\rm if}\ m=m_1;\\
\xi & {\rm if}\ m=m_2.
\end{array}\right.$$
Since $|F|\leqslant \delta-1$, if $\eta(G)\geqslant \delta$, $G'$ has at least one edge with edge-degree $\xi(G)$, which implies $\xi(G')\leqslant \xi(G)$. Moreover, if $G$ is edge-regular, then $\eta(G)\geqslant \delta$ and every edge of $G$ is incident with some edge with edge-degree $\xi$, which implies $\xi(G')< \xi(G)$ if $|F|\geqslant 1$. It follows that $$\label{e2.4}
\xi(G')\leqslant \left\{\begin{array}{ll}
\xi(G) & {\rm if}\ \eta(G)\geqslant \delta;\\
\xi(G)-1 & {\rm if}\ G\ \text{is\ edge-regular\ and $|F|=m\geqslant 1$}.
\end{array}\right.$$
Combining (\[e2.3\]) with (\[e2.4\]), if $m=m_1$ and $\eta(G)\geqslant \delta$ or $m=m_2\geqslant 1$ and $G$ is edge-regular, we have $\la''(G')>\xi(G')$. By Lemma \[lem1.4\], $G'$ is super-$\la'$, and so the conclusions (i) and (ii) hold.
The theorem follows.
\[rem2.5\] [The condition “$\eta(G)\geqslant \delta$" in Theorem \[thm2.4\] is necessary. For example, consider the graph $G$ shown in Figure \[fig1\], $\eta(G)=1<2=\delta$. Since $\xi=\la=\delta=2<4=\la''$, $G$ is super-$\la'$ by Lemma \[lem1.4\]. We should have that $\rho'(G)\geqslant
\la''-\xi-1=\delta-1=1$ by Theorem \[thm2.4\], which shows the removal of any edge from $G$ results in a super-$\la'$ graph. However, $\la''(G-e)=\xi(G-e)=4$, and so $G-e$ is not super-$\la'$ by Lemma \[lem1.4\], which implies $\rho'(G)=0$.]{}
(-2.5,0)(2.5,3.0)
(-2.5,2.1)[3pt]{}[a]{} (-2.5,0.9)[3pt]{}[b]{} (-1.5,3)[3pt]{}[c]{} (-1.5,0)[3pt]{}[d]{} (-.3,3)[3pt]{}[e]{} (-.3,0)[3pt]{}[f]{} (0.7,2.1)[3pt]{}[g]{} (0.7,0.9)[3pt]{}[h]{} (2,2.1)[3pt]{}[i]{} (2,0.9)[3pt]{}[j]{} (2.2,1.5)[$e$]{}
The [*Cartesian product*]{} of graphs $G_1$ and $G_2$ is the graph $G_1\times G_2$ with vertex-set $V(G_1)\times V(G_2)$, two vertices $x_1x_2$ and $y_1y_2$, where $x_1,y_1\in V(G_1)$ and $x_2,y_2\in
V(G_2)$, being adjacent in $G_1\times G_2$ if and only if either $x_1=y_1$ and $x_2y_2\in E(G_2)$, or $x_2=y_2$ and $x_1y_1\in
E(G_1)$. The study on $\la'$ for Cartesian products can be found in [@lcm09; @lcx07; @ou11].
\[rem2.7\]
The graphs $G$ and $H$ shown in Figure \[fig2\] can show that the lower bounds on $\rho'$ given in Theorem \[thm2.4\] are sharp.
In $G$, $X$ and $Y$ are two disjoint subsets of $3t-2$ vertices, and $Z$ is a subset of $Y$ with $t-1$ vertices, where $t\geqslant 2$. There is a perfect matching between $X$ and $Y$ and the subgraphs induced by $X,Y$ and $Z\cup\{x_i,y_i\}$ are all complete graphs, for each $i=1,2,\ldots,t$. It is easy to check that $\eta(G)=\delta(G)=\la(G)=t$, $\la'(G)=\xi(G)=2t-2$, $\la''(G)=3t-2$ and $G$ is super-$\la'$. By Theorem \[thm2.4\], $\rho'(G)\geqslant
\la''(G)-\xi(G)-1=t-1=\delta(G)-1$. Combining this fact with Theorem \[thm1.6\], we have $\rho'(G)=\delta(G)-1$. This example shows that the lower bound on $\rho'$ given in Theorem \[thm2.4\] (i) is sharp.
For the 5-regular graph $H=K_2\times K_3\times K_3$, $\la''(H)=9$ and $\xi(H)=8$, and so $H$ is super-$\la'$ by Lemma \[lem1.4\]. On the one hand, by Theorem \[thm2.4\], $\rho'(H)\geqslant
\la''(H)-\xi(H)=1$. On the other hand, for $F=\{e_1,e_2\}$, $\la''(H-F)=7=\xi(H-F)$, and so $H-F$ is not super-$\la'$ by Lemma \[lem1.4\], which yields $\rho'(H)\leqslant 1$. Hence, $\rho'(H)=\la''(H)-\xi(H)=1$. This example shows that the lower bound on $\rho'$ given in Theorem \[thm2.4\] (ii) is sharp.
(-9.6,-1.5)(3.5,4.8)
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(-9.5,2.4)[$\vdots$]{}(-7,2.4)[$\vdots$]{} (-9.5,0.3)[$\vdots$]{}(-7,0.3)[$\vdots$]{} (-4.35,3.6)[$\cdots$]{}(-4.35,2.35)[$\cdots$]{} (-7,1.05) (-7,4.5) (-7,1.18)[$Z$]{} (-9.5,-0.9) (-9.5,4.7) (-9.5,-0.7)[$X$]{} (-7,-0.9) (-7,4.7) (-7,-0.7)[$Y$]{} (-8.25,-1.4)[$G$]{}
(-1,4.5)[3pt]{}[p1]{}(0.8,4.5)[3pt]{}[p2]{}(2.6,4.5)[3pt]{}[p3]{} (-0.94,4.55)(0.8,4.8)(2.54,4.55) (-1,3.7)[3pt]{}[p4]{}(0.8,3.7)[3pt]{}[p5]{}(2.6,3.7)[3pt]{}[p6]{} (-0.94,3.75)(0.8,4)(2.54,3.75) (-1,2.9)[3pt]{}[p7]{}(0.8,2.9)[3pt]{}[p8]{}(2.6,2.9)[3pt]{}[p9]{} (-0.94,2.95)(0.8,3.2)(2.54,2.95) (-1.06,4.45)(-1.27,3.7)(-1.06,2.95) (0.74,4.45)(0.53,3.7)(0.74,2.95) (2.54,4.45)(2.33,3.7)(2.54,2.95)
(-1,1.6)[3pt]{}[q1]{}(0.8,1.6)[3pt]{}[q2]{}(2.6,1.6)[3pt]{}[q3]{} (-0.94,1.55)(0.8,1.3)(2.54,1.55) (-1,0.8)[3pt]{}[q4]{}(0.8,0.8)[3pt]{}[q5]{}(2.6,0.8)[3pt]{}[q6]{} (-0.94,0.75)(0.8,0.5)(2.54,0.75) (-1,0)[3pt]{}[q7]{}(0.8,0)[3pt]{}[q8]{}(2.6,0)[3pt]{}[q9]{} (-0.94,-0.05)(0.8,-0.3)(2.54,-0.05) (-1.06,1.55)(-1.27,0.8)(-1.06,0.05) (0.74,1.55)(0.53,0.8)(0.74,0.05) (2.54,1.55)(2.33,0.8)(2.54,0.05)
(-0.94,3.65)(-0.7,2.25)(-0.94,0.85) (0.86,3.65)(1.1,2.25)(0.86,0.85) (2.66,3.65)(2.9,2.25)(2.66,0.85) (-0.94,4.45)(-0.4,2.25)(-0.94,0.05) (0.86,4.45)(1.4,2.25)(0.86,0.05) (2.66,4.45)(3.2,2.25)(2.66,0.05) (-1.2,2.25)[$e_{_1}$]{}(3.45,2.25)[$e_{_2}$]{} (0.8,-1.2)[$H=K_2\times K_3\times K_3$]{}
Bounds on $\bm{\rho'}$ for regular graphs
=========================================
The [*girth*]{} of a graph $G$, denoted by $g(G)$, is the length of a shortest cycle in $G$. A graph is said to be [*$C_n$-free*]{} if it contains no cycles of length $n$. In general, $C_3$-free is said [*triangle-free*]{}. To guarantee that $G$ is edge-regular, which is convenient for us to use Theorem \[thm2.4\], we consider regular graphs in this section.
Clearly, any $k$-regular graph contains cycles if $k\geqslant 2$. It is easy to check that $C_4$ and $C_5$ are only two $2$-regular super-$\la'$ graphs. Obviously, $\rho'(C_4)=\rho'(C_5)=1$. In the following discussion, we always assume $k\geqslant 3$ when we mention $k$-regular connected graphs. We first consider $3$-regular graphs, such graphs have even order.
\[lem3.1\] Let $G$ be a $3$-regular super-$\la'$ graph of order $2n$. If $n\geqslant 4$, then the grith $g(G)>4$ and $n\ne 4$.
Since $G$ is a $3$-regular super-$\la'$ graph of order at least $8$, $\la'(G)=\xi(G)=4$, and so $\la(G)=3$ by Lemma \[lem2.1\]. Moreover, every $\la'$-cut of $G$ isolates one edge. If $G$ contains a $C_3$, then let $X=V(C_3)$. If $G-E_G(X)$ isolates a vertex, then $G\cong K_4$, a contradiction with $n\geqslant 4$. Thus, $E_G(X)$ is a 1-extra edge-cut and $\la'(G)\leqslant d_G(X)=3<4=\la'(G)$, a contradiction. If $G$ contains a $C_4$, let $Y=V(C_4)$, then $G-E_G(Y)$ does not isolate a vertex since $G$ contains no triangles, and so $E_G(Y)$ is also a 1-extra edge-cut and $4=\xi(G)=\la'(G)\leqslant d_G(Y)=4$, which implies that $E_G(Y)$ is $\la'$-cut of $G$ and does not isolate one edge since $n\geqslant
4$, which means that $G$ is not super-$\la'$, a contradiction. Thus, the girth $g(G)>4$. Moreover, since any $3$-regular graph with girth greater than 4 has at least 10 vertices, we have $n\geqslant 5$.
\[thm3.2\] Let $G$ be a $3$-regular super-$\la'$ graph of order $2n$. If $n=2$ or $3$, then $\rho'(G)=2$. If $n\geqslant 5$, then $\rho'(G)=1$.
The complete graph $K_4$ and the complete bipartite graph $K_{3,3}$ are the unique $3$-regular super-$\la'$ graphs of order $4$ and $6$, respectively. It is easy to check that $\rho'(K_4)=\rho'(K_{3,3})=2$.
Next, assume $n\geqslant 4$. Then $g(G)>4$ and $n\geqslant 5$ by Lemma \[lem3.1\]. Since $G$ is 3-regular super-$\la'$, $\la'(G)=\xi(G)=4$ and every $\la'$-cut isolates at least one edge. Since $g(G)\geqslant 5$, $G$ is not isomorphic to $G^*_{n,2}$. By Theorem \[thm1.2\], $G$ is $\la''$-connected. By Lemma \[lem1.4\] and Theorem \[thm2.4\] (ii), $\rho'(G)\geqslant
\la''-\xi\geqslant 1$. To prove $\rho'(G)\leqslant 1$, we only need to show that there exists a subset $F\subset E(G)$ with $|F|=2$ such that $G-F$ is not super-$\la'$.
Let $P=(u,v,w)$ be a path of length two in $G$. Since $g(G)>4$, $u$ and $w$ have only common neighbor $v$. Let $\{u_1,u_2,v\}$ and $\{w_1,w_2,v\}$ are the sets of neighbors of $u$ and $w$, respectively. Then either $u_1w_1\not\in E(G)$ or $u_1w_2\not\in
E(G)$ since $g(G)>4$. Assume $u_1w_1\not\in E(G)$ and let $F=\{uu_1,ww_1\}$. Then $\xi(G-F)=3$. Set $X=V(P)$. Then $d_{G-F}(X)=3=\xi(G-F)$. Moreover, it is easy to see that $G[\overline X]$ is connected. Thus, $X$ is a $2$-extra edge-cut of $G-F$, and so $\la''(G-F)\leqslant d_{G-F}(X)=\xi(G-F)$. By Lemma \[lem1.4\], $G-F$ is not super-$\la'$, which yields $\rho'(G)\leqslant 1$. Hence, $\rho'(G)=1$, and so the theorem follows.
The well-known Peterson graph $G$ is a $3$-regular super-$\la'$ graph with girth $g(G)=5$. By Theorem \[thm3.2\], $\rho'(G)=1$.
In general, it is quite difficult to determine the exact value of $\rho'(G)$ of a $k$-regular super-$\la'$ graph $G$ for $k\geqslant
4$. By Theorem \[thm2.3\] for a $k$-regular super-$\la'$ graph $G$, if $G$ is not $\la''$-connected, then $\rho'(G)=k-1$. Thus, we only need to consider $k$-regular $\la''$-graphs. For such a graph $G$, we can establish some bounds on $\rho'(G)$ in terms of $k$.
\[lem3.3\] Let $G$ be $k$-regular $\la''$-optimal graph and $k\geqslant 4$. Then $G$ is super-$\la'$ if and only if $g(G)\geqslant 4$ or $k\geqslant 5$.
Let $G$ be a $k$-regular $\la''$-optimal graph and $k\geqslant 4$. Then $\la''(G)=\xi_2(G)=3k-4>2k-2=\xi(G)$ if and only if $g(G)\geqslant 4$, and $\la''(G)=\xi_2(G)\geqslant 3k-6>2k-2=\xi(G)$ if and only if $k\geqslant 5$. Either of two cases shows that $G$ is super-$\la'$ by Lemma \[lem1.4\].
\[thm3.4\] Let $G$ be a $k$-regular $\la''$-optimal graph and $k\geqslant 4$. If $g(G)\geqslant 4$, then $$k-2\leqslant \rho'(G)\leqslant k-1.$$
Since $G$ is $\la''$-optimal, $G$ is super-$\la'$ by Lemma \[lem3.3\]. Since $G$ is $k$-regular and $g(G)\geqslant 4$, $\la''=3k-4$ and $\xi=2k-2$. By Theorem \[thm2.4\] (ii), $\rho'(G)\geqslant \la''-\xi=k-2$. By Theorem \[thm1.6\], $\rho'(G)\leqslant k-1$.
\[rem3.5\] [The lower bound on $\rho'$ given in Theorem \[thm3.4\] is sharp. For example, the $4$-dimensional cube $Q_4$ (see Figure \[f3\]) is a $4$-regular graph with girth $g=4$ and $\la''(Q_4)=\xi_2(Q_4)=8$. On the one hand, $\rho'(Q_4)\geqslant 2$ by Theorem \[thm3.4\]. On the other hand, let $X$ be the subset of vertices of $Q_4$ whose first coordinates are 0 and $F=\{(0001,1001),(0010,1010),(0100,1100)\}$ (shown by red edges in Figure \[f3\]). Since $\la''(Q_4-F)\leqslant
d_{Q_4-F}(X)=5=\xi(Q_4-F)$, $Q_4-F$ is not super-$\la'$ by Lemma \[lem1.4\], which implies $\rho'(Q_4)\leqslant 2$. Hence, $\rho'(Q_4)=2$. ]{}
(-3,0)(8,4.5) (1,1)[.1]{}[0000]{}(.6,1)[0000]{} (1,3)[.1]{}[0100]{}(.6,3)[0100]{} (3,1)[.1]{}[0001]{}(2.66,1.2)[0001]{} (3,3)[.1]{}[0101]{}(2.66,3.2)[0101]{} (1.9,1.8)[.1]{}[0010]{}(1.5,1.85)[0010]{} (1.9,3.8)[.1]{}[0110]{}(1.5,3.85)[0110]{} (3.9,1.8)[.1]{}[0011]{}(3.55,2)[0011]{} (3.9,3.8)[.1]{}[0111]{}(3.55,4)[0111]{} (5,1)[.1]{}[1001]{}(5.4,.8)[1001]{} (5,3)[.1]{}[1101]{}(5.4,2.8)[1101]{} (7,1)[.1]{}[1000]{}(7.4,.85)[1000]{} (7,3)[.1]{}[1100]{}(7.4,2.85)[1100]{} (5.9,1.8)[.1]{}[1011]{}(6.3,2)[1011]{} (5.9,3.8)[.1]{}[1111]{}(6.3,4)[1111]{} (7.9,1.8)[.1]{}[1010]{}(8.34,1.8)[1010]{} (7.9,3.8)[.1]{}[1110]{}(8.34,3.8)[1110]{} (4,.1)[$Q_4$]{}
For a $k$-regular $\la''$-optimal graph with $g(G)=3$, we can establish an upper bound on $\rho'$ under some conditions. To prove our result, we need the following lemma.
[(Hong [*et al.*]{} [@hmz12])]{}\[lem3.6\] Let $G$ be an $m$-connected graph. Then for any subset $X\subset
V(G)$ with $|X|\geqslant m$ and $|\over{X}| \geqslant m$, there are at least $m$ independent edges in $E_G(X)$.
\[thm3.7\] Let $G$ be a $k$-regular $\la''$-optimal graph with $g(G)=3$ and $k\geqslant 5$. If $G$ is $(k-2)$-connected and not super-$\la''$, then $$\label{e3.1}
k-4\leqslant \rho'(G)\leqslant k-3,$$ and the bounds are best possible.
Since $G$ is $k$-regular $\la''$-optimal, $G$ is super-$\la'$ by Lemma \[lem3.3\], $\la''=3k-6$ and $\xi=2k-2$. By Theorem \[thm2.4\] (ii), $\rho'(G)\geqslant \la''-\xi=k-4$. Thus, we only need to prove $\rho'(G)\leqslant k-3$.
Since $G$ is not super-$\la''$, there exists a $\la''$-fragment $X$ of $G$ such that $|\overline X|\geqslant |X|\geqslant 4$. Let $|X|=t$. If $t<k-2$, then $k>6$. For any $x\in X$, since $d_{G[X]}(x)\leqslant t-1$, $|[\{x\},\over{X}]|=d_G(x)-d_{G[X]}(x)\geqslant k-t+1$, and so $$\label{e3.2}
3k-6=\la''=d_G(X)=\sum_{x\in X}|[\{x\},\over{X}]|\geqslant
t(k-t+1).$$ Since the function $f(t)=t(k-t+1)$ is convex in the integer interval $[3, k-2]$ and reaches the minimum value at two end-points of the interval. It follows that $$\label{e3.3}
f(t)>f(k-2)=f(3)=3k-6\ \ {\rm for}\ k>6.$$ Comparing (\[e3.3\]) with (\[e3.2\]), we obtain a contradiction. Thus, $t\geqslant k-2$. By Lemma \[lem3.6\], there exists a subset $F\subseteq E_G(X)$ consisting of $k-2$ independent edges. If $G-F$ is not $\la''$-connected, then $G-F$ is not super-$\la'$ by Lemma \[lem1.4\]. Assume that $G-F$ is $\la''$-connected. Then $E_{G-F}(X)$ is a $2$-extra edge-cut of $G-F$. Since $$\la''(G-F)\leqslant d_{G-F}(X)=d_G(X)-|F|=2k-4=\xi-2\leqslant \xi(G-F),$$ $G-F$ is not super-$\la'$ by Lemma \[lem1.4\]. Hence $\rho'(G)\leqslant k-3$.
To show these bounds are best possible, we consider the graph $H=K_2\times K_3\times K_3$ and $G=K_4\times K_4$. For the graph $H$, it is $5$-regular $\la''$-optimal, and $\rho'(H)=1$ (see Remark \[rem2.7\]), which shows that the lower bound given in (\[e3.1\]) is sharp when $k=5$. For the graph $G$, it is $6$-regular $\la''$-optimal but not super-$\la''$. For any subset $F\subset E(G)$ with $|F|=3$, $G-F$ is certainly $\la''$-connected and $\la''(G-F)\geqslant \la''-|F|=12-3=9>8\geqslant \xi(G-F)$. By Lemma \[lem1.4\], $G-F$ is super-$\la'$, which yields $\rho'(G)\geqslant 3$. Hence, $\rho'(G)=3$, which shows that the upper bound given in (\[e3.1\]) is sharp.
The theorem follows.
For a $k$-regular super-$\la''$ graph, the lower bound on $\rho'$ can be improved a little, which is stated as the following theorem.
\[thm3.8\] Let $G$ be a $k$-regular super-$\la''$ graph and $k\geqslant 4$. Then
[(i)]{} $\rho'(G)=k-1$ if $k\geqslant 4$ when $g(G)\geqslant 4$;
[(ii)]{} $\rho'(G)\geqslant k-3$ if $k\geqslant 6$ and $\rho'(G)=2$ if $k=5$ when $g(G)=3$.
Since $G$ is super-$\la''$, $G$ is $\la''$-optimal and $\la''\leqslant 3k-4$. If $g(G)\geqslant 4$ or $k\geqslant 5$, then $G$ is super-$\la'$ by Lemma \[lem3.3\]. Let $F$ be any subset of $E(G)$ with $|F|=\la''-\xi+1$ and $G'=G-F$. Since $|F|=\la''-\xi+1\leqslant k-1$, $G'$ is $\la'$-connected by Lemma \[lem2.2\] (i). We first prove that $$\label{e3.5}
\rho'(G)\geqslant \la''-\xi+1\ \text{~if $g(G)\geqslant 4$ or $k\geqslant 5$}.$$ To the end, we need to prove that $G'$ is super-$\la'$. By Lemma \[lem1.4\], we only need to prove that $$\label{e3.6}
\la''(G')>\xi(G')\ \ \text{if $G'$\ is\ $\la''$-connected}.$$
Let $X$ be any $\la''$-fragment of $G'$. Since $d_{G'}(\overline
X)=d_{G'}(X)=\la''(G')$, we can assume $|X|\leqslant |\overline X|$. Since $X$ is a $2$-extra edge-cut of $G$, $d_G(X)\geqslant
\la''(G)=\la''$, and so $$\label{e3.7}
\la''(G')=d_{G'}(X)\geqslant d_G(X)-|F|\geqslant \la''-|F|=\xi-1.$$ On the other hand, since $G$ is edge-regular, we have $$\label{e3.8}
\xi(G')\leqslant \xi(G)-1=\xi-1.$$ Combing (\[e3.7\]) with (\[e3.8\]), in order to prove (\[e3.6\]), we only need to show that at least one of the inequalities (\[e3.7\]) and (\[e3.8\]) is strict.
If $F\not\subseteq E_G(X)$, $d_{G'}(X)> d_G(X)-|F|$, and so the first inequality in (\[e3.7\]) is strict. Assume $F\subset E_G(X)$ below. If $|X|\geqslant 4$, then $E_G(X)$ is not a $\la''$-cut since $G$ is super-$\la''$, which implies $d_G(X)
> \la''$, and so the second inequality in (\[e3.7\]) is strict.
Now, consider $|X| = 3$ and we have the following two subcases.
If $g(G)\geqslant 4$, then $\la''=3k-4$, and so $|F|=\la''-\xi+1=k-1\geqslant 3$. Since $F\subset E_G(X)$, there exists one edge in $G[X]$ which is adjacent to at least two edges of $F$, which implies $\xi(G')\leqslant \xi-2<\xi-1$, that is, the inequality (\[e3.8\]) is strict.
If $g(G)=3$, then $\la''=3k-6$. If $G[X]$ is not a triangle, $d_G(X)=3k-4>3k-6=\la''$, and so the second inequality in (\[e3.7\]) is strict. If $G[X]$ is a triangle, since $|F|=\la''-\xi+1=k-3\geqslant 2$ and $F\subset E_G(X)$, then there exists one edge in $G[X]$ which is adjacent to at least two edges of $F$, which implies $\xi(G')\leqslant \xi-2<\xi-1$, that is, the inequality (\[e3.8\]) holds strictly.
Thus, the inequality (\[e3.6\]) holds, and so the inequality (\[e3.5\]) follows. We now prove the remaining parts of our conclusions.
(i) When $g(G)\geqslant 4$, $\la''=3k-4$. By Theorem \[thm1.6\] and (\[e3.5\]), $k-1\geqslant \rho'(G)\geqslant \la''-\xi+1=k-1$, which implies $\rho'(G)=k-1$.
(ii) When $g(G)=3$, $\la''=3k-6$. By (\[e3.5\]), $\rho'(G)\geqslant \la''-\xi+1=k-3$. If $k=5$, $\rho'(G)\geqslant
2$. Choose a subset $X\subset V(G)$ such that $G[X]$ is a triangle. It is easy to check that $E_G(X)$ is a $\la''$-cut. Let $F$ be a set of three independent edges of $E_G(X)$. Then $\la''(G-F)\leqslant
d_{G-F}(X)=6=\xi(G-F)$. This fact shows that $G-F$ is not super-$\la'$, which implies $\rho'(G)\leqslant 2$. Thus, $\rho'(G)=2$.
The theorem follows.
A graph $G$ is [*transitive*]{} if for any two given vertices $u$ and $v$ in $G$, there is an automorphism $\phi$ of $G$ such that $\phi(u)=v$. A transitive graph is always regular. The studies on extra edge-connected transitive graphs and super extra edge-connected transitive graphs can be found in [@m03; @wl02; @xx02; @yzqg11] etc.
[(Wang and Li [@wl02])]{}\[lem3.9\] Let $G$ be a connected transitive graph of degree $k\geqslant 4$ with girth $g\geqslant 5$. Then $G$ is $\la''$-optimal and $\la''(G)=3k-4$.
[(Yang [*et al.*]{} [@yzqg11])]{}\[lem3.10\] Let $G$ be a $C_4$-free transitive graph of degree $k\geqslant 4$. If $G$ is $\la''$-optimal, then $G$ is super-$\la''$.
Combining Theorem \[thm3.8\] (i) with Lemma \[lem3.9\] and Lemma \[lem3.10\], we have the following corollary immediately.
\[cor3.11\] If $G$ is a connected transitive graph of degree $k\geqslant 4$ with girth $g\geqslant 5$, then $\rho'(G)= k-1$.
\[rem3.12\] [In Corollary \[cor3.11\], the condition “$g\geqslant 5$" is necessary. For example, the connected transitive graph $Q_4$ is $\la''$-optimal and not super-$\la''$, and $\rho'(Q_4)=2$ (see Remark \[rem3.5\]). ]{}
$\bm{\rho'}$ for two families of networks
=========================================
As applications of Theorem \[thm3.8\] (i), in this section, we determine the exact values of $\rho'(G)$ for two families of networks $G(G_0,G_1;M)$ and $G(G_0,G_1,\dots,G_{m-1};\mathscr{M})$ subject to some conditions.
The first family of networks $G(G_0,G_1;M)$ is defined as follows. Let $G_0$ and $G_1$ be two graphs with the same number of vertices. Then $G(G_0,G_1;M)$ is the graph $G$ with vertex-set $V(G)=V(G_0)\cup
V(G_1)$ and edge-set $E(G)=E(G_0)\cup E(G_1)\cup M$, where $M$ is an arbitrary perfect matching between vertices of $G_0$ and $G_1$. Thus the hypercube $Q_n$, the twisted cube $TQ_n$, the crossed cube $CQ_n$, the Möbius cube $MQ_n$ and the locally twisted cube $LTQ_n$ all can be viewed as special cases of $G(G_0,G_1;M)$ (see [@cth03]).
The second family of networks $G(G_0,G_1,\dots,G_{m-1};\mathscr{M})$ is defined as follows. Let $G_0,G_1,\dots,G_{m-1}$ be $m~(\geqslant 3)$ graphs with the same number of vertices. Then $G(G_0,G_1,$ $\dots,G_{m-1};\mathscr{M})$ is the graph $G$ with vertex-set $V(G)
= V(G_0)\cup V(G_1)\cup \cdots\cup V(G_{m-1})$ and edge-set $E(G)=E(G_0)\cup E(G_1)\cup \cdots\cup E(G_{m-1})\cup \mathscr{M}$, where $\mathscr{M}=\cup_{i=0}^{m-1}M_{i,i+1({\rm mod}~m)}$ and $M_{i,i+1({\rm mod}~m)}$ is an arbitrary perfect matching between $V(G_i)$ and $V(G_{i+1({\rm mod}~m)})$. Recursive circulant graphs [@pc94] and the undirected toroidal mesh [@x01] are special cases of this family.
The super edge-connectivity of above two families of networks is studied by Chen [*et al.*]{} [@cth03]. Chen and Tan [@ct07] further studied the restricted edge-connectivity of above two families of networks, and $\la'(G(G_0,G_1;M))$ is also studied by Xu [*et al.*]{} [@xww10]. The 2-extra edge-connectivity of above two families of networks is studied by Wang [*et al.*]{} [@wyl08]. The vulnerability $\rho$ of super edge-connectivity of the two families of networks is discussed by Wang and Lu [@wl12]. In this section, we will further investigate the vulnerability $\rho'$ of the two families of super-$\la'$ networks without triangles.
\[lem4.1\][(see Example 1.3.1 in Xu [@x03])]{} If $G$ is a triangle-free graph of order $n$, then $|E(G)|\leqslant
\frac{n^2}{4}$.
We consider the first family of graphs $G=G(G_0,G_1;M)$ for $k$-regular triangle-free and super-$\la$ graphs $G_0$ and $G_1$. Under these hypothesis, $G$ is $(k+1)$-regular and triangle-free. By Theorem \[thm3.2\], we can assume $k\geqslant 3$. We attempt to use Theorem \[thm3.8\] (i) to determine the exact value of $\rho'(G)$ when $G$ is super-$\la''$. However, there are some such graphs that are not super-$\la''$.
[Let $G_0$ be a $k$-regular triangle-free and super-$\la$ graph of order $n$. Then $G_0$ is $\la'$-connected, $k\geqslant 3$ and $n\geqslant 6$. $G=G_0\times K_2$ can be viewed as $G(G_0,G_0;M)$ for some perfect matching $M$. Assume $n\leqslant 3k-1$ or $\la'(G_0)\leqslant \frac{3k-1}{2}$. If the former happens, then $M$ is a $2$-extra edge-cut, and so $\la''(G_0\times K_2)\leqslant |M|=3k-1$. However, $G_0\times K_2$ is not super-$\la''$ since $n\geqslant 6$. If the latter happens, let $X_0\subset V(G_0)$ such that $E_{G_0}(X_0)$ is a $\la'$-cut of $G_0$, then $G[X_0]\times K_2\subset G$. Let $Y=V(G[X_0]\times
K_2)$. Since $E_G(Y)$ is a $2$-extra edge-cut, $\la''(G_0\times
K_2)\leqslant |E_G(Y)|= 2\la'(G_0)\leqslant 3k-1$. However, $G_0\times K_2$ is not super-$\la''$ since $|Y|\geqslant 4$. ]{}
This example shows that the condition “$\min\{n,\la'(G_0)+\la'(G_1)\}>3k-1$" is necessary to guarantee that $G=G(G_0,G_1;M)$ is super-$\la''$. Thus, we can state our result as follows.
\[thm4.3\] Let $G_i$ be a triangle-free $k$-regular and super-$\la$ graph of order $n$ for each $i=0,1$. If $\min\{n,\la'_0+\la'_1\}>3k-1$, then $G=G(G_0,G_1;M)$ is super-$\la''$ and $\rho'(G)=k$, where $\la'_i=\la'(G_i)$ for each $i=0,1$.
Clearly, $k\geqslant 3$. Since $G$ is $(k+1)$-regular and triangle-free, by Theorem \[thm3.8\] (i), we only need to prove that $G$ is super-$\la''$. Since $M$ is a 2-extra edge-cut of $G$, $\la''(G)$ exists. By Theorem \[thm1.2\], $$\label{e4.1}
\la''(G)\leqslant \xi_2(G)=3k-1.$$ Suppose to the contrary that $G$ is not super-$\la''$. Then there exists a $\la''$-fragment $X$ of $G$ such that $|\overline{X}|\geqslant |X|\geqslant 4$. Since $G$ is triangle-free, $G[X]$ is also triangle-free, and so $|E(G[X])|\leqslant \frac{|X|^2}{4}$ by Lemma \[lem4.1\]. It follows that $$\begin{array}{rl}
3k-1\geqslant d_G(X)=(k+1)|X|-2|E(G[X])|\geqslant (k+1)|X|-\frac{1}{2}|X|^2,
\end{array}$$ that is, $ (|X|-3)(|X|-(2k-1))+1\geqslant 0$, which implies that, since $|X|\geqslant 4$ and $k\geqslant 3$, $$\label{e4.2}
|X|\geqslant 2k-1.$$
We will deduce a contradiction to (\[e4.1\]) by proving that $$\label{e4.3}
\la''(G)>3k-1.$$ To the end, set $V_i=V(G_i)$ and $X_i=X\cap V_i$ for each $i=0,1$. There are two cases.
[*Case 1.*]{} Exactly one of $X_0$ and $X_1$ is empty.
Without loss of generality, assume $X=X_0$. Then $E_G(X)=E_{G_0}(X)\cup [X,V_1]$. By the definition of $G$, $|[X,V_1]|=|X|$, and so $$\begin{aligned}
\label{e4.4}
\la''(G)=d_G(X)=d_{G_0}(X)+|X|.
\end{aligned}$$ It is easy to check that $G_0[V_0\setminus X]$ is connected. Thus, when $2\leqslant |X|\leqslant n-2$, $E_{G_0}(X)$ is a 1-extra edge-cut of $G_0$, and so $d_{G_0}(X)\geqslant \la'_0$. Since $G_0$ is super-$\la$, $\la'_0>\la(G_0)=k$, and so $$\begin{aligned}
\label{e4.5}
d_{G_0}(X)\geqslant \left\{
\begin{array}{ll}
k+1 &\ {\rm if}\ 2\leqslant |X|\leqslant n-2,\\
k &\ {\rm if}\ |X|=n-1,\\
0 &\ {\rm if}\ |X|=n.
\end{array}
\right.\end{aligned}$$ Substituting $n>3k-1$, (\[e4.2\]) and (\[e4.5\]) into (\[e4.4\]) yields the inequality (\[e4.3\]).
[*Case 2.*]{} $X_0\ne \emptyset$ and $X_1\ne \emptyset$.
Assume that one of $G[X_0]$, $G[X_1]$, $G[V_0\setminus X_0]$ and $G[V_1\setminus X_1]$ is not connected. Without loss of generality, assume that $G[X_0]$ has two components $H$ and $T$. Then $[H,V_0\setminus X_0]\cup [T,V_0\setminus X_0]\cup [X_1,V_1\setminus
X_1]\subseteq E_G(X)$, and the first two are edge-cuts of $G_0$, and the last is an edge-cut of $G_1$. Since $G_i$ is super-$\la$, $\la(G_i)=k$ for each $i=0,1$. Thus, $$\la''(G)=|E_G(X)|\geqslant |[H,V_0\setminus X_0]|+|[T,V_0\setminus X_0]|+|[X_1,V_1\setminus
X_1]|\geqslant 3k>3k-1,$$ and so (\[e4.3\]) follows.
Now, we assume that all of $G[X_0]$, $G[X_1]$, $G[V_0\setminus X_0]$ and $G[V_1\setminus X_1]$ are connected. Since $|X|\geqslant 4$, $\max\{|X_0|,|X_1|\}\geqslant 2$. We consider the following two subcases.
[*Subcase $2.1$.*]{} $|X_0|\geqslant 2$ and $|X_1|\geqslant 2$.
In this case, $E_{G_0}(X_0)\cup E_{G_1}(X_1)\subseteq E_G(X)$. For each $i=0,1$, $d_{G_i}(X_i)\geqslant \la'_i$ since $E_{G_i}(X_i)$ is a $1$-extra edge-cut of $G_i$. By our hypothesis, $$\la''(G)=d_G(X)\geqslant d_{G_0}(X_0)+d_{G_1}(X_1)\geqslant \la'_0+\la'_1>3k-1,$$ and so (\[e4.3\]) follows.
[*Subcase $2.2$.*]{} Exact one of $X_0$ and $X_1$ is a single vertex.
Without loss of generality, assume $|X_0|=1$. Then $|X_1|=|X|-1\geqslant 2k-2$ by (\[e4.2\]). Clearly, $$E_G(X)=E_{G_0}(X_0)\cup E_{G_1}(X_1)\cup [X_1,V_0\setminus X_0],$$ $d_{G_0}(X_0)=k$ and $|[X_1,V_0\setminus X_0]|=|X|-2\geqslant
2k-3$, and so $$\begin{aligned}
\label{e4.6}
\la''(G)=d_G(X)\geqslant k+d_{G_1}(X_1)+2k-3.
\end{aligned}$$ If $2\leqslant |X_1|\leqslant n-2$, then $X_1$ is a 1-extra edge-cut of $G_1$, and so $d_{G_1}(X_1)\geqslant \la'_1>\la(G_1)=k$ since $G_1$ is super-$\la$. If $|X_1|=n-1$, then $E_{G_1}(X_1)$ isolates a vertex, and so $d_{G_1}(X_1)=k$. Thus, we always have $d_{G_1}(X_1)\geqslant k$. Substituting this inequality into (\[e4.6\]) yields (\[e4.3\]) since $d_G(X)\geqslant
k+k+2k-3=4k-3>3k-1$ for $k\geqslant 3$.
Under the hypothesis that $G$ is not super-$\la''$, we deduce a contradiction to (\[e4.1\]). Thus, $G$ is super-$\la''$. By Theorem \[thm3.8\] (i), $\rho'(G)=k$, and so the theorem follows.
[(Xu [*et al.*]{} [@xww10])]{}\[lem4.4\] If $G_n\in \{Q_n, TQ_n, CQ_n, MQ_n, LTQ_n\}$, then $\la'(G_n)=2n-2$ and, thus, $G_n$ is $\la'$-optimal for $n\geqslant 2$, and is super-$\la$ for $n\geqslant 3$.
\[cor4.5\] Let $G_n\in \{Q_n, TQ_n, CQ_n, MQ_n, LTQ_n\}$. If $n\geqslant 5$, then $G_n$ is super-$\la''$, super-$\la'$ and $\rho'(G_n)=n-1$.
Let $G_n\in \{Q_n, TQ_n, CQ_n, MQ_n, LTQ_n\}$. Then $G_n$ can be viewed as the graph $G(G_{n-1},G_{n-1};M)$ corresponding to some perfect matching $M$. $G_{n-1}$ is an $(n-1)$-regular and triangle-free graph of order $2^{n-1}$. By Lemma \[lem4.4\], $G_{n-1}$ is super-$\la$ and $\la'(G_{n-1})=2n-4$ for $n\geqslant
4$. Thus, $2\la'(G_{n-1})=4n-8>3(n-1)-1$ and $2^{n-1}>3(n-1)-1$ for $n\geqslant 5$. By Theorem \[thm4.3\], $G_n$ is super-$\la''$ and $\rho'(G_n)=n-1$ if $n\geqslant 5$. Hence, if $n\geqslant 5$, $\la''(G_n)=3n-4>2n-2=\xi(G_n)$ implies $G_n$ is super-$\la'$.
\[rem4.6\] [In Corollary \[cor4.5\], the condition “$n\geqslant 5$" is necessary. For example, $Q_4$ is $\la''$-optimal and not super-$\la''$, and $\rho'(Q_4)=2$ (see Remark \[rem3.5\]). ]{}
We now consider the second family of graphs $G(G_0,G_1,\dots,G_{m-1};\mathscr{M})$. To guarantee that $G$ is triangle-free, we can assume $m\geqslant 4$. Let $I_m=\{0,1,\dots,m-1\}$.
\[thm4.7\] Let $G_i$ be a $k$-regular $k$-edge-connected graph of order $n$ without triangles for each $i\in I_m$. If $k\geqslant 3$, $n>
\lceil\frac{3k+2}{2}\rceil$ and $m\geqslant 4$, then $G=G(G_0,\dots,G_{m-1};\mathscr{M})$ is super-$\la''$ and $\rho'(G)=k+1$.
It is easy to check that $G$ is $(k+2)$-regular and triangle-free. By Theorem \[thm1.2\], $G$ is $\la''$-connected and $$\label{e4.7}
\la''(G)\leqslant \xi_2(G)=3k+2.$$ By Theorem \[thm3.8\] (i), we only need to prove that $G$ is super-$\la''$. Suppose to the contrary that $G$ is not super-$\la''$. Then there exists a $\la''$-fragment $X$ of $G$ such that $|\overline X|\geqslant |X|\geqslant 4$. Since $G$ is triangle-free, $G[X]$ is also triangle-free and $|E(G[X])|\leqslant
\frac{|X|^2}{4}$ by Lemma \[lem4.1\]. It follows that $$\begin{array}{rl}
3k+2\geqslant \la''(G)=d_G(X)=(k+2)|X|-2|E(G[X])|\geqslant (k+2)|X|-\frac{1}{2}|X|^2,
\end{array}$$ that is, $(|X|-3)(|X|-(2k+1))+1\geqslant 0$, which implies, since $|X|\geqslant 4$ and $k\geqslant 3$, $$\label{e4.8}
|X|\geqslant 2k+1.$$
We will deduce a contradiction to (\[e4.7\]) by proving that $$\label{e4.9}
\la''(G)>3k+2.$$
To the end, for each $i\in I_m$, let $$\begin{array}{l}
V_i=V(G_i),\ \ X_i=X\cap V_i,\\
F_i=E_G(X)\cap E(G_i),\ \
F'_i=E_G(X)\cap M_{i,i+1({\rm mod}~m)}.
\end{array}$$ Then $F_i=E_{G_i}(X_i)$. Let $$J=\{j\in I_m:\ X_j\ne \emptyset\}\ {\rm and}\ J'=\{j\in J:\ X_j=V_j\}.$$ Then $|F_j|\geqslant \la(G_j)=k$ for any $j\in J\setminus J'$. Thus, if $|J\setminus J'|\geqslant 4$, then $$\la''(G)=|E_G(X)|\geqslant \sum_{i\in J\setminus J'}|F_i|\geqslant 4k> 3k+2\ \ {\rm for}\ k\geqslant 3,$$ and so (\[e4.9\]) follows. We assume $|J\setminus J'|\leqslant 3$ below. There are two cases.
[*Case*]{} 1. $|J|\leqslant m-1$.
[*Subcase*]{} 1.1. $J'\ne \emptyset$.
Let $\ell\in I_m\setminus J$ and $j\in J'$. Then $X_\ell=\emptyset$ and $X_j=V_j$. Since $j\ne\ell$, without loss of generality, assume $\ell<j$ and let $s=j-\ell$. By the structure of $G$, there exist exactly $n$ disjoint paths of length $s$ between $V_j$ and $V_\ell$ passing through $G_{j-1}$ (maybe $j-1=\ell$), and $n$ disjoint paths of length $m-s$ between $V_j$ and $V_\ell$ passing through $G_{j+1({\rm mod}~m)}$ (maybe $\ell=j+1({\rm mod}~m)$). Each of these paths has at least one edge that is in $E_G(X)$. Since $n>
\lceil\frac{3k+2}{2}\rceil$, we have that $$\la''(G)=|E_G(X)|\geqslant 2n>3k+2,$$ and so (\[e4.9\]) follows.
[*Subcase*]{} 1.2. $J'=\emptyset$. In this subcase, $|J|\leqslant
3$.
If $|J|=1$, say $X_1=X$, since $E_G(X)=F_1\cup F'_0\cup F'_1$ and $|F'_0|=|F'_1|=|X_1|=|X|$, then $|E_G(X)|\geqslant 2|X|+|F_1|$. Combining this fact with (\[e4.8\]), we have that $$\la''(G)=|E_G(X)|\geqslant 2|X|+|F_1|\geqslant 2(2k+1)+k>3k+2,$$ and so (\[e4.9\]) follows.
If $|J|=2$, say $J=\{p,q\}$, then $|p-q|=1$ since $G[X]$ is connected, say $q=p+1$. $F_p\cup F_{p+1}\cup F'_{p-1}\cup
F'_{p+1}\subseteq E_G(X)$. Since $|F_p|\geqslant k$, $|F_{p+1}|\geqslant k$ and $|F'_{p-1}\cup F'_{p+1}|=|X|$. Combining these facts with (\[e4.8\]), we have that $$\la''(G)=|E_G(X)|\geqslant |F_p|+|F_{p+1}|+|X|\geqslant 2k+(2k+1)>3k+2,$$ and so (\[e4.9\]) follows.
If $|J|=3$, without loss of generality, assume $J=\{1,2,3\}$ since $G[X]$ is connected, then $F'_0\ne\emptyset$ and $F'_3\ne\emptyset$ since $m\geqslant 4$. If $|F'_0|=|F'_3|=1$, then $|X_1|=|X_3|=1$. Since $|X|\geqslant 4$, if $|X_1|=|X_3|=1$, then $|X_2|\geqslant 2$, and so $|F'_1|\geqslant 1$ and $|F'_2|\geqslant 1$. Thus, it is always true that $|F'_0|+|F'_1|+|F'_2|+|F'_3|>2$. It follows that $$\la''(G)=|E_G(X)|\geqslant \sum_{j=1}^3|F_j|+\sum_{j=0}^3|F'_j|>
3k+2,$$ and so (\[e4.9\]) follows.
[*Case*]{} 2. $|J|=m$.
In this case, $J'\ne \emptyset$ since $m\geqslant 4$ and $|J\setminus J'|\leqslant 3$. If $|J\setminus J'|\leqslant 2$, then $|J'|\geqslant 2$, and $$|\overline X|=\sum\limits_{j\in J\setminus J'}|V(G_j-X_j)|\leqslant
2(n-1)<2n\leqslant \sum\limits_{j\in J'}|V_j|<|X|,$$ a contradiction to $|\overline X|\geqslant |X|$. Therefore, $|J\setminus J'|=3$. Since $G[X]$ is connected, without loss of generality, let $J\setminus J'=\{1,2,3\}$. Then $F'_0\ne\emptyset$ and $F'_3\ne\emptyset$. Since $|\overline X|\geqslant |X|$, there exists at least two $i\in J\setminus J'$ such that $|X_i|\leqslant
\frac{n}{2}$. Thus, at least one of $|V_1\setminus X_1|$ and $|V_3\setminus X_3|$ is not less than $\frac{n}{2}$, that is, either $|F'_0|\geqslant \frac{n}{2}$ or $|F'_3|\geqslant \frac{n}{2}$. It follows that $$\la''(G)=|E_G(X)|\geqslant \sum_{j=1}^3|F_j|+|F'_0|+|F'_3|> 3k+2,$$ and so (\[e4.9\]) follows.
Under the hypothesis that $G$ is not super-$\la''$, we deduce a contradiction to (\[e4.7\]). Thus, $G$ is super-$\la''$. By Theorem \[thm3.8\] (i), $\rho'(G)=k+1$, and so the theorem follows.
As applications of Theorem \[thm4.7\], we consider two families well-known transitive networks.
Let $G(n,d)$ denote a graph which has the vertex-set $V=\{0,1,\dots,n-1\}$, and two vertices $u$ and $v$ are adjacent if and only if $|u-v|=d^i({\rm mod}~n)$ for any $i$ with $0\leqslant
i\leqslant \lceil\log_d n\rceil-1$. Clearly, $G(d^m,d)$ is a circulant graph, which is $\delta$-regular and $\delta$-connected, where $\delta=2m-1$ if $d=2$ and $\delta=2m$ if $d\ne 2$. For circulant graphs with order between $d^{m}$ and $d^{m+1}$, that is, $G(cd^{m},d)$ with $1<c<d$, $\delta=2m+1$ if $c=2$ and $\delta=2m+2$ if $c>2$, moreover, Park and Chwa [@pc94] showed that $G(cd^{m},d)$ can be recursively constructed, that is, $G(cd^{m},d)=G(G_0,G_1,\dots,G_{d-1};\mathscr{M})$, where $G_i$ is isomorphic to $G(cd^{m-1},d)$ for each $i=0,1,\ldots,d-1$, and so $G(cd^{m},d)$ is called the [*recursive circulant graph*]{}, which is $\delta$-regular and $\delta$-connected. In particular, the graph $G(2^m,4)$ is $2m$-regular $2m$-connected, has the same number of vertices and edges as a hypercube $Q_m$. However, $G(2^m,4)$ with $m\geqslant 3$ is not isomorphic to $Q_m$ since $G(2^m,4)$ has an odd cycle of length larger than $3$. Compared with $Q_m$, $G(2^m,
4)$ achieves noticeable improvements in diameter ($\lceil\frac{3m-1}{4}\rceil$). Thus, the recursive circulant graphs have attracted much research interest in recent ten years (see Park [@p08; @pc94] and references therein). Since, when $c\geqslant 3$, $G(cd^0,d)$ is isomorphic to a cycle of length $c$, $G(cd^m,d)$ contains triangles if $c=3$.
The [*$n$-dimensional undirected toroidal mesh*]{}, denoted by $C(d_1,\dots,d_n)$, is defined as the cartesian products $C_{d_1}\times C_{d_2}\times\cdots\times C_{d_n}$, where $C_{d_i}$ is a cycle of length $d_i~(\geqslant 3)$ for each $i = 1, 2,\dots,
n$ and $n\geqslant 2$. It is known that $C(d_1,\dots,d_n)$ is a $2n$-regular $2n$-edge-connected transitive graph with girth $g =
\min\{4, d_i, 1 \leqslant i \leqslant n\}$. Thus, if $d_i\geqslant
4$ for each $i=1,2,\ldots, n$, then $C(d_1,\dots,d_n)$ is triangle-free. $C(d_1,\dots,d_n)$ can be expressed as $G(G_0,G_1,\dots,G_{d_1-1};\mathscr{M})$, where $G_i$ is isomorphic to $C_{d_2}\times\cdots\times C_{d_n}$ for each $i=0,1,\ldots,d_1-1$.
Since the two families of networks are transitive, by Corollary \[cor3.11\], we can determine the exact values of $\rho'$ when the girth $g\geqslant 5$. By Theorem \[thm4.7\], we have the following two stronger results immediately.
Let $c,d,m$ be three positive integers with $1<c<d$, $c\ne 3$, $d\geqslant 4$, $m\geqslant 2$. Then $G=G(cd^{m},d)$ is super-$\la''$, and $\rho'(G)=2m$ if $c=2$ and $\rho'(G)=2m+1$ if $c\geqslant 4$.
If $n\geqslant 3$ and $d_i\geqslant 4$ for $1 \leqslant i \leqslant
n$, then $G=C(d_1,\dots,d_n)$ is super-$\la''$ and $\rho'(G)=2n-1$.
[s2]{}
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[^1]: E-mail address: [email protected] (Z.-M. Hong)
[^2]: Corresponding author, E-mail address: [email protected] (J.-M. Xu)
[^3]: The work was supported by NNSF of China (No. 61272008).
|
---
abstract: 'The contribution of ground state correlations (GSC) to the non–mesonic weak decay of $^{12}_\Lambda$C and other medium to heavy hypernuclei is studied within a nuclear matter formalism implemented in a local density approximation. We adopt a weak transition potential including the exchange of the complete octets of pseudoscalar and vector mesons as well as a residual strong interaction modeled on the Bonn potential. Leading GSC contributions, at first order in the residual strong interaction, are introduced on the same footing for all isospin channels of one– and two–nucleon induced decays. Together with fermion antisymmetrization, GSC turn out to be important for an accurate determination of the decay widths. Besides opening the two–nucleon stimulated decay channels, for $^{12}_\Lambda$C GSC are responsible for 14% of the rate $\Gamma_1$ while increasing the $\Gamma_{n}/\Gamma_{p}$ ratio by 4%. Our final results for $^{12}_\Lambda$C are: $\Gamma_{\rm NM}=0.98$, $\Gamma_{n}/\Gamma_{p}=0.34$ and $\Gamma_2/\Gamma_{\rm NM}=0.26$. The saturation property of $\Gamma_{\rm NM}$ with increasing hypernuclear mass number is clearly observed. The agreement with data of our predictions for $\Gamma_{\rm NM}$, $\Gamma_n/\Gamma_p$ and $\Gamma_2$ is rather good.'
author:
- 'E. Bauer$^{1,2}$'
- 'G. Garbarino$^3$'
title: 'On the role of ground state correlations in hypernuclear non–mesonic weak decay'
---
Introduction {#intro}
============
The study of nuclear systems with strangeness is a relevant question in modern nuclear and hadronic physics [@snp], which also implies important links with astrophysical processes and observables as well as with QCD, the underlying theory of strong interactions. Various strange nuclear systems can be studied in the laboratory, ranging from hypernuclei and kaonic nuclei to exotic hadronic states such as strangelets, $H$–dibaryons and pentaquark baryons. Strangeness production can also be investigated in relativistic heavy–ion collision experiments, whose main aim is to establish the existence of a quark–gluon plasma. Moreover, the cold and dense matter contained in neutron stars is expected to be composed by strange hadrons, in the form of hyperons and Bose–Einstein condensates of kaons, and eventually by strange quark matter for sufficiently dense systems.
The existence of hypernuclei —bound systems of non–strange and strange baryons— opens up the possibility to study the hyperon–nucleon and hyperon–hyperon interactions in both the strong and weak sectors. In turn, such interactions are important inputs, for instance, when investigating the macroscopic properties (masses and radii) of neutron stars. The best studied hypernuclei contain a single $\Lambda$–hyperon. In a nucleus the $\Lambda$ can decay by emitting a nucleon and a pion (mesonic mode) as it happens in free space, but its (weak) interaction with the nucleons opens new channels which are indicated as non–mesonic decay modes (for recent reviews see Refs. [@ra98; @al02; @Ch08; @Pa07; @Ou05]). These are the dominant decay channels of medium–heavy nuclei, where, on the contrary, the mesonic decay is disfavoured by the Pauli blocking effect on the outgoing nucleon. In particular, one can distinguish between one– and two–body induced decays, $\Lambda N\to nN$ and $\Lambda NN\to nNN$. The hypernuclear lifetime is given in terms of the mesonic ($\Gamma_{\rm M}=\Gamma_{\pi^-}+\Gamma_{\pi^0}$) and non–mesonic decay widths ($\Gamma_{\rm NM}=\Gamma_1+\Gamma_2$) by $\tau=\hbar/\Gamma_{\rm T}= \hbar/[\Gamma_{\rm M}+\Gamma_{\rm NM}]$. The various isospin channels contribute to the one– and two–nucleon induced non–mesonic rates as follows: $\Gamma_1=\Gamma_n+\Gamma_p\equiv \Gamma(\Lambda n\to nn)+\Gamma(\Lambda p\to np)$ and $\Gamma_2=\Gamma_{nn}+\Gamma_{np}+\Gamma_{pp}\equiv
\Gamma(\Lambda nn\to nnn)+\Gamma(\Lambda np\to nnp)+\Gamma(\Lambda pp\to npp)$.
One should note that, strictly speaking, the only observables in hypernuclear weak decay are the lifetime $\tau$, the mesonic rates $\Gamma_{\pi^-}$ and $\Gamma_{\pi^0}$ and the spectra of the emitted particles (nucleons, pions and photons). None of the above non–mesonic partial decay rates ($\Gamma_n$, $\Gamma_p$, $\Gamma_{np}$, etc) is an observable from a quantum–mechanical point of view. Each one of the possible elementary non–mesonic decays occurs in the nuclear environment, thus subsequent final state interactions (FSI) modify the quantum numbers of the weak decay nucleons and new, secondary nucleons are emitted as well: this prevents the measurement of any of the non–mesonic partial decay rates. Instead, the total width $\Gamma_{\rm T}$ can be measured: being an inclusive quantity, for such a measurement one has to detect any of the possible products of either mesonic or non–mesonic decays (typically protons from non–mesonic decays). The fact that the detected particles undergo FSI does not appreciably alters the lifetime measurement, since strong interactions proceeds on a much shorter time scale than weak decays, and $\tau^{\rm measured}=\tau+\tau^{\rm strong}\simeq \tau\equiv
\hbar/\Gamma_{\rm T}$.
In order to achieve a proper knowledge of the various decay mechanisms (in particular of the strangeness–changing baryon–baryon interactions), a meaningful comparison between theory and experiment must be possible. The above discussion shows that such a comparison requires the introduction of non–standard theoretical definitions for the non–mesonic partial decay rates (which, as mentioned, are not quantum–mechanical observables) together with the corresponding experimental methods for determining these rates. In our opinion, this point has not been adequately addressed in previous works and, among others, it has impacted on the well–known puzzle on the ratio $\Gamma_n/\Gamma_p$ between the neutron– and the proton–induced non–mesonic rates.
In order to explain how the total non–mesonic rate can be determined in an experiment, we have to discuss first the measurement of the mesonic rates. The pion and nucleon emitted in a mesonic decay both have a momentum of about $100\, {\rm MeV/c}$. Nucleons of a few MeV kinetic energy cannot be observed as they are below the experimental detection thresholds. Mesonic decays are thus identified by measuring pions ($\pi^-$’s or $\pi^0\to \gamma \gamma$ decays). The mesonic width $\Gamma_{\pi^-}$ ($\Gamma_{\pi^0}$) is determined from the observed $\pi^-$ ($\pi^0\to \gamma \gamma$) energy spectra and the total width $\Gamma_{\rm T}$. For instance, $\Gamma^{\rm exp}_{\pi^-}=(N_{\pi^-}/N_{\rm hyp})\Gamma^{\rm exp}_{\rm T}$, $N_{\pi^-}$ being the total number of detected $\pi^-$’s and $N_{\rm hyp}$ the total number of produced hypernuclei. Both these numbers are corrected for the detection efficiencies and the detector acceptances implied in the measurements. The mesonic rates measured in this way thus include the effect of in–medium pion renormalization. Theoretical models [@mesonic-th; @gal09] also taking into account distorted pion waves obtained mesonic widths in agreement with the experimental values (in particular, the importance of the pion wave–function distortion was first demonstrated in the works of Ref. [@mesonic-th]).
The experimental total non–mesonic rate is then obtained as the difference between the total and the mesonic rates, $\Gamma^{\rm exp}_{\rm NM}=\Gamma^{\rm exp}_{\rm T}-\Gamma^{\rm exp}_{\rm M}$. The experimental determination of $\Gamma_{n}/\Gamma_{p}$ is much more involved. Indeed, this ratio must be extracted from the nucleon emission spectra, and this requires some theoretical input [@ga03; @ga04]. FSI are very important for the non–mesonic processes and nucleons which have or have not suffered FSI are indistinguishable between each other. A theoretical simulation of nucleon FSI is thus needed and, in principle, a coherent sum of both kinds of nucleons must be considered when evaluating the spectra. Generally, FSI are accounted for by an intranuclear cascade model [@ra97], which is a semi–classical scheme.
In the present work we study the non–mesonic weak decay of hypernuclei ranging from $^{11}_\Lambda$B to $^{208}_\Lambda$Pb by using a nuclear matter approach implemented in a local density approximation. All the possible isospin channels for one– and two–body induced mechanisms are included in a microscopic approach based on the evaluation of Goldstone diagrams. The partial decay rates are derived by starting from a two–body weak transition potential. In particular, we investigate the effect of ground state correlations (GSC), i.e., the contribution of nucleon–nucleon correlations in the hypernucleus ground state. Leading order GSC contributions will be introduced on the same ground for one– and two–nucleon induced processes for the first time. The general formalism we adopt was established in Refs. [@ba03; @ba04]. The weak transition potential for the nucleon–nucleon strong interaction contributing to the GSC we adopt a Bonn potential with the exchange of $\pi$, $\rho$, $\sigma$ and $\omega$ mesons.
The paper is organized as follows. In Section \[pref\] we start with general considerations about FSI, the definitions we employ for the weak decay rates as well as the method usually employed for the determination of $\Gamma_n/\Gamma_p$ from data on nucleon spectra. In Section \[miss\] we present and discuss the general framework for the evaluation of the one– and two–nucleon induced decay widths with the inclusion of GSC. In Section \[fsi\] we make some further considerations about the evaluation of the widths and we discuss some former work on the subject. Explicit expressions for the considered GSC diagrams contributing to the one–nucleon induced rates are given in Section \[gngpgsc\] and in Appendix A. Then, in Section \[results\] we present our results and finally in Section \[conclusions\] some conclusions are given.
Preliminary considerations on FSI effects and on the determination of the weak decay rates {#pref}
==========================================================================================
The $\Gamma_{n}/\Gamma_{p}$ ratio is defined as the ratio between the total number of primary (i.e., weak decay) neutron–neutron and neutron–proton pairs, $N^{\rm wd}_{nn}$ and $N^{\rm wd}_{np}$, emerging from the processes $\Lambda n\to nn$ and $\Lambda p\to np$, respectively. Due to nucleon final state interactions and two–body induced decays, the following inequality is expected for the observables $nn$ and $np$ coincidence numbers, $N_{nn}$ and $N_{np}$ [@ga03] [^1]: $$\label{defcos}
\frac{\Gamma_{n}}{\Gamma_{p}} \equiv \frac{N^{\rm wd}_{nn}}{N^{\rm wd}_{np}}
\neq \frac{N_{nn}}{N_{np}}~.$$ Only $N_{nn}/N_{np}$ is a quantum–mechanical observable: generally, its measurement is affected by thresholds on the nucleon energy and the pair opening angle [@Ou05; @KEK; @KEK2]. Theoretical models are thus required to determine the “experimental” value of $\Gamma_n/\Gamma_p$ from a measurement of $N_{nn}/N_{np}$. This unusual procedure to determine $(\Gamma_n/\Gamma_p)^{\rm exp}$ makes complete sense provided different models are at disposal and lead to the same extracted ratio: only in such a case one is allowed to define this value as the experimental result for $\Gamma_n/\Gamma_p$. It is thus important to explore the predictions of alternative models when applied to the analysis of data. In the present section we go deeper into questions of this kind to show some ambiguities which need to be emphasized for a meaningful comparison between theory and experiment.
Let us first illustrate in some detail the procedure normally adopted to extract $(\Gamma_n/\Gamma_p)^{\rm exp}$ from measurements of $N_{nn}/N_{np}$ [@ga03; @ga04]. Each one of the non–mesonic weak decay channel takes place by the emission of two or three primary nucleons. These nucleons propagate within the nuclear environment and cannot be measured. The strong interactions with the surrounding nucleons can change the charge and the energy–momentum of the primary nucleons; some of them can be absorbed by the medium and the emission of additional (secondary) nucleons can occur as well. All these processes are generically designated as final state interactions (FSI): they do not have to be included when calculating the decay rates, but the observable nucleon spectra, i.e., $N_{nn}$ and $N_{np}$, are crucially affected by them. One has to emphasize that, on the contrary, baryon–baryon short range correlations in both the initial and the final states as well as mean field effects on the single particle wave–functions are genuine contributions to the decay rates.
FSI pertain to the same quantum–mechanical problem which starts with the $\Lambda$ decay and ends with the detection of the particles emitted by the hypernucleus. In a strict quantum–mechanical scheme, FSI cannot thus be disentangled from the weak interaction part of the problem: this is an analogous way of expressing the fact that the weak decay rates are not measurable. However, up to now FSI have been simulated by means of semi–classical models, i.e., by intranuclear cascade codes (INC) [@ra97] acting after the weak decay, thus losing quantum–mechanical coherence. In such INC analyses, both one– and two–nucleon induced decays are included as inputs and one proceeds to fit $N_{nn}/N_{np}$ data in order to determine the value of $(\Gamma_n/\Gamma_p)^{\rm exp}$. Technically, this is achieved by applying Eq. (16) of Ref. [@ga04] (see also Eqs. (1) and (2) of Ref. [@ga03] and Eq. (3.14) of Ref. [@Ba06]), which is an exact relation only neglecting quantum coherence among the final, observable nucleons. Note that such a procedure also requires a theoretical estimate for the ratio $\Gamma_2/\Gamma_1$. In other words, present nucleon–nucleon coincidence data only allows us to determine a correlation property between $\Gamma_n/\Gamma_p$ and $\Gamma_2/\Gamma_1$.
In general terms, one could wonder if it is possible to identify those quantum–mechanical contributions whose classical limit leads to a factorization between the weak decay process and the INC rescattering. This is a relevant question since in the theoretical evaluation of the non–mesonic decay rates FSI contributions must not be included; one indeed aims to extract the contribution of the elementary $\Lambda N\to nN$ and $\Lambda nn\to nNN$ processes by studying hypernuclear decay. Unfortunately, the above question does not seem to have a simple solution. Although we make here some considerations about this point, we believe that a complete answer to it goes beyond the present contribution.
Let us illustrate, by using an example, the nature of the problem. Consider the $\Lambda$ self–energy diagram $(a)$ of Fig. \[fsgs2p2h\]. This is a (time–ordered) Goldstone diagram where the weak transition potential $V^{\Lambda N \to NN}$, which is a two–body operator, produces an intermediate $2p1h$ configuration; afterwards, the action of the nucleon–nucleon strong interaction $V^{NN}$ creates a further $1p1h$ pair and leads to a $3p2h$ final state. In terms of amplitudes, $V^{\Lambda N \to NN}$ produces two nucleons, one of which then strongly interacts with another nucleon, ending in the emission of three nucleons. Since the potential $V^{NN}$ acts after $V^{\Lambda N \to NN}$, diagram $(a)$ contains a FSI effect and we argue that it must not be included when evaluating the non–mesonic decay rate. Note that the idea of an interaction taking place after or before another one is a valid statement here as we are working with Goldstone diagrams.
![Goldstone diagrams for FSI $(a)$ and $2p2h$ GSC contributions $(b)$ for three nucleon emission. The dashed and wavy lines stand for the potentials $V^{\Lambda N \to NN}$ and $V^{NN}$, respectively. The diagram $(a)$ has poles on the $2p1h$ and $3p2h$ configurations, while $(b)$ has a single pole on the $3p2h$ configuration. For the present discussion we only consider the $3p2h$ poles indicated by the dotted lines.[]{data-label="fsgs2p2h"}](fig1n)
On the contrary, diagram $(b)$ of Fig. \[fsgs2p2h\] represents a ground state correlation (GSC) effect. It corresponds to an amplitude in which the $\Lambda$ decays by interacting with a correlated nucleon pair. Since the nucleon–nucleon interaction takes place before the action of the weak transition, this diagram must be considered when evaluating the decay rate $\Gamma_{2}$.
Note also that the Goldstone diagrams $(a)$ and $(b)$ are two different time orderings of the same Feynman diagram. If $\Gamma_{2}$ were an observable, it would have to be evaluated by means of Feynman rather than Goldstone diagrams; both diagrams $(a)$ and $(b)$ would contribute to $\Gamma_2$. These diagrams must actually be taken into account when evaluating the observable nucleon spectra. However, here we argue that, since $\Gamma_2$ is not an observable, some of the Goldstone diagram should not be included in the theoretical definition of this rate. The class of diagrams that does not contribute to $\Gamma_2$ depends on the definition one adopts for FSI. Our definition leaves aside those Goldstone diagrams, like diagram $(a)$ in Fig. \[fsgs2p2h\], in which at least one nucleon–nucleon interaction takes place after the weak transition potential. If on the other hand one were to include diagram $(b)$ in the calculation of the widths, then it would not be clear how to identify the diagrams incorporating FSI effects.
A similar analysis to the previous one holds for the one–nucleon induced rates. Summarizing, we assume that one– and two–nucleon induced decay widths, which are not observables, are interpreted in terms of Goldstone diagrams in which no FSI effect is present. All the Goldstone diagrams in which at least one nucleon–nucleon interaction takes place after the weak transition potential must not be included when evaluating the decay rates. Any Goldstone diagram representing a GSC is instead a genuine contribution to the rates. In the calculation of the observable nucleon spectra, a description in terms of Feynman diagrams must instead be employed.
Many–body terms in the non–mesonic decay rates {#miss}
==============================================
Let us consider the one and two–body induced non–mesonic weak decay width for a $\Lambda$–hyperon with four–momentum $k=(k_0,\v{k})$ inside infinite nuclear matter with Fermi momentum $k_F$. In a schematic way, one can write: $$\label{decw}
\Gamma_{1 \, (2)}(k,k_{F}) = \sum_{f} \,
|\bra{f} V^{\Lambda N\to NN} \ket{0}_{k_{F}}|^{2}
\delta (E_{f}-E_{0})~,$$ where $\ket{0}_{k_{F}}$ and $\ket{f}$ are the initial hypernuclear ground state (whose energy is $E_0$) and the possible $2p1h$ or $3p2h$ final states, respectively. The $2p1h$ ($3p2h$) final states define $\Gamma_{1}$ ($\Gamma_{2}$). The final state energy is $E_f$ and $V^{\Lambda N\to NN}$ is the two–body weak transition potential.
The decay rates for a finite hypernucleus are obtained by the local density approximation [@os85], i.e., after averaging the above partial width over the $\Lambda$ momentum distribution in the considered hypernucleus, $|\widetilde{\psi}_{\Lambda}(\v{k})|^2$, and over the local Fermi momentum, $k_{F}(r) = \{3 \pi^{2} \rho(r)/2\}^{1/3}$, $\rho(r)$ being the density profile of the hypernuclear core. One thus has: $$\label{decwpar3}
\Gamma_{1 \, (2)} = \int d \v{k} \, |\widetilde{\psi}_{\Lambda}(\v{k})|^2
\int d \v{r} \, |\psi_{\Lambda}(\v{r})|^2
\Gamma_{1 \, (2)}(\v{k},k_{F}(r))~,$$ where for $\psi_{\Lambda}(\v{r})$, the Fourier transform of $\widetilde{\psi}_{\Lambda}(\v{k})$, we adopt the $1s_{1/2}$ harmonic oscillator wave–function with frequency $\hbar \omega$ ($=10.8$ MeV for $^{12}_\Lambda$C) adjusted to the experimental energy separation between the $s$ and $p$ $\Lambda$–levels in the considered hypernucleus. The $\Lambda$ total energy in Eqs. (\[decw\]) and (\[decwpar3\]) is given by $k_{0}=m_\Lambda+\v{k}^2/(2 m_\Lambda)+V_{\Lambda}$, $V_\Lambda$ ($=-10.8$ MeV for $^{12}_\Lambda$C) being a binding energy term.
Since $V^{\Lambda N\to NN}$ is a two–body operator, the emission of two nucleons is originated either from the Hartree–Fock vacuum or from GSC induced by the nucleon–nucleon interaction. At variance, the emission of three nucleons can be only achieved when $V^{\Lambda N\to NN}$ acts over a GSC. It is therefore convenient to introduce the following hypernuclear ground state wave–function [@ba09]: $$\label{gstate}
\ket{0}_{k_{F}}=\mathcal{N}(k_{F}) \, \left(\ket{\;}
- \sum_{p, h, p', h'} \,
\frac{\bra{p h p' h'} V^{N N} \ket{\;}_{D+E}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}} \,
\ket{p h p' h'}\right) \otimes \ket{p_{\Lambda}}~,$$ where $\ket{\;}$ is the uncorrelated core ground state wave–function, i.e., the Hartree–Fock vacuum, while the second term in the rhs represents $2p2h$ correlations and contains both direct ($D$) and exchange ($E$) matrix elements of the nuclear residual interaction $V^{N N}$. Besides, $\ket{p_{\Lambda}}$ is the normalized state of the $\Lambda$, the particle and hole energies are denoted by $\varepsilon_{i}$ and: $$\label{norconst}
\mathcal{N}(k_{F})=\left( 1 + \sum_{p,
h, p', h'} \, \left|\frac{\bra{p h p' h'}
V^{N N} \ket{\;}_{D+E}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}
\right|^{2} \, \right)^{-1/2}$$ is the ground state normalization function. The particular labeling of Eqs. (\[gstate\]) and (\[norconst\]) is explained in Fig. \[gs2p2h\]. The explicit expression for $\mathcal{N}(k_{F})$ is given in Ref. [@ba09b].
![Direct (D) and exchange (E) Goldstone diagrams for the $2p2h$ GSC induced by the nuclear residual interaction $V^{NN}$.[]{data-label="gs2p2h"}](fig1)
By inserting Eq. (\[gstate\]) into Eq. (\[decw\]), for $\Gamma_{1}$ one obtains: $$\begin{aligned}
\label{decw1}
\Gamma_{1}(\v{k},k_{F}) & = & \mathcal{N}^{\, 2}(k_{F})
\sum_{f} \, \delta (E_{f}-E_{0}) \;
\left|\bra{f} V^{\Lambda N\to NN} \ket{p_{\Lambda}}_{D+E}
\phantom{\frac{A^A}{B^A}} \right. \\
&& \left. -\sum_{p, h, p', h'} \,
\bra{f} V^{\Lambda N\to NN} \ket{p h p' h'; \, p_{\Lambda}}_{D+E}
\frac{\bra{p h p' h'; \, p_{\Lambda}} V^{N N} \ket{p_{\Lambda}}_{D+E}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}\right|^2~, \nonumber\end{aligned}$$ the final states $\ket{f}$ being restricted to $2p1h$ states. For $\Gamma_{2}$ one has: $$\begin{aligned}
\label{decw2}
\Gamma_{2}(\v{k},k_{F}) & = & \mathcal{N}^{\, 2}(k_{F})
\sum_{f} \, \delta (E_{f}-E_{0}) \\
&&\times \left| \sum_{p, h, p', h'} \,
\bra{f} V^{\Lambda N\to NN} \ket{p h p' h'; \, p_{\Lambda}}_{D+E}
%\phantom{\frac{A^A}{B^A}} \right.
%\nonumber \\
%&& \left.
\frac{\bra{p h p' h'; \, p_{\Lambda}} V^{N N} \ket{p_{\Lambda}}_{D+E}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}\right|^{2}~, \nonumber\end{aligned}$$ where the final states are given by $3p2h$ states. Note that all the matrix elements of $V^{NN}$ and $V^{\Lambda N\to NN}$ appear in the antisymmetrized form.
Let us focus now on the kind of diagrams contributing to $\Gamma_{1}$ and $\Gamma_{2}$. This discussion is done in terms of transition amplitudes rather than self–energies.
![Transition amplitudes contributing to $\Gamma_{1}$. A double–line (without arrow) represents the $\Delta(1232)$ resonance.[]{data-label="gam1gsc"}](fig2)
In Fig. \[gam1gsc\] we report some of the most representative transition amplitudes which contribute to $\Gamma_{1}$. All diagrams but $(a)$ are originated by a GSC. Only the contribution of diagram $(a)$ to $\Gamma_{1}$ has been calculated microscopically up to now. The line $(b)$ represents typical $2p2h$ correlations. The contribution $(c)$ is a contact term involving a $\pi \pi NN$ strong vertex, while line $(d)$ represents the contribution of the $\Delta(1232)$ resonance. It should be mentioned that there has been a great deal of controversy around the theoretical determination of the $\Gamma_{n}/\Gamma_{p}$ ratio and the challenging comparison with data. In these discussions, all theoretical efforts have been devoted to the $(a)$ term only; the remaining ones have simply been ignored.
A similar analysis can be done for $\Gamma_{2}$ starting from the amplitudes of Fig. \[gam2gsc\]. Again, only the $(a)$ term has been evaluated up to now in microscopic calculations [@ba04]. The graphs in Figs. \[gam1gsc\] and \[gam2gsc\] are only representative cases. For instance, also the amplitude of Fig. \[gam1-cor\] should be included when calculating $\Gamma_1$. Unlike the other amplitudes of Figures. \[gam1gsc\] and \[gam2gsc\], the one in Fig. \[gam1-cor\] involves a strong interaction $V^{\Lambda N}$ between the $\Lambda$ and a $1p1h$ pair (i.e., a $1p1h$ GSC) and then the usual action of the weak transition potential. Apart from the explicit calculation, such a contribution could in principle be included in an effective way through the calculation of diagram $(a)$ of Fig. \[gam1gsc\] with a suitably chosen weak transition potential $V^{\Lambda N\to NN}$. However, based on the absence of isovector–meson exchange in the strong potential $V^{\Lambda N}$, one may anticipate a small effect of this amplitude. Other amplitudes will provide important contributions. In the graph $(a)$ of Fig. \[gam2gsc\] the weak transition potential can also be connected to a hole line [@ba04]. In addition, since $V^{N N}$ and $V^{\Lambda N\to NN}$ are two–body operators whose matrix elements are antisymmetrized, Pauli exchange terms must be considered as well [@ba09b].
![Transition amplitudes contributing to $\Gamma_{2}$.[]{data-label="gam2gsc"}](fig3)
![Transition amplitude contributing to $\Gamma_{1}$ and involving a strong interaction $V^{\Lambda N}$ between the hyperon and a $1p1h$ pair.[]{data-label="gam1-cor"}](fig_g1_corr)
All the graphs in Figs. \[gam1gsc\] and \[gam2gsc\] have the same initial state, which is the hypernuclear ground state. The final state of the graphs in Fig. \[gam1gsc\] (Fig. \[gam2gsc\]) is a $2p1h$ ($3p2h$) state. To obtain the various decay width, all graphs representing transitions amplitudes with the same initial and final states are added and then squared. For instance, from Fig. \[gam2gsc\] one obtains a total of six direct diagrams: the square of each individual amplitude plus the three interference terms. For the amplitudes in Fig. \[gam1gsc\] there is a total of twenty–eight different direct terms. In addition, antisymmetrization considerably increases the amount of diagrams. From our previous works it is clear to us that a full microscopic evaluation of each term is mandatory for several reasons. First, a raw estimation of a remarkable amount of different diagrams makes the final result quite uncertain. Secondly, there is no ground to evaluate differently the diagrams originated from Fig. \[gam1gsc\] and those from Fig. \[gam2gsc\]: once a microscopic calculation is performed for the square of diagrams $(a)$ of Figs. \[gam1gsc\] and \[gam2gsc\], the same should be done for the remaining contributions, which are all leading order GSC contributions.
In the present work, as a further step towards the calculation of the whole set of diagrams relevant for the non–mesonic decay, the one–nucleon induced widths originated from the sum of the transition amplitudes $(a)$ plus $(b1)$ of Fig. \[gam1gsc\] are evaluated for the first time. Accordingly, the two–nucleon induced rates are instead obtained from the amplitude $(a)$ of Fig. \[gam2gsc\] by following Ref. [@ba09b]. Antisymmetrization is coherently applied to all contributions. Before proceeding with the formal derivation of the decay widths, in the next Section we first point out additional observations on the evaluation of the decay rates and on previous, related work.
Further considerations on the evaluation of the weak decay rates {#fsi}
================================================================
Let us start this discussion by paying attention to the twofold effect of the nuclear residual interaction $V^{NN}$ within the matrix elements of Eq. (\[decw\]). When $V^{NN}$ acts on the uncorrelated hypernuclear ground state $\ket{\,}$, as in Eq. (\[gstate\]), one has a GSC. Alternatively, $V^{NN}$ may introduce medium effects on the weak transition potential $V^{\Lambda N \to NN}$. Both effects must be taken into account when calculating the decay rates. In addition, $V^{NN}$ may modify the final states $\ket{f}=\ket{2p1h}$ or $\ket{3p2h}$: for instance, acting on a $\ket{2p1h}$ final state, it can produce a $\ket{3p2h}$ state, as in Fig. \[fsgs2p2h\](a); this results in a FSI which does not contribute to Eq. (\[decw\]).
Concerning the medium effects previously mentioned, let us discuss some aspects of the work of Ref. [@Ji01]. Here, $V^{NN}$ introduces medium modifications on the mesons propagators appearing in $V^{\Lambda N \to NN}$ through the direct part of the RPA (ring approximation): schematically, in our scheme one simply has to replace $V^{\Lambda N \to NN}$ with $\widetilde{V}^{\Lambda N \to NN}=V^{\Lambda N \to NN}/|1-\Pi V^{NN}|$, where the polarization propagator $\Pi$ contains $1p1h$ and $1\Delta 1h$ contributions in Ref. [@Ji01]. Note that, since only the absolute value of the ring propagator is kept, the modified weak transition potential remains a real function. This approach thus represents a refinement of the weak transition potential and is consistent with Eq. (\[decw\]).
We emphasize that the mere use of diagrams when discussing the formalism developed in Ref. [@Ji01] or the present one could be misleading. For the approximation considered in Ref. [@Ji01], in Eq. (\[decw\]) one has to employ the matrix element $\bra{f} \widetilde{V}^{\Lambda N \to NN} \ket{0}_{D+E}$ of the modified weak transition potential (the corresponding direct and exchange self–energy diagrams are shown in Fig. 2 in Ref. [@Ji01]). By making an expansion of the square of this matrix element in the ring series, the two terms at first order in $V^{NN}$ correspond to a self–energy contribution which matches exactly with the $(a)$ diagram in Fig. \[figfsiab\], where the final state $\ket{f}=\ket{2p1h}$ corresponds either to the upper or the lower bubble. Nevertheless, the same diagram could also be associated to the direct part of the following product of matrix elements: $\bra{f} V^{NN} \ket{i}_{D+E} \, \bra{i} V^{\Lambda N \to NN} \ket{0}_{D+E}$, where $\ket{i}$ is a $2p1h$ intermediate configuration. But, since this product contains a FSI, it is not a correct contribution to the decay rates of Eq. (\[decw\]). Antisymmetry of this product of matrix elements gives rise to a total of eight self–energy diagrams, which are shown in Fig. 4 in Ref. [@ba07b] and used there to calculate the (observable) spectra of the non–mesonic weak decay nucleons. From the analytical point of view, the product $\bra{f} V^{NN} \ket{i}_{D+E} \, \bra{i} V^{\Lambda N \to NN} \ket{0}_{D+E}$ is clearly different from the term at first order in $V^{NN}$ entering $\bra{f} \widetilde{V}^{\Lambda N \to NN} \ket{0}_{D+E}$. When the comparison is done using the full set of direct plus exchange diagrams, FSI and the medium modifications on the weak transition potential are manifestly different effects. Only the latter can be included in the calculation of the decay rates.
![Goldstone diagrams for FSI $(a)$ and $2p2h$ GSC contributions $(b)$ and $(c)$ leading to two nucleon emission. []{data-label="figfsiab"}](figfsin)
As a final remark for this section, we observe that the amplitudes $(a)$ and $(b1)$ of Fig. \[gam1gsc\] produce the self–energy diagrams $(b)$ and $(c)$ of Fig. \[figfsiab\]. They are GSC terms and thus contribute to the decay rates. Conversely, the $(a)$ diagram of Fig. \[figfsiab\] must be left aside in the calculation, unless one considers it as a medium modification on the weak transition potential (but then, other medium modification contributions should be considered simultaneously), as done in Ref. [@Ji01]. The Goldstone diagrams of Fig. \[figfsiab\] are the three possible time orderings of the same Feynman diagram. Again, we stress that the fact that one out of three diagrams in Fig. \[figfsiab\] will not be included in our calculation of the decay rates makes sense since these rates are not observables and thus do not have to be described by Feynman diagrams.
Formal derivation of the decay rates $\bf{\Gamma_{n}}$ and $\bf{\Gamma_{p}}$ including GSC {#gngpgsc}
==========================================================================================
In Fig. \[gam1gsc\] we have shown a set of amplitudes which contribute to the decay rate $\Gamma_{1}$ of Eq. (\[decw\]). Only the amplitude $(a)$ has been evaluated explicitly up to now. In the present work we extend the microscopic approach to include the amplitude $(b1)$, which originates from GSC contributions that we expect to be important.
Before proceeding with the derivation of decay widths, it is convenient to give the expressions for the potentials. The weak transition potential $V^{\Lambda N \to NN}$ and the nuclear residual interaction $V^{NN}$ read: $$\label{intlnnn}
V^{\Lambda N\to NN (NN)} (q) = \sum_{\tau_{\Lambda (N)}=0,1}
{\cal O}_{\tau_{\Lambda (N)}}
{\cal V}_{\tau_{\Lambda (N)}}^{\Lambda N \to NN (NN)} (q)~,$$ where the isospin dependence is given by $$\begin{aligned}
\label{isos} {\cal O}_{\tau_{\Lambda (N)}} =~~~~~
\left\{
\begin{array}{c}1~~~~~\mbox{for}~~\tau_{\Lambda (N)}=0\\
\v{\tau}_1 \cdot \v{\tau}_2~~\mbox{for}~~\tau_{\Lambda (N)}=1~.
\end{array}\right.\end{aligned}$$ The values $0$ and $1$ for $\tau_{\Lambda (N)}$ refer to the isoscalar and isovector parts of the interactions, respectively. The spin and momentum dependence of the weak transition potential is given by: $$\begin{aligned}
\label{intln}
{\cal V}_{\tau_{\Lambda}}^{\Lambda N \to NN} (q) &
= & (G_F m_{\pi}^2) \; \{
S_{\tau_{\Lambda}}(q) \; \v{\sigma}_1 \cdot \v{\hat{q}} +
S'_{\tau_{\Lambda}}(q) \; \v{\sigma}_2 \cdot \v{\hat{q}} +
P_{C, \tau_{\Lambda}}(q) \\
& &
+ P_{L, \tau_{\Lambda}}(q)
\v{\sigma}_1 \cdot \v{\hat{q}} \; \v{\sigma}_2 \cdot
\v{\hat{q}} +
P_{T, \tau_{\Lambda}}(q)
(\v{\sigma}_1 \times \v{\hat{q}}) \cdot (\v{\sigma}_2 \times
\v{\hat{q}}) \nonumber \\
& &
+i S_{V, \tau_{\Lambda}}(q)
\v{(\sigma}_1 \times \v{\sigma}_2) \cdot
\v{\hat{q}} \}~, \nonumber\end{aligned}$$ where the functions $S_{\tau_{\Lambda}}(q)$, $S'_{\tau_{\Lambda}}(q)$, $P_{C, \tau_{\Lambda}}(q)$, $P_{L,\tau_{\Lambda}}(q)$, $P_{T, \tau_{\Lambda}}(q)$ and $S_{V, \tau_{\Lambda}}(q)$, which include short range correlations, are adjusted to reproduce any weak transition potential.
The corresponding expression for the nuclear residual interaction is given by: $$\begin{aligned}
\label{intnn}
{\cal V}_{\tau_N}^{N N} (q)
& = & \frac{f_{\pi}^2}{m_{\pi}^2} \; \{
{\cal V}_{C, \,\tau_{N}}(q) +
{\cal V}_{L, \, \tau_{N}}(q)
\v{\sigma}_1 \cdot \v{\hat{q}} \; \v{\sigma}_2 \cdot
\v{\hat{q}} \\
& & + {\cal V}_{T, \, \tau_{N}}(q)
(\v{\sigma}_1 \times \v{\hat{q}}) \cdot (\v{\sigma}_2 \times
\v{\hat{q}}) \}~, \nonumber\end{aligned}$$ where the functions ${\cal V}_{C, \,\tau_{N}}(q)$, ${\cal V}_{L, \, \tau_{N}}(q)$ and ${\cal V}_{T, \, \tau_{N}}(q)$ are also adjusted to reproduce any nuclear residual interaction.
