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abstract: 'Multiplicity fluctuations in limited segments of momentum space are calculated for a classical pion gas within the statistical model. Results for the grand canonical, canonical, and micro-canonical ensemble are obtained, compared and discussed. We demonstrate that even in the large volume limit correlations between macroscopic subsystems due to energy and momentum conservation persist. Based on the micro-canonical formulation we make qualitative predictions for the rapidity and transverse momentum dependence of multiplicity fluctuations. The resulting effects are of similar magnitude as the predicted enhancement due to a phase transition from a quark-gluon plasma to a hadron gas phase, or due to the critical point of strongly interacting matter, and qualitatively agree with recently published preliminary multiplicity fluctuation data of the NA49 SPS experiment.'
author:
- Michael Hauer
title: |
Multiplicity Fluctuations in Limited Segments of Momentum Space\
in Statistical Models
---
Introduction {#Intro}
============
The statistical model has been, for a long time, successfully applied to fit experimental data on mean hadron multiplicities in heavy ion collision experiments over a wide range of beam energies and system sizes. For recent reviews see [@FOCley; @FOBeca; @FOPBM; @FORafe]. So naturally the question arises whether the statistical model is able to describe | 1 | member_8 |
event-by-event fluctuations of these observables as well. And indeed, a first comparison suggests that this might be possible for the sample of most central events. Global conservation laws, imposed on a statistical system, lead, even in the large volume limit, to suppressed fluctuations. The multiplicity distributions of charged hadrons recently reported [@NA49_fluc] by the NA49 SPS experiment are systematically narrower than a Poissonian reference distribution. This could be interpreted [@MCEvsData] as effects due to energy and charge conservation in a relativistic hadronic gas.
Multiplicity fluctuations are usually quantified by the ratio of the variance of a multiplicity distribution to its mean value, the so-called scaled variance. In statistical models there is a qualitative difference in the properties of mean value and scaled variance. In the case of the mean multiplicity results obtained within the grand canonical ensemble (GCE), canonical ensemble (CE), and micro-canonical ensemble (MCE) approach each other in the large volume limit. One refers here to as the thermodynamic equivalence of these ensembles. It was recently found [@CEfluc1] that corresponding results for the scaled variance are different in different ensembles, and thus this observable is sensitive to conservation laws obeyed by a statistical system.
The growing interest in the | 1 | member_8 |
experimental and theoretical study of fluctuations in strong interactions (see e.g., reviews [@reviewfluc]) is motivated by expectations of anomalies in the vicinity of the onset of deconfinement [@ood] and in the case when the expanding system goes through the transition line between a quark-gluon plasma and a hadron gas phase [@phasetrans]. In particular, a critical point of strongly interacting matter may be accompanied by a characteristic power-law pattern in fluctuations [@critpoint]. A non-monotonic dependence of event-by-event fluctuations on system size and/or center of mass energy in heavy ion collisions would therefore give valuable insight into the phase diagram of strongly interacting matter. Provided the signal survives the subsequent evolution and hadronization of the system (see also [@recomb]). Therefore, in order to asses the discriminating power of proposed measures, for a recent review see [@reviewfluc2], one should firstly study properties of equilibrated sources [@MCEvsData; @res; @VolDep; @vdw] and quantify ‘baseline‘ (or thermal/statistical) fluctuations. Apart from being an important tool in an effort to study a possible critical behavior, the study of fluctuations within the statistical model constitutes also a further test of its validity.
In this paper we make detailed predictions for the momentum space dependence of multiplicity fluctuations. We show | 1 | member_8 |
that energy and momentum conservation lead to a non-trivial dependence of the scaled variance on the location and magnitude of the observed fraction of momentum space. These predictions can be tested against existing and future data from the heavy ion collision experiments at the CERN SPS and BNL RHIC facilities.
The paper is organized as follows: In section \[model\] we briefly introduce our model. In section \[GCECE\] we consider multiplicity distributions in a limited region of momentum space in GCE and CE. For the MCE we follow, in section \[MCE\], the procedure of Ref.[@clt] and show how to calculate the width of the corresponding distributions in the large volume limit. We revisit the so-called ‘acceptance scaling‘ previously suggested as an approximate implementation of experimental acceptance in section \[Results\]. Technical details of the calculations are presented in the Appendix. Concluding remarks and a summary in sections \[Remarks\] and \[Summary\] close the paper.
The Model {#model}
=========
The ideal Boltzmann $\pi^+$ $\pi^-$ $\pi^0$ gas serves as the standard example throughout this paper, while the main subject of investigation is the multiplicity distribution $P(N_{\Omega})$ of particles with momenta inside a certain segment $\Omega$ of momentum space. Calculations are done for the three standard | 1 | member_8 |
ensembles GCE, CE, and MCE. For the sake of argument we will assume that we only want to measure $P(N_{\Omega}^-)$, i.e. the probability distribution of negatively charged pions in a limited segment $\Omega$ of momentum space. Hence $\pi^-$ with momenta inside $\Omega$ are observed, while $\pi^-$ inside the complementary segment $\bar \Omega$ are not observed. $\pi^+$ and $\pi^0$ are never detected. In GCE and CE the presence of $\pi^0$ as a degree of freedom is of no relevance, while in MCE it constitutes a heat bath for the remaining system. For consistency we use the same system throughout this discussion.
In order to keep the model simple, we assume a static homogenous fireball. Our considerations therefore exclude collective motion, i.e. flow, and resulting momentum spectra are purely thermal. We also omit resonance decay contributions in this work. The spectra presented in Fig. \[spectra\] are normalized to the total $\pi^-$ yield in GCE and CE. Thus they are the same in both ensembles. In MCE one expects in the large volume limit only small deviations from Boltzmann spectra. None of the forthcoming arguments are affected by this.
In the following we will use the transverse momentum and rapidity spectra presented in | 1 | member_8 |
Fig. \[spectra\] to construct bins $\Omega_i = \Delta {p_T}_i =
\left[p_{T_i},p_{T_{i+1}} \right]$ (left), or $\Omega_i = \Delta y_i =
\left[y_i,y_{i+1} \right]$ (right), as indicates by the drop-lines.
In section \[GCECE\] we calculate the multiplicity distributions $P(N_{\Omega})$ for arbitrary segments $\Omega$ for the ideal Boltzmann GCE and CE. To characterize the distribution one can calculate its (raw) moments $\langle N_{\Omega}^n \rangle$ from: $$\label{Moments}
\langle N_{\Omega}^n \rangle ~=~ \sum \limits_{N_{\Omega}=0}^{\infty}
~N_{\Omega}^n~ P \left(N_{\Omega} \right)~.$$ A convenient measure for the width of a distribution is the scaled variance: $$\label{ScaledVar}
\omega_{\Omega} ~\equiv~ \frac{\langle N_{\Omega}^2 \rangle - \langle N_{\Omega}
\rangle^2}{\langle N_{\Omega} \rangle} ~.$$ In order to remove simple scaling effects, the bin sizes or segments are chosen such that each bin or segment contains the same fraction $q= \langle
N_{\Omega} \rangle ~/~ \langle N_{4\pi} \rangle$ of the total yield (compare Eq.(\[ScaledVar\])). Here $\langle N_{\Omega} \rangle$ denotes the average particle number in the momentum space segment $\Omega$, and $\langle N_{4\pi}
\rangle$ denotes the average total ($4\pi$ integrated) multiplicity. The effect of finite acceptance can approximately be taken into account by [@CEfluc1]: $$\label{accscaling}
\omega_{q} ~=~ 1 + q \left(\omega_{4\pi}-1 \right)~,$$ where $\omega_{4\pi}$ assumes the ideal situation when all particles are detected, while $\omega_{q}$ assumes that particles are | 1 | member_8 |
detected with probability $q$ regardless of their momentum. Hence Eq.(\[accscaling\]) holds when particles are assumed to be uncorrelated in momentum space. In the limit $q\rightarrow 0$ one observes a random distribution with $\omega_q
\rightarrow 1$, i.e. a Poissonian, while when $q\rightarrow 1$ one sees the real distribution with width $\omega_q \rightarrow \omega_{4\pi}$. In this work we take explicitely correlations due to globally conserved charge (CE), and energy-momentum (MCE) into account and compare the results to Eq.(\[accscaling\]).