In particular, $V^{\Lambda N \to NN}$ is represented by the exchange of the $\pi$, $\eta$, $K$, $\rho$, $\omega$ and $K^*$ mesons, within the formulation of Ref. [@pa97], with strong coupling constants and cut–off parameters deduced from the Nijmegen soft–core interaction NSC97f of Ref. [@st99]. For $V^{N N}$ we have used a Bonn potential [@ma87] in the framework of the parametrization presented in Ref. [@br96], which contains the exchange of $\pi$, $\rho$, $\sigma$ and $\omega$ mesons.
We give now explicit expressions for the partial decay width $\Gamma_{1}(\v{k},k_{F})$ of Eq. (\[decw1\]), which for convenience is expressed in terms of its isospin components $\Gamma_{n}(\v{k},k_{F})$ and $\Gamma_{p}(\v{k},k_{F})$. Let us first rewrite Eq. (\[decw1\]) as follows: $$\label{0gsc}
\Gamma_{n \, (p)}(\v{k},k_{F})=
\Gamma^{0}_{n \, (p)}(\v{k},k_{F})+\Gamma^{0-\rm GSC}_{n \, (p)}(\v{k},k_{F})
+\Gamma^{\rm GSC}_{n \, (p)}(\v{k},k_{F})~,$$ where: $$\begin{aligned}
\label{decw10}
\Gamma^{0}_{n \, (p)}(\v{k},k_{F}) & = & \mathcal{N}^{\, 2}(k_{F})
\sum_{f} \, \delta (E_{f}-E_{0}) \;
\left|\bra{f} V^{\Lambda N\to NN} \ket{p_{\Lambda}}_{D+E}\right|^{2}~, \\
\label{decw10gsc}
\Gamma^{0-\rm GSC}_{n \, (p)}(\v{k},k_{F}) & = & - 2 \mathcal{N}^{\, 2}(k_{F})
\sum_{f} \sum_{p, h, p', h'} \, \delta (E_{f}-E_{0}) \;
\bra{p_{\Lambda}} (V^{\Lambda N\to NN})^{\dagger} \ket{f}_{D+E} \nonumber \\
&&
\times \bra{f} V^{\Lambda N\to NN} \ket{p h p' h'; \, p_{\Lambda}}_{D+E}
\frac{\bra{p h p' h'; \, p_{\Lambda}} V^{N N} \ket{p_{\Lambda}}_{D+E}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}~,\nonumber\\
\label{decw1gsc}
\Gamma^{\rm GSC}_{n \, (p)}(\v{k},k_{F}) & = & \mathcal{N}^{\, 2}(k_{F})
\sum_{f} \sum_{p, h, p', h'} \, \delta (E_{f}-E_{0}) \;
\left|\bra{f} V^{\Lambda N\to NN} \ket{p h p' h'; \, p_{\Lambda}}_{D+E}
\phantom{\frac{A^A}{A^A}} \right. \nonumber \\
&& \left. \times \frac{\bra{p h p' h'; \, p_{\Lambda}} V^{N N} \ket{p_{\Lambda}}_{D+E}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}\right|^{2}~. \nonumber\end{aligned}$$ The first component, $\Gamma^{0}_{n \, (p)}$, is the contribution from the uncorrelated hypernuclear ground state, the third one, $\Gamma^{\rm GSC}_{n \, (p)}$, result from ground state correlations, while $\Gamma^{0-\rm GSC}_{n \, (p)}$ is the interference term between correlated and uncorrelated ground states.
It is now convenient to consider the following decomposition, dictated by the isospin quantum number: $$\begin{aligned}
\label{rpa2}
\Gamma^{0}_{n \, (p)}(\v{k},k_{F})& =
&\sum_{P,Q=D, \, E} \, \Gamma^{PQ}_{n\, (p)}(\v{k},k_{F}) \\
&=&\sum_{P,Q=D, \, E} \, \sum_{\tau_{\Lambda'}, \tau_{\Lambda}=0,1}
{\cal T}^{PQ}_{\tau_{\Lambda'} \tau_{\Lambda}, \; n \, (p)} \;
\Gamma^{PQ}_{\tau_{\Lambda'} \tau_{\Lambda}}(\v{k},k_{F})~, \nonumber \\
\Gamma^{0-\rm GSC}_{n \, (p)}(\v{k},k_{F})& =
&\sum_{P,Q,Q'=D, \, E} \, \Gamma^{PQQ'}_{n\, (p)}(\v{k},k_{F}) \nonumber \\
&=&\sum_{P,Q,Q'=D, \, E} \,\sum_{\tau_{\Lambda'}, \tau_{\Lambda}, \tau_{N}=0,1}
{\cal T}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}, \; n \, (p)} \;
\Gamma^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_{F})~, \nonumber \\
\Gamma^{\rm GSC}_{n \, (p)}(\v{k},k_{F})& =
&\sum_{P',P,Q,Q'=D, \, E} \, \Gamma^{P'PQQ'}_{n\, (p)}(\v{k},k_{F}) \nonumber \\
&=&\sum_{P',P,Q,Q'=D, \, E} \,\sum_{\tau_{N'}, \tau_{\Lambda'}, \tau_{\Lambda}, \tau_{N}=0,1}
{\cal T}^{P'PQQ'}_{\tau_{N'} \tau_{\Lambda'} \tau_{\Lambda} \tau_{N}, \; n \, (p)} \;
\Gamma^{P'PQQ'}_{\tau_{N'} \tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_{F})~, \nonumber\end{aligned}$$ where $P', P, \, Q, \, Q'=$ $D$ or $E$ refer to the direct or exchange character of the matrix elements of Eq. (\[decw10\]). The isospin factors are given by: $$\begin{aligned}
\label{rpa3}
{\cal T}^{PQ}_{\tau_{\Lambda'} \tau_{\Lambda}, \; n \, (p)} & = &
\sum_{f,\, \rm isospin} \, \bra{t_{\Lambda}} {\cal O}_{\tau_{\Lambda'}} \ket{f}_{P}
\bra{f} {\cal O}_{\tau_{\Lambda}} \ket{t_{\Lambda}}_{Q}~, \nonumber \\
{\cal T}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}, \; n \, (p)} & = &
\sum_{f,\, \rm isospin} \, \bra{t_{\Lambda}} {\cal O}_{\tau_{\Lambda'}} \ket{f}_{P}
\bra{f} {\cal O}_{\tau_{\Lambda}} \ket{t_{p} t_{h} t_{p'} t_{h'}, t_{\Lambda}}_{Q} \nonumber \\
&&\times \bra{t_{p} t_{h} t_{p'} t_{h'}, t_{\Lambda}}
{\cal O}_{\tau_{N}} \ket{t_{\Lambda}}_{Q'}~, \nonumber \\
{\cal T}^{P'PQQ'}_{\tau_{N'} \tau_{\Lambda'} \tau_{\Lambda} \tau_{N}, \; n \, (p)} & = &
\sum_{f,\, \rm isospin} \, \bra{t_{\Lambda}} {\cal O}_{\tau_{N'}} \ket{t_{\tilde{p}}
t_{\tilde{h}} t_{\tilde{p}'} t_{\tilde{p}'}, t_{\Lambda}}_{P'}
\bra{t_{\tilde{p}} t_{\tilde{h}} t_{\tilde{p}'}
t_{\tilde{p}'}, t_{\Lambda}} {\cal O}_{\tau_{\Lambda'}} \ket{f}_{P} \nonumber \\
&& \times \bra{f} {\cal O}_{\tau_{\Lambda}} \ket{t_{p} t_{h} t_{p'} t_{h'}, t_{\Lambda}}_{Q}
\bra{t_{p} t_{h} t_{p'} t_{h'}, t_{\Lambda}} {\cal O}_{\tau_{N}} \ket{t_{\Lambda}}_{Q'}~, \nonumber\end{aligned}$$ where the summations run over all the isospin projections $t's$, with the constrain that the emitted particles are $nn$ for $\Gamma_{n}$ and $np$ for $\Gamma_{p}$. For the partial decay widths we instead find: $$\begin{aligned}
\Gamma^{PQ}_{\tau_{\Lambda'} \tau_{\Lambda}}(\v{k},k_{F})
& = & \mathcal{N}^{\, 2}(k_{F}) \, (-1)^{n} \,
\sum_{f} \, \delta (E_{f}-E_{0}) \\
\label{gampq}
&& \times \bra{p_{\Lambda}} ({\cal V}_{\tau_{\Lambda'}}^{\Lambda N \to NN} (q'))^{\dag} \ket{f}_{P}
\bra{f} {\cal V}_{\tau_{\Lambda}}^{\Lambda N \to NN} (q) \ket{p_{\Lambda}}_{Q}~, \nonumber \\
%%
%%
\label{gampqq}
\Gamma^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_{F})
& = & - 2 \, \mathcal{N}^{\, 2}(k_{F}) \, (-1)^{n} \,
\sum_{f} \, \sum_{p, h, p', h'} \, \delta (E_{f}-E_{0}) \\
&& \times \bra{p_{\Lambda}} ({\cal V}_{\tau_{\Lambda'}}^{\Lambda N \to NN} (q'))^{\dag} \ket{f}_{P}
\bra{f} {\cal V}_{\tau_{\Lambda}}^{\Lambda N \to NN} (q)
\ket{p h p' h'; \, p_{\Lambda}}_{Q} \nonumber \\
&&\times \frac{\bra{p h p' h'; \, p_{\Lambda}} {\cal V}_{\tau_N}^{N N} (t) \ket{p_{\Lambda}}_{Q'}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}~, \nonumber \\
%%
%%
\label{gamppqq}
\Gamma^{P'PQQ'}_{\tau_{N'} \tau_{\Lambda'} \tau_{\Lambda} \tau_{N} }(\v{k},k_{F})
& = & \mathcal{N}^{\, 2}(k_{F}) \, (-1)^{n} \,
\sum_{f} \, \sum_{\tilde{p}, \tilde{h}, \tilde{p}', \tilde{h}'} \,
\sum_{p, h, p', h'} \, \delta (E_{f}-E_{0}) \\
&&
\times \frac{\bra{p_{\Lambda}} ({\cal V}_{\tau_{N'}}^{N N}(t'))^{\dag}
\ket{\tilde{p}, \tilde{h}, \tilde{p}', \tilde{h}'; \, p_{\Lambda}}_{P'}}
{\varepsilon_{\tilde{p}}-\varepsilon_{\tilde{h}}+
\varepsilon_{\tilde{p}'}-\varepsilon_{\tilde{h}'}} \nonumber \\
&& \times
\bra{\tilde{p}, \tilde{h}, \tilde{p}', \tilde{h}';
\, p_{\Lambda}} ({\cal V}_{\tau_{\Lambda'}}^{\Lambda N \to NN}(q'))^{\dag}
\ket{f}_{P} \nonumber \\
&& \times \bra{f} {\cal V}_{\tau_{\Lambda}}^{\Lambda N \to NN} (q)
\ket{p h p' h'; \, p_{\Lambda}}_{Q} \nonumber\\
&&\times \frac{\bra{p h p' h'; \, p_{\Lambda}} {\cal V}_{\tau_N}^{N N} (t) \ket{p_{\Lambda}}_{Q'}}
{\varepsilon_{p}-\varepsilon_{h}+\varepsilon_{p'}-\varepsilon_{h'}}~. \nonumber\end{aligned}$$ Note that the values of the energy–momentum carried by the particles and holes lines depends on the topology of the corresponding diagram, while $n$ is the number of crossing between fermionic lines.
Let us now apply the above formalism to a model including the amplitudes $(a)$ and $(b1)$ of Fig. \[gam1gsc\]. Four direct self–energy diagrams correspond to the square of the amplitude sum $(a)+(b1)$; they are given in Fig. \[dirgsc\]. Note that these diagrams admits a single cut, giving rise to a $2p1h$ final state.
![Direct Goldstone diagrams corresponding to the square of the amplitude sum $(a)+(b1)$ of Fig. \[gam1gsc\]. See the decomposition of Eq. (\[rpa2\]).[]{data-label="dirgsc"}](fig4)
The $DD$ diagram contributes to the partial widths $\Gamma^0_{n\, (p)}$ of Eq. (\[rpa2\]). The two $DDD$ diagrams, which have the same numerical value and are interferences between the amplitudes $(a)$ and $(b1)$ of Fig. \[gam1gsc\], are included in the partial widths $\Gamma^{0-\rm GSC}_{n\, (p)}$. Finally, the diagram $DDDD$ contributes to $\Gamma^{\rm GSC}_{n\, (p)}$. Many exchange diagrams are obtained from the antisymmetrized amplitude sum $(a)+(b1)$: one $PQ$ exchange diagram is the partner of the $DD$ one of Fig. \[dirgsc\]; seven $PQQ'$ exchange diagrams are companions of each one of the $DDD$ ones; fifteen $P'PQQ'$ exchange diagrams add to the $DDDD$ one.
Formal expressions for $\Gamma^{0}_{n \, (p)}$ can be found in Ref. [@ba03]. The $\Gamma^{PQQ'}_{n \, (p)}$’s contributing to $\Gamma^{0-\rm GSC}_{n \, (p)}$ (see Eq. (\[rpa2\])) correspond to the diagrams of Fig. \[antgsc\].
![Goldstone diagrams for the partial rates $\Gamma^{PQQ'}_{n \, (p)}$ contributing to Eq. (\[rpa2\]).[]{data-label="antgsc"}](fig5)
By replacing, in Eq. (\[gampqq\]), the sum over momenta by integrals and by performing the energy integrations and the spin summation, the following expression for $\Gamma^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}$ can be obtained: $$\begin{aligned}
\label{pqqgsc}
\Gamma^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_F) & =
& \mathcal{N}^{\, 2}(k_{F})
\frac{1}{4} \frac{(-1)^{n}}{(2 \pi)^8} (G_F m_{\pi}^2)^2
\frac{f_{\pi}^2}{m_{\pi}^2} \\
&&\times \int \int \int \, d \v{q} \, d \v{h} \, d \v{h}' \;
{\cal W}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t) \nonumber \\
&&\times \Theta(k,q,q',t,h,h',k_{F}) \frac{1}{- \varepsilon^{PQQ'}_{2p2h}} \;
\delta(q_0 - (\varepsilon_{\v{h}'+\v{q}}-\varepsilon_{\v{h}'}))~, \nonumber\end{aligned}$$ where $q_0=k_0 - \varepsilon_{\v{k}-\v{q}} - V_N$, $V_N$ being the nucleon binding energy, while the functions ${\cal W}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ and $\Theta(k,q,q',t,h,h',k_{F})$ and the energy denominator $\varepsilon^{PQQ'}_{2p2h}$ are specific of each $PQQ'$ contribution. The function ${\cal W}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ contains the momentum dependence of the nuclear residual interaction and the weak transition potentials and the spin summation, while $\Theta(k,q,q',t,h,h',k_{F})$ is a product of step functions which defines the phase space of particles and holes.
In the present section we present the explicit expression for the direct term $\Gamma^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}$; the other seven ones are displayed in Appendix A. We obtain: $$\begin{aligned}
\label{dddgsc}
\Gamma^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_F) & =
& \mathcal{N}^{\, 2}(k_{F})
\frac{1}{4} \frac{1}{(2 \pi)^8} (G_F m_{\pi}^2)^2
\frac{f_{\pi}^2}{m_{\pi}^2} \\
&&\times \int \int \int \, d \v{q} \, d \v{h} \, d \v{h}' \;
{\cal W}^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q) \;
\nonumber \\
& &
\times \theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}-\v{h}|-k_{F}|)
\theta(k_{F}-|\v{h}|)
\nonumber \\
& & \times \theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|) \nonumber \\
& &
\times \frac{1}{-q_{0}-(\varepsilon_{\v{h}-\v{q}}-\varepsilon_{\v{h}})} \;
\delta(q_0 - (\varepsilon_{\v{h}'+\v{q}}-\varepsilon_{\v{h}'}))~. \nonumber\end{aligned}$$ The expressions for $\Theta(k,q,q',t,h,h',k_{F})$ and $\varepsilon^{DDD}_{2p2h}$ are self–evident. Moreover: $$\begin{aligned}
\label{tdir3}
{\cal W}^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q) & = &
8 \; \{ [S'_{\tau_{\Lambda'}}(q) S'_{\tau_{\Lambda}}(q)
+ P_{C, \tau_{\Lambda'}}(q) P_{C, \tau_{\Lambda}}(q)] {\cal V}_{C, \,\tau_{N}}(q) \\
&&+ [S_{\tau_{\Lambda'}}(q) S_{\tau_{\Lambda}}(q)+P_{L, \tau_{\Lambda'}}(q)
P_{L, \tau_{\Lambda}}(q)]
{\cal V}_{L, \, \tau_{N}}(q) \nonumber \\
&& + 2 \, [S_{V, \tau_{\Lambda'}}(q) S_{V, \tau_{\Lambda}}(q) +
P_{T, \tau_{\Lambda'}}(q) P_{T, \tau_{\Lambda}}(q)]
{\cal V}_{T, \, \tau_{N}}(q) \}~.\end{aligned}$$ Eq. (\[dddgsc\]) can be simplified by introducing the functions: $$\label{lind}
{\cal I}(q_{0},\v{q}) =
\frac{-\pi}{(2 \pi)^{3}} \int \, d \v{h}'
\theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)
\delta(q_0 - \varepsilon_{\v{h}'+\v{q}}+\varepsilon_{\v{h}'})~, \nonumber$$ $$\label{real}
{\cal R}(q_{0},\v{q}) =
\frac{1}{(2 \pi)^{3}} \, {\cal P} \, \int \, d \v{h}
\frac{\theta(|\v{q}-\v{h}|-k_{F}|) \theta(k_{F}-|\v{h}|)}
{q_0 - (\varepsilon_{\v{h}-\v{q}}-\varepsilon_{\v{h}})}~,$$ where ${\cal I}(q_{0},\v{q})$ is the imaginary part of the Lindhard function and the explicit expression for ${\cal R}(q_{0},\v{q})$ is given in Appendix B. Therefore: $$\begin{aligned}
\label{dddgsc2}
\Gamma^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_F) & =
& - \, \frac{\mathcal{N}^{\, 2}(k_{F})}{(2 \pi)^3} (G_F m_{\pi}^2)^2
\frac{f_{\pi}^2}{m_{\pi}^2} \, \int \,
d \v{q} \, \theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \\
&& \times \{ [S'_{\tau_{\Lambda'}}(q) S'_{\tau_{\Lambda}}(q) +
P_{C, \tau_{\Lambda'}}(q) P_{C, \tau_{\Lambda}}(q)]
{\cal V}_{C, \,\tau_{N}}(q) \nonumber \\
&&+ [S_{\tau_{\Lambda'}}(q)
S_{\tau_{\Lambda}}(q)+P_{L, \tau_{\Lambda'}}(q) P_{L, \tau_{\Lambda}}(q)]
{\cal V}_{L, \, \tau_{N}}(q) \nonumber \\
&&+ 2 \, [S_{V, \tau_{\Lambda'}}(q) S_{V, \tau_{\Lambda}}(q) +
P_{T, \tau_{\Lambda'}}(q) P_{T, \tau_{\Lambda}}(q)] {\cal V}_{T, \, \tau_{N}}(q)\}
\nonumber \\
&&\times{\cal R}(-q_{0},\v{q}) {\cal I}(q_{0},\v{q})~. \nonumber\end{aligned}$$
Then one has to perform the isospin summation to obtain $$\Gamma^{DDD}_{n\, (p)}(\v{k},k_{F})
=\sum_{\tau_{\Lambda'}, \tau_{\Lambda}, \tau_{N}=0,1}
{\cal T}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}, \; n \, (p)} \;
\Gamma^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_{F})~.$$ The final result obtained after the local density approximation is therefore: $$\begin{aligned}
\label{dddisop}
\Gamma^{DDD}_{n} & =
& 2 \{\Gamma^{DDD}_{111} + \Gamma^{DDD}_{000} + \Gamma^{DDD}_{010} + \Gamma^{DDD}_{101} \}~, \\
\Gamma^{DDD}_{p} & =
& 2 \{5 \, \Gamma^{DDD}_{111} + \Gamma^{DDD}_{000}
- \Gamma^{DDD}_{010} - \Gamma^{DDD}_{101} \}~. \nonumber\end{aligned}$$ Finally, we present the partial rates corresponding to the diagram $DDDD$ of Fig. \[dirgsc\]. By applying the same procedure used for $\Gamma^{DDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N} }$ to Eq. (\[gamppqq\]) we obtain: $$\begin{aligned}
\label{ddddgsc2}
\Gamma^{DDDD}_{\tau_{N'} \tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_F) & =
& - \, \frac{\mathcal{N}^{\, 2}(k_{F})}{(2 \pi)^2} (G_F m_{\pi}^2)^2
\left(\frac{f_{\pi}^2}{m_{\pi}^2}\right)^{2} \, \int \,
d \v{q} \, \theta(q_0) \\
&&\times \theta(|\v{k}-\v{q}|-k_{F})
\{ (S'_{\tau_{\Lambda'}} S'_{\tau_{\Lambda}} +
P_{C, \tau_{\Lambda'}} P_{C, \tau_{\Lambda}})
{\cal V}^{2}_{C, \,\tau_{N}} \nonumber \\
&&+ (S_{\tau_{\Lambda'}}
S_{\tau_{\Lambda}}+P_{L, \tau_{\Lambda'}} P_{L, \tau_{\Lambda}})
{\cal V}^{2}_{L, \, \tau_{N}} \nonumber \\
& &+ 2 \, ( S_{V, \tau_{\Lambda'}} S_{V, \tau_{\Lambda}} +
P_{T, \tau_{\Lambda'}} P_{T, \tau_{\Lambda}}) {\cal V}^{2}_{T, \, \tau_{N}} \} \nonumber \\
&&\times {\cal R}^{2}(-q_{0},\v{q}) {\cal I}(q_{0},\v{q})~, \nonumber\end{aligned}$$ and $$\begin{aligned}
\label{ddddisop}
\Gamma^{DDDD}_{n} & =
& 4 \{\Gamma^{DDD}_{1111} + \Gamma^{DDD}_{0000} +
\Gamma^{DDD}_{0101} + \Gamma^{DDD}_{1010} \}~, \\
\Gamma^{DDDD}_{p} & =
& 4 \{5 \, \Gamma^{DDD}_{1111} + \Gamma^{DDD}_{0000}
- \Gamma^{DDD}_{0101} - \Gamma^{DDD}_{1010} \}~, \nonumber\end{aligned}$$ after performing the local density approximation.
In this paper the $\Gamma^{P'PQQ'}_{n\, (p)}$ exchange terms will be neglected. Indeed, from our numerical results discussed in the next Section it turns out that already the direct contribution $\Gamma^{DDDD}_{n\, (p)}$ is small and approximately one order of magnitude smaller than $\Gamma^{DDD}_{n\, (p)}$. Moreover, according to the results obtained for the $\Gamma^{PQQ'}_{n\, (p)}$’s, $P'PQQ'$ exchange contributions are expected to be even smaller than the direct term $DDDD$.
Results
=======
In the previous Section we have seen how the neutron– and proton–induced decay widths can be written in the form: $$\begin{aligned}
\label{gamma_np}
\Gamma_{n\, (p)}&=&\Gamma^{0}_{n\, (p)}+\Gamma^{0-\rm GSC}_{n\, (p)}+
\Gamma^{\rm GSC}_{n\, (p)} \\
&\equiv & \sum_{P,Q=D,E}\Gamma^{PQ}_{n\, (p)}
+\sum_{P,Q,Q'=D,E}\Gamma^{PQQ'}_{n\, (p)}
+\sum_{P',P,Q,Q'=D,E}\Gamma^{P'PQQ'}_{n\, (p)} \nonumber~,\end{aligned}$$ $\Gamma^{0}_{n\, (p)}$ being the rates obtained for an uncorrelated hypernuclear ground state, $\Gamma^{\rm GSC}_{n\, (p)}$ the rates originated by ground state correlations and $\Gamma^{0-\rm GSC}_{n\, (p)}$ the rates resulting from the interference between uncorrelated and correlated ground states.
For the present scheme containing the transition amplitudes $(a)$ and $(b1)$ of Fig. \[gam1gsc\], where antisymmetrization is considered for the weak transition potential $V^{\Lambda N\to NN}$ and the nuclear residual interaction $V^{NN}$, we obtained: two contributions to $\Gamma^{0}_{n\, (p)}$, which are $\Gamma^{DD}_{n\, (p)}=\Gamma^{EE}_{n\, (p)}$ and $\Gamma^{DE}_{n\, (p)}=\Gamma^{ED}_{n\, (p)}$ and are generated by the square of amplitude $(a)$; eight different $\Gamma^{PQQ'}_{n\, (p)}$ contributions to $\Gamma^{0-\rm GSC}_{n\, (p)}$, which are interferences between the $(a)$ and $(b1)$ amplitudes; sixteen different $\Gamma^{P'PQQ'}_{n\, (p)}$ contributions to $\Gamma^{\rm GSC}_{n\, (p)}$, which originate from the square of amplitude $(b1)$. An early evaluation of $\Gamma^0_{n\, (p)}$ has been performed in Ref. [@ba03], while $\Gamma^{0-\rm GSC}_{n\, (p)}$ and $\Gamma^{\rm GSC}_{n\, (p)}$ are discussed here for the first time. Among the $\Gamma^{P'PQQ'}_{n\, (p)}$’s, here we only calculate the direct terms $\Gamma^{DDDD}_{n\, (p)}$.
$^{12}_\Lambda$C
----------------
We start by discussing the relevance of the Pauli exchange terms in $\Gamma^{0}_{n\, (p)}$ and $\Gamma^{0-\rm GSC}_{n\, (p)}$. Our results for $\Gamma^{PQ}_{n}$ and $\Gamma^{PQ}_{p}$ are given in Table \[gamm0\] for the decay of $^{12}_\Lambda$C. Note that, for symmetry, $\Gamma^{0}_{n\, (p)}$ are twice the sum of $\Gamma^{DD}_{n\, (p)}$ and $\Gamma^{DE}_{n\, (p)}$. Exchange terms contribute to the uncorrelated rates for neutron–induced (proton–induced) decays by 5.1% (0.3%). Thus, they tend to increase $\Gamma_n/\Gamma_p$ while having a very small effect on $\Gamma_1$.
Channel $~~2\,\Gamma^{DD}~~$ $~~2\,\Gamma^{DE}~~$ $~~\Gamma^{0}~~$
------------------- ---------------------- ---------------------- ------------------
$\Lambda n\to nn$ $0.146$ $0.008$ $0.154$
$\Lambda p\to np$ $0.469$ $0.002$ $0.470$
sum $0.615$ $0.009$ $0.624$
: Direct and exchange $\Gamma^{PQ}_n$ and $\Gamma^{PQ}_p$ terms for $^{12}_\Lambda$C in units of the free $\Lambda$ decay rate, $\Gamma^0= 2.52 \cdot 10^{-6}$ eV. The first column indicates the two different isospin channels and their sum. Note that $\Gamma_{n\, (p)}^{DD}=\Gamma_{n\, (p)}^{EE}$ and $\Gamma_{n\, (p)}^{DE}=\Gamma_{n\, (p)}^{ED}$.[]{data-label="gamm0"}
In Table \[gamm0gsc\] we present predictions for the $\Gamma^{PQQ'}_{n}$ and $\Gamma^{PQQ'}_{p}$ contributions derived from the Goldstone diagrams of Fig. \[antgsc\], again for $^{12}_\Lambda$C. As expected, the direct terms $\Gamma^{DDD}_{n}$ and $\Gamma^{DDD}_{p}$ are the main contributions. Nevertheless, the effect of antisymmetry on the two isospin channels is significant: it increases $\Gamma^{0-\rm GSC}_{n}$ by 34% while decreasing $\Gamma^{0-\rm GSC}_{p}$ by 8%. The overall effect on $\Gamma^{0-\rm GSC}_1=
\Gamma^{0-\rm GSC}_{n}+\Gamma^{0-\rm GSC}_{p}$ is a very small increase, of 2%. We note that, with topologically equivalent diagrams, in Ref. [@ba07b] a similar quasi–cancellation between neutron– and proton–induced decays has been found in nucleon spectra calculations. Moreover, in Ref. [@ba09b] it has been shown that the evaluation of the GSC exchange terms is important for the rate $\Gamma_2$ as well. We emphasize that the exact evaluation of exchange diagrams has been mostly ignored in the literature. It is usually a quite involved (but necessary) task, given the rapidly increasing number of terms one has to consider when going to higher orders in the nuclear residual interaction. Unfortunately, there is no general rule to anticipate the need for the evaluation of exchange terms when the corresponding direct contribution is important.
Channel $~~\Gamma^{DDD}~~$ $~~\Gamma^{DDE}~~$ $~~\Gamma^{DED}~~$ $~~\Gamma^{EDD}~~$
------------------- -------------------- -------------------- -------------------- -------------------- ----------------------
$\Lambda n\to nn$ $0.022$ $-0.002$ $-0.009$ $-0.004$
$\Lambda p\to np$ $0.071$ $0.005$ $-0.027$ $-0.011$
sum $0.093$ $0.003$ $-0.036$ $-0.015$
Channel $\Gamma^{DEE}$ $\Gamma^{EDE}$ $\Gamma^{EED}$ $\Gamma^{EEE}$ $\Gamma^{0-\rm GSC}$
$\Lambda n\to nn$ $0.006$ $0.008$ $0.006$ $0.002$ $0.029$
$\Lambda p\to np$ $-0.008$ $0.009$ $0.025$ $0.002$ $0.066$
sum $-0.003$ $0.017$ $0.031$ $0.004$ $0.095$
: Direct and exchange $\Gamma^{PQQ'}_{n}$ and $\Gamma^{PQQ'}_{p}$ terms for $^{12}_{\Lambda}$C obtained from the diagrams of Fig. \[antgsc\]. The first column indicates the two different isospin channels and their sum.[]{data-label="gamm0gsc"}
In Table \[gamm1gsc\] we present the different contributions to the rates ${\Gamma}_{n}$ and ${\Gamma}_{p}$ of Eq. (\[gamma\_np\]). The uncorrelated parts $\Gamma^{0}_{n}$ and $\Gamma^{0}_{p}$ dominate over the remaining ones: $\Gamma^{0}_{1}=\Gamma^{0}_{n}+\Gamma^{0}_{p}$ constitutes the 86% of the total ${\Gamma}_{1}$. Then, $\Gamma^{0-\rm GSC}_{1}=\Gamma^{0-\rm GSC}_{n}+\Gamma^{0-\rm GSC}_{p}$ and $\Gamma^{\rm GSC}_{1}=\Gamma^{\rm GSC}_{n}+\Gamma^{\rm GSC}_{p}$ represent 13% and 1% of $\Gamma_1$, respectively. We remind the reader that $\Gamma^{\rm GSC}_{n\, (p)}$ are calculated from the direct diagram $DDDD$ in Fig. \[dirgsc\], while $P'PQQ'$ exchange terms are neglected. This omission is justified by the smallness of the direct contributions $\Gamma^{DDDD}_{n\, (p)}$: the neglected exchange part of $\Gamma^{\rm GSC}_{1}$ should contribute to $\Gamma_1$ by less than 1%. Thus, a challenging calculation of the fifteen $P'PQQ'$ exchange diagrams can be reasonably avoided.
Channel $~~\Gamma^{0}~~$ $~~\Gamma^{0-\rm GSC}~~$ $~~\Gamma^{\rm GSC}~~$ $~~\Gamma~~$
------------------- ------------------ -------------------------- ------------------------ --------------
$\Lambda n\to nn$ $0.154$ $0.029$ $0.002$ $0.185$
$\Lambda p\to np$ $0.470$ $0.066$ $0.008$ $0.544$
sum $0.624$ $0.095$ $0.010$ $0.729$
: Predictions for the one–nucleon induced decay rates of Eq. (\[gamma\_np\]) for $^{12}_\Lambda$C. The first column indicates the two different isospin channels and their sum.[]{data-label="gamm1gsc"}
Our predictions for the one– and two–nucleon induced decay rates for $^{12}_\Lambda$C are given in Table \[gamm12\] and compared with the most recent data by KEK [@Ki09] and FINUDA [@Ag09]. For completeness, we report results without and with the inclusion of antisymmetrization and GSC. It should be noted that the hypernuclear ground state normalization function $\mathcal{N}(k_{F})$ of Eq. (\[norconst\]) equally affects ${\Gamma}_{1}$ and ${\Gamma}_{2}$. This function is not identically equal to one only when GSC are present. Therefore, the $\Gamma_{1}$ result without GSC and with exchange terms of Table \[gamm12\], 0.74, is bigger than the prediction for $\Gamma^{0}_1$ of Table \[gamm1gsc\], 0.62, which has been obtained instead by including both GSC and antisymmetrization in the normalization function. This comparison gives an idea of the importance of a proper normalization of the hypernuclear ground state. GSC produces a sizable increase in the value of $\Gamma_{\rm NM}$, thanks to the opening of the two–nucleon induced channel, while $\Gamma_1$ remains practically unaffected. The effect of GSC on the ${\Gamma}_{n}/{\Gamma}_{p}$ ratio is a small increase of 4%, which is due entirely to the exchange terms in $\Gamma^{0-\rm GSC}_n$ and $\Gamma^{0-\rm GSC}_p$ (see Table \[gamm0gsc\]). Antisymmetrization on the contrary introduce an increase of $\Gamma_1$ and a reduction of $\Gamma_2$, and as a result a sizable reduction of $\Gamma_2/\Gamma_1$. We conclude that GSC are important to get agreement with data on $\Gamma_{\rm NM}$, while antisymmetrization is crucial to reproduce the data for $\Gamma_2/\Gamma_1$. Note indeed that only with the set of results including both exchange terms and GSC we can achieve an overall agreement with all data.
Despite this agreement, we have to admit that more refined and systematic theoretical studies should be performed before one can reach definite conclusions from the comparison between theory and experiment. For instance, the result obtained for ${\Gamma}_{\rm NM}$ requires a comment on the eventual inclusion of the full set of diagrams stemming from the amplitudes in Figs. \[gam1gsc\] and \[gam2gsc\] and eventually from other amplitudes. At first glance, one may think that the final outcome from all these diagrams would be a bigger value for ${\Gamma}_{\rm NM}$, thus spoiling the good agreement with data of the present result. This is not necessarily the case, for two reasons. First, the amplitudes $(d1)$ and $(d2)$ in Fig. \[gam1gsc\] and the amplitude $(c)$ in Fig. \[gam2gsc\] originate from $1\Delta1p2h$ GSC. The inclusion of these correlation amplitudes requires the introduction of new terms in the ground state normalization function (\[norconst\]); this leads to a reduction of the individual values for each decay width, including the ones we have obtained above. From the previous studies in Refs. [@ba09; @ba09b] one observes the following property, introduced by ground state normalization: a certain redistribution of the total non–mesonic decay strength among the partial contributions occurs when new self–energy terms are included. Secondly, the presence of several additional self–energy diagrams which are interference terms between amplitudes could also bring to a reduction of the decay rates $\Gamma_1$ and $\Gamma_2$.
Medium and heavy hypernuclei
----------------------------
In order to have a further indication of the reliability of our framework, which adopts the local density approximation to obtain results for finite hypernuclei, we have extended the calculation to medium and heavy $\Lambda$ hypernuclei. All the GSC contributions and the antisymmetrization terms discussed in detail for $^{12}_\Lambda$C have been taken into account. The results we have obtained are given in Table \[m-to-h\] and are compared with recent data in Figure \[fm-to-h\].
The GSC–free rate $\Gamma_1^0$ represents 86% of the rate $\Gamma_1=\Gamma_1^0+\Gamma_1^{0-\rm GSC}+\Gamma_1^{\rm GSC}$ for $^{12}_\Lambda$C. For increasing hypernuclear mass number $A$, this contribution decreases and reaches 81% for $^{208}_\Lambda$Pb. As expected, GSC contributions are thus more important for heavy hypernuclei.
The one– and two–nucleon induced rates increase with $A$ and rapidly saturate. Saturation is expected to begin for those hypernuclei whose radius becomes sensitively larger than the range of the non–mesonic processes. The fact that for $^{40}_\Lambda$Ca and $^{208}_\Lambda$Pb we obtain very similar predictions informs us that in $^{208}_\Lambda$Pb the non–mesonic decay (both one– and two–nucleon stimulated) involve the same nucleon shells which participate in the decay of $^{40}_\Lambda$Ca. Indeed, the $\Lambda$ wave function ($s$ level of the $\Lambda$–nucleus mean potential) is well overlapped to the hypernuclear core already in $^{40}_\Lambda$Ca.
It should be noted that the slight decrease of the non–mesonic rate $\Gamma_{\rm NM}$ going from $^{89}_\Lambda$Y to $^{139}_\Lambda$La is due to the special value of the oscillator parameter $\hbar \omega$ adopted for this hypernucleus. Such a parameter, which is obtained as the difference between the measured $s$ and $p$ $\Lambda$ energy levels in $^{139}_\Lambda$La, is indeed smaller than the values measured for the two neighboring hypernuclei of our calculation, $^{89}_\Lambda$Y and $^{208}_\Lambda$Pb.
The contribution of the two–nucleon induced width is almost independent of the hypernuclear mass number and oscillates between 22 and 26% of $\Gamma_{\rm NM}$. We note from Figure \[fm-to-h\] that the datum recently determined at KEK, $\Gamma_2=0.27\pm 0.13$ [@Ki09], is well reproduced by our calculation. Also the recent determination obtained by FINUDA [@Ag09] of $\Gamma_2/\Gamma_{\rm NM}=0.22\pm 0.08$ for hypernuclei from $^5_\Lambda$He to $^{16}_\Lambda$O is in agreement with our predictions.
Concerning $\Gamma_{\rm NM}$, the agreement of our predictions with data is also rather good. The only exception is the large underestimation of the datum for the $A\simeq 200$ region, which however is also difficult to reconcile with the decay rate measured at KEK for $^{56}_\Lambda$Fe. No known mechanism can be responsible for a large increase in the non–mesonic decay rate when going from $^{56}_\Lambda$Fe to the $A\simeq 200$ region. Concerning the datum for $A\simeq 200$, we have to note that, given the difficulty in employing direct timing methods for heavy hypernuclei, it has been obtained in experiments (performed at COSY, Juelich [@Cosy]) which measured the fission fragments (which are supposed to be generated by the non–mesonic decay) emitted by hypernuclei produced in proton–nucleus reactions. Large uncertainties affect such delayed fission experiments, because of the limited precision of the employed recoil shadow method. The produced hypernuclei cannot be unambiguously identified with this method. It is also possible that mechanisms other than the non–mesonic decay contributed to hypernuclear fission in these experiments. The datum reported in Figure \[fm-to-h\] has been obtained as an average from measurements for hypernuclei produced in proton–Au, proton–Bi and proton–U reactions.
${\rm Hypernucleus}$ $~~~~~\Gamma^0_1~~~~~$ $~~~~~\Gamma_1~~~~~$ $~~~~~\Gamma_2~~~~~$ $~~~~~\Gamma_{\rm NM}~~~~~$
---------------------- ------------------------ ---------------------- ---------------------- -----------------------------
$^{11}_\Lambda$B 0.56 0.64 0.18 0.82
$^{12}_\Lambda$C 0.62 0.73 0.25 0.98
$^{27}_\Lambda$Al 0.80 0.94 0.28 1.22
$^{28}_\Lambda$Si 0.81 0.96 0.29 1.25
$^{40}_\Lambda$Ca 0.87 1.03 0.29 1.33
$^{56}_\Lambda$Fe 0.88 1.06 0.33 1.39
$^{89}_\Lambda$Y 0.87 1.06 0.33 1.39
$^{139}_\Lambda$La 0.86 1.04 0.32 1.36
$^{208}_\Lambda$Pb 0.86 1.06 0.34 1.40
\[m-to-h\]
: Decay rates predicted for medium to heavy hypernuclei.
We think that the results of the evaluation for medium and heavy hypernuclei are encouraging: they give us some confidence in using the local density approximation for obtaining results in finite hypernuclei, even in light systems such as $^{12}_\Lambda$C.
Closing remarks
---------------
Before concluding, we make here some further comments on our calculation. Through our work we wish to emphasize the importance of a detailed many–body treatment of non–mesonic decay. This requires the identification and evaluations of a large number of diagrams, working on a step–by–step basis with the perspective of reaching the condition in which the terms that are not taken into account can be safely neglected. Considering the evolution in the predictions obtained in recent works (see especially Refs. [@ba09; @ba09b]) and here, this stability of results has not been achieved yet, and new many–body terms must be considered. In our opinion, one should explore the dependencies of predictions on the weak transition potential model only after these complicated many–body aspects are properly understood. Finally, one should attempt to reach a detailed agreement with experiment for $\Gamma_{\rm NM}$, $\Gamma_n/\Gamma_p$ and $\Gamma_2/\Gamma_{\rm NM}$ and thus extract sensible information on strangeness–changing baryon interactions. From the experimental side, new and improved data are expected from FINUDA@Daphne [@FI], J–PARC [@jparc; @jparc-exp] and GSI [@GSI]. A direct experimental identification of the two–nucleon induced channels together with the measurement of $\Gamma_2$ is a question of particular importance.
We end this Section with a comment to emphasize the importance of evaluating exchange terms. In our many–body inspired calculation, such terms are considered together with GSC contributions, which are included on the same ground for one– and two–nucleon induced decays. GSC and exchange terms improve by 10% the value of ${\Gamma}_{n}/{\Gamma}_{p}$. Once GSC are included, antisymmetrization turns out to be particularly important for both the one– and the two–nucleon induced channels, reducing ${\Gamma}_{2}$ by 18% and increasing ${\Gamma}_{1}$ by 20%. It would thus be pointless to neglect exchange terms and evaluate only direct ones. Although the introduction of antisymmetry is a difficult task in a many–body framework, one should evaluate all those exchange diagrams which are companions of a direct diagram which one knows to be relevant.
Conclusions
===========
In this contribution we have studied the effects of GSC in the non–mesonic weak decay of $\Lambda$ hypernuclei. A non–relativistic nuclear matter scheme has been adopted together with the local density approximation, for calculations in hypernuclei ranging from $^{11}_\Lambda$B to $^{208}_\Lambda$Pb. All isospin channels contributing to one– and two–nucleon induced decays have been considered. The employed weak transition potential contains the exchange of mesons of the pseudoscalar and vector octets, $\pi$, $\eta$, $K$, $\rho$, $\omega$ and $K^*$. The residual strong interaction, responsible for GSC, has been modeled on a Bonn potential based on $\pi$–, $\rho$–, $\sigma$– and $\omega$–exchange.
By using the Goldstone diagrams technique, GSC have been introduced on the same footing for one– and two–nucleon stimulated decays. The normalization of the hypernuclear ground state introduced by GSC has been taken into account. We have devoted particular attention to those GSC affecting the decay widths $\Gamma_{n}$ and $\Gamma_{p}$. The many–body $\Lambda$ self–energy terms we have considered are originated by the transition amplitudes $(a)$ and $(b1)$ of Fig. \[gam1gsc\] (for one–nucleon induced decays) and by the amplitude $(a)$ of Fig. \[gam2gsc\] (for two–nucleon induced decays). Our approach embodies fermion antisymmetry, i.e., both direct and exchange interactions are considered in the various diagrams. Concerning one–nucleon induced decays, we have evaluated GSC–free rates $\Gamma^0_{n\, (p)}$, generated by amplitude $(a)$, purely GSC terms $\Gamma^{\rm GSC}_{n\, (p)}$, produced by amplitude $(b1)$, and interference terms $\Gamma^{0-\rm GSC}_{n\, (p)}$ between uncorrelated and correlated hypernuclear ground states, i.e., between amplitudes $(a)$ and $(b1)$.
The dominant contribution to $\Gamma_1=\Gamma^0_{1}+\Gamma^{0-\rm GSC}_{1}
+\Gamma^{\rm GSC}_{1}$ turned out to be $\Gamma^0_{1}=\Gamma^0_{n}+\Gamma^0_{p}$. For $^{12}_\Lambda$C, $\Gamma^{0-\rm GSC}_{1}=\Gamma^{0-\rm GSC}_{n}+\Gamma^{0-\rm GSC}_{p}$ and $\Gamma^{\rm GSC}_{1}=\Gamma^{\rm GSC}_{n}+\Gamma^{\rm GSC}_{p}$ represented 13% and 1% of the rate $\Gamma_1$, respectively; GSC are thus responsible for 14% of the one–nucleon induced width (such contribution increases up to 19% for $^{208}_\Lambda$Pb). The above results justify the fact that we have neglected the exchange terms in $\Gamma^{\rm GSC}_{n\, (p)}$. Exchange contributions are rather relevant in the calculation of $\Gamma^{0-\rm GSC}_{n\, (p)}$ (for $^{12}_\Lambda$C, they increase $\Gamma^{0-\rm GSC}_n$ by 34% and decreases $\Gamma^{0-\rm GSC}_p$ by 8%), while only scarcely contribute to $\Gamma^{0}_{n\, (p)}$. GSC and exchange terms together increase the value of ${\Gamma}_{n}/{\Gamma}_{p}$ for $^{12}_\Lambda$C by 10%. Thanks to the opening of the two–nucleon induced channel, GSC produces a sizable increase (of 32% for $^{12}_\Lambda$C when exchange terms are included) in the value of $\Gamma_{\rm NM}=\Gamma_1+\Gamma_2$.
The agreement among our final results and recent data is quite good and clearly demonstrates the necessity of including GSC and antisymmetrization effects. Nevertheless, we believe that a refinement of the present scheme must be pursued. Additional many–body terms should be considered, involving for instance the $\Delta(1232)$ resonance. Only after a certain stability of predictions is reached within such a microscopic approach one should explore the dependencies on the weak transition potential model and determine, through detailed comparison with experiment, sensible information on strangeness–changing baryon interactions.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been partially supported by the CONICET, under contract PIP 6159. This Research is part of the EU Initiative FP7-Project HadronPhysics2 under Project number 227431. We would like to thank A. Ramos and F. Krmpotic for the helpful discussion and the careful reading of the manuscript.
Appendix A {#APPENDB .unnumbered}
==========
In this Appendix we present explicit expressions for the decay rates $\Gamma^{PQQ'}_{n\, (p)}$ with $PQQ' \neq DDD$ associated to the Goldstone diagrams of Fig. \[antgsc\] and contributing to Eq. (\[rpa2\]). In the main text, these widths have been written as: $$\label{iso-sum}
\Gamma^{PQQ'}_{n\, (p)}=
\sum_{\tau_{\Lambda'}, \tau_{\Lambda}, \tau_{N}=0,1}
{\cal T}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}, \; n \, (p)} \;
\Gamma^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_{F})~,$$ where $$\begin{aligned}
\label{appen-gamma}
\Gamma^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(\v{k},k_F) & =
& \mathcal{N}^{\, 2}(k_{F})
\frac{1}{4} \frac{(-1)^{n}}{(2 \pi)^8} (G_F m_{\pi}^2)^2
\frac{f_{\pi}^2}{m_{\pi}^2} \\
&&\times \int \int \int \, d \v{q} \, d \v{h} \, d \v{h}' \;
{\cal W}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t) \nonumber \\
&&\times \Theta(k,q,q',t,h,h',k_{F}) \frac{1}{- \varepsilon^{PQQ'}_{2p2h}} \;
\delta(q_0 - (\varepsilon_{\v{h}'+\v{q}}-\varepsilon_{\v{h}'}))~. \nonumber\end{aligned}$$ The isospin index $\tau_{\Lambda}$ ($\tau_{\Lambda'}$) of the weak transition potential is associated to an energy–momentum $q$ ($q'$), while the nuclear strong interaction isospin index is $\tau_{N}$ and the corresponding energy–momentum $t$. In the following subsections we give the functions ${\cal W}^{PQQ'}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ and $\Theta(k,q,q',t,h,h',k_{F})$, the energy denominator $\varepsilon^{PQQ'}_{2p2h}$ and $n$ (the number of crossing between fermionic lines) for the various cases. Finally, we show the isospin sums of Eq. (\[iso-sum\]).
i) $\bf \Gamma^{DDE}_{n\, (p)}$ {#i-bf-gammadde_n-p .unnumbered}
-------------------------------
The ${\cal W}^{DDE}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=q$ and $t=h'-h+q$, is identical to the ${\cal S}^{ded}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.1) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaDDE}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}-\v{h}|-k_{F}|) \\
&& \times \theta(k_{F}-|\v{h}|)\theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)~, \nonumber\end{aligned}$$ $$\varepsilon^{DDE}_{2p2h}=\varepsilon^{DDD}_{2p2h}
\equiv k_0-\varepsilon_{\v{k}-\v{q}}+\varepsilon_{\v{h}-\v{q}}-
\varepsilon_{\v{h}}-V_N~,$$ and $n=0$. The isospin sums are given by: $$\begin{aligned}
\label{decDDE}
\Gamma^{DDE}_{n}& = & -\Gamma^{DDE}_{111}+\Gamma^{DDE}_{000}+3 \Gamma^{DDE}_{101}
+\Gamma^{DDE}_{110}-\Gamma^{DDE}_{011} +\Gamma^{DDE}_{100} \nonumber \\
&&+3 \Gamma^{DDE}_{001}+\Gamma^{DDE}_{010}~, \nonumber \\
\Gamma^{DDE}_{p}& = & -5 \Gamma^{DDE}_{111}+\Gamma^{DDE}_{000}-3 \Gamma^{DDE}_{101}
+5 \Gamma^{DDE}_{110}+\Gamma^{DDE}_{011} -\Gamma^{DDE}_{100} \nonumber \\
&&+3 \Gamma^{DDE}_{001}-\Gamma^{DDE}_{010}~. \nonumber\end{aligned}$$
ii) $\bf \Gamma^{DED}_{n\, (p)}$ {#ii-bf-gammaded_n-p .unnumbered}
--------------------------------
The ${\cal W}^{DED}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=k-h$ and $t=q$, is identical to the ${\cal S}^{dde}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.3) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaDED}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}-\v{h}|-k_{F}|) \\
&& \times \theta(k_{F}-|\v{h}|)
\theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)~, \nonumber\end{aligned}$$ $$\varepsilon^{DED}_{2p2h}=\varepsilon^{DDD}_{2p2h}~,$$ and $n=0$. The isospin sums are given by: $$\begin{aligned}
\label{decDED}
\Gamma^{DED}_{n}& = & -\Gamma^{DED}_{111}+\Gamma^{DED}_{000}+\Gamma^{DED}_{101}
+3 \Gamma^{DED}_{110}-\Gamma^{DED}_{011}+\Gamma^{DED}_{100} \nonumber \\
&&+3 \Gamma^{DED}_{001}+3\Gamma^{DED}_{010}~, \nonumber \\
\Gamma^{DED}_{p}& = & -5 \Gamma^{DED}_{111}+\Gamma^{DED}_{000}+5 \Gamma^{DED}_{101}
-3 \Gamma^{DED}_{110}+\Gamma^{DED}_{011}-\Gamma^{DED}_{100} \nonumber \\
&&- \Gamma^{DED}_{001}+3\Gamma^{DED}_{010}~. \nonumber\end{aligned}$$
iii) $\bf \Gamma^{EDD}_{n(p)}$ {#iii-bf-gammaedd_np .unnumbered}
------------------------------
The ${\cal W}^{EDD}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=k-q-h'$ and $t=q$, is identical to the ${\cal S}^{dde}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.3) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaEDD}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}-\v{h}|-k_{F}|) \\
&& \times \theta(k_{F}-|\v{h}|) \theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)~, \nonumber\end{aligned}$$ $$\varepsilon^{EDD}_{2p2h}=\varepsilon^{DDD}_{2p2h}~,
%q_{0}+(\varepsilon_{\v{h}-\v{q}}-\varepsilon_{\v{h}})$$ and $n=1$. The isospin sums are given by: $$\begin{aligned}
\label{decEDD}
\Gamma^{EDD}_{n}& = & -\Gamma^{EDD}_{111}+\Gamma^{EDD}_{000}- \Gamma^{EDD}_{101}
+ 3 \Gamma^{EDD}_{110}+\Gamma^{EDD}_{011} +3 \Gamma^{EDD}_{100} \nonumber \\
&&+\Gamma^{EDD}_{001}+\Gamma^{EDD}_{010}~, \nonumber \\
\Gamma^{EDD}_{p}& = & -5 \Gamma^{EDD}_{111}+\Gamma^{EDD}_{000}+\Gamma^{EDD}_{101}
- 3 \Gamma^{EDD}_{110}+ 5 \Gamma^{EDD}_{011}+ 3 \Gamma^{EDD}_{100} \nonumber \\
&& - \Gamma^{EDD}_{001}-\Gamma^{EDD}_{010}~. \nonumber\end{aligned}$$
iv) $\bf \Gamma^{DEE}_{n(p)}$ {#iv-bf-gammadee_np .unnumbered}
-----------------------------
The ${\cal W}^{DEE}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=k-h$ and $t=h-h'-q$, is identical to the ${\cal S}^{eed}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.7) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaDEE}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}-\v{h}|-k_{F}|) \\
& & \times \theta(k_{F}-|\v{h}|)\theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)~, \nonumber\end{aligned}$$ $$\varepsilon^{DEE}_{2p2h}=\varepsilon^{DDD}_{2p2h}~,$$ and $n=1$. The isospin sums are given by: $$\begin{aligned}
\label{decdee}
\Gamma^{DEE}_{n}& = & 5 \Gamma^{DEE}_{111}+\Gamma^{DEE}_{000}+ \Gamma^{DEE}_{101}
+\Gamma^{DEE}_{110}+5\Gamma^{DEE}_{011} +\Gamma^{DEE}_{100} \nonumber \\
&& + \Gamma^{DEE}_{001}+\Gamma^{DEE}_{010}~, \nonumber \\
\Gamma^{DEE}_{p}& = & -2 \Gamma^{DEE}_{111}-4 \Gamma^{DEE}_{101}
+4 \Gamma^{DEE}_{110}+2\Gamma^{DEE}_{011} +2\Gamma^{DEE}_{100} \nonumber\\
&&+2 \Gamma^{DEE}_{001}+2\Gamma^{DEE}_{010}~. \nonumber\end{aligned}$$
v) $\bf \Gamma^{EDE}_{n(p)}$ {#v-bf-gammaede_np .unnumbered}
----------------------------
The ${\cal W}^{EDE}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=k-q-h'$ and $t=h'-h+q$, is identical to the ${\cal S}^{eed}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.7) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaEDE}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}-\v{h}|-k_{F}|) \\
& & \times \theta(k_{F}-|\v{h}|) \theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)~, \nonumber\end{aligned}$$ $$\varepsilon^{EDE}_{2p2h}=\varepsilon^{DDD}_{2p2h}~,$$ and $n=1$. The isospin sums are given by: $$\begin{aligned}
\label{decEDE}
\Gamma^{EDE}_{n}& = & -\Gamma^{EDE}_{111}+\Gamma^{EDE}_{000}+3 \Gamma^{EDE}_{101}
+\Gamma^{EDE}_{110}-\Gamma^{EDE}_{011} +\Gamma^{EDE}_{100} \nonumber\\
&&+ 3 \Gamma^{EDE}_{001}+\Gamma^{EDE}_{010} \nonumber \\
\Gamma^{EDE}_{p}& = & 4 \Gamma^{EDE}_{111}+6 \Gamma^{EDE}_{101}
-4 \Gamma^{EDE}_{110}-2\Gamma^{EDE}_{011}
+2\Gamma^{EDE}_{100}+2\Gamma^{EDE}_{010} \nonumber\end{aligned}$$
vi) $\bf \Gamma^{EED}_{n(p)}$ {#vi-bf-gammaeed_np .unnumbered}
-----------------------------
The ${\cal W}^{EED}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=k-h$ and $t=k-q-h'$, is identical to the ${\cal S}^{ede}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.5) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaEED}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{q}+\v{h}+\v{h}'-\v{k}|-k_{F}|)\\
& & \times \theta(k_{F}-|\v{h}|) \theta(|\v{q}+\v{h}'|-k_{F}|)
\theta(k_{F}-|\v{h}'|)~, \nonumber\end{aligned}$$ $$\varepsilon^{EED}_{2p2h}=k_0-\varepsilon_{\v{h}}+
\varepsilon_{\v{q}+\v{h}+\v{h}'-\v{k}}-\varepsilon_{\v{q}+\v{h'}}-V_N~,$$ and $n=1$. The isospin sum are given by: $$\begin{aligned}
\label{decEED}
\Gamma^{EED}_{n}& = & -\Gamma^{EED}_{111}+\Gamma^{EED}_{000}+ \Gamma^{EED}_{101}
+3\Gamma^{EED}_{110}-\Gamma^{EED}_{011}
+\Gamma^{EED}_{100}~, \nonumber\\
&&+\Gamma^{EED}_{001}+3\Gamma^{EED}_{010} \nonumber \\
\Gamma^{EED}_{p}& = & 4 \Gamma^{EED}_{111}-4 \Gamma^{EED}_{101}
+6 \Gamma^{EED}_{110}-2\Gamma^{EED}_{011}
+2\Gamma^{EED}_{100}+2 \Gamma^{EED}_{001}~. \nonumber\end{aligned}$$
vii) $\bf \Gamma^{EEE}_{n(p)}$ {#vii-bf-gammaeee_np .unnumbered}
------------------------------
The ${\cal W}^{EEE}_{\tau_{\Lambda'} \tau_{\Lambda} \tau_{N}}(q, q', t)$ function, where $q'=k-h$ and $t=h+q-k$, is identical to the ${\cal S}^{eee}_{\tau' \tau_{N} \tau}(q, q', t)$ function in Eq. (A.9) of Ref. [@ba07b]. Moreover: $$\begin{aligned}
\label{thetaEEE}
\Theta(k,q,q',t,h,h',k_{F}) & = &
\theta(q_0) \theta(|\v{k}-\v{q}|-k_{F}) \theta(|\v{h}'+\v{q}|-k_{F}|)\\
& & \times \theta(k_{F}-|\v{h}'|)\theta(k_{F}-|\v{k}-\v{h}+\v{h}'|)
\theta(k_{F}-|\v{h}|)~, \nonumber\end{aligned}$$ $$\varepsilon^{EEE}_{2p2h}=k_0-\varepsilon_{\v{h}}
+\varepsilon_{\v{h}'}-\varepsilon_{\v{k}-\v{h}+\v{h}'}-V_N~,
%\varepsilon_{\v{k}-\v{q}}+
%\varepsilon_{\v{h}'+\v{q}}-\varepsilon_{\v{h}}
%-\varepsilon_{\v{h}'-\v{h}'+\v{k}}~,$$ and $n=0$. The isospin sums are given by: $$\begin{aligned}
\label{deceee}
\Gamma^{EEE}_{n}& = & 5\Gamma^{EEE}_{111}+\Gamma^{EEE}_{000}+\Gamma^{EEE}_{101}
+\Gamma^{EEE}_{110}+5\Gamma^{EEE}_{011}
+\Gamma^{EEE}_{100} \nonumber \\
&&+ \Gamma^{EEE}_{001}+\Gamma^{EEE}_{010}~, \nonumber \\
\Gamma^{EEE}_{p}& = & 7 \Gamma^{EEE}_{111}+\Gamma^{EEE}_{000}+5 \Gamma^{EEE}_{101}
+5 \Gamma^{EEE}_{110}+\Gamma^{EEE}_{011}-\Gamma^{EEE}_{100} \nonumber \\
&&- \Gamma^{EEE}_{001}-\Gamma^{EEE}_{010}~. \nonumber\end{aligned}$$
Appendix B {#APPENDa .unnumbered}
==========
The explicit expressions of the function ${\cal R}(q_{0},\v{q})$ of Eq. (\[real\]) reads: $$\begin{aligned}
\label{real2}
{\cal R}(q_{0},\v{q}) & = & \frac{\pi}{(2 \pi)^{3}} \, \frac{m}{q} \,
\left\{ \frac{m^{2}}{q^{2}} \, \left[2\left(q_0-\frac{q^{2}}{2 m}\right) \frac{q}{m} k_F
+\left(\left(q_0 - \frac{q^{2}}{2 m}\right)^{2}-\frac{q^{2}}{m^{2}} k^{2}_F\right)
\right. \right. \nonumber \\
&&
\times \left. \, \ln \left|\frac{2 m q_0-q^{2}-2 q k_F}
{2 m q_0- q^{2} +2 q k_F}\right|\right] + \theta(2 k_F-q)
\, \left[-\frac{m^{2}}{q^{2}} \,\left(2 q_0 \frac{q}{m}
\left(k_F- \frac{q}{2}\right) \right. \right.
\nonumber \\
&&
\left. +\left(q^{2}_0 - \frac{q^{2}}{m^{2}}\left(k_F- \frac{q}{2}\right)^{2}\right)
\ln \left|\frac{2 m q_0+q^{2}-2 q k_F}
{2 m q_0- q^{2} +2 q k_F}\right|\right) + q \left(\frac{q}{4}-k_F\right)
\nonumber \\
&& \left. \left. \times \ln \left|\frac{2 m q_0-q^{2}+2 q k_F}
{2 m q_0+ q^{2} -2 q k_F}\right|
-q_0 m \ln \left|\frac{q^{2}_0 m^{2}-q^{2}(k_F-q/2)^{2}}
{m^{2} q^{2}_0}\right| \right] \right\}~, \nonumber\end{aligned}$$ where $q=|\v{q}|$ and $m$ is the nucleon mass.
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[^1]: Note that an analogous inequality exists between $\Gamma_n/\Gamma_p$ and the ratio between the total number of emitted neutrons and protons, $N_{n}/N_{p}$ [@ga04]. For the present discussion any of these two expressions is suitable.
|
---
abstract: |
Observations of rapidly rotating solar-like stars show a significant mixture of opposite-polarity magnetic fields within their polar regions. To explain these observations, models describing the surface transport of magnetic flux demand the presence of fast meridional flows. Here, we link sub-surface and surface magnetic flux transport simulations to investigate (i) the impact of meridional circulations with peak velocities of $\le 125{{\,{\rm m\cdot s^{-1}}}}$ on the latitudinal eruption pattern of magnetic flux tubes and (ii) the influence of the resulting butterfly diagrams on polar magnetic field properties. Prior to their eruption, magnetic flux tubes with low field strengths and initial cross sections below $\sim 300{{\,{\rm km}}}$ experience an enhanced poleward deflection through meridional flows (assumed to be poleward at the top of the convection zone and equatorward at the bottom). In particular flux tubes which originate between low and intermediate latitudes within the convective overshoot region are strongly affected. This latitude-dependent poleward deflection of erupting magnetic flux renders the wings of stellar butterfly diagrams distinctively convex. The subsequent evolution of the surface magnetic field shows that the increased number of newly emerging bipoles at higher latitudes promotes the intermingling of opposite polarities of polar magnetic fields. The associated magnetic flux densities are about $20\%$ higher than in the case disregarding the pre-eruptive deflection, which eases the necessity for fast meridional flows predicted by previous investigations. In order to reproduce the observed polar field properties, the rate of the meridional circulation has to be on the order of $100{{\,{\rm m\cdot
s^{-1}}}}$, and the latitudinal range from which magnetic flux tubes originate at the base of the convective zone ($\la 50\degr$) must be larger than in the solar case ($\la 35\degr$).
author:
- |
V. Holzwarth$^{1}$ [^1], D. H. Mackay$^{2}$, M. Jardine$^{1}$\
$^1$School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland\
$^2$School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland
bibliography:
- 'mf1aph.bib'
date: 'Received; accepted 2005'
title: The impact of meridional circulation on stellar butterfly diagrams and polar caps
---
\[firstpage\]
stars: magnetic fields – stars: activity – stars: interior – stars: rotation – stars: spots – stars: imaging
Introduction {#intro}
============
On the Sun, dark spots are exclusively found within an equatorial belt between about $\pm 40\degr$ latitude. In contrast, stellar surface brightness maps, secured with the technique of Doppler imaging [@2001astr.conf..183C and references therein], show that other cool stars frequently have large high-latitude and polar spots, often in conjunction with low-latitude features as well [@2002AN....323..309S and references therein]. Due to specific requirements of the observing technique the targets so far are stars rotating more rapidly than the Sun. Although the time base of surface maps acquired for individual stars is yet not sufficiently long to conclusively discern long-term activity properties, their behaviour seems to digress from the solar 11-year spot/22-year magnetic cycles and polar field properties. In fact, many of the younger and more active stars show no apparent cycle insofar as chromospheric indicators such as CaII H & K can be used as a proxy for magnetic activity [@1995ApJ...438..269B; @Donahue1996]. There is, as yet, also no example of a star undergoing a ‘Maunder minimum’. Different activity and cycle signatures may however be present or more pronounced, such as apparent preferred longitudes in the spot distribution or the ‘flip-flop’ phenomenon .
An important characteristic difference to the solar-like magnetic field distribution is the significant mixture of magnetic flux of opposite polarity within polar regions, as observed with the Zeeman-Doppler Imaging technique [@semel89; @donati97]. In particular the strong flux intermingling in the polar regions is in contrast to the Sun [@2003MNRAS.345.1145D], where the high-latitude magnetic field is essentially unipolar throughout the majority of the activity cycle. On rapidly rotating stars, the associated polar magnetic flux densities are sufficiently high and persistent to cause dark polar caps lasting over a large number of stellar rotation periods [e.g. @jeffers2005abdor].
The magnetic activity signatures of cool stars are ascribed to the emergence of magnetic flux generated by sub-surface dynamo mechanisms. Scenarios for dynamo operation are based on the magneto-hydrodynamic interaction between convective motions and (differential) rotation. Yet there is no complete theory, which unifies the amplification, storage, transport, eruption, and (possibly cyclic) re-generation of magnetic flux consistently; for an extensive review on stellar dynamo theory see . Dynamo mechanisms inside the convective envelope leave characteristic imprints on the observable activity signatures in the stellar atmosphere, which provide constraints for the underlying processes.
In the following, we focus on the transport of magnetic flux both below and on the stellar surface. Magnetic flux tube models describe the evolution of magnetic flux, concentrated in strands of magnetic field lines, inside the convection zone until their eruption on the stellar surface. Although they exclude aspects concerning the (cyclic) generation of magnetic fields, they successfully reproduce characteristic properties of emerging bi-polar spot groups like, in the case of the Sun, their latitude of emergence, their relative velocities and topologies, the asymmetries between the preceding- and following spot group, and Joy’s law . Further applications of the flux eruption model comprise cool stars with different rotation rates, stellar masses, and evolutionary stages as well as components of close binary systems . Frequently observed starspots at higher latitudes of rapid rotators, for example, can be explained by the poleward deflection of rising flux tubes prior to their eruption on the stellar surface . A persistent magnetic flux eruption at high latitudes implies however the availability (and possibly generation) of large amounts of magnetic flux over a similar latitudinal range inside the stellar convection zone, which would be in dissent with the solar case, where the production of magnetic flux is anticipated to be most efficient at low latitudes [e.g. @1999SoPh..184...61C].
Surface flux transport models follow the evolution of the radial magnetic field component on the stellar surface under the combined effects of magnetic flux emergence, differential rotation, meridional flow, and supergranular diffusion. Using empirical properties of bi-polar regions (i.e. latitudinal migration of emergence rates and tilt angles) they successfully reproduce major features of the solar cycle like the reversal of the polar field. Considering moderately rotating stars, @Schrijver2001b showed that an enhancement of the rate of flux emergence produces dark polar caps of a single magnetic polarity, surrounded by a flux ring of opposite polarity. More recently, @Mackay2004 showed that additional enhancements of both the latitudinal range of flux emergence and the meridional flow velocity are required to successfully reproduce a significant mixture of magnetic polarities at high latitudes on rapid rotators.
The surface flux transport models may accurately describe the surface evolution of the magnetic field, but they do not address the question of whether the assumed properties of erupting magnetic flux are consistent with the requirement of high meridional flow velocities. In particular, how the latitudinal distributions of flux eruption are affected by enhanced rates of meridional flows. We therefore link our studies on the pre-eruptive and the post-eruptive evolution of magnetic flux to consistently quantify the impact of meridional circulations on the butterfly diagram and the polar magnetic field properties of rapidly rotating stars. The main aim of this investigation is to validate the assumptions of @Mackay2004 about extended latitudinal ranges of magnetic flux emergence. Yet the relation between observable activity properties and sub-surface transport mechanisms also makes it possible to infer empirical constraints for specific dynamo properties. In Sect. \[erupt\], we investigate the rise of magnetic flux tubes through the convection zone prior to their emergence on the stellar surface to determine how the latitudinal eruption pattern depends on the strength of the circulation. The results are used in Sect. \[butter\] to determine the impact of meridional circulations on stellar butterfly diagrams. In Sect. \[surft\], we use the specified flux emergence latitudes within surface flux transport simulations to determine the requirements for the reproduction of observed stellar magnetic field properties. In Sect. \[disc\], we discuss our results and their implications for possible dynamo scenarios. Our conclusions are summarised in Sect. \[conc\].
Sub-surface evolution and eruption of magnetic flux tubes {#erupt}
=========================================================
Basic scenario
--------------
Magnetic activity signatures in the atmosphere of cool stars are ascribed to the eruption of magnetic flux tubes, which are generated by sub-surface dynamo processes inside the convective envelope . The amplification of the magnetic field is expected to take place at the tachocline, a region of strong shear flows at the interface to the radiative core of the star. Helioseismological observations indicate that the tachocline is slightly prolate [e.g. @1999ApJ...527..445C; @2001MNRAS.324..498B]. At the equator, the bulk of the tachocline is right beneath the convection zone, whereas at higher latitudes a substantial part of it is located inside the convection zone. In the dynamically unstable stratification of the convection zone, magnetic flux is subject to magnetic buoyancy, which leads to its rapid loss through eruption . In the case of the Sun, for example, a magnetic flux tube can traverse the convection zone within several weeks. The generation of high field strengths requires magnetic fields to persist in the amplifying region over time scales comparable with those of supposed dynamo processes. This requirement leads to the conjecture of magnetic flux being stored inside the subadiabatic overshoot region beneath the convection zone . Inside this stably stratified region, magnetic flux tubes are perturbed through overshooting gas plumes penetrating from the convective envelope above. If its magnetic field strength is sufficiently large, a displaced flux tube is liable to a buoyancy-driven instability . Once an unstable, growing flux loop is properly located inside the superadiabatic convection zone, it rapidly rises to the stellar surface. Coriolis forces, induced by the internal plasma flow along the flux tube, cause an asymmetric evolution of the proceeding- and following legs of the tube (relative to the direction of stellar rotation), which twists the upper part of the rising loop [e.g. @1995ApJ...441..886C]. Upon eruption on the stellar photosphere, this twists entails a tilt between the polarity centres of the emerging bipole and the parallels on the stellar surface. Depending on their size and average field strength, the erupted flux tubes produce a spectrum of magnetic activity signatures, from small magnetic knots and pores to bi-polar spot groups and active regions [e.g. @2000sostact.book..S]. Newly emerged surface magnetic flux dynamically disconnects from the sub-surface parts of the parent flux tube ; in the case of the Sun this process is expected to take place after a few days in a depth less than $10{{\,{\rm Mm}}}$ below the surface. The associated upflow of entropy-rich plasma inside the decapitated flux tube, from its anchor rooted in the overshoot region into the upper convection zone, weakens the magnetic field of the tube stumps and promotes their disintegration through turbulent diffusion driven by magneto-convective motions. The dissolving magnetic field may be transported downwards to the stable overshoot region through meridional circulation and convective pumping [e.g. @2001ApJ...549.1183T], providing the seed field for further amplification or a new activity cycle.
There is currently no detailed model for the removal of ‘old’ flux from the lower convection zone prior to the production of magnetic flux of the opposite polarity in the successive cycle. Neither it is clear how, if at all, the transformation of toroidal to poloidal magnetic fields through the twisting of erupting flux loops is related to the re-generation of magnetic flux at the bottom of the convection zone. The lack of models prevents a prediction of the amount and location of magnetic flux as a function of the meridional flow pattern and -velocities. Focusing on the transport of magnetic flux only, we hence presume the existence of appropriate magnetic flux tubes at the bottom of the convection zone of rapidly rotating stars, without further assumptions about their amplification and cyclic re-generation.
Model setup
-----------
### Thin magnetic flux tubes
The first part of the investigation is carried out in the framework of the thin flux tube approximation . For this approximation to be applicable, the radius of the flux tube must be smaller than all other relevant length scales like, for example, the scale height of the pressure or the superadiabaticity, the local radius of curvature of the magnetic field, or the wavelength of perturbations propagating along the tube. Assuming an ideal plasma with infinite conductivity, the magnetic flux over a tube’s circular cross section is conserved.
Due to the relatively short signal timescale across the tube’s diameter, the magnetic flux tube is in instantaneous pressure equilibrium with its environment, with the sum of the gas and magnetic pressure inside the tube balancing the gas pressure of the field-free external plasma, $$p_{e}= p + \frac{B^2}{8\pi}
\ .
\label{pressequi}$$ The total pressure changes smoothly across the tube’s surface [e.g. @1989sun.book.....S]. For magnetic fields inside the convection zone the magnetic pressure is typically smaller than the gas pressure (i.e. $B^2/(8\pi)\ll p$). Since the flux tube is impenetrable for the external plasma, perpendicular motions relative to the tube’s axis cause the external plasma to flow around the tube. The reaction of this distortion on the flux tube dynamics is quantified in terms of a hydrodynamic drag, $$\vec{f}_D
=
\rho_e
\frac{C_D}{\pi a} \left| v_\perp \right| \vec{v}_\perp
\ ,
\label{fdrag}$$ with $\rho_e$ being the density of the external medium. The drag force tries to reduce the perpendicular velocity difference, $\vec{v}_\perp= v_\perp \vec{e}_\perp= \left( \vec{v}_{e} - \vec{v}
\right)_\perp$, between the external and internal motions, $\vec{v}_e$ and $\vec{v}$, respectively. The value of the dimensionless drag coefficient, $C_D$, depends on the radius, $a$, of the tube considered in relation to the spectrum of length scales of the convective motions in its vicinity. The value of the effective turbulent viscosity depends on the specific model description of the convective energy transport. Since it is typically much larger than the molecular viscosity, the drag coefficient is anticipated to be of order one.
In addition to the interaction with its environment, realised through the lateral pressure equilibrium (i.e. magnetic buoyancy) and the hydrodynamic drag, the evolution of a magnetic flux tube also depends on the magnetic curvature force and the Coriolis force. In rapidly rotating stars, the interplay between the last two forces typically causes a deflection of rising flux loops to higher latitudes .
### Stellar stratification and meridional circulation
We consider the evolution of magnetic flux tubes in a $1{{\,{\rm M_{\sun}}}}$ star, described by a spherically symmetric model of the current Sun (i.e. $R_\star= R_{\sun}= 6.96\cdot10^{10}{{\,{\rm cm}}}$). The outer convection zone extends down to about $0.72{{\,{\rm R_{\sun}}}}$ and comprises at its lower boundary a superadiabatically stratified overshoot region of about $10^4{{\,{\rm km}}}$ depth. The stellar rotation period is taken to be $6{{\,{\rm d}}}$ ($\Omega=
1.212\cdot10^{-5}{{\,{\rm s^{-1}}}}$); for the sake of simplicity differential rotation is neglected.
The meridional flow pattern, $\vec{v}_p$, is superposed on the stellar structure, anticipating that its influence on the underlying hydrostatic stratification is negligibly small. We follow the approach of @1988ApJ...333..965V and describe the meridional circulation analytically through the poloidal velocity components $$\begin{aligned}
v_{p,r}
& = &
u_0
\left( 1 + \xi \right)^2
F
\sin^m \theta
\left[ \left( m+2 \right) \cos^2 \theta - \sin^2 \theta \right]
\label{vpr}
\\
v_{p,\theta}
& = &
-
u_0
\frac{G}{\sin \theta}
\left( 1 + \xi \right)^3
\left( 1 - c_1 \xi^n + c_2 \xi^{n+k} \right)
\ ,
\label{vptheta}\end{aligned}$$ where $F (\xi) $ and $G (\theta)$ are univariate functions of the radius, $r$, and the co-latitude, $\theta$, respectively, with $\xi=
R_\star / r - 1$; a more detailed description of this flow model is given in Appdx. \[meriflow\]. The circulation is parametrised through the location of its lower boundary, $r_b$, and the dimensionless quantities $n$, $m$, and $k$, which also define the coefficients $c_1$ and $c_2$ in Eqs. (\[c1coeff\]) and (\[c2coeff\]), respectively. We locate the maximum of the latitudinal flow velocity at co-latitude $\theta_M= 53\degr$ to obtain a solar-like surface flow. Since the radial profiles of stellar meridional circulations are as yet virtually unknown, we adopt for the radial parameters $r_b$ and $k$ the values used by @1988ApJ...333..965V. Thus, the assumed meridional flow pattern is quantified through the set of parameters $r_b= 0.7{{\,{\rm R_\star}}}$, $n= 1.5$, $k= 0.5$, and $m=
0.76$. It is composed of a single-cell circulation (per hemisphere) with a poleward flow at the stellar surface and an equatorward flow at the bottom of the convection zone (Fig. \[lattraj.fig\]). The amplitude of the circulation, $u_0\approx - 2.47\, v_M$ \[cf. Eq.(\[defu0\])\], depends on the choice of the peak flow velocity, $v_M$, on the stellar surface.
### Flux tube equilibria {#fteq}
The magnetic flux tubes start their evolution from a consistently defined initial configuration. For this we assume a mechanical equilibrium, which implies that their orientation is parallel to the equatorial plane and that their radius of curvature, $R_0= r_0 \cos \lambda_0$, is constant, where $r_0$ and $\lambda_0$ are the equilibrium radius and latitude, respectively. In the absence of meridional flows, flux rings in mechanical equilibrium are non-buoyant and the magnetic curvature force (pointing toward the axis of stellar rotation) is balanced by inertia and Coriolis forces (pointing away from the rotation axis). The Coriolis force is caused by a plasma flow inside the flux tube prograde to the stellar rotation. The properties of the mechanical equilibrium are described by ; for different approaches see .
In the presence of an equatorward meridional flow (with the relative direction $\vec{e}_\perp$ perpendicular to the tube axis) the drag force, Eq. (\[fdrag\]), tries to push the flux ring to lower latitudes. For a stationary mechanical equilibrium to exist (Appdx. \[equi\]), the component of the drag parallel to the rotation axis, $\vec{e}_z$, must be balanced by buoyancy, which requires the density contrast $$\frac{\rho_0}{\rho_e}
=
1
-
\frac{C_D}{\pi}
\frac{v_\perp^2}{a_0}
\frac{ \left( \vec{e}_\perp \cdot \vec{e}_z \right) }{ \left(
\vec{g}_{eff} \cdot \vec{e}_z \right) }
<
1
\label{dens}$$ between the external and internal density, where $\vec{g}_{eff}$ is the effective gravitational acceleration defined in Eq. (\[geff\]). The component of the drag force perpendicular to the rotation axis opposes the magnetic curvature force so that the internal flow velocity, $v_0$, required to balance the tension force, is $$v_0
=
\Omega R_0
\left(
\sqrt{ 1 + \frac{c_A^2 - c_D^2}{\left( \Omega R_0 \right)^2} }
-
1
\right)
\ ,
\label{vint}$$ where $c_A$ is the Alfvén velocity and $c_D$ the modification caused by the drag force in the presence of meridional circulation, given in Eq. (\[c2d\]). The Alfvén velocity introduces the magnetic field dependence in the equilibrium condition.
At the assumed equilibrium radius, $r_0= 5.07\cdot10^{10}{{\,{\rm cm}}}$, in the middle of the overshoot region, the stellar stratification is characterised through the pressure scale height $H_p=
5.52\cdot10^9{{\,{\rm cm}}}$, the gravitational acceleration $g=
5.06\cdot10^4{{\,{\rm cm\cdot s^{-2}}}}$, the density $\rho_e= 0.154{{\,{\rm g\cdot
cm^{-3}}}}$, and the superadiabaticity $\delta= -9.77\cdot10^{-7}$. For the meridional circulation defined above, the density contrast and internal flow velocity required for a stationary mechanical equilibrium are shown in Figs. \[dens.fig\] and \[vint.fig\], respectively.
![Density contrast of a magnetic flux ring in mechanical equilibrium at latitude $\lambda_0$ and depth $r_0=
5.07\cdot10^{10}{{\,{\rm cm}}}$. The peak velocity of the meridional flow is $v_M= 0, 25, 50, 75,
100, 125{{\,{\rm m\cdot s^{-1}}}}$ (*top to bottom*) and the radius of the flux tube $a_0= 10^7{{\,{\rm cm}}}$. []{data-label="dens.fig"}](dens){width="\hsize"}
![Flow velocity inside a magnetic flux ring in mechanical equilibrium at latitude $\lambda_0$ and depth $r_0=
5.07\cdot10^{10}{{\,{\rm cm}}}$. The meridional flow velocity is $v_M= 0, 25, 50, 75, 100, 125{{\,{\rm m\cdot
s^{-1}}}}$ (*top to bottom*), the radius of the flux tube $a_0=
10^7{{\,{\rm cm}}}$, and the field strength $B_0= 15\cdot10^4{{\,{\rm G}}}$. Positive (negative) values correspond to flow velocities faster (slower) than the stellar rotation. []{data-label="vint.fig"}](vint){width="\hsize"}
Flux tube simulations
---------------------
The stability properties of magnetic flux tubes depend on the stellar stratification in its environment (including the meridional flow topology) and the magnetic field strength. The determination of instability criteria and characteristic growth times requires a specific linear stability analysis similar to , which is however beyond the scope of the present paper. Investigations in the case of the Sun and other cool stars consistently show that the evolution of flux tubes with initial radii larger than $\sim 1000{{\,{\rm km}}}$ are only marginally affected by the drag force, and that magnetic field strengths $\ga 10^5{{\,{\rm G}}}$ are required to initiate rising flux loops whose properties upon eruption at the surface are in agreement with observations . Here, we consider tubes with initial radii $a_0= 70-1000{{\,{\rm km}}}$ and initial magnetic field strengths $B_0= 10-20\cdot10^4{{\,{\rm G}}}$, to determine the basic influence of these parameters on the latitudinal eruption pattern in the presence of meridional flows. The amount of transported magnetic flux, $\Phi= B_0 \pi a_0^2\sim
10^{19}-6\cdot10^{21}{{\,{\rm Mx}}}$, is comparable to that of small pores and bi-polar spot groups, respectively [@2000sostact.book..S]. We accomplish simulations for the meridional flow pattern defined above with peak flow velocities $v_M\le 125{{\,{\rm m\cdot s^{-1}}}}$.
The simulations start with the perturbation of an equilibrium flux ring located in the middle of the overshoot region within the range of latitudes $\lambda_0= 5-75\degr$. The initial perturbation consists of a superposition of harmonic displacements with low wave numbers and amplitudes of a few percent of the local pressure scale height. The adiabatic evolution of individual flux tubes is followed using a non-linear Lagrangian scheme described in more detail in and @1995ApJ...441..886C. Owing to the lateral pressure balance, Eq. (\[pressequi\]), the summit of a rising flux loop expands as the external pressure decreases. The simulations stop just beneath the stellar surface ($r\ga
0.98{{\,{\rm R_\star}}}$), where the underlying thin flux tube approximation becomes inapplicable. In the upper-most part of the convection zone the rise of the tube summit is dominated by strong magnetic buoyancy so that the tube trajectory is almost radial. The final point of the simulations thus determines the location of the emergence of a bi-polar spot group.
Latitudinal distributions of magnetic flux eruption
---------------------------------------------------
The interplay between Coriolis and magnetic tension force results in a deflection of rising flux loops to higher latitudes. Without a meridional circulation the deflection is less than $\Delta
\lambda= \lambda_e - \lambda_0\la 10\degr$, with $\lambda_e$ being the eruption latitude on the stellar surface. The deflection is greatest for flux tubes which originate from low initial latitudes and decreases for higher starting latitudes. In the presence of meridional flows the poleward deflection is considerably larger and shows a characteristic dependence on the initial latitude of the flux ring (Fig. \[lattraj.fig\]).
![ Latitudinal trajectories of the summit of rising flux loops with initial field strength $B_0= 15\cdot10^4{{\,{\rm G}}}$ and tube radius $a_0=
10^7{{\,{\rm cm}}}$ for maximal flow velocities $v_M= 10, 75$, and $100{{\,{\rm m\cdot s^{-1}}}}$ (*grey shaded lines*). The flow of the meridional circulation (*solid lines*, $n= 1.5, m=
0.76, k= 0.5, r_b= 0.7{{\,{\rm R_\star}}}$) is poleward at the surface and equatorward at the bottom of the convection zone. The shaded region at the bottom of the convection zone marks the overshoot region. []{data-label="lattraj.fig"}](lattraj){width="\hsize"}
Flux tubes originating from low to intermediate latitudes (here, around $\lambda_0\sim 25\degr$) are most affected by poloidal flows, whereas those starting from equatorial or high latitudes are in contrast only slightly more deflected than in the absence of an external circulation (Fig. \[latdist.fig\]).
![ Latitudinal deflection, $\Delta \lambda= \lambda_e - \lambda_0$, of erupting flux tubes in the presence of meridional flows with different peak flow velocities (Panel **a**, for $B_0= 15\cdot10^4{{\,{\rm G}}}$ and $a_0= 10^7{{\,{\rm cm}}}$); initial tube radii (Panel **b**, for $v_M= 100{{\,{\rm m\cdot s^{-1}}}}$ and $B_0= 15\cdot10^4{{\,{\rm G}}}$); initial magnetic field strengths (Panel **b**, for $v_M= 100{{\,{\rm m\cdot
s^{-1}}}}$ and $a_0= 10^7{{\,{\rm cm}}}$). Gaps in the curves for high flow velocities and low magnetic field strengths are due to stable flux ring equilibria for the respective set of parameters. []{data-label="latdist.fig"}](latdist){width="\hsize"}
This latitude-dependent deflection resembles closely the initial density contrast and internal velocity variation (Figs. \[dens.fig\] and \[vint.fig\], respectively), suggesting that the strong deflection is partly caused by the equilibrium conditions.
The latitudinal distribution patterns obtained for different meridional flow velocities are qualitatively similar, with the amplitude of the pattern increasing with the peak flow velocity (Figs.\[latdist.fig\]a and \[dlamvp.fig\]).
![ Dependence of the poleward deflection, $\Delta \lambda= \lambda_e -
\lambda_0$, on the meridional flow velocity, $v_M$. []{data-label="dlamvp.fig"}](dlamvp){width="\hsize"}
Whereas flux tubes with smaller radii show a characteristic deflection pattern with a maximum at intermediate latitudes, flux tubes with radii larger than $200-300{{\,{\rm km}}}$ yield an eruption pattern in which the poleward deflection continuously decreases from low to high initial latitudes (Fig. \[latdist.fig\]b). Flux tubes with larger initial radii show deflections only slightly larger than in the case without a meridional circulation. For magnetic flux tubes with high field strengths the latitudinal deflection is weaker (Fig. \[latdist.fig\]c), since the relative impact of the magnetic buoyancy on the dynamical evolution is larger. Their trajectory through the convection zone is more radial, with minor latitudinal deflections only.