Grand Canonical and Canonical Ensemble {#GCECE}
======================================
Grand Canonical Ensemble
------------------------
In the GCE, both, heat and charge bath are assumed to be infinite. And thus neither charge, energy nor momentum are conserved exactly. Temperature $T$ and charge chemical potential $\mu$ regulate average energy and charge density in a system of volume $V$. Usually it is said that charge, energy and momentum are conserved in the average sense and fluctuations about an equilibrium value are allowed. Apart form Bose and Fermi effects [@Qstats] particles are therefore uncorrelated in momentum space. However this example serves as an illustration for the following CE and MCE calculations. We start by decomposing the Boltzmann single particle partition function $z^-\left(\phi_{N_{\Omega}}\right)$ of $\pi^-$ into two parts, $$\begin{aligned}
z^-\left(\phi_{N_{\Omega}}\right)= z^-_{\Omega} \left(\phi_{N_{\Omega}}
| 1 | member_8 |
\right) + z^-_{\bar \Omega} &=& \frac{gV}{\left(
2\pi\right)^3} \int \limits_{\Omega} d^3 p ~
e^{-\frac{\varepsilon+\mu}{T}}~ e^{i \phi_{N_{\Omega}}} +
\frac{gV}{\left( 2\pi\right)^3} \int \limits_{\bar \Omega} d^3 p ~
e^{-\frac{\varepsilon+\mu}{T}} , \end{aligned}$$ where the single particle energy $\varepsilon = \sqrt{p^2+m^2}$, and $m$, and $g$ are mass and degeneracy factor of $\pi^-$ respectively. Only for momentum states inside the momentum space region $\Omega$ we introduce additionally a Wick-rotated fugacity $\exp \left(i \phi_{N_{\Omega}} \right)$. For the positive and neutral pion (which we do not want to detect in our example) we write: $$\begin{aligned}
z^+ ~=~ \frac{gV}{\left( 2\pi\right)^3} \int d^3 p ~
e^{-\frac{\varepsilon-\mu}{T}}~,
\qquad \qquad \textrm{and} \qquad \qquad
z^0 ~=~ \frac{gV}{\left( 2\pi\right)^3} \int d^3 p ~
e^{-\frac{\varepsilon}{T}}~.\end{aligned}$$ The value of the single particle partition function, for instance of the neutral pion, is given by: $$\label{aveN}
z^0=\langle N^0 \rangle = \frac{gV}{2\pi} m^2 T K_2 \left( \frac{m}{T}\right).$$ For the sake of simplicity we assume equal masses for all pions. To obtain the GCE multiplicity distribution for $N_{\Omega}$ in a momentum space segment $\Omega$ we use the Fourier integral over the generalized GCE partition function $ \mathcal{Z} \left( \phi_{N_{\Omega}}\right)=\exp \left[ z^-_{\Omega}
\left( \phi_{N_{\Omega}} \right) + z^-_{\bar{ \Omega}} + z^+ + z^0 \right] $, normalized by the GCE partition function: $$\begin{aligned}
\label{GCEPDF}
| 1 | member_8 |
P_{gce} \left(N_{\Omega} \right) ~\equiv~ Z^{-1}_{gce} \times \int
\limits_{-\pi}^{\pi} \frac{d\phi_{N_{\Omega}}}{2\pi} ~ e^{-iN_{\Omega}
\phi_{N_{\Omega}} } ~ \mathcal{Z} \left( \phi_{N_{\Omega}}\right) ~=~
\frac{\left(z^-_{\Omega}\right)^{N_{\Omega}}}{N_{\Omega}!}
\exp \left[- z^-_{\Omega} \right]~,\end{aligned}$$ where the system partition function is given by $ Z_{gce} \equiv \mathcal{Z}
\left( \phi_{N_{\Omega}} = 0\right) $, and $z^-_{\Omega} =
z^-_{\Omega} \left(\phi_{N_{\Omega}}=0 \right)$. Independent of the shape or size of $\Omega$ we find a Poissonian for the multiplicity distribution Eq.(\[GCEPDF\]). Thus, using Eq.(\[ScaledVar\]), one finds for the scaled variance $\omega^{gce}_{\Omega} = 1$, since $\langle
N_{\Omega} \rangle = z^-_{\Omega}$, and $\langle N^2_{\Omega}
\rangle = \langle N_{\Omega} \rangle^2 + \langle N_{\Omega} \rangle$.
For Bose and Fermi statistics one does not expect a Poisson distribution and (in particular when the chemical potential is large) deviations from a Poissonian can be large. Thus one expects also deviations from Eq.(\[accscaling\]) when considering only finite acceptance.
Canonical Ensemble
------------------
In the CE the heat bath is still assumed to be infinite, while we remove the charge bath and drop the chemical potential. Thus, we introduce a further Wick-rotated fugacity $\mu/T \rightarrow i \phi_Q $ into the single particle partition functions to account for global (however not in the momentum space segment $\Omega$) conservation of electric charge $Q$. Particles in $\Omega$ are therefore correlated, | 1 | member_8 |
due to the condition of fixed net-charge, with a finite charge bath composed of $\pi^+$ and unobserved $\pi^-$. We again split the single particle partition function for $\pi^-$ into an observed, $z^-_{\Omega}\left(\phi_{N_{\Omega}},\phi_Q\right)$, and an unobserved part, $z^-_{\bar \Omega} \left(\phi_Q\right)$, $$\begin{aligned}
z^-\left(\phi_{N_{\Omega}},\phi_Q\right) = z^-_{\Omega}
\left(\phi_{N_{\Omega}},\phi_Q\right) + z^-_{\bar \Omega}
\left(\phi_Q\right) = \frac{gV}{\left(
2\pi\right)^3} \int \limits_{\Omega} d^3 p ~
e^{-\frac{\varepsilon}{T}} e^{-i \phi_Q} e^{i \phi_{N_{\Omega}}} +
\frac{gV}{\left( 2\pi\right)^3}
\int \limits_{\bar \Omega} d^3 p ~ e^{-\frac{\varepsilon}{T}} e^{-i \phi_Q},\end{aligned}$$ while we do not want to measure $\pi^+$ and $\pi^0$, and thus: $$\begin{aligned}
z^+ \left(\phi_Q\right) ~=~ \frac{gV}{\left(
2\pi\right)^3} \int d^3 p ~ e^{-\frac{\varepsilon}{T}} e^{+ i \phi_Q}~,
\qquad \qquad \textrm{and} \qquad \qquad
z^0 ~=~ \frac{gV}{\left(
2\pi\right)^3} \int d^3 p ~ e^{-\frac{\varepsilon}{T}} .\end{aligned}$$ The normalization of the CE multiplicity distribution is given by the CE system partition function $Z_{ce}$, i.e. the number of all micro states with fixed charge Q, $Z^{ce} = I_Q\left(2z \right) \exp(z^0)$, where $I_Q$ is the modified Bessel function. The multiplicity distribution of $N_{\Omega}$ in a momentum space segment $\Omega$, while charge $Q$ is globally conserved, can be obtained from Fourier integration of the generalized GCE partition function $\mathcal{Z} \left( \phi_{N_{\Omega}}, \phi_Q \right) = \exp \left[ z^-_{\Omega}
\left( \phi_{N_{\Omega}}, \phi_Q \right)~+~ z^-_{\bar{ \Omega}}
\left( \phi_Q \right) | 1 | member_8 |
+ z^+ \left( \phi_Q \right) + z^0\right] $, over both angles $\phi_Q$ and $\phi_{N_{\Omega}}$: $$\begin{aligned}
P_{ce} \left(N_{\Omega}\right) &\equiv& Z_{ce}^{-1} \times
\int \limits_{-\pi}^{\pi} \frac{d\phi_{N_{\Omega}}}{2\pi} \int
\limits_{-\pi}^{\pi} \frac{d\phi_{Q}}{2\pi} ~ e^{-iN_{\Omega}
\phi_{N_{\Omega}} } ~ e^{-i Q \phi_{ Q} } ~\mathcal{Z} \left(
\phi_{N_{\Omega}}, \phi_Q \right) \\
&=& I_Q^{-1}\left(2z \right) \times
\frac{\left(z^-_{\Omega}\right)^{N_{\Omega}}}{N_{\Omega}!} ~ \sum
\limits_{a=0}^{\infty} ~ \frac{\left(z^-_{\bar \Omega}\right)^{a}}{a!} ~
\frac{z^{Q+N_{\Omega}+a}}{\left(Q+N_{\Omega}+a\right)!}~,\end{aligned}$$ where in CE $z^-_{\Omega} = z^-_{\Omega} \left(\phi_{N_{\Omega}}=\phi_Q=0
\right)$, $z^-_{\bar \Omega} = z^-_{\bar \Omega} \left(\phi_Q=0 \right)$, and $z= z^+\left(\phi_Q=0\right)=z^0$. For the respective first two moments one finds from Eq.(\[Moments\]): $$\begin{aligned}
\langle N_{\Omega} \rangle = z^-_{\Omega } ~ \frac{I_{Q+1} \left(2z
\right)}{I_Q \left(2z \right)}~, \qquad \textrm{and} \qquad
\langle N^2_{\Omega} \rangle = \left( z^-_{\Omega }\right)^2 ~\frac{I_{Q+2}
\left(2z \right)}{I_Q \left(2z \right)} + z^-_{\Omega }~ \frac{I_{Q+1}
\left(2z\right)}{I_Q \left(2z \right)}~. \end{aligned}$$ Thus, we obtain the well known canonical suppression of yields [@CEyield; @CEfits; @CEtransport; @RateEqYield] and fluctuations [@CEfluc1; @RateEqFluc]. The result, however, is completely independent of the position of the segment $\Omega$. And therefore the scaled variance, Eq.(\[ScaledVar\]), takes the form: $$\begin{aligned}
\omega^{ce}_{\Omega} = 1 + z^-_{\Omega} ~\left[ \frac{I_{Q+2}
\left(2z\right)}{I_{Q+1} \left(2z \right)} -\frac{I_{Q+1}
\left(2z\right)}{I_Q \left(2z \right)} \right]~, \qquad \textrm{and}
\qquad \omega^{ce}_{4\pi} = 1 + z ~ \left[ \frac{I_{Q+2}
\left(2z\right)}{I_{Q+1} \left(2z \right)} -\frac{I_{Q+1}
\left(2z\right)}{I_Q \left(2z \right)} \right]~,\end{aligned}$$ where $\omega_{\Omega}$ is the width of | 1 | member_8 |
$P_{ce}(N_{\Omega})$, i.e. the multiplicity distribution of $\pi^-$ with momenta inside $\Omega$, while $\omega_{4\pi}$ is the width of the corresponding distribution when $\Omega$ is extended to the full momentum space. It can immediately be seen that this formula is consistent with acceptance scaling, Eq.(\[accscaling\]), $\omega_{\Omega} ~=~ 1 + q \left(\omega_{4\pi} -1 \right)$, if $q \equiv
z^-_{\Omega}/z$. Generally we find $\omega^{ce}_{4\pi} < \omega^{ce}_{\Omega}
< \omega^{gce}=1$. In the limit of $z^-_{\Omega}/z \rightarrow 0$ we approach the Poisson limit of a ‘random‘ distribution with $\omega = 1$, i.e. the observed part of the system is embedded into a much larger charge bath and the GCE is a valid description.
Micro-Canonical Ensemble {#MCE}
========================
For the MCE an analytical solution seems to be out of reach presently, so we use instead the asymptotic solution, applicable to large systems, derived in [@clt]. In order to avoid unnecessary repetition of calculations, we will only give a general outline here, and refer the reader for a detailed discussion to Ref.[@clt]. It should be mentioned that this method be would be also applicable to systems of finite spatial extention, provided the average particle number in a given momentum space bin exceeds roughly $\langle N_{\Omega}\rangle \gtrsim 5$. In this work | 1 | member_8 |
we confine ourselves to large systems and try to asses the general trends.
The basic idea is to define the MCE multiplicity distribution in terms of a joint GCE distribution of multiplicity, charge, energy, momentum, etc. The MCE multiplicity distribution is then given by the (normalized) conditional probability in the GCE to find a number $N_{\Omega}$ of particles in a segment $\Omega$ of momentum space, while electric charge $Q$, energy $E$, and three momentum $\vec P$ are fixed. Therefore we will keep temperature and chemical potentials as parameters to describe our system. Effective temperature and effective chemical potential, i.e. Lagrange multipliers, can be determined by demanding that the GCE partition function is maximized for a certain equilibrium state $(Q,E,\vec P)$. This requirement is entirely consistent [@clt] with the usual textbook definitions of $T$ and $\mu$ in MCE and CE through differentiation of entropy and Helmholtz free energy with respect to conserved quantities. In principle we would have to treat all conservation laws on equal footing [@MCEmagic], and thus introduce Lagrange multipliers for momentum conservation as well. However here we are only interested in a static source, thus $\vec P = \vec 0$, and the relevant parameters are equal to zero.
| 1 | member_8 |
In the large volume limit energy, charge, and particle density in the MCE will correspond to GCE values. This is required by the thermodynamic equivalence of ensembles for mean quantities. MCE and CE partition functions are generally obtained from their GCE counterpart by multiplication with delta-functions, which pick out a set of micro states consistent with a particular conservation law. Here it will be of considerable advantage to use Fourier representations of delta-functions, similar to the treatment in Section \[GCECE\]. This method could be considered to be a Fourier spectral analysis of the generalized GCE partition function [@clt].
The normalized conditional probability distribution of multiplicity $N_{\Omega}$ can be defined by the ratio of the values of two partition functions: $$\label{MCEprob}
P_{mce}(N_{\Omega})~\equiv~ \frac{\textrm{number of all states with $N_{\Omega}$,
$Q$, $E$, and $\vec P = \vec 0$}}{\textrm{number of all states with $Q$,
$E$, and $\vec P= \vec 0$} }~.$$ The real MCE partition function and our modified version are connected as $Z(V,N_{\Omega},Q,E,\vec P) \equiv \mathcal{Z}^{N_{\Omega},Q,E,\vec
P}(V,T,\mu) e^{+E/T} e^{-Q\mu/T}$. In either case the normalization in Eq.(\[MCEprob\]) is given by the partition functions with fixed values of $Q,E,\vec P$, but arbitrary particle number $N_{\Omega}$, hence $Z(V,Q,E,\vec
P) \equiv \sum_{N_{\Omega}=0}^{\infty} Z(V,N_{\Omega},Q,E,\vec P)$, or $\mathcal{Z}^{Q,E,\vec P}(V,T,\mu) | 1 | member_8 |
\equiv \sum_{N_{\Omega}=0}^{\infty}
\mathcal{Z}^{N_{\Omega},Q,E,\vec P}(V,T,\mu)$. However when taking the ratio (\[MCEprob\]) auxiliary parameters chemical potential and temperature drop out: $$\label{MCEprob2}
P_{mce}(N_{\Omega})~ \equiv~ \frac{Z(V,N_{\Omega},Q,E,\vec P)}{Z(V,Q,E,\vec
P)} ~=~ \frac{\mathcal{Z}^{N_{\Omega},Q,E,\vec
P}(V,T,\mu)}{\mathcal{Z}^{Q,E,\vec P}(V,T,\mu)}~.$$ The main difference between the two versions of partition functions is that for $Z(V,N_{\Omega},Q,E,\vec P)$ one is confronted with a heavily oscillating (or even irregular) integrant, while for $\mathcal{Z}^{N_{\Omega},Q,E,\vec
P}(V,T,\mu)$ the integrant becomes ($T$,$\mu$ correctly chosen) very smooth. Thus, introduction of $T$ and $\mu$ allows to derive (and use) the asymptotic solution of Ref.[@clt].
We have a total number of 6 conserved ‘charges‘, and hence we need to solve the 6-dimensional Fourier integral for the numerator in Eq.(\[MCEprob2\])[^1]: $$\begin{aligned}
\label{MCEInt}
\mathcal{Z}^{N_{\Omega},Q,E, \vec P}&=&
\int \limits_{-\pi}^{\pi} \frac{d \phi_{N_{\Omega}}}{2\pi}
\int \limits_{-\pi}^{\pi} \frac{d \phi_Q}{2\pi}
\int \limits_{-\infty}^{\infty} \frac{d \phi_E}{2\pi}
\int \limits_{-\infty}^{\infty} \frac{d \phi_{P_x}}{2\pi}
\int \limits_{-\infty}^{\infty} \frac{d \phi_{P_y}}{2\pi}
\int \limits_{-\infty}^{\infty} \frac{d \phi_{P_z}}{2\pi} \nonumber \\
&\times& e^{-iN_{\Omega} \phi_{N_{\Omega}}}~ e^{-iQ\phi_Q} ~e^{- iE \phi_E}
~e^{-iP_x \phi_{P_x}}~ e^{-iP_y \phi_{P_y}}~ e^{-iP_z \phi_{P_z}} \nonumber \\
&\times& \exp \left[ V \sum_k \psi_k \left( \phi_{N_{\Omega}}\phi_Q, \phi_E,
\phi_{P_x},\phi_{P_y},\phi_{P_z} \right) \right].\end{aligned}$$ The summation in (\[MCEInt\]) should be taken over the single particle partitions $V \psi_k=z_k$ of all considered particle species $k$. The Wick-rotated fugacities $\phi_{Q}$, etc. are related to the individual conservation laws. The distinction between the | 1 | member_8 |
Kronecker delta-function (limits of integration $\left[-\pi,\pi \right]$) for discrete quantities and the Dirac delta-function (limits of integration $\left[-\infty,\infty \right]$) for continuous quantities is important here, however for deriving an asymptotic solution it will not be. To simplify (\[MCEInt\]) we change to shorthand notation for $\phi_j =
(\phi_{N_{\Omega}}\phi_Q, \phi_E, \vec \phi_P)$ and the conserved ‘charge‘ vector $ Q^j = (N_{\Omega},Q,E,\vec P)$. We again split the single particle partition functions in two parts. The first part counts the number of momentum states observable to our detector, while the second part counts momentum states invisible to our detector: $$\begin{aligned}
\psi_{k} \left( \phi_j\right) &=& \frac{g_k}{\left( 2\pi\right)^3} \int
\limits_{ \Omega} d^3 p ~ e^{-\frac{\varepsilon_k - q_k \mu}{T}} ~e^{i q_{k,
\Omega}^j \phi_j}~ +~ \frac{g_k}{\left( 2\pi\right)^3}
\int \limits_{ \bar \Omega} d^3 p~ e^{-\frac{\varepsilon_k-q_k \mu}{T}} ~ e^{i
q_{k,\bar \Omega}^j \phi_j}~.\end{aligned}$$ For the ‘charge‘ vector of all measured particle species $k$ we write $q^j_{k,\Omega} = (1,q_k,\varepsilon_k,\vec p_k)$ for momenta inside $\Omega$, and $q^j_{k, \bar \Omega} = (0,q_k,\varepsilon_k,\vec p_k)$ for momenta outside of $\Omega$. For all unobserved particle species we write $q^j_{k,\Omega}=q^j_{k, \bar \Omega} =(0,q_k,\varepsilon_k,\vec p_k) $. Here $q_k$ is the electrical charge of particle species $k$, and $\varepsilon_k$ and $\vec p_k$ are its energy and momentum vector. In Ref.[@clt], | 1 | member_8 |
where only multiplicity distributions in the full momentum space were considered, the general ‘charge‘ vector took the form $q^j_{k,4\pi} =
(n_k,q_k,\varepsilon_k,\vec p_k)$, where $n_k$ is the multiplicity of this particle. For stable particles $n_k=1$ in case they are observed, and $n_k=0$ if they are not measured, while for unstable particles $n_k$ could also denote the number of measurable decay products.
For large system volume the main contribution to the integral (\[MCEInt\]) comes from a small region around the origin [@VolDep]. Thus we proceed by Taylor expansion of the integrant of (\[MCEInt\]) around $\phi_j=\vec
0$. In this context $ \Psi \left( \phi_j\right) = \sum_k \psi_k \left(
\phi_j\right) $ would be called the cumulant generating function (CGF). Cumulants (expansion terms) are defined by differentiation of the CGF at the origin: $$\label{kappa_n}
\kappa_n^{j_1,j_2,\dots,j_n } ~\equiv~ \left(-i\right)^n\frac{\partial^n
\Psi \left( \phi_j \right) }{\partial \phi_{j_1}
\partial \phi_{j_2} \dots \partial \phi_{j_n} }
\Bigg|_{\phi_j = \vec 0}~.$$ Generally are cumulants tensors of rank $n$ and dimension defined by the number of conserved quantities. Here $\kappa_1$ is a 6 component vector, while $\kappa_2$ is a $6 \times 6$ matrix, etc.