Comments
--------
Thin magnetic flux tubes with weak average field strengths are found to be susceptible to meridional circulations. In particular flux tubes originating from low to intermediate latitudes are subject to significant poleward deflections during their rise to the stellar surface. We conjecture that this latitude-, size-, and field strength-depending deflection causes the bulk of low-flux elements to erupt at higher latitudes than more substantial flux tubes. Compared to stars with weak or no meridional flows, this separation effect causes a bias in the overall surface distribution of erupting magnetic flux toward higher latitudes.
The enhanced susceptibility of smaller magnetic flux elements inside the convection zone to meridional flows is in basic agreement with the properties of magnetic flux features observed on the solar surface. There, smaller flux elements are readily transported to higher latitudes, whereas new active regions and sunspots (in their entity) hardly participate in the poleward motion ($v_M\approx 10{{\,{\rm m\cdot
s^{-1}}}}$). However, after their fragmentation through supergranular motions, the dissolving remnants of initially large flux concentrations are swept polewards as well.
Stellar butterfly diagrams {#butter}
==========================
The latitudinal distribution of erupting flux tubes depends in a non-linear way on the magnitude of the meridional circulation as well as on the magnetic flux and original latitude of the tube inside the overshoot region. In the following, we consider the emergence pattern of flux tubes with an initial magnetic field strength $B_0= 15\cdot 10^{4}{{\,{\rm G}}}$ and an initial radius $a_0= 100{{\,{\rm km}}}$ as representative for the full range of parameters considered in Sect. \[erupt\].
Surface flux transport models require a description of the statistical properties of newly emerging bipoles during an activity cycle, including their emergence rates, latitudes, sizes, fluxes, and tilt angles. Since the observational data base on rapidly rotating solar-like stars is insufficient to create empirical butterfly diagrams, we adopt the solar butterfly diagram as a qualitative template, which we extrapolate in latitude and scale according to the results in Sect. \[erupt\]. The solar butterfly diagram is characterised by emergence latitudes between about $40\degr$ latitude at the start of a cycle and $5\degr$ at its end. The width of the wing of each cycle is about $10\degr$ at the beginning of the cycle and decreases to $5\degr$ at the end. The cycle period is $11{{\,{\rm yr}}}$, excluding one-year overlaps of successive cycles [@vanBal1998; @Mackay2004].
Latitudes of emergence vs. latitudes of origin
----------------------------------------------
Based on the simulation results in Sect. \[erupt\], we associate latitudinal ranges of magnetic flux emergence, $\lambda_e$, with latitudinal ranges of origin, $\lambda_0$, of the parent flux tubes in the overshoot region. For the reference (i.e. solar) butterfly diagram, we refer to the case $v_M= 0{{\,{\rm m\cdot s^{-1}}}}$, since the weak solar meridional circulation with a peak velocity of $\sim 10{{\,{\rm m\cdot s^{-1}}}}$ has little effect on the sub-surface evolution of rising flux tubes. In this case, the relationship between $\lambda_e$ and $\lambda_0$ is virtually linear (Fig. \[fig:plot1\]a, solid line/asterisks),
![Panel **a**: Latitudes of emergence, $\lambda_e$, of magnetic flux tubes as a function of their latitude of origin, $\lambda_0$, inside the convection zone. The initial magnetic field strength of the flux tubes is $B_0=
15\cdot10^{4}{{\,{\rm G}}}$ and the initial radius $a_0= 100{{\,{\rm km}}}$. The flow velocity of the meridional circulation is $v_M= 0{{\,{\rm m\cdot
s^{-1}}}}$ (*solid*), $50{{\,{\rm m\cdot s^{-1}}}}$ (*long-dashed*), $75{{\,{\rm m\cdot s^{-1}}}}$ (*short-dashed*), $100{{\,{\rm m\cdot s^{-1}}}}$ (*dashed-dotted*), and $110{{\,{\rm m\cdot s^{-1}}}}$ (*dashed-tripple dotted*); *lines* represent polynomial fits to the results (*symbols*) of the flux eruption simulations in Sect. \[erupt\]. Panel **b**: Deflection ratio, $\lambda_e (v_M) / \lambda_e (0)$, relative to the case without any meridional circulation. Values larger (smaller) than one indicate relative deflections toward the pole (equator). Note that the abscissa in panel **b** is equal to the ordinate in panel **a**, since the deflection is expressed in terms of latitudes of emergence. []{data-label="fig:plot1"}](newf6){width="\hsize"}
and we ascribe the emergence of bipolar regions on the Sun to magnetic flux tubes which originate from within the range of latitudes $\lambda_0\simeq 35-5\degr$. Further assignments between latitudes of emergence and latitudes of origin are summarised in Table \[lats\].
-------------------- ----------------------- ------------------------- --------------------- --------------------- ----------------------
\[-1.5ex\][Case]{} \[-1.5ex\][$\lambda_0 $v_M= 0{{\,{\rm m/s}}}$ $50{{\,{\rm m/s}}}$ $75{{\,{\rm m/s}}}$ $100{{\,{\rm m/s}}}$
[\degr]$]{}
S 35 – 5 40 – 10 44 – 10 47 – 13 54 – 15
I 47 – 3 50 – 10 54 – 8 47 – 9 60 – 9
L 58 – 3 60 – 10 61 – 9 63 – 9 65 – 9
-------------------- ----------------------- ------------------------- --------------------- --------------------- ----------------------
: Correspondence between the latitudes of origin, $\lambda_0$, and latitudes of emergence, $\lambda_e$, of flux tubes with an initial field strength of $B_0= 15\cdot10^{4}{{\,{\rm G}}}$ and radius $a_0=
100{{\,{\rm km}}}$ (see Fig. \[fig:plot1\]).
\[lats\]
We shall refer to the small ($\lambda_0= 35-5\degr$), intermediate ($47-3\degr$), and large ($58-3\degr$) ranges of originating latitudes as case S, I, and L, respectively.
The wings of the butterfly {#wings}
--------------------------
In contrast to the solar reference case, strong meridional circulations imply non-linear relationships between the eruption latitudes of magnetic flux tubes and the latitudes of their origin (Fig.\[fig:plot1\]a) and, consequently, distortions of the wing shapes of stellar butterfly diagrams. To transform the solar template into a stellar butterfly diagram, we determine the ratio between the latitudes of emergence, $\lambda_e
(v_M)$, *subject to meridional flows* and the latitudes of emergence, $\lambda_e (0)$, of the reference case with *vanishing meridional flow*. The deflection ratios $\lambda_e (v_M) / \lambda_e (0)$ in Fig. \[fig:plot1\]b represent mappings, which describe the non-linear stretching of the solar template. In addition to the poleward displacement of flux emergence at intermediate latitudes also a slight equatorward displacement occurs at low latitudes.
Examples of stellar butterfly diagrams subject to meridional circulations with different peak velocities are shown in Fig.\[fig:plot2\] (for case I).
![Butterfly diagrams over four 11-year activity cycles in the presence of meridional circulations with flow velocities $v_M=
10{{\,{\rm m\cdot s^{-1}}}}$ (Panel **a**); $v_M= 75{{\,{\rm m\cdot s^{-1}}}}$ (Panel **b**); $v_M= 100{{\,{\rm m\cdot s^{-1}}}}$ (Panel **c**). In all cases the generating magnetic flux tubes originate from within the range of latitudes $\lambda_0= 47-3\degr$. The pre-eruptive poleward deflection caused by the meridional circulation pushes the upper boundary of the butterfly diagram to higher latitudes, from $\lambda_e\approx 50\degr$ for the weak flow (Panel **a**) to $\lambda_e\approx 60\degr$ for the strong flow (Panel **c**).[]{data-label="fig:plot2"}](duncan2_small){width="\hsize"}
Figure \[fig:plot2\]a depicts a stellar butterfly diagram subject to a weak meridional circulation (i.e. the solar template linearly stretched to the latitudinal range $50-10\degr$), whereas the Figs.\[fig:plot2\]b & c show the result of the non-linear transformation of the latitudinal pattern of emerging bipoles caused by higher flow velocities. The diagrams show that for strong meridional circulations the number of bipoles emerging at higher latitudes is increased: the higher the flow velocity, the more convex is the temporal evolution of the stellar butterfly diagram. Qualitatively similar results are obtained for the cases S and L.
Surface evolution and intermingling of magnetic bipoles {#surft}
=======================================================
Basic scenario and empirical constraints
----------------------------------------
After the dynamical disconnection of newly emerged bipoles from their parent flux tubes, the small-scale evolution of magnetic surface features is governed by local magneto-convective motions and interactions with ambient magnetic flux concentrations. Advection of flux with the same (opposite) polarity through the convective turnover of supergranular cells leads to an enhancement (annihilation) of magnetic flux. Averaged over length scales larger than supergranular cells (in the case of the Sun about $30{{\,{\rm Mm}}}$) the evolution of the magnetic field resembles a dispersion process [@Leighton1964]. The lifetime of individual magnetic surface features depends on the diffusion time scale. The evolution of the global magnetic field topology, in turn, depends on the relation between the lifetimes of merging flux features and the characteristic transport timescales of the differential rotation and the meridional circulation. Differential rotation shears and diverges bipolar spot groups spread out over a range of latitudes. The increasing distance between the centres of opposite flux polarities reduces the rate of mutual flux annihilation, whereas the associated decrease of the tilt angle reduces the bias in the latitudinal distribution of magnetic polarities (see also Appdx. \[single\]). Merging magnetic surface features are transported to higher latitudes by the poleward directed meridional circulation and pile-up in polar regions. Owing to the biased latitudinal polarity distribution, the polar flux is dominated by one polarity during half of the magnetic cycle. With the reversal of the polarity of newly emerging bipolar groups after half a magnetic cycle (Hale’s law), magnetic flux of the opposite polarity is predominantly transported toward the pole, which annihilates and substitutes the previous flux conglomeration. This alternating process implies a stable oscillation of the net surface flux over successive activity cycles, without a persistent accumulation of magnetic flux. For a review about the development and limitations of surface flux transport model see @2005LRSP....2....5S.
In contrast to the case of the Sun, where the polar magnetic field is virtually unipolar, with magnetic flux densities of about $10{{\,{\rm Mx\cdot cm^{-2}}}}$, the surface magnetic fields observed on young active stars exhibit a high degree of intermingling of magnetic flux polarities [@donati97abdor95; @donati99abdor96; @donati03] and dark polar caps [@2002AN....323..309S], indicative for strong magnetic field strengths. @Schrijver2001b found that flux emergence rates thirty times larger than on the Sun entail polar flux densities of $300-500{{\,{\rm Mx
\cdot cm^{-2}}}}$, which are deemed to be sufficiently large to suppress convective upflows of energy, entailing the formation of dark spots [e.g. @1982smhd.book.....P]. In their simulations, the polar magnetic flux is unipolar and surrounded at lower latitudes by a ring of magnetic flux of the opposite polarity. Focusing on the intermingling of polar magnetic flux, @Mackay2004 identified two key parameters for the generation of high degrees of polarity mixture: magnetic bipoles have to emerge at latitudes up to $50-70\degr$, and the peak value of the meridional flow has to be over $100{{\,{\rm m\cdot s^{-1}}}}$.
The simulations of @Mackay2004 disregard the feedback of strong meridional circulations on the latitudinal pattern of flux eruption. This feedback is now provided through the consistently determined stellar butterfly diagrams in Sect. \[butter\]. In the following, we use the @Mackay2004-model to investigate the impact of the convex wing structures on the polar magnetic field properties.
Surface flux transport model
----------------------------
The surface flux transport simulations of @Mackay2004 follow the radial magnetic field component on the stellar surface regarding the combined influence of flux emergence, differential rotation, meridional flow, and supergranular diffusion. Using spherical harmonics up to degree 63, the spatial resolution is about $30{{\,{\rm Mm}}}$. The simulations are based on the extrapolation of solar transport and flux emerging properties [@Gaizauskas1983; @Wang1989; @Harvey1993; @Schrijver1994; @Tain1999] to more rapidly rotating stars [see also @Schrijver2001b].
The strength and profile of the differential rotation of rapidly rotating stars is similar to the solar case [e.g. @cameron02diffrot]. We therefore adopt the solar differential rotation profile given by @Snodgrass1983, which implies the characteristic shear timescale $\tau_\Omega= 0.25{{\,{\rm yr}}}$. The meridional flow profile is given by $v_p (\lambda)= - v_M \sin
\left( \pi \lambda / \lambda_p \right)$, whereby the flow velocity vanishes near the pole above $\lambda_p= 75\degr$ [@Hathaway1996]. The characteristic timescale for the latitudinal flux transport is $\tau_\mathrm{mf}= R_\odot/v_M$. Lacking appropriate empirical constraints, it is assumed that the supergranular convective profile and turn-over timescales of rapidly rotating stars are solar-like, implying a diffusion coefficient of $D= 450{{\,{\rm km^2\cdot s^{-1}}}}$ [@vanBal1998] and a diffusion time scale for magnetic features with length scale $l$ of $\tau_D=
l^2/D$ ($= 34{{\,{\rm yr}}}$ for $l= R_\odot$).
Following @Schrijver2001b, we take the net amount of magnetic flux emerging during each cycle to be thirty times the solar value, that is $3.15\cdot10^{26}{{\,{\rm Mx}}}$. New flux is injected into the evolving surface magnetic field distribution at random longitudes in the form of 13200 bipolar magnetic regions, whose average tilt angles vary as $\lambda_e/2$ [@Wang1989]. In each successive cycle the polarity of the preceding and following flux of the bipoles alternates according to Hale’s law. The simulations cover a time span of four activity cycles to verify that stable oscillations of the polar field from one cycle to the next are obtained. For further details about model assumptions, input parameters, and initial conditions see @Mackay2004 [ Sect. 2–4].
In the following, we focus on the total (unsigned) magnetic flux, $$\Phi_\mathrm{tot} (t)
=
\int | B_r (R,\theta,\phi,t) | d S
\ ,
\label{phitotdef}$$ the total (unsigned) *polar* magnetic flux in each hemisphere, $$\Phi_\mathrm{[np,sp]} (t)
=
\int | B_r (R,\theta_{[\theta<20\degr,\theta>160\degr]},\phi,t) | d S
\ ,
\label{phipolardef}$$ the imbalance of positive and negative magnetic flux in the polar regions, $$\delta_\mathrm{[n,s]} (t)
=
\int B_r (R,\theta_{[\theta<20\degr,\theta>160\degr]},\phi,t) d S
\ ,
\label{phiinbaldef}$$ and the polar magnetic flux in the *northern* hemisphere, $$\Phi_{[+,-]} (t)
=
\int B_r (R,\theta< 20\degr,\phi,t)_{[B_r>0,B_r<0]} d S
\ .
\label{phinorthdef}$$ The principal quantities of the analysis will be mean magnetic flux densities, that is the respective magnetic flux divided by the surface over which it is calculated [see @Mackay2004]. The last quantity, Eq. (\[phinorthdef\]) is a measure for the degree of intermingling of opposite polarity elements at high latitudes.
Surface magnetic field properties of rapid rotators {#simus}
---------------------------------------------------
We consider weak and strong meridional flows of the case I to detail our results (Fig. \[fig:plot3\]).
{width="\hsize"}
For both flow velocities the mean surface flux density, $\Phi_\mathrm{tot}/(4\pi R^2)$, is in phase with the activity cycle emergence rate (Fig. \[fig:plot3\]a & f). If the meridional flow velocity is high, then more magnetic flux is pushed to higher latitudes, where the proximity of opposite bipole polarities increases the flux annihilation rate. In this case, the mean surface flux densities are consequently smaller than for slow meridional circulations. The resulting variation of the mean *polar* flux density (i.e.beyond $70\degr$ latitude) is shown in Figs. \[fig:plot3\]b & g. In the case of weak meridional flows this quantity shows a noticeable phase shift relative to the mean surface flux density, with a *cycle-averaged* polar flux density of $\sim 192{{\,{\rm G}}}$. In the case of fast flows the polar field rises and falls in phase, with a significantly higher cycle-averaged polar flux density of $\sim
270{{\,{\rm G}}}$. Breaking up the polar flux density imbalance (Figs. \[fig:plot3\]c & h) according to the contributions of positive and negative flux (Fig. \[fig:plot3\]d & i) shows that for slow meridional flow velocities the polar field is unipolar, with a field reversal midway through the cycle. For fast flow velocities, in contrast, both magnetic polarities are present throughout the cycle and rise and fall in strength together. The ratio between the positive and negative flux densities in the polar caps is thus in both cases different (Fig. \[fig:plot3\]e & j). For weak flows there is hardly any intermingling of flux of opposite polarities at high latitudes, except for a limited time span during field reversal, when little magnetic flux is located within the polar cap. For strong meridional flows, in contrast, the intermingling of magnetic flux with opposite polarities is about $65\%$, averaged over an activity cycle.
Snapshots of maps showing the radial magnetic field component (Fig. \[fig:plot6\]) illustrate that in the case of weak flows the magnetic flux in polar regions is dominated by one polarity only, whereas in the case of fast flows the field is intermingled.
{width="\hsize"}
### Parameter study
For the latitudinal ranges I and L, the degree of intermingling of opposite polarities within the polar regions exceeds $20\%$ for meridional flow velocities beyond about $40{{\,{\rm m\cdot s^{-1}}}}$ (Fig.\[fig:plot4\]a).
![Panel **a**: Cycle-averaged ratio between the positive and negative magnetic flux within the northern polar cap as a function of the meridional flow velocity, $v_M$, for originating latitudes $\lambda_0= 35-5\degr$ (*triangles*); $47-3\degr$ (*diamonds*); and $57-3\degr$ (*asterisks*). Panel **b**: Cycle-averaged polar flux densities. []{data-label="fig:plot4"}](newf10){width="\hsize"}
The fact that in case I the intermingling is higher than in case L indicates that merely increasing both the meridional flow velocity and the latitudinal range of flux origin does not produce per se higher degrees of intermingling. Instead, there exists an optimal combination of latitudes of origin and meridional flow velocities for which the intermingling within polar caps peaks.
The associated polar flux densities are on the order of $200-300{{\,{\rm Mx\cdot cm^{-2}}}}$ (Fig. \[fig:plot4\]b), with values increasing (decreasing) with the meridional flow velocity above (below) $v_M\sim 50{{\,{\rm m\cdot s^{-1}}}}$. This behaviour is due to the different transport- and annihilation- timescales (see Appdx. \[single\]). In the case of strong meridional flows, the advection of magnetic flux to higher latitudes is more efficient than the decrease caused by flux annihilation as the magnetic surface features are transported poleward. The same argument holds for the increase of the polar flux density with larger latitudinal ranges of flux origin, since for bipoles emerging at higher latitudes the shorter migration time towards the pole implies less flux annihilation. For meridional flow velocities around $100{{\,{\rm m\cdot s^{-1}}}}$ the polar flux densities become sufficiently large ($\sim 300{{\,{\rm G}}}$) to suppress convection.
In case S, the polar magnetic flux is unipolar throughout the activity cycle, without a significant intermingling of opposite polarities. The average polar flux densities of $\approx 300{{\,{\rm G}}}$ are almost independent of the meridional flow velocity.
In summary, we find that a combination of intermediate ranges of originating latitudes (here, $\lambda_0 = 47-3\degr$) with fast meridional flow velocities ($v_M\sim 100{{\,{\rm m\cdot s^{-1}}}}$) is optimal for the production of both strong intermingling of polarities and high field strengths in polar regions. This suggests that in rapidly rotating solar-like stars magnetic flux originates from latitudinal ranges somewhat larger than in the case of the Sun.
### Effect of the pre-eruptive poleward deflection
@Mackay2004 disregard the influence of meridional circulations on the sub-surface evolution of magnetic flux. In their investigation the reference (i.e. solar) butterfly diagram is linearly stretched to different latitudinal ranges, leaving the structure of the wings unaltered. According to our results in Sect. \[wings\], the eruption pattern *within* a given range of emergence latitudes depends on the meridional flow velocity, with more bipoles emerging at higher latitudes the faster the flow velocity. We compare the two approaches in the case of a meridional circulation with a flow velocity of $100{{\,{\rm m\cdot s^{-1}}}}$. Keeping other model parameters unchanged, including the pre-eruptive poleward deflection entails about $20\%$ higher polar flux densities than in the original approach of @Mackay2004 This result applies to all ranges of generating latitudes (Table \[comp\]).
------------------------------------ ---------------- ----------------
$\bar{B}\ [G]$ $\bar{B}\ [G]$
\[-1.5ex\][$\lambda_e\ [\degr]$]{} incl. defl. excl. defl.
50 – 10 248 200
60 – 10 266 218
65 – 10 290 236
------------------------------------ ---------------- ----------------
: Comparison of the cycle-averaged polar magnetic field strength, $\bar{B}$, including and excluding the effect of pre-eruptive deflection of magnetic flux tubes for a meridional circulation with flow velocity $v_M= 100{{\,{\rm m\cdot s ^{-1}}}}$. The latter values are taken from @Mackay2004.
\[comp\]
The reason for this increase lies in the more convex structure of the wings of the butterfly (cf. Fig. \[fig:plot2\]). The larger number of emerging bipoles at higher latitudes over an activity cycle implies shorter transport times (on average) toward the pole during which less flux annihilation occurs.
Discussion {#disc}
==========
The present work relates observed properties of magnetic flux on the stellar surface to the sub-surface evolution of the originating magnetic fields and, therefore, to the underlying dynamo processes inside the convective envelope. Our results extend the findings of @Mackay2004, who identified high latitudes of magnetic flux emergence and high meridional flow velocities as the key ingredients for a strong mixing of opposite polarities in polar regions of rapidly rotating stars. We find that the enhanced pre-eruptive poleward deflection of rising flux tubes inside the convection zone provides a consistent explanation for the required larger latitudinal range for newly emerging bipoles.
Within the framework of our model assumptions, a solar-like butterfly diagram with flux emergence up to $40\degr$ latitudes is generated by erupting magnetic flux tubes, which originate at the bottom of the convection zone from latitudes $\la 35\degr$. Owing to the slight prolateness of the solar tachocline [e.g. @1999ApJ...527..445C], these low latitudes coincide in the case of the Sun well with the region of strong shear flows, efficient magnetic field amplification, and field storage. For more rapidly rotating stars, the butterfly structures producing the optimal levels of polar flux intermingling are generated by erupting flux tubes, which originate from higher latitudes, $\la 50\degr$, as well. This suggest the existence of a larger latitudinal range of efficient dynamo operation than in the case of the Sun, implying a possibly less prolate tachocline.
Based on the results of @Schrijver2001b, we assumed a magnetic flux emergence rate thirty times larger than the solar value to produce polar flux densities sufficiently strong to entail dark caps. Dynamo operation is generally expected to be more efficient at higher rotation rates, mainly entailing an increase of the surface filling factor of magnetic features [e.g. @1991LNP...380..389S]. The larger latitudinal range of dynamo operation suggested by our results complies with this expectation of a larger amount of magnetic flux generated during an activity cycle. Considering a possible concomitant increase of the average magnetic field strength, we note that strong-field flux tubes are less susceptible to meridional circulations and therefore hamper the formation of high flux densities and high degrees of flux intermingling in polar regions. We also find that flux tubes with initial radii $a_0\ga 1000{{\,{\rm km}}}$, which are expected to lead to the formation of average sunspots, are only marginally affected by meridional flows. Since the magnitude of the drag force scales $\propto v_\perp^2 / a$, a meridional flow velocity about three times faster would correspond roughly to a flux tube radius about ten times smaller, entailing a noticeable change of the eruption latitude. In view of the uncertainties of the sub-surface flow profile, this estimate indicates that also larger magnetic flux elements could be affected by meridional circulations.
By using the solar butterfly diagram as an empirical template, we circumvent the treatment of the magnetic field amplification and dissolving mechanisms inside the stellar convection zone, which as yet lack a consistent theoretical description. This approach, however, implies that the activity signatures of rapidly rotating stars are due to a solar-like dynamo. The assumption of an $\alpha\Omega$-dynamo mechanism has important implications, for example, on the period, and even the mere existence, of the activity cycle. The $11{{\,{\rm yr}}}$ activity cycle assumed here sets the time scale for the pile-up of magnetic flux in polar regions. If we, for example, increase the cycle length, then we would obtain qualitatively similar results by increasing the total amount of magnetic flux emerging per cycle. A conclusive parameter study of different ratios between eruption, transport, and diffusion time scales is however beyond the scope of this work; a discussion of the impact of selected surface transport model parameters is given by @Mackay2004.
Our work supplements earlier investigations, which considered the evolution of the weak solar magnetic field subject to a prescribed meridional flow pattern and a dynamo wave at the bottom of the convection zone . All approaches, however, suffer from the same lack of knowledge about the sub-surface profiles of meridional circulation and their dependence on the rotation, mass, structure, and evolutionary stage of the star. Owing to the density difference between the top and the bottom of the convection zone, the equatorward flow is slower than the surface flow, and the larger the radial extent of the equatorward flow region, the lower the associated velocities will be. The latitudinal flow velocity at the bottom of the convection zone possibly has an influence on the propagation velocity of dynamo waves and, consequently, on the period of the activity cycle. But in absence of an appropriate theory, this impact cannot be quantified. The circulation model applied here assumes a ratio of 2:1 between the upper convection zone flowing poleward and the lower convection zone flowing toward the equator. A different radial profile would not only affect the equator- and poleward drift of rising flux tubes on their way to the surface, but also their equilibrium and stability properties inside the overshoot region. The former aspect has to be investigated in a parameter study considering both meridional flow and differential rotation profiles as well as different stellar rotation periods. Furthermore, specifically determined tilt angles and eruption time scales of newly emerging bipoles have to be consistently included in the surface flux transport model.
Compared with a mere linear stretching of the solar butterfly diagram to higher latitudes [e.g. @Mackay2004], the additional convexity of the butterfly wings promotes the intermingling of polar flux even more, which eases the necessity for high poleward flow velocities. Although the meridional flow velocities of $\sim 100{{\,{\rm m\cdot s^{-1}}}}$ are lower than the values suggested by @Mackay2004, they are yet still about ten times larger than in the case of the Sun. Whereas in the case of the Sun, with $v_M\approx 11{{\,{\rm m\cdot s^{-1}}}}$, the bias in the latitudinal size distribution of newly emerging bipoles is probably neither discernible nor relevant, convex butterfly structures and the size distributions of emerging dipoles may become observable on rapidly rotating stars once the temporal and spatial resolutions are sufficiently high. Strong meridional circulations on the order of $100{{\,{\rm m\cdot s^{-1}}}}$ are also expected to cause a shift of the stellar eigenfrequencies, in addition to the splitting of eigenfrequencies caused by differential rotation (M Roth, priv. comm.) Depending on the sub-surface profile of the flow, a meridional circulation of $100{{\,{\rm m\cdot s^{-1}}}}$ would cause in the case of the Sun a frequency shift of a few tenth of $\mu$Hz. Albeit small, this effect would be observable on the Sun. Given sufficiently long and complete time sequences, astroseismological observations may provide constraints on the magnitude (and possible structure) of stellar meridional flows.
Conclusion {#conc}
==========
Strong meridional circulations enhance the poleward deflection of rising magnetic flux tubes inside stellar convection zones. Flux tubes which comprise small amounts of magnetic flux and which originate between low to intermediate latitudes within the overshoot region are particularly susceptible to meridional flows. The resulting latitudinal distribution of newly emerging bipoles renders the wing structure of stellar butterfly diagrams distinctively more convex than in the solar case. The larger amount of magnetic flux emerging at higher latitudes supports the formation of high magnetic flux densities and a significant intermingling of opposite-polarity magnetic flux in polar regions, which is in agreement with recent observational results of rapidly rotating solar-like stars.
The strong pre-eruptive deflection of magnetic flux tubes provides a consistent explanation for the required high latitudes of flux emergence identified by previous investigations [@Mackay2004]. The synergetic combination of pre-eruptive flux tube deflection and post-eruptive bipole transport yields higher values for the average polar flux densities than in the previous approach, and thus eases the necessity for high meridional flow velocities. The functional dependence of the polar field properties on the extent and structure of the stellar butterfly diagram makes it possible to conjecture about potential regions of efficient dynamo operation at the bottom of the convection zone. In rapidly rotating cool stars, we suppose the latitudinal range of magnetic flux production to extend to higher latitudes ($\la 50\degr$) than in the case of the Sun ($\la 35\degr$).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank M Roth and M Schüssler for helpful comments and F Moreno-Insertis, P Caligari, and A van Ballegooijen, who initially developed the codes used in this paper. VH and DHM acknowledge financial support for their research through PPARC grands.
Meridional flow model {#meriflow}
=====================
We use the analytical model of @1988ApJ...333..965V for a parametrised description of the meridional circulation inside the convection zone . The poloidal flow pattern, $\vec{v}_p = \left( \nabla \times \Psi
\vec{e}_\phi \right) / \rho_e$, is expressed in terms of the scalar function, $\Psi$, to ensure that the stationary continuity equation is consistently fulfilled. Under the assumption of a separation of variables, the stream function $$\Psi \left(r,\theta\right) r \sin \theta
=
u_0 R_\star^2 \rho_e (r) \cdot F(r) \cdot G \left(\theta\right)
\ ,
\label{stream}$$ is expressed in terms of the univariate functions $$\begin{aligned}
F (r)
& = &
\left(
-
\frac{1}{n+1}
+
\frac{c_1}{2n+1} \xi^n
-
\frac{c_2}{2n + k + 1} \xi^{n+k}
\right)
\xi
\label{fxi}
\\
G (\theta)
& = &
\sin^{m+2} \theta \cos \theta
\ ,
\label{gtheta}\end{aligned}$$ with the rescaled radial coordinate $$\xi = \frac{R_\star}{r} - 1
\ .
\label{xi}$$ The coefficients $$\begin{aligned}
c_1
& = &
\frac{ \left( 2n+1 \right) \left( n+k \right)}{\left( n+1 \right) k}
\xi_b^{-n}
\label{c1coeff}
\\
c_2
& = &
\frac{ \left( 2n+k+1 \right) n}{\left( n+1 \right) k}
\xi_b^{-\left(n+k\right)}
\label{c2coeff}\end{aligned}$$ depend on the location of the lower boundary, $\xi_b= R_\star / r_b -
1$, of the flow pattern, with $r_b$ being a free model parameter. The poloidal components of the flow velocity are $$\begin{aligned}
v_{p,r}
& = &
u_0
\left( \frac{R_\star}{r} \right)^2
\frac{F}{\sin \theta}
\frac{\partial}{\partial \theta} G
\label{vprappdx}
\\
v_{p,\theta}
& = &
-
u_0 R_\star^2
\frac{G}{r \sin \theta}
\left(
\frac{\partial}{\partial r} F + F \frac{d \ln \rho_e}{d r}
\right)
\\
& = &
-
u_0
\frac{G}{\sin \theta}
\left( 1 + \xi \right)^3
\left[
\left( 1 - \frac{n-\mathcal{H} (\xi)}{n+1} \right)
\right.
\nonumber \\ & & {}
\left.
-
c_1 \left( 1 - \frac{n-\mathcal{H} (\xi)}{2n+1} \right) \xi^n
\right.
\nonumber \\ & & {}
\left.
+
c_2 \left( 1 - \frac{n-\mathcal{H} (\xi)}{2n+k+1} \right) \xi^{n+k}
\right]
\label{vpthetappdx}\end{aligned}$$ with $$\mathcal{H}
=
- \frac{r^2 \xi}{R_\star} \frac{d \ln \rho_e}{d r}
=
\frac{r^2 \xi}{R_\star H_\rho}
\ ,
\label{densstrat}$$ and the local density scale height $H_\rho= - \left( d \ln \rho_e / d r
\right)$. If the density stratification is approximated through the expression $\rho_e (r)= C \xi^n$, then it is $\mathcal{H}= n$ and Eq.(\[vpthetappdx\]) results in Eq. (\[vptheta\]). For an adiabatically stratified convection zone with a ratio of specific heats $\gamma= 5/3$, it is $n= 1.5$ [@1988ApJ...333..965V]. The maximum of the latitudinal flow velocity is located at co-latitude $\theta_M$, specified by the condition $\tan \theta_M = \sqrt{m + 1}$. We use the value $m= 0.76$ to obtain a solar-like value $\theta_M\simeq
53\degr$. The flow pattern is composed of a downflow at high latitudes (i.e. $v_{p,r}< 0$ for $0< \theta< 58\degr$) and an upflow at low latitudes (i.e. $v_{p,r}> 0$ for $58\degr< \theta< 90\degr$). The latitudinal velocity component changes its direction at about $0.8
R_\star$, that is $v_{p,\theta}> 0$ (equatorward) for $0.7\le r/R_\star
\la 0.8$ and $v_{p,\theta}< 0$ (poleward) for $0.8\la
r/R_\star \le 1$ (Fig. \[profile.fig\]).
![ Radial (*left*) and latitudinal (*right*) velocity components of the meridional flow for the parameter $n= 1.5, m= 0.76, k= 0.5$, and $r_b= 0.7 R_\star$. The maximum of the latitudinal flow velocity on the stellar surface is located at $\theta_M= 53^o$. At the radius $\sim 0.8 R_\star$, the latitudinal flow pattern changes its sign from poleward (above) to equatorward (below). []{data-label="profile.fig"}](profile){width="\hsize"}
At the surface, $r= R_\star$ it is $\xi= 0$ and $v_{p,\theta} = - u_0
\sin^{m+1} \theta \cos \theta$. The amplitude, $$u_0
=
-
\frac{v_M}{\sin^{m+1} \theta_M \cos \theta_M}
\label{defu0}$$ of the circulation is determined through the maximal flow velocity, $v_M$.
Flux tube equilibrium {#equi}
=====================
The mechanical equilibrium in the presence of meridional flows is determined in the frame of reference co-rotating with the star. For a homogeneous flux tube (i.e. vanishing derivatives with respect to the arc length), the stationary equation of motion is $$\rho
\left( v^2 - \frac{B^2}{4\pi \rho} \right)
\frac{1}{R}
\vec{n}
=
-
\nabla \left( p + \frac{B^2}{8\pi} \right)
+
\rho
\vec{g}_{eff}
+
2
\rho
\left( \vec{v} \times \vec{\Omega} \right)
\ ,
\label{eqmb}$$ where $\vec{v}$ is the internal flow velocity, $\vec{B}$ the magnetic field strength, $\rho$ the density, $p$ the gas pressure, $R$ the local radius of curvature, $\vec{n}$ the normal vector of the tube, $\vec{\Omega}= \Omega \vec{e}_z$ the stellar rotation vector, and $$\vec{g}_{eff}
=
\vec{g}
-
\vec{\Omega} \times \left( \vec{\Omega} \times \vec{r} \right)
=
g (r)
\left(
\vec{e}_r
+
\vec{e}_R \frac{\Omega^2 r}{g} \sin \theta
\right)
\label{geff}$$ the effective gravitational acceleration comprising the centrifugal contribution, which (for the model assumptions specified in Sect. \[fteq\]) inside the overshoot region is of the order $\mathcal{O}
\left( \Omega^2 r / g \right) \sim 10^{-4}$. Anticipating the meridional circulation to be of minor importance for the stellar structure, the external stratification is approximately hydrostatic, $$\nabla p_e
=
\rho_e
\vec{g}_{eff}
\ .
\label{eqmwob}$$ Using the lateral pressure balance, Eq. (\[pressequi\]), the difference between Eqs. (\[eqmb\]) and (\[eqmwob\]) yields $$\left( v^2 - \frac{B^2}{4\pi \rho} \right) \kappa \vec{n}
=
\left( 1 - \frac{\rho_e}{\rho} \right)
\vec{g}_{eff}
+
2
\left( \vec{v} \times \vec{\Omega} \right)
+
\vec{f}_D
\ ,
\label{eqmft}$$ with the hydrodynamic drag force, $\vec{f}_D$, accounting for the influence of the distorted external flow on the dynamics of the flux tube. For toroidal magnetic flux tubes with a constant radius of curvature, the component of Eq. (\[eqmft\]) tangential to the tube axis (i.e.the azimuthal component) vanishes per se. The bi-normal component (i.e. parallel to axis of stellar rotation, $\vec{e}_z$) implies the condition $$\left( 1 - \frac{\rho_e}{\rho} \right)
\left( \vec{g}_{eff} \cdot \vec{e}_z \right)
=
-
\frac{1}{\rho}
\left( \vec{f}_D \cdot \vec{e}_z \right)
\ .
\label{paracomp}$$ With the definition for $\vec{f}_D$ in Eq. (\[fdrag\]) this yields the density contrast given in Eq. (\[dens\]), which is required to balance the influence of the drag force along the z-axis through buoyancy. The component perpendicular to the stellar rotation axis ($\vec{e}_R=
-\vec{n}$), $$\left( v^2 - \frac{B^2}{4\pi \rho} \right) \frac{1}{R}
+
\left( 1 - \frac{\rho_e}{\rho} \right)
\left( \vec{g}_{eff} \cdot \vec{e}_R \right)
+
2 v \Omega
=
-
\frac{1}{\rho}
\left( \vec{f}_D \cdot \vec{e}_R \right)
\label{perpcomp}$$ shows that a meridional flow toward the equator in the overshoot region supports the outward directed inertia and Coriolis force in balancing the magnetic tension force of the curved flux ring, where $R= r \cos
\lambda$ is the constant radius of curvature. From the alternative form of Eq. (\[perpcomp\]), $$\frac{v^2}{R}
+
2 v \Omega
+
\frac{c_D^2}{R}
=
\frac{c_A^2}{R}
\label{velo}$$ follows the internal flow velocity given in Eq. (\[vint\]), which is required to keep the flux ring in mechanical equilibrium. The contribution by the drag force is formally expressed in terms of the velocity $$\frac{c_D^2}{R_0}
=
\left( 1 - \frac{\rho_e}{\rho_0} \right)
\left( \vec{g}_{eff} \cdot \vec{e}_R \right)
+
\frac{\rho_e}{\rho_0}
\frac{C_D}{\pi}
\frac{v_\perp^2}{a_0}
\left( \vec{e}_\perp \cdot \vec{e}_R \right)
\label{c2d}$$ whereas the Alfvén velocity $c_A^2= B^2/(4\pi\rho)$ contains the dependence on the magnetic field strength.
The particular susceptibility of magnetic flux rings located at low to intermediate latitudes (cf. Figs. \[dens.fig\] & \[vint.fig\]) is likely a characteristic feature of the equilibrium properties. Neglecting centrifugal forces, the latitudinal variation of the density contrast, Eq. (\[dens\]), is proportional to $v_\perp^2 \cot
\lambda$. Albeit the actual meridional flow profile inside the stars is unknown, the increase and decrease of the latitudinal flow velocity from the pole down the equator is to yield a peak at intermediate latitudes. In combination with the factor $\cot \lambda$, which continuously increases toward the equator, the peak of the density contrast will be located at low to mid latitudes.
Evolution of single bipoles {#single}
===========================
The surface flux transport simulations in Sect. \[simus\] show that the intermediate range of originating latitudes (Case I, cf. Table \[lats\]) causes a higher degree of intermingling than the two cases S and L. To investigate this aspect in more detail, we analyse the relevant mechanisms in the simplified case of a single bipole evolving across the surface under the effect of meridional flow, differential rotation and supergranular diffusion. For a meridional flow velocity of $v_M= 100{{\,{\rm m\cdot s^{-1}}}}$, the butterfly diagrams in Case S, I, and L have the *mean* latitude of bipole emergence $\bar{\lambda}_e= 21\degr$, $39\degr$, and $45\degr$, respectively. In each of the three cases, a single bipole is inserted onto a (magnetically empty) stellar surface at the mean latitude of emergence. The initial tilt angle is $\bar{\lambda}_e/2$, whereas the initial total magnetic flux is in all cases the same ($1.5\cdot10^{23}{{\,{\rm Mx}}}$). The temporal evolution of the polar flux density and ratio of opposite polarities is shown Fig. \[fig:plot5\].
![Panel **a**: North polar flux density for a single bipole inserted at a mean latitude of emergence, $\bar{\lambda}= 21\degr$ (*dashed*), $39\degr$ (*dotted*), and $44\degr$ (*solid*). Panel **b**: Ratio between positive and negative magnetic flux in the polar region; the closer to one, the stronger the intermingling. []{data-label="fig:plot5"}](newc1){width="\hsize"}
The higher the (mean) latitude of emergence, the sooner the bipole reaches the polar region, and the stronger is the remaining polar flux, due to the shorter time span available for flux annihilation as it is transported to the poles. The trailing flux region (here, of positive polarity) enters the polar region before the leading region (of negative polarity); when the latitude of emergence is small the timing when both polarities enter the polar region becomes similar. In contrast, the higher the latitude of emergence the less intermingling between magnetic flux of opposite polarity is obtained. As the entire flux is pushed into the polar region, the flux ratio eventually approaches unity. Using the area under each curve as a measure for the degree of intermingling, the results for the two lower mean latitudes of emergence are similar, whereas the degree of intermingling for $\bar{\lambda}_e= 45\degr$ is overall considerably lower.
Since each bipole obeys Joy’s law, the trailing flux region is located at somewhat higher latitudes than the preceding flux region of the opposite polarity. The meridional circulation then pushes both magnetic flux regions in this form toward the pole. However, during the poleward transport the differential rotation however rotates the bipole. This causes more flux of each polarity to be located at a common latitude, which is a requirement for intermingling to occur once the bipole is pushed into the polar region. If a bipole emerges at high latitudes, then the differential rotation has not enough time to act before its entry into the polar region. Hence a higher degree of intermingling is obtained if the initial latitude of emergence is low. This is in contrast to the polar flux density, which is found to decrease with the initial latitudes of emergence. Consequently, there is a particular range of latitudes of emergence and poleward meridional flows, which results in both high degrees of intermingling and strong polar flux densities. Optimal values occur when the timescales of both meridional flow and differential rotation are similar.
\[lastpage\]
[^1]: E-mail: [email protected]
|
---
abstract: 'By employing angle-resolved photoemission spectroscopy combined with first-principles calculations, we performed a systematic investigation on the electronic structure of LaBi, which exhibits extremely large magnetoresistance (XMR), and is theoretically predicted to possess band anticrossing with nontrivial topological properties. Here, the observations of the Fermi-surface topology and band dispersions are similar to previous studies on LaSb \[Phys. Rev. Lett. **117**, 127204 (2016)\], a topologically trivial XMR semimetal, except the existence of a band inversion along the $\Gamma$-$X$ direction, with one massless and one gapped Dirac-like surface state at the $X$ and $\Gamma$ points, respectively. The odd number of massless Dirac cones suggests that LaBi is analogous to the time-reversal $Z_2$ nontrivial topological insulator. These findings open up a new series for exploring novel topological states and investigating their evolution from the perspective of topological phase transition within the family of rare-earth monopnictides.'
author:
- 'R. Lou'
- 'B.-B. Fu'
- 'Q. N. Xu'
- 'P.-J. Guo'
- 'L.-Y. Kong'
- 'L.-K. Zeng'
- 'J.-Z. Ma'
- 'P. Richard'
- 'C. Fang'
- 'Y.-B. Huang'
- 'S.-S. Sun'
- 'Q. Wang'
- 'L. Wang'
- 'Y.-G. Shi'
- 'H. C. Lei'
- 'K. Liu'
- 'H. M. Weng'
- 'T. Qian'
- 'H. Ding'
- 'S.-C. Wang'
title: Evidence of topological insulator state in the semimetal LaBi
---
Exploring exotic topological states has sparked extensive research interest, both theoretically and experimentally, due to their promising potential in low consumption spintronics devices [@Hasan2010; @Qi2011; @Weng2014]. In the past decade, since the discovery of quantum spin Hall effect in graphene [@KaneGraphene], remarkable achievements have been reached, including the findings of two-dimensional (2D) [@Bernevig2006; @Konig2007; @Knez2011] and three-dimensional (3D) topological insulators (TIs) [@Chen2009], node-line semimetals [@Burkov2011; @Yu2015], topological crystalline insulators [@Fu2011; @Heish2012NC], and Dirac and Weyl semimetals [@Wang2012; @Liu2014; @Weng2015Weyl; @Lv2015; @Huang2015; @Xu2015Weyl]. Strikingly, although the TIs have ignited the whole field for years, the realization of novel massless surface Dirac fermions, $i.e.$, topological surface states (SSs), is still pretty exciting due to the great prospects in tunability of the topological characteristics through easily accessible manipulations [@Ando2013]. Among the extensive efforts, the study on the topological phase transition (TPT) can also effectively facilitate the exploration and investigation of topological SSs and even the Dirac semimetals [@Sato2011; @Xu2011; @Lou2015].
{width="1\columnwidth"}
{width="1.77\columnwidth"}
Recently, the discovery of simple rock salt rare-earth monopnictides Ln$X$ (Ln = La, Y, Nd, or Ce; $X$ = Sb/Bi) [@Tafti2015; @Sun2016; @Kumar2016; @Yu2016; @Pavlosiuk2016; @Wakeham2016; @Alidoust2016] has renewed the platform for searching these novel topological states. The most remarkable signature of this series is the extremely large magnetoresistance (XMR) with a resistivity plateau at low temperatures, which is proposed as the consequence of breaking time-reversal symmetry [@Tafti2015]. However, the theoretical predictions for the topological properties vary in the Ln$X$ series [@Zeng2015]. In addition, similar fingerprints have also been reported in several semimetals including TmPn$_2$ (Tm = Ta/Nb, Pn = As/Sb) [@WangK2014; @Shen2016; @WuD2016; @XuC2016; @WangY2016; @WangZ2016], ZrSiS [@Singha2016; @Ali2016; @WangX2016], WTe$_2$ [@Ali2014], Cd$_3$As$_2$ [@FengJ2015; @Liang2015], TaAs [@HuangX2015], and NbP [@Shekhar2015]. The topological origin of the large magnetoresistance and resistivity plateau is still under debate. Furthermore, the rich and interesting topological phases predicted in the Ln$X$ family offer an unprecedented opportunity for investigating the novel topological properties and TPT in a relatively simple system without convoluted bulk and SSs [@Zeng2015; @JiangJ2015; @Pletikosic2014; @Schoop2016; @Lou2016ZST]. To our knowledge, although there have been several angle-resolved photoemission spectroscopy (ARPES) studies on this series of compounds [@Alidoust2016; @Niu2016; @WuY2016; @Nayak2016; @Neupane2016] stimulated by previous investigations on LaSb [@Lou2016LaSb], the experimental data and understandings are still ambiguous to fully determine the existence of nontrivial band topology in other compounds of the Ln$X$ family.
In this paper, we report systematic ARPES measurements and first-principles calculations on LaBi single crystals. We identify two hole-like Fermi surfaces (FSs) at the Brillouin zone (BZ) center $\Gamma$ and one electron-like FS at the BZ boundary $X$, similar to that of LaSb. Furthermore, we find that LaBi exhibits a clear nontrivial band anticrossing along the $\Gamma$-$X$ direction, with the presence of one massless and one gapped Dirac-like SSs around $X$ and $\Gamma$, respectively. Based on the observation of band inversion and odd number of massless Dirac cones, our results unambiguously demonstrate the $Z_2$-type nontrivial band topology of LaBi. The diverse topological phases among the Ln$X$ series of compounds provide an excellent platform for exploring novel topological states and investigating their evolution from the perspective of TPT.
Single crystals of LaBi were grown by the In flux method. The residual resistance ratio \[RRR = $R$(300 K)/$R$(2 K) = 204\] and large magnetoresistance (MR = 3.8$\times$10$^4$ % at 2 K under 14-T magnetic field) indicate the high quality of samples used in this paper [@Sun2016]. The Vienna $\emph{ab initio}$ simulation package is employed for most of the first-principles calculations. The generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) type is used for the exchange-correlation potential [@Perdew1996]. Spin-orbit coupling is taken into account. The $k$-point grid in the self-consistent process is 11$\times$11$\times$11. To get the tight-binding (TB) model Hamiltonian, we use the package WANNIER90 to obtain maximally localized Wannier functions of $d$ and $f$ orbits of La and $p$ orbit of Bi. ARPES measurements were performed at the Dreamline beamline of the Shanghai Synchrotron Radiation Facility with a Scienta D80 analyzer, at the Surface and Interface Spectroscopy beamline of Swiss Light Source using a Scienta R4000 analyzer, and at the beamline 13U of the National Synchrotron Radiation Laboratory equipped with a Scienta R4000 analyzer. The energy and angular resolutions were set to 15 meV and 0.2$^{\circ}$, respectively. Fresh surfaces for ARPES measurements were obtained by cleaving the samples $\emph{in situ}$ along the (001) plane in a vacuum better than 5$\times$10$^{-11}$ Torr. All data shown in this work were recorded at $T$ = 30 K.
LaBi crystallizes in a NaCl-type crystal structure (face-center cubic) with space group $Fm$-3$m$, in which Bi is located at the face center adjacent to La atoms, as illustrated in Fig. 1(a). The schematic bulk Brillouin zone (BZ) and the (001)-projected 2D surface BZ are presented in Fig. 1(b). One bulk $X$ and two bulk $L$ points are projected to the surface $\bar{\Gamma}$ and $\bar{M}$ points, respectively; two bulk $X$ points are projected to the surface $\bar{X}$ point. Figure 1(c) demonstrates the FS topologies of LaBi recorded with photon energy $h\nu$ = 84 eV, close to the $k_z$ = $\pi$ plane according to our photon energy dependent study discussed below. One can obtain remarkable consistency between the experimental FSs and theoretical calculations with the PBE functional [@GuoP2016], as shown in Fig. 1(d), including one elliptical electron pocket at $X$ elongated along the $\Gamma$-$X$ direction, and two hole pockets centered at $\Gamma$ with the intersecting-elliptical one enclosing the circular one. Further examining the measured FSs carefully, we can observe additional FSs around $X$ perpendicular to the elongated pocket along the $\Gamma$-$X$ direction, which is a common feature in the Ln$X$ series of compounds; this could be possibly interpreted by the band folding effect as a consequence of breaking translational symmetry from bulk to the (001) surface [@Lou2016LaSb]. Moreover, the $k_z$ broadening effect caused by the short escape length of the photoelectrons excited by the vacuum ultraviolet light in our ARPES experiments may also be an alternative explanation [@Kumigashira1997; @Kumigashira1998; @Strocov2003; @Niu2016].
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In order to illuminate the underlying topological properties of the measured electronic structure of LaBi, we have investigated the near-$E_F$ band dispersions along the high-symmetry lines $\Gamma$-$X$ and $X$-$W$-$X$, with photon energies $h\nu$ = 53 and 84 eV, close to the $k_z$ = 0 and $\pi$ planes, respectively. Figures 2(a) and 2(b) show the measured band structure and corresponding second derivative intensity plot along the $\Gamma$-$X$ direction recorded with $h\nu$ = 53 eV, respectively. The overall bulk band dispersions are similar to our previous study on LaSb [@Lou2016LaSb], except that the outer hole band around $\Gamma$ exhibits a shoulder at $\sim$-0.33 eV when gradually leveling off from $\Gamma$ to $X$. Additionally, unlike the full parabolic electron band centered at $X$ in LaSb, the one in LaBi curves upward towards $X$ around the shoulder, forming a hole band with a top at $\sim$-0.14 eV at $X$. The calculated band structure using the PBE functional can reproduce these bulk features very well and indicate that LaBi is a topologically nontrivial material with band anticrossing along the $\Gamma$-$X$ direction [@GuoP2016]. Using the detailed energy distribution curve (EDC) analysis presented in Fig. 2(j), we clearly observe a band gap around the shoulder, further demonstrating the existence of nontrivial band inversion in LaBi. More ARPES spectra showing much clearer features around the shoulder and the electron band centered at $X$ measured on another piece of sample can be found in the Supplemental Material [@Supplementary].
Besides the expected bulk bands, there are some additional Dirac-like surface bands around the $\Gamma$ and $X$ points. To avoid the complexity introduced by the convoluted bulk states, we investigate the electronic structure along the $X$-$W$-$X$ direction with $h\nu$ = 53 and 84 eV in Figs. 2(c)-2(i). Owing to the projection over a wide range of $k_z$ [@Niu2016], the Dirac-like surface band around the bulk $\Gamma$ point in the $k_z$ = 0 ($\pi$) plane is projected to the bulk $X$ point in the $k_z$ = $\pi$ (0) plane, further validated by the surrounding two dispersive hole bands around the right $X$ points in Figs. 2(c)-2(f), which are the contributions from the bulk $\Gamma$ point. Thus the surface bands located at $\Gamma$ points in Figs. 2(a) and 2(b) are identical to those around the right $X$ points in Figs. 2(c)-2(f). Accordingly, it seems that the total number of Dirac points (DPs) below $E_F$ is even, $i.e.$, 2, which is inconsistent with the odd number of band anticrossings in LaBi.
To examine this topological characteristic, we perform EDC and momentum distribution curve (MDC) studies around the right and left $X$ points in Fig. 2(e), and present the results in Figs. 2(g) and 2(i), respectively. Unlike the massless Dirac-like surface band with a DP at $\sim$-0.14 eV in Fig. 2(i) (see detailed EDC plot in the Supplemental Material [@Supplementary]), which coincides with the valance-band maximum at the $X$ points in Figs. 2(a) and 2(b), we clearly resolve an energy gap in Fig. 2(g) separating the upper and lower linearly dispersive bands. Correspondingly, as seen in Fig. 2(h), the EDC at the right $X$ point in Fig. 2(e) exhibits a shoulder beside the prominent peak. We use two Lorentzian curves to fit this single EDC and the fitting result is shown as the superimposed blue curve. By extracting the peak positions of Lorentzian curves, we can obtain a surface band gap of $\sim$23 meV, suggesting the existence of a massive Dirac cone at the bulk $\Gamma$ point. This exotic mass acquisition of Dirac fermions in LaBi was also observed recently by Wu $\emph{et al}$. [@WuY2016], demonstrating the novelty and complexity of the topological properties in this compound. Since the SS at $\Gamma$ is proved to be gapped, the total number of massless Dirac cones below $E_F$ is odd. Therefore, by summarizing the bulk and surface band dispersions along the high-symmetry lines, we conclude that LaBi is analogous to the time-reversal $Z_2$ nontrivial TI [@Moore2007; @Fu2007], while the massive Dirac fermion at $\Gamma$ leaves an elusive issue in understanding the novel topological characteristics in LaBi.
The detailed photon energy dependent study summarized in Fig. 3 is performed from 20 to 100 eV to demonstrate the 2D nature of the SSs. We show the FS mapping data in the $k_y$-$k_z$ plane at $E_F$ and $E$ = -0.14 eV, corresponding to the DP at $X$, with different photon energies in Figs. 3(a) and 3(c), respectively. According to the photon energy dependence measurements, we estimate the inner potential of LaBi to 14 eV. The $k_z$ = 0 and $\pi$ planes at the BZ center correspond to $h\nu$ = 51 and 82 eV, respectively, while at the BZ boundary the values are $h\nu$ = 53 and 84 eV, respectively. As illustrated in Fig. 3(a), the two selected photon energies ($h\nu$ = 53 and 84 eV) for the investigations on the electronic structure at $k_z$ $\sim$ 0 and $\pi$ planes, respectively, are reasonable. We can clearly identify the two SSs around the $\Gamma$ and $X$ points showing no obvious dispersion along the $k_z$ direction from the $k_y$-$k_z$ maps in Figs. 3(a) and 3(c). We focus on the massless Dirac-like SS around $X$ and plot the MDCs at $E_F$ and $E$ = -0.14 eV in Figs. 3(b) and 3(d), respectively. The $k_z$ independence of the Fermi crossings and DP provide further proof for the 2D surface band nature of this SS.
{width="0.85\columnwidth"}
We also calculate the (001) surface band structure for a semi-infinite slab along the $z$ direction by using the Green’s-function method, which is based on the TB Hamiltonian. As shown in Fig. 4, although one cannot unambiguously determine the SSs around $\bar{\Gamma}$ due to the convoluted bulk states, the Dirac-like SSs around $\bar{X}$ are consistent with our experimental results. The additional surface bands should relate to the topologically trivial origin.
We notice that there are two recent ARPES studies focusing on the surface band structure and topological properties of LaBi [@Niu2016; @Nayak2016], both reporting the presence of two separated Dirac cones at $X$. Nevertheless, the good consistency between our experimental band dispersions and slab model calculations demonstrate the existence of only one Dirac cone at $X$. Further, we perform a similar semi-infinite slab calculation terminated by the surface on the other side of cleavage, and present the result into the Supplemental Material [@Supplementary]. One can resolve two well-separated Dirac cones located at $\bar{X}$ from this surface termination. Although both sides of the cleavage should have identical atomic arrangement, the different calculated surface band structures make this issue much more controversial [@NbAsHasan]. Therefore, we speculate that the distinct observations at $X$ might result from the different surface terminations and periodic potentials after cleavage. We believe that further studies should be invested on the topological characteristics of LaBi both theoretically and experimentally.
To conclude, we have performed ARPES experiments and first-principles calculations to investigate the electronic structure and intrinsic topological characteristics of LaBi. A nontrivial band anticrossing is obviously observed along the $\Gamma$-$X$ direction. We further identify two distinct topological SSs. One is gapless around $X$ and the other is massive around $\Gamma$ with a $\sim$23-meV gap. Such an exotic mass acquisition of Dirac fermions requires future studies to clarify. Developed theoretical calculations need to be formulated to resolve the surface band at $\Gamma$ from the convoluted bulk states. The presence of band inversion and odd number of massless Dirac cones provide compelling evidence that LaBi is analogous to the time-reversal $Z_2$ TI. The rich and novel topological states in LaBi and other compounds of the Ln$X$ family offer an excellent opportunity for investigating the evolution of topological properties from the perspective of TPT, facilitating the future search of new topological phases.
This work was supported by the Ministry of Science and Technology of China (Programs No. 2012CB921701, No. 2013CB921700, No. 2015CB921000, No. 2016YFA0300300, No. 2016YFA0300504, No. 2016YFA0300600, No. 2016YFA0302400, and No. 2016YFA0401000), the National Natural Science Foundation of China (Grants No. 11274381, No. 11274362, No. 11474340, No. 11234014, No. 11274367, No. 11474330, No. 11574394, and No. 11674371), and the Chinese Academy of Sciences (CAS) (Project No. XDB07000000). RL and KL were supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (RUC) (Grants No. 17XNH055, No. 14XNLQ03, No. 15XNLF06, and No. 15XNLQ07). Computational resources have been provided by the Physical Laboratory of High Performance Computing at RUC. The FSs were prepared with the XCRYSDEN program [@Kokalj2003]. YH was supported by the CAS Pioneer Hundred Talents Program.
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abstract: 'In this work, the classical and the quantum capacitances are calculated for a Fabry-Pérot interferometer operating in the integer quantized Hall regime. We first consider a rotationally symmetric electrostatic confinement potential and obtain the widths and the spatial distribution of the insulating (incompressible) circular strips using a charge density profile stemming from self-consistent calculations. Modelling the electrical circuit of capacitors composed of metallic gates and incompressible/compressible strips, we investigate the conditions to observe Aharonov-Bohm (quantum mechanical phase dependent) and Coulomb Blockade (capacitive coupling dependent) effects reflected in conductance oscillations. In a last step, we solve the Schrödinger and the Poisson equations self-consistently in a numerical manner taking into account realistic experimental geometries. We find that, describing the conductance oscillations either by Aharanov-Bohm or Coulomb Blockade strongly depends on sample properties also other than size, therefore, determining the origin of these oscillations requires further experimental and theoretical investigation.'
address:
- 'Isik University, Department of Physics, 34980 Istanbul, Turkey'
- 'Istanbul Yeni Yuzyil University, Vocational School of Health Services, 34010 Istanbul, Turkey'
- 'Physics Department, Faculty of Letters and Sciences, Mimar Sinan Fine Arts University, 34380 Sisli, Istanbul, Turkey'
author:
- 'O. Kilicoglu'
- 'D. Eksi'
- 'A. Siddiki'
title: 'A realistic quantum capacitance model for quantum Hall edge state based Fabry-Pérot interferometers'
---
July 2016
Introduction
============
The charge transport measurements performed at quantized Hall effect (QHE) based particle and quasi-particle interferometers provide information on electronic and statistical properties of the particles [@Halperin:2011; @marcus:01; @Goldman:2005; @Heiblum:2006; @Rosenow:2009]. In these experiments conductance through a quantum dot (QD) is measured as a function of external magnetic field $B$ or the gate potential(s) $V_g$ defining the electrostatic confinement. Interestingly, conductance presents oscillations as a function of both $B$ and $V_g$. The peak-to-peak periodicity of the oscillations, $\Delta B$ and $\Delta V_g$ respectively, strongly depend on various properties of the devices. On one hand, the side gate (SG) defined Fabry-Pérot type interferometers with large interference areas $(A > 5 ~\mu$m$^2)$ present usual Aharonov-Bohm (AB) periodicity as a function of the external magnetic field, i.e. the number of enclosed magnetic flux increase linearly with $B$. Hence the $\Delta B$ periodicity is constant in $B$. Meanwhile, if a voltage is applied to the top-gate (TG) or the voltage on SG is changed, $\Delta V_g$ varies inverse linearly with $B$. On the other hand, small samples $(A < 3 ~\mu$m$^2)$ show an opposite behavior. In this situation, $\Delta B$ varies linearly with $B$, i.e. inversely proportional to filling factor $\nu$, and $\Delta V_g$ is independent of the $B$ field. In mainstream theoretical works, this behavior is interpreted as a result of charging effects and therefore is called the Coulomb blockade or Coulomb dominated (CD) regime [@Halperin:2011; @marcus:01; @Rosenow:2009]. However, other interpretations are also available in the literature, based on interactions and time-dependent calculations [@Goldman:2005; @Salman:2013; @Cicek:2012].
The importance of charging effects on the AB interference has already been discussed some two decades ago by Beenakker within the single particle approximation [@Beenakker:1991]. They found that, the conductance oscillation period as a function of magnetic field $\Delta B$ is modified by the charging effects, namely by the capacitance $C$ of the QD. It is shown that, the Coulomb blockade of the Aharonov-Bohm effect occurs, whenever charging energy $e^2/C$ becomes comparable or larger than the energy separation of the magnetic field quantized levels. Later, Evans and co-workers calculated the geometrical capacitance of a similar system also taking into account the direct electron-electron interaction, namely they considered the formation of metal-like compressible and insulator-like incompressible strips [@Evans:1993]. They elucidated the experimental findings of McEuen et al, where magneto-transport through a relatively small QD (lithographic size of 500 nm x 750 nm, which confines less than 100 electrons) is investigated [@McEuen:1992]. In the experimental work, peaks in the conductance were reported and are attributed to charging of the metallic-like compressible island.
Recently, there has been theoretical and experimental reports which essentially emphasises the importance of so-called “Quantum capacitance". The classical (or equivalently called geometrical) capacitance takes into account only the direct Coulomb interaction and a homogeneous dielectric constant $\varepsilon$. Whereas, in calculating the quantum capacitance $C_{\rm q}$ one should also take into account density of states (DOS), Pauli exclusion principle and correlation effects. As we will summarise below, its essence relies on the fact that the quantum capacitance is just proportional to thermodynamical density of states $D_{\rm T}(\mu)$ (TDOS) at the chemical potential $\mu$ [@Gerhardts:2005]. Therefore, systems having a gap at Fermi energy $E_{\rm F}$ in the limit of zero temperature, i.e. $D_{\rm T\rightarrow 0}(E_{\rm F})=0$, the quantum capacitance also vanishes. Since the geometrical $C_{\rm geo}$ and the quantum capacitances are added in series the total capacitance also reads to zero. Hence, to understand capacitive effects (like charging) at gapped systems it is important to investigate the contribution of quantum capacitance, which becomes the dominant counter-part at low temperatures and high $B$ fields.
We organise our paper as follows, first we briefly discuss how the dielectric function is modified due to dimensionality affected DOS considering a two dimensional electron system (2DES). There we also touch the situation where the 2DES is subject to high magnetic fields. Next, we consider a toy model to calculate the capacitance of a rotationally symmetric QD to find the $B$ intervals where the total capacitance is dominated either by the geometric or quantum capacitances, following Beenaker. Afterwards, we show that the oscillation period $\Delta B$ observed at the conduction decreases with increasing field strength only when the geometrical capacitance is taken into account. In contrast, we observe that the period strongly depends on the field strength if one takes into account the quantum capacitance. Our discussion is followed by a Section, where we compare our results which now include quantum capacitances with a previous similar work by Evans et al. There we show that inclusion of quantum effects strongly alter the oscillations at the conductance also in $\Delta V_g$. In Section \[sec:cap\_halperin\], we calculate the total capacitance of a realistic device and seek for the parameter regimes where the conductance oscillations can be determined either by charging effects (CD) or by interference (AB) effects. In this Section, we first consider only the electrostatic effects and then include self-consistent calculations. We find that although it is possible to observe CD oscillations in real experiments, it is strongly constrained by the sample parameters.
Geometrical and Quantum Capacitances of a homogeneous 2DEG
==========================================================
In general, capacitance is a measure of energy to be paid to charge a device which is composed of two [*metals*]{} separated by an [*insultor*]{} [@Serway]. Here, of course we used the words metal and insulator in a hand-waving way. The energy necessary to charge such a device is given by $E_C=Q.V=Q^2.C$, where $V$ is the potential difference between the metals, $Q$ the charge and $C$ the capacitance. Considering the simplest classical device in three dimensions, two metal plates perpendicular to each other and separated by an insulator, the capacitance is given by the area of the metal plate $A$, the distance between metal plates and the dielectric [*constant*]{} $\kappa$ of the insulator, C=. However, as we will briefly discuss below, all our arguments above impose couple of assumptions which are not valid in low-dimensions also including quantum mechanical interactions beyond the direct Coulomb interaction.
First, it is useful to check the common assumptions imposed on the dielectric [*function*]{} $\varepsilon(\vec r,t)$ [@Kittel]. Here, $\vec r$ determines the spatial coordinate and $t$ is the time. Equivalently the dielectric function can be defined by its Fourier adjuncts, $\vec k$ and $\omega$, which are momentum and frequency respectively. The dielectric function can be regarded as the response of a material subject to an external electric field $\vec{E}(\vec r,t)$. Now lets first assume that our material is simultaneously responding to the external field, hence assume that the dielectric function is time (or equally frequency) independent, i.e $\varepsilon(\vec k,\omega)=\varepsilon(\vec k,0)$. Assuming a spatial homogeneity results in a constant dielectric function $\kappa$. To make the connection between the dielectric function and energy dispersion of the system we now lift the assumption on spatial homogeneity [@Ashcroft]. Then, it is easy to show that the response of the material to the external field is described by (k)=\^[ext]{}(k). Here we implicitly made the assumption that, the screened potential $\phi(\vec k)$ and the external potential $\phi^{\rm ext}(\vec k)$ are linear. Then, a material can be called metal in the limit $\varepsilon(\vec k)\rightarrow \infty$ ($k\rightarrow 0$). We also assumed that the field is uniform and the potential varies slowly on the scale of the particle separation. Under the assumption of uniformity an insulator can be defined as a material where its dielectric function is finite and small compared to a metal. Now the problem is reduced to find the exact form of $\varepsilon(\vec k)$ including dimensional and quantum effects. In the following we will confine ourself to the lowest order mean field approximation to describe quantum mechanical corrections to $\varepsilon(\vec k)$, namely the Thomas-Fermi approximation.
If one solves the related time-independent Schrödinger equation and keep the assumption of slowly varying potential the momentum dependent eigenenergies are described as, (k)=-e(r),where $m$ is the bare electron mass and $e$ is the charge of an electron. One step further in including quantum mechanical effects is to calculate the position dependent electron number density utilising the Fermi-Dirac distribution so that, n\_[el]{}(r)= , [\[TFA\_density\]]{}where $\xi=\beta[\epsilon(\vec k)-\mu]$ comprises the thermal energy $1/\beta=k_{\rm B}T$, where $T$ is temperature and $k_{\rm B}$ is Boltzmann constant. Then the dielectric function that includes energy dispersion and also the Fermi-Dirac statistics can be written as, (k)=1+, where $n_0$ is the equilibrium density without the external field. Namely, the Thomas-Fermi dielectric function.
The above equation includes two important information about the system at hand, first in the $k\rightarrow 0$ limit it describes a metal properly and, second via $\frac{\partial n_0}{\partial \mu}$ the effects arise from dimensionality and spin degree of freedom are explicitly included. Note that, the thermodynamic of states TDOS $D_{T}(\mu,T)$ equals to $\frac{\partial n_0}{\partial \mu}$ and describes how the levels are occupied at a given temperature and number of particles. It is also useful to define a wave vector $k_0=\sqrt{4\pi e^2\frac{\partial n_0}{\partial \mu}}$, which will be helpful in defining a length scale to check the validity of our assumptions, below. Then, the dielectric function reads [@Gerhardts:2003], (k)=1+.Given the definition of capacitance in general, it is now possible to define a quantum mechanically corrected counter part, which now also includes the statistical properties of the system, C\_q=e\^2D\_T(,T),and is defined per area. Then the total capacitance reads, \[eq:totalC\]1/C\_T=1/C\_[geo]{}+1/C\_q
It is important to note that, the dielectric function is now a thermodynamic quantity. Namely, it is not only a function of material properties but also temperature, statistics of the particles and how the quantum mechanical states are occupied. Thus, it is indispensable to take into account the limitations of the above expression. One can assign a special role to the wave vector $k_0=2\pi/\lambda_T$, where $\lambda_T$ is a thermal quantity, thermal wavelength. In the limit of zero temperature, $1/k_0$ is strongly related with the Fermi wavelength, which essentially limits validity of above screening argument by the number of particles involved, when utilised. Therefore, if one uses the dielectric function, as a mean field quantity one should also take into account whether if it is valid when the number of particles is sufficiently low. Hence, we will keep our eye on the validity regimes of our assumptions, when considering “small" ensembles such as quantum dots and narrow incompressible strips.
Homogeneous 2DEG in the absence of an external $B$ field
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Now we are equipped with the minimally corrected dielectric function which comprises also the necessary quantum mechanical and statistical information. In the beginning of this subsection, we will just refer to the well known text book results that describe the TDOS of a homogeneous 2DEG and connect quantum capacitance with the dielectric function.
It is well known that, if one can neglect the boundary effects and also the finiteness of the particle number density (i.e. in the thermodynamical limit) one can write the DOS of a 2DES as, D(E)=D\_0= , here $m$ is the bare mass of an electron in vacuum, and will be replaced by an effective mass $m^*$ when considering a semi-conductor heterostructure (homogeneous insulator) later. In the limit of $T\rightarrow0$ and utilising the fact that $E_F=\mu$ in 2D Eq. (\[TFA\_density\]), reduces to n\_[el]{}(x,y)=D\_0\[E\_F-e(x,y)\], which is a linear relation if, $e\phi(x,y)<E_F$, satisfying our previous assumption.
Recall that, the dielectric function can be expressed in terms of TDOS and dielectric constant of the material $\kappa$ in the limit of zero temperature, namely, (k)=1+ =1+,where $a^*_B=\kappa\hbar^2/(me^2)$ is the effective Bohr radius (For GaAs $a^*_b=9.81$ nm).
Hence, we can write a relation between capacitance per area and the TDOS of a homogeneous unbounded 2DES via dielectric function as, C\_q=e\^2D\_0.
![(color online) \[fig1\] (a) The Thermodynamic density of states (TDOSs) calculated according to Eq.(\[TermoDynamic\_DensityOfState\]) considering different $g^*$ values and assuming a homogeneous 2DEG, for increasing $B$ ($=2/\nu$) (b) The total capacitances which are obtained from Eq.(\[capacitance\_geo\_qua\]), as a function of the filling factor. Strong dips occur at integer filling factors, since TDOS approaches zero. Note that for $g^*=0$ such behaviour occurs only at even integers. The inset emphasises the considerable difference between Zeeman split and non-split cases. ](Fig1){width="0.9\columnwidth"}
At this point, we would like to remind the reader the close relation between *compressibility* and *capacitance*, both being thermodynamical quantities. Recall that, capacitance is nothing but the energy required to add an additional charge (per area), which is given by the TDOS. As a well known text book result, at finite temperature, the compressibility is given by: =n\_[el]{}\^[-2]{}D\_T(). One sees that if the TDOS vanishes both the capacitance and compressibility vanishes. In other words, once TDOS becomes zero, the system under investigation becomes incompressible and capacitance reads zero. Such a situation can be obtained, if the system at hand has gap at the Fermi energy. An external $B$ field applied perpendicularly to a 2DES provides this opportunity, which we are going to discuss next.
Homogeneous 2DEG in the presence of a finite external $B$ field
---------------------------------------------------------------
Once the relation between the TDOS and capacitance is obtained it is almost straight forward to obtain the magnetic field dependent quantum capacitance per area for a homogeneous 2DES subject to homogeneous perpendicular $B$ field as, $$\label{capacitance_geo_qua}
C_q(B)=e^2D_{\rm T}(B),$$ where $D_{\rm T}(B)$ is the TDOS of a homogeneous electron system. Since, the experiments are performed at high mobility samples it is reasonable to assume that the collision broadening of the Landau levels is negligible, hence, the TDOS is given by, $$\label{TermoDynamic_DensityOfState}
D_T(B)={\frac{g_s}{2\pi\ell^2_B}}\sum_{n=0} ^{\infty} \frac{\beta}{4\cosh^2(\beta [E_n - \mu]/2)}.$$ Here, $g_s$ is a pre-factor determining the spin degeneracy, $E_n=\hbar\omega_c(n+1/2)$ is the cyclotron energy ($\omega_c=\frac{eB}{m^*}$), and $n$ indexes the spin degenerate Landau levels. The area of the available states is determined by the magnetic length $\ell_B(=\sqrt{\hbar/eB})$. In a later step we will also take into account Zeeman splitting in an effective field approach. However, we will not consider correlation effects while we are mainly interested in the integer quantised Hall effect, which is believed to be a single particle effect.
One should make it clear that, disorder is an important parameter from the experimental point of view. As mentioned above, the experiments presenting interference effects are conducted on very high quality samples, which has mean free path much much longer than the size of the quantum dot. Since we only consider the oscillations emanating from the dot, it is we think that the assumption of Dirac like DOS is acceptable, also given the fact that TDOS is also broadened due to temperature. On the other hand, long-range potential fluctuations arising from disorder are at the order of microns. Therefore, it is realistic to ignore the contribution from the disorder both for the DOS broadening and potential fluctuation point of view. It is useful to introduce a dimensionless parameter, which essentially counts the number of Landau levels below $E_F$, the filling factor =g\_s\_B\^2|[n]{}\_[el]{},where $\bar{n}_{\rm el}$ is the average electron number density. The filling factor is an integer when $E_F$ falls in between two following energy levels. The corresponding thermodynamic quantity is then defined as: (,T)=g\_s\_[n=0]{}\^f(E\_n;,T) .
Fig. \[fig1\]a, plots the magnetic field dependent TDOS as a function of inverse filling factor (i.e. $\propto B$) at low temperatures. If Zeeman splitting is neglected ($g^*=0$, solid line) one observes zeros only at even integer filling factors, whereas if one also considers Zeeman splitting with the bulk Landé $g^*(=-0.44$, broken lines) or the exchange enhanced $g^*(=-5.2$, dotted lines), one observes additional zeros also at odd filling factors. However, the difference in TDOS between the bulk and exchange enhanced is rather small and is observed only at non-integer filling factors. In Fig. \[fig1\]b, we show the corresponding total capacitance calculated from Eq. (\[eq:totalC\]), where the geometric capacitance is assumed to be $e^2D_0/23.6$ and at the default temperature $kT/E_0=1/350$ ($E_0=n_0/D_0$ and $T\sim 200-500$ mK). Following the variations at the magnetic field dependency of the TDOS, one observes strong oscillations at the total capacitance. Moreover, it is clearly seen that the quantum capacitance is the dominating term and total capacitance diverges to infinity at the integer filling factors in the $T\rightarrow 0$ limit. In this situation the compressibility becomes zero, hence 2DES becomes incompressible. Of course, at finite temperature the compressibility of the system increases exponentially. However, in the presence of disorder and with localised states at the tails of Landau levels this exponential increase is limited by the number of available localised states.
In the above discussion, we considered an unconfined (homogeneous) 2DES and calculated the total capacitance. We observed that, the quantum counter-part is the dominating term, when the filling factor is an integer. This result, in fact is expected since once all the levels below the Fermi energy are occupied the energy required to add a particle to the system becomes relatively large.
Next, we will utilise the local version of the above formalism considering a QD which is defined by electrostatics and is capacitively coupled to metallic gates. There we will also assume that the confinement potential is slowly varying at the length scale of wave width ($\propto \ell_B$), namely where TFA is still valid.
![\[fig2\] \[afifhoca\_density\_ISwidth\_capacitances\] (a) The schematic presentation of incompressible and compressible strips in the presence of a perpendicular magnetic field, considering a rotationally symmetric QD. (b) Variation of the electron distribution calculated from eq. (\[afifhoca\_density\]), regarding two different sample widths and steepness values. (c) Density distribution as a function of position considering various steepness. (d) The widths of the incompressible strips taking into account the Zeeman splitting (with the parameters extracted from experiments, $g^*=-0.44$, $t=10 a^*_B$, $\ell_d=10a^*_B$ and $a^*_B\approx10$ nm, for GaAs). The bulk electron density is set to be $n_0=2.8\times10^{11}$ cm$^{-2}$, similar to experimental values. ](Fig2){width="0.9\columnwidth"}
Toy model\[sec:toy\_model\]
===========================
In this Section we will calculate both the geometrical and quantum capacitances of a toy FPI operating under integer quantised Hall conditions, within a semi-classical Hartree type approximation. In principle, both the classical and quantum capacitances can be calculated numerically, either by direct diagonalisation or by density functional methods. However, the real QD structures utilised to measure conductance oscillations are relatively large, hence, the number of particles confined to these systems usually exceed thousands. Therefore, both of the techniques are unable to span the related Hilbert space of electrons or quasi-particles, i.e. Khom-Sham particles. Here, we employ a rather simpler method based on a Hartree-type mean field approximation, namely the Thomas-Fermi-Dirac approximation (TFDA) [@Bilgec:2010]. The assumptions of TFDA can be summarised as follows: First, the total potential $\phi(r)$, which is composed of confinement and interaction potentials, varies slowly on the quantum mechanical length scales, such as the magnetic length. Hence, the electron wave functions can be replaced by Dirac-delta functions and the corresponding energy eigenvalues are given by [@Goldman-spin:2007] $$E_{n,j}=E_n+g^*\mu_B B S_z+\phi(r_j),$$ where $S_z$ is the spin index with $\pm1/2$ and $r_j=(2m^*\hbar/eB)^{1/2}$ is the radius of the drifting cyclotron radius encircling $j$ flux quanta and $m^*=0.067~m_e$, for GaAs, where $m_e$ is the bare electron mass at rest. Second, the spin effects such as Zeeman splitting and exchange potential are taken into account within the Dirac approximation, namely the exchange potential is obtained from density functional approximation where the density is calculated within the Thomas-Fermi approximation. By this approach we can simply replace the bulk Landé $g$ factor of the material, by an exchange enhanced effective factor $g^*$ and determine the effective Zeeman gap. It is worthy to note that, our approximation does not include correlations effects. However, since we are only dealing with the integer quantised Hall effect and the correlation effects are thought to be suppressed, higher order many-body effects will be neglected.
To calculate both the geometric and quantum capacitances, one essentially needs to know the spatial and magnetic field dependency of the compressible/incompressible regions [@Chklovskii:1992]. Here, we will follow the work by Chklovskii et al, where direct the Coulomb interaction between electrons and quantising perpendicular magnetic field $B$ yield formation of compressible and incompressible strips. In their work, it is analytically shown that once the electro-statically confined 2DES is subject to strong $B$ fields, the electrostatic stability condition results in formation of co-existing electronic regions with different and peculiar screening properties. It is proposed that, starting from the edges of the system, there exists an electron depleted region due to the repulsive force generated by the side gates that define the confinement. The length of the depleted region $\ell_d$ is determined by the potential applied to the metallic gates. This depleted region is followed by a metal-like region, where the Fermi energy is pinned to the lowest Landau level with high DOS. In this region, electron density varies spatially, however, the total potential is (almost) constant due to good screening, similar to a metal, and is called the compressible region. Proceeding to inner parts of the sample, one observes a region where the Fermi energy falls in between two Landau levels, hence, due to vanishing DOS at the Fermi energy, the 2DES becomes insulator-like (locally) and this region is called incompressible. Fig. \[fig2\]a, depicts a schematic presentation of such a system together with the relevant capacitances which are denoted by $C_I$, $C_L$ and $C_{IL}$. These capacitances will be calculated considering both geometrical and quantum mechanical effects, in the following subsections.
Electron density profile, incompressible strips and capacitances
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To obtain the magnetic field and confinement potential dependent spatial distribution of the compressible (CS) and incompressible strips (IS) one has to first obtain the charge density distribution. In the original work of Chklovskii et al[@Chklovskii:1992], the confinement potential is generated by in-plane metallic gates and donors, which essentially yields a smooth electron density at the edges of the sample. However, self-consistent calculations both in 2D and 3D show that the electron density is rather steep than that of the analytical ones. The deviation is a result of the assumed unrealistic geometry (i.e. in-plane gates and charges) together with the assumption of infinite DOS of the electronic system as if it was a perfect metal ($\varepsilon(\vec k)\rightarrow \infty$). The self-consistent calculations suggest that the electron density distribution can be described by [@Ali:2013], $$\label{afifhoca_density}
n(\vec r)=n_0(1-e^{[-(\vec r-\ell_d)/t]}),$$ where $\vec r$ is the radial coordinate and $t$ determines the electron poor region in front of the gates fixing the electron density gradient which vanishes for $|r|>R$ and $n_0$ is the bulk electron density. Fig. \[fig2\]b depicts the calculated density distribution as a function of radial coordinate considering two sample sizes and two edge potential steepness. Our choice of these parameters are based on the experimental realisation of the samples: In typical experiments two different sizes are considered, meanwhile edges are defined ether by gates (smooth confinement) or by chemical etching (sharp confinement).
We observe that, for the small sample with radius $R\sim 1.2~\mu$m and steep edge $t=1~a_B^*$, the electron density reaches its bulk value rather close to the boundary. Meanwhile, for a smoother edge profile ($t=10~a_B^*$) the bulk value is reached only for a narrow radial interval. For the large sample with radius $R\sim 2.5~\mu$m, the difference between steep and smooth edge profiles is rather small, in the sense that the bulk electron density covers most of the sample and edge effects can be neglected in a first order approximation. Fig. \[fig2\]c, provides additional density profiles where the edge potential varies from relatively steep to relatively smooth profile. In a previous work, Salman et. al, reported that the samples defined by chemical etching present steep profiles at the edges, whereas, in-plane gate defined samples show a smoother profile, by 3D self-consistent calculations. However, the exact shape of the edge density profile strongly depends on the couple of parameters, such as the distance of the 2DES and the dopants from the surface, the etching depth, amount of surface charges, area of the top and length of the side gates etc. Therefore, each sample may have a different edge profile, hence, one should perform numerical simulations to obtain a realistic density distribution. In any case, as a rule of thumb we will consider small $t<3$ values to mimic etched and larger $t$ values for gate defined samples. A detailed analysis of various edge profiles can be found in Salman paper.
In the next step one has to obtain the distribution of insulator-like (incompressible) strips to calculate the geometrical capacitances as a function of both $B$ and steepness $t$.
![\[fig3\] The widths of incompressible strips having four different filling factors as a function of magnetic field, calculated from Eq. \[IS\_width\] using the self-consistent density profile, i.e Eq \[afifhoca\_density\]. The solid lines denote $g^*=-0.44$, whereas broken lines depict $g^*=5.2$. (a) The steep edged QD, $t=a^*_B$. (b) The smooth edged QD, $t=10a^*_B$. In both cases depletion length is set to be $10a^*_B$. ](Fig3){width=".84\columnwidth"}
The spatial positions and the widths of the incompressible strips can be obtained analytically utilizing the formulation provided in Ref. [@Chklovskii:1992]. The width of the $k^{\rm th}$ strip, where $k=\nu$ is an integer, is given by [@Chklovskii:1992] $$\label{IS_width}
W_{k}^2={\frac{2\kappa\Delta E}{\pi^2e^2 \frac{dn(r)}{dr }|_{r=r_k}}},$$ here $\Delta E$ is the single particle energy gap, either due to Zeeman splitting (=$g^*\mu_BB$, with $\mu_B$ being the effective Bohr magneton and $\nu$ is an odd-integer) or due to Landau splitting (=$\hbar\omega_c-g^*\mu_BB$ and $\nu$ is an even-integer) and $\kappa~(=12.4)$ is the dielectric constant of the heterostructure. Fig. \[fig2\]d shows the evolution of the strip widths as a function of external $B$ field, considering a chemically etched sample with bulk Landé $g^*$ factor. For all incompressible strips, one observes that their width decrease by decreasing $B$ field, with different rates depending on the filling fraction involved. The odd-integer strips become narrower faster than the even-integer strips, which is a consequence of the smaller energy gap of the Zeeman splitting compared to Landau splitting.
As mentioned previously, now we should check whether if the strip widths are larger than the thermodynamic length scale, $\lambda_T$, which is the Fermi wavelength at $T=0$. The horizontal broken line in Fig. \[fig2\]d depicts the Fermi wavelength, we can see that $\nu=1$ incompressible strip is larger than $\lambda_F$ for $B\gtrsim8$ T. Below this value, the strip becomes thermodynamically compressible. Let’s consider the classical case, if the separation between plates $d$ becomes too small, such a capacitor will become leaky, while it is possible to have charge transfer from one plate to the other, regardless of the insulator in between. In such a situation diffusion takes place between the two surrounding compressible regions ($\nu<1$ and $\nu>1$), in thermodynamical terms. The same argument holds also for other filling factors at different field values. Therefore we should not threat incompressible strips as an insulator, since, when $W_k.\lambda_T\lesssim 1$, the strip is no longer incompressible thermodynamically [@Siddiki:2011; @Wild:2010; @Metin:2013].