The parts of the integrant related to discrete quantities, i.e. $N_{\Omega}$ and $Q$, are now not $2\pi$ periodic anymore | 1 | member_8 |
(while in Eq.(\[MCEInt\]) they are), but superpositions of oscillating and decaying parts. Thus we extent the limits of integration to $\pm \infty$, what introduces a negligible error. Eq.(\[MCEInt\]) therefore simplifies to: $$\begin{aligned}
\label{MCEIntApprox}
\mathcal{Z}^{Q^j} &\simeq& \left[ \prod_{j=1}^{6} \int
\limits_{-\infty}^{\infty} \frac{d\phi_j}{\left( 2\pi \right)} \right] ~\exp
\Big[ -iQ^j\phi_j
~+~V \sum_{n=0}^{\infty} \frac{i^n}{n!} \; \kappa_n^{j_1,j_2,\dots,j_n } \;
\phi_{j_1} \phi_{j_2} \dots \phi_{j_n} \Big] ~.\end{aligned}$$ Summation over repeated indices is implied. Existence and finiteness of the first three cumulants provided, any such integral can be shown to converge to a multivariate normal distribution in the large volume limit: $$\label{MCE_MultNormal}
\mathcal{Z}^{Q^j} ~\simeq~ Z_{gce} \frac{\exp \left(-\frac{\xi^j \; \xi_{ j}}{2}
\right)}{\left(2\pi V \right)^{6/2} \det| \sigma| }~,$$ where $Z_{gce}\equiv \exp \left[V \kappa_0 \right]$ is the GCE partition function, $\kappa_0$ is the cumulant of $0^{th}$ order, $\xi^j=\left( Q^k - V
\kappa_1^k \right) \left( \sigma^{-1}\right)_{k}^{\;\;j} V^{-1/2}~$ is a measure for the distance of a particular macro state $Q^k$ to the peak $V
\kappa_1^k$ of the joint distribution, and $\sigma$ is the square root of the second rank tensor $\kappa_2$, see [@clt] for details.
The normalization in Eq.(\[MCEprob2\]) can essentially be found in two ways. The first way would be to integrate the distribution (\[MCE\_MultNormal\]) over all possible values of multiplicity $N_{\Omega}$, while | 1 | member_8 |
all other variables are set to their peak values, e.g. $Q=V\kappa_1^Q$, $E=V\kappa_1^E$, $\vec P = \vec 0$. The second and more practical way is to use an approximation similar to Eq.(\[MCE\_MultNormal\]) to describe the macro state $Q^j = (Q,E,\vec P)$. The normalization in Eq.(\[MCEprob2\]), $\mathcal{Z}^{E,Q,\vec P}$, is then given by the 5-dimensional integral, similar to Eq.(\[MCEInt\]), without the integration over $\phi_{N_{\Omega}}$. The 1-dimensional slice along $N_{\Omega}$, i.e. the conditional distribution of particle number $N_{\Omega}$, while charge, energy and momentum are fixed to $Q,E,\vec P = \vec 0$, can then be shown [@clt] to converge to a Gaussian in the large volume limit: $$\begin{aligned}
\label{PMCE}
P_{mce}(N_{\Omega} ) ~ \simeq~ \frac{1}{\left(2\pi ~\omega^{mce}_{\Omega}
~ \langle N_{\Omega} \rangle \right)^{1/2}} ~ \exp \left(- \frac{ \left(
N_{\Omega} - \langle N_{\Omega} \rangle \right)^2}{2
~ \omega^{mce}_{\Omega} ~ \langle N_{\Omega} \rangle } \right) ~. \end{aligned}$$ The scaled variance $\omega^{mce}_{\Omega}$ is given by the ratio of the two determinants of the two relevant second rank cumulants, $\kappa_2$ and $\tilde
\kappa_2$, of the two partition functions $\mathcal{Z}^{N_{\Omega},E,Q,\vec
P}$ and $\mathcal{Z}^{E,Q,\vec P}$, hence[^2]: $$\label{SimpleOmega}
\omega^{mce}_{\Omega} = \frac{ \det | \kappa_2| }{
\kappa_1^{N_{\Omega}} \;\det| \tilde \kappa_2| }~.$$ The asymptotic ($V\rightarrow \infty$) scaled variance can therefore be written in the form of Eq.(28) in | 1 | member_8 |
[@clt]. Considering only the asymptotic solution we need to investigate only the first two cumulants ($n=1,2$) in detail. We will first discuss the structure of $\kappa_1$ and $\kappa_2$, and then deduce a few properties of Eq.(\[SimpleOmega\]).
The first order cumulant $\kappa_1$ of $\mathcal{Z}^{N_{\Omega},Q,E,\vec
P}$ gives GCE expectation values for particle density $\kappa_1^{N_{\Omega}}$, charge density $\kappa_1^{Q}$, energy density $\kappa_1^{E}$, and expectation values of momentum $\kappa_1^{p_x}$, etc. Since we are only interested in a static source we find due to the antisymmetric momentum integral (see Appendix \[Calc\]) $\kappa_1^{p_x} = \kappa_1^{p_y} = \kappa_1^{p_z}=0$. The general form of the first cumulant $\kappa_1$ is then: $$\begin{aligned}
\kappa_1 =
\begin{pmatrix}
\kappa_1^{N_{\Omega}}, & \kappa_1^{Q}, & \kappa_1^{E}, & 0, & 0, & 0
\end{pmatrix}~. \label{vector}\end{aligned}$$
The second cumulant $\kappa_2$ of $\mathcal{Z}^{N_{\Omega},Q,E,\vec
P}$ contains information about correlations due to different conserved quantities. A detailed discussion of correlation terms only involving Abelian charges and/or energy, e.g. $\kappa_2^{Q,Q}$, $\kappa_2^{Q,E}$, and $\kappa_2^{E,E}$, can be found in [@clt]. Again, due to the antisymmetric nature of the momentum integral, all cumulant entries involving an odd order in one of the momenta, e.g. $\kappa_2^{E,p_x}$, $\kappa_2^{p_x,p_y}$, or $\kappa_2^{Q,p_x}$ are equal to zero. The general second order cumulant $\kappa_2$ thus reads: $$\begin{aligned}
\kappa_2 =
\begin{pmatrix}
\kappa_2^{N_{\Omega},N_{\Omega}} & | 1 | member_8 |
\kappa_2^{N_{\Omega},Q} &
\kappa_2^{N_{\Omega},E} & \kappa_2^{N_{\Omega},p_x} &
\kappa_2^{N_{\Omega},p_y} & \kappa_2^{N_{\Omega},p_z} \\
\kappa_2^{Q,N_{\Omega}} & \kappa_2^{Q,Q} & \kappa_2^{Q,E} & 0 & 0 & 0 \\
\kappa_2^{E,N_{\Omega}} & \kappa_2^{E,Q} & \kappa_2^{E,E} & 0 & 0 & 0 \\
\kappa_2^{p_x,N_{\Omega}} & 0 & 0 & \kappa_2^{p_x,p_x} & 0 & 0 \\
\kappa_2^{p_y,N_{\Omega}} & 0 & 0 & 0 & \kappa_2^{p_y,p_y} & 0 \\
\kappa_2^{p_z,N_{\Omega}} & 0 & 0 & 0 & 0 & \kappa_2^{p_z,p_z}
\end{pmatrix}~. \label{matrix}\end{aligned}$$ Please note that by construction, Eq.(\[kappa\_n\]), the matrix (\[matrix\]) is symmetric, hence $\kappa_2^{N_{\Omega},Q} =
\kappa_2^{Q,N_{\Omega}}$, etc.
The second matrix $\tilde \kappa_2$, now related to the partition function $\mathcal{Z}^{Q,E,\vec P}$, is obtained from $\kappa_2$, Eq.(\[matrix\]), by crossing out the first row and first column. In the following we are going to make use of the fact that one can express the determinant of a matrix $A$ by: $$\label{calcdet}
\det |A| ~=~ \sum \limits_{j=1}^n \left( -1\right)^{j+k}
A_{j,k} M_{j,k} ~,$$ where $A_{j,k}$ is the matrix element $j,k$ of a general non-singular $n\times
n$ matrix $A$, and $ M_{j,k}$ is its complementary minor. A simple consequence of Eq.(\[calcdet\]) is: $$\label{normdet}
\det |\tilde \kappa_2| = \kappa_2^{p_x,p_x} ~\kappa_2^{p_y,p_y} ~\kappa_2^{p_z,p_z}
\left[ \kappa_2^{E,E}~\kappa_2^{Q,Q}- \left(\kappa_2^{E,Q}\right)^2 \right] ~=~
\left( \kappa_2^{p_x,p_x} \right)^3 \det |\hat \kappa_2|,$$ where $\kappa_2^{p_x,p_x} =\kappa_2^{p_y,p_y} =\kappa_2^{p_z,p_z} $, | 1 | member_8 |
due to spherical symmetry in momentum space, and $\hat \kappa_2$ is just a $2\times2$ matrix involving only terms containing $E$ and $Q$. In case correlations between particle number and conserved momenta are vanishing, i.e. $\kappa_2^{N_{4\pi},p_x} = 0$, or $\kappa_2^{N_{\Omega},p_x} = 0$, then, similarly to Eq.(\[normdet\]), the determinant of $\kappa_2$ factorizes into a product of correlation terms $(\kappa_2^{p_x,p_x})^3$ and the determinant of a $3\times3$ sub-matrix involving only terms containing $E$, $Q$, and $N$. Hence in taking the ratio Eq.(\[SimpleOmega\]) one notes, that in this case momentum conservation will not affect multiplicity fluctuations in the large volume limit [@clt]. In this work, however we do not necessarily find $ \kappa_2^{N_{\Omega},p_x} = 0 $, as we only integrate over a limited segment $\Omega$ of momentum space, and taking momentum conservation into account may affect the result.