Another limit bounding the widths of the incompressible strips from below is the magnetic length, which is essentially the width of the wave function. In this case one can think of the incompressible strip as a barrier and once the wave functions of the compressible regions on each side of the barrier overlap, scattering takes place. Note that, only in the limit of $T\rightarrow0$ and without any disorder the scattering is zero, i.e. at any finite temperature and real sample with finite mobility, the scattering probability is finite due to finite TDOS at $E_F$. In Fig. \[fig2\]d, we also show the magnetic length as a function of $B$, which increases with $B$. As an example, for $\nu=1$ below 2.5 T it is possible to have tunnelling (or scattering) between the two compressible regions surrounding
From Eq. \[IS\_width\], one can clearly see that, the widths of the incompressible strips are essentially determined by the energy gap together with the strong density gradient dependency. Next, we investigate the effect of the edge steepness and the effect of effective $g^*$ on the magnetic field dependency of the incompressible strip widths. Fig. \[fig3\], plots the widths of incompressible strips as a function of $B$, assuming two $g^*$ (solid lines, -0.44 and broken lines 5.2) and considering step edge, Fig. \[fig3\]a, and smooth edge, Fig. \[fig3\]b. The first observation is, when the Zeeman splitting is large the strips stay incompressible for larger $B$ intervals, which is more pronounced for the smooth edge profile. Meanwhile, if the edge is steeper then the the possibility to observe incompressible strips at the edges are suppressed. Interestingly, to observe two incompressible strips at the edge having different filling factors is not possible for edge profiles considered here. This implies that, only the inner strip is thermodynamically incompressible and is able to decouple the surrounding compressible regions. Here, the decoupling term is used both thermodynamically and electro-dynamically.
In the next discussion we will utilise the above electrostatic picture to obtain the geometric capacitances of a QD and determine quantum capacitance dominated $B$ field intervals also considering the spatial distribution of the incompressible strips.
![ \[fig4\] Geometric capacitances of a small QD with a radius of $R=120a^*_B$ (a,b) and a large QD $R=250a^*_B$ (c,d). (a) and (c) demonstrates a steep edge, $t=a^*_B$, whereas (b), (d) depicts smooth edge, $t=10a^*_B$. In all cases solid lines denote $g^*=-0.44$ and broken lines denote $g^*=5.2$, with $\ell_d=10a^*_B$.](Fig4){width=".9\columnwidth"}
### Geometrical capacitances from electrostatics
Once the widths of the insulating strips are obtained as a function of magnetic field, it is rather straight forward to calculate the geometrical capacitances. Here, we will only refer to the capacitances for similar structures proposed in the literature, namely to ones proposed by Evans et al and Halperin et al, to compare our findings with the existing results. For the sake of simplicity, we will first consider the two lowest Zeeman split Landau level, for which a single incompressible strip exists. Including higher levels result in capacitances in series and yield relatively complicated electric circuits, can be obtained with some tedious arithmetical calculations. However, as discussed, the outer most strips become transparent to radial electric field. Since, their widths become smaller than thermodynamic and the magnetic length at lower fields, hence, these insulating strips do not contribute to total geometric capacitance.
As a first step, we calculate the capacitance between the inner compressible region ($\nu>1$) and the outer compressible region $\nu<1$ separated by the $\nu=1$ incompressible strip, depicted as $C_{IL}$ in Fig. \[fig2\]a. Assuming that the width of the incompressible strip is narrow compared to the radius of the compressible region, one can determine the charge distribution in the close neighbourhood of the incompressible strip using two parallel conducting strips. Since the distance from the surface to the 2DES is larger than the widths of the incompressible strips then capacitance is given by [@Evans:1993], $$\label{Evans_C}
C_{IL}={\frac{\kappa L_{\rm IS}}{2\pi^2}}\ln(\frac{4d}{W_1}),$$ where $L_{\rm IS}$ is the perimeter of the incompressible strip with $\nu=1$, $W_1$ is its width and $d$ is the distance between the top gate and the 2DES, taken to be 100 nm in accordance with common experimental structures. Similarly, the capacitance between the outer compressible strip with $\nu<1$ and the surrounding gate reads, $$\label{Evans_C1}
C_L={\frac{\kappa L_{\rm CS}}{2\pi^2}}\ln(\frac{4d}{\ell_d}),$$ where $L_{\rm CS}$ denotes the outer perimeter of the compressible strip and remains unchanged with changing $B$, namely $L_{\rm CS}=2\pi R$. Note that the argument of the natural logarithm is also constant in $B$, since both the distance from the top gate and the depletion length are fixed. Finally, the capacitance between the top gate and the inner compressible region is given by $$\label{Evans_C2}
C_I={\kappa \frac{A_1}{d}}.$$ Here, $A_1$ is the area of the inner metallic-like island with $\nu<1$. The total geometric capacitance of the equivalent electrical circuit shown in Fig. \[fig2\]a which is composed of capacitors described above reads, $$\label{Evans_CT}
C_{geo}=C_I+\frac{C_{IL}C_L}{(C_{IL}+C_L)},$$ and is plotted in Fig. \[fig4\] together with $C_{IL}$, $C_L$ and $C_I$ as a function of $B$ considering various sample parameters. The upper and lower panels differ in edge steepness, whereas left and right panels show different sample sizes. In all plots the bulk effective $g^*$ is shown by solid lines, meanwhile the exchange enhanced effective $g^*$ is depicted by broken lines. The first observation is that the smaller sample has an order of magnitude smaller capacitance compared to large sample. The second observation is, the total geometric capacitance as well as $C_{IL}$ and $C_L$ increases by decreasing $B$ field. The $B$ dependency of the curves is mainly affected by the edge steepness rather than sample size. Capacitances of the smooth edge sample almost linearly increase with decreasing field and the steep edge samples follow the inverse behaviour of the strip widths as a function of the field. We should remind that, in the above discussion we considered a $B$ interval in which a single incompressible strip existed with $\nu=1$. However for lower fields, it might be possible to have co-existing incompressible strips with different filling factors, depending on the edge profile. In such a situation, one should follow simply the addition of serially connected capacitors to obtain related capacitances.
In the left panel of Fig. \[fig5\] we show the total geometric capacitances as a function of $B$ field, depicted by the solid line, calculated according to the addition rule for two different $g^*$. It is observed that, $C_{geo}$ increases stepwise for filling factors greater than $\nu=1$, where the amplitude depends on $g^*$. The capacitance is constant for $\nu<1$, reflecting constant area of the compressible island at the QD. For $g^*=-0.44$, both the even and odd integer filling factors with $\nu<5$ present stepwise increase of the capacitance. For higher filling factors stepwise increase is smeared out, while the energy gap becomes comparable with the thermal energy and incompressibility vanishes. At the larger $g^*$, the behaviour is repeated, however, since the Zeeman gap is relatively larger at $\nu=4$, we observe that two incompressible strips ($\nu=4$ and $\nu=3$) contribute to total capacitance and hence the geometric capacitance is decreased at $\nu=4$. Similar observation is also valid for other even integer filling factors, however, less pronounced.
So far we have only calculated the geometrical capacitances resulting from the co-existence of compressible and incompressible strips. However, as mentioned in the Introduction, one can also define a capacitance that depends on the density of states of the system. Since, to add a particle to a system with finite DOS requires some work to be done on the system. Notably, for 2DES without an external $B$ field the DOS is constant $D_0=m^*/\pi\hbar^2$, hence, the number of particles to be added is constrained by the area of the system. However, in the presence of an external field the DOS varies as a function of the field strength yielding a field dependent capacitance. In the next subsection we will calculate this quantum capacitance and investigate its influence on charging energy, later. Fig. \[fig5\], left panel depicts the total quantum capacitances by broken lines together with the total capacitance composed of both geometric and quantum counter parts. We observe that, $C_q$ vanishes at integer filling factors for $\nu<4$ and is at least an order of magnitude smaller compared to $C_{geo}$, hence, the total capacitance is dominated by the quantum counter part. In particular, when there exists an incompressible strip both the total and the quantum capacitances vanish. In between the integer filling factors, the geometric capacitance also contributes to the total capacitance.
In the next section we will investigate the effect of the total capacitance on Aharonov-Bohm interference induced conductance oscillations following the pioneering work of Beenakker et. al.
Coulomb blockade of Aharonov-Bohm oscillations
===============================================
More than three decades ago, Coulomb blockade of the Aharonov-Bohm oscillations were predicted for a QD operating under integer quantized Hall conditions [@Beenakker:1991]. There, using energy arguments and the electrostatic stability conditions it is shown that the charging energy of the QD enhances the periodicity, when conductance is measured as a function of the $B$ field. In the original AB interference experiments, the field does not penetrate the path of the electrons, hence the $\Delta B$ periodicity is unaffected by the interactions. However, in our case the flux also exists where the interfering electrons reside. Therefore, it is expected that the $\Delta B$ period should be also effected by interactions, namely by the charging energy E\_C= e\^2/C. Given the charging energy, the energy level spacing $\Delta E$ is renormalised and can be expressed as $\Delta E^*=\Delta E+E_C$ and one can write the renormalised periodicity, $$\label{Magnetoconductance_star}
\Delta B^*=\Delta B(1+\gamma),$$ where phase shift $\gamma$ equals $=e^2/{C\Delta E}$ and unperturbed periodicity is $\Delta B=h/eA$, $A$ being the enclosed area of the QD. In Fig. \[fig5\]b and Fig. \[fig5\]d, we show the calculated phase shifts from the geometric capacitances and also taking into account the quantum capacitances. The phase shift, when only the geometric capacitance is considered is at the order of $10^{-3}$ for $g^*=5.2$, meanwhile is at most $6 \%$ for bulk $g^*$ [@Rosenow:2009]. Hence, is not able to explain shifts reported at the recent experiments, while in experiments the shifts are at the order of unity. In the next step, we also calculated $\gamma_{geo+q}$, the shift obtained by considering both the geometric and quantum capacitances, and show in the same figure. We observe that, the quantum capacitance increases $\gamma_{geo+q}$ at integer filling factors dramatically and the phase shift becomes at the order of unity.
At a first glance, one can think that taking into account quantum capacitances can explain the experimental findings. However, note that order of unity phase shifts only occur at a very limited $B$ field interval, in contrast, experimental findings report that such oscillations can be observed throughout the inter plateau interval. In addition, most of the experiments report AB oscillations at out of the plateau (to be specific, at the lower field part) intervals, where $\gamma_{geo+q}$ is still less than $10 \%$ for $g^*=5.2$. Finally, only taking into account the phase shifts can not explain the unexpected periodicity in $V_g$.
In the next Section, we will utilise the formulation developed by Evans et al to investigate the effect of quantum capacitance on the gate voltage periodicity [@Evans:1993].
![ \[fig5\] (a, b, c) Geometric, quantum and total capacitances of a small QD with a radius of $R=120a^*_B$ without (a) and with spin splitting, i.e. $g^*=-0.44$ (b) and $g^*=5.2$ (c). (d,e, f) Calculated phase corrections $\gamma$ according to Eq. \[Magnetoconductance\_star\].](Fig5){width=".94\columnwidth"}
![ \[fig6\] The capacitance dependent charging voltage $V_g(C_2)$ calculated using the density profile given in Eq. \[afifhoca\_density\] and incompressible strip width defined by Eq. \[IS\_width\]. (a) In the case when the outer conductor is charged. (b) Same as (a) while charging the inner conductor. (c) Physically observable conductance oscillations due to charging effects. ](Fig6){width=".93\columnwidth"}
![ \[fig7\] The conductance oscillations as a function of the field. Left panel (a,b,c) shows the oscillation period considering a small smooth edge ($t=a^*_B$) sample, whereas right panel depicts the same quantity for a steep edge ($t=10a^*_B$) sample, at different filling factors.](nfig5){width=".93\columnwidth"}
![ \[fig8\] The conductance oscillations as a function of the field, where quantum capacitance is also taken into account considering a small sample. At the upper panel, simulation results considering a smooth edged dot is shown, whereas lower panel depicts the same quantity for a steep edged dot. It is seen that the steepness has an observable difference between two edge profiles at both samples. This already points the fact that at small samples steepness is effective in determining oscillation periods. In accordance with experimental findings. ](nfig7){width=".89\columnwidth"}
![ \[katman\_sonuclar\_gateler\] The field dependency of the magnetic field periodicity, depending on the edge potential profile. (a) Only considering the geometric capacitances. (b) Also including quantum capacitances. ](Fig9){width=".8\columnwidth"}
CONDUCTANCE PEAK STRUCTURE DERIVED FROM THE ELECTROSTATIC MODEL \[sec:cap\_halperin\]
=====================================================================================
Here, we utilise the previously calculated capacitances of the QD shown in Fig. \[fig2\]a and obtain the periodicity of the Coulomb Blockade oscillations considering a model including only three mutual capacitances as a function of the gate potential. The capacitance $C=C_{IL}$ is between two conducting regions, and $C_{1}=C_{I}$, $C_{2}=C_{L}$ are the capacitances to the gate. The details of the model can be found in the original work by Evans et al., however, our model differs from the original work where the $B$ field dependency of the capacitances even for the geometric case is neglected. Here, we explicitly calculate the $B$ field dependency of the incompressible strip widths and also include TDOS enriched total capacitance.
Now we briefly summarise the electrostatic argumentation of the original work. At $B=0$, there is no insulating strip, hence, if the capacitances of the compressible electron island to the leads can be neglected, the electrostatic energy is given by [@Evans:1993] $$\label{Electrostatic_energy}
U_{0}(N)=\frac{(Ne)^{2}}{2C_{g}}-{NeV_{g}},$$ where, $C_{g}(=C_{1}+C_{2})$ is a constant, since the area of dot is independent of the magnetic field. Here, $N$ is the total number of the electrons in the system, and $V_{g}$ is potential difference between source and drain leads. In the presence of an external field incompressible strips form and transferring an electron from the inner compressible region to outer compressible region requires extra energy, at low temperatures and small bias voltage $V_{g}$ this energy is $$\label{Energy_different}
U(N+1)-U(N)=E_{F},$$ where $E_{F}$ is the Fermi energy and $U$ is the electrostatic energy described by,
$$\label{Polarization_charges_with_energy}
U=\frac{p^{2}}{2C_{1}}+\frac{q^{2}}{2C_{2}}+\frac{r^{2}}{2C}+\frac{s^{2}}{2C_{LD}}-V_{g}(p+q),$$
here $p,q,r$ and $s$ are the polarisation charges given as, $$\label{eq_a}
en_{1}=p+s-r,$$ $$\label{eq_b}
en_{2}=q+r,$$ $$\label{eq_c}
V_{g}=\frac{p}{C_{1}}-\frac{s}{C_{LD}}$$ and $$\label{eq_d}
V_{g}=\frac{q}{C_{2}}-\frac{r}{C}-\frac{s}{C_{LD}},$$ here, $n_{1}$ is electron number in the outer conducting area, $n_{2}$ is electron number in the inner conducting area and total electron number is $N=n_{1}+n_{2}$. Using these four equations one can obtain the polarisation charges from the Kirchhoff law, energy and charge conservation and finally express the $B$ field dependent conductance oscillations. We assume that the lead capacitances are zero, hence polarisation charge $s$ also vanishes. First, to obtain the energy of system, we determine the polarisation charges as a function of $n_{1}$ and $n_{2}$, using Eq. ([\[eq\_a\]]{}-[\[eq\_d\]]{}), $$\label{pol_p}
p=\frac{eCC_{1}(n_{1}+n_{2})+eC_{1}C_{2}n_{1}}{CC_{1}+CC_{2}+C_{1}C_{2}},$$ $$\label{pol_q}
q=\frac{eC_{2}C(n_{1}+n_{2})+eC_{1}C_{2}n_{2}}{CC_{1}+CC_{2}+C_{1}C_{2}},$$ and $$\label{pol_r}
r=\frac{eC(C_{1}n_{2}-C_{2}n_{1})}{CC_{1}+CC_{2}+C_{1}C_{2}}.$$ In the second step, we obtain the total electrostatic energy replacing the above calculated polarisation charges in Eq. (\[Polarization\_charges\_with\_energy\]),
$$\label{total_electrostatic_energy}
U=-eV_{g}N+\frac{e^{2}N^{2}}{2C_{g}}+ \frac{e^{2}(n_{1}C_{2}-n_{2}C_{1})^{2}}{2C_{g}(CC_{g}+C_{1}C_{2})}.$$
In the case of zero magnetic field, we can consider the QD as a single conducting region, corresponding to the limit $C\rightarrow\infty$ and Eq. (\[Polarization\_charges\_with\_energy\]) then reduces to Eq. (\[total\_electrostatic\_energy\]). The first two terms in Eq. (\[total\_electrostatic\_energy\]) are independent of the charge distribution between the two conductors and depend only on $N$. The third term therefore determines the charge distribution inside the dot. The equilibrium value of $n_{1}$ can be found by minimising the third term in Eq. (\[total\_electrostatic\_energy\]) with respect to variations in $n_{1}$, holding $N$ fixed, $$\label{n1_ust}
n'_{1}=\frac{C_{1}N}{C_{g}},$$ $n'_{2}$’ can also be determined similarly. Rewriting Eq. (\[Electrostatic\_energy\]) using $n'_{1}$ and $n'_{2}$ and terms we obtain the explicit form of the $V_g$ as a function of capacitances, which now also depends implicitly on $B$, $$\label{Vg_oscillations}
V_{g}=-\frac{E_{F}}{e}+\frac{(2N+1)e}{2C_{g}}+ \frac{eC^{2}_{2}}{2C_{g}(C_{g}C+C_{1}C_{2})}+\frac{eC_{2}(n'_{2}-n_{2})}{C_{g}C+C_{1}C_{2}}$$
For the sake of consistency, we will first repeat the calculation of Evans et al., however, in our calculations we will include the $B$ field dependency of the incompressible strip widths and the area of the compressible regions obtained as previously. Fig. \[fig6\] plots the capacitance dependency of the voltage (conductance) oscillations, in Fig. \[fig6\]a the case where the an electron is brought to inner conductor, whereas in Fig. \[fig6\]b an electron is placed at the outer conductor is shown. The total voltage oscillations are depicted in Fig. \[fig6\]c. For comparison we refer to Fig.4, where they constructed the voltage oscillations by assuming $C_2$ is a monotonous function of the $B$ field. We observe that, our self-consistent calculations that also take into account the widths of the incompressible strips modify their picture slightly. Namely, the conductance oscillations are no longer triangular shaped and sharp, but smoothened due to $B$ field dependency.
Next, we investigate the conductance oscillations as a function of $B$ field considering two different edge profiles for a small QD ($R=120a^*_B$), where only the geometric capacitances are taken into account. The left panel of Fig. \[fig7\] plots the conductance oscillations at different magnetic field intervals for a smooth edged QD, where we observe that the period $\Delta B$ strongly depends on $B$. For the steep edge sample (right panel, Fig. \[fig7\]), the period is an order of magnitude smaller and is also strongly $B$ dependent. These two behaviours can be understood quite easily: For the smooth edged sample the incompressible strips reside closer to the bulk of the QD and the area covered by the compressible region in the bulk is small. For the steep edged sample, the opposite behaviour is expected where incompressible strips reside closer to the edges and area is larger. As a direct consequence of this simple areal dependency the $\Delta B$ would be large for the smooth edged sample compared to steep edged. However, they will present similar $B$ field dependency. Comparing Fig. \[fig7\] with Fig. \[fig6\], we can conclude that by changing the field one only transfers charge from outer compressible strip to the inner compressible area
Now we are in a position to include quantum capacitances to our calculations. Fig. \[fig8\], presents the same quantity as shown in Fig. \[fig7\], however, here we also include the quantum capacitance to our calculations. The effect of edge steepness is not as pronounced strongly, since the quantum capacitance is now the dominating parameter. The geometric capacitance only depends on the areas of compressible and incompressible regions (or strips). In contrast, the quantum capacitance is mainly determined by the amount of available TDOS, which essentially determines the oscillation period. Remarkably, at the geometric case the slope of $V_g/B$ is always same, whereas once the quantum capacitance is taken into account this slope changes sign in proximity of bulk integer filling factors. To investigate the magnetic field periodicity $\Delta B$ as a function of $B$ we calculated the change of $\Delta B$ considering both different edge profiles and considering only geometric and both geometric and quantum capacitances, Fig.\[katman\_sonuclar\_gateler\]. For the geometric capacitance, the period $\Delta B$ decreases linearly with increasing field. Interestingly the value of periods differ as much as an order of magnitude. Meanwhile, once the quantum capacitance is taken into account the regardless of the steepness the period is at the same order of magnitude. As mentioned previously, while approaching an incompressible bulk ($B\sim 3.8$ T) both from left (low $B$) or right (high $B$) the value of $\Delta B$ decreases in a non-linear manner. In the close proximity of integer bulk the period remains approximately unchanged. However, once the very centre of the dot becomes compressible once more with a smaller filling factor, the period increases rapidly until the total area of compressible and incompressible strips equate. Clearly stating, if the TDOS (which depends on area) of incompressible and compressible areas contribute quantum capacitance equally the phase shift in $\Delta B$ saturates. For higher fields the compressible are at the centre increases and leads to a lower charging energy, decreasing the $\Delta B$ period until the incompressible strip surrounding the central compressible region becomes larger in area.
CONCLUSION
==========
The long standing debate on the explanation of the observed conduction oscillations at 2D electron systems subject to high perpendicular fields is tackled by many outstanding physicists. It is claimed within the main stream approach that the observed oscillations are due to charging effects, whereas other possible mechanisms are also provided, as mentioned in Introduction. Here, we investigated the limitations of charging effect explanation also considering a reasonable model by handling the capacitive coupling proposition via calculating both the geometric and quantum capacitances also taking into account experimental device properties. We found that, without taking into account quantum capacitance, which is determined by TDOS, and the experimental properties of the device it is barely possible to provide a realistic and comprehensive explanation of experimental findings, utilising naive single particle edge state picture.
Our approach is promising in couple of senses: First we can handle the classical electrostatic charging problem of a quantum dot subject to perpendicular magnetic fields in a self consistent manner by calculating the widths and positions of incompressible (insulating) strips. Second, we calculated quantum capacitance of the device considering the existence of compressible and incompressible regions as a function of magnetic field and sample properties, such as area and edge profile of the devices. Up to our knowledge, no similar approach has been reported at the literature. On one hand, our findings reveal many of the unexplained observations, such as the variation of the $\Delta B$ period, including the constant regime, and the effect of geometric and lithographic properties of the sample. Our results show that, the commonly accepted theory of the Fabry-Pérot interferometers may become questionable once real experimental conditions are taken into account. On the other hand, in experiments different size qdots are defined by electrostatic means, e.g. directly gate [@marcus:01] or trench gating [@Goldman:2005], hence we claim that our formulation can explain both experimental geometries and also the cross-over between them.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to express our gratitude to B. Halperin, M. Heiblum, B. Rosenow, K. von Klitzing, E. J Heller and R. R. Gerhardts both for their fruitful discussions and sincere criticism. This work is financially supported by the scientific and research council of Turkey (TÜBİTAK) under grant no: TBAG-112T264 and 211T264.
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abstract: 'The ALICE experiment has measured at mid-rapidity electrons from heavy-flavour decays in pp collisions at and 7 TeV, and in Pb-Pb collisions at . In pp collisions, electrons from charm-hadron and from beauty-hadron decays are identified by applying cuts on displaced vertices. Alternatively the relative yield of electrons from beauty decays to those from heavy-flavour decays is extracted using electron-hadron correlations. Results are compared to pQCD-based calculations. In Pb-Pb collisions, the $p_{\rm T}$ dependence of the nuclear modification factor of electrons from heavy-flavour decays is presented in two centrality classes. The status of the analysis of electrons from beauty decays in Pb-Pb collisions is reported, in view of the measurement of the corresponding nuclear modification factor.'
address: |
Physikalisches Institut, Universität Heidelberg,\
Heidelberg, 69120, Germany
author:
- MinJung Kweon for the ALICE Collaboration
title: 'Measurement of electrons from heavy-flavour decays in pp and Pb-Pb collisions with ALICE at the LHC'
---
Heavy quark ,ALICE ,nuclear modification factor ,semileptonic heavy-flavour decays
Introduction
============
In ultra-relativistic heavy-ion collisions, heavy quarks are, due to their large mass, mainly produced via initial hard parton scatterings. Therefore, heavy-flavour particles are sensitive to the full evolution of the strongly-interacting partonic medium expected to be formed in the collisions [@sqgpform]. According to perturbative QCD (pQCD), the energy-loss of a parton in the medium depends on its colour charge and on the quark mass [@phymotive]. The modifications of heavy-flavour particle momentum distributions in Pb-Pb collisions with respect to the ones in pp collisions represent a sensitive probe for the medium properties and the underlying mechanism of in medium parton energy loss. The measurement of heavy-flavour production in pp collisions provides a precision test of pQCD calculations and a crucial baseline for the interpretation of the results in Pb-Pb collisions.
The nuclear modification factor, $R_{\rm AA}$, is defined as $R_{\rm AA}(p_{\rm T})$ = $\frac{1}{<T_{\rm AA}>}
\frac{{\rm d}N_{\rm AA}/{\rm d}p_{\rm T}}{{\rm d}\sigma_{\rm pp}/{\rm d}p_{\rm T}}$, where $<T_{\rm AA}>$ is the average nuclear overlap function for a given collision centrality class, ${{\rm d}N_{\rm AA}/{\rm d}p_{\rm T}}$ and ${{\rm d}\sigma_{\rm pp}/{\rm d}p_{\rm T}}$ correspond to the $p_{\rm T}$-differential electron yield in nucleus-nucleus collisions and the $p_{\rm T}$-differential cross section in pp collisions respectively.
The ALICE experiment [@jinst] is the dedicated heavy-ion experiment at the LHC. The ALICE detector has electron identification capability down to very low $p_{\rm T}$ at mid-rapidity which allows us to study heavy-flavours via their semi-electronic decays (branching ratio $\sim$ 10%). The central barrel detectors of ALICE have a very good spatial resolution to separate secondary vertices. Thus, electrons produced from beauty decays can be separated from those originated from charm decays.
Analysis
========
Tracks were reconstructed in the central rapidity region with the Time Projection Chamber (TPC) and the Inner Tracking System (ITS). Track momentum resolution is better than 4% for and the resolution of the transverse impact parameter d$_{\rm 0}$ is better than 75 $\mu$m for $p_{\rm T}$ $>$ 1 GeV/$c$ [@hfqm2011]. For the tracks which fulfill the track quality criteria, electron selection cuts [@alicehfe] were applied using the signals in the TPC, the Time-Of-Flight detector (TOF), the Transition Radiation Detector (TRD) and the Electromagnetic Calorimeter (EMCal). TPC and TOF were used for electron selection in Pb-Pb data. For pp data, two different strategies were used. In addition to TPC, the first one employs the TOF and the TRD whereas the second one makes use of the EMCal.
Tracks compatible with the electron hypothesis within 3 $\sigma$ from the time of flight measured by the TOF were selected, thus rejecting kaons up to a momentum of $\sim$ 1.5 GeV/$c$ and protons up to $\sim$ 3 GeV/$c$. Using charge deposited in the TRD, the electron likelihood of a track was calculated. A momentum dependent cut was applied on the likelihood to have a constant electron efficiency of 80%, providing excellent separation of electrons from pions up to for pp collisions. Alternatively, the ratio $E/p$ of the energy deposited in the EMCal and the measured momentum was calculated, and tracks within 3 $\sigma$ from the $E/p$ peak were selected. Further hadron rejection was done by applying a cut on TPC specific energy deposit expressed as the distance to the expected energy deposit of electrons, normalized by the energy-loss resolution. The remaining hadron contamination, which is below 3 (10)% up to for pp (Pb-Pb) data, was calculated using a fit of the TPC d$E$/d$x$ in momentum slices and subtracted.
The inclusive electron spectrum consists of background electrons from various sources in addition to the signal (electrons from heavy-flavour decays). The main background electrons are originating from Dalitz decays of light-neutral mesons and dielectron decays of light-vector mesons, and from the conversions of decay photons in the beam pipe and the innermost layer of the ITS. They were calculated based on ALICE measured $\pi^{0}$ and $\eta$ spectrum [@alice_pieta] and $m_{\rm T}$-scaled spectra for other light-flavour mesons using PYTHIA [@pythia] decay kinematics. The background electrons from dielectron decays of heavy quarkonia were calculated based on measurements at the LHC [@alicejpsi; @cmsjpsi; @cmsupsilon], and those from real and virtual QCD photon decays were estimated based on Next-to-Leading Order (NLO) pQCD calculation [@nlophoton]. At high electron $p_{\rm T}$, contributions from hard scattering processes are important and become dominant [@alicehfe]. The $p_{\rm T}$ spectrum of electrons from heavy-flavour decays was obtained by subtracting the background cocktail from the inclusive electron $p_{\rm T}$ spectrum.
In pp collisions, the measurement of the $p_{\rm T}$-differential cross section of electrons from the beauty decays was done by applying an additional cut on the d$_{\rm 0}$ of identified electron tracks [@aliceb2e]. Since the electrons from beauty decays have larger d$_{\rm 0}$ compared to those of background electrons due to their large mean proper decay length a cut on the minimum d$_{\rm 0}$ of the electron candidate tracks enhances the signal to background ratio. The remaining background electrons were estimated based on other ALICE measurements ($\pi^{0}$ $p_{\rm T}$ spectra [@alice_pieta] and D-mesons $p_{\rm T}$ spectra [@alice_d2h]), and subtracted. After corrections for geometrical acceptance and efficiency, the electron yield from beauty decays per minimum bias collision was normalized using the cross section of minimum bias pp collisions.
Alternatively, the relative beauty contribution to the heavy-flavour electron yields was measured based on characteristic azimuthal angular correlations of electrons from heavy-flavour decays and charged hadrons (e-h). It was extracted by fitting the correlation distribution with Monte Carlo templates obtained using PYTHIA [@pythia] .
![\[fig2\] Production cross section of electrons from beauty and charm-hadron decays and their ratio in pp collisions at compared to FONLL pQCD calculations [@fonll].](Fig1.pdf){width="18.1pc"}
![\[fig2\] Production cross section of electrons from beauty and charm-hadron decays and their ratio in pp collisions at compared to FONLL pQCD calculations [@fonll].](Fig2.pdf){width="16.5pc"}
![\[fig3\]Ratio of electrons from beauty decays to those from heavy-flavour decays measured via e-h correlations, compared to FONLL calculations [@fonll].](Fig3.pdf){width="15pc"}
Results in pp collisions at $\sqrt s$ = 7 TeV and $\sqrt s$ = 2.76 TeV
======================================================================
The results were obtained from a sample of minimum bias pp collisions (2.6 ${\rm nb}^{-1}$ for heavy-flavour decay electrons and 2.1 ${\rm nb}^{-1}$ for beauty-decay electrons) recorded in 2010 at $\sqrt s$ = 7 TeV and a sample of EMCal triggered pp collisions ($6.2\times10^{5}$ events) recorded in 2011 at $\sqrt s$ = 2.76 TeV.
The $p_{\rm T}$-differential invariant cross section of electrons from heavy-flavour decays is measured for $p_{\rm T}$ above in the rapidity interval $|y|<$ 0.5 at $\sqrt s$ = 7 TeV (Figure \[fig1\]). Systematic uncertainties on the measured electron spectrum amounts to $\sim$ 10($\sim$ 20)% for $p_{\rm T}<$($>$) 3 GeV/$c$ and to $\sim$ 10($\sim$ 20)% on the electron cocktail [@alicehfe]. Figure \[fig2\] presents the invariant production cross section of electrons from beauty decays in $|y|<$ 0.8 obtained with the analysis based on the d$_{\rm 0}$ cut, and the calculated electron spectrum from charm decays at $\sqrt s$ = 7 TeV. Electrons from beauty decays take over those from charm decays and become the dominant source for $p_{\rm T}$ $>$ 4 GeV/$c$. The relative beauty contribution to the heavy-flavour electron yields was measured using e-h correlations at (Figure \[fig3\]). In all analyses, Fixed-Order plus Next-to-Leading Log (FONLL) calculations [@fonll] are in agreement with the measurements within uncertainties.
Results in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
=============================================================
The results were obtained from a sample of minimum bias Pb-Pb collisions ($6.2\times10^{5}$ events) recorded in fall 2010 at $\sqrt{s_{\rm NN}}$= 2.76 TeV. The Silicon Pixel Detector (SPD) and the two scintillator hodoscopes (V0) provide the minimum bias trigger, and events are classified according to their centrality based on percentiles of the distribution of the sum of the amplitudes in the V0 detectors [@pbpbevt]. The inclusive electron spectra and background electron cocktails, calculated based on the charged pion spectra measured by ALICE, were obtained for six centrality classes. The systematic uncertainties are dominated by particle identification ($\sim$ 35%) on .
![\[fig5\]Fit of d$_{\rm 0}$ distribution using MC templates.](Fig4.pdf){width="15.5pc"}
![\[fig5\]Fit of d$_{\rm 0}$ distribution using MC templates.](Fig5.pdf){width="16.7pc"}
The $p_{\rm T}$ dependence of the nuclear modification factor $R_{\rm AA}$ of background-subtracted electrons for the centrality ranges 0-10% and 60-80% in $|y|<$ 0.8 has been calculated with respect to pp reference spectra (Figure \[fig4\]). The reference cross section in pp collisions at $\sqrt s$ = 2.76 TeV was obtained by applying a $p_{\rm T}$-dependent scaling factor, calculated based on FONLL predictions [@fonll], to the cross section measured at A factor 1.5-4 suppression is observed for 3.5 $<p_{\rm T}<$ 6 GeV/$c$, where heavy-flavour decays are dominant. It suggests us a strong energy-loss of heavy quarks in the medium produced in central Pb-Pb collisions.
The measurement of electrons from beauty decays is under study by fitting the measured d$_{\rm 0}$ distribution with Monte Carlo templates from individual sources (top of Figure \[fig5\]). Differences between the fit and the data are consistent with statistical variations (bottom of Figure \[fig5\]). Higher statistics are being analyzed for both peripheral and central collisions. An analysis approach similar to that used for pp data is also being pursued with the Pb-Pb data.
Summary
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The $p_{\rm T}$-differential cross section of electrons from heavy-flavour decays, , were measured in pp collisions at $\sqrt{s}$ = 2.76 and 7 TeV. pQCD-based calculations are in agreement with measurements within uncertainties. The electron spectra subtracted by the known background electrons were measured in collisions at . The nuclear modification factor implies strong energy-loss of heavy quarks in central Pb-Pb collisions. The measurement of electrons from beauty decays in Pb-Pb collisions is under way.
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K. Adcox et al. \[PHENIX collaboration\] Nucl. Phys. **A757** 184 (2005). N. Armesto et al. Phys. Rev. **D71** 054027 (2005). K. Aamodt et al. \[ALICE Collaboration\] J. Instrum. **3** S08002 (2008). A. Dainese (for the ALICE Collaboration). J. Phys. G: Nucl. Part. Phys. **38** 124032 (2011). B. Abelev et al. \[ALICE Collaboration\] Phys. Rev. **D**, submitted. arXiv:1205.5423. B. Abelev et al. \[ALICE Collaboration\] arXiv:1205.5724. T. Sjöstrand, S. Mrenna, P. Skands, J. High Energy Phys. 05026 (2006). K. Aamodt et al. \[ALICE Collaboration\] Phys. Lett. **B704** 442 (2011). V. Khachatryan et al. \[CMS Collaboration\] Eur. Phys. J. **C71** 1575 (2011). V. Khachatryan et al. \[CMS Collaboration\] Phys. Rev. **D83** 112004 (2011). W. Vogelsang, private communication. B. Abelev et al. \[ALICE Collaboration\] Phys. Rev. Lett., submitted. arXiv:1208.1902. K. Aamodt et al. \[ALICE Collaboration\] JHEP **01** 128 (2012). M. Cacciari, M. Greco, P. Nason, JHEP **9805** 007 (1998); private communication. K. Aamodt et al. \[ALICE Collaboration\] Phys. Rev. Lett. **106** 032301 (2011). Y. Pachmayer (for the ALICE Collaboration) J. Phys. G: Nucl. Part. Phys. **38** 124186 (2011).
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abstract: 'We describe simple tricks to compute the Manin black products with the operads $\mathcal{A}ss$, $\mathcal{C}om$ and $pre\mathcal{L}ie$.'
author:
- Ruggero Bandiera
title: A trick to compute certain Manin products of operads
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{}
\[theorem\][Definition]{} \[theorem\][Example]{}
\[theorem\][Remark]{}
Introduction {#introduction .unnumbered}
============
The aim of this paper is to describe a simple method to compute the Manin black products with the operads ${\mathcal{A}}ss$, ${\mathcal{C}}om$ and $pre{\mathcal{L}}ie$. For instance, this allows the computation of ${\mathcal{A}}ss\bullet{\mathcal{C}}om$ and $pre{\mathcal{L}}ie\bullet pre{\mathcal{L}}ie$ (Examples \[ex:ass\] and \[ex:prelie\]), answering some questions posed by Loday [@L] (these had beeen answered already [@bai2; @GK]). While simple methods to compute $pre{\mathcal{L}}ie\bullet-$ were already known [@bai; @bai3], to our knowledge the results are new for the functors ${\mathcal{A}}ss\bullet-$, ${\mathcal{C}}om\bullet-$. We illustrate the method by several explicit computations: among these, we show that in the square diagram of operads introduced by Chapton [@cha] $$\xymatrix{ pre{\mathcal{L}}ie\ar[r]&{\mathcal{D}}end\ar[r]& {\mathcal{Z}}inb\\ {\mathcal{L}}ie\ar[r]\ar[u]&{\mathcal{A}}ss\ar[r]\ar[u]&{\mathcal{C}}om\ar[u]\\{\mathcal{L}}eib\ar[r]\ar[u]&di{\mathcal{A}}ss\ar[r]\ar[u]&{\mathcal{P}}erm\ar[u] }$$ where it is well known that the top row is the Manin black product of the middle one with $pre{\mathcal{L}}ie$ and the bottom row is the Manin white product of the middle one with ${\mathcal{P}}erm$, it is also true that the top row is the Manin *white* product of the middle one with ${\mathcal{Z}}inb$ and the bottom row is the Manin *black* product of the middle one with ${\mathcal{L}}eib$.
We explain the method by considering the case of $({\mathcal{A}}ss,\cup)$, where we denote by $\cup$ the generating associative product. It is convenient to consider first the (Koszul) dual computation of the Manin white product ${\mathcal{A}}ss\circ-$. Roughly, given an operad ${\mathcal{O}}$, which will be always an operad in vector spaces over a field $\mathbb{K}$ and moreover binary, quadratic and finitely generated by non-symmetric operations $\cdot_i$, $i=1,\ldots,p$, symmetric operations $\bullet_j$, $j=1,\ldots,q$, and anti-symmetric operations $[-,-]_k$, $k=1,\ldots,r$, the operad ${\mathcal{A}}ss\circ{\mathcal{O}}$ is generated by the tensor product operations $\cup\otimes\cdot_i$, $\cup\otimes\cdot_i^{op}$, $\cup\otimes\bullet_j$, $\cup\otimes[-,-]_k$ (all of which are non-symmetric, and where $\cdot_i^{op}$ are the opposite products) together with the relations holding in the tensor product $A{\otimes}V$ of a generic ${\mathcal{A}}ss$-algebra $(A,\cup)$ and a generic ${\mathcal{O}}$-algebra $(V,\cdot_i,\bullet_j,\ast_k)$. We define a functor ${\operatorname}{Ass}_\circ(-):\mathbf{Op}\to\mathbf{Op}$ by replacing the generic ${\mathcal{A}}ss$-algebra $(A,\cup)$ in the above definition of ${\mathcal{A}}ss\circ-$ with the dg associative algebra $(C^*(\Delta_1;\mathbb{K}),\cup)$ of non-degenerate cochains on the $1$-simplex with the usual cup product, and show that ${\operatorname}{Ass}_\circ({\mathcal{O}})={\mathcal{A}}ss\circ{\mathcal{O}}$, essentially because the only relations satisfied in $(C^*(\Delta_1;\mathbb{K}),\cup)$ are the associativity relations.
We notice that besides the relation of ${\mathcal{A}}ss\circ{\mathcal{O}}$-algebra, the tensor product operations (which we will simply call the cup products in the body of the paper) on $C^*(\Delta_1;V)=C^*(\Delta_1;\mathbb{K})\otimes V$ satisfy the Leibniz identity with respect to the differential: moreover, for any $X\subset\Delta_1$ (that is, the boundary or one of the vertices) the subcomplex of relative cochains $C^*(\Delta_1,X;V)\subset C^*(\Delta_1;V)$ is a dg ${\mathcal{A}}ss\circ{\mathcal{O}}$-ideal. We reassume the above properties by saying that $C^*(\Delta_1;V)$ with the tensor product operations (cup products) is a local dg ${\mathcal{A}}ss\circ{\mathcal{O}}$-algebra.
Conversely, our trick to compute ${\mathcal{A}}ss\bullet{\mathcal{O}}$ consists in imposing a local dg ${\mathcal{O}}$-algebra structure $(C^*(\Delta_1;V),\cdot_i,\bullet_j,[-,-]_k)$ on the complex $C^*(\Delta_1;V)$. Notice that the space $C^*(\Delta_1;V)$ splits into the direct sum of three copies of $V$, we write $C^*(\Delta_1;V)=V_0\oplus V_1\oplus V_{01}$, where $V_0$ (resp.: $V_1$) is the copy corresponding to the left (resp.: right) vertex and $V_{01}$ is the copy corresponding to the $1$-dimensional cell. The locality assumption and the Leibniz relation with respect to the differential imply that the whole ${\mathcal{O}}$-algebra structure on $C^*(\Delta_1;V)$ is determined by the products $\cdot_i:V_0\otimes V_{01}\to V_{01}$, $\cdot_i:V_{01}\otimes V_0\to V_{01}$, $\bullet_j:V_0\otimes V_{01}\to V_{01}$, $[-,-]_k:V_0\otimes V_{01}\to V_{01}$, that is, by the datum of $2p+q+r$ non-symmetric operations $\prec_i,\succ_i,\circ_j,\ast_k$ on $V$: then the relations of ${\mathcal{O}}$-algebra on $C^*(\Delta_1;V)$ are equivalent to certain relations on the products $\prec_i,\succ_i,\circ_j,\ast_k$, and this defines a functor ${\operatorname}{Ass}_\bullet(-):\mathbf{Op}\to\mathbf{Op}$. We remark that the computation of this functor is completely mechanical. Finally, this is exactly the same as the functor ${\mathcal{A}}ss\bullet-$: this will be proved at the very end of the paper, by showing that the functors $\xymatrix{{\operatorname}{Ass}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Ass}_\circ(-)\ar@<2pt>[l]}$ form an adjoint pair (notice how the counit is obvious: given an ${\mathcal{O}}$-algebra structure on $V$, there is a local dg ${\operatorname}{Ass}_\circ({\mathcal{O}})$-algebra structure on $C^*(\Delta_1;V)$ via the tensor product operations, and by defninition this is the same as an ${\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))$-algebra structure on $V$).
The trick to compute ${\mathcal{C}}om\bullet-$ and $pre{\mathcal{L}}ie\bullet-$ (and dually ${\mathcal{L}}ie\circ-$ and ${\mathcal{P}}erm\circ-$) is similar: we replace the dg associative algebra $(C^*(\Delta_1;\mathbb{K}),\cup)$ in the above discussion with the dg Lie algebra $(C^*(\Delta_1;\mathbb{K}),[-,-]=\cup-\cup^{op})$ in the first case, and with the dg (right) permutative algebra $(C^*(\Delta_1,v_l;\mathbb{K}),\cup)$ (where $v_l\subset\Delta_1$ is the left vertex) in the second case.
The author is grateful to Domenico Fiorenza for some useful discussions.
Preliminary remarks. {#preliminary-remarks. .unnumbered}
--------------------
The author is not an actual expert on operads (and is afraid this might be painfully evident throughout the reading to anyone who is), accordingly, we will think of operads näively, as defined by the corresponding type of algebras. Moreover, although the constructions seem to make sense in more general settings (for instance, dg operads), we limit ourselves to work in the category $\mathbf{Op}$ of finitely generated binary quadratic symmetric operads on vector spaces over a field $\mathbb{K}$.
Given a vector space $V$, we denote by $V^{{\otimes}n}$, $V^{\odot n}$, $V^{\wedge n}$, $n\geq0$, the tensor powers, symmetric powers and exterior powers of $V$ respectively. We shall always denote by $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$ an operad in $\mathbf{Op}$ generated by non-symmetric products $\cdot_i$, $i=1,\ldots,p$, commutative products $\bullet_j$, $j=1,\ldots,q$, and anti-commutative brackets $[-,-]_k$, $k=1,\ldots,r$: the relations shall be omitted from the notation. For the definitions of the Koszul duality functor $-^!:\mathbf{Op}\to\mathbf{Op}$ and the Manin white and black products $-\circ-,-\bullet-:\mathbf{Op}\times\mathbf{Op}\to\mathbf{Op}$ we refer to [@V; @LV]. For the definitions of the various operads we will consider, where not already specified, we refer to [@LV; @zinb].
We denote by $\Delta_1$ the standard $1$-simplex, which shall be represented as an arrow $\to$, by $v_i\subset\Delta_1$, $i=l,r$, the left and right vertex respectively, and by ${\partial}\Delta_1=v_l\sqcup v_r\subset\Delta_1$ the boundary. Given a vector space $V$, we shall denote by $C^\ast(\Delta_1;V)$ (resp.: $C^*(\Delta_1{\partial}\Delta_1;V)$, $C^*(\Delta_1,v_i;V)$, $i=l,r$) the usual complex of non-degenerate (resp.: relative) cochains with coefficients in $V$. We shall depict a $0$-cochain in $C^*(\Delta_1;V)$ as $_x\to_y$, $x,y\in V$ (moreover, we write $_x\to$ for $_x\to_0$ and $\to_y$ for $_0\to_y$), and a $1$-cochain as $\xrightarrow{x}$, similarly in the relative cases. The differential $d:C^0(\Delta_1;V)\to C^1(\Delta_1;V)$ is the usual one $d(_x\to_y)=\xrightarrow{y-x}$.
A trick to compute Manin black products
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In this section we shall introduce, and compute in several cases, endofunctors $\mathbf{Op}\to\mathbf{Op}$ which we denote by ${\operatorname}{Ass}_\bullet(-)$, ${\operatorname}{Com}_\bullet(-)$, ${\operatorname}{preLie}_\bullet(-)$, later we shall prove that these coincide with the functors ${\mathcal{A}}ss\bullet-$, ${\mathcal{C}}om\bullet-$, $pre{\mathcal{L}}ie\bullet-$ respectively.
\[def:ManinBlack\] Given an operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$ in $\mathbf{Op}$ and a vector space $V$, a structure of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$-algebra on $V$ is the datum of operations $$\begin{gathered}
\cdot'_i:C^\ast(\Delta_1;V)^{{\otimes}2}\to C^*(\Delta_1;V),\quad\bullet'_j:C^\ast(\Delta_1;V)^{\odot2}\to C^*(\Delta_1;V)\\\mbox{and}\quad[-,-]'_k:C^\ast(\Delta_1;V)^{\wedge2}\to C^*(\Delta_1;V)\quad\mbox{such that}\end{gathered}$$
1. \[item:dg\] $(C^*(\Delta_1;V),d,\cdot'_i,\bullet'_j,[-,-]'_k)$ is a dg ${\mathcal{O}}$-algebra structure on $C^*(\Delta_1;V)$, and moreover the following locality assumption holds:\
2. \[item:locality\] for all closed subsets $X\subset\Delta_1$ (i.e., $X=v_l,v_r,{\partial}\Delta_1$, cf. the preliminary remarks) the complex of relative cochains $C^*(\Delta_1,X;V)\subset C^*(\Delta_1;V)$ is a dg ${\mathcal{O}}$-ideal.
The functor ${\operatorname}{preLie}_\bullet(-)$ is defined similarly.
A ${\operatorname}{preLie}_\bullet({\mathcal{O}})$-algebra structure on $V$ is the datum of a dg ${\mathcal{O}}$-algebra structure $(C^*(\Delta_1,v_l;V),d,\cdot'_i,\bullet'_j,[-,-]'_k)$ on $C^*(\Delta_1,v_l;V)$ (we notice that in this case the locality assumption, that is, the fact that $C^*(\Delta_1,{\partial}\Delta_1;V)\subset C^*(\Delta_1,v_l;V)$ is a dg ${\mathcal{O}}$-ideal, is satisfied for trivial degree reasons).
\[rem:prelie\] This actually defines ${\operatorname}{preLie}_\bullet(-)={\operatorname}{preLie}_{r,\bullet}(-)$, by replacing the vertex $v_l$ with the one $v_r$ in the previous definition we get a second endofunctor ${\operatorname}{preLie}_{l,\bullet}(-)$ : more on this in Remark \[rem:leftcase\].
It is not immediately obvious that this defines endofunctors $\mathbf{Op}\to\mathbf{Op}$: we consider first the case of ${\operatorname}{Ass}_\bullet(-)$. Given the datum $(C^\ast(\Delta_1;V),d,\cdot_i',\bullet_j',[-,-]_k')$ of a local dg ${\mathcal{O}}$-algebra structure on $(C^\ast(\Delta_1;V),d)$, we notice that the locality assumption implies $_x\to\cdot_i'\to_y\:=\:\to_x\cdot_i'\,\,_y\!\to=0$, thus, applying the differential $d$ and Leibniz rule we see that $_x\to\cdot_i'\xrightarrow{y}\:=\:\xrightarrow{x}\cdot_i'\to_y$, and similarly $\to_x\cdot_i'\xrightarrow{y}\:=\:\xrightarrow{x}\cdot_i'\,\,_y\to$. We define non-symmetric products $\prec_i,\succ_i:V^{{\otimes}2}\to V$ on $V$ by $$_x\to\cdot_i'\xrightarrow{y}\,\,=:\,\,\xrightarrow{x\prec_iy}\,\,:=\,\,\xrightarrow{x}\cdot_i'\to_y,\qquad\to_x\cdot_i'\xrightarrow{y}\,\,=:\,\,-\xrightarrow{y\succ_i x}\,\,:=\,\,\xrightarrow{x}\cdot_i'\,\,_y\to.$$ Always by locality and Leibniz rule we have a product $\cdot_i:V^{{\otimes}2}\to V$ defined equivalently by $\to_x\cdot_i'\to_y=\to_{x\cdot_iy}$ or $_x\to\cdot_i'\,_y\to=_{x\cdot_i y}\to$, and moreover $x\cdot_i y= x\prec_i y - y\succ_i x$. In fact, $$\xrightarrow{x\cdot_iy}=d\left(\to_x\cdot_i'\to_y\right)=\xrightarrow{x}\cdot_i'\to_y+\to_x\cdot_i'\xrightarrow{y}=\xrightarrow{x\prec_iy-y\succ_ix}=-d\left(_x\to\cdot_i'\,_y\to\right).$$ In the same way, there are non-symmetric products $\circ_j,\ast_k:V^{{\otimes}2}\to V$ on $V$, defined by the formulas $$\begin{gathered}
_x\to\bullet_j'\xrightarrow{y}\,\,=\,\,\xrightarrow{x}\bullet_j'\to_y\,\,=\,\,\xrightarrow{y}\bullet_j'\,\,_x\to\,\,=\,\,\to_y\bullet_j'\xrightarrow{x}\,\,=:\,\,\xrightarrow{x\circ_j y},\\ \left[ _x\to,\xrightarrow{y}\right]_k'\,\,=\,\, \left[ \xrightarrow{x},\to_y\right]_k' \,\,=\,\, -\left[\xrightarrow{y},\,_x\to\right]_k' \,\,=\,\, -\left[ \to_y,\xrightarrow{x}\right]_k'\,\,=:\,\, \xrightarrow{x\ast_k y},\end{gathered}$$ products $\bullet_j:V^{\odot 2}\to V$, $[-,-]_k:V^{\wedge2}\to V$ defined by $$\to_x\bullet_j'\to_y=\to_{x\bullet_j y},\qquad_x\to\bullet_j'\,_y\to=_{x\bullet_jy}\to,\qquad \left[ \to_x,\to_y\right]_k'=\to_{[x,y]_k},\qquad\left[ _x\to,\,_y\to \right]_k'=_{[x,y]_k}\to,$$ and moreover by Leibniz rule $$x\bullet_j y = x\circ_j y + y \circ_j x, \qquad [x,y]_k=x\ast_k y - y\ast_k x.$$
*Warning:* the previous formulas will be used extensively troughout the computations of this section without further mention.
In other words, the datum of an ${\operatorname}{Ass}_\bullet({\mathcal{O}})$-algebra structure on $V$ is the same as the one of operations $\prec_i,\succ_i,\circ_j,\ast_k:V^{{\otimes}2}\to V$ on $V$, inducing operations $\cdot_i',\bullet_j',[-,-]_k'$ on $C^\ast(\Delta_1;V)$ via the previous formulas: the requirement that these make the latter into a (by construction, local dg) ${\mathcal{O}}$-algebra translates into a finite set of terniary relations on the operations $\prec_i,\succ_i,\circ_j,\ast_k$. More precisely, for every relation $R(x,y,z)=0$ satisfied in an ${\mathcal{O}}$-algebra, we get six (in general not independent) relations in the operad ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ as in the next lemma.
\[lem:rel\] Given operations $\prec_i,\succ_i,\circ_j,\ast_k$ on $V$ inducing operations $\cdot_i',\bullet_j',[-,-]_k'$ on $C^\ast(\Delta_1;V)$ as above, then the latter is a local dg ${\mathcal{O}}$-algebra if and only if for every relation $R(x,y,z)=0$ in the operad ${\mathcal{O}}$ the six relations $$\begin{gathered}
0=R(\xrightarrow{x},\to_y,\,_z\to)=R(\xrightarrow{x},\,_y\to,\to_z)=R(\to_x,\xrightarrow{y},\,_z\to)=\\=R(_x\to,\xrightarrow{y},\to_z)=R(\to_x,\,_y\to,\xrightarrow{z})=R(_x\to,\to_y,\xrightarrow{z})=0,\end{gathered}$$ are satisfied.
The only if part is clear. For the if part we have to show that the given relations imply all the others. First of all, we notice that the only relations not trivially satisfied are those in total degree 0 or 1. Moreover, in total degree 0 all necessary relations follow by the locality assumption but the ones $R(_x\to,\,_y\to,\,_z\to)=0=R(\to_x,\to_y,\to_z)$. In total degree 1, all the necessary relations are satisfied by hypothesis but the ones $$\begin{gathered}
0=R(_x\to,\,_y\to,\xrightarrow{z})=R(\to_x,\to_y,\xrightarrow{z})=R(_x\to,\xrightarrow{y},\,_z\to)=\\=R(\to_x,\xrightarrow{y},\to_z)=R(\xrightarrow{x},\,_y\to,\,_z\to)=R(\xrightarrow{x},\to_y,\to_z)=0.\end{gathered}$$ For instance, to prove the first one we apply $d$ to the relation $0=R(_x\to,\,_y\to,\to_z)$ (which follows by locality), and by Leibniz rule and the hypothesis of the lemma $$0 = -R(\xrightarrow{x},\,_y\to,\to_z)-R(_x\to,\xrightarrow{y},\to_z)+R(_x\to,\,_y\to,\xrightarrow{z})=R(_x\to,\,_y\to,\xrightarrow{z}).$$ The others are proved similarly. Finally, if we denote by $R(x,y,z)$ the relation $R$ computed in the operations $\cdot_i$, $\bullet_j$, $[-,-]_k$ on $V$, we have $R(\to_x,\to_y,\to_z)=\to_{R(x,y,z)}$, and by Leibniz rule $$\xrightarrow{R(x,y,z)}=d\left(\to_{R(x,y,z)}\right)=R(\xrightarrow{x},\to_y,\to_z)+R(\to_x,\xrightarrow{y},\to_z)+R(\to_x,\to_y,\xrightarrow{z})=0,$$ hence $R(\to_x,\to_y,\to_z)=0$, and $R(_x\to,\,_y\to,\,_z\to)=0$ is proved similarly.
\[ex:ass\] We consider the operad ${\mathcal{L}}ie$ of Lie algebras. By the previous argument, an ${\operatorname}{Ass}_\bullet({\mathcal{L}}ie)$-algebra structure on $V$ is the datum of $\ast:V^{{\otimes}2}\to V$ such that $$\begin{gathered}
\left[\left[_x\to,\to_y\right],\xrightarrow{z}\right]=0=\left[ _x\to,\left[\to_y,\xrightarrow{z}\right]\right]+\left[\left[_x\to,\xrightarrow{z}\right],\to_y\right]=\\=\left[_x\to,\xrightarrow{-z\ast y}\right]+\left[\xrightarrow{x\ast z},\to_y\right]=\xrightarrow{-x\ast(z\ast y)+(x\ast z)\ast y},\qquad\forall x,y,z\in V.\end{gathered}$$ In other words, $\ast$ is an associative product on $V$. The other five relations to be checked in the previous lemma follow from this one by symmetry of the Jacobi identity, thus, we see that ${\operatorname}{Ass}_\bullet({\mathcal{L}}ie)={\mathcal{A}}ss$, the operad of associative algebras.
We may repeat the previous considerations in the case of ${\operatorname}{preLie}_\bullet(-)$, showing that a dg ${\mathcal{O}}$-algebra structure $(C^\ast(\Delta_1,v_l;V),d,\cdot_i'.\bullet_j',[-,-]_k')$ on the complex $C^\ast(\Delta_1,v_l;V)$ is determined, via the previous formulas, by operations $\prec_i,\succ_i,\circ_j,\ast_k:V^{{\otimes}2}\to V$ on $V$. The analog of Lemma \[lem:rel\] is the following
\[lem:relpl\]○ Given operations $\prec_i,\succ_i,\circ_j,\ast_k$ on $V$ inducing operations $\cdot_i',\bullet_j',[-,-]_k'$ on $C^\ast(\Delta_1,v_l;V)$ as above, then the latter is a local dg ${\mathcal{O}}$-algebra if and only if for every relation $R(x,y,z)=0$ in the operad ${\mathcal{O}}$ the three relations $$R(\xrightarrow{x},\to_y,\to_z)=R(\to_x,\xrightarrow{y},\to_z)=R(\to_x,\to_y,\xrightarrow{z})=0,$$ are satisfied.
It follows from the proof of Lemma \[lem:rel\].
\[ex:prelie\] A ${\operatorname}{preLie}_\bullet({\mathcal{L}}ie)$-algebra structure on $V$ is the datum of $\ast:V^{{\otimes}2}\to V$ such that, where as usual $[x,y]=x\ast y - y \ast x$, $$\begin{gathered}
\left[\xrightarrow{x},\left[\to_y,\to_z\right]\right]=\left[\xrightarrow{x},\to_{[y,z]} \right]=\xrightarrow{x\ast[y,z]}=\\=\left[ \left[\xrightarrow{x},\to_y\right],\to_z\right]+\left[\to_y,\left[\xrightarrow{x},\to_z\right]\right]=\left[\xrightarrow{x\ast y},\to_z\right]+\left[\to_y,\xrightarrow{x\ast z}\right]=\xrightarrow{(x\ast y)\ast z- (x\ast z)\ast y},\qquad\forall x,y,z\in V.\end{gathered}$$ In other words, $\ast$ is a right pre-Lie product on $V$. Since the remaining two relations to be checked in the previous lemma follow from this one by symmetry of the Jacobi identity, we see that ${\operatorname}{preLie}_\bullet({\mathcal{L}}ie)=pre{\mathcal{L}}ie$, the operad of (right) pre-Lie algebras.
\[rem:leftcase\] As was said in Remark \[rem:prelie\] the above construction is actually the one of ${\operatorname}{preLie}_\bullet({\mathcal{L}}ie)={\operatorname}{preLie}_{r,\bullet}({\mathcal{L}}ie)$: applying the functor ${\operatorname}{preLie}_{l,\bullet}(-)$ introduced there we find by similar computations as in the above example that ${\operatorname}{preLie}_{l,\bullet}({\mathcal{L}}ie)$ is the operad of *left* pre-Lie algebras. More in general, it is easy to see that ${\operatorname}{preLie}_{l,\bullet}({\mathcal{O}})$ is always the opposite of the operad ${\operatorname}{preLie}_{r,\bullet}({\mathcal{O}})={\operatorname}{preLie}_{\bullet}({\mathcal{O}})$ for any ${\mathcal{O}}\in\mathbf{Op}$. We won’t insist further on this point, and restrict our computations to the right case.
By the proof of Lemma \[lem:rel\] we have morphisms of operads ${\mathcal{O}}\to {\operatorname}{Ass}_\bullet({\mathcal{O}})$, ${\mathcal{O}}\to {\operatorname}{preLie}_\bullet({\mathcal{O}})$, sending an ${\operatorname}{Ass}_\bullet({\mathcal{O}})$-algebra structure $(V,\prec_i,\succ_i,\circ_j,\ast_k)$ on $V$ to the associated ${\mathcal{O}}$-algebra structure $$(V\:\:,\:\:x\cdot_iy=x\prec_i y-y\succ_ix\:\:,\:\:x\bullet_j y=x\circ_j y+y\circ_j x\:\:,\:\:[x,y]_k=x\ast_k y-y\ast_k x),$$ and similarly in the other case.
A ${\operatorname}{Com}_\bullet({\mathcal{O}})$-algebra is an ${\operatorname}{Ass}_\bullet({\mathcal{O}})$-algebra with trivial associated ${\mathcal{O}}$-algebra. In other words, the operad ${\operatorname}{Com}_\bullet({\mathcal{O}})$ is generated by non-symmetric products $\star_i$, anti-commutative brackets $\{-,-\}_j$ and commutative products $\circledast_k$, together with the relations obtained from the ones of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$-algebra by further imposing the identities $$x\prec_i y= y\succ_i x=: x\star_i y,\qquad x\circ_j y=-y\circ_j x =:\{x,y\}_j,\qquad x\ast_k y= y\ast_k x=: x\circledast_k y.$$
We see immediately by Example \[ex:ass\] that ${\operatorname}{Com}_\bullet({\mathcal{L}}ie)={\mathcal{C}}om$, the operad of commutative and associative algebras.
\[th:black\] For any operad ${\mathcal{O}}$ in $\mathbf{Op}$, we have natural isomorphisms ${\operatorname}{Ass}_\bullet({\mathcal{O}})={\mathcal{A}}ss\bullet{\mathcal{O}}$, ${\operatorname}{Com}_\bullet({\mathcal{O}})={\mathcal{C}}om\bullet{\mathcal{O}}$, ${\operatorname}{preLie}_\bullet({\mathcal{O}})=pre{\mathcal{L}}ie\bullet{\mathcal{O}}$.
The proof is postponed to the next section (it follows from theorems \[th:asswhite\], \[th:permwhite\], \[th:liewhite\] and \[th:adj\]). In the remaining of this section we shall illustrate the result by several explicit computations.
\[rem:curlyeqprec\] In some of the following computations it is conventient to choose a different basis for the operations of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ and ${\operatorname}{preLie}_\bullet({\mathcal{O}})$, by replacing the products $\succ_i$ with the opposite products $\curlyeqsucc_i:=-\succ^{op}_i$: with the previous notations $\to_x\cdot'_i\xrightarrow{y}=:\xrightarrow{x\curlyeqsucc_i y}:=\xrightarrow{x}\cdot'_i\,_y\to$.
Given non-negative integers $p,q,r$, we denote by ${\mathcal{M}}ag_{p,q,r}$ the operad generated by $p$ magmatic non-symmetric operations, $q$ magmatic commutative operations and $r$ magmatic anti-commutative operations with no relations among them. The previous arguments show that ${\operatorname}{Ass}_\bullet({\mathcal{M}}ag_{p,q,r} )={\operatorname}{preLie}_\bullet ({\mathcal{M}}ag_{p,q,r})={\mathcal{M}}ag_{2p+q+r,0,0}$, whereas ${\operatorname}{Com}_\bullet({\mathcal{M}}ag_{p,q,r})={\mathcal{M}}ag_{p,r,q}$.
We introduce a further notation.
\[def:+x\] Given operads $\mathcal{O},\mathcal{P}\in\mathbf{Op}$, we denote by $\mathcal{O}+\mathcal{P}$ the operad defined by saying that an $(\mathcal{O}+\mathcal{P})$-algebra structure on $V$ is the datum of both an ${\mathcal{O}}$-algebra structure and a $\mathcal{P}$-algebra structure on $V$ with no relations among them, while we denote by ${\mathcal{O}}\times\mathcal{P}$ the operad defined in the same way and by further imposing that every triple product involving an operation from ${\mathcal{O}}$ and one from $\mathcal{P}$ vanishes. It is easy to see that the functors $-+-,-\times-:\mathbf{Op}\to\mathbf{Op}$ are Koszul dual in the sense that $({\mathcal{O}}+{\mathcal{P}})^!={\mathcal{O}}^!\times{\mathcal{P}}^!$ for any pair of operads ${\mathcal{O}},{\mathcal{P}}\in\mathbf{Op}$.
We consider the operad $({\mathcal{A}}ss,\cdot)$ of associative algebras. Given an ${\operatorname}{Ass}_\bullet({\mathcal{A}}ss)$-algebra structure $(V,\prec,\succ)$ on a vector space, we shall denote the associator of the associated product $\cdot'$ on $C^\ast(\Delta_1;V)$ by $A_{\cdot'}(-,-,-)$. Straightforward computations show that $$0=A_{\cdot'}(_x\to,\to_y,\xrightarrow{z})\:=\: \left(_x\to\cdot'\to_y\right)\cdot'\xrightarrow{z}-\,_x\to\cdot'(\to_y\cdot'\xrightarrow{z}) \:=\: 0+\,_x\to\cdot'\xrightarrow{z\succ y}\:=\:\xrightarrow{x\prec(z\succ y)},$$ $$\begin{aligned}
\nonumber 0=A_{\cdot'}\left(\xrightarrow{}_x,{}_y\xrightarrow{},\xrightarrow{z} \right) &=& \xrightarrow{(y\prec z)\succ x}, \\
\nonumber 0=A_{\cdot'}\left({}_x\xrightarrow{},\xrightarrow{y},\xrightarrow{}_z \right) &=& \xrightarrow{(x\prec y)\prec z-x\prec(y\prec z)}, \\
\nonumber 0=A_{\cdot'}\left(\xrightarrow{}_x,\xrightarrow{y},{}_z\xrightarrow{} \right) &=& \xrightarrow{z\succ(y\succ x)-(z\succ y)\succ x}, \\
\nonumber 0=A_{\cdot'}\left(\xrightarrow{x},{}_y\xrightarrow{},\xrightarrow{}_z \right) &=& \xrightarrow{-(y\succ x)\prec z},\\ \nonumber 0=A_{\cdot'}\left(\xrightarrow{x},\xrightarrow{}_y,{}_z\xrightarrow{} \right) &=& \xrightarrow{-z\succ(x\prec y)}. \end{aligned}$$ Conversely, according to Lemma \[lem:rel\], the above six relations on $\prec,\succ$ imply the vanishing of $A_{\cdot'}(-,-,-)$ on $C^*(\Delta_1;V)$. Hence, we see that an ${\operatorname}{Ass}_\bullet( {\mathcal{A}}ss)$-algebra structure on $V$ is the datum of two non-symmetric products $\prec,\succ:V^{{\otimes}2}\to V$ which are associative and such that moreover $$x\prec(y\succ z)=(x\prec y)\succ z =x\succ(y\prec z)=(x\succ y)\prec z =0,\qquad\forall x,y,z\in V.$$ With the notations from the previous definition we found that that ${\operatorname}{Ass}_\bullet( {\mathcal{A}}ss)={\mathcal{A}}ss\times{\mathcal{A}}ss$. By further imposing $x\prec y-y\succ x =: x\cdot y= 0$, that is, $x \prec y = y\succ x=:x\star y$, we see that the operad ${\operatorname}{Com}_\bullet({\mathcal{A}}ss)$ is generated by a non symmetric product $\star$ such that $(x\star y)\star z=x\star (y\star z)=0$: in other words ${\operatorname}{Com}_\bullet({\mathcal{A}}ss)= nil{\mathcal{A}}ss$, the operad of two step nilpotent associative algebras, in accord with Theorem \[th:black\] and the computations in [@GK]. To compute the relations of ${\operatorname}{preLie}_\bullet({\mathcal{A}}ss)$ it is convenient to replace the generating set of operations $\prec,\succ$ with the one $\prec,\curlyeqsucc=-\succ^{op}$ as explained in Remark \[rem:curlyeqprec\], then we get the following relations according to Lemma \[lem:relpl\] (notice that $x\cdot y:=x\prec y-y\succ x=x\prec y + x\curlyeqsucc y$) $$\begin{aligned}
\nonumber 0=A_{\cdot'}\left(\xrightarrow{x},\xrightarrow{}_y,\xrightarrow{}_z \right) &=& \xrightarrow{(x\prec y)\prec z-x\prec(y\cdot z)}, \\
\nonumber 0=A_{\cdot'}\left(\xrightarrow{}_x,\xrightarrow{y},\xrightarrow{}_z \right) & = & \xrightarrow{(x\curlyeqsucc y)\prec z-x\curlyeqsucc(y\prec z)}, \\ \nonumber 0=A_{\cdot'}\left(\xrightarrow{}_x,\xrightarrow{}_y,\xrightarrow{z} \right) &=& \xrightarrow{(x\cdot y)\curlyeqsucc z - x\curlyeqsucc(y\curlyeqsucc z)}, \end{aligned}$$ which are exactly the dendirform relations for the products $\prec,\curlyeqsucc$, thus ${\operatorname}{preLie}_\bullet({\mathcal{A}}ss)={\mathcal{D}}end$, the operad of dendriform algebras, which as well known is also ${\mathcal{D}}end=pre{\mathcal{L}}ie\bullet{\mathcal{A}}ss$.
Next we consider the operad $({\mathcal{C}}om,\bullet)$ of commutative and associative algebras. Given an ${\operatorname}{Ass}_\bullet({\mathcal{C}}om)$ algebra structure $(V,\circ)$ on a vector space, together with the asociated commutative product $\bullet'$ on $C^*(\Delta_1;V)$, similar computations as in the previous example show that the vanishing of the associator $A_{\bullet'}(-,-,-)$ is equivalent to the identities $(x\circ y)\circ z = x\circ(y\circ z)=0$, thus recovering once again ${\operatorname}{Com}_\bullet({\mathcal{A}}ss)= {\mathcal{C}}om\bullet {\mathcal{A}}ss = nil{\mathcal{A}}ss$. If we further impose that the commutative product $x\bullet y = x\circ y + y\circ x$ on $V$ vanishes, we see that ${\operatorname}{Com}_\bullet({\mathcal{C}}om)$ is the operad generated by an anti-commutative bracket $\{x,y\}:=x\circ y=-y\circ x$ such that $\{ \{ x,y \}, z \}=0$, or in other words ${\operatorname}{Com}_\bullet({\mathcal{C}}om)=nil{\mathcal{L}}ie$, the operad of two step nilpotent Lie algebras. Finally, a ${\operatorname}{preLie}_\bullet({\mathcal{C}}om)$-algebra structure on $V$ is the datum of a non-symmetric product $\circ:V^{{\otimes}2}\to V$ such that, where as usual we denote by $x\bullet y=x\circ y + y\circ x$, $$0=A_{\bullet'}(\xrightarrow{x},\to_y,\to_z)=\xrightarrow{x\circ(y\bullet z)-(x\circ y) \circ z},\qquad\forall x,y,z\in V,$$ and a second relation coming from Lemma \[lem:relpl\], namely, $(x\circ y)\circ z= (x\circ z)\circ y$, already follows from this one. We find that ${\operatorname}{preLie}_\bullet({\mathcal{C}}om)={\mathcal{Z}}inb$, the operad of (right) Zinbiel algebra, according to the well known fact $pre{\mathcal{L}}ie\bullet {\mathcal{C}}om={\mathcal{Z}}inb$ [@V].
We consider the operad $pre{\mathcal{L}}ie$ of (right) pre-Lie algebras. Given an ${\operatorname}{Ass}_\bullet(pre{\mathcal{L}}ie)$-algebra structure $(V,\prec,\succ)$ on $V$, together with the associated dg pre-Lie algebra $(C^\ast(\Delta_1;V),d,\cdot')$, the associator $A_{\cdot'}(-,-,-)$ on $C^\ast(\Delta_1;V)$ is computed as in Example \[ex:ass\]. Writing the generating relation in $pre{\mathcal{L}}ie$ as $0=R(\alpha,\beta,\gamma)=A_{\cdot'}(\alpha,\beta,\gamma)-A_{\cdot'}(\alpha,\gamma,\beta)$, $\forall \alpha,\beta,\gamma\in C^*(\Delta_1;V)$, we find that according to Lemma \[lem:rel\] this is equivalent to the following relations on $\prec,\succ$ (the remaining three follow from these ones by symmetry of $R$) $$0 = R(_x\to,\to_y,\xrightarrow{z})=\xrightarrow{x\prec(z\succ y)-(x\prec z)\prec y+x\prec(z\prec y)}$$ $$0 = R(\to_x,\,_y\to,\xrightarrow{z})=\xrightarrow{(y \prec z)\succ x -y\succ(z\succ x)+(y\succ z)\succ x}$$ $$0 = R(\xrightarrow{x},\,_y\to,\to_z)=\xrightarrow{-(y\succ x)\prec z+ y\succ(x\prec z)}$$
These are precisely the relations of dendriform algebra, thus once again ${\operatorname}{Ass}_\bullet(pre{\mathcal{L}}ie)={\mathcal{D}}end$. If we impose that the right pre-Lie product $x \cdot y = x\prec y - y\succ x$ on $V$ vanishes, we get as well known [@LV] the Zinbiel relation for $x\star y:=x\prec y=y\succ x$, that is, once again ${\operatorname}{Com}_\bullet(pre{\mathcal{L}}ie)={\mathcal{Z}}inb$. Finally, in a ${\operatorname}{preLie}_\bullet(pre{\mathcal{L}}ie)$-algebra we get the relations (the remaining one follows from the first one and simmetry of $R$) $$0 = R(\to_x,\to_y,\xrightarrow{z})=\xrightarrow{-z\succ (x\cdot y)-(z\succ y)\succ x+(z\succ x)\prec y-(z\prec y)\succ x}$$ $$0 = R(\xrightarrow{x},\to_y,\to_z)=\xrightarrow{(x\prec y)\prec z -(x\prec z)\prec y- x\prec(y\cdot z -z\cdot y)}$$ The reader will recognize the relations of $L$-dendriform algebra [@bai2; @V]. We recall the proof of the following fact, giving the two forgetful functors from the category of $(pre{\mathcal{L}}ie\bullet pre{\mathcal{L}}ie)$-algebras to the one of $pre{\mathcal{L}}ie$-algebras.
Let $V$ be a vector space, together with non symmetric products $\prec,\succ:V^{{\otimes}2}\to V$ such that the following relations, where we put $x\cdot y:= x\prec y-y\succ x$, $x\triangleleft y := x\prec y + x\succ y$, $[x,y]:=x\cdot y-y\cdot x=x\triangleleft y-y\triangleleft x$, are verified $$x\prec [y,z]=(x\prec y)\prec z - (x\prec z)\prec y,\qquad (x\succ y)\prec z=x\succ(y\cdot z)+(x\triangleleft z)\succ y,\qquad\forall x,y,z\in V.$$ Then the products $\triangleleft,\cdot$ are right pre-Lie products on $V$ and $[-,-]$ is a Lie bracket.
The relations in the claim of the lemma imply the pre-Lie relation for $\cdot$ according to the previous computations and the proof of Lemma \[lem:rel\]. The fact that $[-,-]$ is a Lie bracket follows. Finally, we may write the second relation as $A_{\succ}(x,z,y)=(x\succ y)\prec z - (x\prec z)\succ y -x\succ(y\prec z)$: substituting in $$A_{\triangleleft}(x,y,z)=A_{\prec}(x,y,z)+A_{\succ}(x,y,z)+(x\prec y)\succ z-x\prec(y\succ z)+(x\succ y)\prec z-x\succ(y\prec z)$$ we find that $$\begin{gathered}
A_{\triangleleft}(x,y,z)= A_\prec(x,y,z)-x\prec(y\succ z)+\\+(x\succ z)\prec y-(x\prec y)\succ z-x\succ(z\prec y)+(x\prec y)\succ z+(x\succ y)\prec z-x\succ(y\prec z).
\end{gathered}$$ The bottom row is clearly symmetric in $y$ and $z$, while the same statement for the right hand side of the top row is just another way of writing the first relation in the claim of the lemma.
\[ex:leib\] An interesting example is the operad ${\mathcal{L}}eib$ of (right) Leibniz algebras: this is the operad generated by a single non symmetric product $\cdot$ and the relation $0=R_\cdot(x,y,z):=(x\cdot y)\cdot z-x\cdot(y\cdot z)-(x\cdot z)\cdot y$. From Lemma \[lem:rel\] we get the following five indipendent relations on $\prec,\succ$ $$\begin{gathered}
0=R_{\cdot'}(_x\to,\to_y,\xrightarrow{z})\:=\: \left(_x\to\cdot'\to_y\right)\cdot'\xrightarrow{z}-\,_x\to\cdot'(\to_y\cdot'\xrightarrow{z})-(_x\to\cdot'\xrightarrow{z})\cdot'\to_y \:= \\ 0+\,_x\to\cdot'\xrightarrow{z\succ y}-\xrightarrow{x\prec z}\cdot'\to_y\:=\:\xrightarrow{x\prec(z\succ y)-(x\prec z)\prec y},\end{gathered}$$
$$\begin{aligned}
\nonumber 0=R_{\cdot'}\left(\xrightarrow{}_x,{}_y\xrightarrow{},\xrightarrow{z} \right) &=& \xrightarrow{(y\prec z)\succ x-y\succ(z\succ x)}, \\
\nonumber 0=R_{\cdot'}\left({}_x\xrightarrow{},\xrightarrow{y},\xrightarrow{}_z \right) &=& \xrightarrow{(x\prec y)\prec z-x\prec(y\prec z)}, \\
\nonumber 0=R_{\cdot'}\left(\xrightarrow{}_x,\xrightarrow{y},{}_z\xrightarrow{} \right) &=& \xrightarrow{z\succ(y\succ x)-(z\succ y)\succ x}, \\
\nonumber 0=R_{\cdot'}\left(\xrightarrow{x},{}_y\xrightarrow{},\xrightarrow{}_z \right) &=& \xrightarrow{-(y\succ x)\prec z+y\succ(x\prec z)},
\end{aligned}$$
and the remaining one is equivalent to the last one. These are the precisely the relations of diassociative algebra on $(V,\prec,\succ)$, in other words we found that ${\operatorname}{Ass}_\bullet({\mathcal{L}}eib)=di{\mathcal{A}}ss$, the operad of diassociative algebras. If we further impose $0=x\cdot y=x\prec y -y\succ x$ and we put as usual $x\star y :=x\prec y=y\succ x$, the previous relations reduce to the two independent ones $(x\star y)\star z=x\star(y\star z)$, $(x\star y)\star z=x\star(z\star y)$: in other words, we found that ${\operatorname}{Com}_\bullet({\mathcal{L}}eib)={\mathcal{P}}erm$, the operad of (right) permutative algebras. Finally, we consider the operad ${\operatorname}{preLie}_\bullet({\mathcal{L}}eib)$: this is generated by non-symmetric products $\prec,\succ$ and the relations, where as usual $x\cdot y= x\prec y- y\succ x$, $$\begin{aligned}
\nonumber 0=R_{\cdot'}\left(\xrightarrow{x},\xrightarrow{}_y,\xrightarrow{}_z \right) &=& \xrightarrow{(x\prec y)\prec z-(x\prec z)\prec y-x\prec(y\cdot z)}, \\
\nonumber 0=R_{\cdot'}\left(\xrightarrow{}_x,\xrightarrow{y},\xrightarrow{}_z \right) &=& \xrightarrow{-(y\succ x)\prec z+(y\prec z)\succ x+y\succ(x\cdot z)}, \\
\nonumber 0=R_{\cdot'}\left(\xrightarrow{}_x,\xrightarrow{}_y,\xrightarrow{z} \right) &=& \xrightarrow{-z\succ(x\cdot y)-(z\succ y)\succ x+(z\succ x)\prec y}.\end{aligned}$$ These are equivalent to the following relations for $\prec,\succ$ $$\begin{gathered}
x\prec(y\cdot z)=(x\prec y)\prec z -(x\prec z)\prec y,\qquad (x\succ y)\succ z=(x\prec y)\succ z, \\ x\succ(y\cdot z)=(x\succ y)\prec z-(x\prec z)\succ y. \end{gathered}$$
\[ex:pois\] We consider the operad $({\mathcal{P}}ois,\bullet, [-,-])$ of commutative Poisson algebras. By the previous computations, an $\operatorname{Ass}_\bullet({\mathcal{P}}ois)$-algebra structure on $V$ is the datum $(V,\circ,\ast)$ of an ${\operatorname}{Ass}_\bullet({\mathcal{C}}om)=nil{\mathcal{A}}ss$-algebra structure $(V,\circ)$ and an ${\operatorname}{Ass}_\bullet({\mathcal{L}}ie)={\mathcal{A}}ss$-algebra structure $(V,\ast)$ satisfying the additional relations induced, as in Lemma \[lem:rel\], by the Poisson identity $0=[x,y\bullet z]-[x,y]\bullet z - y\bullet[x,z]$ in the operad ${\mathcal{P}}ois$. These additional relations are easily computed, for instance $$\begin{gathered}
0=\left[\xrightarrow{x},\,_y\to\bullet'\to_z\right]' - \left[\xrightarrow{x},\,_y\to\right]'\bullet'\to_z-\,_y\to\bullet'\left[\xrightarrow{x},\to_z\right]' = \\=\,0\,-\xrightarrow{-y\ast x}\cdot'\to_z -\,_y\to\cdot'\xrightarrow{x\ast z}\:=\:\xrightarrow{(y\ast x)\circ z - y\circ (x\ast z)}.
\end{gathered}$$ Proceeding in this way we find the following relations for $\circ,\ast$ $$(x\ast y)\circ z= x\ast(y\circ z)=x\circ(y\ast z)= (x\circ y)\ast z.$$ Similarly, a $\operatorname{Com}_\bullet({\mathcal{P}}ois)$-algebra structure on $V$ is the datum $(V,\{-,-\},\circledast)$ of a $\operatorname{Com}_\bullet({\mathcal{C}}om)=nil{\mathcal{L}}ie$-algebra structure $(V,\{-,-\})$ and a $\operatorname{Com}_\bullet({\mathcal{L}}ie)={\mathcal{C}}om$-algebra structure $(V,\circledast)$ such that moreover the above identities hold. Consider the one $\{x,y\}\circledast z=\{x\circledast y,z \} $: as the left and right hand side are respectively symmetric and anti-symmetric in $x$ and $y$ both vanish, thus we have the relations $$\{x\circledast y,z \}=x\circledast\{y,z\}=0,\qquad\forall x,y,z\in V.$$ In other words, we found $\operatorname{Com}_\bullet({\mathcal{P}}ois)=nil{\mathcal{L}}ie\times{\mathcal{C}}om$. Finally, a ${\operatorname}{preLie}_\bullet({\mathcal{P}}ois)$-algebra structure on $V$ is the datum $(V,\circ,\ast)$ of a ${\operatorname}{preLie}_\bullet({\mathcal{C}}om)={\mathcal{Z}}inb$-algebra structure $(V,\circ)$ and a ${\operatorname}{preLie}_\bullet({\mathcal{L}}ie)=pre{\mathcal{L}}ie$-algebra structure $(V,\ast)$, such that moreover, where as usual we denote by $x\bullet y =x\circ y+ y\circ x$ and $[x,y]=x\ast y-y\ast x$ the associated ${\mathcal{C}}om$- and ${\mathcal{L}}ie$-algebra structures on $V$ respectively, $$\begin{gathered}
0=\left[\xrightarrow{x},\to_y\bullet'\to_z\right]' - \left[\xrightarrow{x},\,\to_y\right]'\bullet'\to_z-\to_y\bullet'\left[\xrightarrow{x},\to_z\right]' = \\=\left[\xrightarrow{x},\to_{y\bullet z}\right]'-\xrightarrow{x\ast y}\bullet'\to_z -\to_y\bullet'\xrightarrow{x\ast z}=\xrightarrow{x\ast(y\bullet z) -(x\ast y)\circ z - (x\ast z)\circ y},
\end{gathered}$$ $$\begin{gathered}
0=\left[\to_x,\xrightarrow{y}\bullet'\to_z\right]' - \left[\to_x,\xrightarrow{y}\right]'\bullet'\to_z-\xrightarrow{y}\bullet'\large[\to_x,\to_z\large]' = \\=\left[\to_x,\xrightarrow{y\circ z}\right]'-\xrightarrow{-y\ast x}\bullet'\to_z -\xrightarrow{y}\bullet'\to_{[x,z]}=\xrightarrow{-(y\circ z)\ast x+(y\ast x)\circ z - y\circ [x,z]}.
\end{gathered}$$ We conclude that ${\operatorname}{preLie}_\bullet({\mathcal{P}}ois)=pre{\mathcal{P}}ois$, Aguiar’s operad of right pre-Poisson algebras [@A], in accord with the fact [@U] that $pre{\mathcal{L}}ie\bullet{\mathcal{P}}ois = pre{\mathcal{P}}ois$.
\[ex:perm\] We consider the operad ${\mathcal{P}}erm$: recall that a ${\mathcal{P}}erm$-algebra structure, or (right) permutative algebra structure, on $V$ is the datum of an associative product $\cdot$ which is commutative on the right hand side whenever there are three or more variables, or, in other words, satisfies the relations $(x\cdot y)\cdot z=x\cdot(y\cdot z)=x\cdot(z\cdot y)$. Thus, an ${\operatorname}{Ass}_\bullet({\mathcal{P}}erm)$-algebra structure on $V$ is the datum of an ${\operatorname}{Ass}_\bullet({\mathcal{A}}ss)={\mathcal{A}}ss\times{\mathcal{A}}ss$ algebra structure $(V,\prec,\prec)$, as in Example \[ex:ass\], such that moreover $$0 \:=\: _x\to\cdot'\left( \to_y\cdot'\xrightarrow{z}\right)\,-\,_x\to\cdot'\left( \xrightarrow{z}\cdot'\to_y\right) =\xrightarrow{-x\prec(z\prec y)-x\prec(z\prec y)}=\xrightarrow{-x\prec(y\prec z)}$$ $$0 \:=\: \to_x\cdot'\left( _y\to\cdot'\xrightarrow{z}\right)-\to_x\cdot'\left( \xrightarrow{z}\cdot'\,_y\to \right) =\xrightarrow{-(y\prec z)\prec x-(y\prec z)\prec x}=\xrightarrow{-(y\prec z)\prec x}.$$ In other words, ${\operatorname}{Ass}_\bullet({\mathcal{P}}erm)=nil{\mathcal{A}}ss\times nil{\mathcal{A}}ss$. Further imposing $0=x\prec y-y\prec x=x\cdot y$, we find that ${\operatorname}{Com}_\bullet({\mathcal{P}}erm)=nil{\mathcal{A}}ss$. Finally, as in Remark \[rem:curlyeqprec\] and Example \[ex:ass\], it is convenient in the compuation of ${\operatorname}{preLie}_\bullet({\mathcal{P}}erm)$ to replace the product $\prec$ by the opposite one $\curlyeqsucc:=-\prec^{op}$, satisfying the formulas $\to_x\cdot'\xrightarrow{y}=\xrightarrow{x}\cdot'\,_y\to=:\xrightarrow{x\curlyeqsucc y}$. In fact, as we saw in \[ex:ass\] the relations of ${\operatorname}{preLie}_\bullet({\mathcal{P}}erm)$-algebra translate in the relations of ${\operatorname}{preLie}_\bullet({\mathcal{A}}ss)={\mathcal{D}}end$-algebra for the products $\prec,\curlyeqsucc$, and we get the additional relations (where as usual $x\cdot y=x\prec y-y\prec x=x\prec y+ x\curlyeqsucc y$) $$0= \xrightarrow{x}\cdot'\left( \to_y\cdot'\to_z\right)-\xrightarrow{x}\cdot'\left(\to_z\cdot'\to_y\right)=\xrightarrow{x\prec(y\cdot z) - x\prec(z\cdot y)}$$ $$0= \to_x\cdot'(\xrightarrow{y}\cdot'\to_z)-\to_x\cdot'\left( \to_z\cdot'\xrightarrow{y}\right)= \xrightarrow{x\curlyeqsucc(y\prec z)-x\curlyeqsucc(z\curlyeqsucc y)}.$$ This is in accord with [@V Theorem 21].
Given an operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$, an ${\mathcal{O}}_{adm}$-algebra structure on $V$ is the datum of non-symmetric operations $\prec_i,\succ_i,\circ_j.\ast_k:V^{{\otimes}2}\to V$ such that the induced operations $x\cdot_i y:=x\prec_i y+x\succ_iy$, $x\bullet_j y= x\circ_j y+y\circ_j x$, $[x,y]_k=x\ast_k y-y\ast_k x$ define an ${\mathcal{O}}$-algebra structure on $(V,\cdot_i,\bullet_j,[-,-]_k)$. This defines the operad $({\mathcal{O}}_{adm},\prec_i,\succ_i,\circ_j,\ast_k)$ of ${\mathcal{O}}$ admissible algebras.
\[ex:lieadm\] We consider the operad ${\mathcal{L}}ie_{adm}$ of Lie admissible algebras, that is, the operad generated by a non-symmetric operation $\cdot$ such that the commutator is a Lie bracket. We shall write the only relation in ${\mathcal{L}}ie_{adm}$ as $0=R_\cdot(x_1,x_2,x_3)=\sum_{\sigma\in S_3}\varepsilon(\sigma)A_\cdot(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})$, where $\varepsilon(\sigma)$ is the sign of the permutation $\sigma$ and as usual $A_\cdot(-,-,-)$ is the asociator of $\cdot$. Given an $\operatorname{Ass}_\bullet({\mathcal{L}}ie_{adm})$-algebra structure $(V,\prec,\succ)$ on $V$, together with the associated ${\mathcal{L}}ie_{adm}$-algebra structure on $C^*(\Delta_1;V)$, the associator $A_{\cdot'}(-,-,-)$ is computed as in Example \[ex:ass\]. By Lemma \[lem:rel\] the vanishing of $R_\cdot'(-,-,-)$ is equivalent to the following relation on $\prec,\succ$ (the others follow from this one by symmetry of $R$) $$\begin{gathered}
0= R_{\cdot'}(_x\to,\to_y,\xrightarrow{z})=A_{\cdot'}(_x\to,\to_y,\xrightarrow{z})+ A_{\cdot'}(\to_y,\xrightarrow{z},\,_x\to)+A_{\cdot'}(\xrightarrow{z},\,_x\to,\to_y)-\\-A_{\cdot'}(\to_y,\,_x\to,\xrightarrow{z})-A_{\cdot'}(_x\to,\xrightarrow{z},\to_y)-A_{\cdot'}(\xrightarrow{z},\to_y,\,_x\to)=\end{gathered}$$ $$=\xrightarrow{x\prec(z\succ y) + x\succ(z\succ y)-(x\succ z)\succ y - (x\succ z)\prec y-(x\prec z)\succ y-(x\prec z)\prec y -x\prec(z\prec y)+x\succ(z\prec y)}$$ This is exactly the relation saying that the product $x\cup y:=x\prec y + x\succ y$ on $V$ (so denoted to distinguish it from the Lie admissible product $x\cdot y=x\prec y-y\succ x$) is associative. Hence, we found ${\operatorname}{Ass}_\bullet({\mathcal{L}}ie_{adm})={\mathcal{A}}ss_{adm}$. Further imposing $x\prec y=y\succ x=:x\star y$, we see that $x\cup y= x\star y + y\star x$ is an associative and commutative product, and thus that ${\operatorname}{Com}_\bullet({\mathcal{L}}ie_{adm})={\mathcal{C}}om_{adm}$. For the operad $(\operatorname{preLie}_\bullet({\mathcal{L}}ie_{adm}),\prec,\succ)$ we find the relation (and the others follow by symmetry of $R$), where as usual $x\cdot y=x\prec y-y\succ x$, $$\begin{gathered}
0= R_{\cdot'}(\to_x,\to_y,\xrightarrow{z})=A_{\cdot'}(\to_x,\to_y,\xrightarrow{z})+ A_{\cdot'}(\to_y,\xrightarrow{z},\to_x)+A_{\cdot'}(\xrightarrow{z},\to_x,\to_y)-\\-A_{\cdot'}(\to_y,\to_x,\xrightarrow{z})-A_{\cdot'}(\to_x,\xrightarrow{z},\to_y)-A_{\cdot'}(\xrightarrow{z},\to_y,\to_x)= \end{gathered}$$ $$=\xrightarrow{-z\succ(x\cdot y) - (z\succ y)\succ x - (z\succ y)\prec x + (z\prec x)\succ y + (z\prec x)\prec y-z\prec(x\cdot y)+z\succ (y\cdot x) + (z\succ x)\succ y + (z\succ x)\prec y -(z\prec y)\succ x -(z\prec y)\prec x+ z\prec(y\cdot x)}$$ Putting $x\cup y = x\prec y + x\succ y$, the reader will check that the above relation can be rewritten as $0=A_\cup(z,x,y)-A_\cup(z,y,x)$, and thus we found ${\operatorname}{preLie}_\bullet({\mathcal{L}}ie_{adm})=pre{\mathcal{L}}ie_{adm}$.
The latter example suggests ${\mathcal{O}}\bullet{\mathcal{L}}ie_{adm}={\mathcal{O}}_{adm}$ for every operad ${\mathcal{O}}$ in $\mathbf{Op}$.
\[ex:postlie\] We consider the operad $(post{\mathcal{L}}ie,\cdot,[-,-])$ of post-Lie algebras: this is the operad generated by a non-symmetric product $\cdot$ and a Lie bracket $[-,-]$ satisfying the relations $0=R_1(x,y,z)=[x,y]\cdot z-[x,y\cdot z]-[x\cdot z,y]$ and $0=R_2(x,y,z)=x\cdot(y\cdot z +z\cdot y + [z,y])-(x\cdot y)\cdot z+(x\cdot z)\cdot y$. In the operad $({\operatorname}{Ass}_\bullet({\mathcal{O}}),\prec,\succ,\ast)$ we get the associativity relation for the product $\ast$ and the six independent relations $$0=R_1(_x\to,\xrightarrow{y},\to_z)=\xrightarrow{(x\ast y)\prec z-x\ast(y\prec z)},$$ $$0=R_1(\to_x,\xrightarrow{y},_z\to)=\xrightarrow{z\succ(y\ast x)-(z\succ y)\ast x},$$ $$0=R_1(_x\to,\to_y,\xrightarrow{z})=\xrightarrow{x\ast(z\succ y)-(x\prec z)\ast y},$$ $$0=R_2(_x\to,\xrightarrow{y},\to_z)=\xrightarrow{x\prec(y\prec z+z\prec y+y\ast z)-(x\prec y)\prec z},$$ $$0=R_2(\to_x,\xrightarrow{y},_z\to)= \xrightarrow{(z\succ y+ y\prec z +z\star y)\succ x-z\succ(y\succ x)},$$ $$0=R_2( \xrightarrow{x},_y\to,\to_z)=\xrightarrow{(y\succ x)\prec z-y\succ(x\prec z)}.$$ These are the relations of dendriform trialgebra [@LR], hence we found ${\operatorname}{Ass}_\bullet(post{\mathcal{L}}ie)=\mathcal{T}ridend$. If we further impose $x\prec y= y\succ x=: y\star x$, $x\star y=y\star x=:x\circledast y$, we find that the operad $({\operatorname}{Com}_\bullet(post{\mathcal{L}}ie),\star,\circledast)$ is defined by the following relations $$(x\circledast y)\circledast z=x\circledast(y\circledast z),\quad(x\circledast y)\star z=x\circledast(y\star z),\quad(x\star y)\star z =x\star(y\star z+z\star y +y\circledast z).$$ We shall denote this operad by $post{\mathcal{C}}om:={\operatorname}{Com}_\bullet(post{\mathcal{L}}ie)$. We leave to the interested reader the computation of ${\operatorname}{preLie}_\bullet(post{\mathcal{L}}ie)$. We have morphism of operads $$({\mathcal{L}}ie,\{-,-\})\to(post{\mathcal{L}}ie,\cdot,[-,-]):\{-,-\}\:\:\longrightarrow\:\:\cdot\:-\:\cdot^{op}\:+\:[-,-],$$ $$({\mathcal{A}}ss,\cup)\to(\mathcal{T}ridend,\prec,\succ,\ast):\cup\:\:\longrightarrow\:\:\prec+\succ+\ast,$$ $$({\mathcal{C}}om,\bullet)\to(post{\mathcal{C}}om,\star,\circledast):\bullet\:\:\longrightarrow\:\: \star+\star^{op}+\circledast,$$ in the first (as well as the second) case this is well known, the other two follow by functoriality.
We consider the operad generated by a Lie bracket $[-,-]$ and a right Leibniz product $\cdot$ satisfying the additional relations $$[x,y]\cdot z=[x,y\cdot z]+[x\cdot z,y],\qquad x\cdot[y,z]=x\cdot(y\cdot z).$$ It can be checked that this is the Koszul dual $(post{\mathcal{C}}om^!, \cdot,[-,-])$ of the operad $post{\mathcal{C}}om:={\operatorname}{Com}_\bullet(post{\mathcal{L}}ie)$ from the previous example. To aid in the computations, we notice that this is the same, up to changing the sign of the operation $\cdot$, as the operad we obtain from $(post{\mathcal{L}}ie,\cdot,[-,-])$ by further imposing the right Leibniz identity for $\cdot$. By the previous example and Example \[ex:leib\], we see that the operad ${\operatorname}{Ass}_\bullet(post{\mathcal{C}}om^!)$ is the same as the one we obtain from the operad $\mathcal{T}ridend$ of dendriform trialgebras by further imposing the diassociativity relations for $\prec,\succ$, and then changing the signs of $\prec,\succ$: the reader will readily verify that this is the operad ${\operatorname}{Ass}_\bullet(post{\mathcal{C}}om^!)=\mathcal{T}riass$ of triassociative algebras by Loday and Ronco [@LR]. Likewise, ${\operatorname}{Com}_\bullet(post{\mathcal{C}}om^!)$ is the operad we obtain from $(post{\mathcal{C}}om,\star,\circledast)$, defined as in the previous example, by further imposing that $\star$ is a right permutative product, and then changing the sign of $\star$: the reader will readily verify that this is the same as the operad ${\mathcal{C}}omtrias$ of commutative trialgebras [@Vpos].
A Koszul dual trick to compute Manin white products
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We denote by $-\circ-:\mathbf{Op}\times\mathbf{Op}\to\mathbf{Op}$ the Manin white product of operads [@V], it gives $\mathbf{Op}$ a structure of symmetric monoidal category, and by $-^!:\mathbf{Op}\to\mathbf{Op}$ the Koszul duality functor. Recall that the Manin white and black products are Koszul dual, in the sense that $({\mathcal{O}}\bullet{\mathcal{P}})^!={\mathcal{O}}^!\circ{\mathcal{P}}^!$ for every pair of operads ${\mathcal{O}}$ and ${\mathcal{P}}$ in $\mathbf{Op}$, and moreover for every operad ${\mathcal{O}}$ the functors $\xymatrix{{\mathcal{O}}\bullet-:\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\mathcal{O}}^!\circ-\ar@<2pt>[l]}$ form an adjoint pair. recall that ${\mathcal{A}}ss^!={\mathcal{A}}ss$, ${\mathcal{C}}om^!={\mathcal{L}}ie$ and $pre{\mathcal{L}}ie^!={\mathcal{P}}erm$. In the previous section we introduced easily computable functors ${\operatorname}{Ass}_\bullet(-)$, ${\operatorname}{Com}_\bullet(-),{\operatorname}{preLie}_\bullet(-):\mathbf{Op}\to\mathbf{Op}$ and claimed that they coincide with the respective Manin black products ${\mathcal{A}}ss\bullet-,{\mathcal{C}}om\bullet-, pre{\mathcal{L}}ie\bullet-$. The aim of this section is to introduce the respective right adjoint functors ${\operatorname}{Ass}_\circ(-),{\operatorname}{Lie}_\circ(-),{\operatorname}{Perm}_\circ(-):\mathbf{Op}\to\mathbf{Op}$ (the adjointness relation will be shown at the very end of the section) and prove that they in fact coincide with the Manin white products ${\mathcal{A}}ss\circ-,{\mathcal{L}}ie\circ-,{\mathcal{P}}erm\circ-$: this will also complete the proof of Theorem \[th:black\].
We consider first the case of $\operatorname{Ass}_\circ(-)$. Given an (as usual, binary, quadratic and finitely generated) operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$, the operad ${\operatorname}{Ass}_\circ({\mathcal{O}})$ is generated by non symmetric operations $\prec_i,\succ_i,\circ_j,\ast_k$. We want a morphism of operads $\mu_{{\mathcal{O}}}:{\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))\to{\mathcal{O}}$ corresponding to the counit of the adjunction $\xymatrix{{\operatorname}{Ass}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Ass}_\circ(-)\ar@<2pt>[l]}$: in other words, by definition of ${\operatorname}{Ass}_\bullet(-)$, given an ${\mathcal{O}}$-algebra structure $(V,\cdot_i,\bullet_j,[-,-]_k)$ on a vector space $V$ we want an induced local dg ${\operatorname}{Ass}_\circ({\mathcal{O}})$-algebra structure on the complex $C^*(\Delta_1;V)=C^*(\Delta_1;\mathbb{K})\otimes V$. Denoting by $\cup$ the usual cup product of cochains on $C^*(\Delta_1;\mathbb{K})$, the ${\operatorname}{Ass}_\circ({\mathcal{O}})$-algebra structure on $C^*(\Delta_1;V)$ will be given by the tensor product operations $\prec_i=\cup\otimes\cdot_i$, $\succ_i=\cup\otimes\cdot_i^{op}$, $\circ_j=\cup{\otimes}\bullet_j$, $\ast_k=\cup{\otimes}[-,-]_k$: explicitly, with the same notations as in the previous section, $$\to_x\prec_i\to_y\:=\:\to_{x\cdot_i y}\:=\:\to_y\succ_i\to_x,\qquad \to_x\circ_j\to_y\:=\:\to_{x\bullet_j y},\qquad\to_x\ast_k\to_y\:=\: \to_{[x,y]_k},$$ $$_x\to\prec_i\,_y\to\:=\:_{x\cdot_i y}\to\:=\:_y\to\succ_i\,_x\to,\qquad _x\to\circ_j\,_y\to\:=\:_{x\bullet_j y}\to,\qquad_x\to\ast_k\,_y\to\:=\: _{[x,y]_k}\to,$$ $$_x\to\prec_i\xrightarrow{y}=_y\to\succ_i\xrightarrow{x}\:=\:\xrightarrow{x\cdot_i y}\:=\:\xrightarrow{x}\prec_i\to_y= \xrightarrow{y}\succ_i\to_{x},$$ $$_x\to\circ_j\xrightarrow{y}\:=\:\xrightarrow{x\bullet_j y}\:=\:\xrightarrow{x}\circ_j\to_y,\qquad _x\to\ast_k\xrightarrow{y}\:=\:\xrightarrow{[x,y]_k}\:=\:\xrightarrow{x}\ast_k\to_y,$$ and the remaining products vanish. It is immediately seen that these operations satisfy the Leibniz identity with respect to $d$ and the locality assumption from Definition \[def:ManinBlack\]. We will generically call the operations $\prec_i,\succ_i,\circ_j,\ast_k$ on $C^*(\Delta_1;V)$ the cup products.
Given an operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$ as usual, the operad ${\operatorname}{Ass}_\circ({\mathcal{O}})$ is generated by non-symmetric operations $\prec_i,\succ_i,\circ_j,\ast_k$, together with the larger set of relations making $C^*(\Delta_1;V)$ with the cup products a local dg ${\operatorname}{Ass}_\circ({\mathcal{O}})$-algebra for every ${\mathcal{O}}$-algebra $V$.
Next we want to describe a generating set of relations of ${\operatorname}{Ass}_\circ({\mathcal{O}})$. Given a terniary operation $R(-,-,-):C^*(\Delta_1;V)^{{\otimes}3}\to C^*(\Delta_1;V)$ in the cup products, we write $R=\sum_{\sigma\in S_3}R_\sigma$, where $S_3$ is the symmetric group and $R_\sigma$ is spanned by triple products permuting the variables according to $\sigma$. As in the proof of Lemma \[lem:rel\], we see that the relation $R(-,-,-)=0$ holds in $C^*(\Delta_1;V)$ if and only if $$\begin{gathered}
0=R(\xrightarrow{x},\to_y,\,_z\to)=R(\xrightarrow{x},\,_y\to,\to_z)=R(\to_x,\xrightarrow{y},\,_z\to)=\\=R(_x\to,\xrightarrow{y},\to_z)=R(\to_x,\,_y\to,\xrightarrow{z})=R(_x\to,\to_y,\xrightarrow{z})=0\end{gathered}$$ for all $x,y,z\in V$. We claim that this is true if and only if $R_\sigma(-,-,-)=0$ holds in $C^*(\Delta_1;V)$ for every $\sigma\in S_3$, where the [“]{}if” implication is obvious. To fix the ideas, we consider the identical permutation ${\operatorname{id}}\in S_3$ and prove $R(-,-,-)=0{\Rightarrow}R_{{\operatorname{id}}}(-,-,-)=0$: the remaining implications are proved similarly. We obtain the desired conclusion again by Lemma \[lem:rel\], the point is that the only non-vanishing triple products that we can form out of the cochains $_1\to,\to_1,\xrightarrow{1}\in C^*(\Delta_1;\mathbb{K})$ are $(_1\to\cup\xrightarrow{1})\cup\to_1\:=\:\xrightarrow{1}\:=\:_1\to\cup(\xrightarrow{1}\cup\to_1)$: this shows at once that $R_{{\operatorname{id}}}(\xrightarrow{x},\to_y,\,_z\to)=R_{{\operatorname{id}}}(\xrightarrow{x},\,_y\to,\to_z)=R_{{\operatorname{id}}}(\to_x,\xrightarrow{y},\,_z\to)=R_{{\operatorname{id}}}(\to_x,\,_y\to,\xrightarrow{z})=R_{{\operatorname{id}}}(_x\to,\to_y,\xrightarrow{z})=0$, whereas $R_{{\operatorname{id}}}(_x\to,\xrightarrow{y},\to_z)=R(_x\to,\xrightarrow{y},\to_z)=\xrightarrow{R'(x,y,z)}=0$ by hypothesis, where $R'(-,-,-):V^{{\otimes}3}\to V$ is a terniary operation in ${\mathcal{O}}$; finally, if this holds for a generic ${\mathcal{O}}$-algebra $V$, we conclude that $R'(-,-,-)=0$ is a relation in the operad ${\mathcal{O}}$. We sum up the previous discussion in the following lemma.
\[lem:relwhite\] The operad ${\operatorname}{Ass}_\circ({\mathcal{O}})$ has a generating set of relations $R(-,-,-)=0$ which are non-symmetric (where a relation is non-symmetric if $R=R_{{\operatorname{id}}}$), consisting of all the possible splittings, as in the following $$R(_x\to,\xrightarrow{y},\to_z)=\xrightarrow{R'(x,y,z)}=0,$$ of a relation $R'(-,-,-)=0$ in ${\mathcal{O}}$ via the cup products on $C^*(\Delta_1;V)$ (where $V$ is a generic ${\mathcal{O}}$-algebra).
It is best to illustrate the previous lemma by some examples.
We consider the operad $({\mathcal{C}}om,\bullet)$: in this case $R'$ as in the previous lemma has to be non-symmetric itself, and since the space of non-symmetric relators of ${\mathcal{C}}om$ is one-dimensional, spanned by the associativity relation, we only get the splitting $$(_x\to \circ \xrightarrow{y})\circ\to_y - _x\to\circ(\xrightarrow{y}\circ\to_z) =\xrightarrow{(x\bullet y)\bullet z-x\bullet (y\bullet z)}=0,$$ telling us that the cup product $\circ$ is associative: thus ${\operatorname}{Ass}_\circ({\mathcal{C}}om)={\mathcal{A}}ss$ as expected. In the case of the operad $({\mathcal{L}}ie,[-,-])$, again $R'$ as in the previous Lemma has to be non-symmetric, and since the space of relators of the operad ${\mathcal{L}}ie$ is one-dimensional generated by the Jacobi identity, and this can’t be written in a non-symmetric form, we find ${\operatorname}{Ass}_\circ({\mathcal{L}}ie)={\mathcal{M}}ag_{1,0,0}$, in accord with our computation of the Koszul dual ${\operatorname}{Ass}_\bullet({\mathcal{C}}om)=nil{\mathcal{A}}ss$.
Whereas the computation of the functor ${\operatorname}{Ass}_\bullet(-)$ is completely mechanical, as we apply Lemma \[lem:rel\] to a generating set of relations of ${\mathcal{O}}$ to get a generating set of relations of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$, the previous lemma is not as effective in the compution of ${\operatorname}{Ass}_\circ({\mathcal{O}})$: we illustrate this fact by considering the operad $({\mathcal{P}}ois,\bullet,[-,-])$. By the previous example, the operad ${\operatorname}{Ass}_\circ({\mathcal{P}}ois)$ is generated by an associative product $\circ$ and a magmatic product $\ast$. Since there is no way to split the Poisson identity $[x\bullet y, z] -x\bullet[y,z] - [x,z]\bullet y=0$ as in the previous lemma, as this can’t be written in a non-symmetric form, we may be tempted to conclude that ${\operatorname}{Ass}_\circ({\mathcal{P}}ois)={\mathcal{A}}ss+{\mathcal{M}}ag_{1,0,0}$, but this is in contrast with our computation of the Koszul dual ${\operatorname}{Ass}_\bullet({\mathcal{P}}ois)$ in Example \[ex:pois\]. In fact, looking closely we find the splitting $$\begin{gathered}
(_x\to\circ \xrightarrow{y})\ast\to_z-_x\to\circ(\xrightarrow{y}\ast\to_z) + (_x\to\ast\xrightarrow{y})\circ\to_z-_x\to\ast(\xrightarrow{y}\circ\to_z) = \\ = \xrightarrow{[x\bullet y,z ]-x\bullet[y,z]+[x,y]\bullet z-[x,y\bullet z]}=\xrightarrow{[x,z]\bullet y - y\bullet[x,z]}=0.\end{gathered}$$ So we find the relation $(\alpha\circ\beta)\ast\gamma-\alpha\circ(\beta\ast\gamma)+(\alpha\ast\beta)\circ\gamma-\alpha\ast(\beta\circ\gamma)=0$, $\forall \alpha,\beta,\gamma\in C^*(\Delta_1;V)$, in the operad ${\operatorname}{Ass}_\circ({\mathcal{P}}ois)$. Together with the associativity relation for $\circ$, these generate the relations of ${\operatorname}{Ass}_\circ({\mathcal{P}}ois)$: the easiest way to see this is to check that the operad ${\operatorname}{Ass}_\circ({\mathcal{P}}ois)$ defined in this way is in fact the Koszul dual of ${\operatorname}{Ass}_\bullet({\mathcal{P}}ois)$ from \[ex:pois\]. Returning to the initial remark, the difficulty in applying Lemma \[lem:relwhite\] is that we have to look for the splittings of any relation $R'$ in ${\mathcal{O}}$, and we can’t limit ourselves to let $R'$ vary in a generating set of these relations, so there is no telling us in general if we are missing some of the possible splittings. As in the previous computation, the easiest way around this, and what we will do in practice in the following examples, is to take advantage of the computation of the Koszul dual ${\operatorname}{Ass}_\bullet({\mathcal{O}}^!)$ from the previous section. Of course, this depends on the yet to be given proofs of theorems \[th:black\], \[th:asswhite\],\[th:permwhite\] and \[th:liewhite\].
We consider the operad $({\mathcal{A}}ss,\cdot)$: in this case we know from Example \[ex:ass\] that ${\operatorname}{Ass}_\circ({\mathcal{A}}ss)=({\mathcal{A}}ss\bullet{\mathcal{A}}ss)^!=({\mathcal{A}}ss\times{\mathcal{A}}ss)^!={\mathcal{A}}ss+{\mathcal{A}}ss$ (with the functors $-\times-$ and $-+-$ as in Definition \[def:+x\]), and in fact we find the relations $$(_x\to\prec\xrightarrow{y})\prec\to_z-_x\to\prec(\xrightarrow{y}\prec \to_z)=\xrightarrow{(x\cdot y)\cdot z-x\cdot(y\cdot z)}=0,$$ $$(_x\to\succ\xrightarrow{y})\succ\to_z-_x\to\succ(\xrightarrow{y}\succ \to_z)=\xrightarrow{z\cdot(y\cdot x)-(z\cdot y)\cdot x}=0,$$ in the operad $({\operatorname}{Ass}_\circ({\mathcal{A}}ss),\prec,\succ)$.
We consider the operad $(pre{\mathcal{L}}ie, \cdot)$. The right pre-Lie relation $(x\cdot y)\cdot z-x\cdot(y\cdot z)-(x\cdot z)\cdot y+x\cdot(z\cdot y)=0$ can’t be splitted as in Lemma \[lem:relwhite\], for instance because there is no argument remaining inside every parenthesis. As the previous remark illustrates, this alone wouldn’t be enough to conclude ${\operatorname}{Ass}_\circ(pre{\mathcal{L}}ie)={\mathcal{M}}ag_{2,0,0}$: on the other hand, this is true since it agrees with the computation of the Koszul dual ${\operatorname}{Ass}_\bullet({\mathcal{P}}erm)=nil{\mathcal{A}}ss\times nil{\mathcal{A}}ss$ in Example \[ex:perm\].
We consider the operad $({\mathcal{L}}eib,\cdot)$. Again, the right Leibniz relation $(x\cdot y)\cdot z-x\cdot(y\cdot z)-(x\cdot z)\cdot y=0$ can’t be splitted as in Lemma \[lem:relwhite\], on the other hand, this imply the relation $x\cdot(y\cdot z+z\cdot y) = (x\cdot y)\cdot z-(x\cdot z)\cdot y +(x\cdot z)\cdot y-(x\cdot y)\cdot z=0$ in ${\mathcal{L}}eib$: for the latter, we find the splittings $$_x\to\prec (\xrightarrow{y}\prec\to_z + \xrightarrow{y}\succ\to_z ) =\xrightarrow{x\cdot(y\cdot z+z\cdot y)}=0,$$ $$(_x\to\prec\xrightarrow{y} + \,_x\to\succ\xrightarrow{y})\succ\to_z = \xrightarrow{z\cdot (x\cdot y + y\cdot x)}=0,$$ hence the relations $\alpha\prec(\beta\prec\gamma+\beta\succ\gamma)=0=(\alpha\prec\beta +\alpha\succ\beta)\succ\gamma$ in $({\operatorname}{Ass}_\circ({\mathcal{L}}eib),\prec,\succ)$. We leave to the reader to check that this is a generating set of relations, for instance by computing the Koszul dual ${\operatorname}{Ass}_\bullet({\mathcal{Z}}inb)$ with the method of the previous section.
As a final example, we consider $({\mathcal{Z}}inb,\cdot)$. In this case, we know form Example \[ex:leib\] that ${\operatorname}{Ass}_\circ({\mathcal{Z}}inb)=({\mathcal{A}}ss\bullet{\mathcal{L}}eib)^!=di{\mathcal{A}}ss^!={\mathcal{D}}end$: in fact, we get the dendriform relations on the operad $({\operatorname}{Ass}_\circ({\mathcal{Z}}inb),\prec,\succ)$, corresponding to the splittings as in Lemma \[lem:relwhite\] (notice that the relation $(x\cdot y)\cdot z-(x\cdot z)\cdot y=0$ holds in ${\mathcal{Z}}inb$) $$_x\to\prec( \xrightarrow{y}\prec\to_z+\xrightarrow{y}\succ\to_z)-(_x\to\prec\xrightarrow{y})\prec\to_z=\xrightarrow{x\cdot(y\cdot z+ z\cdot y)-(x\cdot y)\cdot z}=0$$ $$(_x\to\prec\xrightarrow{y}+\,_x\to\succ\xrightarrow{y})\succ\to_z -\,_x\to\succ(\xrightarrow{y}\succ\to_z)=\xrightarrow{z\cdot(x\cdot y+y\cdot x)-(z\cdot y)\cdot x}=0$$ $$(_x\to\succ\xrightarrow{y})\prec\to_z -\, _x\to\succ(\xrightarrow{y}\prec\to_z)=\xrightarrow{(y\cdot x)\cdot z-(y\cdot z)\cdot x}=0.$$
\[th:asswhite\] There is a natural isomorphism ${\operatorname}{Ass}_\circ(-)\xrightarrow{\cong}{\mathcal{A}}ss\circ-$ of functors $\mathbf{Op}\to\mathbf{Op}$.
We denote by $({\mathcal{A}}ss,\cup)$ the associative operad with its generating product. Given an operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$ as usual, the operad ${\mathcal{A}}ss\circ{\mathcal{O}}$ is generated by the tensor product operations (which are non-symmetric) $\cup\otimes\cdot_i, \cup\otimes\cdot_i^{op}$, $\cup\otimes\bullet_j$, $\cup\otimes[-,-]_k$, together with the larger set of relations holding in the tensor product $A\otimes V$ of a generic ${\mathcal{A}}ss$-algebra $(A,\cup)$ and a generic ${\mathcal{O}}$-algebra $(V,\cdot_i,\bullet_j,[-,-]_k)$. The isomorphism of operads ${\operatorname}{Ass}_\circ({\mathcal{O}})\leftrightarrow {\mathcal{A}}ss\circ{\mathcal{O}}$ is given by $\prec_i\leftrightarrow \cup\otimes\cdot_i$, $\succ_i\leftrightarrow\cup\otimes\cdot_i^{op}$, $\circ_j\leftrightarrow\cup\otimes\bullet_j$ and finally $\ast_k\leftrightarrow\cup\otimes[-,-]_k$. This defines a morphism of operads ${\mathcal{A}}ss\circ{\mathcal{O}}\to{\operatorname}{Ass}_\circ({\mathcal{O}})$: in fact, by construction of the cup products $\prec_i,\succ_i,\circ_j,\ast_k$, these satisfy the relations of ${\mathcal{A}}ss\circ{\mathcal{O}}$-algebra in the tensor product $C^*(\Delta_1;V)$ of the ${\mathcal{A}}ss$-algebra $(C^*(\Delta_1;\mathbb{K}),\cup)$ (where we may forget the gradation) and the generic ${\mathcal{O}}$-algebra $V$. Conversely, we must show that if we evaluate a generating relation $R(-,-,-)=0$ of ${\operatorname}{Ass}_\circ({\mathcal{O}})$, given as in Lemma \[lem:relwhite\], in the operations $\cup\otimes\cdot_i, \cup\otimes\cdot_i^{op}$, $\cup\otimes\bullet_j$, $\cup\otimes[-,-]_k$, we get a relation of the operad ${\mathcal{A}}ss\circ{\mathcal{O}}$: but this is also clear, since given a generic ${\mathcal{A}}ss$-algebra $(A,\cup)$ we find $R(a\otimes x,b\otimes y,c\otimes z)= (a\cup b\cup c)\otimes R'(x,y,z)=0$, $\forall a\otimes x,b\otimes y,c\otimes z\in A\otimes V$, where $a\cup b\cup c:=(a \cup b)\cup c=a\cup(b\cup c)$.
The construction of the functor ${\operatorname}{Perm}_\circ(-):\mathbf{Op}\to\mathbf{Op}$ is similar. Given a generic ${\mathcal{O}}$-algebra $V$, the cup products on the complex $C^*(\Delta_1,v_l;V)$ are defined by the same formulas as in the associative case.
Given an operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$ as usual, the operad ${\operatorname}{Perm}_\circ({\mathcal{O}})$ is generated by non-symmetric operations $\prec_i,\succ_i,\circ_j,\ast_k$, together with the larger set of relations making $C^*(\Delta_1,v_l;V)$ with the cup products a local dg ${\operatorname}{Perm}_\circ({\mathcal{O}})$-algebra for every ${\mathcal{O}}$-algebra $V$.
We want to describe a generating set of relations of ${\operatorname}{Perm}_\circ({\mathcal{O}})$ as in the associative case. We denote the elements of the symmetric group $S_3=\{{\operatorname{id}},(123),(132),(12),(13),(23)\}$ according to their cycle decomposition. Given a terniary operation $R(-,-,-):C^*(\Delta_1,v_l;V)^{\otimes 3}\to C^*(\Delta_1,v_l;V)$ in the cup products, we write $R=R_1+R_2+R_3$, where $R_1:=R_{{\operatorname{id}}}+R_{(23)}$, $R_2:=R_{(123)}+ R_{(12)}$ and $R_3:=R_{(132)}+R_{(13)}$. By Lemma \[lem:relpl\], the relation $R(-,-,-)=0$ holds in $C^*(\Delta_1,v_l;V)$ if and only if $$R(\xrightarrow{x},\to_y,\to_z)=R(\to_x,\xrightarrow{y},\to_z)=R(\to_x,\to_y,\xrightarrow{z})=0$$ for all $x,y,z\in V$. We claim that this is true if and only if so is $R_i(-,-,-)=0$ for $i=1,2,3$: again, we limit ourselves to show $R(-,-,-)=0{\Rightarrow}R_1(-,-,-)=0$. This time, the point is that the only way to form a non-vanishing triple product out of $\xrightarrow{1},\to_1,\to_1\in (C^*(\Delta_1,v_l;\mathbb{K}),\cup)$ is by putting the degree one cochain $\xrightarrow{1}$ on the left: this implies $R_1(\to_x,\xrightarrow{y},\to_z)=R_1(\to_x,\to_y,\xrightarrow{z})=0$, whereas $R_1(\xrightarrow{x},\to_y,\to_z)=R(\xrightarrow{x},\to_y,\to_z)=\xrightarrow{R'(x,y,z)}=0$ by hypothesis, where $R'(-,-,-)=0$ is a relation of ${\mathcal{O}}$ since $V$ is generic, so we may conclude again by Lemma \[lem:relpl\]. To sum up
\[lem:relpm\] The operad ${\operatorname}{Perm}_\circ({\mathcal{O}})$ has a generating set of relations $R(-,-,-)=0$ of the form $R=R_{{\operatorname{id}}}+R_{(23)}$, consisting of all the possible splittings, as in the following $$R(\xrightarrow{x},\to_y,\to_z)=\xrightarrow{R'(x,y,z)}=0,$$ of a relation $R'(-,-,-)=0$ in ${\mathcal{O}}$ via the cup products on $C^*(\Delta_1,v_l;V)$.
\[th:permwhite\] There is a natural isomorphism ${\operatorname}{Perm}_\circ(-)\xrightarrow{\cong}{\mathcal{P}}erm\circ-$ of functors $\mathbf{Op}\to\mathbf{Op}$.
This is done as in Theorem \[th:asswhite\] by noticing that the relations of ${\mathcal{P}}erm\circ{\mathcal{O}}$-algebra are satisfied by the cup products on the tensor product $C^*(\Delta_1,v_l;V)$ of the right permutative algebra $(C^*(\Delta_1,v_l;\mathbb{K}),\cup)$ and a generic ${\mathcal{O}}$-algebra $V$. Conversely, given a generating relation $R(-,-,-)=0$ of ${\operatorname}{Perm}_\circ({\mathcal{O}})$ as in the previous lemma, it is straightforward to check that this holds more in general in the tensor product $A\otimes V$ of a generic ${\mathcal{P}}erm$-algebra $A$ and a generic ${\mathcal{O}}$-algebra $V$.
We consider the operad $({\mathcal{L}}ie,[-,-])$: by the previous theorem ${\operatorname}{Perm}_\circ({\mathcal{L}}ie)={\mathcal{P}}erm\circ{\mathcal{L}}ie={\mathcal{L}}eib$, where the right Leibniz relation corresponds to the splitting as in Lemma \[lem:relpm\] $$(\xrightarrow{x}\ast \to_y)\ast\to_z-\xrightarrow{x}\ast(\to_y\ast\to_z)-(\xrightarrow{x}\ast\to_z)\ast\to_y=\xrightarrow{[[x,y],z]-[x,[y,z]]-[[x,z],y]}=0.$$
We consider the operad $({\mathcal{P}}ois,\bullet,[-,-])$. This is generated by a right permutative product $\circ$ and a right Leibniz product $\ast$: we have the following splittings of the Poisson identity $$\begin{gathered}
0=\xrightarrow{[x\bullet y,z]-x\bullet [y,z]-[x,z]\bullet y} = \left( \xrightarrow{x}\circ\to_y\right)\ast\to_z -\xrightarrow{x}\circ(\to_y\ast\to_z)-\left( \xrightarrow{x}\ast\to_z\right)\circ\to{y}=\\= \left( \xrightarrow{x}\circ\to_y\right)\ast\to_z +\xrightarrow{x}\circ(\to_z\ast\to_y)-\left( \xrightarrow{x}\ast\to_z\right)\circ\to{y}, \end{gathered}$$ $$\begin{gathered}
0=\xrightarrow{[x,y\bullet z]-[x,y]\bullet z-[x,z]\bullet y}= \xrightarrow{x}\ast\left( \to_y\circ\to_z\right) -\left(\xrightarrow{x}\ast\to_y\right)\circ\to_z-\left(\xrightarrow{x}\ast\to_z\right)\circ\to_y =\\ = \xrightarrow{x}\ast\left( \to_z\circ\to_y\right) -\left(\xrightarrow{x}\ast\to_y\right)\circ\to_z-\left(\xrightarrow{x}\ast\to_z\right)\circ\to_y,\end{gathered}$$ corresponding to the three independent relations $$(\alpha\circ \beta)\ast\gamma=\alpha\circ(\beta\ast\gamma)+(\alpha\ast\gamma)\circ\beta,\quad\alpha\ast(\beta\circ\gamma)=(\alpha\ast\beta)\circ\gamma+(\alpha\ast\gamma)\circ\beta,\quad \alpha\circ(\beta\ast\gamma)=-\alpha\circ(\gamma\ast\beta),$$ in the operad ${\operatorname}{Perm}_\circ({\mathcal{P}}ois)$. In fact, these are the relations defining Aguiar’s operad ${\mathcal{P}}erm\circ{\mathcal{P}}ois=(pre{\mathcal{L}}ie\bullet{\mathcal{P}}ois)^!$ of dual pre-Poisson algebras [@A; @U].
We consider the operad $({\mathcal{A}}ss,\cdot)$: then we know ${\mathcal{P}}erm\circ{\mathcal{A}}ss=di{\mathcal{A}}ss$. In this case, the relations we find as in Lemma \[lem:relpm\] turn out to be the diassociative relations for the generating products $\prec$, $\curlyeqsucc:=\succ^{op}$ of ${\operatorname}{Ass}_\circ({\mathcal{O}})$: in fact, we find $$0=\xrightarrow{(x\cdot y)\cdot z-x\cdot(y\cdot z)}=(\xrightarrow{x}\prec\to_y)\prec\to_z -\xrightarrow{x}\prec(\to_y\prec\to_z)=(\xrightarrow{x}\prec\to_y)\prec\to_z -\xrightarrow{x}\prec(\to_z\succ\to_y),$$ $$0=\xrightarrow{z\cdot(y\cdot x)-(z\cdot y)\cdot x}=(\xrightarrow{x}\succ\to_y)\succ\to_z -\xrightarrow{x}\succ(\to_y\succ\to_z)=\\=(\xrightarrow{x}\succ\to_y)\succ\to_z -\xrightarrow{x}\succ(\to_z\prec\to_y),$$ $$0=\xrightarrow{(y\cdot x)\cdot z-y\cdot(x\cdot z)} =(\xrightarrow{x}\succ\to_y)\prec\to_x-(\xrightarrow{x}\prec\to_z)\succ\to_y.$$ The reader may compare this with the computation of ${\operatorname}{preLie}_\bullet({\mathcal{A}}ss)$ in Example \[ex:ass\], as well as what we said in Remark \[rem:curlyeqprec\].
We consider the operad $(pre{\mathcal{L}}ie,\cdot)$. We split the right pre-Lie identity as $$0=\xrightarrow{(x\cdot y)\cdot z-x\cdot(y\cdot z)-(x\cdot z)\cdot y +x\cdot(z\cdot y)}= (\xrightarrow{x}\prec\to_y)\prec\to_z-\xrightarrow{x}\prec\to_{y\cdot z}-(\xrightarrow{x}\prec\to_z)\prec\to_y+\xrightarrow{x}\prec\to_{z\cdot y},$$ Since we may split $\to_{y\cdot z}$ both as $\to_y\prec\to_z=\to_{y\cdot z}=\to_z\succ\to_y$, and similarly for $\to_{z\cdot y}$, we get the two independent relations $(\alpha\prec\beta)\prec\gamma-\alpha\prec(\beta\prec\gamma)=(\alpha\prec\gamma)\prec\beta-\alpha\prec(\gamma\prec\beta)$ and $\alpha\prec(\beta\prec\gamma)=\alpha\prec(\gamma\succ\beta)$ in the operad ${\operatorname}{Perm}_\circ(pre{\mathcal{L}}ie)$. We also have the splitting $$0=\xrightarrow{z\cdot (y\cdot x)-(z\cdot y)\cdot x-z\cdot(x\cdot y) +(z\cdot x)\cdot y}= (\xrightarrow{x}\succ\to_y)\succ\to_z-\xrightarrow{x}\succ\to_{z\cdot y}-(\xrightarrow{x}\prec\to_y)\succ\to_z+(\xrightarrow{x}\succ\to_{z})\prec\to_y,$$ giving the relations $(\alpha\succ\beta)\succ\gamma-\alpha\succ(\beta\succ\gamma)=(\alpha\prec\beta)\succ\gamma-(\alpha\succ\gamma)\prec\beta$ and $\alpha\succ(\beta\succ\gamma)=\alpha\succ(\gamma\prec\beta)$. We leave to the reader to check that this is a generating set of relations, and in fact the operad $({\operatorname}{Perm}_\circ(pre{\mathcal{L}}ie),\prec,\curlyeqsucc:=\succ^{op})$ with the relations $$\begin{aligned}
\nonumber
(\alpha\prec\beta)\prec\gamma-\alpha\prec(\beta\prec\gamma)\:=\:(\alpha\prec\gamma)\prec\beta-\alpha\prec(\gamma\prec\beta), & \alpha\prec(\beta\prec\gamma)\:=\:\alpha\prec(\beta\curlyeqsucc\gamma), \\ \nonumber (\alpha\curlyeqsucc\beta)\curlyeqsucc\gamma-\alpha\curlyeqsucc(\beta\curlyeqsucc\gamma)\:=\:(\alpha\curlyeqsucc\gamma)\prec\beta-\alpha\curlyeqsucc(\gamma\prec\beta), & (\alpha\curlyeqsucc\beta)\curlyeqsucc\gamma \:=\:(\alpha\prec\beta)\curlyeqsucc\gamma,\end{aligned}$$ is the Koszul dual of the operad ${\operatorname}{preLie}_\bullet({\mathcal{P}}erm)$ from Example \[ex:perm\].
We finally come to the definition of ${\operatorname}{Lie}_\circ(-):\mathbf{Op}\to\mathbf{Op}$. In this case, we consider the complex $C^*(\Delta_1;\mathbb{K})$ as a ${\mathcal{L}}ie$-algebra via the bracket $[-,-]=\cup-\cup^{op}$, then given a generic ${\mathcal{O}}$-algebra $(V,\cdot_i,\bullet_j,\ast_k)$ we shall call the tensor product operations $\star_i := [-,-]\otimes\cdot_i=(\cup-\cup^{op})\otimes\cdot_i=\prec_i-\succ_i^{op}$, $\{-,-\}_j:=[-,-]\otimes\bullet_j=\circ_j-\circ_j^{op}$, $\circledast_k:=[-,-]\otimes[-,-]_k=\ast_k+\ast_k^{op}$ the cup brackets on $C^*(\Delta_1;V)$.
Given an operad $({\mathcal{O}},\cdot_i,\bullet_j,[-,-]_k)$ as usual, the operad ${\operatorname}{Lie}_\circ({\mathcal{O}})$ is generated by non-symmetric operations $\star_i$, anti-symmetric operations $\{-,-\}_j$ and symmetric operations $\circledast_k$ together with the larger set of relations making $C^*(\Delta_1;V)$ with the cup brackets a local dg ${\operatorname}{Lie}_\circ({\mathcal{O}})$-algebra for every ${\mathcal{O}}$-algebra $V$.
\[th:liewhite\] There is a natural isomorphism ${\operatorname}{Lie}_\circ(-)\xrightarrow{\cong}{\mathcal{L}}ie\circ-$ of functors $\mathbf{Op}\to\mathbf{Op}$.
This follows from Theorem \[th:asswhite\]: in fact, by construction ${\operatorname}{Lie}_\circ({\mathcal{O}})$ is the suboperad of ${\operatorname}{Ass}_\circ({\mathcal{O}})$ generated by the operations $\star_i=\prec_i-\succ_i^{op}$, $\{-,-\}_j=\circ_j-\circ_j^{op}$, $\circledast_k=\ast_k+\ast_k^{op}$, and similarly ${\mathcal{L}}ie\circ{\mathcal{O}}$ is the suboperad of ${\mathcal{A}}ss\circ{\mathcal{O}}$ generated by the operations $(\cup-\cup^{op})\otimes\cdot_i$, $(\cup-\cup^{op})\otimes\bullet_j$, $(\cup-\cup^{op})\otimes[-,-]_k$.
\[ex:liewhite\] The computations from the previous section, together with the above theorem and the fact that ${\mathcal{L}}ie\circ{\mathcal{O}}=({\mathcal{C}}om\bullet{\mathcal{O}}^!)^!$, imply for instance ${\operatorname}{Lie}_\circ({\mathcal{A}}ss)={\operatorname}{Lie}_\circ(pre{\mathcal{L}}ie)={\mathcal{M}}ag_{1,0,0}$, ${\operatorname}{Lie}_\circ({\mathcal{L}}ie)={\mathcal{M}}ag_{0,1,0}$, ${\operatorname}{Lie}_\circ({\mathcal{P}}ois)={\mathcal{L}}ie+{\mathcal{M}}ag_{0,1,0}$, ${\operatorname}{Lie}_\circ({\mathcal{P}}erm)={\mathcal{L}}eib$, ${\operatorname}{Lie}_\circ({\mathcal{Z}}inb)=pre{\mathcal{L}}ie$. To illustrate the latter, we show that the (right) pre-Lie relation holds in the tensor product $L\otimes V$ of a generic Lie algebra $(L,[-,-])$ and a generic ${\mathcal{Z}}inb$-algebra $(V,\cdot)$, equipped with the tensor product operation $\star:=[-,-]\otimes\cdot$. The associator of $\star$ is given by $A_\star(l{\otimes}x,m{\otimes}y,n{\otimes}z)=[[l,m],n]{\otimes}(x\cdot y)\cdot z\,-\,[l,[m,n]]{\otimes}x\cdot(y\cdot z)$: to show that this is graded symmetric in the last two arguments, we compute $$[[l,m],n]{\otimes}(x\cdot y)\cdot z\,-\,[l,[m,n]]{\otimes}x\cdot(y\cdot z)\,-\,[[l,n],m]{\otimes}(x\cdot z)\cdot y\,+\,[l,[n,m]]{\otimes}x\cdot(z\cdot y) =$$ $$=[l,[m,n]]{\otimes}\left( (x\cdot y)\cdot z-x\cdot(y\cdot z)-x\cdot(z\cdot y) \right)\,+\,[[l,n],m]{\otimes}\left( (x\cdot y)\cdot z -(x\cdot z)\cdot y \right)\:=\:0.$$
We are finally ready to complete the proof of Theorem \[th:black\]: this will follow from the following theorem and theorems \[th:asswhite\], \[th:permwhite\], \[th:liewhite\].
\[th:adj\] The following $\xymatrix{{\operatorname}{Ass}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Ass}_\circ(-)\ar@<2pt>[l]}$, $\xymatrix{{\operatorname}{Com}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Lie}_\circ(-)\ar@<2pt>[l]}$, $\xymatrix{{\operatorname}{preLie}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Perm}_\circ(-)\ar@<2pt>[l]}$ are pairs of adjoint functors.
We consider the case of $\xymatrix{{\operatorname}{Ass}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Ass}_\circ(-)\ar@<2pt>[l]}$ in detail.
So far we used overlapping notations for the generating sets of operations of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ and ${\operatorname}{Ass}_\circ({\mathcal{O}})$: this isn’t practical to prove adjointness, thus, only for this proof, we will have to change notations; moreover, we will use slightly different generating sets than the ones we used before. We consider first the case of a non-symmetric operation $\cdot_i$ of ${\mathcal{O}}$. Corresponding to $\cdot_i$ there are two generating operations of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ which we denote by $\underline{\cdot_i}$ and $\overline{\cdot_i}$ respectively: we use the generating set of operations from Remark \[rem:curlyeqprec\], explicitly, $\underline{\cdot_i}$ and $\overline{\cdot_i}$ are defined by the formulas $_x\to\cdot_i\xrightarrow{y}\:=\:\xrightarrow{x\underline{\cdot_i} y}\:=\:\xrightarrow{x}\cdot_i\to_y$ and $\to_x\cdot_i\xrightarrow{y}\:=\:\xrightarrow{x\overline{\cdot_i}y}\:=\:\xrightarrow{x}\cdot_i\,_y\to$ (with the previous notations from remark \[rem:curlyeqprec\] we have $\underline{\cdot_i}=\prec_i$ and $\overline{\cdot_i}=\curlyeqsucc_i=-\succ_i^{op}$). Likewise, corresponding to $\cdot_i$ there are two generating operations of ${\operatorname}{Ass}_\circ({\mathcal{O}})$ which we denote by $\cup\otimes\cdot_i$ and $\cup^{op}\otimes\cdot_i$ respectively: explicitly, these are defined by the formulas $_x\to(\cup\otimes\cdot_i)\xrightarrow{y}=\xrightarrow{x\cdot_i y}=\xrightarrow{x}(\cup\otimes\cdot_i)\to_y$ and $\to_x(\cup^{op}\otimes\cdot_i)\xrightarrow{y}=\xrightarrow{x\cdot_i y}=\xrightarrow{x}(\cup^{op}\otimes\cdot_i)\,_y\to$ (with the previous notations from this section we have $\cup\otimes\cdot_i=\prec_i$ and $\cup^{op}\otimes\cdot_i=\succ_i^{op}$). Finally, corresponding to $\cdot_i$ there are four generating operations of the operad ${\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$, which we denote by $\cup\otimes\underline{\cdot_i}$, $\cup^{op}\otimes\underline{\cdot_i}$, $\cup\otimes\overline{\cdot_i}$ and $\cup^{op}\otimes\overline{\cdot_i}$ respectively, and four generating operations of ${\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))$, which we denote by $\underline{\cup\otimes\cdot_i}$, $\overline{\cup\otimes\cdot_i}$, $\underline{\cup^{op}\otimes\cdot_i}$ and $\overline{\cup^{op}\otimes\cdot_i}$ respectively. Similarly, to a generating symmetric operation $\bullet_j$ of ${\mathcal{O}}$ correspond a generating (non-symmetric) operation $\underline{\bullet_j}$ of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ and a generating (non-symmetric) operation $\cup\otimes\bullet_j$ of ${\operatorname}{Ass}_\circ({\mathcal{O}})$, as well as two generating (non-symmetric) operations $\cup\otimes\underline{\bullet_j}$, $\cup^{op}\otimes\underline{\bullet_j}$ of ${\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$ and two generating (non-symmetric) operations $\underline{\cup\otimes\bullet_j}$, $\overline{\cup\otimes\bullet_j}$ of ${\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))$. Finally, to a generating antisymmetric operation $[-,-]_k$ of ${\mathcal{O}}$ correspond generating (non-symmetric) operations $\underline{[-,-]_k}$ and $\cup\otimes[-,-]_k$ of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ and ${\operatorname}{Ass}_\circ({\mathcal{O}})$ respectively, as well as generating operations $\cup\otimes\underline{[-,-]_k}$, $\cup^{op}\otimes\underline{[-,-]_k}$ of ${\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$ and $\underline{\cup\otimes[-,-]_k}$, $\overline{\cup\otimes[-,-]_k}$ of ${\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))$ respectively.
Having established the previous notations, we will prove the theorem by explicitly exihibiting the unit $\varepsilon_{\mathcal{O}}:{\mathcal{O}}\to{\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$ and the counit $\mu_{{\mathcal{O}}}:{\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))\to{\mathcal{O}}$ of the adjunction. The construction of the counit is implicit in the very definition of the functors ${\operatorname}{Ass}_\circ(-)$ and ${\operatorname}{Ass}_\bullet(-)$: given an ${\mathcal{O}}$-algebra $V$, the tensor product operations induce a local dg ${\operatorname}{Ass}_\circ({\mathcal{O}})$-algebra structure on the complex of $V$-valued cochains $C^*(\Delta_1;V)=C^*(\Delta_1;\mathbb{K})\otimes V$, but by definition this is the same as an ${\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))$-algebra structure on the space $V$. Unraveling the definitions, we find that the counit $\mu_{{\mathcal{O}}}:{\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))\to{\mathcal{O}}$ is explicitly given by $$\underline{\cup\otimes\cdot_i},\:\overline{\cup^{op}\otimes\cdot_i}\to\cdot_i,\qquad\overline{\cup\otimes\cdot_i},\: \underline{\cup^{op}\otimes\cdot_i}\to0,$$ $$\underline{\cup\otimes\bullet_j}\to\bullet_j,\qquad\overline{\cup\otimes\bullet_j}\to0,\qquad\underline{\cup\otimes[-,-]_k}\to[-,-]_k,\qquad\overline{\cup\otimes[-,-]_k}\to0.$$ As already remarked, this defines a morphism of operads $\mu_{{\mathcal{O}}}:{\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}}))\to{\mathcal{O}}$ by construction of the functors ${\operatorname}{Ass}_\circ(-)$ and ${\operatorname}{Ass}_\bullet(-)$. It remains to define the unit $\varepsilon_{\mathcal{O}}:{\mathcal{O}}\to{\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$, this is explicitly given by $$\cdot_i\to \cup\otimes\underline{\cdot_i}+\cup^{op}\otimes\overline{\cdot_i}, \qquad\bullet_j\to\cup\otimes\underline{\bullet_j}+(\cup\otimes\underline{\bullet_j})^{op},\qquad[-,-]_k\to\cup\otimes\underline{[-,-]_k}-(\cup\otimes\underline{[-,-]_k})^{op}.$$ We have to show that this is a morphism of operads. Given a generic ${\operatorname}{Ass}_\bullet({\mathcal{O}})$-algebra $(V,\underline{\cdot_i},\overline{\cdot_i},\underline{\bullet_j},\underline{[-,-]_k})$, the complex $C^*(\Delta_1;V)$ carries both an ${\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$-algebra structure via the tensor product operations and an ${\mathcal{O}}$-algebra structure $(C^*(\Delta_1;V),\cdot_i,\bullet_j,[-,-]_k)$ by definition of ${\operatorname}{Ass}_\bullet(-)$. An easy verification shows $\alpha(\cup\otimes\underline{\cdot_i}+\cup^{op}\otimes\overline{\cdot_i})\beta=\alpha\cdot_i\beta$ for all $\alpha,\beta\in C^*(\Delta_1;V)$. For instance, $$\to_x (\cup\otimes\underline{\cdot_i}+\cup^{op}\otimes\overline{\cdot_i}) \xrightarrow{y}=\to_x(\cup^{op}\otimes\overline{\cdot_i})\xrightarrow{y}=\xrightarrow{x\overline{\cdot_i} y}=\to_x\cdot_i\xrightarrow{y}.$$ Similarly, one verifies $\alpha(\cup\otimes\underline{\bullet_j}+(\cup\otimes\underline{\bullet_j})^{op})\beta=\alpha\bullet_j\beta$ and $\alpha(\cup\otimes\underline{[-,-]_k}-(\cup\otimes\underline{[-,-]_k})^{op})\beta=[\alpha,\beta]_k$ for all $\alpha,\beta\in C^*(\Delta_1;V)$. Finally, given a relation $R(-,-,-)=0$ of ${\mathcal{O}}$, this induces a relation $R'(-,-,-)=0$ of ${\operatorname}{Ass}_\bullet({\mathcal{O}})$ as in Lemma \[lem:rel\] $$R(_x\to,\xrightarrow{y},\to_z)=\xrightarrow{R'(x,y,z)}=0.$$ On the other hand, by the previous considerations $R(-,-,-)$ in the left hand side can be computed equivalently either in the products $\cdot_i,\bullet_j,[-,-]_k$ of the ${\mathcal{O}}$-algebra structure or in the products $\cup\otimes\underline{\cdot_i}+\cup^{op}\otimes\overline{\cdot_i}, \:\:\cup\otimes\underline{\bullet_j}+(\cup\otimes\underline{\bullet_j})^{op},\:\:\cup\otimes\underline{[-,-]_k}-(\cup\otimes\underline{[-,-]_k})^{op}$ of the ${\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$-algebra structure, which shows that $\varepsilon_{\mathcal{O}}$ sends an ${\mathcal{O}}$-algebra relation to an ${\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}}))$-algebra relation, and is thus a well defined morphism of operads.
To complete the proof, it remains to show that $\varepsilon_{-}$ and $\mu_{-}$ satisfy the conditions to be the unit and the counit of an adjunction. More precisely, we have to show that the compositions $${\operatorname}{Ass}_\bullet({\mathcal{O}})\xrightarrow{{\operatorname}{Ass}_\bullet(\varepsilon_{\mathcal{O}})}{\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\mathcal{O}})))\xrightarrow{\mu_{{\operatorname}{Ass}_\bullet({\mathcal{O}})}}{\operatorname}{Ass}_\bullet({\mathcal{O}}),$$ $${\operatorname}{Ass}_\circ({\mathcal{O}})\xrightarrow{\varepsilon_{{\operatorname}{Ass}_\circ({\mathcal{O}})}}{\operatorname}{Ass}_\circ({\operatorname}{Ass}_\bullet({\operatorname}{Ass}_\circ({\mathcal{O}})))\xrightarrow{{\operatorname}{Ass}_\circ(\mu_{\mathcal{O}})}{\operatorname}{Ass}_\circ({\mathcal{O}}),$$ are the respective identities. We consider the first one: notice that given a morphism $f:{\mathcal{O}}\to{\mathcal{P}}$ of operads, the morhism ${\operatorname}{Ass}_\bullet(f)$ is defined by ${\operatorname}{Ass}_\bullet(f)(\underline{\cdot_i})=\underline{f(\cdot_i)}$, ${\operatorname}{Ass}_\bullet(f)(\overline{\cdot_i})=\overline{f(\cdot_i)}$, ${\operatorname}{Ass}_\bullet(f)(\underline{\bullet_j})=\underline{f(\bullet_j)}$ and ${\operatorname}{Ass}_\bullet(f)(\underline{[-,-]_k})=\underline{f([-,-]_k)}$. Now it is easy to compute the first composition using the previous formulas (for a non-symmetric operation $\#$ we notice that $\underline{(\#^{op})}=(\overline{\#})^{op}$, in the following computation we apply this for $\#=\cup\otimes\underline{\bullet_j}$ and $\#=\cup\otimes\underline{[-,-]_k}$) $$\underline{\cdot_i}\to\underline{\cup\otimes\underline{\cdot_i}}+\underline{\cup^{op}\otimes\overline{\cdot_i}}\to\underline{\cdot_i}+0,\qquad\overline{\cdot_i}\to\overline{\cup\otimes\underline{\cdot_i}}+\overline{\cup^{op}\otimes\overline{\cdot_i}}\to0+\overline{\cdot_i},$$$$\underline{\bullet_j}\to\underline{\cup\otimes\underline{\bullet_j}+(\cup\otimes\underline{\bullet_j})^{op}}=\underline{\cup\otimes\underline{\bullet_j}}+(\overline{\cup\otimes\underline{\bullet_j}})^{op}\to\underline{\bullet_j}+0,$$$$\underline{[-,-]_k}\to\underline{\cup\otimes\underline{[-,-]_k}-(\cup\otimes\underline{[-,-]_k})^{op}}=\underline{\cup\otimes\underline{[-,-]_k}}-(\overline{\cup\otimes\underline{[-,-]_k}})^{op}\to\underline{[-,-]_k}-0.$$ Next we consider the second composition. We have ${\operatorname}{Ass}_\circ(f)(\cup\otimes\cdot_i)=\cup\otimes f(\cdot_i)$, ${\operatorname}{Ass}_\circ(f)(\cup^{op}\otimes \cdot_i)=\cup^{op}\otimes f(\cdot_i)$, and similarly for the other cases. As desired, the second composition is $$\cup\otimes\cdot_i\to\cup\otimes\underline{\cup\otimes\cdot_i} +\cup^{op}\otimes\overline{\cup\otimes\cdot_i}\to\cup\otimes\cdot_i+0,$$$$\cup^{op}\otimes\cdot_i\to\cup\otimes\underline{\cup^{op}\otimes\cdot_i} +\cup^{op}\otimes\overline{\cup^{op}\otimes\cdot_i}\to0+\cup^{op}\otimes\cdot_i,$$ $$\cup\otimes\bullet_j\to\cup\otimes\underline{\cup\otimes\bullet_j} +\cup^{op}\otimes\overline{\cup\otimes\bullet_j}\to\cup\otimes\bullet_j+0,$$ $$\cup\otimes[-,-]_k\to\cup\otimes\underline{\cup\otimes[-,-]_k} +\cup^{op}\otimes\overline{\cup\otimes[-,-]_k}\to\cup\otimes[-,-]_k+0.$$
The remaining cases of $\xymatrix{{\operatorname}{Com}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Lie}_\circ(-)\ar@<2pt>[l]}$ and $\xymatrix{{\operatorname}{preLie}_\bullet(-):\mathbf{Op}\ar@<2pt>[r]& \mathbf{Op}:{\operatorname}{Perm}_\circ(-)\ar@<2pt>[l]}$ are proved by the same argument: in the former, we notice that we may identify ${\operatorname}{Com}_\bullet({\operatorname}{Lie}_\circ({\mathcal{O}}))$, ${\mathcal{O}}$ and ${\operatorname}{Lie}_\circ({\operatorname}{Com}_\bullet({\mathcal{O}}))$ with quotients of the same free operad $({\mathcal{M}}ag_{p,q,r},\cdot_i,\bullet_j,[-.-]_k)$ in such a way that the unit and the counit are the identities on the generating operations.
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abstract: |
The exact distribution of the square sum of Dirichlet random variables is given by two different univariate integral representations. Alternatively, three representations by orthogonal series with Jacobi or Legendre polynomials are derived. As a special case the distribution of the square sum $U_n^2$ of spacings - also called Greenwood’s statistic - is obtained. Nine quantiles of $nU_n^2-1$ are tabulated with eight digits for $n$ from 10 to 100.\
*Key words and phrases:* Greenwood’s statistic, square sum of spacings, Dirichlet distribution, distribution of the sample coefficient of variation from gamma random variables, exact distributions, tabulated quantiles, goodness of fit tests, tests for exponentiality, tests for gamma distributions.\
*MSC 2010 subject classifications:* 62E15, 62H10
author:
- |
T. Royen\
Fachhochschule Bingen, University of Applied Sciences,\
Berlinstraße 109, D-55411 Bingen, Germany\
E-mail:
title: 'THE DISTRIBUTION OF THE SQUARE SUM OF DIRICHLET RANDOM VARIABLES AND A TABLE WITH QUANTILES OF GREENWOOD’S STATISTIC'
---
Introduction and Notation
=========================
Greenwood’s statistic is defined by
$$U^2_{k+1}= \sum^{k+1}_{i=1}\left( U_{i:k}- U_{i-1 : k}\right)^2$$
where the $U_{i : k}$ denote the successive ordered values of a random sample $U_1, \ldots, U_k$ from a uniform distribution on $(0,1)$ with $U_{0 : k}= 0$ and $ U_{k+1 : k}=1$. If $F_0$ denotes any completely specified continuous cdf then, with a random sample $X_1, \ldots X_k$ and $U_i=F_0 (X_i)$, the statistic $U_{k+1}^ 2$ can be used for a test of the hypothesis that the unknown cdf $F$ of $X$ coincides with $F_0$. For the efficiency of this test see e.g. Pyke (1965). For the application of Greenwood’s statistic by a test for exponentiality see e.g. Subhash and Gupta (1988).\
The computation of the distribution of $U_n^2$ has a long history beginning with Moran (1947) and (1951), Gardner (1952) followed by Hill (1979), Burrows (1979), Currie (1981), Stephens (1981) Currie and Stephens (1986), Does et al. (1988), Ghosh and Jammalamadaka (1998) and (2000). The exact methods of Burrows, Currie and Stephens use iterated integrations up to $n = 20$. The subsequent papers are concerned with approximations mainly by Edgeworth expansions and saddle-point methods. It is well known that the distribution of the standardized $U_n^2$ converges very slowly to a normal distribution. For the asymptotic normality of a more general class of statistics, based on spacings, see Darling (1953). A comparison of the approximate quantiles with the available “exact” quantiles for small sample sizes e.g. in Ghosh and Jammalamadaka (2000) shows rather unsatisfying discrepancies. However, thanks to the increasing computing power of computer algebra systems, computations are available based on exact representations of the cdf of Greenwood’s statistic. Thus errors are reduced solely to those of the numerical evaluation.\
In this paper more generally some exact representations are given for the distribution of the square sum $U_n^2$ of Dirichlet random variables derived from gamma distributed variables of order $\alpha$. This statistic can be used as a competitor with the maximum likelihood statistic for the test of a specified value $\alpha_0$ of the unknown order parameter $\alpha$ of a gamma distribution with any unknown scale parameter. For the maximum likelihood estimation of $\alpha$ see Provost (1988) and chapter 17 in Kotz et al. (1994). Greenwood’s distribution is obtained by the special case $\alpha= 1$. In section 4 the cdf of $U_n$ is represented by orthogonal series with Jacobi or Legendre polynomials. In section 5 two different univariate integral representations for the cdf of $U_n^2$ are found, derived by the Fourier and Laplace inversion formula. Theorem 2, the main result, provides also the more general case with gamma random variables of different orders $\alpha_i$ by a univariate integral over a product of parabolic cylinder functions. Theoretically, the integral representations look more elegant since no special coefficients have to be computed before. For the orthogonal series a sequence of moments must be computed, but the computing effort can be reduced to two linear recursion formulas. The eight digit quantiles of Greenwood’s statistic for $n$ from 10 to 100 in section 6 were computed by the orthogonal series in (\[eq\_4.6\]) and again by (\[eq\_4.7\]) . In spite of the very high computing accuracy, required to achieve the eight digit quantile values, this method works more quickly than the integral representations in section 5.
Now let $X_1, \ldots, X_n$ be i.i.d. random variables with a gamma pdf $$\label{eq_1.1}
g_a (x) = \left( \Gamma (\alpha)\right)^{-1} x^{a - 1} e^{- x} \ \textrm{and} \ G_a (x) = \int_0^x g_a (\xi) d \xi \, , \, \alpha > 0 \, , \ x > 0 \, .$$ The “sample triangle” in $\mathbb{R}^n$ generated by $X_1, \ldots, X_n$ , has the vertices $(0, \ldots , 0), (X_1, \ldots , X_n), \ ( \bar{X}, \ldots , \bar{X})$, with the sample mean $\bar{X}$ and the angle $\Phi$ between the vectors ($X_1, \ldots, X_n$ ) and $(1, \ldots, 1)$. We use the further notations $$\begin{aligned}
Y &=& \sum_{i=1}^n X_i = n \bar{X} \, , \quad \ Z=R^2= \sum_{i=1}^n X_i^2 \, , \nonumber \\
U &=& \frac{R}{Y}= \frac{1}{\sqrt{n} \cos \Phi}\, , \ \ Q=R^2 - n \bar{X}^2 = n S^2 \, . \label{eq_1.2}\end{aligned}$$ The random variables $Y_i = X_i /Y , \ i = 1\, , \ldots \, , n-1$ have the joint Dirichlet density $$\frac{\Gamma (\alpha n)}{(\Gamma (\alpha))^n} \left( 1 - \textstyle{\sum_{i=1}^{n-1}} y_i \right)^{\alpha - 1} \textstyle{\prod_{i=1}^{n-1}}y_i^{\alpha-1} \, .$$ Thus, the distribution of the square sum is closely related to the distribution of the squared sample coefficient of variation $$S^2 /\bar{X}^2= \tan^2 (\Phi) = n U_n^2 -1 \, . \label{eq_1.3}$$ By Laha (1954) and Lukacs (1955) the independence of $\bar{X}$ and $S/\bar{X}$ was shown to be a characterization of the gamma distribution. We shall use the independence of $U_n$ and $Y=n \bar{X}$.
Apart from some special notations as e.g. in (\[eq\_1.1\]) the cdf, pdf, char. function (cf) and the Laplace transform (Lt) of any continuous random variable $Z$ will be denoted by $F_Z\, , \ f_Z \, , \ \hat{f}_Z$ and $f_Z^\ast$ respectively. $E_Z (\ldots )$ means expectation with regard to $Z$. The difference operator is denoted by $\Delta$, i.e. $\Delta a_k = a_{k+1}- a_k$ for any sequence $(a_k)$ and $(c)_k = \prod_{i=0}^{k-1}(c+i)$. Formulas from Abramowitz and Stegun are denoted by A.S. and their number.
The distribution of $R$ and $Q$
===============================
These distributions were already given in Royen (2007a), (2007b). The coefficients, needed for a series representation of the cdf $F_R$ of $R$, are also used in the subsequent sections. Therefore, this representation is derived here more concisely.
For a given power series $f(z) = \textstyle{\sum_{k=0}^\infty} a_k z^k , a_0 >0$, the coefficients $a_{n,k}$ of $\left(f (z)\right)^n$ can be computed by two steps: Setting $$\log \left(f (z)\right) = \log\left(a_0\right)+ \textstyle{\sum_{k=1}^\infty} b_k z^k / k\, ,$$ the relation $$\begin{aligned}
{}
f' (z)&=& \textstyle{\sum_{k= 0}^\infty} (k+1)a_{k+1}z^k =
f(z) \textstyle{\sum_{k=0}^\infty} b_{k+1} z^k \nonumber\\
&=& \textstyle{\sum_{k=0}^\infty} \left( \textstyle{\sum_{j= 0}^k} a_j b_{k+1-j}\right)z^k \ \mbox{ implies}\nonumber\\
b_{k+1}&=& a_0^{-1}\left((k+1)a_{k+1}- \textstyle{\sum_{j=1}^k} a_j b_{k+1-j}\right)\, , k \in \mathbb{N}_0 \, , \ \mbox{and}\nonumber\\
a_{n,0}&=& a_0^n \, , \ \ a_{n, k+1}= \textstyle{\frac{n}{k+1} \sum_{j=0}^k} a_{n,j}b_{k+1-j}\, , k \in \mathbb{N}_0 \ . \label{eq_2.1}\end{aligned}$$
Let be $C=\cos (\Phi)$. The moments $$\gamma_{\alpha n +k}:= E \left( C^{\alpha n+k} \right) \label{eq_2.2}$$ depend on $\alpha$ and $n$ not only by $\alpha n$, but we write $\gamma_{\alpha n+k}$ for simplicity instead more precisely $\gamma_{\alpha ,n, \alpha n+k}$. With $U^{-1} = \sqrt{n}C$ it follows from the independence of $U$ and $\bar{X}$ that $$\begin{aligned}
F_R (r) & = & E_U \left( G_{\alpha n} (r U^{-1}) \right)\nonumber \\
& = & E_C \left( G_{\alpha n} (r C\sqrt{n}) \right) = \frac{(r \sqrt{n})^{\alpha n}}{\Gamma (\alpha n)} \sum_{k= 0}^\infty \frac{\gamma_{\alpha n+k}}{\alpha n +k} \frac{(-r \sqrt{n})^k}{k!}\, . \label{eq_2.3}\end{aligned}$$ On the other hand we have with $$\begin{aligned}
a_{\alpha, k} &=& \frac{
\Gamma \left( (\alpha+k)/2\right)}{2 \Gamma (\alpha) k!}\ \mbox{the Lt} \nonumber \\[.8ex]
f_Z^\ast (t) &=& \left( f_{X^2}^\ast (t)\right)^n = \left( t^{-\alpha/2} \textstyle{\sum_{k=0}^\infty} (-1)^k a_{\alpha , k} t^{- k/2}\right)^n \nonumber \\[1.5ex]
&=& t^{-\alpha n/2} \textstyle{\sum_{k=0}^\infty} (-1)^k a_{\alpha , n,k} t^{- k/2} \ , \label{eq_2.4}
\end{aligned}$$ where the $a_{\alpha , n,k}$ are computed from the $a_{\alpha ,k}$ by the above procedure leading to (\[eq\_2.1\]). With $Z = R^2$ we obtain from (\[eq\_2.4\]) by inversion, followed by integration $$F_R (r) = r^{\alpha n} \sum_{k=0}^\infty \frac{a_{\alpha,n,k}}{\Gamma \left( 1+ (\alpha n + k)/2\right)}(-r)^k \, , \label{eq_2.5}$$ and by comparison with (\[eq\_2.3\]) $$\gamma_{\alpha n + k}= \frac{2 \Gamma (\alpha n)k!}{\Gamma \left( (\alpha n+k)/2\right)n^{(\alpha n +k)/2 }} a_{\alpha,n,k}\ . \label{eq_2.6}$$ Finally, from the density $$\begin{aligned}
f_R (r) &=& E_\Phi \left( \sqrt{n}\cos \Phi \cdot g_{\alpha n} ( r \sqrt{n} \cos \Phi ) \right)\\
&=& \sqrt{n}g_{\alpha n} (r \sqrt{n}) E_\Phi \left( (\cos \Phi)^{\alpha n} e^{r \sqrt{n} (1-\cos \Phi )} \right)\\
&=& \sqrt{n} g_{an} (r\sqrt{n}) \sum_{k=0}^\infty E_\Phi \left( (\cos \Phi)^{\alpha n}(1-\cos \Phi)^k \right)\frac{(r \sqrt{n})^k}{k!}\\
&=& \sqrt{n} \sum_{k=0}^\infty \textstyle{\alpha n + k -1 \choose k} (-\Delta)^k \gamma_{\alpha n} \cdot g_{\alpha n + k}(r \sqrt{n})\end{aligned}$$ we obtain $F_R$ as a convex combination of gamma distribution functions: $$F_R (r) = \sum_{k=0}^\infty p_{\alpha,n,k} G_{\alpha n +k} (r \sqrt{n}) \, , \ \ p_{\alpha,n,k} = \textstyle{\alpha n + k -1 \choose k} (-\Delta)^k \gamma_{\alpha n} \, . \label{eq_2.7}$$ With gamma$(\alpha n+k)$-distributed random variables $Y_{\alpha n+k}$ and a random index $K$ with $P\{ K=k\} = p_{\alpha,n,k}$ this can be restated more concisely as a stochastic representation: $$R \ \textrm{is distributed as } n^{-1/2} Y_{\alpha n + K} \ . \label{eq_2.8}$$ The distribution of $Q=nS^2$ could be derived from the distribution of $U_n^2$ , but we have directly with $$K_\beta (x,y) = \sum_{k=0}^\infty \frac{He_k (y)}{\Gamma \left( \beta + k/2 \right)} \frac{(-x)^k}{k!} \, ,$$ where $He_k$ denotes the Hermite polynomials (A.S.22.2.15), and the parabolic cylinder functions $D_{-\alpha}$ the integral representation $$\!\!\!\!\!\!\!\!\!\!\!\! F_Q(x) = \frac{1}{\sqrt{2 \pi}}\left( \frac{x}{2} \right)^{\alpha n/2}\!\!\! \int_{- \infty}^{\infty} K_{\alpha n/2+1} \textstyle( {\sqrt{nx/2}, y} ) \! \left( D_{-\alpha} (- y/\sqrt{n})\right)^n e^{-y^2/4} dy \, , \label{eq_2.9}$$ which was given in Royen (2007b) with slightly different notations. Further representations of $F_Q$ are found in Royen (2007a).
Some Laplace transforms
=======================
If $X$ has a gamma density $g_\alpha$, then the bivariate Lt of $(X, X^2)$ is given by $$\begin{aligned}
E \left( e^{-sX-tX^2}\right) &=& \frac{1}{t^{\alpha/2}\Gamma (\alpha)} \int_0^\infty x^{\alpha-1} e^{-x^2} e^{-x(1+s)/\sqrt{t}}dx \nonumber \\
&=& \frac{1} {2t^{\alpha/2}\Gamma (\alpha)} \sum_{k=0}^\infty
\frac{\Gamma
\left( (\alpha + k)/2\right)}{k!}
\left( - \frac{1+s}{\sqrt{t}} \right)^k \nonumber \\
&=& \frac{1}{(2t)^{\alpha/2}}\exp
\left( \frac{(1+s)^2}{8t} \right)D_{-\alpha} \left( \frac{1+s}{\sqrt{2t}} \right) \label{eq_3.1}\end{aligned}$$ with the parabolic cylinder function $D_{-\alpha}$ from (A.S.19.3.1). If $\alpha \in \mathbb{N}$ then $$\!\!\!\!\!\! 2^{-\alpha/2}\exp
\left( \frac{z^2}{8} \right)
\!\! D_{-\alpha} \!\! \left( \frac{z}{\sqrt{2}} \right) =
%
{\frac{\sqrt{\pi}}{2 \Gamma (\alpha)}}
\left( - \frac{d}{dz}\right)^{\alpha-1}
\!\! \left( \exp \left( \frac{z^2}{4} \right) er\!f\!c
\left( \frac{z}{2} \right) \right)
\label{eq_3.2}$$ and in particular $$\exp \left( \frac{z^2}{4}\right) D_{-1}(z) = \sqrt{\frac{\pi}{2}} \exp \left( \frac{z^2}{2}\right) er\!f\!c \left( \frac{z}{\sqrt{2}} \right) \, .$$ >From (\[eq\_1.2\]), (\[eq\_2.2\]), (\[eq\_2.7\]) we obtain $$\begin{aligned}
F_R(r) &=& P\{ UY \leq r \}= \int_0^{r\sqrt{n}} F_U (ry^{-1}) g_{\alpha n} (y) dy \\
&=& \frac{1}{\Gamma (\alpha n)}\int_0^{r\sqrt{n}} \left( 1- F_C ( n^{-1/2} r^{-1} y ) \right) y^{\alpha n -1} e^{-y}dy \\
&=&\frac{(r\sqrt{n})^{\alpha n}}{\Gamma (\alpha n)}\int_0^1 (1 - F_C (x)) x^{\alpha n-1}e^{-xr\sqrt{n}} dx \, .\end{aligned}$$ Thus, $$\Gamma (\alpha n) s^{-\alpha n}F_R \left( n^{-1/2} s \right) =
\Gamma (\alpha n) s^{-\alpha n} \sum_{k=0}^\infty p_{\alpha , n , k} G_{\alpha n + k}(s)
\label{eq_3.3}$$ is the Lt of $x^{\alpha n-1} (1 - F_C (x)), 0 < x \leq1$. $(F_C(x) = 0 \ \ if \ x \leq n^{-1/2})$.\
Also the Lt of $C= \cos (\Phi)$ can be represented by means of the probabilities $p_{\alpha , n , k}$ from (\[eq\_2.7\]). If $Z_1$ denotes a r.v. with density $e^{-z}$, independent of $U$, and $X_{\alpha n -1}$ a r.v. with a beta$(1, \alpha n -1)$-distribution, independent of $R$, then $$\frac{E (X_{\alpha n -1}^k)}{E(Z_1^k)}= \frac{(\alpha n-1) B (k + 1 , \alpha n -1 )}{k!}=
\frac{\Gamma (\alpha n)}{\Gamma (\alpha n + k)}= \frac{1}{(\alpha n)_k}\, .$$ Comparing this with $E(R^k) = E(UY)^k = E(U^k) E(Y^k) = E(U^k) (\alpha n)_k$ it follows that $UZ_1$ and $RX_{\alpha n-1}$ have the same moments which determine the distribution completely. Thus $$UZ_1 \textrm{ is distributed as }RX_{\alpha n -1}
\label{eq_3.4}$$ and consequently the Lt $f_C^\ast$ of $C = \cos (\Phi)$ is given by $$\begin{aligned}
f_C^\ast(s) &=& E \left(e^{-sC} \right)
= E \left(e^{-s/ (\sqrt{n} \cdot U)} \right) = P \{ UZ_1 > sn^{-1/2}\} \nonumber \\
&=& P \{ RX_{\alpha n-1} > sn^{-1/2}\}
= P \{ X_{\alpha n-1}Y_{\alpha n+K}> s\} \nonumber \\[1mm]
&=& \textstyle{\sum_{k=0}^\infty} p_{\alpha, n, k} \int_s ^\infty (1-sy^{-1})^{\alpha n-1}g_{\alpha n+k}(y)dy \nonumber \\[1mm]
&=&\textstyle{ \sum_{k=0}^\infty} \left( (- \Delta)^k \gamma_{\alpha n} \right) \frac{1}{k! \Gamma (\alpha n)} \int_s^\infty (y-s)^{\alpha n-1}y^k e^{-y}dy \nonumber \\[1mm]
&=& e^{-s }\textstyle{\sum_{k=0}^\infty} \left( (- \Delta)^k \gamma_{\alpha n} \right) \frac{1}{k!}\int_0^\infty (y+s)^k g_{\alpha n} (y) dy \nonumber \\[1mm]
&=& e^{-s} \textstyle{\sum_{k=0}^\infty} \left( (- \Delta)^k \gamma_{\alpha n} \right) \textstyle{\sum_{j=0}^k} {\alpha n +k-j-1 \choose k-j} s{^j}/{j!} \nonumber \\[1mm]
&=& e^{-s} \textstyle{\left( 1 + \sum_{k=1}^\infty p_{\alpha, n, k} {\alpha n + k - 1 \choose k}^{-1}
\sum_{j=1}^k {\alpha n + k -j - 1 \choose k - j} s^j/j! \right)} . \label{eq_3.5}\end{aligned}$$
>From the binomial series we obtain the mgf of $\log (\sqrt{n}U)$ : $$\begin{aligned}
E (\sqrt{n}U)^s &=& E (\cos (\Phi))^{-s} = E \left( \frac{(\cos (\Phi))^{\alpha n}}{(1-(1-\cos (\Phi)))^{\alpha n+s}} \right) \nonumber \\[1ex]
&=& \textstyle{\sum_{k=0}^\infty} ((-\Delta)^k \gamma_{\alpha n}) \frac{(\alpha n + s)_k}{k!} = \textstyle{\sum_{k=0}^\infty}
p_{\alpha, n, k} \frac{(\alpha n + s)_k}{(\alpha n)_k} \, , \label{eq_3.6}
\end{aligned}$$ following also from (\[eq\_2.8\]) and $\hat{f}_{\log Y_{\alpha n + K}} = \hat{f}_{\log Y} \hat{f}_{\log (\sqrt{n}U)}$.\
In particular for $\alpha=1$ the “missing moments” $\gamma_m$, $m=1,\ldots n-1$, are given by $$\gamma_m = \gamma_n + \textstyle{\sum_{k=1}^\infty} p_{1, n, k} \frac{(n-m)_k}{(n)_k} \ \textrm{ or }\
\gamma_{m-1} = \textstyle{\sum_{k=0}^\infty} (-\Delta)^k \gamma_m \, , \ m=n, n-1, \ldots 2\, . \label{eq_3.7}$$ The inversion formula, derived from the Bernstein polynomials, (see e.g. Feller (1971) would already provide $$F_C(x) = \lim_{n \to \infty} \textstyle{\sum_{k \leq nx}} {n \choose k} (-\Delta)^{n-k} \gamma_k \, ,\label{eq_3.8}$$ but for a satisfying accuracy a very large $n$ would be required. Therefore, some series expansions with orthogonal polynomials will be given in the subsequent section.