Finally it should be stressed that this procedure can be easily generalized to account for Bose or Fermi statistics. Also phenomenological phase space suppression (enhancement) factors $\gamma_q$ [@gammaQfirst] or $\gamma_s$ [@gammaSfirst] could be straightforwardly included. However, without proper implementation of the effect of additional correlations due to resonance decay and collective motion, i.e. flow, it seems of little value to do too strict calculations for experimentally measurable | 1 | member_8 |
distributions. We thus return to the pion gas example from section \[GCECE\] and restrict the discussion to simple momentum space cuts in rapidity, transverse momentum, and azimuthal angle, see also the Appendix for details.
Results {#Results}
=======
Multiplicity fluctuations in the full momentum space
----------------------------------------------------
Let us firstly recall basic properties of multiplicity fluctuations of negative particles in the full momentum space ($4\pi$ fluctuations) in the three standard ensembles, of the Boltzmann pion gas considered here.
Multiplicity fluctuations in the CE are suppressed due to exact charge conservation. For a neutral ($Q=0$) system one finds in the large volume limit $\omega_{4\pi}^{ce} = 0.5$ [@CEfluc1]. Further suppression of fluctuations arise from additionally enforcing exact energy conservation in the MCE. Here one finds $\omega_{4\pi}^{mce} \approx 0.25$ for a Boltzmann pion gas at $T\approx 160MeV$. In the GCE, since no conservation laws are enforced, we always find a Poisson distribution with width $\omega_{4\pi}^{gce} =1$.
Since charge conservation in CE links the distributions of negatively charged particles to the one of their positive counterparts, i.e. $P(N_-) = P(N_+-Q)$, the relative width of $P(N_-)$ increases (decreases) as we move the electric charge density to positive (negative) values [@CEfluc2]. This can be easily be seen from | 1 | member_8 |
Eq.(\[matrix\]) by crossing out all rows and columns containing energy and momentum and calculating the asymptotic scaled variance of negatively charged particles, $\omega^{ce}_{4\pi}$, from Eq.(\[SimpleOmega\]), $$\omega_{4\pi}^{ce}~=~ \frac{\kappa_2^{N_{4\pi},N_{4\pi}}\kappa_2^{Q,Q}-\left(
\kappa_2^{N_{4\pi},Q} \right)^2}{\kappa_1^{N_{4\pi}}~ \kappa_2^{Q,Q}} ~=~
\frac{\exp \left( \frac{\mu}{T} \right)}{2\cosh \left(
\frac{\mu}{T}\right)}~.$$ The same effect is present in the MCE, however the calculation is slightly longer.
Results for $4\pi$ multiplicity fluctuations of negatively charged particles in a Boltzmann pion gas at $T=160MeV$ and different charge densities are summarized in Table \[table\]. Additionally estimates, based on our previously employed ‘uncorrelated particle‘ approach, Eq(\[accscaling\]), for multiplicity fluctuations with limited acceptance are given.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\quad \omega^{gce}_{4\pi} \quad$ $\quad \omega^{ce}_{4\pi} \quad$ $ $ \quad \omega^{gce}_{q=1/9} \quad$ $\quad \omega^{ce}_{q=1/9} \quad$ $\quad \omega^{mce}_{q=1/9} \quad$
\quad \omega^{mce}_{4\pi} \quad$
-------------------- ----------------------------------- ---------------------------------- ---------------------------------- ------------------------------------- ----------------------------------- ------------------------------------
$\mu=0$ $1$ $0.5$ $0.235$ $1$ $0.944$ $0.915$
$\mu=-\frac{m}{2}$ $1$ $0.294$ $0.147$ $1$ $0.922$ $0.905$
$\mu=+\frac{m}{2}$ $1$ $0.706$ $0.353$ $1$ $0.967$ $0.928$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Multiplicity fluctuation of $\pi^-$ in a classical pion gas in the large volume limit in the three standard ensembles at $T=160MeV$ for different charge densities. The index ‘$4\pi$‘ denotes fluctuations in the full momentum space, while the index ‘$q=1/9$‘ assumes acceptance scaling, Eq.(\[accscaling\]). The ratio $n_-/n_{tot}$ equals to $0.33$ for $\mu=0$, $0.48$ for $\mu=-m/2$, | 1 | member_8 |
and $0.20$ for $\mu=+m/2$.[]{data-label="table"}
Despite the fact that $\omega_{4\pi}$ is very different in GCE, CE, or MCE and also rather sensitive to the charge density, the estimates for limited acceptance ($q=1/9$) based on Eq.(\[accscaling\]) vary only by a few %. In order to decisively distinguish predictions for different ensembles a large value of $q$ would be needed.
Multiplicity fluctuations in limited segments of momentum space
---------------------------------------------------------------
In Section \[GCECE\] we have seen that in the Boltzmann CE multiplicity fluctuations observed in a limited segment of phase space are insensitive to the position of this segment. The dependence on the size of the segment can thus be taken into account by use of acceptance scaling Eq.(\[accscaling\]). To balance charge a particle can be produced or annihilated anywhere in momentum space. And due to a infinitely large heat and momentum bath in the CE no momentum state is essentially preferred.
In the MCE this dependence is qualitatively different. When using the MCE formulation particles are correlated due to the constraints of exactly conserved energy and momentum, even in the large volume limit. Fluctuations in a macroscopic subsystem are strongly affected by correlations with the remainder of the system.
In Fig. \[dodp\] we | 1 | member_8 |
---
abstract: 'The profiles of the chromo-electric field generated by static quark-antiquark, $Q{\bar Q}$ and three-quark, $QQQ$ sources are calculated in Coulomb gauge. Using a variational ansatz for the ground state, we show that a flux tube-like structure emerges and combines to the “Y”-shape field profile for three static quarks. The properties of the chromo-electric field are, however, not expected to be the same as those of the full action density or the Wilson line and the differences are discussed.'
author:
- 'Patrick O. Bowman and Adam P. Szczepaniak'
title: ' Chromo-Electric flux tubes '
---
Introduction
============
An intuitive picture of quark-gluon dynamics emerges in the Coulomb gauge, $\na\cdot \A^a = 0$ [@clee; @zw1; @zw2]. In this case QCD is represented as a many-body system of strongly interacting physical quarks, antiquarks and gluons. In particular the gluon degrees of freedom have only the two transverse polarizations and in the non-interacting limit reduce to the physical massless plane wave states. In the interacting theory gluonic states, just like any other colored objects, are expected to be non-propagating, [*i.e.*]{} confined on the hadronic scale. The non-propagating nature of colored states follows from the infrared enhanced dispersion relations which can be set | 1 | member_9 |
up in the Coulomb gauge [@zw2; @as1; @as2; @as3].
In the Coulomb gauge the $A^0$ component of the 4-vector potential results in an instantaneous interaction (potential) between color charges. Unlike QED, where the corresponding potential is a function only of the relative distance between the electric charges, in QCD it is a functional of the transverse gluon components, ${\bf A}$ [@clee]. Thus the numerical value of the potential cannot be obtained without knowing the correct wave functional of the state and its dependence on the gluon coordinates. So in QCD the chromo-electric field is expected to be non-local and to depend on the global distribution of charges, which set up the gluon wave functional.
Even though the exact solution to the general many-body problem is unavailable it is often possible to obtain good approximations if the dominant correlations can be identified. In Coulomb gauge QCD (in the Schrödinger field representation) the domain of the transverse gluon field, $\A$ is bounded and non-flat, and is referred to as the Gribov region. It is expected that the strong interaction between static charges originates from the long-range modes near the boundary of the Gribov region, the so called Gribov horizon. For example it | 1 | member_9 |
has been recently shown that center vortices, when transformed to the Coulomb gauge, indeed reside on the Gribov horizon [@goz].
The curvature of the Gribov region contributes to matrix elements via the functional measure determined by the determinant of the Faddeev-Popov operator. This determinant prevents analytical calculations of functional integrals, however it has been shown that its effect can be approximated by imposing appropriate boundary conditions on the gluon wave functional [@as4; @hr]. This wave functional is in turn constrained by minimizing the expectation value of the energy density which leads to a set of coupled self-consistent Dyson equations [@as1; @sw]. Once the wave functional is determined it is possible to calculate the distribution of the chromo-electric field in the system. This is the main subject of this paper.
In the following we study the chromo-electric field in the presence of the static quark-antiquark and three-quark systems, prototypes for a meson and a baryon respectively. Recent lattice computations indicate that the gluonic field near the static $Q-{\bar Q}$ state forms flux tubes. There are also indications that for the $QQQ$ state the fields arrange in the so called “Y”-shape [@tak; @latY], although some work supports the “$\Delta$”-shape [@latD]. String-like behavior | 1 | member_9 |
has been observed in the chromo-electric field in Ref. [@sho] and the “Y”-shape interaction advocated in Ref. [@kuzsim]. A recent reevaluation of the center-vortex model also supports the “Y”-shape [@corn].
In the following section we summarize the relevant elements of the Coulomb gauge formalism and discuss the approximations used. This is followed by numerical results and outlook of future studies. There is a fundamental difference between lattice gauge flux tubes corresponding to the distribution for the action density and the chromo-electric field profiles. In the context of the potential energy of the sources, this difference was emphasized in Zwanziger, Greensite and Olejnik [@goz; @zwancon]. We discuss those in Section. IV.
Chromo-electric Coulomb field in the presence of static charges
===============================================================
The Coulomb gauge Hamiltonian
-----------------------------
The Yang-Mills Coulomb gauge Hamiltonian in the Schrödinger representation, $H=H(\Pi, A)$ is given by $$H = \frac{1}{2} \int d\x \left[ \bm{\Pi}^a(\x) \cdot \bm{\Pi}^a(\x)
+ \B^a(\x)\cdot \B^a(\x) \right] + {\hat V}_C. \label{h}$$ The gluon field satisfies the Coulomb gauge condition, $\na \cdot \A^a(\x)=0$, for all color components $a=1\cdots N_c^2-1$. The conjugate momenta, $\bm{\Pi}^a(\x)=
-i\partial/\partial {\A^a(\x)}$ obey the canonical commutation relation, $[\Pi^{i,a}(\x), A^{j,b}(\y)] = -i\delta_{ab}
\delta^{ij}_T(\na_\x)\delta(\x-\y)$, with $\delta^{ij}_T(\na) =
\delta_{ij} - \nabla_i \nabla_i/\na^2$. The canonical momenta also | 1 | member_9 |
correspond to the negative of the transverse component of the chromo-electric field, $\bm{\Pi}^a(\x) = -
\E^a_T(\x)$, $\na \cdot \E^a_T = 0$. The chromo-magnetic field, $\B$ contains linear and quadratic terms in $\A$. It will also be convenient to transform to the momentum space components of the fields by $$\A^a(\k) = \int d\x\A^a(\x) e^{-i\k\cdot x},$$ and similarly for $\bm{\Pi}^a(\k)$. The Coulomb potential ${\hat V}_C$ may be expressed in terms of the longitudinal component of the chromo-electric field, $${\hat V}_C = \frac{1}{2}\int d\x {\bf E}^a(\x) {\bf E}^a(\x),$$ with $${\bf E}^a(\x) = \int d\y d\z { \frac{\x - \y}{4\pi|\x - \y|^3}}
\left[ \frac{g}{1 - \lambda}\right]^{ab}_{\y,\z} \rho^b(\z). \label{e}$$ Here $(1-\lambda)$ is the Faddeev-Popov (FP) operator which in the configuration-color space is determined by, $$[\lambda]^{ab}_{\x,\y} = \int {\frac{d\p}{(2\pi)^3}} {\frac{d\q}{(2\pi)^3}}
e^{i\p\cdot\x} e^{-i\q\cdot\y}
\lambda_{ab}(\p,\q),$$ where $$\lambda_{ab}(\p,\q) = ig f_{acb} {\frac{\A^c(\p-\q)\cdot \q}{\q^2}},$$ $f$ are the $SU(N_c)$ structure constants, and $g$ is the bare coupling. In [Eq. (\[e\])]{}, $\rho$ is the color charge density given by $$\rho^a(\x) = \psi^{\dag}(\x)T^a \psi(\x) + f_{abc} \A^b(\x)\cdot
\bm{\Pi}^c(\x),$$ with the two terms representing the quark and the gluon contribution, respectively; the former is replaced by a $c$-number for static quarks. Without light flavors there is no other dependence on the quark degrees of | 1 | member_9 |
freedom. The energy of the static $Q{\bar Q}$ or $QQQ$ systems measured with respect to the state with no sources is thus given by the Coulomb term and is determined by the expectation value of the longitudinal component of the chromo-electric field.
It is the dependence of the chromo-electric field and the Coulomb interaction on the static vector potential (through $\lambda$) that produces the differences between QCD and QED. In QED the kernel in the bracket in [Eq. (\[e\])]{} reduces to $[\cdots] \to
\delta(\y-\z)$ and the Abelian expression for the electric field emerges. In QCD the chromo-electric field and the Coulomb potential are enhanced due to long-wavelength transverse gluon modes on the Gribov horizon where the FP operator vanishes. The combination of two effects on the Gribov horizon: enhancement of $(1 - \lambda)^{-1}$ in the longitudinal electric field and vanishing of the functional norm, which is proportional to $\det(1-\lambda)$, leads to finite, albeit large, expectation values of the static interaction between color charges. In [Eq. (\[h\])]{} we have omitted the FP measure since, as mentioned earlier in Ref. [@as4], its effect can be approximately accounted for by imposing specific boundary conditions on the ground state wave functional.
Since the chromo-electric | 1 | member_9 |
field depends on the distribution of the transverse vector potential it is necessary to know the wave functional of the system. A self-consistent variational ansatz can be chosen in a Gaussian form,
$$\Psi[A] = \exp\left( - \frac{1}{2}
\int {\frac{d\p}{(2\pi)^3}} \omega(p)
\A^a(\p)\cdot \A^a(-\p) \right). \label{varia}$$
The parameter $\omega(p)$ ($p\equiv |\p|$) is determined by minimizing the expectation value of the energy density of the vacuum ([*i.e.*]{} without sources). The boundary condition referred to above corresponds to setting $\omega(0) \equiv \mu$ to be finite, which plays the role of $\Lambda_{QCD}$, [*i.e*]{} it controls the position of the Landau pole. Minimizing the energy density of the vacuum leads to a set of coupled self-consistent integral equations: one for $\omega$, one for the expectation value of the inverse of the FP operator, $d(p)$, $$\begin{gathered}
(2\pi)^3 \delta(\p-\q)\delta_{ab} d(p)
\equiv \\
\int d\x d\y e^{-i\p\cdot\x} e^{i\q\cdot\y}
\langle \Psi| \left[ \frac{g}{1 - \lambda}\right]^{ab}_{\x,\y}
|\Psi \rangle / {\langle \Psi|\Psi \rangle },
\label{d}\end{gathered}$$ and one for the expectation value of the square of the inverse of the FP operator, which appears in the matrix elements of $V_C$, $$\begin{gathered}
(2\pi)^3 \delta(\k-\q)\delta_{ab} f(p)d^2(p)
\equiv \\
\int d\x d\y e^{-i\p\cdot\x} e^{i\q\cdot\y}
\langle \Psi| \left[ \left(\frac{g}{1 - \lambda}\right)^2
\right]^{ab}_{\x,\y}
|\Psi \rangle / {\langle \Psi|\Psi | 1 | member_9 |
\rangle }.
\label{ff}\end{gathered}$$ The approximation $f=1$ ignores the dispersion in the expectation value of the inverse of the FP operator, $$\Bigg\langle \left[ \frac{g}{1 - \lambda} \right]^2 \Bigg\rangle
\to \Bigg\langle \frac{g}{1 - \lambda} \Bigg\rangle^2. \label{disp}$$ This approximation has been extensively used, [*e.g.*]{} in Refs. [@zw1; @zwan]. The three Dyson equations were analyzed in Ref. [@as1] where it was found that the solution of $\omega$ can be well approximated by the simple function $\omega(p) = \theta( \mu - p) \mu + \theta( p -\mu)
p$. The renormalization scale $\mu$, being the only parameter in the theory, can constrained by the long range part of the Coulomb kernel $\langle V_C \rangle \propto fd^2$. We will discuss this more in the subsection below. The low momentum, $p<\mu$ dependence of $d(p)$ and of the Coulomb potential $V_C(p) = f(p)d(p)^2$ is well approximated by a power-law, $$d(p) = d(\mu) \left( \frac{\mu}{p}\right)^\alpha, f(p) = f(\mu)
\left( \frac{\mu}{p}\right)^\beta \label{df}$$ with $\alpha \sim 0.5$ and $\beta \sim 1$. The exponents are bounded by $ 2\alpha + \beta \le 2$ and the upper limit corresponds to the linearly rising confining potential. At large momentum, $p >> \mu$, as expected from asymptotic freedom, both $d$ and $f$ are proportional to $1/\log^\gamma(p)$, | 1 | member_9 |
with $\gamma = O(1)$. Adding static sources does not modify the parameters of the vacuum gluon distribution, [*e.g.*]{} $\omega(p)$. This is because the vacuum energy is an extensive quantity while sources contribute a finite amount to the total energy. Thus we can use the three functions $\omega$, $d$ and $f$ calculated in the absence of the sources to compute the expectation value of the chromo-electric field in the presence of static sources. The ansantz state obtained by applying quark sources to the variational vacuum of Eq. (\[varia\]) does not, however optimize the state with sources.
The field lines in the $Q{\bar Q}$ and $QQQ$ systems
------------------------------------------------------
For a quark and an antiquark at positions $\x_q \equiv \R/2 =
R{\hat \z}/2$ and $\x_{\bar q} = -\R/2 = -R{\hat \z}/2$, respectively, and the gluon field distributed according to $\Psi[\A]$, the expectation value of the square of the magnitude of the chromo-electric field measured at position $\x$ is given by
$$\begin{gathered}
\langle \E^2(\x,\R) \rangle = { \frac{C_F}{(4\pi)^2}}
\sum_{\z_1=\pm \R/2} \sum_{\z_2 =\pm \R/2} \pm
\int d\y_1 d\y_2 \\
\times{ \frac{ (\x - \y_1)\cdot (\x-\y_2)}{|\x - \y_1|^3
|\x - \y_2|^3}} E(\z_1,\y_1;\z_2,\y_2), \label{qq}\end{gathered}$$
where the $+ (-)$ sign is for the $\z_1 =(\ne) \z_2$ contributions, and | 1 | member_9 |
$$E(\z_1,\y_1;\z_2,\y_2)
\equiv \frac{\langle \Psi |
\left[ \frac{g}{1 - \lambda}\right]_{\z_1,\y_1}
\left[ \frac{g}{1 - \lambda}\right]_{\y_2,\z_2}|\Psi \rangle }
{ \langle \Psi|\Psi \rangle }. \label{ee}$$ The color factors leading to $C_F$ can be extracted from the expectation value in [Eq. (\[ee\])]{} (the ground state expectation value of the inverse of two FP operators is an identity in the adjoint representation). In the Abelian limit, $E(\z_1 \cdots \y_2) \to \delta(\y_1 -
\z_1)\delta(\y_2 - \z_2)$ and Eq. (\[qq\]) gives the dipole field distribution, $\langle \E^2 \rangle_{QED}$. One should note that [Eq. (\[qq\])]{} contains the two self energies. These self energies are necessary to produce the correct asymptotic behavior at $x >>
R$ for charge-neutral systems, (in QED and QCD) [*i.e*]{} $\E^2$ has to fall-off at least as $1/\x^4$ at large distances from the sources.