Some representations of the distribution of $U$ by orthogonal polynomials
=========================================================================
The square integrability of the densities $f_C$ and $f_U$ is equivalent and guarantees the uniform convergence of the following orthogonal series. The condition (\[eq\_4.4\]) of the subsequent lemma is sufficient for the square integrability of $f_U$. \
Let $Y_1, \ldots Y_n$ be random variables with $\sum_{i=1}^n Y_i=1$ and a joint Dirichlet distribution with the parameters $\alpha_1, \ldots \alpha_n$. Then the cdf $F_U$ of $U= \left( \sum_{i=1}^n Y_i^2 \right)^{1/2}$ satisfies the following relations:
\
$$\begin{gathered}
F_U \left( n^{1/2} + \varepsilon \right)
\simeq \frac{\Gamma (\alpha)}{\prod_{i=1}^n \Gamma (\alpha_i)}
\frac{(2 \pi)^{(n-1)/2}} {\Gamma\left(( n+1)/2\right)} n^{(3n-1)/4-\alpha} \cdot \varepsilon^{(n-1)/2} , \label{eq_4.1} \\[1.5ex]
\qquad \qquad \quad \ \alpha = \textstyle{\sum_{i=1}^n} \alpha_i\, , \quad
\varepsilon \to 0 \, . \nonumber\end{gathered}$$ \
$$\begin{gathered}
\sum_{i=1}^n
P \left\{ Y_i \geq 1- \varepsilon - \frac{\varepsilon^2}{2 (n-1)} \right\} < 1 - F_U (1 - \varepsilon) \leq \nonumber\\
\qquad \leq
\sum_{i=1}^n
P\left\{ Y_i \geq \frac{1}{2} + \frac{1}{2}
(1 - 4 \varepsilon + 2 \varepsilon^2)^{1/2}\right\} \, , \label{eq_4.2}\\
%
0 < \varepsilon < 1-2^{-1/2} , \textrm{where the } Y_i \textrm{ are beta} (\alpha_i , \alpha - \alpha_i) \textrm{-distributed.} \notag\\
%
1 - F_U (1 - \varepsilon) \simeq
\frac{
\# \left\{ i | \alpha_i = \alpha_{\mathrm{max}} \right\}} {B (\alpha_{\mathrm{max}} , \alpha - \alpha_{\mathrm{max}})}
\cdot \frac
{\varepsilon^{\alpha - \alpha_{\mathrm{max} } }}
{\alpha - \alpha_{\mathrm{max} }} \, , \ \ \varepsilon \to 0 \, .\label{eq_4.3}\\[0.5ex]
\!\!\!\! \textrm{The density } f_U \textrm{ is square integrable if } n > 2 \textrm{ and } \alpha - \alpha_{\mathrm{max}} > 1/2 \, . \label{eq_4.4}
%\end{gathered}$$ Remark: For identical $\alpha_i=\alpha$ the $\alpha$ in the lemma has to be replaced by $\alpha n$ and we obtain the conditions $$\begin{gathered}
f_U \textrm{ is square integrable if } n > \mathrm{max} \left( 2, 1 + \frac{1}{2 \alpha} \right) ,\nonumber \\
f_U \!\left( n^{-1/2}\right) = f_U (1)= 0 \textrm{ if } n > \mathrm{max} \left( 3, 1 + \frac{1}{\alpha} \right) \, . \label{eq_4.5}\end{gathered}$$
With $x_i = (1 + \varepsilon_i) \bar{x} = (1 + \varepsilon_i) yn^{-1}$ we have $\tan^2(\varphi) = \frac{1}{n} \sum_{i=1}^n \varepsilon_i^2$ and consequently with $t \to 0$ : $$\begin{gathered}
P \{ \tan (\Phi) \leq t \} = \left( \textstyle{\prod_{i=1}^n} \Gamma (\alpha_i) \right)^{-1} \int_{\tan (\Phi) \leq t} \textstyle{\prod_{i=1}^n} e^{-x_i}x_i^{\alpha_i-1}dx_i \simeq \\
\simeq b_{n-1} \left( \textstyle{\prod_{i=1}^n} \Gamma (\alpha_i) \right)^{-1} \int_0^\infty e^{-y} (yn^{-1})^{\alpha-n} (yn^{-1/2})^{n-1} n^{-1/2}
dy \cdot t^{n-1}=\\
= b_{n-1} \left( \textstyle{\prod_{i=1}^n} \Gamma (\alpha_i) \right)^{-1} \Gamma (\alpha) n^{n/2-\alpha} t^{n-1} \, ,\end{gathered}$$ where $b_{n-1}$ denotes the volume $\pi^{(n-1)/2} \bigl( \Gamma \left((n+1)/2 \right)\bigr)^{-1}$ of the $(n\!-\!1)$-unit ball. Thus, (\[eq\_4.1\]) follows from $$P \left\{ U \leq n^{-1/2}+ \varepsilon \right\} = P \left\{ \tan^2 (\Phi) \leq 2 \varepsilon \sqrt{n} + n\varepsilon^2 \right\} \simeq P \left\{ \tan(\Phi) \leq n^{1/4} \sqrt{2 \varepsilon} \right\} \, .$$ Now let be $Y_n = \mathrm{max } \, Y_i$ and $0 < \delta < {1/2}$. Then $$\begin{gathered}
Y_n = 1- \delta \Rightarrow \textstyle{\sum_{i=1}^{n-1}} Y_i = \delta \Rightarrow
\frac{\delta^2}{n-1} \leq \textstyle{\sum_{i=1}^{n-1}} Y_i^2 \leq \delta^2 \Rightarrow \\
\left( (1-\delta)^2 + \frac{\delta^2}{n-1} \right)^{1/2} \leq U \leq \left( (1-\delta)^2 + \delta^2 \right)^{1/2} \, .\end{gathered}$$ Conversely, from $Y_n = \mathrm{max } \, Y_i$ and $U=1-\varepsilon \, , \ \varepsilon < 1-2^{-1/2}$, it follows that $Y_n = 1- \delta$ with $$\begin{gathered}
\varepsilon + \frac{1}{2 (n-1)} \varepsilon^2 <
\delta_1 (\varepsilon) := \frac{n-1}{n}-\frac{n-1}{n} \left( 1- \frac{2n}{n-1} \varepsilon (1 - \varepsilon/2) \right)^{1/2} \leq \\
\leq \delta \leq \delta_2 (\varepsilon) := \frac{1}{2} - \frac{1}{2} \Big( 1 - 4 \varepsilon (1 - \varepsilon/2) \Big)^{1/2} \, .\end{gathered}$$ Therefore, $$\mathrm{max } \, Y_i \ge 1 - \delta_1 (\varepsilon) \Rightarrow U \ge 1- \varepsilon \Rightarrow \mathrm{max } \, Y_i \ge 1- \delta_2 (\varepsilon) \, ,$$ which implies (\[eq\_4.2\]). The asymptotic relation (\[eq\_4.3\]) is an immediate consequence of (\[eq\_4.2\]).\
For small values $\alpha - \alpha_\mathrm{max}$ the density $f_U$ can increase near its end-point $u =1$. Then already $$1 - F_U (1 - \varepsilon) = O \left( \varepsilon^{\alpha - \alpha_\mathrm{max} }\right) \, , \ \varepsilon \to 0 \, , \textrm{ entails } f_U (1 - \varepsilon) = O \left( \varepsilon^{\alpha - \alpha_\mathrm{max} -1 }\right) \, ,$$ which implies (\[eq\_4.4\]). \
Now with the Jacobi polynomials $G_k (\alpha n , \alpha n \textrm{; } x)$ from (A.S.22.3.3), belonging to the weight function $w (x) = x^{\alpha n -1}, 0 < x \le 1$, and the shifted Legendre polynomials $P_k^\ast (x) = P_k (2x-1) = {2k \choose k} G_k (1 , 1; x)$, (A.S.22.2.11), (A.S.22.5.2), we obtain with (\[eq\_2.2\]), (\[eq\_2.6\]) and $$\int_0^1 \left( 1 - F_C (x) \right) x^{\alpha n +j-1} dx = \frac{\gamma_{\alpha n +j}}{\alpha n +j}$$ the following two representations of the cdf $F_U (u)= 1 - F_C \left( n^{-1/2} u^{-1}\right)$:
Let $U$ be defined as in *(\[eq\_1.3\])*. Then $$\begin{gathered}
\!\!\!\!\!\!\!\!\!\!\!\!\!\! F_U (u) = \label{eq_4.6} \\
\! \!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{k=0}^\infty (-1)^k (\alpha n + k) {\alpha n + 2k \choose k}
\!\!\left( \sum_{j=0}^k ( -1)^j \, \frac{(\alpha n + j)_k}{j! (k-j)! } \frac{\gamma_{\alpha n +j}}{\alpha n +j} \right)
G_k (\alpha n , \alpha n ; n^{-1/2} u^{-1})\nonumber \, ,\end{gathered}$$ which is uniformly and absolutely convergent on compact intervals $\lbrack a ; b\rbrack \subset \left(1/\sqrt{n}; 1\right)$ under the first condition in *(\[eq\_4.5\])*. Besides, $$\begin{gathered}
\!\!\!\!\!\!\!\!\!\!\!\!\!\! F_U (u) = \left( n^{1/2} u\right)^{\alpha n-1} \cdot \label{eq_4.7}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!
%\textstyle
\sum_{k=0}^\infty (-1)^k (2 k + 1)
\left( %\textstyle
{\sum_{j=0}^k} (-1)^j
\displaystyle{\frac{(k - j + 1)_{2j}}{j!j!}}
\displaystyle{\frac{\gamma_{\alpha n +j}}{\alpha n +j}} \right)
\!\!P_k^\ast \! \left( n^{-1/2} u^{-1} \right) \, , \nonumber\end{gathered}$$ which is uniformly and absolutely convergent at least for $n \!\! >$max$\left( 2, 1 \!+ \!\frac{1}{2 \alpha}, \frac{3}{2 \alpha} \right)$.
There remains only an explanation for the additional condition $n \!> 3/(2\alpha)$ in the $2^{\textrm{nd}}$ formula. The $2^{\textrm{nd}}$ series is directly obtained from the corresponding series for the function $x^{\alpha n -1} (1-F_C (x)),\, 0 < x < 1$, by the substitution $x= n^{-1/2} u^{-1}$. The square integrability of the density $f_C (x)$ on $\lbrack n^{-1/2}; 1 \rbrack$ is equivalent to the square integrability of $f_U$. The additional condition for $n$ is sufficient for the square integrability of $x^{\alpha n -2}$. Hence, the derivative of the above function is square integrable under the given condition.
In particular for $\alpha = 1$, it follows with the moments $\gamma_j$ from (\[eq\_3.7\]) that $$\begin{gathered}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!F_U \!\!\! ~\left( n^{-1/2} x^{-1}\right) =
1 - F_C (x) = \nonumber \\
%\textstyle
\!\!\!\!\!\!\!\!\!\!\!\!\!\! \sum_{k=0}^\infty (-1)^k (2k +1)
\left( %\textstyle
\sum_{j=0}^k (-1)^j \displaystyle{\frac{(k-j+1)_{2j}}{j!j!}
\frac{\gamma_{j+1}}{j+1}} \right) \!\!P_k^\ast (x) \, . \label{eq_4.8}\end{gathered}$$ However, it should be noted that the moments $\gamma_j , j=1, \ldots, n-1$, are computed by infinite series, whereas the coefficients in the orthogonal series (\[eq\_4.6\]), (\[eq\_4.7\]) are given by finite calculations only. This is important since a high number of digits for the moments $\gamma_{\alpha n +j}$ is required to obtain sufficiently accurate values of $F_U$.\
From (\[eq\_1.2\]) we can also obtain the moments $$\mu_{2k} = E (U^{2k}) = \frac{E (Z^k)}{E (Y^{2k})} = \frac{k!}{(\alpha n)_{2k}} \sum_{(k)} \prod_{i=1}^n \frac{(\alpha)_{2k_i}}{k_i!} \, , \label{eq_4.9}$$ where $\sum_{(k)}$ means summation over all decompositions $k_1 + \ldots + k_n = k $, $k_1, \ldots , k_n \in \mathbb{N}_0$ . Since only partial sums are used in the computing procedure (\[eq\_2.1\]), these moments can be computed by the same method, starting with the divergent series $$\sum_{k=0}^\infty (\alpha)_{2k} \frac{z^k}{k!} \, .$$ The square integrability of the densities $f_U$ and $f_{U^2}$ is equivalent. Therefore, we get a further convergent orthogonal series $$\begin{gathered}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!1 - F_{U^2} (x) = \label{eq_4.10}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!%\textstyle
\sum_{k=0}^\infty (-1)^k (2k +1)
\left( %\textstyle
\sum_{j=0}^k(-1)^j
\displaystyle{ \frac{(k-j+1)_{2j}}{j!j!}
\frac{\mu_{2(j+1)}}{j+1}}\right) \!\!P_k^\ast (x) \, , \
n^{-1} \le x \le 1 \, , \nonumber\end{gathered}$$ but the extreme magnitude of the sums in (\[eq\_4.9\]) is unfavourable.\
A generalization of the orthogonal series in this section to $U$ computed from $n$ completely independent gamma random variables with not identical order parameters is feasible. However, the required moments would have to be computed from a product of $n$ generating power series instead from just an $n^{\textrm{th}}$ power.
Integral representations of the cumulative distribution function of $U^2$
=========================================================================
Under the $2^{\textrm{nd}}$ condition in (\[eq\_4.5\]) the density of $U_n^2 - n^{-1}$ vanishes at its end-points and is of bounded variation. Then the cdf is representable by an integrated Fourier-sine series , i.e. $$P \{ U_n^2 -n^{-1} \le x \} = \frac{2}{\pi} \sum_{k=1}^\infty \Im \! \left( \hat{f}_{U_n^2 -n^{-1}} (t_k)\right)
\frac{1-\cos (t_k x)}{k} , \
t_k = \frac{k \pi}{1-n^{-1}} \, , \label{eq_5.1}$$ which is absolutely und uniformly convergent with terms at least of order $o(k^{-2})$.\
In a similar way a series for $P \left \{ \log ( \sqrt{n}U ) \le x \right\}$ can be derived from (\[eq\_3.6\]), replacing $s$ by $it_k = ik\pi/\log (\sqrt{n})$. However, an accurate computation of the coefficients is difficult for large $k$. Also a direct use of the Fourier inversion formula is not recommended because of the comparatively slow decrease of the cf of $\log (\sqrt{n}U)$.
Before we return to (\[eq\_5.1\]) a univariate integral representation of is given by means of parabolic cylinder functions $D_{-\alpha}$ (A.S.19.3.1).
Let $X_1, \ldots , X_n$ be completely independent random variables with gamma densities $(\Gamma (\alpha_j))^{-1} e^{-x} x^{\alpha_j-1}$, $\alpha = \alpha_1 + \ldots + \alpha_n$ and $U_n^2 = \left( \textstyle\sum_{i=1}^n X_i \right)^{-2} \textstyle\sum_{i=1}^n X_i^2$. Then the cdf of $U_n^2$ is given by $$\begin{gathered}
% \!\!\\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!P \{ U_n^2 \le x \} =
\pi^{-3/2} \sqrt{2} \Gamma (\alpha) x^{(\alpha-1)/2} \cdot \\
\!\!\!\!\!\!\! \!\!\!\!\!\!\!\int_0^\infty \Re\left(\left( {\textstyle \prod_{j=1}^n \textrm{exp} \!\left( -\frac{is^2}{4} \right) \!D_{-\alpha_j}(i \sqrt{i} s )} \right)
\!\textrm{exp} \!\left( {\textstyle \frac{is^2}{4x}} \right) \! D_{-\alpha} (\sqrt{-i}sx^{-1/2}) \sqrt{i} \right) \!ds \, ,
\end{gathered}$$ at least for $ \alpha > 1$.
[Remarks: The theorem should be true for all $\alpha>0$, but the additional condition implies the absolute convergence of the integral and facilitates the proof. The cdf of Greenwood’s statistic is obtained by identical $\alpha_j =1$ and $\alpha = n$. For the relation of $D_{-1}$ to the error function see (\[eq\_3.2\]). Splitting the integral into integrals over intervals between successive zeros of the integrand provides an alternating series which enables a good control of the remainder in many cases. E.g. by integration over the intervals (0; 2.978235860548) and (2.978235860548; 3.004569229995) we find the inequalities]{} $$0.98999999977 < P \{ 60 U_{60}^2 < 2.7167772982\} < 0.99000000060 \, .$$ With increasing $n$ a higher computing accuracy is required to compensate the extinction of the leading digits, in particular for the upper tail of this distribution.
With (A.S.19.3.1), (A.S.19.12.3) and (A.S.13.5.1) we find for all positive $\alpha$ that $$\begin{gathered}
D_{-\alpha} (i \sqrt{i}s) = O (s^{\mathrm{max} (-\alpha , \alpha - 1)}) \, , \ s \to \infty \, ,\ \label{eq_5.2}\\
\ \nonumber\\
\exp ( -is^2/4) D_{-\alpha}(\sqrt{-i}s) \simeq \exp (i \alpha \pi/4) s^{-\alpha}\, , \ s \to \infty \, , \label{eq_5.3}
\\[1.5ex]
\textrm{(see also (A.S.19.8.1)) }. \nonumber\end{gathered}$$ Hence, the above integrand is absolutely $\displaystyle{O \! \left( s^{-k-2\sum_{\alpha_ j < 1 \slash 2}{\alpha_j}}\right)}$, where $k =$$ \# \{ \alpha_j | \alpha_j \! \ge \! 1/2 \}$, and the integral is absolutely convergent if max $\alpha_j \ge $$ 1/2$ or $\alpha > 1/2$.\
Moreover, to justify a change of the order of integration in (\[eq\_5.6\]) according to Fubini, $$\begin{gathered}
\int_0^\infty \int_0^\infty \left( \textstyle\prod_{j=1}^n | D_{-\alpha_j} (i \sqrt{i}s) | \right) | D_{-\alpha} ( x^{-1/2} s \sqrt{-i}) | x^{(\alpha - 1)/2} e^{-tx} dsdx =\\
%
\int_0^1 \left( \int_0^\infty \left( \textstyle\prod_{j=1}^n | D_{-\alpha_j} (i \sqrt{i}s) | \right) | D_{-\alpha} ( x^{-1/2} s \sqrt{-i}) | ds \right) x^{(\alpha - 1)/2} e^{-tx} dx \ +\\
%
\int_1^\infty \left( \int_0^\infty \left( \textstyle\prod_{j=1}^n | D_{-\alpha_j} (i \sqrt{i}s \sqrt{x}) | \right) | D_{-\alpha} (s \sqrt{-i})| ds \right)
x^{\alpha/2} e^{-tx} dx <\infty\end{gathered}$$ is verified by (\[eq\_5.2\]) and (\[eq\_5.3\]).\
The Lt of $$\begin{gathered}
\frac{x^{(\alpha-1)/2}}{\sqrt{2\pi}}
\exp \! \left( - \frac{ x^{-1} s^2 e^{-2i\phi} }{ 4 } \right)
D_{-\alpha} \! \left( x^{-1/2} se^{-i\phi} \right)\nonumber\\
\textrm{ with regard to $x$ is given by} \nonumber\\
%
\exp\! \left( -se^{-i\phi} \sqrt{2t} \right)
(2t)^{-(\alpha+1)/2}\, , \ t >0 \, , \ 0 \le \phi < \pi/4 \, . \label{eq_5.4}\end{gathered}$$ Due to dominated convergence this relation holds also for the limit $\phi = \pi/4$, i.e. $$\begin{gathered}
\frac{x^{(\alpha-1)/2}}{\sqrt{2\pi}}\exp \! \left( \frac{is^2}{4x} \right) D_{-\alpha} \! \left(x^{-1/2} s \sqrt{-i} \right) \nonumber\\
\textrm{ has the Lt } \exp (is \sqrt{2it}) (2t)^{-(\alpha+1)/2}\, . \label{eq_5.5}\end{gathered}$$ Therefore, $$\begin{gathered}
\frac{x^{(\alpha-1)/2}}{\sqrt{2\pi}} \cdot \nonumber \\
\int_0^\infty \left(
\prod_{j=1}^n
\exp \! \left( - \frac{is^2}{4} \right) D_{-\alpha_j}
\! \left( i \sqrt{i}s \right)
\right)
\exp \! \left( \frac{is^2}{4x} \right) D_{-\alpha}
\! \left( x^{-1/2} s \sqrt{-i} \right) \sqrt{i}\ ds \nonumber \\
\textrm{ has the Lt}
\label{eq_5.6}\\
\int_0^\infty
\left(
\prod_{j=1}^n
\exp \! \left( - \frac{is^2}{4} \right) D_{-\alpha_j}
\! \left( i \sqrt{i}s \right) \right)
\exp (is\sqrt{2it}) (2t)^{-(\alpha +1)/2} \sqrt{i}\ ds \, . \nonumber\end{gathered}$$ By (A.S.19.12.3) in conjunction with (A.S.13.1.5) we obtain $$(2t)^{-\alpha/2} \exp \left( \frac{(1+is)^2}{8t} \right) D_{-\alpha}
\left( \frac{1+is}{\sqrt{2t}}\right) = O \left( | 1 + is |^{-\alpha} \right) \, , \ s \to \infty \, ,$$ since $\Re \Bigl( (1+is)^2 \Bigr) < 0 \ if \ |s| > 1$.\
From the Lt in (\[eq\_3.1\]) we get the Lt $f_{Y,Z}^\ast (s,t)$ of the joint density $f_{Y,Z}$ of $Y = \textstyle\sum_{j=1}^n X_j$ and $Z = \textstyle\sum_{j=1}^n X_j^2$ , and find by the Laplace inversion formula with regard to $s$ the Lt of $f_{Y,Z}$ with regard to $Z$ as $$\begin{gathered}
\frac{1}{2\pi} \int_{-\infty}^\infty
\left( \prod_{j=1}^n \exp \!
\left( \frac{(1+is)^2}{8t} \right) D_{-\alpha_j} \!
\left( \frac{(1+is)}{\sqrt{2t}} \right) \right)
(2t)^{-\alpha/2} e^{isy}ds = \\
%
\frac{1}{2\pi} \int_{-\infty}^\infty
\left( \prod_{j=1}^n \exp \!
\left( \frac{1}{4} \left( \frac{1}{\sqrt{2t}} + is \right)^2 \right) \!\! D_{-\alpha_j}\!
\left( \frac{1}{\sqrt{2t}} + is \right)
\right)\!\!
(2t)^{-(\alpha-1)/2} e^{is\sqrt{2t}y}ds =\\
%
e^{-y} \frac{1}{2\pi i} \int_{\frac{1}{\sqrt{2t}}-i\infty}^{\frac{1}{\sqrt{2t}}+i\infty}
\Big( \prod_{j=1}^n
%\exp \left(\frac{\zeta^2}{4}\right)
e^{\zeta^2/4}
D_{-\alpha_j} (\zeta) \Big)
(2t)^{-(\alpha-1)/2} e^{\zeta \sqrt{2t}y} d\zeta =\\
%
e^{-y} \frac{1}{2\pi} \int_{-\infty}^\infty
\Big( \prod_{j=1}^n
%\exp \left(-\frac{s^2}{4}\right)
e^{-s^2/4} D_{\alpha_j} (is) \Big)
(2t)^{-(\alpha-1)/2} e^{is\sqrt{2t}y} ds \, .\end{gathered}$$
Then the Lt of the conditional density of $ZY^{-2} | Y = y$ is $$\begin{gathered}
\frac{\Gamma (\alpha)}{2\pi} \int_{-\infty}^\infty
\Big( \prod_{j=1}^n
% \exp \left( -\frac{s^2}{4} \right)
e^{-s^2/4} D_{-\alpha_j} (is)
\Big)
(2t)^{-(\alpha-1)/2}
e^{is\sqrt{2t}} ds =\\
%
\frac{\Gamma (\alpha)}{\pi} \int_0^\infty \Re
\left( \Big( \prod_{j=1}^n
%\exp\left( -\frac{s^2}{4} \right)
e^{-s^2/4} D_{-\alpha_j} (is)
\Big) (2t)^{-(\alpha-1)/2}
e^{is\sqrt{2t}}
\right) ds \, ,\end{gathered}$$ independent of $y$. Hence, the Lt of the cdf of $U_n^2$ is $$\begin{gathered}
\frac{2}{\pi} \Gamma (\alpha) \int_0^\infty \Re
\left( \Big( \prod_{j=1}^n
% \exp \left( -\frac{s^2}{4} \right)
e^{-s^2/4} D_{-\alpha_j} (is)
\Big) (2t)^{-(\alpha+1)/2}
e^{is\sqrt{2t}}
\right) ds = \nonumber \\
%
\ \label{eq_5.8}\\
%
\frac{2}{\pi} \Gamma (\alpha) \int_0^\infty \Re
\left( \Big(\prod_{j=1}^n
%\exp \left( -i \frac{s^2}{4} \right)
e^{-i s^2/4} D_{-\alpha_j} (i\sqrt{i}s) \Big)
(2t)^{-(\alpha+1)/2}
e^{is\sqrt{2it}} \sqrt{i}
\right) ds \, . \nonumber\end{gathered}$$ Now, theorem 2 follows by comparison of (\[eq\_5.8\]) with (\[eq\_5.6\]).
With a sufficient number of terms the partial sums of the integrand in the following theorem show a rather smooth behaviour within the “main part”, also for larger $n$. The part of the integrand oscillating around zero becomes absolutely small and is more distant from zero. Its contribution to the integral becomes neglectable. For simplicity we give here only the formula derived from identically distributed gamma random variables. The generalization to different order parameters $\alpha_j$ is obvious.
Let $X_1, \ldots , X_n$ be i.i.d. random variables with a gamma density $(\Gamma(\alpha))^{-1} e^{-x} x^{\alpha-1}$ and $U_n^2 = \left( \textstyle\sum_{i=1}^n X_i \right)^{-2} \textstyle\sum_{i=1}^n X_i^2 $. Then the cdf of $U_n^2 - n^{-1}$ is given by $$\begin{gathered}
P \left\{ U_n^2 - n^{-1} \le x \right\} = \nonumber \\
%
\frac{\Gamma (\alpha n +1 )}
{\pi^2 ( \alpha n/e)^{\alpha n} }
\lim_{K \to \infty} \int_{-\infty}^\infty \sum_{k=1}^K
\Im \left\{
\left( \psi (s,t_k)\right)^n
\exp \left( - i \alpha n s - i (n- 1)^{-1} k \pi \right)
\right\} \cdot \nonumber \\
%
\quad \frac{1- \cos \bigl( (1-n^{-1})^{-1} k \pi x \bigr)}{k} ds \, , \label{eq_5.9} \\
\ \nonumber \\
%
\psi (s,t) = \left( \frac{i}{2t} \right)^{\alpha/2}
\exp \! \left(\frac{i(1 - is)^2}{8t} \right) D_{-\alpha} \!
\left( \sqrt{i} \frac{1- is}{\sqrt{2t}}\right) \, , \nonumber \\
t_k = \frac{k\pi}{\alpha^2 n (n-1)}\, , n > \mathrm{max}
\left( 3,1 + \frac{1}{\alpha}\right) \, , \ 0 \le x \le 1 - \frac{1}{n} \, . \nonumber\end{gathered}$$
The limit can be taken under the integral at least for $$\left[ \frac{n}{2} \right] > \mathrm{max} \left( 2, 1 + \frac{1}{2\alpha} , \frac{3}{2\alpha} \right) \, .
\label{eq_5.10}$$ In particular with $\alpha = 1$ and $t_k = k\pi/\big(n (n-1)\big)$ we have at least for $n\ge 6$: $$\begin{gathered}
\!\!\!\!\!\!\!\!\!\!\!\!P \left\{ U_n^2 - n^{-1} \le x \right\} = \frac{n!}{\pi^2 \left( n/e \right)^n} \ \cdot \nonumber \\
%
\!\!\!\!\!\!\!\!\!\!\!\!\int_{-\infty}^\infty \sum_{k=1}^\infty
%
\Im \left\{
\left(
\left( \frac{i}{2t_k} \right)^{1/2} D_{-1}\!\!
\left( \sqrt{i} \frac{1- is}{\sqrt{2t_k}}\right)
\exp \! \left( i \left(
\frac{(1 - is)^2}{8t_k}-t_k -s \right)
\right)
\right)^n
\right\} \, .\end{gathered}$$ $$\frac{1- \cos \bigl( (1-n^{-1})^{-1} k \pi x \bigr)}{k} ds \, . \label{eq_5.11}$$ Remark: The condition (\[eq\_5.10\]) is too restrictive but enables the use of Parseval’s identity in the following proof. The absolute integrability of the cf $\hat{f}_{Y,Z}$ is sufficient for the application of the Fourier inversion formula but not necessary.
From (\[eq\_3.1\]) we obtain the characteristic functions $$\begin{gathered}
\psi (s,t) = E \bigl( \exp ( is X + it X^2) \bigr) =
\left( \frac{i}{2t}\right)^{\alpha/2}
\exp \left( \frac{i (1-is)^2}{8t}\right) D_{-\alpha}
\left( \sqrt{i} \frac{1-is}{\sqrt{2t}} \right) \\
\textrm{and } \ \hat{f}_{Y,Z} (s,t) = (\psi (s,t))^n \, .\end{gathered}$$ With $\Re \bigl( i ( 1 - is)^2 (2t)^{-1} \bigr) = st^{-1}$, $t >0$ and $\alpha n > 1$ the absolute convergence of $$\int_{-\infty}^0 (\psi (s,t))^ne^{-isy} ds$$ follows from (A.S.19.12.3) and (A.S.13.5.1) and the absolute convergence of $$\int_0^{\infty} (\psi (s,t))^ne^{-isy} ds$$ is obtained by the asymptotic relation (A.S.19.8.1).\
Since the conditional distribution of $ZY^{-2}| Y=y$ doesn’t depend on $y$ we can choose $y = E(Y) = \alpha n$ and obtain the cf of $U_n^2 -n^{-1}$: $$\hat{f}_{U_n^2 -n^{-1}} (t) = ( 2 \pi g_{\alpha n} (\alpha n) )^{-1}
\exp (-itn^{-1}) \int _{-\infty}^\infty (\psi (s,(\alpha n)^{-2} t))^n
\exp (-i \alpha ns) ds \, .$$ Together with (\[eq\_5.1\]) this entails (\[eq\_5.9\]).\
Now, a sufficient condition for the square integrability of $\hat{f}_{Y,Z}$ is given. The square integrability of the densities $f_U$ and $f_{U^2}$ is equivalent because of $$\int_{n^{-1}}^1 \left( f_{U^2} (x)\right)^2 dx=
\int_{n^{-\frac{1}{2}}}^1 (2u)^{-1} \left( f_U (u)\right)^2 du\, .$$ According to (\[eq\_4.5\]) $f_U$ is square integrable if $n > \mathrm{max} \left( 2, 1 +(2 \alpha)^{-1}\right)$. From the identity of the distributions of $U^2$ and $ZY^{-2} | Y = y$ it follows $$\begin{gathered}
f_{U^2} (w) = \frac{d}{dw} P \left\{ Z \le wy^2 | Y = y \right\} =
y^2 \frac{f_{Y,Z} (y,wy^2)}{g_{\alpha n} (y)}
\\ \textrm{with } g_{\alpha n} (y) = (\Gamma (\alpha n))^{-1} \exp (-y) y^{\alpha n-1} \, .\end{gathered}$$ With the additional assumption $n> 3/(2 \alpha)$ we obtain $$\begin{gathered}
\int_0^\infty \left(
\int_{n^{-1} y^2}^{y^2}
\left( f_{Y,Z} (y,z) \right)^2 dz
\right) dy =\\
%
\int_0^\infty \left(
\int_{n^{-1} y^2}^{y^2}
\left( \left( g_{\alpha n} (y) \right)^{-1} f_{Y,Z} (y,z)\right)^2 dz
\right)
\left( g_{\alpha n} (y) \right)^2 dy =\\
%
\int_0^\infty \left(
y^4 \int_{n^{-1}}^1 \left( \left( g_{\alpha n} (y) \right)^{-1} f_{Y,Z} (y,wy^2)\right)^2 dw\right)
\left( y^{-1} g_{\alpha n} (y) \right)^2 dy =\\
%
\int_{n^{-1}}^1 \left( f_{U^2} (w) \right)^2 dw \cdot \int_0^\infty \left( y^{-1} g_{\alpha n} (y) \right)^{2} dy < \infty \, .\end{gathered}$$ Thus, due to Parseval’s identity, the cf $\hat{f}_{Y,Z}$ is square integrable and it follows $$\int_{-\infty}^{\infty} \int_{-\infty}^\infty | \psi (s,t) |^n dsdt < \infty \textrm{ at least for } \left[ \frac{n}{2} \right] > \mathrm{max} \left( 2,1 + \frac{1}{2\alpha} , \frac{3}{2 \alpha} \right) \, .$$ >From (\[eq\_4.5\]) it follows also that the density $f_U$ vanishes at its end-points for these $n$.\
Then the change of summation and integration in (\[eq\_5.9\]) is justified by $$\begin{gathered}
\sum_{k \ge K} \frac{|\psi (s,t_k)|^n}{k} =
O \left( \int_{t_K}^\infty | \psi (s,t) |^n dt \right)\!, \
K \to \infty \, , \textrm{ and}\\
\int_{-\infty}^\infty \left( \int_{t_1}^\infty | \psi (s,t) |^n dt \right) ds < \infty \ .\end{gathered}$$ \
Some $p$-quantiles of $nU_n^2 - 1 $
====================================
It should be noted that $n$ is the number of squares. In the application for a goodness of fit test the sample size is $k=n-1$.\
[p[0.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}p[1.2cm]{}]{}
n: &p=0.005 &p=0.010 &p=0.025 &p=0.050 &p=0.500 &p=0.950 &p=0.975 &p=0.990 &p=0.995\
\
10 & 0.17789466 &0.20866627 &0.26014429 &0.31111296 &0.72416820 &1.64511506 &1.93318559 &2.33028205 &2.64025028\
11 &0.19730682 &0.22863245 &0.28069377 &0.33196840 &0.74212800 &1.64417262 &1.92622396 &2.31678132 & 2.62372056\
12 & 0.21512589 &0.24685519 & 0.29933241 &0.35078097 &0.75769056 &1.64103896 &1.91685253 &2.29998228 &2.60265097\
13 &0.23155755 &0.26358997 &0.31635339 &0.36787930 &0.77132393 &1.63644125 &1.90598495 &2.28124294 &2.57887125\
14 &0.24678209 & 0.27904048 &0.33198974 &0.38352158 &0.78337931 &1.63086853 &1.89422669 &2.26144430 &2.55358516\
\[3mm\] 15 &0.26094814 &0.29337145 &0.34642980 &0.39791268 &0.79412572 &1.62465699 &1.88198346 &2.24116547 &2.52758847\
16 &0.27417918 &0.30671949 &0.35982670 &0.41121857 &0.80377307 &1.61804138 &1.86952924 &2.22079055 &2.50140936\
17 &0.28657943 & 0.31919848 &0.37230685 &0.42357520 &0.81248776 &1.61118729 &1.85705008 &2.20057436 &2.47539868\
18 &0.29823733 &0.33090418 &0.38397570 &0.43509525 &0.82040353 &1.60421247 &1.84467227 &2.18068462 & 2.44978800\
19 &0.30922844 &0.34191790 &0.39492200 &0.44587314 &0.82762921 &1.59720104 &1.83248093 &2.16122966 &2.42472760\
20 &0.31961776 &0.35230915 &0.40522118 &0.45598866 &0.83425434 &1.59021315 &1.82053247 &2.14227701 &2.40031181\
21 &0.32946153 &0.36213779 &0.41493779 &0.46550982 &0.84035330 &1.58329171 &1.80886317 &2.12386603 &2.37659630\
22 &0.33880877 &0.37145573 &0.42412748 &0.47449503 &0.84598846 &1.57646707 &1.79749507 &2.10601650 &2.35360984\
23 &0.34770234 &0.38030822 &0.43283857 & 0.48299478 &0.85121248 &1.56976041 &1.78644011 &2.08873466 &2.33136257\
24 &0.35618001 &0.38873496 &0.44111322 &0.49105303 &0.85607024 &1.56318610 &1.77570311 &2.07201741 &2.30985179\
\[3mm\] 25 &0.36427515 &0.39677092 &0.44898846 &0.49870821 &0.86060015 &1.55675354 &1.76528382 &2.05585528 &2.28906598\
26 &0.37201740 &0.40444708 &0.45649693 &0.50599415 &0.86483536 &1.55046837 &1.75517847 &2.04023458 &2.26898774\
27 &0.37943320 &0.41179099 &0.46366760 &0.51294074 &0.86880461 &1.54433350 &1.74538083 &2.02513884 &2.24959586\
28 &0.38654622 &0.41882726 &0.47052622 &0.51957452 &0.87253294 &1.53834976 &1.73588302 &2.01054993 &2.23086677\
29 &0.39337772 &0.42557795 &0.47709583 &0.52591915 &0.87604231 &1.53251648 &1.72667611 &1.99644881 &2.21277557\
\[3mm\] 30 &0.39994689 &0.43206291 &0.48339710 &0.53199583 &0.87935202 &1.52683189 &1.71775048 &1.98281610 &2.19529685\
31 &0.40627105 &0.43830006 &0.48944863 &0.53782356 &0.88247913 &1.52129341 &1.70909620 &1.96963242 &2.17840515\
32 &0.41236596 &0.44430562 &0.49526722 &0.54341952 &0.88543880 &1.51589788 &1.70070319 &1.95687868 &2.16207537\
33 &0.41824592 &0.45009432 &0.50086810 &0.54879921 &0.88824450 &1.51064174 &1.69256144 &1.94453628 &2.14628301\
34 &0.42392401 &0.45567960 &0.50626510 &0.55397668 &0.89090828 &1.50552119 &1.68466105 &1.93258721 &2.13100433\
\[3mm\] 35 &0.42941215 &0.46107370 &0.51147084 &0.55896474 &0.89344094 &1.50053226 &1.67699239 &1.92101411 &2.11621649\
36 &0.43472131 & 0.46628784 &0.51649683 &0.56377506 &0.89585221 &1.49567088 &1.66954609 &1.90980037 &2.10189756\
37 &0.43986154 &0.47133233 &0.52135362 &0.56841830 &0.89815084 &1.49093298 &1.66231313 &1.89893010 &2.08802660\
38 &0.44484211 &0.47621662 &0.52605089 &0.57290423 &0.90034476 &1.48631451 &1.65528479 &1.88838815 &2.07458363\
39 &0.44967158 &0.48094944 &0.53059755 &0.57724183 &0.90244115 &1.48181147 &1.64845273 &1.87816010 &2.06154966\
\[3mm\] 40 &0.45435784 &0.48553886 &0.53500180 &0.58143937 &0.90444655 &1.47741992 &1.64180898 &1.86823224 &2.04890661\
41 &0.45890822 &0.48999231 &0.53927124 &0.58550446 &0.90636692 &1.47313604 &1.63534588 &1.85859156 &2.03663735\
42 &0.46332950 &0.49431672 &0.54341286 &0.58944415 &0.90820768 &1.46895611 &1.62905616 &1.84922570 &2.02472560\
43 &0.46762802 &0.49851849 &0.54743319 &0.59326496 &0.90997382 &1.46487652 &1.62293287 &1.84012296 &2.01315594\
44 &0.47180964 &0.50260359 &0.55133827 &0.59697294 &0.91166990 &1.46089377 &1.61696938 &1.83127222 &2.00191375\
\[3mm\] 45 &0.47587985 &0.50657758 &0.55513370 &0.60057371 &0.91330010 &1.45700451 &1.61115938 &1.82266295 &1.99098517\
46 &0.47984376 &0.51044565 &0.55882473 &0.60407248 &0.91486830 &1.45320550 &1.60549687 &1.81428516 &1.98035706\
47 &0.48370616 &0.51421263 &0.56241624 &0.60747414 &0.91637805 &1.44949360 &1.59997614 &1.80612938 &1.97001698\
48 &0.48747154 &0.51788306 &0.56591280 &0.61078322 &0.91783264 &1.44586584 &1.59459172 &1.79818663 &1.95995310\
49 &0.49114409 &0.52146118 &0.56931867 &0.61400398 &0.91923511 &1.44231932 &1.58933845 &1.79044838 &1.95015426\
\[3mm\] 50 &0.49472776 &0.52495098 &0.57263786 &0.61714039 &0.92058828 &1.43885128 &1.58421139 &1.78290656 &1.94060981\
51 &0.49822626 &0.52835619 &0.57587411 &0.62019618 &0.92189476 &1.43545906 &1.57920584 &1.77555347 &1.93130970\
52 &0.50164308 &0.53168034 & 0.57903095 &0.62317484 &0.92315700 &1.43214014 &1.57431733 &1.76838184 &1.92224435\
53 &0.50498151 &0.53492672 &0.58211170 &0.62607966 &0.92437724 &1.42889205 &1.56954159 &1.76138474 &1.91340469\
54 &0.50824465 &0.53809847 &0.58511947 &0.62891372 &0.92555761 &1.42571247 &1.56487456 &1.75455558 &1.90478209\
n: &p=0.005 &p=0.010 &p=0.025 &p=0.050 &p=0.500 &p=0.950 &p=0.975 &p=0.990 &p=0.995\
\
55 &0.51143542 &0.54119852 &0.58805721 &0.63167993 &0.92670007 &1.42259915 &1.56031236 &1.74788810 &1.89636834\
56 &0.51455659 &0.54422965 &0.59092769 &0.63438104 &0.92780645 &1.41954994 &1.55585131 &1.74137635 &1.88815565\
57 &0.51761077 &0.54719450 &0.59373352 &0.63701962 &0.92887849 &1.41656277 &1.55148788 &1.73501467 &1.88013661\
58 &0.52060045 &0.55009554 &0.59647719 &0.63959812 &0.92991779 &1.41363566 &1.54721870 &1.72879765 &1.87230415\
59 &0.52352797 &0.55293513 &0.59916103 &0.64211884 &0.93092586 &1.41076673 &1.54304057 &1.72272016 &1.86465156\
\[3mm\] 60 &0.52639556 &0.55571550 &0.60178727 &0.64458398 &0.93190412 &1.40795414 &1.53895042 &1.71677730 &1.85717243\
61 &0.52920533 &0.55843878 &0.60435801 &0.64699559 &0.93285390 &1.40519615 &1.53494533 &1.71096439 &1.84986065\
62 & 0.53195929 &0.56110696 &0.60687524 &0.64935565 &0.93377645 &1.40249108 &1.53102248 &1.70527699 &1.84271041\
63 &0.53465935 &0.56372196 &0.60934085 &0.65166600 &0.93467296 &1.39983733 &1.52717920 &1.69971082 &1.83571616\
64 &0.53730733 &0.56628561 &0.61175665 &0.65392842 &0.93554453 &1.39723335 &1.52341293 &1.69426184 &1.82887259\
65 &0.53990496 &0.56879962 &0.61412434 &0.65614458 &0.93639222 &1.39467765 &1.51972122 &1.68892617 &1.82217464\
66 &0.54245388 &0.57126564 &0.61644556 &0.65831607 &0.93721701 &1.39216880 &1.51610172 &1.68370008 &1.81561748\
67 &0.54495567 &0.57368524 &0.61872186 &0.66044443 &0.93801984 &1.38970545 &1.51255218 &1.67858004 &1.80919648\
68 &0.54741183 &0.57605993 &0.62095471 &0.66253108 &0.93880159 &1.38728626 &1.50907044 &1.67356265 &1.80290723\
69 &0.54982380 &0.57839112 &0.62314553 &0.66457742 &0.93956310 &1.38490998 &1.50565444 &1.66864466 &1.79674550\
\[3mm\] 70 &0.55219293 &0.58068019 &0.62529566 &0.66658476 &0.94030516 &1.38257538 &1.50230220 &1.66382296 &1.79070723\
71 & 0.55452055 & 0.58292844 &0.62740638 &0.66855434 &0.94102852 &1.38028129 &1.49901181 &1.65909457 &1.78478855\
72 &0.55680790 &0.58513712 &0.62947893 &0.67048737 &0.94173389 &1.37802659 &1.49578145 &1.65445663 &1.77898576\
73 &0.55905619 &0.58730741 &0.63151447 &0.67238500 &0.94242196 &1.37581019 &1.49260938 &1.64990641 &1.77329528\
74 &0.56126655 &0.58944047 &0.63351413 &0.67424831 &0.94309335 &1.37363103 &1.48949389 &1.64544127 &1.76771370\
\[3mm\] 75 &0.56344010 &0.59153738 &0.63547899 &0.67607835 &0.94374869 &1.37148812 &1.48643338 &1.64105870 &1.76223775\
76 &0.56557788 &0.59359919 &0.63741007 &0.67787613 &0.94438855 &1.36938049 &1.48342629 &1.63675627 &1.75686429\
77 &0.56768091 &0.59562692 &0.63930836 &0.67964260 &0.94501349 &1.36730719 &1.48047113 &1.63253166 &1.75159031\
78 &0.56975017 &0.59762152 &0.64117481 &0.68137869 &0.94562403 &1.36526732 &1.47756645 &1.62838263 &1.74641289\
79 &0.57178658 &0.59958393 &0.64301032 &0.68308527 &0.94622067 &1.36326002 &1.47471087 &1.62430704 &1.74132927\
\[3mm\] 80 & 0.57379104 &0.60151503 &0.64481577 &0.68476320 &0.94680390 &1.36128444 &1.47190306 &1.62030282 &1.73633675\
81 &0.57576441 &0.60341568 &0.64659200 &0.68641328 &0.94737416 &1.35933977 &1.46914173 &1.61636798 &1.73143278\
82 &0.57770753 &0.60528669 &0.64833980 &0.68803629 &0.94793190 &1.35742523 &1.46642564 &1.61250061 &1.72661487\
83 &0.57962119 &0.60712887 &0.65005995 &0.68963298 &0.94847753 &1.35554007 &1.46375360 &1.60869887 &1.72188063\
84 &0.58150615 &0.60894297 &0.65175320 &0.69120408 &0.94901145 &1.35368355 &1.46112447 &1.60496099 &1.71722779\
\[3mm\] 85 &0.58336317 &0.61072973 &0.65342025 &0.69275028 &0.94953403 &1.35185497 &1.45853712 &1.60128526 &1.71265412\
86 &0.58519295 &0.61248984 &0.65506180 &0.69427223 &0.95004564 &1.35005365 &1.45599050 &1.59767002 &1.70815751\
87 &0.58699617 &0.61422400 &0.65667851 &0.69577058 &0.95054664 &1.34827892 &1.45348357 &1.59411371 &1.70373589\
88 &0.58877351 &0.61593285 &0.65827102 &0.69724596 &0.95103734 &1.34653015 &1.45101532 &1.59061477 &1.69938729\
89 &0.59052559 &0.61761702 &0.65983995 &0.69869894 &0.95151808 &1.34480673 &1.44858480 &1.58717175 &1.69510981\
\[3mm\] 90 &0.59225304 &0.61927713 &0.66138588 &0.70013012 &0.95198916 &1.34310806 &1.44619108 &1.58378320 &1.69090160\
91 &0.59395645 &0.62091377 &0.66290939 &0.70154003 &0.95245086 &1.34143355 &1.44383326 &1.58044777 &1.68676089\
92 &0.59563640 &0.62252750 &0.66441103 &0.70292921 &0.95290349 &1.33978265 &1.44151047 &1.57716411 &1.68268597\
93 &0.59729344 &0.62411887 &0.66589134 &0.70429818 &0.95334729 &1.33815482 &1.43922186 &1.57393094 &1.67867517\
94 &0.59892811 &0.62568842 &0.66735083 &0.70564743 &0.95378254 &1.33654953 &1.43696663 &1.57074703 &1.67472691\
\[3mm\] 95 &0.60054092 &0.62723664 &0.66879000 &0.70697744 &0.95420949 &1.33496628 &1.43474398 &1.56761117 &1.67083964\
96 &0.60213239 &0.62876405 &0.67020932 &0.70828867 &0.95462836 &1.33340457 &1.43255316 &1.56452220 &1.66701186\
97 &0.60370298 &0.63027111 &0.67160927 &0.70958158 &0.95503940 &1.33186393 &1.43039344 &1.56147901 &1.66324213\
98 &0.60525319 &0.63175829 &0.67299029 &0.71085658 &0.95544282 &1.33034389 &1.42826408 &1.55848051 &1.65952906\
99 &0.60678345 &0.63322604 &0.67435281 &0.71211411 &0.95583885 &1.32884400 &1.42616442 &1.55552565 &1.65587129\
\[3mm\] 100 &0.60829421 &0.63467480 &0.67569726 &0.71335456 &0.95622767 &1.32736383 &1.42409376 &1.55261341 &1.65226752
References {#references .unnumbered}
==========
: Abramowitz, M. and Stegun, I. (1968) *Handbbook of Mathematical Functions*, Dover Publicatons Inc., New York.
: Burrows, P.M. (1979) Selected percentage points of Greenwood’s statistic, *Journal of the Royal Statistical Society, Series A*, **142**, 256-258.
: Currie, I.D. (1981) Further percentage points of Greenwood’s statistic, *Journal of the Royal Statistical Society, Series A*, **144**, 360-363.
: Currie, I.D. and Stephens, M.A. (1986) Relations between statistics for testing exponentiality and uniformity, *The Canadian Journal of Statistics*, **14**, 177-180.
: Darling, D.A. (1953) On a class of problems related to the random division of an interval, *Annals of Mathematical Statistics* **24**, 239-253.
: Does, R.J.M.M. et al. (1988) Approximating the distribution of Greenwood’s statistic, *Statistica Neerlandica* **42**, 153-161.
: Feller, W. (1971) *An Introduction to Probability Theory and Its Applications*, Vol.II, 2nd ed., John Wiley & Sons, New York.
: Gardner, A. (1952) Greenwood’s “Problem of intervals”: an exact solution for $n = 3$, *Journal of the Royal Statistical Society, Series B*, **14**, 135-139.
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: Ghosh, K. and Jammalamadaka, S.R. (2000) Some recent results on inferences based on spacings, in: Puri, M.L. (Ed.) Asymptotics in Statistics and Probability: Papers in Honor of George Gregory Roussas, VSP Utrecht, 185-196.
: Hill, I.D. (1979) Approximating the distribution of Greenwood’s statistic with Johnson distributions, *Journal of the Royal Statistical Society, Series A*, **142**, 378-380.
: Kotz, S. et al. (1994) Continuous Univariate Distributions, Vol.1, 2nd ed., chap. 17, John Wiley & Sons, Inc., New York.
: Laha, R.G. (1954) On a characterization of the gamma distribution, *Annals of Mathematical Statistics* **25**, 784-787.
: Lukacs, E. (1955) A characterization of the gamma distribution, *Annals of Mathematical Statistics* **26**, 319-324.
: Moran, P.A.P. (1947) The random division of an interval, Journal of the Royal Statistical Society, Series B, **9**, 92-98 (Corrigendum (1981) *Journal of the Royal Statistical Society, Series A*, **144**, 388).
: Moran, P.A.P: (1951) The random division of an interval II, *Journal of the Royal Statistical Society, Series B*, **13**, 147-150.
: Provost, S.B. (1988) The exact density of a statistic related to the shape parameter of a gamma random variate, *Metrika* **35**, 191-196.
: Pyke, R. (1965) Spacings (With Discussion), *Journal of the Royal Statistical Society, Series B*, **27**, 395-449.
: Royen, T. (2007a) Exact distribution of the sample variance from a gamma parent distribution, arxiv:0704.1415 \[math.ST\].
: Royen, T. (2007b) On the Laplace transform of some quadratic forms and the exact distribution of the sample variance from a gamma or uniform parent distribution, arxiv.0710.5749 \[math.ST\].
: Stephens, M.A. (1981) Further percentage points for Greenwood’s statistic, *Journal of the Royal Statistical Society, Series A*, **144**, 364-366.
: Subhash, C.K. and Gupta, R.P. (1988) A Monte Carlo study of some asymptotic optimal tests of exponentiality against positive aging, *Communications in Statistics - Simulation and Computation* **17**, 803-811.
|
---
abstract: |
A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time $-\infty$, called non-standardness. For example, two dyadic filtrations with trivial tail $\sigma$-field are not necessarily isomorphic. Yet, any essentially separable filtration indexed by the non-positive integers becomes standard when sufficiently many integers are skipped.
In this paper, we focus on the non standard filtrations which become standard if (and only if) infinitely many integers are skipped. We call them filtrations at the threshold of standardness, since they are as close to standardardness as they can be although they are non-standard.
Two class of filtrations are studied, first the filtrations of the split-words processes, second some filtrations inspired by an unpublished example of B. Tsirelson. They provide examples which disproves some naive intuitions. For example, it is possible to have a standard filtration extracted from a non-standard one with no intermediate (for extraction) filtration at the threshold of standardness. It is also possible to have a filtration which provides a standard filtration on the even times but a non-standard filtration on the odd times.
author:
- 'Gaël Ceillier, Christophe Leuridan'
title: Filtrations at the threshold of standardness
---
**MSC 2010:** 60G05, 60J10.
**keywords:** filtrations, standardness, split-words processes.
Introduction {#introduction .unnumbered}
============
The notion of standardness has been introduced by A. Vershik [@Vershik] in the context of decreasing sequences of measurable partitions indexed by the non-negative integers. Vershik’s definition and characterizations of standardness have been translated their original ergodic theoretic formulation into a probabilistic language by M. Émery and W. Schachermayer [@Emery-Schachermayer]. In this framework, the objects of focus are the filtrations indexed by non-positive integers. These are the non-decreasing sequences $(\fc_n)_{n \le 0}$ of sub-$\sigma$-fields of a probability space $(\Omega,\ac,\P)$.
All the sub-$\sigma$-fields of $\ac$ that we will consider are assumed to be complete and essentially separable with respect to $\P$. By definition, a sub-$\sigma$-field of $(\Omega,\ac,\P)$ is separable if it can be generated as a complete $\sigma$-field by a sequence of events, or equivalently, by some real random variable. One can check that a sub-$\sigma$-field $\bc \subset \ac$ is separable if and only if the Hilbert space $L^2(\Omega,\bc,\P)$ is separable.
Almost all filtrations that we will consider in this study have the following property: for each $n$, $\fc_n$ is generated by $\fc_{n-1}$ and by some random variable $U_n$ which is independent of $\fc_{n-1}$ and uniformly distributed on some finite set with $r_n$ elements, for some sequence $(r_n)_{n \le 0}$ of positive integers. Such filtrations are called $(r_n)_{n \le 0}$-adic.
For such filtrations, as shown by Vershik [@Vershik], standardness turns out to be tantamount to a simpler, much more intuitive property: an $(r_n)$-adic filtration $\fc$ is standard if and only if $\fc$ is of product type, that is, $\fc$ is the natural filtration of some process $V = (V_n)_{n\le 0}$ where the $V_n$ are independent random variables; in this case, it is easy to see that the process $V$ can be chosen with the same law as $U= (U_n)_{n\le 0}$. So, at first reading, ‘standard’ can be replaced with ‘of product type’ in this introduction.
Although intuitive, the notion of product-type filtrations is not as simple as one could believe. For example, the assumption that the tail $\sigma$-field $\fc_{-\infty} = \bigcap_{n \le 0} \fc_n$ is trivial, and the property $\fc_n = \fc_{n-1} \vee \sigma(U_n)$ for every $n \le 0$ do not ensure that $(\fc_n)_{n \le 0}$ is generated by $(U_n)_{n \le 0}$. In the standard case, $(\fc_n)_{n \le 0}$ can be generated by some other sequence $(V_n)_{n \le 0}$ of independent random variables which has the same law as $(U_n)_{n \le 0}$. In the non-standard case, no sequence of independent random variables can generate the filtration $(\fc_n)_{n \le 0}$.
The first examples of such a situation were given by Vershik [@Vershik]. By modifying and generalizing one of these examples, M. Smorodinsky [@Smorodinsky] and Émery and Schachermayer [@Emery-Schachermayer] introduced the split-words processes.
The law of a split-words process depends on an alphabet $A$, endowed with some probability measure, and a decreasing sequence $(\ell_n)_{n \le 0}$ of positive integers (the lengths of the words) such that $\ell_0=1$ and the ratios $r_n = \ell_{n-1}/\ell_n$ are integers. For the sake of simplicity, we consider here only finite alphabets endowed with the uniform measure.
A split-words process is an inhomogeneous Markov process $((X_n,U_n))_{n \le 0}$ such that for every $n \le 0$:
- $(X_n,U_n)$ is uniform on $A^{\ell_n} \times \odc 1,r_n \fdc$.
- $U_n$ is independent of $\fc^{(X,U)}_{n-1}$.
- if one splits the word $X_{n-1}$ (of length $\ell_{n-1} = r_n\ell_n$) into $r_n$ subwords of lengths $\ell_n$, then $X_n$ is the $U_n$-th subword of $X_{n-1}$.
Such a process is well-defined since the sequence of uniform laws on the sets $A^{\ell_n} \times \odc 1,r_n \fdc$ is an entrance law for the transition probabilities given above. By construction, the natural filtration $\fc^{X,U}$ of $((X_n,U_n))_{n \le 0}$ is $(r_n)_{n \le 0}$-adic. One can check that the tail $\sigma$-field $\fc^{X,U}_{-\infty}$ is trivial. Thus, it is natural to ask whether $\fc^{X,U}$ is standard or not.
Whether a split-words process with lengths $(\ell_n)_{n \le 0}$ generates a standard filtration or not is completely characterised : the filtration is non-standard if and only if $$\sum_{n}\frac{ \ln(r_{n})}{ \ell_{n}} < +\infty \hfill \quad (\Delta).$$ Note that this condition does not depend on the alphabet $A$.
In this statement, the ‘if’ part and a partial converse have been proved by Vershik [@Vershik] (in a very similar framework) and by S. Laurent [@Laurent-thesis]. The ‘only if’ part has been proved by D. Heicklen [@Heicklen] (in Vershik’s framework) and by G. Ceillier [@Ceillier]. The generalization to arbitrary alphabets has been performed by Laurent in [@Laurent-standardness]: the characterisations and all the results below still hold are when the alphabet is a Polish space endowed with some probability measure.
Although these examples are rather simple to construct, proving the non-standardness requires sharp tools like Vershik’s standardness criterion [@Vershik; @Emery-Schachermayer]. One can also use the I-cosiness criterion of Émery and Schachermayer [@Emery-Schachermayer] which may be seen as more intuitive by probabilists. Actually, Laurent proved directly that both criteria are actually equivalent. Moreover, applying these criteria to the examples above leads to rather technical estimations.
Another question concerns what happens to a filtration when time is accelerated by extracting a subsequence. Clearly, every subsequence of a standard filtration is still standard. But Vershik’s lacunary isomorphism theorem [@Vershik] states that from [*any*]{} filtration $(\fc_n)_{n \le 0}$ such that $\fc_0$ is essentially separable and $\fc_{-\infty}$ is trivial, one can extract a filtration $(\fc_{\phi(n)})_{n \le 0}$ which is standard. This striking fact is mind-boggling for anyone who is interested by the boundary between standardness and non-standardness. A natural question arises:
when $(\fc_n)_{n \le 0}$ is not standard, how close to identity the increasing map $\phi$ (from $\zzf_-$ to $\zzf_-$) provided by the lacunary isomorphism theorem can be?
Of course, as standardness is an asymptotic property, the extracting map $\phi$ has to skip an infinity of times integers (equivalently, $\phi(n)-n \to -\infty$ as $n \to -\infty$).
In [@Vershik], Vershik provides an example of a non-standard dyadic filtration $(\fc_n)_{n \le 0}$ such that $(\fc_{2n})_{n \le 0}$ is standard. Gorbulsky also provides such an example in [@Gorbulsky].
Using the fact that the family of split-words filtrations is stable by extracting subsequences, Ceillier exhibits in [@Ceillier] an example of a non-standard filtration $(\fc_n)_{n \le 0}$ which is as close to standardness as it can be: every subsequence $(\fc_{\phi(n)})_{n \le 0}$ is standard as soon as $\phi$ skips an infinity of integers.
This paper is devoted to the filtrations sharing this property. We call them filtrations [*at the threshold of standardness*]{}.
Main results and organization of the paper {#main-results-and-organization-of-the-paper .unnumbered}
------------------------------------------
Some definitions and classical facts used in the paper are recalled in an annex, at the end of the paper. In the sections \[split-words\] and \[Tsirelson\] which are the core of the paper, two class of filtrations are studied, first the filtrations of the split-words processes, second some filtrations inspired by an unpublished example of B. Tsirelson.
#### The case of split-words filtrations
The first part deals with split-words filtrations.
First, we characterise the filtrations at the threshold of standardness among the split-words filtrations.
\[characterisation of the threshold\] A split-words filtration with lengths $(\ell_n)_{n \le 0}$ is at the threshold of standardness if and only if $$\sum_{n \le 0} \frac{\ln(r_{n})}{\ell_{n}} < +\infty \hfill \quad (\Delta)$$ and $$\inf_{n \le 0} \frac{\ln(r_{n}r_{n-1})}{\ell_{n}} >0 \hfill \quad (\star).$$
Next, we characterise (among the split-words filtrations) the filtrations that cannot be extracted from any split-words filtration at the threshold of standardness.
\[not extracted from a filtration at the threshold\] If $$\sum_{n \le 0} \frac{\ln(r_{n})}{\ell_{n}} = +\infty
\hfill \quad (\neg \Delta)$$ and $$\lim_{n \to -\infty} \frac{\ln(r_{n})}{\ell_{n}} = 0
\hfill \quad (\Box),$$ then any split-words filtration with lengths $(\ell_n)_{n \le 0}$ is standard but cannot be extracted from a split-words filtration at the threshold of standardness.
One could think that the threshold of standardness is a kind of boundary between standardness and non-standardness. Yet, the situation is not so simple. Indeed, proposition \[not extracted from a filtration at the threshold\] provides an example (example \[no intermediate at the threshold\]) of two split-words filtrations, where
- the first one is non-standard,
- the second one is standard,
- the second one is extracted from the first one,
- yet, no intermediate filtration (for extraction) is at the threshold of standardness.
Furthermore, we provide an example of a non-standard split-words filtration from which no filtration at the threshold of standardness can be extracted (example \[no extracted filtration at the threshold\]). The proof relies on theorem \[various types\] below.
Recall that, given any filtration $(\F_n)_{n \le 0}$ and an infinite subset $B$ of $\Z^-$, the extracted filtration $(\F_n)_{n \in B}$ is standard if and only if the complement $B^c = \zzf_- \setminus B$ is large enough in a certain way. Here, the meaning of “large enough” depends on the filtration $\F$ considered. When $\F$ is at the threshold of standardness, “large enough” means exactly “infinite”. But various types of transition from non-standardness to standardness are possible, and the next theorem provides some other possible conditions.
\[various types\] Let $(\alpha_n)_{n \le 0}$ be any sequence of non-negative real numbers. There exists a split-words filtration $(\fc_n)_{n \le 0}$ such that for every infinite subset $B$ of $\zzf_-$, the extracted filtration $(\fc_n)_{n \in B}$ is standard if and only if $$\sum_{n \in B^c} \alpha_n = +\infty \text{ or } \sum_{n \le 0}
\one_{[n \notin B,\,n+1\notin B]} = +\infty.$$
Theorem \[various types\] immediately provides other interesting examples. For example, it may happen that $(\fc_{2n})_{n \le 0}$ is standard while $(\fc_{2n-1})_{n \le 0}$ is not, or vice versa. When this phenomenon occurs, we will say that the filtration $(\fc_{n})_{n \le 0}$ “interlinks” standardness and non-standardness.
Repeated interlinking is possible. By suitably slowing time suitably in a filtration at the threshold of standardness (example \[repeated interlinking\]), one gets can a filtration $(\fc_{n})_{n \le 0}$ such that $(\fc_{2n})_{n \le 0}$, $(\fc_{4n})_{n \le 0}$, $(\fc_{8n})_{n \le 0}$,... are non-standard, whereas $(\fc_{2n+1})_{n \le 0}$, $(\fc_{4n+2})_{n \le 0}$, $(\fc_{8n+4})_{n \le 0}$,... are standard.
#### Improving on an example of Tsirelson
In a second part, we study another type of filtrations inspired by a construction of Tsirelson in unpublished notes [@Tsirelson].
Tsirelson has constructed an inhomogeneous discrete Markov process $(Z_n)_{n \le 0}$ such that the random variables $(Z_{2n})_{n \le 0}$ are independent and such that the natural filtration $(\fc^Z_n)_{n \le 0}$ is non-standard although its tail $\sigma$-field is trivial. This example is illuminating since “simple” reasons explain why the standardness criteria do not hold and no technical estimates are required. Tsirelson’s construction relies on a particular structure of the triples $(Z_{2n-2},Z_{2n-1},Z_{2n})$ that we explain. We call “bricks” these triples.
In this paper, we give a modified and simpler construction which provides stronger results by requiring more on the bricks: in our construction, for every $n \le 0$, $Z_{2n-2}$ is a deterministic function of $Z_{2n-1}$ and $Z_{2n-1}$ is a deterministic function of $(Z_{2n-2},Z_{2n})$, hence the filtration $(\fc^Z_{2n})_{n \le 0}$ is generated by the sequence $(Z_{2n})_{n \le 0}$ of independent random variables. Yet, $(\fc^Z_{2n-1})_{n \le 0}$ is not standard. Thus the filtration $\fc^Z$ “interlinks” standardness and non-standardness. Actually, we have a complete characterisation of the standard filtrations among the filtrations extracted from $\fc^Z$.
\[existence theorem\] There exists a Markov process $(Z_n)_{n \le 0}$ such that
- for each for $n \le 0$, $Z_n$ takes its values in some finite set $F_n$.
- the random variables $(Z_{2n})_{n \le 0}$ are independent.
- for each for $n \le 0$, $Z_{2n-1}$ is a measurable deterministic function of $(Z_{2n-2},Z_{2n})$.
- the filtration $(Z_n)_{n \le 0}$ is $(r_n)_{n \le 0}$-adic for some sequence $(r_n)_{n \le 0}$.
- for any infinite subset $D$ of $\zzf_-$, the filtration $(\fc^Z_{n})_{n \in D}$ is standard if and only if $2n-1 \notin D$ for infinitely many $n \le 0$.
In particular, the filtration $(\fc^Z_{2n-1})_{n \le 0}$ is at the threshold of standardness.
In this theorem, the statement that $(\fc^Z_{2n-1})_{n \le 0}$ is at the threshold of standardness cannot be deduced from the standardness of $(\fc^Z_{2n})_{n \le 0}$ and the non-standardness of $(\fc^Z_{2n-1})_{n \le 0}$ only. Indeed, the example of repeated interlinking mentioned above (see example \[repeated interlinking\] in section \[split-words\]) provides a counterexample (modulo a time-translation). The proof that $(\fc^Z_{2n-1})_{n \le 0}$ is at the threshold of standardness actually uses the fact that $(Z_n)_{n \le 0}$ is an inhomogeneous Markov process.
The case of split-words filtrations
===================================
\[split-words\]
In the whole section, excepted in subsection \[interlinking\], $\fc = (\fc_n)_{n \le 0}$ denotes a split-words filtration associated to a finite alphabet $A$ (endowed with the uniform measure) and a decreasing sequence $(\ell_n)_{n \le 0}$ of positive integers (the lengths) such that $\ell_0=1$ and the ratios $r_n = \ell_{n-1}/\ell_n$ are integers.
First, we prove the characterisation at the threshold of standardness among the split-words filtrations stated in proposition \[characterisation of the threshold\].
Proof of proposition \[characterisation of the threshold\]
----------------------------------------------------------
[**Preliminary observations:**]{} let $B$ be an infinite subset of $\Z^-$ such that $B^c$ is infinite. Then the filtration $(\F_n)_{n \in B}$ is a split-words filtration with lengths $(\ell_n)_{n \in B}$. The ratios between successive lengths are the integers $(R_n)_{n\in B}$ given by $$R_n = \ell_{m(n)}/\ell_n \text{ where } m(n) = \sup\{k<n~:~ k \in B\}.$$ Set $B_1=B\cap (1+B)$ and $B_2=B\backslash(1+B).$ Then $B_2$ is infinite and
- for $n \in B_1$, $R_n=r_n$,
- for $n \in B_2$, $R_n\ge r_n r_{n-1}$.
Furthermore, if $B^c$ does not contain two consecutive integers, then for any $n \in B_2$, one has $n-2\in B$ since $n-1\notin B$, thus $m(n)=n-2$ and $R_n= r_n r_{n-1}$.
[**Proof of the “if” part:**]{} assume that $$\sum_{n\le 0} \frac{\log(r_n)}{\ell_n}<+\infty
\text{ and } \inf_{n\le 0} \frac{\log(r_n r_{n-1})}{\ell_n} >0.$$ The first condition ($\Delta$) ensures that $\F$ is not standard. Let $B$ be an infinite subset of $\Z^-$ such that $B^c$ is infinite. One has $$\sum_{n\in B} \frac{\log(R_n)}{\ell_n}
\ge \sum_{n\in B_2} \frac{\log(R_n)}{\ell_n}
\ge \sum_{n\in B_2} \frac{\log(r_n r_{n-1})}{\ell_n} = +\infty,$$ since $B_2$ is infinite and $\inf\{ (\log(r_n r_{n-1}))/\ell_n~;~n\le 0\} >0.$ Thus, the split-words filtration $(\fc_n)_{n \in B}$ is standard since the sequence of lengths $(\ell_n)_{n \in B}$ fulfils condition $\neg (\Delta)$. Therefore $\F$ is at the threshold of standardness.
[**Proof of the “only if” part:**]{} condition $(\Delta)$, which is equivalent to the non-standardness of $\F$, is necessary for $\F$ to be at the threshold of standardness. Let us show that if $(\Delta)$ and $\neg(\star)$ hold, then $\F$ is not at the threshold of standardness. Since the reals $\log(r_n r_{n-1})/\ell_n$ are positive, condition $\neg (\star)$ induces the existence of a subsequence $(\log(r_{\phi(n)} r_{\phi(n)-1})/\ell_{\phi(n)})_{n \le 0}$ such that $$\forall n \in \Z^-,\quad \frac{\log(r_{\phi(n)} r_{\phi(n)-1})}{\ell_{\phi(n)}}
\le 2^n \quad \text{ and } \quad \phi(n-1)\le \phi(n)-2.$$
Set $B=(\phi(\Z^-)-1)^c$. Let us show that the filtration $(\F_n)_{n\in B}$ is not standard. By construction, $\phi(\Z^-)$ is infinite and does not contain two consecutive integers. Hence $B$ and $(B)^c$ are both infinite and $B_2=B\backslash (B+1)=\phi(\Z^-)$. Moreover, according to the preliminary observations, $R_n=r_n$ for every $n \in B_1$ and $R_n=r_n r_{n-1}$ for every $n \in B_2$ since $B^c$ does not contain two consecutive integers. Thus $$\begin{aligned}
\sum_{n\in B} \frac{\log(R_n)}{\ell_n} &=& \sum_{n\in B_1} \frac{\log(r_n)}{\ell_n} + \sum_{n\in \phi(\Z^-)} \frac{\log(r_n r_{n-1})}{\ell_n}\\
&\le& \sum_{n \le 0} \frac{\log(r_n)}{\ell_n} + \sum_{m\le 0} 2^m\\
&<& +\infty.\end{aligned}$$ Therefore $(\F_n)_{n \in B}$ is not standard. Thus $\F$ is not at the threshold of standardness.
Proof of proposition \[not extracted from a filtration at the threshold\] and example
-------------------------------------------------------------------------------------
Assume that $(\neg\Delta)$ and $(\Box)$ hold and that $\F$ is extracted from some split-words filtration $\H$ with lengths $(\ell'_n)_{n \le 0}$, namely $\F_n = \H_{\phi(n)}$ for every $n \le 0$, for some increasing map $\phi$ from $\Z^-$ to $\Z^-$. Then for every $n \le 0$, $\ell_n=\ell'_{\phi(n)}$ and $r_n = r'_{\phi(n)} \cdots r'_{\phi(n-1)+1}$ where $r'_k = \ell'_{k-1}/\ell'_{k}$. Let us show that $\H$ cannot be at the threshold of standardness.
Condition $(\neg\Delta)$ ensures that $\F$ is standard. If $\phi$ skips only finitely many integers, then $\H$ is standard and the conclusion holds. Otherwise, $\phi(u_n-1)\le \phi(u_n)-2$ for infinitely many $n$, and for those $n$, $$\frac{\log(r'_{\phi(n)}r'_{\phi(n)-1})}{\ell'_{\phi(n)}}
\le \frac{\log(r'_{\phi(n)} \cdots r'_{\phi(n-1)+1})}{\ell'_{\phi(n)}}
= \frac{\log r_n}{\ell_n}.$$ Thus, $(\Box)$ implies that $$\inf_{k\le 0} \frac{\log(r'_k r'_{k-1})}{\ell'_k}=0.$$ Since the sequence $(r'_n)_{n \le 0}$ does not fulfill condition $(\star)$, $\H$ is not at the threshold of standardness.
\[no intermediate at the threshold\] Define the sequence of lengths $(\ell_n)_{n \le 0}$ by $\ell_0=1$, $\ell_{-1}=2$ and, for every $n\le-1$, $$\ell_{n-1}=\ell_n 2^{\lfloor \ell_n/|n| \rfloor},$$ where $\lfloor x\rfloor$ denotes the integer part of $x$.
A recursion shows that for every $n \leq 0$, $\ell_n$ is a power of $2$, and that $\ell_n \ge 2^{|n|} \ge |n|$, hence $r_n=\ell_{n-1}/\ell_n\ge 2$. Moreover, for every $n \le -1$, $$\frac{\log_2(r_n)}{\ell_n} =
\frac{{\lfloor \ell_n/|n| \rfloor}}{\ell_n} \in
\left[\frac{1}{2|n|},\frac{1}{|n|}\right].$$ Therefore, $(\neg \Delta)$ and $(\Box)$ hold, hence $\F$ is standard but cannot be extracted from any split-words filtration at the threshold of standardness.
Yet, since each $\ell_n$ is a power of $2$, $\F$ is extracted from the dyadic split-words filtration $\H$, which is not standard. Since every filtration extracted from $\H$ is a split-words filtration, one can deduce that there is no intermediate filtration (for extraction) between $\H$ and $\F$.
Remark: there are trivial examples of standard split-words filtrations which cannot be extracted from any split-words filtration at the threshold of standardness. For example, consider any split-words filtrations such that $\neg(\Delta)$ holds and such that $r_n$ is a prime number for every $n \le 0$. The last condition prevents the filtration from being extracted from [*any other*]{} split-words filtration. Yet, it still could be extracted from some filtration at the threshold of standardness which is not a split-words filtration.
Proof of theorem \[various types\]
----------------------------------
Replacing $\alpha_{n}$ by $\min(\max(\alpha_{n},1/|n+2|^2),1)$ for $n\le-3$ does not change the nature of the series $\sum_{k \in B^c} \alpha_{k}$, hence we may assume that for $n\le-3$, $$1/|n+2|^2 \le \alpha_{n} \le 1.$$ Set $\ell_0=1$, $\ell_{-1}=2$, $\ell_{-2}=8$, $\ell_{-3}=64$, $\ell_{-4}=2^{11}=2048$ and $\ell_{n-2}=2^{\lfloor\alpha_{n-1}\ell_n\rfloor}$ for every $n\le -3$, where $\lfloor x\rfloor$ denotes the integer part of $x$. We begin with two technical lemmas.
\[minorations\] For every $n \le -1$, $\ell_n \ge |n|^3$ and $\ell_{n} \ge 2|n+1|^2 \ell_{n+1}$.
The proof of lemma \[minorations\] is done by induction. One checks that the above inequalities hold for $-4 \le n \le -1$.
Fix some $n \le -3$. Assume that the inequalities hold for $n+1$, $n$ and $n-1$. Then $$\begin{aligned}
\log_2\ell_{n-2} - \log_2\ell_{n-1}
&=& \lfloor\alpha_{n-1}\ell_{n}\rfloor - \lfloor\alpha_n\ell_{n+1}\rfloor\\
&\ge& \alpha_{n-1}\ell_{n}-1-\alpha_n\ell_{n+1}\\
&\ge& \frac{\ell_{n}}{|n+1|^2}-\ell_{n+1}-1\\
&\ge& \ell_{n+1}-1 \text{ (since } \ell_{n} \ge 2|n+1|^2 \ell_{n+1})\\
&\ge& |n+1|^3-1 \text{ (since } \ell_{n+1} \ge |n+1|^3),\end{aligned}$$ hence $$\ell_{n-2}/\ell_{n-1} \ge 2^{|n+1|^3-1} \ge 2|n-1|^2 \text{ (since }n \le -3).$$ Since $\ell_{n-1} \ge |n-1|^3$, one has $$\ell_{n-2} \ge 2|n-1|^2\ell_{n-1} \ge 2|n-1|^5 \ge |n-2|^3
\text{ (since }n \le -3).$$ Thus the inequalities hold for $n-2$. The proof is complete.
\[estimates\] For every $n \le -4$, $$\begin{aligned}
\frac{\log_2 \ell_{n-1}}{\ell_n} &\le& \frac{1}{2|n+1|^2},\\
\frac{\alpha_{n-1}}{2} \le \frac{\log_2 \ell_{n-2}}{\ell_n}
&\le& \alpha_{n-1},\\
\frac{\log_2 \ell_{n-3}}{\ell_n} &\ge& 1.\end{aligned}$$
Fix $n \le -4$. The assumptions made on the sequence $(\alpha_k)_{k \le 0}$, and lemma \[minorations\] entail $\ell_n\alpha_{n-1}\ge |n|^3/|n+1|^2\ge 1,$ thus $\alpha_{n-1}\ell_n/2 \le \lfloor\alpha_{n-1}\ell_{n}\rfloor
\le \alpha_{n-1}\ell_n$. Thus, the recursion formula $\ell_{n-2} = 2^{\lfloor\alpha_{n-1}\ell_n\rfloor}$ yields $$\frac{\alpha_{n-1}}{2}\le \frac{\log_2 \ell_{n-2}}{\ell_n} \le \alpha_{n-1}.$$ Since $n \le -4$, the same inequalities hold for $n+1$ and $n-1$, hence by lemma \[minorations\] $$\frac{\log_2 \ell_{n-1}}{\ell_n}
\le \alpha_n \frac{\ell_{n+1}}{\ell_{n}} \le
\frac{\ell_{n+1}}{\ell_{n}} \le \frac{1}{2|n+1|^2},$$ and $$\frac{\log_2 \ell_{n-3}}{\ell_n} \ge \frac{\alpha_{n-2}}{2}
\frac{\ell_{n-1}}{\ell_{n}} \ge \frac{1}{2|n|^2}2|n|^2 = 1.$$ The proof is complete.
We now prove theorem \[various types\].
Let us check that the split-words filtration associated to the to the lengths $(\ell_n)_{n \le 0}$ fulfills the properties of the previous proposition.
Let $B$ be an infinite subset of $\Z^-$ such that $B^c$ is infinite. Since replacing $B$ by $B \setminus \{-2,-1,0\}$ does not change the nature of the filtration $(\F_n)_{n\in B}$, one may assume that $B \subset ]-\infty,-3]$.
Set $m(n) = \sup\{k<n~:~ k \in B\}$ for every $n \le 0$. Then $(\ell_{m(n)}/\ell_n)_{n \in B}$ is the sequence of ratios associated to the lengths $(\ell_n)_{n \in B}$. Since $(\Delta)$ characterises standardness of split-words filtrations, $$(\F_n)_{n \in B} \text{ is standard } \Longleftrightarrow
\sum_{n \in B} \frac{\log_2 (\ell_{m(n)}/\ell_n)}{\ell_n} = +\infty
\Longleftrightarrow\sum_{n \in B} \frac{\log_2 \ell_{m(n)}}{\ell_n} = +\infty,$$ where the last equivalence follows from the convergence of the series $\sum_n \log_2\ell_n/\ell_n$ since $\ell_n \ge 2^{|n|}$ for every $n \le 0$.
Let us split $B$ into three subsets:
- $B_1=\{ n\in B : m(n)=n-1\}$,
- $B_2=\{ n\in B : m(n)=n-2\}$,
- $B_3=\{ n\in B : m(n)\le n-3\}$.
Then $$\sum_{n \in B} \frac{\log_2 \ell_{m(n)}}{\ell_n}
= \sum_{n \in B_1} \frac{\log_2 \ell_{n-1}}{\ell_n}
+ \sum_{n \in B_2} \frac{\log_2 \ell_{n-2}}{\ell_n} +
\sum_{n \in B_3} \frac{\log_2 \ell_{m(n)}}{\ell_n}.$$ The inequality $\ell_{m(n)} \ge \ell_{n-3}$ for $n \in B_3$ and lemma \[estimates\] show that in the right-hand side,
- the first sum (over $B_1$) is always finite;
- the middle sum (over $B_2$) has the same nature as $\sum_{n \in B_2} \alpha_n$;
- the last sum (over $B_3$) is finite if and only if $B_3$ is finite.
When $B_3$ is finite, any pair of consecutive integers excepted a finite number of them contain at least one element of $B$. Hence, $(B_2 -1)$ only differs from $B^c$ by a finite set of integers. Thus the sum $\sum_{n \in B_2} \alpha_n$ has the same nature as $\sum_{n \in B^c} \alpha_n$. Theorem \[various types\] follows.
Some applications of theorem \[various types\]
----------------------------------------------
Choosing particular sequences $(\alpha_n)_{n \le 0}$ in theorem \[various types\] provides interesting examples of non-standard filtrations. In what follows, $\F$ denotes the filtration associated the sequence $(\alpha_n)_{n \le 0}$ given by theorem \[various types\].
If $\alpha_n=1$ for every $n$, then $\F$ is at the threshold of standardness.
If $\alpha_{n}=0$ for every even $n$ and $\alpha_{n}=1$ for every odd $n$, then $(\F_{2n})_{n \le 0}$ is standard whereas $(\F_{2n-1})_{n \le 0}$ is not.
If the series $\sum \alpha_n$ converges, then for every infinite subset $B$ of $\Z^-$, the extracted filtration $(\F_n)_{n\in B}$ is standard if and only if $(B \cup (B-1))^c$ is infinite. In particular, the filtrations $(\F_{2n})_{n \le 0}$ and $(\F_{2n-1})_{n \le 0}$ are at the threshold of standardness.
\[no extracted filtration at the threshold\] If $\alpha_n\sim 1/|n|$ as $n$ goes to $-\infty$, then $\F$ is not standard and no filtration at the threshold of standardness can be extracted from $\F$.
The non-standardness of $\F$ is immediate by theorem \[various types\].
Call $\mu$ the non-finite positive measure on $\Z^-$ defined by $$\mu(B) = \sum_{n \in B} \alpha_n \text{ for } B \subset \Z^-.$$
Let $(\F_n)_{n \in B}$ be any non-standard filtration extracted from $\F$. We show that $(\F_n)_{n \in B}$ cannot be at the threshold of standardness by constructing a subset $B'$ of $B$ such that and $(\F_n)_{n \in B'}$ is not standard although $B \setminus B'$ is infinite .
By to theorem \[various types\], we know that $\mu(B^c) < +\infty$ and $$n \notin B \text{ and } n+1 \notin B
\text{ only for finitely many } n \in \Z^-.$$ Since $\mu(B^c)$ is finite, the elements of $B^c$ get rarer and rarer as $n \to -\infty$. In particular, the set $A = (B-1) \cap B \cap (B+1)$ is infinite.
We get $B'$ from $B$ by removing a “small” infinite subset of $A$. Namely, we set $B'= B \backslash A'$ where $A'$ is an infinite subset of $A$ which does not contain two consecutive integers and chosen such that $\mu(A') < +\infty$. By construction, $B \backslash B' = A'$ is infinite and $\mu((B')^c) < +\infty$ since $(B')^c = B^c \cup A'$. Thus $B'$ is an infinite subset of $B$.
Using the definition of $A$ and the fact that $A'$ does not contain two consecutive integers and by construction of $A$, one checks that $(B'\cup(B'-1))=(B\cup(B-1))$, therefore $(B'\cup(B'-1))^c$ is infinite.
Thus $(\F_n)_{n \in B'}$ is not standard, which shows that $(\F_n)_{n \in B}$ is not at the threshold of standardness.
Interlinking standardness and non standardness
----------------------------------------------
\[interlinking\]
Given any filtration $(\fc_n)_{n \le 0}$, a simple way to get a “slowed” filtration is to repeat each $\fc_n$ some finite number of times, which may depend of $n$. We now show that this procedure does not change the nature of the filtration.
\[slowed filtrations\] Let $(\F_n)_{n \le 0}$ be any filtration and $\phi$ an increasing map from $\Z^-$ to $\Z^-$ such that $\phi(0)=0$. For every $n \le 0$, set $\G_n=\F_k$ if $\phi(k-1)+1 \le n \le \phi(k)$. Then:
- $(\G_n)_{n \le 0}$ is a filtration,
- $(\F_n)_{n \le 0}$ is extracted from $(\G_n)_{n \le 0}$,
- $(\G_n)_{n \le 0}$ is standard if and only if $(\F_n)_{n \le 0}$ is standard.
By construction, $\G_{\phi(k)} = \F_k$ for every $k \le 0$ and the sequence $(\G_n)_{n \le 0}$ is constant on every interval $\odc \phi(k-1)+1,\phi(k) \fdc$. The first two points follow.
The “only if” part of the third point is immediate since $\F$ is extracted from $\G$.
Assume that $\F$ is standard. Then, up to a enlargement of the probability space, one may assume that $\F$ is immersed in some product-type filtration $\H$. Define a slowed filtration by $\K_n=\H_k$ if $\phi(k-1)+1 \le n \le \phi(k)$. Then $\K$ is still a product-type filtration To prove that $\G$ is immersed in $\K$, we have to check that for every $n \le -1$, $\G_{n+1}$ and $\K_n$ are independent conditionally on $\G_n$. This holds in any case since:
- when $\phi(k-1)+1 \le n \le \phi(k)-1$, $\G_{n+1}=\F_k$, $\K_n=H_k$ and $\G_n=\F_k$;
- when $n=\phi(k)$, $\G_{n+1}=\F_{k+1}$, $\K_n=H_k$ and $\G_n=\F_k$.
Hence $\G$ is standard.
\[repeated interlinking\] Assume that $(\F_n)_{n \le 0}$ is at the threshold of standardness. Set $\phi(0)=0$, $\phi(-1)=-1$ and, for every $k \le 0$, $\phi(2k) = -2^{|k|}$ and $\phi(2k-1) = -2^{|k|}-1$. Let $\G$ be the slowed filtration obtained from $\F$ as above. Then for any $d \ge 1$, the filtration $(\G_{2^dn})_{n \le 0}$ is not standard, whereas the filtration $(\G_{2^dn-2^{d-1}})_{n \le 0}$ is standard.
Fix $d \ge 1$. The filtrations $(\G_{2^dn})_{n \le -2}$ and $(\G_{2^dn-2^{d-1}})_{n \le -1}$ can be obtained from $(\F_{n})_{n \le -2d-2}$ and $(\F_{2n-1})_{n \le -d}$ by time-translations and by the slowing procedure just introduced. And truncations, time-translations and slowing procedure preserve the nature of the filtrations.
Improving on an example of Tsirelson
====================================
\[Tsirelson\]
In some non-published notes, Tsirelson gives a method to construct an inhomogeneous Markov process $(X_n)_{n \le 0}$ such that the natural filtration $(\fc^X_n)_{n \le 0}$ is easily proved to be non-standard, although the tail $\sigma$-field $\F^X_{-\infty}$ is trivial and the random variables $(X_{2n})_{n \le 0}$ are independent.
In Tsirelson’s construction, each triple $(X_{2n-2},X_{2n-1},X_{2n})$ has a particular structure that we will explain soon. since the sequence $(X_n)_{n \le 0}$ is obtained by gluing the triples $(X_{2n-2},X_{2n-1},X_{2n})$ in a Markovian way, we call [*Tsirelson’s bricks*]{} these triples.
The basic Tsirelson’s brick
---------------------------
Informally, the basic brick in Tsirelson’s construction is a triple of uniform random variables $X_{0},X_{1},X_{2}$ with values in some finite sets $F_{0},F_{1},F_{2}$ such that for some $\alpha \in [0,1[$,
- the set $F_{2}$ is arbitrarily large, and the set $F_{0}$ is much larger;
- the triple $(X_{0},X_{1},X_{2})$ is Markov;
- the random variables $X_{0}$ and $X_{2}$ are independent;
- any two different values of $X_{0}$ lead to different values of $X_{2}$ with probability $\ge 1-\alpha$.
We now explain what the last requirement means.