The infrared, $|\x| \sim |\R| >> 1/\mu$ enhancement in QCD arises from the expectation value of the inverse of the FP operator. If $\langle \E^2(\x,\R) \rangle$ is integrated over $\x$ one obtains the expectation value of the Coulomb energy of the $Q{\bar Q}$ source. The mutual interaction energy is given by, $$\begin{aligned}
V_C(\R) &= {1\over 2}\int d\x \langle \E^2(\x,\R) \rangle \nonumber \\
&\hspace{-5mm}= -C_F {{\langle \Psi |
\left[ {g\over {1 - \lambda}} | 1 | member_9 |
\left(-{1\over \na^2}\right)
{g\over {1 - \lambda}}\right]_{{\R\over 2},-{\R\over 2}}
|\Psi \rangle }
/{ \langle \Psi|\Psi \rangle }
}, \nonumber \\
&\hspace{-5mm} = -C_F \int {{d\p}\over {(2\pi)^3}} {{d^2(p) f(p)}\over {p^2}}
e^{i\p\cdot \R},
\label{vc} \end{aligned}$$ and the net self-energy contribution is, $$\begin{aligned}
\Sigma &= C_F {{\langle \Psi |
\left[ {g\over {1 - \lambda}} \left(-{1\over \na^2}\right)
{g\over {1 - \lambda}}\right]_{\pm {\R\over 2},\pm {\R\over 2}}
|\Psi \rangle }
/{ \langle \Psi|\Psi \rangle } }, \nonumber \\
&= C_F \int {{d\p} \over {(2\pi)^3}} {{d^2(p) f(p)}\over {p^2}}.
\label{sigma} \end{aligned}$$ In lattice simulations it has been shown [@Greensite] that the Coulomb energy and the phenomenological static $Q\bar Q$ potential obtained from the Wilson loop are different. In particular it was found that the Coulomb potential string tension is about three times larger than the phenomenological string tension. This is in agreement with the “no confinement without Coulomb Confinement” scenario discussed by Zwanziger [@zwancon]. It is simple to understand the origin of the difference. Even if $|\Psi[\A]\rangle $ were the true vacuum state (without sources) of the Coulomb gauge QCD Hamiltonian (here we approximate it by a variational ansatz) the state $|Q{\bar Q},R\rangle \equiv Q(\R/2){\bar Q}(-\R/2) |\Psi[A]\rangle$ is no longer an eigenstate. For example ${\hat V}_C$ acting on $|Q{\bar
| 1 | member_9 |
Q},R\rangle$ excites any number of gluons and couples them to the quark sources. The Coulomb energy was defined as the expectation value, $V_C$ in $|Q{\bar Q},R\rangle$ minus the vacuum energy and it is therefore different from the phenomenological static potential energy which corresponds to the total energy (measured with respect to the vacuum) of the true eigenstate of the Hamiltonian with a $Q{\bar Q}$ pair. If one defines [@goz] $$\begin{aligned}
G(R,T) &\equiv \langle Q{\bar Q},R|e^{-(H-E_0)T}|Q{\bar Q},R\rangle
\nonumber \\
& = \sum_n |\langle Q {\bar Q},R,n|Q{\bar Q},R\rangle|^2 e^{-(E_n -
E_0) T},\end{aligned}$$ then the Coulomb potential on the lattice can be calculated from $$V_C(R) = \lim_{T=0} -{d\over {dT}} \log(G(R,T)),$$ and the phenomenological potential from $$V(R) = \lim_{T=\infty} -{d\over {dT}} \log(G(R,T)).$$ Thus one should be comparing $V_C(R)$ in [Eq. (\[vc\])]{} to the lattice Coulomb potential and not to the phenomenological potential obtained from the Wilson loop. Finally, one could try to optimize the state with sources, [*e.g.*]{} by adding gluonic components. In this case terms in the Hamiltonian beyond the Coulomb term would contribute to the energy of the system and one could compare with the true (Wilson loop) static energy. In our previous studies, where we extracted numerical values for $\mu$ and | 1 | member_9 |
the critical exponents $\alpha,\beta,\gamma$ ([*cf.*]{} [Eq. (\[df\])]{}) we have instead compared $V_C$ to the phenomenological, Wilson potential [@as1]. In what follows we will use the larger value of the string tension, to be in agreement with Ref. [@goz].
If the two exponents $\alpha$ and $\beta$, which determine the infrared behavior of $d(p)$ and $f(p)$ respectively, satisfy $2\alpha + \beta > 2$, then the self energy in [Eq. (\[sigma\])]{} is divergent and so is the [*rhs*]{} of [Eq. (\[vc\])]{}. This reflects the long-range behavior of the effective confining potential generated by self-interactions between the gluons that make up the Coulomb operator. For the colorless $Q\bar Q$ system the total energy which is the sum of $V_C$ and $\Sigma$, is finite as it should be. For a colored system, [*e.g.*]{} a quark-quark source, the sign of $V_C$ changes, there is no cancellation between the infrared singularities, and in the confined phase the system would be un-physical with infinite energy. The integral determining the self energy also becomes divergent in the UV, since for $p\to \infty$ the product $d^2(p)f(p)$ only falls-off logarithmically. Modulo these logarithmic corrections this UV divergence is the same as in the Abelian theory and can be removed by renormalizing | 1 | member_9 |
the quark charge.
It follows from translational invariance of the matrix element in [Eq. (\[ee\])]{}, that $E$ depends only on the relative coordinates, $\z_1 - \y_1$ and $\z_2 - \y_2$. We therefore introduce the momentum space representation, $$\begin{aligned}
E(\z_1,\y_1;\z_2,\y_2) & = & \int {{d\p} \over {(2\pi)^3}}
{{d\q} \over {(2\pi)^3}} e^{i\p \cdot (\z_1 - \y_1)
-i \q\cdot (\z_2 - \y_2) } \nonumber \\
& & \qquad\times d(p) d(q) E(\p;\q), \label{ft}\end{aligned}$$ and define $F_\L(\l) \equiv E(\l+\L/2; \l-\L/2)$ with $\l \equiv (\p + \q)/2$ and $\L \equiv \p - \q$. The Dyson equation for $F$ can be derived in the rainbow-ladder approximation which, as shown in Ref. [@as1; @as2], sums up the dominant infrared and ultra-violet contributions to the expectation value of the inverse of two FP operators,
$$F_\L(\l)
= 1 + N_c \int {{d\k}\over {(2\pi)^3 }}
{{\left[ (\k-\L/2)\delta_T(\k + \l)(\k+\L/2) \right] }
\over {2\omega(\k + \l)}}
{{d(\k - \L/2)}\over {(\k - \L/2)^2}}
{{d(\k + \L/2)}\over {(\k + \L/2)^2}} F_\L(\k), \label{F}$$
It follows from [Eq. (\[ft\])]{} that $\L$ and $\l$ are conjugate to the [*center of mass*]{}, $\R \equiv [ (\z_1 - \y_1) + (\z_2 - \y_2)
]/2$ and the [*relative*]{}, $\r \equiv [ (\z_1 - \y_1) - (\z_2 - \y_2)
]$ coordinate | 1 | member_9 |
respectively. The Dyson equation for $F_L$ is UV divergent if for $p/\mu >> 1$, and $d(p) \ge \log^{1/2}(p^2)$. This divergence can be removed by the Coulomb operator renormalization constant. The renormalized equation is obtained from the once-subtracted equation $F_\L(\l) - F_{\L_0}(\l_0)$. For example, if the subtraction is chosen at $|\l_0| = \mu$ and $\L_0 = {\bf 0}$, the renormalized coupling $F_0(\mu)$ can be fixed from the Coulomb potential. After integrating [Eq. (\[qq\])]{} (over $\x$) one obtains $\delta(\y_1 -
\y_2)$ multiplying $E(\z_1, \cdots, \y_2)$. Therefore, it follows from [Eq. (\[ft\])]{} that $V_C(\R)$ is determined by $F_0(\l)$ and $F_0(\l) = F_0(l) = f(l)$ with $f$ defined in [Eq. (\[ff\])]{}.
In [Eq. (\[F\])]{} $\L$ is a parameter, [*i.e.*]{} the Dyson equation does not involve self-consistency in $\L$. We have just shown that as $\L \to {\bf 0}$, $F_\L(\l)$ has a finite limit: it is given by $f$. For large $L=|\L|$ ($L/\mu >> 1$), due to asymptotic freedom, $F_\L$ is expected to vanish logarithmically, $F_\L \to d^2(L) \propto 1/\log(L^2)$. We do not attempt here to solve [Eq. (\[F\])]{}, instead we use a simple interpolation formula between the $L=0$ and $L\to \infty$ limits,
$$F_\L(\l) =
f(\l) \theta(\mu - |\l|) \theta(\mu -|{\L\over 2}|) + \left[ 1 | 1 | member_9 |
-
\theta(\mu - |\l|)\theta(\mu - |{\L\over 2}|) \right]
\sim f\left( {\p + \q}\over 2 \right) \theta(\mu - p) \theta( \mu - q)
+ \left[ 1 - \theta(p) \theta(q) \right], \label{f}$$
[*i.e.*]{} in the term in the bracket we ignore the short distance logarithmic corrections. It is easy to show that if logarithmic corrections are ignored then the short-range, $p,q > \mu$ contribution to the energy density is the same as in the Abelian case. Since we are mainly interested in the long range behavior of the chromo-electric field, in the following we shall ignore contributions from the region $p,q>\mu$ all together. In the long-range approximation, $x, R >> 1/\mu$ the expectation value of $\E^2$ is then given by,
$$\langle \E^2(\x,\R) \rangle =
{{C_F}\over {(4\pi)^2}}
\sum_{ij=1}^2 \xi^{Q\bar Q}_{ij} \int d\r f_L(r)
{{\z_i - \x - \r/2} \over {|\z_i - \x - \r/2|}}
\cdot {{\z_j - \x + \r/2} \over {|\z_j - \x + \r/2|}}
d'_L(\z_i - \x - \r/2) d'_L(\z_j - \x + \r/2).