Fix two distinct values in $F_{0}$, namely $x'_0$ and $x''_0$. Choose randomly but not necessarily independently $x'_1$ and $x''_1$ in $F_{1}$ according to the laws $\lc(X_1|X_0=x'_0)$ and $\lc(X_1|X_0=x''_0)$. Then choose randomly but not necessarily independently $x'_2$ and $x''_2$ in $F_{2}$ according to the laws $\lc(X_2|X_1=x'_1)$ and $\lc(X_2|X_1=x''_1)$. Then the values $x'_2$ and $x''_2$ must be different with probability $\ge 1-\alpha$, whatever was the strategy used to make the different choices.
More precisely, note $\rho_2$ the discrete metric on $F_2$: for all $x'_2$ and $x''_2$ in $F_2$, $$\begin{aligned}
\rho_2(x'_2,x''_2)=1 & \text{ if } x'_2 \ne x''_2,\\
\rho_2(x'_2,x''_2)=0 & \text{ if } x'_2 = x''_2.\end{aligned}$$ For all $x'_1$ and $x''_1$ in $F_1$, note $\rho_1(x'_1,x''_1)$ the Kantorovitch-Rubinstein distance between the laws $\lc(X_2|X_1=x'_1)$ and $\lc(X_2|X_1=x''_1)$. By definition, $$\rho_1(x'_1,x''_1) = \inf\{\eef[\rho_2(X'_2,X''_2)]\ ; X'_2 \leadsto
\lc(X_2|X_1=x'_1), X''_2 \leadsto \lc(X_2|X_1=x''_1) \}.$$ Since $\rho_2$ is the discrete metric on $F_2$, $\rho_1(x'_1,x''_1)$ is actually the total variation distance between $\lc(X_2|X_1=x'_1)$ and $\lc(X_2|X_1=x''_1)$.
By the same way, for all $x'_0$ and $x''_0$ in $F_0$, denote by $\rho_0(x'_0,x''_0)$ the Kantorovitch-Rubinstein distance between the laws $\lc(X_1|X_0=x'_0)$ and $\lc(X_1|X_0=x''_0)$. The last requirement means that $\rho_0(x'_0,x''_0) \ge 1-\alpha$ when $x'_0 \ne x''_0$. This condition is used by Tsirelson to negate Vershik’s criterion.
Here is another formulation, which is closer to the I-cosiness criterion recalled in section \[annex\]: for any non-anticipative coupling of two copies $(X'_{0},X'_{1},X'_{2})$ and $(X''_{0},X''_{1},X''_{2})$ of $(X_{0},X_{1},X_{2})$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$, $$\bar{\P}[X'_{2} \ne X''_{2}|\sigma(X'_{0},X''_{0})] \ge 1-\alpha
\text{ on the event } [X'_{0} \ne X''_{0}].$$ Here, the expression “non-anticipative” means that the filtrations generated by the processes $X'$ and by $X''$ are immersed in the natural filtration of $(X',X'')$. In particular, $X'_{1}$ and $X''_{0}$ are independent conditionally on $X'_{0}$ (the couple $(X'_{0},X''_{0})$ gives no more information on $X'_{1}$ than $X'_{0}$ does). Similarly, $X'_{2}$ and $(X''_{0},X''_{1})$ are independent conditionally on $(X'_{0},X'_{1})$. And the same holds when the roles of $X'$ and $X''$ are exchanged.
Let us give a formal definition.
Fix $\alpha \in ]0,1[$. Let $F_{0},F_{1},F_{2}$ be finite sets. We will say that a triple $(Z_{0},Z_{1},Z_{2})$ of uniform random variables with values in $F_{0},F_{1},F_{2}$ is a Tsirelson’s $\alpha$-brick if
- the triple $(Z_{0},Z_1,Z_{2})$ is Markov.
- $Z_{0}$ and $Z_{2}$ are independent.
- for any non-anticipative coupling of two copies $(X'_{0},X'_{1},X'_{2})$ and $(X''_{0},X''_{1},X''_{2})$ of $(X_{0},X_{1},X_{2})$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$, $$\bar{\P}[X'_{2} \ne X''_{2}|\sigma(X'_{0},X''_{0})] \ge 1-\alpha
\text{ on the event } [X'_{0} \ne X''_{0}].$$
Tsirelson’s example of a brick
------------------------------
Tsirelson gives an example of such a brick which is enlightening.
Let $p$ be a prime number, and $\zzf_p$ be the finite field with $p$ elements. Note $F_{0}$ the set of all two-dimensional linear subspaces of $(\zzf_p)^5$, $F_{1}$ the set of all one-dimensional affine subspaces of $(\zzf_p)^5$ and $F_{2} = (\zzf_p)^5$. Then the size of $F_2$ is $|F_{2}| = p^5$ whereas $$|F_{0}| = \frac{(p^5-1)(p^5-p)}{(p^2-1)(p^2-p)}
= (p^4+p^3+p^2+p+1)(p^2+1).$$ Indeed, the number of couples of independent vectors in $(\zzf_p)^5$ is $(p^5-1)(p^5-p)$, but any linear plane in $(\zzf_p)^5$ can be generated by $(p^2-1)(p^2-p)$ of these couples.
Tsirelson constructs a Markovian triple $(X_{0},X_{1},X_{2})$ as follows:
- choose uniformly $X_{0}$ in $F_{0}$ ;
- given $X_{0}$, choose uniformly $X_{1}$ among the affine lines whose direction are included in the linear plane $X_{0}$ ;
- given $X_{0}$ and $X_{1}$, choose uniformly $X_{2}$ on the affine line $X_{1}$.
One can check that $X_{2}$ is uniform on $F_{2}$, and independent of $X_{0}$.
Now, let $(X'_{0},X'_{1},X'_{2})$ and $(X''_{0},X''_{1},X''_{2})$ be any non-anticipative coupling of two copies of $(X_{0},X_{1},X_{2})$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$. Then, conditionally on $(X'_{0},X''_{0},X'_{1},X''_{1})$, the law of $X'_{2}$ is uniform on the line $X'_{1}$ and the law of $X''_{2}$ is uniform on the $X''_{1}$. Since two distinct lines have at most one common point, one has $$\bar{\P}[X'_2 = X''_2|\sigma(X'_{0},X''_{0},X'_{1},X''_{1})] \le
\one_{[X'_{1} = X''_{1}]} + \frac{1}{p} \one_{[X'_{1} \ne X''_{1}]},$$ hence $$\bar{\P}[X'_2 \ne X''_2|\sigma(X'_{0},X''_{0},X'_{1},X''_{1})] \ge
\frac{p-1}{p} \one_{[X'_{1} \ne X''_{1}]}.$$ Similarly, conditionally on $(X'_{0},X''_{0})$, the law of $X'_{1}$ is uniform on the set of all affine lines which are parallel to $X'_{0}$ and the law of $X''_{1}$ is uniform on the set of all affine lines which are parallel to $X''_{0}$. But the affine lines $X'_{1}$ and $X''_{1}$ must have the same direction to be equal. Since each linear plane in $(\zzf_p)^5$ contains $p+1$ linear lines whereas two distinct planes contain at most one common line, $$\bar{\P}[X'_{1} = X''_{1}|\sigma(X'_{0},X''_{0})] \le
\one_{[X'_{0} = X''_{0}]} + \frac{1}{p+1} \one_{[X'_{0} \ne X''_{0}]},$$ hence $$\bar{\P}[X'_{1} \ne X''_{1}|\sigma(X'_{0},X''_{0})] \ge
\frac{p}{p+1} \one_{[X'_{0} \ne X''_{0}]}.$$ Putting things together, one gets $$\begin{aligned}
\bar{\P}[X'_2 \ne X''_2|\sigma(X'_{0},X''_{0})]
&\ge& \frac{p-1}{p} \P[X'_{1} \ne X''_{1}|\sigma(X'_{0},X''_{0})] \\
&\ge& \frac{p-1}{p+1} \one_{[X'_{0} \ne X''_{0}]}.\end{aligned}$$ Hence, $(X_{0},X_{1},X_{2})$ is a Tsirelson’s $\alpha$-brick with $\alpha = 2/(p+1)$.
Assembling bricks together
--------------------------
The next step is to construct a non-homogeneous Markov process $(X_n)_{n \le 0}$ such that for each $n \le 0$, the subprocess $(X_{2n-2},X_{2n-1},X_{2n})$ is an Tsirelson’s $\alpha_n$-brick, where the $]0,1[$-valued sequence $(\alpha_n)_{n \le 0}$ fulfills $$\sum_{n \le 0} \alpha_n < +\infty.$$ The next theorem achieves Tsirelson’s construction.
\[Tsirelson’s construction\] Let $(X_n)_{n \le 0}$ be a sequence of uniform random variables with values in finite sets $(F_n)_{n \le 0}$ and $(\alpha_n)_{n \le 0}$ be an $]0,1[$-valued sequence such that the series $\sum_n \alpha_n$ converges. Assume that
- the sets $F_{2n}$ are not singles,
- $(X_n)_{n \le 0}$ is a non-homogeneous Markov process,
- for each $n \le 0$, the subprocess $(X_{2n-2},X_{2n-1},X_{2n})$ is a Tsirelson’s $\alpha_n$-brick.
Then the natural filtration $\F^X$ is not standard. Moreover, if the tail $\sigma$-field $\fc^X_{-\infty}$ is trivial, then $|F_{2n}| \to +\infty$ as $n \to -\infty$.
First, we show that $X_0$ does not fulfills the I-cosiness criterion (see section \[annex\]). Indeed, set $$c = \prod_{k \le 0}(1-\alpha_k) > 0$$ and consider any non-anticipative coupling $(X'_n)_{n \le 0}$ and $(X''_n)_{n \le 0}$ of the process $(X_n)_{n \le 0}$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$. By assumption, for every $n \le 0$, $$\bar{\P}[X'_{2n} \ne X''_{2n}|\sigma(X'_{2n-2},X''_{2n-2})] \ge
(1-\alpha_n) \one_{[X'_{2n-2} \ne X''_{2n-2}]}.$$ By induction, for every $n \le 0$, $$\begin{aligned}
\bar{\P}[X'_0 \ne X''_0|\sigma(X'_{2n},X''_{2n})]
\ge \Big(\prod_{k=n+1}^0 (1-\alpha_k) \Big) \one_{[X'_{2n} \ne X''_{2n}]}
\ge c \one_{[X'_{2n} \ne X''_{2n}]} \end{aligned}$$ If, for some $N \le 0$, the $\sigma$-fields $\F^{X'}_{2N}$ and $\F^{X''}_{2N}$ are independent, then $$\bar{\P}[X'_0 \ne X''_0] \ge c \bar{\P}[X'_{2N} \ne X''_{2N}]
= c(1-|F_{2n}|^{-1}) \ge c/2.$$ Hence $\bar{\P}[X'_0 \ne X''_0]$ is bounded away from $0$, which negates the I-cosiness criterion. The non-standardness of $\F^X$ follows.
The second part of the theorem directly follows from the next proposition, applied to the sequence $(Y_n)_{n \le 0} = (X_{2n})_{n \le 0}$.
\[necessity of large sets\] Let $(\gamma_n)_{n \le 0}$ be a sequence of positive constants such that $$\prod_{n \le 0}\gamma_n > 0.$$ Let $(Y_n)_{n \le 0}$ be a family of random variables which are uniformly distributed on finite sets $(E_n)_{n \le 0}$. Let $(Y'_n)_{n \le 0}$ and $(Y''_n)_{n \le 0}$ be independent copies of the process $(Y_n)_{n \le 0}$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$. Assume that $\fc^Y_{-\infty}$ is trivial and that for every $n \le 0$, $$\bar{\P}[Y'_{n} \ne Y''_{n}|\sigma(Y'_{n-1},Y''_{n-1})] \ge \gamma_n
\one_{[Y'_{n-1} \ne Y''_{n-1}]}.$$ Then $|E_n| \to 1$ or $|E_n| \to +\infty$ as $n \to -\infty$.
By the independence of $(Y'_n)_{n \le 0}$ and $(Y''_n)_{n \le 0}$, the following exchange properties apply (see [@Weizsacker]) $$\begin{aligned}
\bigcap_{m \le 0} \bigcap_{n \le 0} \left( \fc^{Y'}_m \vee \fc^{Y''}_n \right)
&=& \bigcap_{m \le 0}
\left( \fc^{Y'}_m \vee \Big( \bigcap_{n \le 0} \fc^{Y''}_n \Big) \right) \\
&=& \bigcap_{m \le 0}
\left( \fc^{Y'}_m \vee \fc^{Y''}_{-\infty} \right) \\
&=& \left( \bigcap_{m \le 0} \fc^{Y'}_m \right) \vee \fc^{Y''}_{-\infty} \\
&=& \fc^{Y'}_{-\infty} \vee \fc^{Y''}_{-\infty}.\end{aligned}$$ Using that $\fc^{Y'}_m \vee \fc^{Y''}_n$ is non-decreasing with respect to $m$ and $n$, one gets $$(\fc^{Y'} \vee \fc^{Y''})_{-\infty} =
\bigcap_{n \le 0} \left( \fc^{Y'}_n \vee \fc^{Y''}_n \right) =
\bigcap_{m \le 0} \bigcap_{n \le 0} \left( \fc^{Y'}_m \vee \fc^{Y''}_n \right).$$ Hence the tail $\sigma$-field $(\fc^{Y'} \vee \fc^{Y''})_{-\infty}$ is trivial. Thus the asymptotic event $$\liminf_{n \to -\infty} [Y'_n \ne Y''_n]$$ has probability $0$ or $1$.
But a recursion shows that for every $n \le 0$ $$\bar{\P} \Big( \bigcap_{n \le k \le 0} [Y'_{k} \ne Y''_{k}] \Big|
\sigma(Y'_{n},Y''_{n}) \Big)
\ge \Big( \prod_{n+1 \le k \le 0} \gamma_k \Big) \one_{[Y'_{n} \ne Y''_{n}]}.$$ By taking expectations, $$\bar{\P} \Big( \bigcap_{n \le k \le 0} [Y'_{k} \ne Y''_{k}] \Big) \ge
\big(1-|E_n|^{-1} \big) \prod_{n+1 \le k \le 0} \gamma_k.$$
If $|E_n| \ge 2$ for infinitely many $n \le 0$, then $$\bar{\P} \Big( \bigcap_{k \le 0} [Y'_{k} \ne Y''_{k}] \Big) \ge
\frac{1}{2} \prod_{k \le 0} \gamma_k > 0.$$ Thus $|E_n| \ge 2$ for every $n \le 0$ and $$\bar{\P}(\liminf_{n \to -\infty} [Y'_n \ne Y''_n])=1.$$ But by Fatou’s lemma, $$\bar{\P}(\liminf_{n \to -\infty} [Y'_n \ne Y''_n]) \le
\liminf_{n \to -\infty} \bar{\P}[Y'_n \ne Y''_n].$$ Hence $1-|E_n|^{-1} = \bar{\P}[Y'_n \ne Y''_n] \to 1$ thus $|E_n| \to +\infty$ as $n \to -\infty$.
Choosing the size of the sets $F_n$
-----------------------------------
The last theorem explains the necessity to have bricks $(Z_{0},Z_{1},Z_{2})$ such that the set $F_{2}$ of all possible values of $Z_2$ is arbitrarily large, and the set $F_{0}$ of all possible values of $Z_0$ is much larger. In Tsirelson’s example, the size of $F_{2}$ is $p^5$ where $p$ is a prime number, whereas the size of $F_{0}$ is $(p^4+p^3+p^2+p+1)(p^2+1)$.
Such bricks provided cannot be glued together since the size of $F_{2}$ is not a power of a prime number: it has at least two prime divisors since the greatest common divisor of $p^4+p^3+p^2+p+1$ and $p^2+1$ is $1$. Replacing $\zzf_p$ by a more general finite field would not change anything since the size of any finite field is necessarily a power of a prime number. Fortunately, a slight modification solve this problem.
A first way to solve the problem is to choose a prime number $q$ such that $q^5$ is slightly smaller than $(p^4+p^3+p^2+p+1)(p^2+1)$ and to call $F_{0}$ a subset with size $q^5$ of all two-dimensional linear subspaces of $(\zzf_p)^5$. After this modification, the law of $Z_{1}$ (a random line choose uniformly along the affine lines which are parallel to the linear plane $Z_{0}$) will no longer be an uniform law, but the law of $Z_{1}$ plays no particular role in the construction.
A second solution is to replace the affine [*lines*]{} by the affine [*planes*]{} in the definition of $Z_{1}$ and $F_{1}$. In this last solution, $Z_{0}$ is a deterministic function of $Z_{1}$ (namely, the vector plane is the direction of the affine plane) and $Z_{1}$ is a deterministic function of $(Z_{0},Z_{2})$ (namely, $Z_{1}$ is the only affine plane which is parallel to $Z_{0}$ and contains $Z_{2}$). These two additional properties have many advantages. First, the construction and the proofs are even simpler. Next, we will use them to get stronger results.
From now on, we will consider only bricks having these two additional properties.
Strong bricks
-------------
Let us give a rigorous definition.
Fix $\alpha \in ]0,1[$ and two positive integers $r_{1},r_{2}$. Let $F_{0},F_{1},F_{2}$ be finite sets. We will say that a triple $(Z_{0},Z_{1},Z_{2})$ of uniform random variables with values in $F_{0},F_{1},F_{2}$ is a strong $(r_{1},r_{2})$-adic $\alpha$-brick if
- $Z_{0}$ and $Z_{2}$ are independent.
- $Z_{1}$ is a deterministic function of $(Z_{0},Z_{2})$;
- $Z_{0}$ is a deterministic function of $Z_{1}$;
- the conditional law of $Z_{1}$ given $Z_{0}$ is uniform on some finite random set of size $r_{1}$;
- the conditional law of $Z_{2}$ given $Z_{1}$ is uniform on some finite random set of size $r_{2}$;
- for every distinct elements $z'_{1}$ and $z''_{1}$ in $F_{1}$, $$~\label{bad coupling}
\sum_{z \in F_{2}} \min \big(\P[Z_{2}=z|Z_{1}=z'_{1}],\P[Z_{2}=z|Z_{1}=z''_{1}] \big)
\le \alpha.$$
The next lemma shows that the definition of strong bricks is more restrictive that the definition of Tsirelson’s bricks.
\[strong bricks are bricks\] If $(Z_{0},Z_{1},Z_{2})$ is a strong $\alpha$-brick, then for any non-anticipative coupling $(Z'_{0},Z'_{1},Z'_{2})$ and $(Z''_{0},Z''_{1},Z''_{2})$ of $(Z_{0},Z_{1},Z_{2})$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$, $$\bar{\P}[Z'_2 \ne Z''_2|\sigma(Z'_{1},Z''_{1})] \ge
(1-\alpha) \one_{[Z'_{1} \ne Z''_{1}]} \ge
(1-\alpha) \one_{[Z'_{0} \ne Z''_{0}]}.$$ Thus, $(Z_{0},Z_{1},Z_{2})$ is a Tsirelson’s $\alpha$-brick
The triple $(Z_{0},Z_{1},Z_{2})$ is Markov since $Z_{0}$ is a function of $Z_{1}$.
Now, let $(Z'_{0},Z'_{1},Z'_{2})$ and $(Z''_{0},Z''_{1},Z''_{2})$ be any non-anticipative coupling of $(Z_{0},Z_{1},Z_{2})$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P})$. Set $\gc = \sigma(Z'_{0},Z'_{1},Z''_{0},Z''_{1})$. By the non-anticipative and the Markov properties, $$\lc(Z'_{2}|\gc) = \lc(Z'_{2}|\sigma(Z'_{0},Z'_{1}))
= \lc(Z'_{2}|\sigma(Z'_{1}))$$ and the same holds with $Z''$.
Thus for any distinct values $z',z''$ in $F_{1}$, one has, on the event $[Z'_{1}=z'\ ;\ Z''_{1}=z'']$, $$\begin{aligned}
\P[Z'_{2} = Z''_{2}|\gc]
&=& \sum_{z \in F_{2}} \P[Z'_{2} = z\ ;\ Z''_{2}=z|\gc]\\
&\le& \sum_{z \in F_{2}} \P[Z'_{2n} = z|\gc] \wedge \P[Z''_{2}=z|\gc]\\
&=& \sum_{z \in F_{2}} \P[Z'_{2} = z|Z'_{1}=z'] \wedge \P[Z''_{2}=z|Z''_{1}=z'']\\
&=& \sum_{z \in F_{2}} \P[Z_{2} = z|Z_{1}=z'] \wedge \P[Z_{2}=z|Z_{1}=z'']\\
&\le& \alpha.\end{aligned}$$ Hence $$\P[Z'_{2} = Z''_{2}|\gc]
\le \alpha \one_{[Z'_{1} \ne Z''_{1}]} + \one_{[Z'_{1} = Z''_{1}]}.$$ Taking complements, one gets $$\P[Z'_{2} \ne Z''_{2}|\gc]
\ge (1-\alpha) \one_{[Z'_{1} \ne Z''_{1}]}.$$ The last inequality follows from the inclusion $[Z'_{0} \ne Z''_{0}] \subset [Z'_{1} \ne Z''_{1}]$.
As we now see, the definition of a strong brick provides constraints on the size of the sets $F_{0},F_{1},F_{2}$.
\[constraints\] [**(Properties of bricks)**]{} Fix $\alpha \in ]0,1[$ and two positive integers $r_{1},r_{2}$. Let $F_{0},F_{1},F_{2}$ be finite sets. Assume the existence of a triple $(Z_{0},Z_{1},Z_{2})$ of uniform random variables with values in $F_{0},F_{1},F_{2}$ such that $(Z_{0},Z_{1},Z_{2})$ is a $(r_{1},r_{2})$-adic $\alpha$-brick. Let $f : F_1 \to F_0$ and $g : F_0 \times F_2 \to F_1$ be the maps such that $f(Z_1)=Z_0$ and $g(Z_0,Z_2)=Z_1$. Then:
1. the map $f$ is $r_1$ to one and the map $g$ is $r_2$ to one. More precisely, for every $z_1 \in F_1$, $g^{-1}(\{z_1\}) = \{f(z_1)\} \times S(z_1)$ where $S(z_1)$ is a subset of $F_2$ of size $r_2$.
2. for every $z_1 \in F_{1}$, the law of $Z_2$ conditionally on $Z_1=z_1$ is uniform on $S(z_1)$.
3. for each $z_0 \in F_0$, the subsets $S(z_1)$ for $z_1 \in f^{-1}(\{z_0\})$ form a partition of $F_2$ in $r_1$ blocks.
4. $|F_1| = r_1|F_0|$, $|F_0 \times F_2| = r_2|F_1|$ and $|F_2| = r_{1}r_{2}$.
5. for every distinct elements $z'_1$ and $z''_1$ in $F_{1}$, $|S(z'_1) \cap S(z''_1)| \le \alpha r_2$.
6. if $|F_0| \ge 2$, then $r_2 \ge 1/\alpha$.
By hypothesis, for every $(z_0,z_1,z_2) \in F_0 \times F_1 \times F_2$, $$\P[Z_0=z_0\ ;\ Z_1=z_1] = \frac{1}{|F_1|} \one_{[z_0=f(z_1)]}.$$ Hence $$\P[Z_1=z_1|Z_0=z_0]
= \frac{1}{|f^{-1}(\{z_0\})|} \one_{f^{-1}(\{z_0\})}(z_1),$$ which shows that $|f^{-1}(\{z_0\})| = r_1$.
By the same way, $$\P[Z_0=z_0\ ;\ Z_1=z_1\ ;\ Z_2=z_2] = \frac{1}{|F_0 \times F_2|}
\one_{[z_1=g(z_0,z_2)]}.$$ Hence $$\P[Z_0=z_0\ ;\ Z_2=z_2|Z_1=z_1] =
\frac{1}{|g^{-1}(\{z_1\})|} \one_{g^{-1}(\{z_1\})}(z_0,z_2),$$ which shows that $|g^{-1}(\{z_1\})| = r_2$.
Since $(Z_0,Z_2)$ is uniform on $F_0 \times F_2$, the equalities $Z_0 = f(Z_1)$ and $Z_1 = g(Z_0,Z_2)$ shows that $z_0 = f(g(z_0,z_2))$ for every $(z_0,z_2) \in F_0 \times F_2$. Hence, for every $z_1 \in F_1$, if $(z_0,z_2) \in g^{-1}(\{z_1\})$ then $z_0 = f(z_1)$. This shows that $g^{-1}(\{z_1\}) = \{f(z_1)\} \times S(z_1)$ where $S(z_1)$ is some subset of $F_2$.
Thus, for every $(z_1,z_2) \in F_1 \times F_2$, $$\P[Z_2=z_2|Z_1=z_1] = \P[Z_0=f(z_1)\ ;\ Z_2=z_2|Z_1=z_1] =
\frac{1}{|S(z_1)|} \one_{S(z_1)}(z_2).$$ Hence the law of $Z_2$ conditionally on $Z_1=z_1$ is uniform on $S(z_1)$ which has size $r_2$. This completes the proof of the first two points.
The third and fourth points follow.
Fix two distinct elements $z'_1$ and $z''_1$ in $F_{1}$. Then for every $z \in F_{2}$, $$\min \big(\P[Z_{2}=z|Z_{1}=z'_1],\P[Z_{2}=z|Z_{1}=z''_1]
= \frac{1}{r_2} \min (\one_{S(z'_1)}(z),\one_{S(z''_1)}(z)).$$ Summing over $z$ and using the inequality \[bad coupling\], one gets $$|S(z'_1) \cap S(z''_1)| \le \alpha r_2,$$ which is the fifth point.
If $|F_0| \ge 2$, then one can choose two distinct elements $z'_0$ and $z''_0$ in $F_0$. Let $z_2 \in F_2$, $z'_1 = g(z'_0,z_2)$ and $z''_1 = g(z'_0,z_2)$. Then $z'_1$ and $z''_1$ are distinct elements in $F_{1}$ since $f(z'_1)=z'_0$ and $f(z''_1)=z''_0$ are distinct. But $z_2$ belongs to $S(z'_1)$ since $$\P[Z_1=z'_1|Z_2=z_2] = \P[Z_0=z'_0|Z_2=z_2]= |F_0|^{-1},$$ and $z_2$ also belongs to $S(z''_1)$. Hence $1 \le |S(z'_1) \cap S(z''_1)| \le \alpha r_2$. which shows the sixth point.
Getting bricks
--------------
The next lemma provides a general method to get bricks.
\[general method\] [**(Method to get bricks)**]{}
Fix $\alpha \in ]0,1[$ and two positive integers $r_{1},r_{2}$.
Let $F_{0},F_{2}$ be finite sets such that $F_{2}$ has size $r_{1}r_{2}$.
Let $Z_{0}$ and $Z_{2}$ be independent random variables, uniformly distributed in $F_{0}$ and $F_{2}$.
Let $(\Pi_z)_{z \in F_{0}}$ be a family of partitions of $F_{2}$ indexed by $F_{0}$ such that
- each partition $\Pi_z$ has $r_{1}$ blocks $S_{z,1},\ldots,S_{z,r_{1}}$;
- each block has $r_{2}$ elements.
- for any distinct $(z',i')$ and $(z'',i'')$ in $F_{0} \times \odc 1,r_{1} \fdc$, $|S_{z',i'} \cap S_{z'',i''}| \le \alpha r_{2}$.
(This “transversality condition” forces the partitions to be all different and says that two blocks chosen in any two different partitions have a small intersection.)
Define a random variable with values in $F_{1} = F_{0} \times \odc 1,r_{1} \fdc$ by $Z_{1} = (Z_{0},J)$, where $J$ is the index of the only block of $\Pi_{Z_{0}}$ which contains $Z_{2}$ (that is to say $Z_{2} \in S_{Z_{0},J})$.
Then $(Z_{0},Z_{1},Z_{2})$ is a $(r_{1},r_{2})$-adic $\alpha$-brick.
The first statement is obvious.
For every $z_{0} \in F_{0}$, $j \in \odc 1,r_{1} \fdc$ and $z_2 \in F_{2}$, $$\begin{aligned}
\P[Z_{0}=z_{0}\ ;\ J=j\ ;\ Z_{2}=z_2]
&=& \one_{[z_2 \in S_{z_{0},j}]}\ \P[Z_{0}=z_{0}\ ;\ Z_{2}=z_2]\\
&=& \one_{[z_2 \in S_{z_{0},j}]}\ \times \frac{1}{|F_{0}|}\
\times \frac{1}{r_{1}r_{2}}.\end{aligned}$$ Summing over $z_2$ yields $$\P[Z_{0}=z_{0}\ ;\ J=j] = \frac{1}{|F_{0}|}\ \times \frac{1}{r_{1}}.$$ By division, one gets $$\P[Z_{2}=z_2\ |\ Z_{0}=z_{0}\ ;\ J=j] = \one_{[z_2 \in S_{z_{0},j}]}
\ \times \frac{1}{r_{1}}.$$ The last two equalities show that $J$ is independent of $Z_{0}$ and uniform on $\odc 1,r_{1} \fdc$, and that given $(Z_{0},J)$, $Z_{2}$ is uniform on the block $S_{Z_{0},J}$. This proves the third and the fourth statement.
Let $z'_1$ and $z''_1$ be distinct elements in $F_{1}$. Conditionally on $[Z_{1}=z'_1]$, the law $Z_{2}$ is uniform on the block $S_{z'_1}$. Conditionally on $[Z_{1}=z''_1]$, the law $Z_{2}$ is uniform on the block $S_{z''_1}$. Thus $$\sum_{z \in F_{2}} \P[Z_{2}=z|Z_{1}=z'_1] \wedge \P[Z_{2}=z|Z_{1}=z''_1]
= \sum_{z \in S_{z'_1} \cap S_{z''_1}} \frac{1}{r_2} \le \alpha.$$ The last statement follows.
Examples of bricks
------------------
Algebra helps us to construct many partitions on a given set such that each partition has a fix number of blocks, each block has a fix number of elements and any two blocks chosen in any two different partitions have a small intersection.
Let $q$ be any power of a prime number. Let $K$ be the field with $q$ elements, and $L$ the field with $q^2$ elements. Since $L$ is a quadratic extension of $K$, $L$ is isomorphic to $K^2$ as a vector space on $K$. Actually, one only needs to have a bijection between $K^2$ and $L$.
[**First example**]{}
We set $r_{1}=r_{2}=q^4$, $F_{0} = L^8$ (identified with the set $\mc_4(K)$ of all $4 \times 4$ matrices with entries in $K$) and $F_{2} = K^8$ identified with $K^4 \times K^4$.
To each matrix $A \in \mc_4(K)$, one can associate the partition of $K^8$ given by all four-dimensional affine subspaces of $K^8$ with equations $y=Ax+b$ where $b$ ranges over $K^4$. Each of these subspaces has size $q^4$. But two subspaces of equations $y=A'x+b'$ and $y=A''x+b''$ intersect in at most $q^3$ points (a three dimensional affine subspace) when $A' \ne A''$. Hence these partitions provide a $(q^4,q^4)$-adic $1/q$-brick.
[**Second example**]{}
We set $r_{1}=r_{2}=q$, $F_{0} = L^2$ (identified with $K^4$) and $F_{2} = K^2$.
To each quadruple $(a,b,c,d) \in K^4$, one can associate the partition of $K^2$ given by the $q$ graphs of equations $y=ax^4+bx^3+cx^3+dx+e$ where $e$ ranges over $K$. Each of these graphs has size $q$. But two graphs with different $(a,b,c,d,e) \in K^4$ intersect in at most $4$ points. Hence, if $p \ge 5$ these partitions provide a $(q,q)$-adic $4/q$-brick.
[**Gluing bricks together**]{}
In both exemples above, the family of partitions provides bricks which can be glued as follows. Let $q$ be any power of a prime number. For each $n \le 0$, call $K_n$ the field with $q_n=q^{2^{|n|}}$ elements. Set $$\forall n \le 0,\ F_{2n} = K_n^8,\ r_{2n-1} = r_{2n} = q_n^4,
\alpha_n = 1/q_n \text{ and }
F_{2n-1} = F_{2n-2} \times \odc 1,r_{2n-1} \fdc$$ or $$\forall n \le 0,\ F_{2n} = K_n^2,\ r_{2n-1} = r_{2n} = q_n,
\alpha_n = 4/q_n\text{ and }
F_{2n-1} = F_{2n-2} \times \odc 1,r_{2n-1} \fdc.$$ Start with a sequence of independent random variables $(Z_{2n})_{n \le 0}$. For each $n \le 0$, consider the partitions of $F_{2n}$ provided by the first or the second example and define $Z_{2n-1}$ from $Z_{2n-2}$ and $Z_{2n}$ as in lemma \[general method\]. By construction, $(Z_{2n},Z_{2n-1},Z_{2n})$ is an $(r_{2n-1},r_{2n})$-adic $\alpha_n$-brick.
The next theorem shows that the process $(Z_n)_{n \le 0}$ thus defined provides an example which proves the existence stated in theorem \[existence theorem\].
Proof of theorem \[existence theorem\]
--------------------------------------
Theorem \[existence theorem\] directly follows from the construction above and from the theorem below.
\[simple construction\] Let $(\alpha_n)_{n \le 0}$ be a sequence of reals in $]0,1[$ such that the series $\sum_n \alpha_n$ converges. Let $(Z_n)_{n \le 0}$ be any sequence of random variables taking values in some finite sets $(F_n)_{n \le 0}$ of size $\ge 2$. Assume that
- the random variables $(Z_{2n})_{n \le 0}$ are independent;
- for each $n \le 0$, $(Z_{2n-2},Z_{2n-1},Z_{2n})$ is an $(r_{2n-1},r_{2n})$-adic $\alpha_n$-brick.
Then
- $(Z_n)_{n \le 0}$ is a Markov process which generates a $(r_n)$-adic filtration;
- for every infinite subset $D$ of $\zzf_-$, $(\fc^Z_{n})_{n \in D}$ is standard if and only if $2n-1 \notin D$ for infinitely many $n \le 0$.
In particular, the filtration $(\fc^Z_{2n-1})_{n \le 0}$ is at the threshold of standardness.
We now prove the statements.
[**Proof that $(Z_n)_{n \le 0}$ is a Markov process and generates a $(r_n)$-adic filtration**]{}
First, note that the filtration $(\fc^Z_{2n})_{n \le 0}$ is generated by the independent random variables $(Z_{2n})_{n \le 0}$ since for every $n \le 0$, $Z_{2n-1}$ is a deterministic function of $(Z_{2n-2},Z_{2n})$. Hence, for every $n \le 0$, $$\fc^Z_{2n-2} = \sigma(Z_{2n-2}) \vee \fc^Z_{2n-4},$$ Moreover, since $Z_{2n-2}$ is a deterministic function of $Z_{2n-1}$, $$\fc^Z_{2n-1} = \sigma(Z_{2n-1}) \vee \fc^Z_{2n-2}= \sigma(Z_{2n-1})
\vee \fc^Z_{2n-4}.$$ By independence of $(Z_{-2n-2},Z_{2n-1},Z_{-2n})$ and $\fc^Z_{2n-4}$, we get $$\lc(Z_{2n-1}|\fc^Z_{2n-2}) = \lc(Z_{2n-1}|\sigma(Z_{2n-2})),$$ $$\lc(Z_{2n}|\fc^Z_{2n-1}) = \lc(Z_{2n}|\sigma(Z_{2n-1})).$$ The Markov property follows. But for every $n \le 0$, $(Z_{2n-2},Z_{2n-1},Z_{2n})$ is an $(r_{2n-1},r_{2n})$-adic $\alpha_n$-brick. The $(r_n)$-adic character of $\fc^Z$ follows.
[**Proof that $(\fc^Z_n)_{n \in D}$ is not standard when $D$ contains all but finitely many odd negative integers**]{}
First, we show that $(\fc^Z_{2n-1})_{n \le 0}$ is not standard. To do this, we check that the random variable $Z_{-1}$ does not satisfy the I-cosiness criterion. Note that $(\fc^Z_{2n-1})_{n \le 0}$ is the natural filtration of $(Z_{2n-1})_{n \le 0}$ only since for every $n \le 0$, $Z_{2n-2}$ is some deterministic function $f_n$ of $Z_{2n-1}$.
Let $(Z'_{2n-1})_{n \le 0}$ and $(Z'_{2n-1})_{n \le 0}$ be two copies of the process $(Z_{2n-1})_{n \le 0}$, defined on some probability space $(\bar{\Omega},\bar{\ac},\bar{\P)}$. Set $Z'_{2n-2} = f_n(Z'_{2n-1})$ and $Z''_{2n-2} = f_n(Z''_{2n-1})$ for every $n \le 0$. Then $(Z'_{n})_{n \le 0}$ and $(Z''_{n})_{n \le 0}$ are copies of the process $(Z_n)_{n \le 0}$. Moreover, $(\fc^{Z'}_{2n-1})_{n \le 0}$ and $(\fc^{Z''}_{2n-1})_{n \le 0}$ are the natural filtrations of $(Z'_{2n-1})_{n \le 0}$ and $(Z''_{2n-1})_{n \le 0}$.
Assume that these filtrations are immersed in some filtration $(\gc_{2n-1})_{n \le 0}$. Then, for every $n \le -1$, $$\lc(Z'_{2n+1}|\gc_{2n-1}) = \lc(Z'_{2n+1}|\fc^{Z'}_{2n-1})
= \lc(Z'_{2n+1}|\sigma(Z'_{2n-1})),$$ and since $Z'_{2n}$ is a deterministic function of $Z'_{2n+1}$, $$\lc(Z'_{2n}|\gc_{2n-1})
= \lc(Z'_{2n}|\sigma(Z'_{2n-1})).$$ The same holds with the process $Z''$.
For any distinct values $z',z''$ in $F_{2n-1}$, one has on the event $[Z'_{2n-1}=z'\ ;\ Z''_{2n-1}=z'']$, $$\begin{aligned}
\P[Z'_{2n} = Z''_{2n}|\gc_{2n-1}]
&=& \sum_{z \in F_{2n}} \P[Z'_{2n} = z\ ;\ Z''_{2n}=z|\gc_{2n-1}]\\
&\le& \sum_{z \in F_{2n}} \P[Z'_{2n} = z|\gc_{2n-1}] \wedge \P[Z''_{2n}=z|\gc_{2n-1}]\\
&=& \sum_{z \in F_{2n}} \P[Z'_{2n} = z|Z'_{2n-1}=z'] \wedge
\P[Z''_{2n}=z|Z''_{2n-1}=z'']\\
&=& \sum_{z \in F_{2n}} \P[Z_{2n} = z|Z_{2n-1}=z'] \wedge \P[Z_{2n}=z|Z_{2n-1}=z'']\\
&\le& \alpha_n.\end{aligned}$$ Hence, since $[Z'_{2n+1} = Z''_{2n+1}] \subset [Z'_{2n} = Z''_{2n}]$, $$\begin{aligned}
\bar{\P}[Z'_{2n+1} = Z''_{2n+1}|\gc_{2n-1}]
&\le& \bar{\P}[Z'_{2n} = Z''_{2n}|\gc_{2n-1}]\\
&\le& \alpha_n\one_{[Z'_{2n-1} \ne Z''_{2n-1}]} + \one_{[Z'_{2n-1} = Z''_{2n-1}]}.\end{aligned}$$ Taking the complements, one gets $$\bar{\P}[Z'_{2n+1} \ne Z''_{2n+1}|\gc_{2n-1}] \ge
(1-\alpha_n) \one_{[Z'_{2n-1} \ne Z''_{2n-1}]}.$$ A simple recursion yields $$\P[Z'_{-1} \ne Z''_{-1}|\gc_{2n-1}] \
\ge \prod_{n \le k \le -1} (1-\alpha_k)\ \one_{[Z'_{2n-1} \ne Z''_{2n-1}]}.$$ Taking the expectations, one gets $$\P[Z'_{-1} \ne Z''_{-1}] \
\ge \prod_{n \le k \le -1} (1-\alpha_k)\ \P[Z'_{2n-1} \ne Z''_{2n-1}].$$ Assume now that that for some $N > -\infty$, the $\sigma$-fields $\fc'_{2N-1}$ and $\fc''_{2N-1}$ are independent. Then for every $n \le N$, $$\P[Z'_{2n-1} \ne Z''_{2n-1}] = 1-\frac{1}{|F_{2n-1}|} \ge \frac{1}{2},$$ since $Z'_{2n-1}$ and $Z''_{2n-1}$ are independent and uniform on $F_{2n-1}$. Going to the limit yields $$\P[Z'_{-1} \ne Z''_{-1}] \ge \frac{1}{2} \prod_{k \le -1} (1-\alpha_k) > 0,$$ which shows that $Z_{-1}$ does not satisfy the I-cosiness criterion.
Thus $(\fc^Z_{2n-1})_{n \le 0}$ is not standard. Thus, if $D$ is any subset of $\zzf_-$ which contains all odd negative integers, the filtration $(\fc^Z_n)_{n \in D}$ is not standard (since standardness is preserved by extraction). This conclusion still holds when $D$ contains all but finitely many odd negative integers (since standardness is an asymptotic property).
[**Proof that $(\fc^Z_n)_{n \in D}$ is standard when $D$ skips infinitely many odd negative integers**]{}
Since standardness is preserved by extraction, one only needs to consider the case where $D$ contains all even non-positive numbers. In this case, the filtration $(\fc^Z_n)_{n \in D}$ is generated by $(Z_n)_{n \in D}$ only. Indeed, if $n$ is any integer in $\zzf_- \setminus D$, then $n$ is odd, hence $n-1 \in D$, $n+1 \in D$ and $Z_n$ is a function of $(Z_{n-1},Z_{n+1})$.
For each $n \le 0$, the conditional law $\lc(Z_n|\fc^Z_{n-1})=\lc(Z_n|Z_{n-1})$ is (almost surely) uniform on some random subset of $F_n$ with $r_n$ elements. By fixing a total order on the set $F_n$, one can construct an uniform random variable uniform $U_n$ on $\odc 1,r_n \fdc$, independent of $\fc^Z_{n-1}$, such that $Z_n$ is a function of $Z_{n-1}$ and $U_n$. Set $Y_n = Z_n$ if $n-1 \in D$ (which may happen only for even $n$) and $Y_n = U_n$ otherwise. Then $Y_n$ is $\fc^Z_n$-measurable. This shows that $\fc^Y_n \subset \fc^Z_n$ for every $n \in D$.
Let us prove the reverse inclusion. Fix $n \in D$, and call $m \le n$ the integer such that $m-1 \notin D$ but $k \in D$ for all $k \in \odc m,n \fdc$. Then $Z_n$ is $\fc^Y_n$-measurable as a function of $Y_m=Z_m,Y_{m+1}=U_{m+1},\ldots,Y_n=U_n$.
Last, for every $n \in D$, $Y_n$ is independent of $\fc^Z_{n-1}$ if $n-1 \in D$ and $Y_n$ is independent of $\fc^Z_{n-2}$ otherwise. This shows the independence of the random variables $(Y_n)_{n \in D}$. Hence the filtration $(\fc^Z_n)_{n \in D}$ is of product type, which completes the proof.
Annex: some basic facts on standardness
=======================================
\[annex\]
We summarize here the main definitions and results used in this paper. A complete exposition can be found in [@Emery-Schachermayer].
Recall that we work with filtrations indexed by the non-positive integers on a probability space $(\Omega,\ac,\P)$, and that all the sub-$\sigma$-fields of $\ac$ that we consider here are assumed to be complete and essentially separable with respect to $\P$.
Most of the time, the probability measure $\P$ is not explicitly mentioned when we deal with filtrations. Yet, it actually plays an important role and the true object of study are filtered probability spaces $(\Omega,\ac,\P,(\fc_n)_{n \le 0})$.
Isomorphisms of filtered probability spaces
-------------------------------------------
The definition of isomorphism is not as simple as one could expect.
Let $\F = (\F_n)_{n \le 0}$ and $\F' = (\F_n')_{n \le 0}$ be filtrations on $(\Omega,\mathcal{A},\P)$ and $(\Omega',\mathcal{A}',\P')$.
An isomorphism of filtered probability spaces from $(\Omega,\mathcal{A},\P,\F)$ into $(\Omega',\mathcal{A}',\P',\F')$ is a bijective (linear) application from the space ${\bf L}^0(\Omega,\F_{\infty},\P)$ of the real random variables on $(\Omega, \F_{\infty}, \P)$ into ${\bf L}^0(\Omega',\F_{\infty}',\P')$ which preserves the laws of the random variables, commutes with Borelian applications, and sends $\F$ on $\F'$.
By definition, saying that an isomorphism $\Psi$ sends $\F$ on $\F'$ means that for every $n \le 0$, the random variables $\Psi(X)$ for $X \in
{\bf L}^0(\Omega,\F_{n},\P)$ generate $\fc'_{n}$. Saying that $\Psi$ commutes with Borelian applications means that for every sequence $(X_n)_{n \ge 1}$ of real random variables on $(\Omega,\mathcal{A},\P)$, and every Borelian application $F : \R^\infty \to \R$, $$\Psi \big( F \circ (X_n)_{n \ge 1} \Big) = F \circ (\Psi(X_n))_{n \ge 1}.$$ In particular, this equality holds when $F$ is given by $F((x_n)_{n \ge 1}) = \alpha_1 x_1 + \alpha_2 x_2$ with $(\alpha_1,\alpha_2)
\in \R^2$, which shows that $\Psi$ is linear.
Of course, any bimeasurable application $\psi$ from $(\Omega,\F_{\infty})$ to $(\Omega',\F_{\infty}')$ which sends $\P$ on $\P'$ induces an isomorphism $\Psi$ from $(\Omega,\mathcal{A},\P,\F)$ into $(\Omega',\mathcal{A}',\P',\F')$, defined by $\Psi(X) = X \circ \phi^{-1}$. Yet, the converse is not true: an isomorphism of filtered spaces from $(\Omega,\F_{\infty})$ to $(\Omega',\F_{\infty}')$ is not necessarily associated to some bimeasurable application from $\Omega$ to $\Omega'$ which sends $\P$ on $\P'$.
As a matter of fact, the most interesting objects associated to a probability space $(\Omega,\mathcal{A},\P)$ are the random variables and not the elements of $\Omega$. Note that for any sequence $(X_n)_{n \le 0}$ of random variables defined on $(\Omega,\mathcal{A},\P)$, the filtrations which are isomorphic to the natural filtration of $(X_n)_{n \le 0}$ are exactly the filtrations of the copies of $(X_n)_{n \le 0}$ on arbitrary probability spaces.
Immersion of filtrations
------------------------
Let $\F = (\F_n)_{n \le 0}$ and $\G = (\G_n)_{n \le 0}$ be filtrations on $(\Omega,\mathcal{A},\P)$.
One says that $\F$ is immersed into $\G$, if, for every $n \le 0$, $\F_n \subset \G_n$ and $\F_n$ is independent of $\G_{n-1}$ conditionally on $\F_{n-1}$. Equivalently, $\F$ is immersed into $\G$ if and only if every martingale in $\F$ is still a martingale in $\G$.
Immersion is stronger than mere inclusion. If $\F$ is immersed into $\G$, the additional information contained in $\G$ cannot give information on $\F$ in advance: intuitively, the independence of $\F_n$ and $\G_{n-1}$ conditionally on $\F_{n-1}$ means that $\G_{n-1}$ gives no more information on $\F_n$ than $\F_{n-1}$ does.
The notion of immersion is implicitely present in many usual situations. For instance, when one considers a Markov process $X$ [*in some filtration $\gc$*]{}, it means that the natural filtration of $X$ is immersed in $\gc$.
Immersibility and standardness
------------------------------
The notion of immersion can be weakened to provide a notion invariant by isomorphism.
Let $\F = (\F_n)_{n \le 0}$ and $\G' = (\G_n')_{n \le 0}$ be filtrations on $(\Omega,\mathcal{A},\P)$ and $(\Omega',\mathcal{A}',\P')$. One says that $\F$ is immersible into $\G'$ if there exists a filtration $\F'$ on $(\Omega',\mathcal{A}',\P')$, isomorphic to $\F$, such that $\F'$ is immersed into $\G'$.
We can now define the standardness of filtrations.
A filtration is standard if it is immersible into a product-type filtration.
Because of Kolmogorov’s 0-1 law, any filtration must have a trivial tail $\sigma$-field in order to be standard, but this necessary condition is not sufficient. In [@Vershik], Vershik established two different characterisations of standardness in the context of decreasing sequences of measurable partitions, which were extended and reformulated into a probabilistic language and called Vershik’s “first level” and “second level” criteria by Émery and Schachermayer [@Emery-Schachermayer]. Émery and Schachermayer also introduced a new standardness criterion, namely the I-cosiness criterion.
I-cosiness criterion
--------------------
Let $\F = (\F_n)_{n \le 0}$ be a filtration on $(\Omega,\mathcal{A},\P)$.
Let $R$ be any $\fc_0$-measurable real random variable $R$. One says that $R$ satisfies I-cosiness criterion for $(\F_n)_{n \le 0}$ (to abbreviate, we say that I($R$) holds) if for any positive real number $\delta$, there exists a probability space $(\overline{\Omega},\overline{\mathcal{A}},\overline{\P})$ supplied with two filtrations $\F'$ and $\F''$ such that:
- the filtrations $\F'$ and $\F''$ are isomorphic to the filtration $\F$;
- the filtrations $\F'$ and $\F''$ are immersed into $\F' \vee \F''$;
- there exists an integer $n_0<0$ such that the $\sigma$-fields $\F'_{n_0}$ and $\F''_{n_0}$ are independent;
- the copies $R'$ and $R''$ of $R$ given by the isomorphisms of the first condition are such that $\overline{\P}[|R'-R''| \ge \delta]
\le \delta$.
One says that $\F$ is I-cosy when I($R$) holds for every $R \in L^0(\Omega,\fc_0,\P)$.
The definition of I-cosiness was implicitly used by Smorodinsky in [@Smorodinsky] to prove that the dyadic split-words filtration is not standard (although Smorodinsky uses a different terminology). The I stands for independence, to distinguish I-cosiness from other variants of cosiness.
Intuitively, the conditions defining I($R$) mean that one can couple two copies of $\F$ in a non-anticipative way so that old enough independent initial conditions have weak influence on the final value of $R$.
Laurent noticed that if $I(R)$ holds, then $I(\phi(R))$ holds for every Borel function $\phi$ from $\R$ to $\R$. Hence, to prove that $\F$ is I-cosy, it is sufficient to check that $I(R)$ for [*one*]{} real random variable generating $\F_0$.
It is also sufficient and sometimes handful to check $I(R)$ for all random variables with values in an arbitrary finite set, with the discrete distance $\one_{[R' \ne R'']}$ replacing $|R'-R''|$ in the definition of $I(R)$.
I-cosiness provides a standardness criterion.
\[I-cosiness criterion\] [**(Émery and Schachermayer [@Emery-Schachermayer])**]{} $\F$ is standard if and only if $\F$ is I-cosy.
[cc]{}
G. Ceillier, [*The filtration of the split-words process*]{}. Probability Theory and Related Fields, 2012, [**153**]{}, no 1-2, 269–292 (2012).
M. [É]{}mery and W. Schachermayer. [*On [V]{}ershik’s standardness criterion and [T]{}sirelson’s notion of cosiness*]{}. Séminaire de [P]{}robabilités, XXXV, LNM [**1755**]{}, 265–305 (2001).
A. Gorbulsky, [*About one property of entropy of a decreasing sequence of measurable partitions*]{}. Nauchnykh Seminarov POMI [**256**]{}, 19–24 (1999); translation in J. Math. Sci. (New York) [**107**]{} no. 5, 4157–4160 (2001).
D. Heicklen, [*Bernoullis are standard when entropy is not an obstruction*]{}. Israel Journal of Mathematics, [**107**]{}, 141–155 (1998).
S. Laurent, [*Filtrations à temps discret négatif*]{}. PhD thesis, Université Louis Pasteur, Institut de Recherche en Mathématique Avancée, Strasbourg (2004).
S. Laurent, [*On Vershikian and I-cosy random variables and filtrations*]{}. (Russian summary) Teor. Veroyatn. Primen. 55, no. 1, 104–132 (2010); translation in Theory Probab. Appl. 55, no. 1, 54–76 (2011).
S. Laurent, [*Standardness and I-cosiness*]{}. Séminaire de Probabilités XLIII, LNM [**2006**]{}, 127–186 (2010).
M. Smorodinsky, [*Processes with no standard extension*]{}. Israel Journal of Mathematics, [**107**]{}, 327–331 (1998) .
M. Tsirelson. [*About Yor’s problem*]{}. Unpublished notes.
A. Vershik, [*Theory of decreasing sequences of measurable partitions*]{}. Algebra i Analiz, [**6**]{}:4 (1994), 1–68. English Tranlation: St. Petersburg Mathematical Journal, [**6**]{}:4 (1995), 705–761.
H. von Weizsäcker, [*Exchanging the order of taking suprema and countable intersections of $\sigma$-algebras*]{}. Ann. Inst. H. Poincaré Sect. B, [**19**]{}, no. 1, 91–100 (1983).
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---
abstract: 'New neutrino degrees of freedom allow for more sources of CP-invariance violation (CPV). We explore the requirements for accessing CP-odd mixing parameters in the so-called $3+1$ scenario, where one assumes the existence of one extra, mostly sterile neutrino degree of freedom, heavier than the other three mass eigenstates. As a first step, we concentrate on the $\nu_{e}\to\nu_{\mu}$ appearance channel in a hypothetical, upgraded version of the $\nu$STORM proposal. We establish that the optimal baseline for CPV studies depends strongly on the value of $\Delta m^2_{14}$ – the new mass-squared difference – and that the ability to observe CPV depends significantly on whether the experiment is performed at the optimal baseline. Even at the optimal baseline, it is very challenging to see CPV in $3+1$ scenarios if one considers only one appearance channel. Full exploration of CPV in short-baseline experiments will require precision measurements of tau-appearance, a challenge significantly beyond what is currently being explored by the experimental neutrino community.'
author:
- André de Gouvêa
- 'Kevin J. Kelly'
- Andrew Kobach
bibliography:
- 'SterileBib.bib'
title: 'CP-Invariance Violation at Short-Baseline Experiments in $3+1$ Neutrino Scenarios'
---
[NUHEP-TH/14-10]{}
Introduction
============
The existence of Standard Model (SM) gauge singlet fermions – sterile neutrinos – is a very simple and attractive extension to our understanding of fundamental particle physics. Sterile fermions may play a central role when it comes to addressing several of the current outstanding questions, including the dark matter puzzle and the origin of nonzero neutrino masses [@Abazajian:2012ys].
At the renormalizable level, the only allowed interactions of sterile neutrinos with SM degrees of freedom are those described by Yukawa operators containing left-handed fermions, the Higgs doublet, and the sterile neutrinos. Phenomenologically, this implies that observable sterile-neutrino effects are mostly mediated by the mixing between the active neutrinos ($\nu_e, \nu_{\mu}, \nu_{\tau}$) and the sterile neutrino states. If the sterile neutrino masses are very low (say, below 10 eV), their properties can, almost exclusively,[^1] be explored via neutrino oscillation experiments.
Over the last couple of decades, distinct experiments have revealed anomalies that are not consistent with the SM augmented by massive active neutrinos [@Aguilar:2001ty; @AguilarArevalo:2008rc; @Aguilar-Arevalo:2013pmq; @Mention:2011rk; @Mueller:2011nm; @Frekers:2011zz]. These can be interpreted as evidence for more than three neutrinos, with a new oscillation length proportional to a new mass-squared difference around 1 eV$^2$ (for recent analyses, see [@Giunti:2012tn; @Conrad:2012qt; @Kopp:2013vaa; @Giunti:2013aea]). Given that the number of active neutrinos is known to be three [@Agashe:2014kda], the extra degrees of freedom must be sterile neutrinos. While our understanding of these so-called short-baseline anomalies remains clouded, there are several experimental proposals aimed at definitively testing the sterile-neutrino interpretation [@deGouvea:2013onf]. It is possible that, in five to ten years, experiments will reveal, beyond reasonable doubt, the existence of new neutrino degrees of freedom. Such a monumental discovery would qualitatively impact our understanding of fundamental physics and would invite a new generation of short-baseline neutrino oscillation experiments capable of exploring the new-physics sector.
Among the properties of the newly-discovered neutrino states would be their couplings to the SM particles, including the probabilities that these would act as $\nu_e,~ \nu_{\mu},~ \nu_{\tau}$, and the relative phases among the new elements of the augmented leptonic mixing matrix. Even if there were only one new neutrino state, new sources of CP-invariance violation (CPV) would become accessible. Given our current understanding of CPV and the potential importance of this phenomenon to some of the basic contemporary particle physics questions, it would be imperative to understand whether, and under what circumstances, these new CPV phenomena are experimentally accessible.
Here, we discuss the challenges associated with studying CPV in the new-physics sector, assuming that next-generation short-baseline experiments confirm the existence of new neutrino states with parameters that are approximately consistent with those indicated by the sterile-neutrino interpretation to the current short-baseline anomalies. We restrict our discussion to the case of only one accessible new neutrino state. According to [@Kopp:2013vaa], the assumption that there are at least two accessible states might be a better fit to the short-baseline data. However, if there are two (or more) sterile neutrinos, CPV present in the interference between the two (or more) new oscillation frequencies may already have manifested itself in the current generation of short-baseline experiments [@Karagiorgi:2006jf; @Giunti:2012tn; @Conrad:2012qt; @Kopp:2013vaa; @Giunti:2013aea], and is hence more-or-less straight forward to observe. On the other hand, if only one new neutrino state is accessible, CPV will manifest itself in the interference between the new mass-squared difference and the known atmospheric and solar ones, a phenomenon which depends only on a few new-physics parameters and may turn out to be much more challenging to explore experimentally. These interference effects are very small and virtually impossible to observe in current and proposed experimental setups, which can safely neglect them. We discuss this further in Sections \[parameterization\] and \[sec:exp\].
In Sec. \[parameterization\], we discuss $3+1$ oscillations, concentrating on experimental circumstances where only two of the three independent oscillation frequencies are accessible. We present the relevant oscillation probabilities and discuss which parameters can be measured and what are the different sources of CPV. In Sec. \[sec:exp\], we discuss the requirements for observing $3+1$ CPV in short baseline experiments, and explore the capabilities of a concrete next-next-generation experimental setup, inspired by the $\nu$STORM proposal [@Adey:2014rfv], to study CPV in a high-statistics, high-resolution short-baseline experiment. In Sec. \[sec:conclusion\], we summarize our results and offer some concluding thoughts.
Neutrino Oscillations at Short Baselines {#parameterization}
========================================
Under the hypothesis that there are four neutrino states $\nu_{i}$, $i=1,2,3,4$, $P_{\alpha\beta}(E_{\nu},L)$ – the probability that a $\nu_\alpha$ flavor eigenstate with energy $E_{\nu}$ is detected as a $\nu_\beta$ flavor eigenstate, $\alpha,\beta=e,\mu,\tau$, after it propagates a distance $L$ – is given by the absolute value squared of the oscillation amplitude $\mathcal{A}_{\alpha\beta}$. For $\alpha \neq \beta$, $$\label{Amplitude}
\mathcal{A}_{\alpha\beta} = U_{\alpha 2}U_{\beta 2}^* \left(e^{-i\Delta_{12}}-1\right) + U_{\alpha 3}U_{\beta3 }^* \left(e^{-i\Delta_{13}}-1\right) + U_{\alpha 4} U_{\beta 4}^* \left(e^{-i\Delta_{14}}-1\right).$$ Here, $\Delta_{ij} \equiv 2.54(\Delta m_{ij}^2/1 \text{ eV}^2)(L/\text{km})(\text{GeV}/E_\nu)$ and $\Delta m_{ij}^2 \equiv m_j^2 - m_i^2$, where $m_i$ are the neutrino masses, $i,j=1,2,3,4$. $U_{\alpha i}$ are the elements of the unitary $4\times 4$ neutrino mixing matrix, $\alpha=e,\mu,\tau,s$, where $s$ stands for the sterile neutrino. Note that the $U_{si}$ elements are not accessible to experiments, assuming there are no interactions directly sensitive to the sterile neutrino state.
We presume that the matrix elements $U_{\alpha i}$ and the values of $\Delta m_{ij}^2$ are such that they fit the existing long-baseline neutrino data for $i=1,2,3$, $\alpha=e,\mu,\tau$ [@Capozzi:2013csa; @Forero:2014bxa; @Gonzalez-Garcia:2014bfa]. We further assume that next-generation short-baseline neutrino oscillation experiments will confirm the existence of one new mass-squared difference, $|\Delta m_{14}|^2\sim |\Delta m_{24}^2|\sim|\Delta m_{34}^2|\gg |\Delta m^2_{13}|,$ $\Delta m^2_{12}$, consistent with the sterile neutrino interpretation of the short-baseline anomalies [@Giunti:2012tn; @Conrad:2012qt; @Kopp:2013vaa; @Giunti:2013aea]. Hence, we assume $$\label{MassRange}
\Delta m_{14}^2 \in [0.1, 10]\text{ eV}^2,$$ and will only consider the mass ordering where $m_4^2\gg m_3^2,m_2^2,m_1^2$. The effective mixing angle $|U_{e4}||U_{\mu 4}|$ is assumed to lie within the range $$\label{MixAngleRange}
|U_{e4}||U_{\mu_4}| \in [0.01, 0.15].$$ Note that this assumption is consistent with $|U_{e4}||U_{\mu_4}|\sim |U_{e3}|^2\simeq 0.02$.
We parameterize the elements of the $4\times 4$ unitary transformation $U$ as (ignoring potentially physical, but irrelevant-for-oscillations, Majorana phases) $$\begin{aligned}
\label{Ue2}U_{e2} &=& s_{12}c_{13}c_{14}, \\
U_{e3} &=& e^{-i\delta}c_{14}s_{13}, \\
U_{e4} &=& s_{14}e^{-i\delta_1}, \\
U_{\mu 2} &=& c_{24}\left(c_{12}c_{23}-e^{i\delta}s_{12}s_{13}s_{23}\right) - e^{i(\delta_1-\delta_2)}c_{13}s_{12}s_{14}s_{24}, \\
\label{Umu3}U_{\mu 3} &=& c_{13}c_{24}s_{23}-e^{i(\delta_1 -\delta_2 - \delta)}s_{13}s_{14}s_{24},\\
U_{\mu 4} &=& s_{24}c_{14}e^{-i\delta_2},\\
U_{\tau 2} &=& c_{34}\left(-e^{i\delta}c_{23}s_{12}s_{13}-c_{12}s_{23}\right)-e^{i\delta_1}c_{13}c_{24}s_{12}s_{14}s_{34}\nonumber \\
&&-e^{i\delta_2}\left(c_{12}c_{23}-e^{i\delta}s_{12}s_{13}s_{23}\right)s_{24}s_{34,}\\
\label{Utau3}U_{\tau 3} &=& c_{13}c_{23}c_{34}-e^{i(\delta_1-\delta)}c_{24}s_{13}s_{14}s_{34}-e^{i\delta_2}c_{13}s_{23}s_{24}s_{34},\\
\label{Utau4}U_{\tau 4} &=& s_{34}c_{14}c_{24},\end{aligned}$$ where $s_{ij} \equiv \sin{\theta_{ij}},$ $ c_{ij} \equiv \cos{\theta_{ij}}$, $(i,j = 1, 2, 3, 4)$. The matrix elements depend on six mixing angles ($\theta_{12},\theta_{13},\theta_{23},\theta_{14},\theta_{24},\theta_{34}$) and three CP-odd phases ($\delta,\delta_1,\delta_2$). The elements not listed here can be determined by imposing unitarity conditions on $U$.
If one of the new mixing angles – $\theta_{14},$ $\theta_{24},$ $\theta_{34}$ – were to vanish, one of the new CP-odd phases – $\delta_1$ and $\delta_2$, or combinations thereof – would become non-physical, as expected. A similar phenomenon would be observed if any of the mass-squared differences were to vanish. While we know (or assume) that all $\Delta m^2_{ij}\neq 0$, their effects might still be unobservable. Since the mass-squared differences are quite hierarchical – $\Delta m^2_{12}\ll |\Delta m^2_{13}|\ll \Delta m^2_{14}$ – we examine this issue in more detail.
We will consider experiments that probe $P_{\alpha\beta}$ when $E_\nu= \mathcal{O}(1\text{ GeV})$ and baselines $L=\mathcal{O}(1\text{ km})$, i.e., $L/E_\nu \sim 1$ km/GeV, such that $\Delta_{14}\sim 1$.[^2] Under these circumstances, $\Delta_{12}=\mathcal{O}(10^{-5})$ and $|\Delta_{13}|=\mathcal{O}(10^{-3})$. With the above information in mind, we revisit Eq. (\[Amplitude\]), taking into account that $(e^{-i\Delta_{12,13}} - 1) \simeq -i\Delta_{12,13}$. To illustrate the relative size of terms in Eq. (\[Amplitude\]), we define $(\mathcal{R}_{\alpha\beta})_{ij}$ as the ratio of the “$1i$” to the “$1j$” contribution to $\mathcal{A}_{\alpha\beta},$[^3] $$\label{Ratio}
\left(\mathcal{R}_{\alpha\beta}\right)_{ij} \sim \frac{|U_{\alpha i}U^*_{\beta i}|\Delta m^2_{1i}}{|U_{\alpha j}U^*_{\beta j}|\Delta m^2_{1j}},$$ for $i,j=2,3,4$. For all $\alpha$ and $\beta$, $\left(\mathcal{R}_{\alpha\beta}\right)_{23}$ and $\left(\mathcal{R}_{\alpha\beta}\right)_{24}$ are small. For example, even though $|U_{e3}| = \sin{\theta_{13}} \simeq 0.15$ is small compared to $|U_{e2}| \simeq 0.55$, the ratio between $\Delta m^2_{12}$ and $\Delta m^2_{13}$ is such that $\left(\mathcal{R}_{e\mu}\right)_{23} \simeq 0.1$. Additionally, considering a new mass splitting in agreement with Eq. (\[MassRange\]) and mixing angles in agreement with Eq. (\[MixAngleRange\]), the ratio $\left(\mathcal{R}_{e\mu}\right)_{24} \in [10^{-5}, 10^{-2}]$. Thus, it is practical to set $\Delta_{12} = 0$, which is an approximation we make henceforth. Furthermore, since we only consider $|U_{e4}U^*_{\mu_4}|\Delta m^2_{14}\gtrsim 8\times 10^{-3}$ eV$^2$, which is approximately four times larger than $|\Delta m^2_{13}|$, $\left(\mathcal{R}_{e\mu}\right)_{34}\lesssim 10^{-1}$ is also small. In summary, if the oscillation interpretation of the short-baseline anomalies is correct, in experiments performed at $L$ and $E_{\nu}$ values where $\Delta_{14}\sim 1$, solar contributions are irrelevant and atmospheric contributions are small, at least in the $e\mu$ sector. If $|U_{\tau 4}|\sim|U_{\mu 4}|,$ $|U_{e4}|$,[^4] the same is approximately true of the $e\tau$ and $\mu\tau$ sectors, even when one takes into account that $|U_{\tau 3}|$ is several times larger than $|U_{e3}|$.
In the limit $\Delta m^2_{12}\to 0$, we “lose” the angle $\theta_{12}$ and the CP-odd phase $\delta$,[^5] and the oscillation probabilities depend on five angles ($\theta_{13}, \theta_{23}, \theta_{14}, \theta_{24}, \theta_{34}$) and two independent CP-odd phases, which we define as $\psi_s \equiv \delta_1 - \delta$ and $\phi_s \equiv (\delta_1 - \delta) - \delta_2$. Taking, in addition, the limit $\Delta m^2_{13}\to 0$, the oscillation probabilities depend on three angles ($\theta_{14},\theta_{24},\theta_{34}$) and zero physical CP-odd phases.[^6] The latter limit is the one usually considered in the analyses of short-baseline experiments [@Adey:2014rfv; @Giunti:2012tn; @Conrad:2012qt; @Kopp:2013vaa; @Giunti:2013aea].
When assuming there is only one relevant sterile neutrino, therefore, the study of CPV at short-baseline experiments requires sensitivity to the small $\Delta_{13}$ effects. Since disappearance channels are CP-invariant as a consequence of the CPT-theorem, we concentrate on the appearance channels.[^7] Taking advantage of what is known (or assumed) about the mixing parameters, we can further simplify the oscillation expressions. In detail, we approximate $U_{\mu 3} = c_{13}c_{24}s_{23}$ and $U_{\tau 3} = c_{13}c_{23}c_{34}$, since the subleading terms in Eqs. (\[Umu3\]) and (\[Utau3\]) are $\mathcal{O}(10^{-2})$, so the appearance probabilities can be written as $$\begin{aligned}
\label{probmutau}
P_{\mu\tau} &\simeq& 4c_{14}^4 s_{24}^2 c_{24}^2 s_{34}^2 \sin^2{\left(\frac{\Delta_{14}}{2}\right)} \nonumber \\
&+&8 c_{13}^2 s_{23} c_{23} c_{14}^2 s_{24} c_{24}^2 s_{34} c_{34}\sin{\left(\frac{\Delta_{13}}{2}\right)}\sin{\left(\frac{\Delta_{14}}{2}\right)}\cos{\left(\frac{\Delta_{14}}{2}+\psi_s-\phi_s\right)} \nonumber \\
&+&4c_{13}^4 s_{23}^2 c_{23}^2 c_{24}^2 c_{34}^2 \sin^2{\left(\frac{\Delta_{13}}{2}\right)},
\\
\label{probetau}
P_{e\tau} &\simeq& 4s_{14}^2 c_{14}^2 c_{24}^2 s_{34}^2 \sin^2{\left(\frac{\Delta_{14}}{2}\right)} \nonumber \\
&+&8 s_{13}c_{13}c_{23} s_{14} c_{14}^2 c_{24} s_{34} \sin{\left(\frac{\Delta_{13}}{2}\right)}\sin{\left(\frac{\Delta_{14}}{2}\right)}\cos{\left(\frac{\Delta_{14}}{2}+\psi_s\right)} \nonumber \\
&+&4 s_{13}^2 c_{13}^2 c_{23}^2 c_{14}^2 c_{34}^2\sin^2{\left(\frac{\Delta_{13}}{2}\right)},
\\
\label{probemu}
P_{e\mu} &\simeq& 4s_{14}^2 c_{14}^2 s_{24}^2 \sin^2{\left(\frac{\Delta_{14}}{2}\right)}\nonumber \\
&+& 8 s_{13}^2 c_{13} s_{23} s_{14} c_{14}^2 s_{24} c_{24} \sin{\left(\frac{\Delta_{13}}{2}\right)} \sin{\left(\frac{\Delta_{14}}{2}\right)} \cos{\left(\frac{\Delta_{14}}{2} + \phi_s\right)}\nonumber \\
&+& 4 s_{13}^2 c_{13}^2 s_{23}^2 c_{14}^2 c_{24}^2 \sin^2{\left(\frac{\Delta_{13}}{2}\right)}.\end{aligned}$$ The CP-conjugate and T-conjugate channels are obtained by changing the sign of the CP-odd phases $\psi_s$ and $\phi_s$, i.e., $P_{\bar{\alpha}\bar{\beta}}(\phi_s,\psi_s)=P_{\alpha\beta}(-\phi_s,-\psi_s)$ and $P_{\beta\alpha}(\phi_s,\psi_s)=P_{\alpha\beta}(-\phi_s,-\psi_s)$. The explicit CPV effects that render $P_{\alpha\beta}\neq P_{\bar{\alpha}\bar{\beta}}$ are contained in the interference between the $\Delta_{13}$ and the $\Delta_{14}$ terms in Eq. (\[Amplitude\]). In each $P_{\alpha\beta}$, the “14-squared” term is dominant, the interference term is the next-to-leading term, followed by the “13-squared” term, which is smallest. The measurement of two different appearance channels is required in order to determine the two independent CP-odd phases and, in principle, the measurement of the third appearance channel would serve as a nontrivial test of the $3+1$ hypothesis.
We assume that experiments will reveal that neither $\theta_{14}$ nor $\theta_{24}$ is very small, but anticipate learning very little about $\theta_{34}$, which is linked to $U_{\tau 4}$ and tau-appearance. Furthermore, working with taus is extremely challenging. It requires “detection” center-of-mass energies larger than the tau mass, and detectors capable of identifying taus with nonzero efficiency. Henceforth, we utilize exclusively the appearance oscillation probability $P_{e\mu}$, returning to taus in the concluding statements. In the range of values for $\theta_{14}$ and $\theta_{24}$ satisfying Eq. (\[MixAngleRange\]), the oscillation probability in Eq. (\[probemu\]) is approximately degenerate under interchange of $\theta_{14} \leftrightarrow \theta_{24}$; thus, we choose to simplify the parameterization by taking $\theta_s \equiv \theta_{14} = \theta_{24}$, and rewrite Eq. (\[probemu\]) as $$\begin{aligned}
\label{probemus}
P_{e\mu} &\simeq& 4s_{s}^4 c_{s}^2 \sin^2{\left(\frac{\Delta_{14}}{2}\right)}\nonumber \\
&+& 8 s_{13}^2 c_{13} s_{23} s_{s}^2 c_{s}^3 \sin{\left(\frac{\Delta_{13}}{2}\right)} \sin{\left(\frac{\Delta_{14}}{2}\right)} \cos{\left(\frac{\Delta_{14}}{2} + \phi_s\right)}\nonumber \\
&+& 4 s_{13}^2 c_{13}^2 s_{23}^2 c_{s}^4 \sin^2{\left(\frac{\Delta_{13}}{2}\right)},\end{aligned}$$ where $c_s \equiv \cos{\theta_s}$ and $s_s \equiv \sin{\theta_s}$. Our results are not sensitive to this assumption. Instead, the combined analyses of $\nu_e$ or $\nu_{\mu}$ appearance and $\nu_{\mu}$ or $\nu_e$ disappearance can distinguish $\theta_{14}$ effects from those of $\theta_{24}$. We do not pursue such an analysis here. Finally, Eq. (\[probemus\]) (and Eq. (\[probemu\])) is invariant under $\Delta m_{13}^2 \to -\Delta m^2_{13}$ and $\phi_s \to \phi_s + \pi$. For this reason, we assume henceforth that the sign of $\Delta m^2_{13}$ is positive. Our results are still valid if the mass-hierarchy turns out to be inverted, but for the shifted value of $\phi_s$.
Experimental Sensitivity to CP-Violating Phases {#sec:exp}
===============================================
We investigate the capability of next-next-generation experimental efforts to see $3+1$ CPV by simulating short-baseline experiments based on the $\nu$STORM proposal [@Adey:2014rfv]. According to the discussion in Section \[parameterization\], $3+1$ CPV effects are quite small and therefore require large statistics, excellent control of systematics, and very good energy resolution. All of these are potentially within reach of future neutrino experiments with beams from muon decay in flight. Other ideas for future experiments should be explored, including pion-decay-at-rest “beams,” similar to, for example, DAE$\delta$ALUS [@Alonso:2010fs].
The $\nu$STORM proposal is designed to measure the values of $|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$ in the $\nu_e \rightarrow \nu_\mu$ appearance channel for a $3+1$ scenario [@Adey:2014rfv]. Unlike pion-decay-in-flight long-baseline experiments that investigate the process $\nu_\mu \rightarrow \nu_e$, $\nu$STORM uses $\nu_e$ and $\overline{\nu}_\mu$ from the decay of stored $\mu^+$ to produce two neutrino beams. A detector with a strong magnetic field allows for $\mathcal{O}(1\%)$ energy resolution and powerful discrimination between detecting $\mu^+$ from $\overline{\nu}_\mu\rightarrow \overline{\nu}_\mu$ and $\mu^-$ from $\nu_e \rightarrow \nu_\mu$, which dramatically reduces beam-related backgrounds.
First, we reproduce the $\nu$STORM analysis in [@Adey:2014rfv] using similar flux, cross section, detector design, background rate, signal and background efficiency, and systematic uncertainties (1% and 10% associated with signal and background normalizations, respectively) [@Kyberd:2012iz; @Formaggio:2013kya]. This analysis is performed in the limit $\Delta m^2_{13}\to0$. Our results, depicted by the solid line in Fig. \[nustormplot\], agree with those from [@Adey:2014rfv] and illustrate that the $\nu$STORM experiment with a 1.3 kt detector at $L=2$ km would be able to constrain $4|U_{e4}|^2|U_{\mu4}|^2 < \mathcal{O}(10^{-4} - 10^{-3})$ at 99% CL for $\Delta m_{14}^2 \gtrsim 0.5 \text{ eV}^2$ after 10 years of running. If instead the $3+1$ scenario were confirmed by the $\nu$STORM experiment, the precision with which $\nu$STORM could measure $|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$ strongly depends on their physical values, as shown in Table \[table:nustormprecision\].
-- ------------------------------------------ --------------------------- -----------------------
$\nu$STORM $\nu$STORM+
Precision Precision
$4|U_{e4}|^2|U_{\mu4}|^2=3\times10^{-4}$ $\mathcal{O}(100\%)$ $\mathcal{O}(15\%)$
$\Delta m_{14}^2 = 1.0 \text{ eV}^2$ $\mathcal{O}(100\%)$ $\mathcal{O}(1\%)$
$4|U_{e4}|^2|U_{\mu4}|^2=4\times10^{-3}$ $\mathcal{O}(25\%-100\%)$ $\mathcal{O}(5\%)$
$\Delta m_{14}^2 = 1.0 \text{ eV}^2$ $\mathcal{O}(30\%)$ $\mathcal{O}(0.5\%)$
$4|U_{e4}|^2|U_{\mu4}|^2=2\times10^{-2}$ $\mathcal{O}(20\%)$ $\mathcal{O}(1\%)$
$\Delta m_{14}^2 = 1.0 \text{ eV}^2$ $\mathcal{O}(15\%)$ $\mathcal{O}(0.1\%)$
$4|U_{e4}|^2|U_{\mu4}|^2=5\times10^{-2}$ $\mathcal{O}(10\%)$ $\mathcal{O}(1\%)$
$\Delta m_{14}^2 = 1.0 \text{ eV}^2$ $\mathcal{O}(10\%)$ $\mathcal{O}(0.1\%)$
$4|U_{e4}|^2|U_{\mu4}|^2=4\times10^{-3}$ $\mathcal{O}(20\%)$ $\mathcal{O}(1\%)$
$\Delta m_{14}^2 = 5.0 \text{ eV}^2$ $\mathcal{O}(1\%)$ $\mathcal{O}(0.05\%)$
$4|U_{e4}|^2|U_{\mu4}|^2=2\times10^{-2}$ $\mathcal{O}(100\%)$ $\mathcal{O}(5\%)$
$\Delta m_{14}^2 = 0.35 \text{ eV}^2$ $\mathcal{O}(100\%)$ $\mathcal{O}(0.5\%)$
-- ------------------------------------------ --------------------------- -----------------------
: The position of the six colored points in Fig. \[nustormplot\] and the approximate 95% CL expected precisions with which $\nu$STORM and $\nu$STORM+ can measure $4|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$. While the baseline of $\nu$STORM is 2 km, the baseline of $\nu$STORM+ is optimized to measure CPV by requiring $\Delta m_{14}^2 L = 11.5$ eV$^2\cdot$km with 1000 times more statistics, for the same baseline, than $\nu$STORM. []{data-label="table:nustormprecision"}
![The solid line is the 99% CL sensitivity using the $\nu_e \rightarrow \nu_\mu$ channel at $\nu$STORM, assuming $\Delta m^2_{13}=0$, in agreement with [@Adey:2014rfv]. The dashed and dotted lines correspond to the 99% CL sensitivity if atmospheric effects are taken into account (see Eq. (\[probemus\])), when $\phi_s=-\pi/2$ and $\phi_s=\pi/2$, respectively. The six numbered points correspond to values of $|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$ used in this analysis to discuss the measurement of the value of $\phi_s$. []{data-label="nustormplot"}](nuSTORMexclusion.pdf){width="70.00000%"}
To illustrate the effects of CPV at $\nu$STORM, we recalculate the exclusion limits with the full expression in Eq. (\[probemus\]), when $\phi_s= -\pi/2$ and $\pi/2$.[^8] These results are depicted by the dashed and dotted lines in Fig. \[nustormplot\], respectively. Small differences in the limits occur; the difference between them is of the same order, around one percent, as the effects of the systematic uncertainties outlined in [@Adey:2014rfv]. To measure CPV, more statistics and an optimal choice of the baseline are required.
In order to investigate what is necessary to measure CPV in the $3+1$ scenario, we consider a dramatically upgraded version of the $\nu$STORM proposal, increasing the data sample – for the same baseline – by a factor of 1000 with respect to [@Adey:2014rfv]. This could be achieved if, for example, the beam flux were ten times larger ($\sim 10^{22}$ protons on target) over 10 years and the detector mass were 130 kt. We will refer to this experiment as $\nu$STORM+. While the proposed beam power and detector mass are outside the realm of possibilities today, they are not entirely outlandish. For comparison purposes, the proposed, and recently approved, India-Based Neutrino Observatory is a 51 kt magnetized iron calorimeter [@Kaur:2014rfa]. On the other hand, the proton driver for the proposed Neutrino Factory is planned to deliver $10^{22}$ protons on target per $10^7$ seconds [@Weng:2006uz]. $\nu$STORM+ would accumulate a large enough data sample such that the values of $|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$ would be measured very precisely, as displayed in Table \[table:nustormprecision\], at which point the value of $\phi_s$ would begin to induce observable changes to the oscillation probability.
To analyze our simulated data, we make use of the $\chi^2$ function $$\chi^2\left(\Delta m^2_{14},\theta_s,\phi_s | \Delta m^{2\star}_{14}, \theta_s^{\star}, \phi_s^{\star}\right) = \displaystyle\sum_i^{\text{bins}} \frac{\left[N^\text{data}_i(\Delta m^{2\star}_{14},\theta_s^{\star},\phi_s^{\star}) - N^\text{hyp}_i(\Delta m^2_{14}, \theta_s, \phi_s)\right]^2}{N^\text{hyp}_i(\Delta m^2_{14}, \theta_s, \phi_s)},
\label{Chi2}$$ where $N^\text{data}_i$ and $N^\text{hyp}_i$ are the measured and expected number of events in energy bin $i$, respectively, and $\Delta m^{2\star}_{14},\theta^{\star}_s,\phi^{\star}_s$ and $\Delta m^2_{14},\theta_s,\phi_s$ are physical (i.e. input) and hypothetical values of the new mass-squared difference, mixing angle, and CP-odd phase, respectively. Strictly speaking, the $\chi^2$ function also depends on $\theta_{13},\theta_{23},$ and $\Delta m_{13}^2$. We assume, however, that $\theta_{13},\theta_{23},$ and $\Delta m_{13}^2$ will be measured with sufficient precision such that the $\chi^2$ function above, once marginalized over $\theta_{13},\theta_{23},$ and $\Delta m_{13}^2$, is sufficiently indistinguishable from its expression for the best-fit values of $\theta_{13},\theta_{23},$ and $\Delta m_{13}^2$, which we take to be the ones in [@Agashe:2014kda]. Indeed, if one includes the current central values and uncertainties on these parameters [@Capozzi:2013csa; @Forero:2014bxa; @Gonzalez-Garcia:2014bfa], the oscillation probability $P_{e\mu}$ and the $\chi^2$ function change only at the sub-percent level. We can safely presume that current and future experiments will increase the precision with which these parameters are known before $\nu$STORM$+$ exists. We also considered the possibility that the uncertainties are significantly larger – 5% and 50% associated with the normalization of the signal and background, respectively. The effect of inflating the uncertainties, as far as all results presented henceforth, is negligible.
The value of $N_i$ is determined by integrating, over the bin width, $$\label{dNdE}
\frac{dN}{dE} = \Delta t \cdot \Phi(E) \cdot \sigma(E) \cdot \epsilon(E) \cdot P_{e\mu}(E).$$ Here $\Phi$, $\sigma$, $\epsilon$, $P_{e\mu}$, and $\Delta t$ are the flux, cross section, efficiency, $\nu_e \rightarrow \nu_\mu$ oscillation probability, and the amount of time the experiment runs, respectively. From Eqs. (\[Chi2\]) and (\[dNdE\]), we see that $\chi^2$ depends linearly on $\Delta t,$ $\Phi(E),$ $\sigma(E)$, and $\epsilon(E)$.
We can understand, semi-quantitatively, the sensitivity to CPV by analyzing Eq. (\[Chi2\]) in more detail and making a few simplifying assumptions. We consider that $N_i$ can be approximated by evaluating $dN/dE$ at the value of the energy corresponding to the center of bin $i$, $E_i$. Taking into account that $\Delta_{13} \ll 1$, and, only for the sake of this discussion, fixing $\Delta m^2_{14}$ and $\theta_s$ to their input values, i.e., setting $\Delta m^2_{14}=\Delta m^{2\star}_{14},\theta_s=\theta_s^{\star}$, $$\label{Chi2Simp}
\chi^2\left(\phi_s | \phi^{\star}_s\right) \propto \Delta t\sum_{i}^\text{bins} L^2\Phi(E_i)\sigma(E_i)\epsilon(E_i)\left(\frac{s_s^2c_s^4}{s_s^2+ALc_{s}\cos{\left(\frac{\Delta_{14}}{2}+\phi_s\right)}/\sin{\left(\frac{\Delta_{14}}{2}\right)}+BL^2c_s^4}\right)f_i(\phi_s,\phi^{\star}_s),$$ where $$f_i(\phi_s,\phi^{\star}_s)=\left[\cos{\left(\frac{\Delta_{14}}{2}+\phi_s\right)}-\cos{\left(\frac{\Delta_{14}}{2}+\phi^{\star}_s\right)}\right]^2,$$ $A\sim \mathcal{O}(10^{-4}\text{ km}^{-1})$, and $B\sim \mathcal{O}(10^{-8}\text{ km}^{-2})$. Even though we are interested in $L\lesssim 100$ km, we preserve the term involving $B$ in the event that the $L/E$-dependent coefficient of $A$ is vanishingly small.
The bins with the largest number of events $N_i$, and therefore the most statistical power, will contribute the most to the value of $\chi^2$. These correspond to the peaks of $P_{e\mu}(E)$, approximately where $\sin{\left(\frac{\Delta_{14}}{2}\right)} = \pm1$. Fixing $\sin{\left(\frac{\Delta_{14}}{2}\right)} = \pm 1$, we can further simplify Eq. (\[Chi2Simp\]) down to its dominant contributions $$\label{Chi2Dom}
\chi^2\left(\phi_s | \phi^{\star}_s\right) \propto \Delta t \sum_{s{\left(\frac{\Delta_{14}}{2}\right)} = \pm 1} L^2\Phi(E_i)\sigma(E_i)\epsilon(E_i)\left(\frac{s_s^2c_s^4}{s_s^2+CLc_s+BL^2c_s^4}\right)g(\phi_s,\phi^{\star}_s),$$ where the sum is restricted to the bins where the approximation $\sin{\left(\frac{\Delta_{14}}{2}\right)} = \pm1$ is good, $|C| \lesssim \mathcal{O}(10^{-4}\text{ km}^{-1})$ and $$g(\phi_s,\phi^{\star}_s) = \left(\sin{\phi_s}-\sin{\phi^{\star}_s}\right)^2.$$ Eqs. (\[Chi2Simp\]) and (\[Chi2Dom\]) allow one to conclude the following:
- Because the flux, $\Phi$, scales like $1/L^2$ and $B$ and $C$ are small numbers, the most significant dependence on $L$ comes in the product $\Delta m_{14}^2 L$. Therefore, the ability to measure $\phi_s$ is, to a good approximation, greatest for some constant value of the product $\Delta m_{14}^2 L$.
- The sensitivity for measuring the value of $\phi_s$ is linearly dependent on the power of the beam, size of the detector, and the amount of time that the experiment runs.
- The value of $\phi_s$ is easiest to measure when $\theta_s \sim 0.18$, i.e., $4|U_{e4}|^2|U_{\mu4}|^2\sim 4\times10^{-3}$. Around this value, $\chi^2$ falls off slowly for $\theta_s > 0.18$ and falls off rapidly for $\theta_s < 0.10$. This is apparent by analyzing the term $(s_s^2c_s^4)/(s_s^2+CLc_s+BL^2c_s^4)$, which contains all the $\theta_s$ dependence in Eq. (\[Chi2Dom\]) and recognizing that $B$ and $C$ are small numbers.
- It is easiest to measure the CP-odd phase if $\phi_s = \pm \pi/2$. This can be seen in $g(\phi_s,\phi^{\star}_s)$, the maximal variance of this is for $\phi^{\star}_s = \pi/2$ and $\phi_s = -\pi/2$, or vice-versa.
We note that these observations do not depend on the details of the beam nor the detector but rather stem from the form of the oscillation probability.
The best strategy for choosing a baseline $L$ to maximize sensitivity for measuring $|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$ is to require that the highest-energy (first) oscillation maximum is within the energy range associate with the experiment. If so, the signal yield mostly depends on $|U_{e4}|^2|U_{\mu4}|^2$, and the peak of the measured signal distribution mostly depends on $\Delta m_{14}^2$. On the other hand, the best strategy for measuring the effects of CPV – the subject of our study – is to arrange for multiple oscillations to occur within the measured neutrino energy range. These may or may not include the first oscillation maximum.
Based on the discussion of the $\chi^2$ function, the best value of $\Delta m_{14}^2L$ can be estimated independent from the physical value of $\Delta m_{14}^2$. For the $\nu$STORM+ flux shape, signal efficiency, and background rate, we find that choosing the product $\Delta m_{14}^2 L$ equal to $11.5$ eV$^2\cdot$km optimizes the sensitivity to $\phi_s$. This result is obtained by calculating the maximal $\Delta \chi^2$ for a particular set of parameters and varying $L$. We find, for all Points 1-6 in the parameter space, that the greatest sensitivity for measuring the effects of CPV is for a fixed value of $\Delta m_{14}^2 L$. If a different beam profile were chosen, then this value would change. We illustrate this fact by concentrating on Points 2, 5, and 6, where $\Delta m_{14}^2 = 1.0$ eV$^2$, 5.0 eV$^2$, and 0.35 eV$^2$, respectively. Fig. \[probs\] depicts the “data” corresponding to Points 2, 5, and 6 for $L=11.5$ km, 2.3 km, and 33 km, respectively, so $\Delta m_{14}^2 L = 11.5$ eV$^2\cdot$km for all three panels, and $\phi_s=\pm\pi/2$. It is easy to see that, while the number of events and the relative CPV effects are quite different, the “shapes” corresponding to the three points are almost identical. As we will show immediately, the sensitivity to $\phi_s$ is almost identical for these three scenarios.
Our “measurements” of $\phi_s$ are depicted in Fig. \[phisensitivity\], for the six values of $4|U_{e4}|^2|U_{\mu4}|^2$ and $\Delta m_{14}^2$ in Fig. \[nustormplot\] and Table \[table:nustormprecision\], for $\Delta m_{14}^2 L = 11.5$ eV$^2\cdot$km and $\phi_s = \pi/2$. We make use of Eq. (\[Chi2\]) and compute $\Delta\chi^2$ after numerically marginalizing over $\Delta m^2_{14}$ and $\theta_s$, for each Point. In the Appendix, we present constant $\chi^2$ contours in the two-dimensional $\phi_s\times \Delta m^2_{14}$ and $\phi_s\times \theta_s$ planes, for Point 2. As advertised, if $\theta_s$ is not much larger nor smaller than 0.18, i.e., $4|U_{e4}|^2|U_{\mu4}|^4\sim4\times10^{-3}$, the value of $\Delta\chi^2$ changes minimally for different values of $\Delta m_{14}^2$, as long as $\Delta m_{14}^2 L = 11.5$ eV$^2\cdot$km is satisfied. Qualitatively, for different values of $\phi_s$, one obtains similar results, though the overall sensitivity for measuring $\phi_s$ is reduced. Also depicted in Fig. \[phisensitivity\] (dotted line), is the measurement of $\phi_s$ in the case nature agrees with Point 2 but assuming $\nu$STORM+ is performed at the proposed $\nu$STORM baseline. It is apparent that the optimal choice of baseline is very significant.
![The expected values of $\Delta \chi^2$ at $\nu$STORM+ when $\phi_s=\pi/2$ for the six points shown in Fig. \[nustormplot\] and Table \[table:nustormprecision\]. The dotted line correspond to Point 2, assuming $\nu$STORM+ is performed at the $\nu$STORM baseline.[]{data-label="phisensitivity"}](CP_phihalfpi.pdf){width="70.00000%"}
$\nu$STORM+ can establish, for all points defined in Fig. \[nustormplot\], that CP-invariance is violated if the CP-odd phase is $\pi/2$ only at the one-sigma level, i.e., it would constrain $\phi_s\in [0,\pi]$ at the one-sigma level and can rule out $\phi_s=-\pi/2$ at around the two-sigma level (Fig. \[phisensitivity\]). As discussed above, the expected uncertainties are larger for different values of $\phi_s$. More power to establish CPV might come from combining other information on $\Delta m^2_{14}$, $\theta_s$, and $\phi_s$. Such information may come from the disappearance channel, combining neutrino running with antineutrino running, or combining searches for $\nu_e\to \nu_{\mu}$ with those for $\nu_{\mu}\to\nu_e$. The latter could be pursued, for example, at different experiments making use of a well-characterized $\nu_{\mu}$ source, including pion decay at rest [@Alonso:2010fs]. The $\nu$STORM+ disappearance data from the process $P_{\bar{\mu}\bar{\mu}}$ would be available concurrently with those from $P_{e\mu}$. While they provide no information on $\theta_s$ – $\nu_{\mu}$ disappearance is mostly dependent on $\theta_{24}$, providing virtually no information on $\theta_{14}$ – they do provide a different measurement of $\Delta m^2_{14}$, mostly independent from $\phi_s$. Performing a joint appearance and disappearance analysis is beyond the scope of this paper.[^9] Nonetheless, we estimate the consequences of combining the two data sets by “measuring” $\Delta m^2_{14}$ using disappearance data and applying the result as a prior to the $\chi^2$ analyses described in detail in this section. For all Points, except Point 6, we find that one can exclude $\phi_s=0$ at a level somewhere between two and three sigma ($4\lesssim \Delta \chi^2\lesssim 10$, depending on the Point). An improved measurement of $\theta_s$ might prove at least as fruitful. In this case, however, in order to make use of disappearance data, one needs to combine both $\nu_{\mu}$ and $\nu_e$ disappearance in order to determine both $\theta_{14}$ and $\theta_{24}$. A precise measurement of $\nu_e$ disappearance would require, for example, a short-baseline reactor experiment (see, for example, [@Ashenfelter:2013oaa]) or a radioactive-source experiment (see, for example, [@Bungau:2012ys]).
Conclusion {#sec:conclusion}
==========
The discovery of new, light neutrino degrees of freedom would qualitatively impact our understanding of fundamental physics. A new wave of oscillation experiments, aimed at exploring the physics at the new oscillation length(s), will be required in order to explore the new-physics sector.
New neutrino degrees of freedom allow for more sources of CP-invariance violation (CPV). Here, we explore the requirements for accessing CP-odd mixing parameters in the so-called $3+1$ scenario, where one assumes the existence of one extra, mostly sterile neutrino degree of freedom, significantly heavier than the other three mass eigenstates. CPV is present in the interference term between the solar and atmospheric oscillation lengths, proportional to $\Delta m^2_{12}$ and $\Delta m^2_{13}$ respectively, and the new shorter oscillation length, proportional to $\Delta m^2_{14}$. We concentrate on short-baseline experiments, engineered such that $\Delta m^2_{14}\sim E_{\nu}/L$, and argue that solar effects, due to the fact that $\theta_{13}$ is not too small, can be safely neglected. We also show that, if new neutrino states are indeed discovered in the next round of short-baseline experiments, atmospheric effects are small, rendering the study of CPV most challenging. Our results confirm that, for the on-going and planned short-baseline experiments, it is safe to approximate $\Delta m^2_{13}=\Delta m^2_{12}=0$ when discussing the $3+N$ oscillation hypotheses, for $N\ge 1$.
As a first step towards understanding how to measure CPV in short-baseline experiments, we concentrate on the $\nu_{e}\to\nu_{\mu}$ appearance channel in a hypothetical, upgraded version of the $\nu$STORM proposal, $\nu$STORM+. Using only appearance data, we establish that the optimal baseline for CPV studies depends strongly on the value of $\Delta m^2_{14}$ and, in turn, the ability of $\nu$STORM+ to observe CPV depends significantly on whether the experiment is performed at the optimal baseline.
Our results, assuming a set-up one thousand times more powerful than that of $\nu$STORM ($\nu$STORM+), are depicted in Fig. \[phisensitivity\]. Even at the optimal baselines, it will be very challenging to see CPV in $3+1$ scenarios if one considers only one appearance channel. Significantly better results are expected if one includes more information. Some is already accessible at $\nu$STORM+, including more information on $\Delta m^2_{14}$ from the $\nu_{\mu}$ disappearance channel. Other possibilities include combining the neutrino and the antineutrino appearance channels by changing the charge of the muons in the storage ring, or combining $\nu_{e}\to\nu_{\mu}$ data with those from a different experiment capable of precision measurements of $\nu_{\mu}\to\nu_{e}$, the T-conjugate channel.
Even in the simple $3+1$ scenario, CPV effects beyond those studied here can be easily accommodated. The study of the tau-appearance channel ($\nu_{\mu}\to \nu_{\tau}$ or $\nu_{e}\to \nu_{\tau}$) is required for exploring the second new CP-odd “Dirac” phase contained in the extended $4\times 4$ mixing matrix. As of right now, if there is a new mass-squared difference of order 1 eV$^2$, very little is known about the $\nu_{\tau}$ content of the fourth neutrino mass eigenstate. Searches for tau-appearance – let alone precision measurements of tau-appearance – are extremely challenging and will require new, dedicated experimental efforts that go significantly beyond what is currently being explored by the experimental neutrino community.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is sponsored in part by the DOE grant \#DE-FG02-91ER40684.
Determining the New Oscillation Parameters: Point 2
===================================================
Fig. \[AppendixFig\] depicts the experimental sensitivity to the oscillation parameters at the $\nu$STORM$+$ experiment, outlined in Section \[sec:exp\], for physical values (i.e. input values) of the parameters $\Delta m_{14}^2 = 1.0$ eV$^2$, $4|U_{e4}|^2|U_{\mu 4}|^2 = 4\times 10^{-3}$ (or $\theta_s = 0.18$), and $\phi_s = \pi/2$ (Point 2 in Fig. \[nustormplot\] and Table \[table:nustormprecision\]). In each plot, the third variable is marginalized over when calculating $\Delta \chi^2$ contours. The blue, yellow, and red contours correspond to $68\%$, $95\%$, and $99\%$ CL sensitivity, respectively. Stars indicate the input values of $\phi_s$, $\theta_s$, and $\Delta m_{14}^2$.
[^1]: Other options include neutrinoless double-beta decay and precision measurements of $\beta$-decay energy spectra. See, for example, [@deGouvea:2006gz; @Formaggio:2011jg; @Rodejohann:2011mu; @Esmaili:2012vg].
[^2]: The other possibility is to aim at the atmospheric oscillation, $L/E_\nu \sim 1000$ km/GeV, such that $|\Delta_{13}|\sim 1$. We do not consider this case here. When $|\Delta_{13}|\sim 1$, the fast oscillations associated with the (mostly) sterile neutrino tend to average out, rendering the study of CPV very challenging, because the effects of CPV are most apparent when comparing different values of $L/E$. If the new oscillations do “average out,” then this does not necessarily remove the possibility of indirectly exploring CPV phenomena; the combination of results from multiple experiments can be used to measure CPV effects in a $3+1$ scenario [@Hollander:2014iha; @Klop:2014ima].
[^3]: The approximation $\Delta_{14}\sim1$ implies that the “14” term is of order $|U_{\alpha 4}U_{\beta4}^*|$ and $L/E_{\nu}\sim(\Delta m^2_{14})^{-1}$.
[^4]: Currently, there is very little experimental information regarding the $\tau$ sector.
[^5]: When $\Delta_{12}=0$ in Eq. (\[Amplitude\]), one is not sensitive to any $U_{\alpha 2}$ and hence the value of $\theta_{12}$. This, in turn, implies that the amplitude is consistent with any value of $\theta_{12}$, including $\theta_{12}=0$. When $\theta_{12}=0$, one of the CP-odd phases is unphysical.
[^6]: In general, in the limit where $j$ mass-squared splittings vanish, some of the observables that parameterize $U$ become unphysical. The total number of angles $N_\theta$ and phases $N_\delta$ that determine $P_{\alpha\beta}$ in this case are $$\begin{aligned}
\label{NAngles}N_\theta &=& n(n-1)/2 -j(j+1)/2,\\
\label{NPhases}N_\delta &=& (n-1)(n-2)/2 - j(j+1)/2,\end{aligned}$$ assuming there are $n$ neutrino states.
[^7]: One could try to infer that CP-invariance is violated by comparing different disappearance channels and fitting them to the $3+1$ oscillation hypothesis. We do not explore this possibility here.
[^8]: While we use Eq. (\[probemus\]), we convert $\theta_s$ into $|U_{e4}||U_{\mu4}|$ for comparison purposes: $\sin^2\theta_s\cos\theta_s\equiv|U_{e4}||U_{\mu4}|$.
[^9]: One challenge is that, once appearance and disappearance data are combined, the $\theta_{14}=\theta_{24}=\theta_s$ choice can no longer be made, and one is required to explore the three dimensional $\theta_{14},\theta_{24},\Delta m^2_{14}$ new-physics parameter space. This renders the discussion much more cumbersome.
|
---
abstract: 'The forward–backward asymmetry in , which must be zero in the center-of-mass system if charge symmetry is respected, has been measured to be $[17.2 \pm 8 {\rm (stat)} \pm 5.5 {\rm (sys)}] \times 10^{-4}$, at an incident neutron energy of 279.5 MeV. This charge symmetry breaking observable was extracted by fitting the data with GEANT-based simulations and is compared to recent chiral effective field theory calculations, with implications regarding the value of the $u \ d$ quark mass difference.'
author:
- 'A.K. Opper'
- 'E. Korkmaz'
- 'D.A. Hutcheon'
- 'R. Abegg'
- 'C.A. Davis'
- 'R.W. Finlay'
- 'P.W. Green'
- 'L.G. Greeniaus'
- 'D.V. Jordan'
- 'J.A. Niskanen'
- 'G.V. O’Rielly'
- 'T.A. Porcelli'
- 'S.D. Reitzner'
- 'P.L. Walden'
- 'S. Yen'
bibliography:
- 'e704\_prl.bib'
title: 'Charge symmetry breaking in $ \bm {n p \rightarrow d \pi^0}$ '
---
In the quark model, the breaking of charge independence and charge symmetry arises from the mass difference of the $up$ and $down$ current quarks and the electromagnetic interaction between quarks. The basic $np$ interaction is particularly sensitive to such fundamental effects since the “background" Coulomb force is absent in this system. Indeed, charge symmetry breaking (CSB) has been unambiguously observed [@tri-csb; @tri-new; @iucf-csb] in $np$ elastic scattering at three different energies. Measurement of CSB in the inelastic reaction complements the existing data in that it is sensitive to contributions that are absent in the elastic channel. Furthermore, this reaction is unique as a testing ground for effective field theory calculations addressing the important issue of isospin symmetry violation in pion-nucleon scattering. The observable of interest in is the center-of-mass forward–backward asymmetry, $A_{\rm fb}$, which we define as $$A_{\rm fb}(\theta ) \equiv
{ {\sigma (\theta ) - \sigma (\pi - \theta )} \over
{\sigma (\theta ) + \sigma (\pi - \theta )} }
\label{eq:Afb-def-eq}$$ where $\theta$ is the angle between the incident beam and the scattered deuteron. Note that the asymmetry must be zero if charge symmetry is conserved. We report on a measurement of this asymmetry at a neutron energy a few MeV above the reaction threshold (275.06 MeV), and compare our result to recent theoretical predictions [@Ni99; @vKNM] bearing on such fundamental questions as the $u \ d$ quark mass difference and our understanding of QCD dynamics and symmetries in low-energy hadronic interactions.
The Experiment\[expt-sec\]
==========================
The experiment was performed at TRIUMF with a 279.5 MeV neutron beam, a liquid hydrogen target, and the [<span style="font-variant:small-caps;">SASP</span>]{} magnetic spectrometer [@Wa99] positioned at $0^{\circ}$. With these near threshold kinematics and the large acceptance of [<span style="font-variant:small-caps;">SASP</span>]{}, the full deuteron distribution from was detected in one setting of the spectrometer thereby eliminating many systematic uncertainties. These deuterons form a distinct kinematic locus in momentum vs laboratory scattering angle, which is shown in fig. \[locus-fig\] for the collected data.
The [<span style="font-variant:small-caps;">TRIUMF</span>]{} <span style="font-variant:small-caps;">CHARGEX</span> facility [@He87] produced the neutron beam by passing a high intensity proton beam through a thin $\rm ^7Li$ target. A sweeping magnet deflected the primary proton beam into a well- shielded dump. The liquid hydrogen target ($\rm LH_2$) was centered 92 cm downstream from the $\rm ^7Li$ target and was contained within a flat cylindrical volume, 10 cm in diameter with a nominal thickness of 2 cm. Two sets of veto counters (FEV1, FEV2) and a trigger counter set (FET) were each composed of a pair of plastic scintillators positioned above one another. This allowed more stable operation in the high (few MHz) particle rate environment. The thick veto scintillators were upstream of the and shadowed it. The FET counters were positioned immediately downstream of the .
Three multi-wire proportional chambers, positioned upstream of the [<span style="font-variant:small-caps;">SASP</span>]{} entrance (FECs i.e. Front-End Chambers), provided tracking information for charged particles. Each FEC consisted of a pair of orthogonal wire planes. The first and last FECs, separated by 33 cm, were mounted to measure vertical and horizontal coordinates. The third FEC was positioned midway between the other two and rotated $40^\circ$ with respect to them for efficiency measurements and to aid in multi-hit track reconstruction. Particle tracking near the [<span style="font-variant:small-caps;">SASP</span>]{} focal plane was provided by two vertical-drift chambers (VDCs). Three sets of scintillators, downstream from the VDCs, provided timing and particle ID information as well as sufficient redundancy to determine the efficiencies of all focal plane area detectors.
Measurements of $np$ elastic scattering with incident neutron beams that filled the same target space and produced protons that spanned the momentum distribution of the reaction provided a stringent test of the description of the spectrometer acceptance. Further details on the apparatus and other technical aspects of the measurement are found in reference [@Hu01].
Extraction of $\mathbf{A_{\rm {\bf fb} } }$ \[extraction-sec\]
===============================================================
Close to threshold, the cross-section in the center-of-mass frame is given by $$\frac{d\sigma}{d\Omega} (\theta) = A_{0} +
A_{1}P_{1}(\cos\theta) + A_{2}P_{2}(\cos\theta),
\label{eq:x-section-eq}$$ where $P_1$ and $P_2$ are Legendre polynomials. The $A_0$ and $A_2$ coefficients were previously measured [@Hu91] at a number of energies within 10 MeV above threshold. The presence of charge symmetry breaking is reflected in the $A_{1}$ term as it is odd in $\cos\theta$. In this standard parametrization, the angle integrated form of $A_{\rm fb}$ is given by $A_{\rm fb} = \frac{1}{2}A_{1}/A_{0}$.
For a given beam energy, $\cos\theta$ varies linearly with the longitudinal component of deuteron momentum in the laboratory reference frame. Ideally, the $\cos\theta$ distribution would be found by a suitable, simple projection of the data of fig. \[locus-fig\]. However, the measured deuteron locus is distorted by energy loss, multiple scattering, energy spread of the beam, and spectrometer acceptance making a direct extraction of impossible. Instead, the data were binned according to laboratory momentum and angle (as in fig. \[locus-fig\]) and compared to a model which represented the background due to $\rm C(n,d)$ reactions as a low-order polynomial and generated the locus of $\rm H(n,d)\pi^0 $ events by Monte Carlo simulation of the beam, target, reaction cross section, spectrometer and detectors.
The simulation was based on <span style="font-variant:small-caps;">GEANT3</span>. It began with a proton beam incident on the $\rm^7Li$ target and included energy loss by the proton beam as well as the angular and energy distribution of neutrons from the $\rm^7Li(p,n)$ reaction. Production of deuterons according to the distribution of equation \[eq:x-section-eq\] was allowed in the target and other hydrogenous material such as scintillators and their wrapping. Standard [<span style="font-variant:small-caps;">GEANT</span>]{} tracking options were adopted for deuteron energy loss and multiple scattering but the reaction losses, which amount to 1–2% and are momentum dependent, were parametrized from data on deuteron elastic and reaction cross-sections from hydrogen and carbon [@deut_cross]. Tracking through the [<span style="font-variant:small-caps;">SASP</span>]{} dipole used a field map obtained at 875 Amp and scaled up to the operating current of 905 Amp. Data were acquired in 10 different periods spanning two years and the simulation accounted for measured detector efficiencies, scintillator thresholds, missing FEC wires, and known changes in target thickness in a manner consistent with the actual running periods.
To reduce the possibility of psychological bias in matching simulation to data, a blind analysis technique was used which incorporated a hidden offset to the $A_1/A_0$ asymmetry parameter of the generator. The collaborators developing the simulation and extracting the observable did not know the value of the offset until all consistency checks had been satisfied.
Systematic Effects \[sys-sec\]
===============================
The acceptance of [<span style="font-variant:small-caps;">SASP</span>]{} is a function of the initial target position and direction of the deuteron as well as its momentum. Non–uniformities in the momentum acceptance of [<span style="font-variant:small-caps;">SASP</span>]{} would systematically produce a false asymmetry and had to be limited. High-statistics data from $np$ elastic scattering were collected and compared to model simulations to determine a fiducial volume of uniform acceptance. For these calibration measurements, the [<span style="font-variant:small-caps;">SASP</span>]{} magnets were set to their values for the running, but the primary beam energy was adjusted so that the elastically scattered protons had a momentum deviation $\delta = (p - p_0)/p_0 = -4, 0$ or $+4\% $ compared to the central momentum of the deuterons of interest. Projections of the $np$ elastic data direction for position slices were formed, and the ratios of yields at $ -$4% vs +4% were formed for both data and simulation; see fig. \[accep-fig\]. The analysis software acceptance cuts in position and direction were then limited to the regions common to both data and simulation which were uniform in momentum to the statistical precision of the data.
Simulation vs simulation comparisons were carried out to determine how strongly experimental parameters and other effects were correlated with $A_1/A_0$. For example, momentum dependent deuteron reaction losses and detection efficiencies are obvious mechanisms which can mimic the effect of a non–zero $A_1/A_0$. Combining each correlation with the independently-determined uncertainty of its parameter gave the systematic contributions shown in Table \[error\_bud-tab\]. However, for the target thickness, the proton beam energy ($T_{\rm beam}$) and the central momentum of [<span style="font-variant:small-caps;">SASP</span>]{} ($p_0$) the independent information was not a sufficient constraint. Therefore, these three parameters, along with $A_1/A_0$, were treated as free parameters and their values extracted from fitting the data. To this end, simulations were made and $\chi^2$ calculated for 81 points in a four-dimensional space, in which each of the four free parameters was stepped above and below a nominal value. $\chi^2$ minimization techniques [@num_rec] were then used to obtain the values of the parameters at the global $\chi^2$ minimum, while the local curvature of the $\chi^2$ surface gave their errors and mutual correlations.
Uncert $ \times (10^{-4})$
---------------------------------------- ----------------------------
FEV threshold 2.5
Separation between front and rear FECs 2.5
Longitudinal position of $ \rm ^7Li$ 2.5
$A_2/A_0 $ 2
Deuteron reaction losses 1.5
Detection efficiencies 1.5
Primary beam energy spread 1
Neutron angle 1
Background 1
FET threshold 0.5
Total 5.5
: Systematic Error Contributions to \[error\_bud-tab\]
Results and Discussion \[result-sec\]
=====================================
As a test of the model, the $\chi^2$ calculations and fitting were repeated on subsets of the data and simulated data, selected according to whether the reaction occurred in the Top or Bottom part of the target. A second test divided events into those originating in the Left or Right part of the target. The best fit values and errors of $A_1/A_0$ (after removal of the offset) and the other three parameters are presented in Table \[params\_subspaces-tab\].
The root mean square (rms) systematic error for the full acceptance and the four subspaces is $\sim 2.7\%$ with the standard binning scheme of 50 bins in $\delta$ and 20 bins in $\theta$, indicating a substantial discrepancy between data and the simulation. Pixel by pixel examination of the contribution to $\chi^2$ revealed a systematic difference in the profile of the locus along lines of steepest ascent. The sign of the differences tended to be positive at the peak and negative at both the “inner” and “outer” margins of the locus, possibly due to inadequate treatment of deuteron scattering in the simulation.
A change in $A_1/A_0$ will not change the ratio of counts in peak vs margins of the locus because it multiplies $\cos(\theta_{cm})$. In contrast, the thickness, $p_0$, and $\rm T_{beam}$ all shift or broaden the locus and thus are sensitive to the ratio of locus counts at the peak vs margins. It is reasonable to expect further rebinning to remove sensitivity to unimportant details of the simulation without losing sensitivity to $A_1/A_0$. We repeated the $\chi^2$ grid search using 20 bins in $\delta$ and 10 bins in $\theta$, and again with 10 bins in $\delta$ and 5 bins in $\theta$. As expected, the fractional error dropped to 2.1% and 1.4%, respectively, with $A_1/A_0$ remaining consistent within errors. A more sophisticated binning scheme which treated the locus as a set of “elliptical” and “radial” bins on top of rectangular background bins produced an rms error of 2.1% for 2500 background bins and 36 locus bins, and an rms error of 1.2% for 100 background bins and 6 locus bins. In all fits and binning schemes the best fit values of the asymmetry in the acceptance subspaces agreed within errors with the value for the full acceptance, which is $(34.4 \pm 16) \times 10^{-4}$, implying $A_{\rm fb} = [17.2 \pm 8 {\rm (stat)} \pm 5.5 {\rm (sys)}] \times 10^{-4}$.
------ --------------- ----------------- ------------------- ----------------------
$(A_1/A_0)$ relative relative relative
$(10^{-4})$ $LH_2$ (mm) $p_0$ (MeV/c) $T_{\rm beam}$ (MeV)
full 34.4 $\pm$ 16 0.94 $\pm$ 0.05 0.365 $\pm$ 0.015 0.048 $\pm$ 0.001
b 30 $\pm$ 26 0.39 $\pm$ 0.09 0.547 $\pm$ 0.025 0.086 $\pm$ 0.002
t 20 $\pm$ 20 1.14 $\pm$ 0.07 0.236 $\pm$ 0.018 0.021 $\pm$ 0.002
l 29 $\pm$ 23 1.21 $\pm$ 0.08 0.273 $\pm$ 0.021 0.042 $ \pm$ 0.002
r 15 $\pm$ 22 0.75 $\pm$ 0.08 0.427 $\pm$ 0.021 0.051 $\pm$ 0.002
------ --------------- ----------------- ------------------- ----------------------
: Stability of the four free parameters over target subspaces; b = bottom; t = top; l = left; r = right. \[params\_subspaces-tab\]
Theoretical predictions of $A_{\rm fb}$ have been made by Niskanen [@Ni99] using a meson-exchange coupled-channel model which showed that the major contribution by far is due to $\pi \eta$ (and $\pi \eta'$) mixing in both the exchange and produced (outgoing) meson. At our energy the prediction is $A_{\rm fb}= -28 \times 10^{-4}$, when accepted values are used for the $\eta NN$ coupling constant and the $\pi^0 \eta$ mixing matrix element ($ g_{\eta NN}^2/4\pi = 3.68 $ from meson exchange $NN$ potential models [@Du83] and $\langle \pi^0| {\cal H }| \eta \rangle = -0.0059$ $\rm GeV^2$ from analysis of $\eta$ decay data [@Co86]). More recently, $A_{\rm fb}$ was revisited [@vKNM] within the framework of chiral effective field theory where the issue of charge symmetery breaking in the rescattering amplitude of the exchanged pion was addressed. The resulting additional contribution to is then expressed in terms of two parameters $\delta m_N$ and $\bar{\delta}m_N$ representing contributions from the $u \ d$ quark mass difference and from electromagnetic effects within the nucleon, respectively. Specifically, at our energy, is expressed as
$$A_{\rm fb} = -0.28\% \times \left[
\left( { {g_{\eta NN}} \over {\sqrt{4\pi (3.68)}}} \right)
\left( { {\langle \pi^0| {\cal H }| \eta \rangle} \over
{ -0.0059 {\rm \ GeV^2} } } \right)
- { {0.87} \over {{\rm MeV} } } (\delta m_N -
{ {\bar{\delta}m_N} \over 2}) \right]
\label{Afb-cont-eq}$$
where the first term arises from $\pi \eta $ mixing and the second from $\pi^0 \ N $ scattering. With the introduction of the new term, changes sign and becomes positive with an estimated upper value around $+69 \times 10^{-4}$, when large but reasonable values of $\delta m_N$ and $\bar{\delta}m_N$ are used [@vKNM]. Our positive experimental result strongly suggests, therefore, that such isospin violating $\pi^0 N$ interactions as outlined in reference [@vKNM] are indeed significant.
The parameters $\delta m_N$ and $\bar{\delta}m_N$ are also constrained by the proton-neutron mass difference as $$\Delta_N = m_n - m_p = \delta m_N + \bar{\delta}m_N = 1.29 \ {\rm MeV.}
\label{mn-mp-eq}$$ When our result is combined with equations \[Afb-cont-eq\] and \[mn-mp-eq\], and the values given above for the $\eta NN$ coupling constant and $\pi \eta$ mixing matrix element, we find that $\delta m_N = 1.66 \pm 0.27$ MeV and $\bar{\delta}m_N = -0.36 \pm 0.27$ MeV, assuming no theoretical uncertainties. We emphasize, however, that this last exercise is only meant to illustrate the significance and potential important implications of our result. Further theoretical studies are currently underway [@int-csb] to accommodate simultaneously the new CSB result of our study and that of a recent cross-section measurement of the isospin forbidden reaction $dd \rightarrow \alpha \pi^0$ [@St03].
This work was supported by grants from The Natural Sciences and Engineering Research Council of Canada, The National Science Foundation, and The Ohio Supercomputer Center. TRIUMF is operated under a grant from the National Research Council of Canada.
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**A bridge between liquids and socio-economic systems:**
**the key role of interaction strengths**
**Bertrand M. Roehner $ ^1 $**
**Institute for Theoretical and High Energy Physics**
**University Paris 7**
[**Abstract**]{}One distinctive and pervasive aspect of social systems is the fact that they comprise several kinds of agents. Thus, in order to draw parallels with physical systems one is lead to consider binary (or multi-component) compounds. Recent views about the mixing of liquids in solutions gained from neutron and X-ray scattering show these systems to have a number of similarities with socio-economic systems. It appears that such phenomena as rearrangement of bonds in a solution, gas condensation, selective evaporation of molecules can be transposed in a natural way to socio-economic phenomena. These connections provide a novel perspective for looking at social systems which we illustrate through some examples. For instance, we interpret suicide as an escape phenomenon and in order to test that interpretation we consider social systems characterized by very low levels of social interaction. For those systems suicide rates are found to be 10 to 100 times higher than in the general population. Another interesting parallel concerns the phase transition which occurs when locusts gather together to form swarms which may contain several billion insects. What hinders the thorough investigation of such cases from the standpoint of collective phenomena that we advocate is the lack or inadequacy of statistical data for, up to now, they were collected for completely different purposes. Most essential for further progress are statistics which would permit to estimate the strength of social ties and interactions. Once adequate data become available, rapid advance may be expected.
May 8, 2004
*Preliminary version, comments are welcome*
1: [email protected] Postal address where correspondence should be sent: B. Roehner, LPTHE, University Paris 7, 2 place Jussieu, F-75005 Paris, France. E-mail: [email protected] FAX: 33 1 44 27 79 90
Over the past decade econophysics and sociophysics have been developed by theoretical physicists who mainly came from statistical physics. Recent research in econophysics comprised a large body of empirical inquiries on topics which so far had been largely ignored by economists or sociologists. In addition, a number of theoretical tools developed in polymer physics, spin glass studies, Ising model simulations or discrete scaling were tentatively applied to problems in economics and sociology. This paper develops the idea that there is a connection between some of the phenomena studied in statistical physics and processes which occur in human societies. If persuasive, that argument would strongly support the claim (which is at the core of econophysics) made by physicists that the insight they have gained in studying physical systems can indeed be of value in the social sciences as well. Apart from this broad contention, our investigation will also tell us which phenomena are most likely to provide a good starting point for studying social systems in a fruitful way.
In the course of this paper we will see that it is the liquid state which seems to provide the best bridge to social systems. This is easy to understand intuitively. Crystallized solids have a structure whose regularity and symmetries have no match in social systems. On the other hand gases are characterized by a complete lack of structure which does not match social networks. With their non trivial and adaptive intermolecular interactions, liquids and more specifically solutions offer a better analog to socio-economic systems. Glasses, that is to say solids without crystal structure, could also be possible candidates but in the present paper we restrict our attention to solutions. Subsequently we give other, more technical, arguments in favor of a parallel between solutions and social systems.
Unfortunately, the liquid state is probably the less well understood. It has been suspected for a long time (see for instance Moelwyn-Hughes 1957) that the departure from ideal (or even regular) solution behavior is due to the formation of complex molecular assemblages even for non-ionic solutions. This was the central assumption on which Dolezalek’s theory was based; however it is only in recent decades, that neutron and X-ray scattering as well as infra-red spectroscopy provided a more accurate picture of such molecular clusters. The new picture which progressively emerged from these studies gave us an insight into microscopic mechanisms at molecular level. It is at this level that the parallel with social systems becomes closer and more natural. That is why, throughout this paper, we try to stick to molecular mechanisms and refrain from using such concepts as entropy, energy or temperature which become meaningful only at macroscopic level. So far, these concepts have no clear equivalent in social systems which, by the way, is not surprising for in a similarly way at molecular level the only notions which make sense are those of distance and molecular attraction, stretching and vibrations, molecular assemblage, and so on. In the first part of this paper I describe some physical phenomena involving solutions in terms which can be easily transposed to social systems; in the second, I invite the reader to take the plunge and outline some social parallels.
The paper proceeds as follows. In the second section, I recall that the key variable which accounts for a whole range of phenomena as diverse as boiling temperature, vapor pressure, surface tension, or viscosity is the strength of the intermolecular interaction. This is particularly true in the liquid state. That observation is a strong incentive to develop methods for measuring the strength of social ties. Once the prime importance of intermolecular coupling has been recalled, I describe what I call a paradigm experiment because it provides so to say a blueprint for future studies of social systems. This and similar experiments with solutions suggest that seeing the mixing of two liquids merely from the point of view of entropy as an irreversible operation which increases disorder prevents us from seeing the major role played by amalgamating and combining, two mechanisms which play a key role in both biological and social phenomena. In section 4, I consider the phenomenon of suicide in situations in which one has good reason to expect a low level of social interaction and, accordingly, high suicide rates. Then I devote a few words to social or biological situations which are similar to gas condensation or solvation. Needless to say, each of these phenomena would deserve a more detailed study. Our objective in this paper is to draw a possible agenda for future research rather than to offer detailed case-studies.
**Part I Physical background**
Studying social phenomena is often frustrating because for each law or regularity that one tentatively tries to propose there are usually many exceptions and outliers. The situation is fairly similar in physical chemistry. No model has a broad validity and exceptions abound even for the most basic effects. In that sense physical chemists are certainly better prepared to cope with social systems than for instance particle physicists or solid state physicists. Our background presentation in the first part is entirely based on experimental evidence; the main reason for avoiding theoretical concepts is the fact that they cannot be easily adapted to social systems. The discussions I have had with the colleagues in my lab convinced me that even experienced theoretical physicists may not necessarily be familiar with those facts and interpretations; however, this first part may be safely skipped by physical chemists.
In statistical physics we know that, at least in principle, the properties of a system may be derived from its Hamiltonian. However, for systems like liquids this can only be done with great difficulty and often only numerically. The point we want to make in this section is much simpler. We show that the behavior of a system to a large extent depends only on the [*strength*]{} of the interaction; its precise form, whether it is a coupling between ions, permanent dipoles, induced dipoles or a mix of those interactions does not really matter from an experimental point of view. In what follows the interaction strength will be our key parameter. It has by the way a simple geometrical interpretation in the sense that there is a close relationship between interaction strength and length of intermolecular bonds.
Before looking at more elaborate properties we first consider the state, whether solid, liquid or gas, in which a compound is to be found at room temperature. At first sight our claim that it is determined by the coupling strength could seem utterly wrong. Compare for instance $ \hbox{H}_{\hbox{2}}\qb{O} $ and $ \qb{H}_{\qb{2}}\qb{S} $; while the first is a liquid, the second is a gas (boiling temperature = $ -60 \qd $) Yet, these molecules could be expected to be fairly similar for sulfur and oxygen are in the same column of Mendeleev’s periodic table. Perhaps an even more striking apparent counter-example is ethanol and methyl ether which both correspond to the formula $ \qb{C}_{\qb{2}}\qb{H}_{\qb{6}}\qb{O} $. Whereas the first is a liquid (boiling point = $ 78 \qd $), the second is a gas (bp=$-24 \qd $). Thanks to the thorough and patient work of physical chemists, we now know that the interactions between $ \hbox{H}_{\hbox{2}}\qb{O} $ molecules is indeed much stronger than the one between $ \qb{H}_{\qb{2}}\qb{S} $ molecules and similarly for ethanol and methyl ether molecules. It would take us too far away from the main purpose of this paper to explain the reasons of these differences. The main point we want to emphasize is that, in contrast to mainstream ideas in the 1910s (see for instance Holmes 1913) overall properties such a size, mass or composition of molecules turned out to be completely inadequate to explain their properties; that could be done only once experimental methods had been developed which could provide reliable estimates of intermolecular coupling.
Building on this knowledge we present in Fig.1a some data which illustrate the key role of the coupling strength.
[**Fig.1a: Boiling temperature as a function of intermolecular attraction**]{}. [For alkanes $ \qb{C}_n\qb{H}_{2n+2} $ with a linear chain, which are represented by dots surrounded by a square, the inter-molecular attraction is proportional to the number of the hydrogen atoms and hence also to the molecular weight $ M=14n+2 $. The trend portrayed by the solid line means that for longer carbon chains more thermal agitation is required in order to break the intermolecular bonds. The dots represent hydrocarbons $ \qb{C}_n\qb{H}_{p} $ whose intermolecular forces, are slightly different due for instance to branched carbon chain which results in boiling temperature differences of the order of 10%. The stars correspond to compounds whose molecular coupling are of a different nature, either much weaker (argon) or much stronger (ammonia, ethanol, water)]{}. [*Source: Lide (2001)*]{}.
Apart from the outliers to which we come back below, the graph concerns only hydrocarbons and more particularly alkanes: $ \qb{C}_n\qb{H}_{2n+2} $. As is well known, alkane molecules interact only trough dipole-induced forces, the so-called London dispersion forces. These fairly weak forces exist between any pair of atoms. As a result the interaction between two alkane molecules is basically proportional to the length of the carbon chain that is to say to the number $ n $ or in other words to the molecular weight of the alkane. The square correspond to experimental data for linear alkane chains, whereas the dots correspond to other hydrocarbons. Of particular interest are the dots which correspond to isomers (same molecular weight) of a given alkane. These isomers have ramified carbon chains, a feature which to some extent changes the London forces between the molecules and results in differences of the order of 10% (when temperatures are expressed in Kelvin degrees). The stars show a number of cases characterized by different kinds of interactions. As we know both water and ethanol, $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{2}} \ql \qb{O}\qb{H} $ have a dipole $ \qb{O}_{-}\qb{H}_{+} $ which causes a fairly strong interaction trough so-called hydrogen bonds. At the bottom of the graph the single atoms of argon have almost no interaction at all which results in a very low boiling point close to $ -200 \qd $. Incidentally, it can be observed that there is a close relationship between bond strength and bond length. For instance if we assume a potential corresponding to a ion-ion attraction and a hard core repulsive force proportional to $ 1/ r^p,\ (p\sim 9) $ the strength $ s $ of the bond is related to the distance $ R $ between the two ions by the relation: $$s=\left. { d^2 V \over dr^2 }\right|_{r=R}
= (p-1){ e^2 \over 4\pi \epsilon _0 }{1 \over R^3 }$$
In order to show that the previous argument extends to many other physical properties we show in Fig.1b that the enthalpy of vaporization and the viscosity of alkanes is again determined by the strength of the intermolecular forces that is to say in this case by the molecular weight. As it would be pointless to compare the viscosity of gases with that of liquids we restricted the latter curve to the alkanes which are liquid at room temperature.
[**Fig.1b: Latent heat of vaporization and viscosity as a function of inter-molecular attraction for alkanes**]{}. [As explained in Fig.1a, there is a direct relationship between attraction strength and molecular weight. The solid line corresponds to the 20 first alkanes (except $ n=15 $ which is missing in data tables); it describes the empirical relationship: $ L_s (\qb{C}_n\qb{H}_{2n+2})=
1.1+1.7 n $. The broken line represents the viscosity; it is restricted to the alkanes which are liquid at room temperature, namely $ n=7,\ldots ,16 $ ($ n=15 $ is again missing)]{}. [*Sources: Lide (2001), Moelwyn-Hughes (1961, p.702)*]{}.
The relationships displayed by the curves in Fig.1a,b have a clear intuitive interpretation. The stronger the interaction, the better the molecules are held together and the more kinetic energy it takes to disrupt the molecular assemblages that make up solids or liquids. In the same way, in a liquid with a strong interaction only the fastest molecules will be able to escape which translates into a low vapor pressure. As to viscosity, a strong interaction will make neighboring layers to stick more closely together which for liquids results in a higher viscosity.
In order to test this way of reasoning let us see if we can use it in order to predict the relationship between interaction strength and other physical properties such as for instance the speed of sound. For sound to propagate, successive layers must be put into motion. Due to inertia, in order to put one layer into motion the main factor is the weight of the molecule. If there are strong bonds between molecules two things will happen. In a given layer, the inertia effect will be increased because strongly coupled molecules will somewhat behave as a cluster of molecules that is to say a super molecule of greater molecular weight. The transmission of the perturbation from one layer to the next will be facilitated.
Which one of these effects will prevail is not obvious. Observation shows that the first effect prevails in gases. Thus, by comparing the speed of sound in methane, $ \qb{C}\qb{H}_{\qb{4}} $, and propane, $ \qb{C}_{\qb{3}}\qb{H}_{\qb{8}} $, we see that it is smaller in propane even after the factor $ \sqrt{M} $ due to the molecular weight has been corrected for. In other words, for gases the speed of sound decreases with stronger interactions. On the contrary, in liquids and solids it is the second effect which prevails as illustrated by the two following examples. (i) The speed of sound in pentane, $ \qb{C}_{\qb{5}}\qb{H}_{\qb{12}} $ is 1012 m/s (at $ 25 \qd $ and 1 bar); thus, on account of its higher molecular mass one would expect a smaller velocity for heptane, $ \qb{C}_{\qb{7}}\qb{H}_{\qb{16}} $, yet it is higher at 1129 m/s. (ii) As one knows, diamond, a solid with very strong interatomic bonds has a velocity of sound of 12,000 m/s, one of the highest to be observed in any substance.
The previous discussion shows that even in cases where the physical consequences of a strong interaction are less transparent than in the cases of Fig.1a,b, this factor nevertheless plays an essential role. In the next section we examine the role of interaction strengths in the mixing of two liquids.
The mixing of two liquids is described in Fig.2a.
.
To call this a paradigm experiment could seem an inflated expression for such a modest experiment. Nevertheless, we will see that it carries a number of important ideas. Accurate experiments of this kind were carried out in the early 20th century in particular by the German physicist Emil Bose (1907). The experiment described in Fig.2a does not aim at precision, its objective is rather to give an intuitive feeling of the phenomenon. The upper line in Fig.2a shows that after being mixed the volume of a solution of water and ethanol, $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{2}} \ql \qb{O}\qb{H} $ decreases. The experiment provides a rough estimate of the contraction which is of the order of 2%. The lower line shows that the mixing phenomenon is exothermic. Naturally, the quantity of heat which is released would not be the same for another alcohol such as methanol, $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{O}\qb{H} $, or propanol, $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{2}} \ql \qb{C}\qb{H}_{\qb{2}}
\ql \qb{O}\qb{H} $. A more detailed picture is given in Fig 2b.
[**Fig. 2b: Heats of mixing: water - alcohols**]{}. [Methanol, ethanol and propanol are the first three alcohols: $ \qb{C}_n\qb{H}_{2n+1}\qb{OH},\ n=1,2,3 $. Usually, for instance for acetone-chloroform or methanol-ethanol, the corresponding curves are symmetrical with respect to molar concentration (as indicated by the thin line curve); this points to a a connection between the shape of the curves and the structure of molecular assemblages: the dissymmetry shows that several water molecules surround each alcohol molecule. Note that when the proportion of water becomes too small, the mixing with propanol becomes endothermic.]{} [*Source: Bose (1907), Landolt-Börnstein (1976).*]{}.
Bose’s results furthermore show that when the experiment is carried out at $ 43 \qd $ the mixing becomes endothermic as soon as the molar proportion of propanol becomes higher than 15%.
Having presented the facts, let us now see what can be learned from them and why this experiment is of interest for social phenomena. We will proceed from macroscopic to microscopic level. From what has been said in section 1, it is obvious that the contraction and temperature increase are related. The contraction shows that intermolecular attraction in the solution is on average stronger than in the pure compounds. As the molecules rearrange themselves in line with the new interactions, energy is released in the same way as when an expanded spring returns to its equilibrium length. What has thermodynamic to say about the mixing of liquids? Because the mixing is exothermic we know that the solution is not an ideal solution. The heat of mixing is given in terms of partial pressures by a formula first proposed by Nernst (Bose 1907, p. 621): $$Q_{\hbox{mix}} = -RT^2 { d \over dT} \left[ x\log{ p_a \over p'_a }
+ (1-x)\log{p_b \over p'_b } \right] \qn{3.1}$$
where $ T $ denotes the Kelvin temperature, $ x $ the molar proportion of water, $ p_a,\ p'_a $ the pressure of vapor over pure water and over the solution respectively and $ p_b,\ p'_b $ the same partial pressures for alcohol. When Raoult’s law applies $ p'_a = xp_a $ and similarly $ p'_b = (1-x)p_b $ which means that the terms between square brackets become independent of $ T $ and, as a result, $ Q_{\hbox{mix}} =0 $. This is the ideal solution case. For non-ideal solutions, the application of the above formula requires detailed input information about how the partial pressures depend upon temperature. An interesting question is to understand the location of the peaks in Fig.2b. Why, for instance is $ Q_{\hbox{mix}} $ maximum for a proportion of 5 molecules of water for one molecule of ethanol? This leads us to examine the phenomenon at molecular level. Thanks to the extensive work done by physical chemists we now have a better understanding of the mixing of liquids at molecular level. Of cardinal importance in the case of ethanol and water is the fact that the ethanol molecule $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{2}} \ql \qb{O}\qb{H} $ comprises two sections which react to water molecules in very different ways. The $ \qb{OH} $ segment holds a dipole $ \qb{O(-)H(+)} $ which can link up with the $ \qb{O(-)H(+)} $ dipoles of the water molecules. The segment $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{2}} $, on the other hand, is similar to ethane $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{3}} $ and we know that ethane like any alkane is miscible in water in very small proportion only. As a result, this segment could seem to be irrelevant as far as the mixing with water is concerned. This, however, would be a simplistic view. Neutron and X-ray scattering experiments have shown that water molecules form a kind of net around alkane molecules in solution (Schmid 2001, Baumert et al. 2003). The molecular assemblage between water and ethanol molecules is described schematically in Fig. 3.
. [In contrast to methane $ \qb{C}\qb{H}_{\qb{4}} $ which features only weak London dispersion forces, the molecule of ethanol comprises two segments (i) the alkane-like segment $ \qb{C}\qb{H}_{\qb{3}} \ql \qb{C}\qb{H}_{\qb{2}} $ (ii) the water-like end $ \qb{OH} $. For that reason, ethanol displays a dual behavior: like methane, it attracts an hydration shell of water and like water it forms strong hydrogen bonds. According to some recent studies (Dill et al. 2003, p. 581) there may be as many as 17 water molecules in the first hydration shell. The precise shape of the molecular assemblage is of little importance for the purpose of this paper; what matters is the fact that it is a highly ordered arrangement]{}. [*Sources: Baumert et al (2003), Dill et al. (2003), Dixit et al. (2002), Guo et al. (2003), Israelachvili et al. (1996)*]{}.
The solid lines represent the dipole-dipole bonds while the dashed lines represent the weaker bonds between induced dipoles. One should bear in mind that these bonds and indeed the whole assemblage are rearranged on a picosecond time scale. In other words any representation such as the one in Fig. 3 can be nothing but an average view of a rapidly changing structure. In dynamical terms the mixing can be described by the following reaction: $$x\qb{H}_{\qb{2}}\qb{O} + y\qb{C}_{\qb{2}}\qb{H}_{\qb{5}}\qb{O} \
\qh{\longrightarrow}{\longleftarrow} \
\left( x\qb{H}_{\qb{2}}\qb{O}, y\qb{C}_{\qb{2}}\qb{H}_{\qb{5}} \qb{O} \right)
+ Q_{\hbox{mix}}$$
where, according to Fig. 3, $ x $ is of the order of 4-5 and $ y $ of the order of one. Specific details, and in particular the exact values of $ x $ and $ y $ are anyway unimportant for the present discussion. The central point is that the mixing creates a new molecular assemblage and that this structural rearrangement basically results in an entropy [*reduction*]{}. Furthermore, it is of interest to observe that while the solution has a greater cohesion than the initial compounds, it is also characterized by a greater molecular agitation. The simultaneous occurrence of higher cohesion and increased agitation is also observed in biological and social systems. It is particularly spectacular in the formation of swarms of locusts which we briefly discuss below.
We said that Fig. 3 is based on scattering experiments which permit direct observation. However, it can also be justified by indirect arguments. We mention that way of reasoning because it was the main approach in use during several decades between 1905 and 1945. By measuring specific physical variables for various compounds and by comparing these observations, physical chemists tried to gain a better understanding of the processes taking place at molecular level (see for instance Holmes 1913 or Earp and Gladsstone 1935). In the present case, this kind of reasoning can be illustrated as follows. In order to see if our point about the roles of the two segments in the ethanol molecule is correct, it is natural to examine what happens when one of the segments becomes preponderant. When the hydrocarbon chain becomes longer one would expect the alkane character of the molecule to become more pronounced. This is indeed confirmed by comparative observation for ascending members of the alcohol family, $ \qb{C}_n\qb{H}_{2n+2}\qb{O} $. For $ n=1,2,3 $ the alcohols are soluble in water in all proportion which indicates the formation of strong links. However, starting with $ n=5 $, the solubility is greatly reduced which shows that the alkane character becomes predominant; for $ n=4,5,6 $ solubility is 0.11 g/ mole, 0.03g/mole and 0.06g/mole respectively. The solubility of the $ n=6 $ alcohol, namely hexanol, is in fact of the same order of magnitude as the solubility of the corresponding alcane, namely hexane, which is 0.001 g/mole. The alkane-type behavior of high order alcohols is also observed at the level of boiling temperatures. Whereas there is a huge difference of about $ 220 \qdk $ between the boiling temperatures of methanol and ethane, boiling temperatures of higher order alcohols tend asymptotically toward those of the alkanes; thus for $ n=10 $ the difference is reduced to less than $ 50 \qdk $. Conversely a behavior which becomes close to that of water is observed when the role of the OH segment becomes predominant. Whereas for $ n \geq 2 $ all alcohols are soluble in hexane, methanol ($ n=1 $) is only slightly soluble at 0.12g/mole. Similarly the role of the OH segment can be expected to be enhanced in di-alcohols or tri-alcohols, that is to say molecules that contain two or three OH segments. Thus glycerol, $$\qb{H}_{\qb{2}}\!\!\! \qh{\qb{C}}{\qh{|}{\qb{OH}}} \ql
\qh{\qb{C}}{\qh{|}{\qb{OH}}}\!\!\! \qb{H} \ \ \ql
\qh{\qb{C}}{\qh{|}{\qb{OH}}}\!\!\! \qb{H}_{\qb{2}}$$
is soluble in water in all proportions but is almost not soluble in hexane .
Attempts to bridge the gap between statistical physics and socio-economic phenomena usually come up against two difficulties. The first one, which is not often mentioned, is the ergodic hypothesis to which we come back later. The second is the fact that the notions of entropy, energy or temperature which are so central in physics have no obvious counterpart in social phenomena. That is why we carefully avoided using these notions. All the mechanisms described in this first part can be transposed to biological or social phenomena. This is the purpose of the second part of the paper.
**Part IISocial phenomena in the light of physics**
The main challenge in this part is to identify those (if any) social phenomena which can be better understood in the light of the notions presented in the first part. Needless to say, many kinds of social phenomena do not fall into this category. For instance, cultural or gender studies draw on notions which have no parallels in physics. There are however many important socio-economic phenomena which can be interpreted along the lines used in Part I. One can mention the following. In the first part we emphasized the connection between interaction strength and molecular rates of escape from a liquid. Any system whose members are held together by some cohesion forces but may occasionally escape from the system would provide a possible parallel. Table 1 provides a number of examples.
**Table 1 Retention dynamics in various institutions**
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$$\matrix{
\tvi
\hbox{Institution} \hfill &
\hbox{Intra-institutional bonds} \hfill & \hbox{Type of escape} \hfill \cr
\noalign{\hrule}
\qth
\hbox{High school,} \hfill &
\hbox{Links with students, teachers, professors;} \hfill & \hbox{Dropout} \hfill \cr
\hbox{college, university} \hfill &
\hbox{attraction of qualified jobs} \hfill & \hbox{} \hfill \cr
\hbox{} \hfill & \hbox{} \hfill & \hbox{} \hfill \cr
\hbox{Faith community,} \hfill &
\hbox{Links with rest of congregation,} \hfill &
\hbox{Decline in attendance} \hfill \cr
\hbox{religious order} \hfill &
\hbox{common faith} \hfill & \hbox{} \hfill \cr
\hbox{} \hfill & \hbox{} \hfill & \hbox{} \hfill \cr
\hbox{Army} \hfill &
\hbox{Patriotism, discipline, renumeration} \hfill & \hbox{Desertion} \hfill \cr
\hbox{} \hfill & \hbox{} \hfill & \hbox{} \hfill \cr
\hbox{Nation} \hfill &
\hbox{Family ties,} \hfill & \hbox{Immigration} \hfill \cr
\hbox{} \hfill & \hbox{attraction of home country} \hfill & \hbox{} \hfill \cr
\hbox{} \hfill & \hbox{} \hfill & \hbox{} \hfill \cr
\hbox{Society} \hfill &
\qtb
\hbox{Family ties, links with friends} \hfill & \hbox{Suicide} \hfill \cr
\noalign{\hrule}
}$$
1.5mm Notes: The fact that one often observes a conjunction of substantial high school dropout rates with high suicide rates among teens seems to show that the interactions which account for these effects overlap to some extent. As an illustration, for the Oglala Sioux who live on Pine Ridge Reservation, South Dakota, dropout and suicide rates among teens are 6 and 4 times higher respectively than in the general population; for American Indians overall, the dropout and teen suicide rates are 35 percent and 37 per 100,000, respectively 3 and 2.5 times higher than in the general population. In the second column we attempted to list some of the possible bonds that keep an individual attached to a given institution. This list, however, is more based on common sense than on genuine measurements. As a matter of fact, we do not yet know what is the respective importance of these links. For instance, we know that family ties are important in suicide, but we do not have a clear picture of the respective role of short-range versus long-range ties. What makes reliable measurements difficult is the fact that the level of exogenous shocks (which represent thermal agitation) is usually time-dependent and has therefore to be controlled for. Sources: Reyhner (1992), Olson (2003), http://www.re-member.org
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For instance, if the system is a college or a university the retention rate of freshmen provides an indication about the balance between group cohesion and centrifugal forces. If the system is an army, the desertion or AWOL (Absent WithOut Leave) rates provide a global estimate of the resultant of many forces such as for instance patriotism, fear of being punished, conservation instinct, and so on. All the cases mentioned in table 1 would provide interesting testing fields for the interpretation that we advocated. Unfortunately, for most of them only scarce or fragmentary data are available. By a slight but natural extension it is possible to include suicide in the present category. The likelihood of not committing suicide represents a kind of retention rate. The main incentive for including suicide is the fact that in this case at least there are numerous statistical data. Naturally, suicide has been studied by sociologists for decades without any reference to physics; one may therefore wonder what difference it makes to adopt the present perspective. Instead of analyzing suicide statistics almost indiscriminately, the present perspective leads us to focus on situations where social ties are either very strong or very weak. By so doing we will be in a better position to grasp the key mechanisms (as opposed to incidental circumstances) of the phenomenon. The next section provides an introduction to this approach. The second physical phenomenon for which there are some natural biological and social parallels is the condensation of a gas, that is to say the transition from a state in which the molecules have low interactions to a state where they form an entity characterized by a substantial interaction and cohesion. Macromolecules, bacterias, protozoa, insects, animals or humans in certain conditions display a tendency to self-aggregation. Instead of considering each of these cases as separate it may help our understanding to look at them from a unified standpoint. We briefly discuss two cases which belong to this category of phenomena. The third physical phenomenon for which there is a natural sociological extension is the mixing of to liquids. Amalgamation of different populations is a mechanism of fundamental importance. Under that heading one can consider the amalgamation of populations of peasants, merchants and craftsmen. Through the links of cooperation and exchange that they establish, cohesion and productivity are greatly enhanced. Another important mechanism of amalgamation is the so-called melting pot mechanism by which a group of immigrants becomes integrated. Again, one may ask what benefit can be gained from considering these phenomena from the standpoint of statistical mechanics. In a physical solution the new bonds between solute and solvent are established in a matter of seconds if the solution is mixed up by an external device, but it will take much longer if one has to rely on diffusion for the mixing process. Similarly, the time scale required by the amalgamation process very much depends upon the magnitude of the “mixing”. It may take one or two generations in a city, but much longer in a mountainous region where population density is low and contacts are rare. This parallel shows that in order to understand the dynamics of bond formation one must adopt an adequate time scale. For urban integration, 50 years may be an acceptable time period, whereas for low density regions two or three centuries would be more suitable. In short, through the analogy with physical phenomena we get a better understanding of how to set up the inquiry.
In the late 19th century there have been numerous studies about suicide in all European countries. The following references (arranged in chronological order) constitute a select sample of the publications of that period, along with some more recent ones: Boismont (1865), LeRoy (1870), Cristau (1874), Morselli (1879), Legoyt (1881), Masarick (1881), Nagle (1882), Durkheim (1897), Krose (1906), Bayet (1922), Douglas (1967), Baudelot et al. (1984). All these studies of course took advantage of the fact that thanks to the development of census offices extensive demographical statistics became available in all industrialized countries. Among the aforementioned authors, the contribution of Emile Durkheim stands out because, in contrast to most other authors, he was not interested in why individual people commit suicide but from the start considered suicide as a social phenomenon. In the very first sections of his book, he makes clear that to understand suicide one should examine the web of connections and affiliations each individual has with the people around him. For Durkheim it is the failure of family, church, community of neighbors to provide effective forces of social integration which is at the heart of the problem. In short, Durkheim’s perspective is very close to the standpoint of statistical physics that we presented in part I. Unfortunately, his message has been largely discarded and forgotten, to the point that nowadays most studies center on individual psychological causes.
In support of his thesis Durkheim presents a great wealth of data for many different countries. However, for his argument to become really compelling and conclusive one would need a way to measure the strength of social ties in an objective and quantitative way. Instead Durkheim relies on common sense and intuition with the result that his proofs remain somewhat tautological. For instance, even if it is natural to admit that bachelors have fewer social (and especially family) ties than people who are married with several children, estimates based on an objective criterion would be needed. Otherwise the observation that suicide rates are higher among bachelors cannot be quite conclusive. We must confess that our own methodology will have the same defect, only to some extent mitigated by the fact that the situations that we consider are so extreme that in order to make sense our “common sense” estimates need only to have the right order of magnitude. This is why we focus on situations characterized by low levels of interaction. In the following we consider three situations of that kind.
Schizophrenia is a severe mental illness characterized by a variety of symptoms including lost of contact with reality and social withdrawal. People with schizophrenia may avoid others or act as though others do not exist; for example they may avoid eye contact with others or may lack interest in participating in group activities. Clearly this is a situation where interpersonal links are severely weakened. It turns out that suicide rates among people with schizophrenia are 10 to 15 times higher than in the general population: a typical figure is 200 per 100,000 as compared to 15 per 100,000 in the general population.
Persons who are arrested and jailed see links with family, friends, colleagues or neighbors suddenly severed. Of course, once in jail for some time, inmates are likely to build new ties for instance with other inmates, guardians, lawyers, chaplains or other persons who may assist them. One would expect, therefore, that it is in the first days in jail that the disaggregation of social ties is the most severely felt. This prediction is matched by observation. Indeed, it turns out that suicide rates are particularly high during the first few days in jail. A study performed in 1986 about jail suicide in the US found that 51% of the suicides which occur in jail (as opposed to prison which in the US designates facilities for stays of over one year) happen in the first 24 hours of incarceration. Thanks to official data which are available on the Internet for New York State (New York State 1998: Crime and Justice Annual Report, http://criminaljustice. state.ny.us) we are able to compute an order of magnitude of the suicide rate in short-term detention facilities technically known as “lockups” where detainees usually stay for less than 72 hours before being transferred to county jails. The reasoning goes as follows. On a single day of 1998 the average number of detainees in lockups was 473 (151 for New York City and 322 for upstate New York). Naturally, these detainees were not the same throughout but this is irrelevant for the present calculation. Over the whole year there were 6 suicides (2 in New York City and 4 upstate) which gives a rate of $ 6/473 = 1268 \hbox{ per } 10^5 $. If the same calculation is done for each year between 1990 and 1999 one gets an average suicide rate of $ 903 $ per $ 10^5 $. Because, a great majority of inmates are males, this figure should be compared to the suicide rate of men in the general population of New York State which for the period 1990-1998 was $ 13.0 $ per $ 10^5 $. The suicide rate in the first 6 days of detention was therefore 69 times higher, This order of magnitude is consisted with results obtained by other studies which analyzed suicide rate in the first days of detention (table 2). As detainees form new links in jail, the suicide rate progressively declines. In county jail where inmates usually stay for periods of less than one year, the rate is about 10 times higher than in the general population. In state prisons, where inmates stay for periods of more than one year, the rate is almost the same as in the general male population.
Table 2 provides also data for some other countries. These data do not distinguish between short-term and long-term facilities. Most of the figures are between 100 and 200 which is consistent with the rates observed in US county jails.
**Table 2 Suicide rates among inmates**
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$$\matrix{
\tvi
&\hbox{Type of} \hfill &
\hbox{Time elapsed} \hfill & \hbox{Location} \hfill & \hbox{Time} &
\hbox{Suicide} \cr
&\hbox{institution} \hfill &
\hbox{since} \hfill & \hbox{} \hfill & \hbox{interval} &
\hbox{rate} \cr
\qtb
&\hbox{} \hfill &
\hbox{incarceration} \ (T) \hfill & \hbox{} \hfill & \hbox{} &
\hbox{[per 100,000]} \cr
\noalign{\hrule}
\qth
1&\hbox{Lockup} \hfill & T< 72\ \hbox{hours} \hfill &
\hbox{New York State} \hfill & 1990-1999 & 900 \cr
2&\hbox{Lockup} \hfill & T< 72\ \hbox{hours} \hfill &
\hbox{South Dakota} \hfill & 1984 & 2975 \cr
3&\hbox{Jail} \hfill & 72\ \hbox{hours}< T< 1\ \hbox{year}\hfill &
\hbox{Texas} \hfill & 1981& 137 \cr
4&\hbox{Jail} \hfill & 72\ \hbox{hours}< T< 1\ \hbox{year} \hfill &
\hbox{South Carolina} \hfill & 1984& 166\cr
5&\hbox{Jail} \hfill & 72\ \hbox{hours}< T< 1\ \hbox{year} \hfill &
\hbox{US} \hfill & 1986 & 107 \cr
6&\hbox{Jail} \hfill & 72\ \hbox{hours}< T< 1\ \hbox{year} \hfill &
\hbox{New York State} \hfill & 1986-1987 & 112\cr
7&\hbox{Prison} \hfill & 1\ \hbox{year} < T \hfill &
\hbox{US} \hfill & 1984-1993 & 21 \cr
\hbox{} \hfill & \hfill & \hbox{} \hfill & & \cr
8&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{Belgium} \hfill & 1872 & 190\cr
9&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{England} \hfill & 1872 & 112 \cr
10&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{Saxony} \hfill & 1872& 860\cr
11&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{Canada} \hfill & 1984-1992 & 125 \cr
12&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{New Zealand} \hfill & 1988-2002 & 123 \cr
13&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{England} \hfill & 1990-2000 & 112\cr
14&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{France} \hfill & 1991-1992 & 158 \cr
15&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{Australia} \hfill & 1997-1999 & 175\cr
16&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{Canada} \hfill & 1997-2001 & 102 \cr
17&\hbox{Not spec.} \hfill & \hbox{Not spec.} \hfill &
\hbox{Scotland} \hfill & 1997-2001 & 227 \cr
\qtb
&\hbox{\bf Average (8-17)} \hfill & \hbox{} \hfill &
\hbox{} \hfill & & \hbox{\bf 218}\cr
\noalign{\hrule}
}$$
1.5mm Notes: As a useful yardstick one can use the suicide rate among males in the United States between 1979 and 1998 which was about 20 per 100,000. Suicide rates of inmates are highly dependent upon the time they have spent in prison since their incarceration. A detailed study based on 339 suicides that occurred in the US in 1986 found that 51 percent of the suicides occurred in the first 24 hours of incarceration. This observation is consistent with the interpretation of suicide as resulting from a severing of social ties. In the statistics published in other countries than the United States, the time of incarceration is not specified. However, since inmates incarcerated for less than one year are in greater number than those incarcerated for longer durations, one would expect the former to predominate. Therefore it is not surprising that the order of magnitude of suicide rates is more or less the same everywhere (one exception is Saxony). Sources: 1: DCJS Report (tables 7,9,11); 2: Hayes et al. (p. 4); 3,4,5: Hayes et al. (p. 52-53); 6: DCJS Report (table 1), Hayes et al. (table 2); 7: http://www.mces.org/Suicide\_ Prisons\_ Jails.html; 8-10: Legoyt; 11: Correctional Service; 12: Corrections Department, http://www.corrections.govt.nz; 13: Her Majesty Prison Service; 14: Baron-Laforet, Bourgoin; 15-17: same as 12.
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How accurate and reliable are the data given in table 2? This is certainly an essential question. An official report (Hayes and Rowan 1998) found under-reporting of jail suicide in 1986 to be of the order 40 percent nationally, but with great differences between states. Thus, in New York State no under-reporting was identified (which is why we selected this state to compute the previous estimate), whereas in Alabama, Louisiana, Pennsylvania or Tennessee under-reporting was over 50 percent. As there are no reasons for and indeed no mention of over-reporting we can at least be assured that the figures which are made public provide trustworthy lower bounds.
In conclusion, the phenomena of suicide among people with schizophrenia or inmates seem to provide spectacular illustrations of the effect of a weakening of social ties on suicide (escape) rates. However these situations may be seen with good reason as somewhat artificial in the sense that these groups are subject to illness or special living conditions imposed from outside. This is why we now turn to situations which can be considered as more “natural” in the sense that they concern entire societies and occur every time “traditional” societies come into contact with societies which are technically more advanced.
Every time the social framework of a society undergoes overwhelming changes there is a time of transition during which the old structures no longer work or exist and those which are better adapted to the new situation have not yet emerged. As a result, one would expect such periods of transition to be characterized by a low level of social interaction. When two liquids mix the time it takes for the pattern of forces to rearrange and for the molecular structure to be reordered is probably to be counted in microseconds at the molecular level and in seconds at macroscopic level (provided the two liquids get mixed). In a society the transition may take decades. The figures in table 3 show that these situations are characterized by a substantial increase in suicide rates up to levels which are 3 to 4 times higher than in stable societies. These phenomena have many facets: familial, communal, demographic, economic, political, etc. In the rest of this section we focus our attention on the case of Micronesia for which one has fairly good statistical data.
**Table 3 Suicide rates in populations in a state of transition**
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$$\matrix{
\tvi
&\hbox{Population} \hfill &
\hbox{Gender or age} \hfill & \hbox{Location} \hfill & \hbox{Time} &
\hbox{Suicide} \cr
&\hbox{} \hfill &
\hbox{specification} \hfill & \hbox{} \hfill & \hbox{interval} &
\hbox{rate} \cr
\qtb
&\hbox{} \hfill &
\hbox{} \hfill & \hbox{} \hfill & \hbox{} &
\hbox{[per 100,000]} \cr
\noalign{\hrule}
\qth
1&\hbox{Blackfoot} \hfill & \hfill &
\hbox{U.S.} \hfill & 1960-1969& 130\cr
2&\hbox{Cheyenne} \hfill & \hfill &
\hbox{U.S.} \hfill & 1960-1968 & 48 \cr
3&\hbox{Papago} \hfill & \hfill &
\hbox{U.S.} \hfill & 1960-1970& 100 \cr
4&\hbox{Indians} \hfill & 10-19 &
\hbox{Canada} \hfill & 1986-1990 & 65\cr
5&\hbox{Natives} \hfill & \hbox{Male} &
\hbox{Micronesia} \hfill & 1975-1990 & 50 \cr
6&\hbox{Natives} \hfill & \hbox{Male}, 15-24 &
\hbox{Micronesia} \hfill & 1978-1987 & 129 \cr
\qtb
7&\hbox{Natives} \hfill & \hbox{Male}, 15-24 &
\hbox{Chuuk Islands} \hfill & 1978-1987 & 200 \cr
\noalign{\hrule}
}$$
1.5mm Notes: As a matter of comparison the average suicide rate in the United States over the period 1979-1998 was 12.2 per 100,000 for both genders, 19.5 per 100,000 for males and 20.7 for males aged 15-24. The area considered in cases 5 and 6 comprises the Federal State of Micronesia (Chuuk, Kosrae, Pohnpei, Yap), the Marshall Islands and Palau. Blackfoot, Cheyenne and Papago are three tribes of American Indians. Basically, either for North American Indians or in Micronesia the suicide rate of young adults is at least 4 times higher than in the general population. Sources: 1-3: Lester; 4: http://www.hc-sc.gc.ca; 5-7: Rubinstein (2002)
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Micronesia as it is defined by Donald Rubinstein (1994, 2002) from which we borrow most of the following information comprises the Marshall Islands, the Carolinas which now form the Federated States of Micronesia and the Northern Marianas. Most of these islands were occupied by Japan in 1914 and some of them were colonized by Japanese farmers. After World War II they became American Territories until they acceded to some form of autonomy in the late 1980s. In the 1950s and 1960s nuclear tests and missile tests were conducted at Bikini, Eniwetok and Kwajalein located in the Marshall Islands. While alcohol consumption by the islanders had been regulated or prohibited under Japanese rule, these restrictions were lifted in the 1960s. Moreover, thousands of US Peace Corps Volunteers arrived, schools were built in every island and the economy began to shift from a subsistence economy based on family gardening and fishing to an economy based on imported products and wage labor. Men played a central role in these two activities and their place in society was more affected than women’s activities which centered around preparing food and taking care of the house and children. Yet, on Fig. 4 it can be seen that suicide rates increased for men as well as for women even if the latter did not exceed the level observed in industrialized countries.
How can we integrate and interpret these various changes in terms of social interaction? As we already noted, to do this in a satisfactory way would require either quantitative information about the frequency and intensity of interpersonal contacts and links, or a methodology (an equivalent to infra-red spectroscopy) that would enable us to estimate interaction strengths. However, we can make two observations which are fairly revealing at least quantitatively. If interaction strength really plays a key role in this phenomenon, one would predict that the individuals who are most at risk are those who do not have strong family connections. That is true for suicide in a general way, but one would expect the effect to be much stronger in a situation where community bonds have been weakened. In such a situation the transition period between childhood, characterized by strong ties with parents, and adulthood, characterized by strong ties with one’s own wife and children, is a particularly critical moment. In short, one would expect a high suicide rate for young adults. This is indeed what observation shows (table 3).
[**Fig.4: Suicide rate in Micronesia**]{}. [The area covered by these data comprises the island of Palau, the Federated States of Micronesia (which includes the Chuuk islands) and the Marshall Islands. Concomitantly with the shift from a subsistence economy to one based on imported food and wage labor, there has been a huge increase in suicide rates for both men and women. The dotted horizontal lines show the American average suicide rates for men and women. The insert shows that it is the 15-24 age group which has by far the highest rate, a pattern which strongly differs from the age pattern in industrialized countries where it suicide rate increases with age. It is of interest to note that in Micronesia the high school dropout rate is around 40 percent that is to say four times more than in the United States]{}. [*Source: Rubinstein (1994, 2002), Hezel (2001)*]{}.
This effect is expected in any society in transition, but in Micronesia it is to some extent amplified because of two local factors. As it is (or at least was) considered taboo for a sexually mature boy to sleep in the same house as his sisters, teenagers used to move to community men’s houses which they would share with extended family relatives. But in many islands these community houses are no longer kept up with the result that young men do not have places where they can conveniently stay at this critical juncture. The second aggravating circumstance is related to the custom of adoption. In Micronesia, adoption was a widespread custom and a central pillar of family life. Like godfathers or godmothers but in a much stronger sense, the adoptive parents were a major constituent of the enlarged family. However, this institution has been imperiled both by the decay of traditional culture and by the rapid population growth. With a large number of children in each family there is much less incentive for adopting. The clear effect of the decay of adoption was to reduce the links between teenagers and adults. Micronesia consists of more than 100 islands. As they have been affected to different degrees by social change we are in a good position for making comparative observations. Rubinstein (2002) mentions that suicide is less frequent in rural outer islands where traditional ways of life have to some extent be maintained. For instance, in the central island of Chuuk the average suicide rate of young males (15-24) was $ 207 $ per $ 10^5 $, whereas 500 kilometers to the east in the Pohnpei and Kosrae islands it was only $ 93 $ per $ 10^5 $.
At this point, we do not know if the factors that we have just described are really the main mechanisms that account for the high suicide rates. In order to confirm (or confute) these conjectures other similar cases must be investigated in a comparative perspective.
Particularly abundant rain in an area of central or northern Africa may result in a population of locusts which is in greater number than in ordinary years. It seems that once the density of locusts reaches a critical level a phase transition occurs which leads to the formation of swarms of locust which may contains billions of insects and have a duration of several years. In this process, locusts undergo slight physical changes (change in color, development of a gland for the release of a specific pheromone) which lead entomologists to describe them as a distinct subspecies (which they call a phase), namely [*locust gregaria*]{} as opposed to [*locust solitaria*]{} (Faure 1932). In order to better understand the factors which determine the transition from one phase to the other Jacobus Faure raised population of locusts in adjacent cages. It is worth reporting how he describes a cage containing [*locust solitaria*]{} in one half and [*locust gregaria*]{} in the other. “Whereas an individual isolated in one half of a cage leads a life of lazy idleness, its fellows in the other compartment separated by a gauze partition only, live a life of intense activity that could perhaps best be described as a frenzied effort to escape from some relentless, inwardly pursuing force.” This observation suggests that the level of interaction and activity is greatly increased in the swarm state.
After having established that it is the crowding that leads to the formation of swarms, Faure went on to show that inter-specific crowding has the same effect as intra-specific crowding, In other words, the enhanced interaction that leads to swarm formation seems to be unconnected to a particular species of locusts.
The second example that we wish to mention of a transition analog to a condensation of a gas into a liquid is linked with the phenomenon of territorial conquest by clans of nomads that occurred repeatedly in central Asia. At least 5 episodes have been recorded by historians between the first and 17th century. In their “gaseous” state the clans wander more or less randomly over the vast land expanses of the steppe and have only minimal interaction with one another. Then, all of a sudden for no obvious reason, the various clans will gather around a leader whether it is Attila, Genghis Khan or Tamerlane. On such episodes our historical information is mostly limited to anecdotal evidence. We largely ignore the reasons which trigger these events ; however, once started, it may have worked like a chain reaction in the sense that any battle won by the new leader strengthened and broadened his force of attraction and brought clans closer together. The physical analog would consist in following an isotherm on a (volume per mole, pressure) phase transition diagram; as one moves from the region of high volume per mole (i.e. low density) to lower volume per mole, one first sees some droplets of liquid form, and then, progressively, an ever larger proportion of gas becomes liquid until eventually all the gas is liquid. If this mechanism is correct, it means that the episodes of territorial conquest experienced by the nomadic tribes of Central Asia have been triggered by a major population increase.
What was the threshold density of population, in other words what is the upper density limit for nomadic people living on their livestock and occasional hunting. This is certainly an important parameter and in order to find its order of magnitude the most reliable procedure is probably to look at present day population density in regions where this mode of subsistence is still in effect. The following figures are for 1994 (i) The density of the vast area which comprises the Republic of Mongolia, Inner Mongolia (part of China) and Buryatia (a part of historical Mongolia now located in Russia) was 8.4 inhabitants per square kilometer. (ii) The density of Xinjiang in western China was 9.7 (iii) The density of the Chinese province of Qinhai which is located between Xinjiang and Tibet was 6.5. In all these cases the density of the grazing livestock was between two to three times the human density. To sum up, one can conclude that the upper density limit was around 8 people per square kilometer with about 2.5 head of livestock for each person. It should be noted that this limit is an upper bound but that the actual critical population density may have been somewhat lower. Once more population data about the previous historical episodes become available, it will be possible to say if our conjecture that they were driven by a common mechanism is indeed confirmed.
The integration of new immigrants is easier in large cities than in rural towns. Why? It is of course easy to provide a number of “anthropological" reasons such as better economic opportunities or the fact that the inhabitants of large cities are more used to the presence of various immigrants than the people in rural towns. Here we want to see whether our parallel with solutions can tell us something about this question. Two observations can be made in that respect. Roughly speaking, solubility requires that the molecules of the solute come in between the molecules of solvent and that new links are established between the former and the latter. For this to occur the interactions between solvent molecules should not be too strong whereas the solvent-solute interactions should be as strong as possible. Translated into sociological language, the fact that integration is easier in cities means that the interactions in cities (solvent) are weaker than in towns or that immigrant-city interactions are greater than immigrant-town interactions ( both can conditions can be fulfilled simultaneously). Intuitively, these conditions seem to agree with common sense, but once again we are hampered by our inability to [*measure*]{} the strength of interactions. The previous discussion is not completely realistic because it considers a binary solution whereas in social situations there are several components. Although physical data handbooks contain less information about multicomponent solutions than about binary solutions, one aspect appears very clearly: multicomponent solutions are much less selective than binary components. What do we mean by the expression “less selective”? The heat of mixing curves for water-alcohol (Fig.1b) resemble resonance curves; the fact that there is a sharp fall on both sides shows that energetically the solvation is much less favorable as soon as one leaves the peak region; in other words the solubility is fairly selective. In contrast, for a multicomponent solution the heat of mixing curves are almost flat which shows that the solvation has a low selectivity. Intuitively, this is easy to understand for if the solution already comprises various molecules characterized by different kinds of bonds, it will be easy for any new molecule to link itself to one of them. This property of multicomponent solutions is illustrated in a more quantitative way in the following table.
**Table 4 Relative selectivity of solvation according to number of components**
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$$\matrix{
\tvi
&\hbox{2 components} & \hbox{3 components} & \hbox{5 components} \cr
\noalign{\hrule}
\tvi
\hbox{Coeff. of var. of Q / Coeff. of var. of proportion} \hfill & 77\% &
24\% & 12\% \cr
\noalign{\hrule}
}$$
1.5mm Notes: The table tells us that a solution with several components is less “selective” than a binary solution. The percentages give the ratio of the coefficient of variation (i.e. standard deviation divided by mean) of the heats of mixing, $ Q $, relative to the coefficient of variation of changes in the proportion of one of the components. As an illustration, for water + ethanol (considered in Fig.1b) the ratio is for instance equal to 0.98. The 2-component figure is an average over three cases: water + methanol, water + ethanol, ethanol + toluene; the 3-component figure is the average of the two following cases: benzene + cyclohexane + hexane (proportion changes refer to hexane), benzene + cyclohexane + n-heptane (proportion changes refer to heptane); the 5-component case is: benzene + hexane + toluene + cyclohexane + heptane (proportion changes refer to heptane). Source: Landolt-Börnstein (1976, p. 536-540).
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The more components there are, the less changes in the molar proportion of one of the components perturbed the solution, as reflected in the fact that the heat of mixing becomes almost independent of the proportion.
Although I did not propose any model, time is not yet ripe for that, I hope this paper will help us to see a number of socio-economic phenomena in a more unified and less anthropocentric way. The potential usefulness of the parallels developed in this paper is that it gives us the incentive to compare phenomena which at first sight seem to have little in common. In this concluding section I would like firstly to discuss the question of ergodicity, an important theoretical issue on which depends the applicability of statistical mechanics, and secondly to suggest an agenda for future research.
The success of statistical mechanics is entirely based on the fact that ensemble averages can be identified with time averages. On the theoretical side we compute the most probable configurations of the system on the basis of a collection of similar systems characterized by the same initial conditions and macroscopic constraints. The classical example is a gas contained in a container in a state of equilibrium. Strictly speaking, the probability of finding all the molecules in one half of the container is not null, but it is overwhelmingly smaller than the probability of the situations where the molecules are uniformly distributed (except for small random fluctuations). The assumption that the time the system under observation spends in each macrostate is proportional to the probability of this state rests on the hypothesis not only that the system randomly explores all accessible microstates, but that it explores them “quickly enough”. As we have seen, for molecules in a gas or in a liquid the typical duration of a given configuration is of the order of one picosecond which means that within the time it takes to make a measurement the system explores over $ 10^{12} $ configurations.
No matter how we define the configuration space, it is obvious that it will be explored much more slowly in the case of socio-economic systems. For instance, on stock markets probably one of the economic systems with the highest transition rate, there are on average less than 10 transactions per second even for the most heavily traded stocks. For other socio-economic systems the number of transitions may be smaller by several orders of magnitude. This has at least two consequences. (i) The time it may take for equilibrium to be reached may be large compared to the time scale of human observation. (ii) Consequently, there is a substantial probability of seeing the system in some metastable state rather than in its “true” equilibrium state. As a matter of fact, this problem is not specific to social systems; it also exists for some physical systems such as selenium, sulfur or tin which have different allotropic forms. For instance the transition from white tin to gray tin is supposed to occur at $ 13 \qd $, but it may take centuries for a plate of white tin to decompose into gray tin even at temperatures as low as $ -18 \qd $ (Kariya et al. 2000, http://www.natmus.dk). Most of the tools developed in statistical mechanics are not well suited to such systems.
One may wonder whether the three manifestations of suicide that we examined can be accounted for by the same mechanism in spite of the fact that they correspond to very different time scales ranging from a few hours to several decades. To try to build a model at this point would probably be premature. If the model “explains” observed suicide rates in terms of social bonds that we cannot estimate in an independent way, this would be no more than a form of circular reasoning. This shows that one of our most urgent tasks is to develop methods for estimating the strength of social interaction. That this objective has so far been largely ignored by sociologists may seem surprising. How, for instance, is it possible to understand revolutions which basically consist in a rearrangement of social networks, if one has no real means for assessing the strength of social ties?
What approach can we think of for that purpose? In physics, experimental methods for measuring the strength of molecular bonds involve infra-red spectroscopy, ultra-sonic spectrography or X-ray/neutron scattering. The three methods are similar in their principle. A wave is sent through the medium and the various ways in which it is affected are recorded and used to probe various characteristics of the medium. The first method is slightly different from the two others in the sense that it relies on the absorption which occurs when the frequency of the source coincides with the stretching or vibration modes of the molecular structure. So far we do not know much about the eigenfrequencies (if any) of socio-economic systems which means that for the time being this is not a straightforward approach. The two other methods have a broader applicability. For instance, in ultra-sonic spectrography an ultra-sonic wave is transmitted through the medium, its velocity and attenuation are recorded, and from these measurements one may derive many properties of the medium, for instance its density, the size of emulsion droplets or the existence of a temperature gradient. At this point, an important requirement needs to be emphasized which is crucial if one wants to extend this approach to the social sciences. In order to be able to use light or sound waves as probes, one must already know how specific characteristics of the medium are likely to affect these signals. In other words, it is only once we have gained some kind of understanding (even if it is only an empirical understanding) of how a given signal is affected by a society that we can use it as a probe. But once this condition is fulfilled, this probe will allow more detailed and systematic explorations. In short, this is a kind of cumulative process where any new knowledge about social interactions is used as a springboard for further explorations.
There are many social signals which should enable us to transpose the previous approach to the social sciences. For instance, the way an epidemic propagates in a society can reveal a lot about the interactions that take place in a population. As an example, one can mention the fact that if two population groups show highly different levels of prevalence for a sexually transmissible disease such as gonorrhea or chlamydia, one can be almost sure that the two groups have little interactions through marriage or non-marital sexual contacts. Such assessments can be checked by confronting them against inter-marriage rates; alternatively, they can replace such statistics in those countries which for some reason do not record that kind of data. Naturally, this example concerns only one particular aspect of social interaction. One would have to develop similar approaches for other aspects as well. The transmission of rumors, innovations or new fashions can provide other possible probes.
In this paper, I tried to convince the reader of the key role of interaction strength in social phenomena. I am convinced that once we get a clearer picture of this factor many socio-economic phenomena will become more transparent and, to some extent, more predictable.
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