\label{el}$$
$\xi^{Q\bar Q}_{ij} = 1$ for $i=j$ and $-1$ for $i\ne j$, $\z_{1,(2)} = (-)\R/2$, $$d'_L(r) \equiv {2\over {\pi}}\int^\mu_0 p dp j_1(rp) d(p), \label{dpl}$$ is the derivative of $d_L$ w.r.t. $r$, $$f_L(r) = {1\over {2\pi^2}} \int^\mu_0 dp | 1 | member_9 |
p^2 f(p) j_0(pr), \label{fl}$$ and $j_0, j_1$ are Bessel’s functions. We note that the expression in [Eq. (\[el\])]{} is not necessarily positive. In the limit $f(p)=1$, the matrix element of the square of the inverse of the FP operator is approximated by the square of matrix elements (cf. [Eq. (\[disp\])]{}) and $\langle \E^2\rangle$ becomes positive.
The expression for $\langle \E^2\rangle$ for the three quark system is derived by taking the expectation value of the Coulomb operator, ${\hat V}_C$ in a color-singlet state $\epsilon_{ijk}Q_i(\z_1) Q_j(\z_2)
Q_k(\z_3) |\Psi[\A]\rangle$, which gives
$$\langle \E^2(\x,\R_i) \rangle = {{C_F}\over {(4\pi)^2}}
\sum_{ij=1}^3 \xi^{QQQ}_{ij} \int d\r f_L(r) {{ \z_i - \x - \r/2 } \over
{ |\z_i - \x - \r/2| } }
\cdot
{{ \z_j - \x + \r/2 } \over
{ |\z_j - \x + \r/2| }} d'_L(\z_i - \x - \r/2)
d'_L(\z_j - \x + \r/2)$$
where $\xi^{QQQ}_{ij} = 1$ if $i = j$ and $\xi^{QQQ}_{ij} = -1/2$ if $i \ne j$. We note that the energy density for the $QQQ$ system comes from two-body correlations between the $QQ$ pairs.
Numerical results
===================
We first consider the simple approximation to the expectation value of the Coulomb kernel of [Eq. (\[disp\])]{} in which $f(p)=1$. If one | 1 | member_9 |
wishes to have the confining potential grow linearly at large distances then it is necessary to set $\alpha = 1$, [*i.e.*]{} $d(p) \propto \mu/p$ for $p/\mu < 1$. In this case, assuming that the long-range behavior of the potential is of the form $V_C(r) = b_C r$, we obtain from [Eq. (\[vc\])]{}, $$b_C = C_F d^2(\mu)\mu^2/(8\pi). \label{bc}$$ We use the Coulomb string tension $b_C = 0.6 \mbox{ GeV}^2$. For the $Q\bar Q$ system the long-range contribution to the electric fields is then given by, $$\begin{gathered}
\langle \E^2(\x,\R) \rangle =
{{2b_C}\over {\pi^3}}
\biggl[
{{(\R/2 - \x)} \over {|\R/2 -
\x|^2}}\left(1-j_0(\mu|\R/2-\x|)\right) \\
+ (\x \to -\x) \biggr]^2. \label{elnof}\end{gathered}$$
![\[fig1\] $R^2 \langle \E^2(x) \rangle$ in units of $2b_C/\pi^3(\hbar c)^2 $ as a function of the distance $x$ along the $Q{\bar Q}$ axis. We employ the $f(p) = 1$ approximation. The quark and the antiquark are located at $R/2=5\mbox{ fm}$ and $-R/2 = -5\mbox{ fm}$ respectively. The renormalization scale $\mu=1.1\mbox{ GeV}$ is calculated from [Eq. (\[bc\])]{} using $d(\mu) = 3.5$ from Ref. [@as1]. The dashed line is the contribution from the two self energies, the dash-dotted line represents mutual interactions and the solid line is the total.](Fig1a.eps){width="3.in"}
In Fig. 1 we show the Coulomb | 1 | member_9 |
energy density as a function of position on the $Q{\bar Q}$ axis, $\x = x\hat{\R}$, for $R=|\R|=10\mbox{ fm}$. The small oscillations come from the sharp-cutoff introduced by the $\theta$-functions in [Eq. (\[f\])]{} which produces the Bessel’s functions in [Eq. (\[elnof\])]{}. For a smooth cutoff, [*e.g.*]{} with $\theta(\mu - p) \to \exp(-p/\mu)$ in [Eq. (\[elnof\])]{} one should replace $1 - j_0(\mu|\R/2-\x|)$ by $1 - \arctan(\mu|\R/2-\x|)/\mu|\R/2-\x|$. The cut-off is also responsible for the rapid variations near the quark positions, $\x = \pm R/2$.
We note that for large separations between the quarks, $R >> 1/\mu$ and $x << R$, the Coulomb energy density behaves as expected from dimensional analysis, $$\langle \E^2(\x,R\mu\to \infty) \rangle \to {{32 b_C}\over {\pi^3R^2}}.
\label{short}$$ which is consistent with linear confinement, [*i.e.*]{} if $\langle \E^2(\x,R\mu\to \infty) \rangle$ is integrated over $\x$ in the region $|\x|<R$ on obtains $V_C(R) \propto R$.
At large distances $x >> R >> 1/\mu$ we obtain $$\langle \E^2(|\x|/R \to \infty,R\mu\to \infty) \rangle \to
{{2 b_C R^2}\over {\pi^3\x^4}}. \label{inf}$$ If there were a finite correlation length one would expect $\langle \E^2(|\x|/R \to \infty,R\mu\to \infty)\rangle$ to fall-off exponentially with $|\x|$ [@DDS] and not as a power-law. The power-law behavior obtained in Eq. (\[inf\]) is again related to | 1 | member_9 |
the difference between the $|Q{\bar Q}, R\rangle$ state used here, which is built by adding quark sources to the vacuum and the true ground state of the $Q{\bar Q}$ system as discussed in Sec. IIA. In other words the profile of the chromo-electric field distribution for such a state is not expected to agree with the profile of the flux-tube or action density. To illustrate this difference, in Fig. 2 we plot the energy density as a function of the magnitude of the distance transverse to the $Q\bar Q$ axis, $x_\perp = |\x_\perp|$, $\R\cdot \x = \R\cdot \x_\perp = 0$.
![\[fig1b\] $R^2 \langle \E^2(x) \rangle$ in units of $2b_C/\pi^3(\hbar c)^2 $ as a function of the distance $x$ transverse to the $Q{\bar Q}$ axis. The units and the setting are as in Fig. 1.](Fig1b.eps){width="3.in"}
Finally, in Fig. 3, we show the contour plot of the energy density as a function of the position in the $xz$ plane with quark and antiquark on the $z$ axis at $R/2$ and $-R/2$ respectively.
![\[fig2\] $R^2 \langle \E^2(x) \rangle$ as a function of position in the $xz$ plane. The units and the same setting as in Fig. 1. ](Fig2.eps){width="3.in"}
It is clear from Figs. | 1 | member_9 |
2 and 3 that a flux tube like structure emerges and from [Eq. (\[short\])]{} that it has the correct scaling as a function of the $Q{\bar Q}$ separation but, as discussed above it does not have a finite correlation length (large $x$ behavior).
The field distribution for the $QQQ$ system in the $f_L(p) =1 $ approximation is equal to the sum of three terms each representing a contribution from a $QQ$ pair. We place each of the three quarks in a corner of an equilateral triangle, $\z_i$, $i=1,\cdots3$ $$\begin{gathered}
\langle \E^2(\x,\R_i) \rangle = \frac{C_F}{32\pi^2}
\bigl[ (\D_1 - \D_2)^2 \\
+ (\D_1 - \D_3)^2 + (\D_2 - \D_3)^2 \bigr],\end{gathered}$$ where $$\D_i = {{ \z_j - \x + \r/2 } \over
{ |\z_j - \x + \r/2| }} d'_L(\z_i - \x - \r/2).$$
The contour plot of of energy density in this case is shown in Fig. 4. Even though the field originates from the two-particle correlations the net field seems to form into a “Y”-shape structure. This structure has also recently been seen to emerge in Euclidean lattice simulations.
![\[fig3\] $R^2 \langle \E^2(x) \rangle$ as a function of position in the $xz$ plane. The units and the same setting as in | 1 | member_9 |
Fig. 1. The upper panel show the total field distribution and the lower the distribution from mutual interaction (no self-energies) only. ](Fig3.eps){width="2.5in"}
Finally, to study the effects of $f_L(p)$, in Fig. 5 we show the predictions for the $Q{\bar Q}$ field distribution given by [Eq. (\[vc\])]{} where we use $d(p)$ and $f(p)$ in the form given by [Eq. (\[df\])]{} with $\alpha = 1/2$ and $\beta=1$ and normalized such that $V(R) \to bR$ at large distances. Furthermore, to remove the oscillations introduced by the momentum space cutoff, we now cut the small $x$ region in coordinate space, by i) extending the upper limits of integration in Eqs. (\[dpl\]) and (\[fl\]) to infinity and ii) cutting off the position space functions at short distances, $$d'_L(r) = {2\over {\pi\alpha}} \sin(\pi\alpha/2)
\Gamma(2 -\alpha) \theta(r\mu - 1)
{ {\mu^2} \over {(\mu r)^{2-\alpha}}}$$ $$f_L(r) = {1\over {2\pi^2}} \sin(\pi\beta/2)
\Gamma(2-\beta) \theta(r \mu - 1)
{ {\mu^3} \over {(\mu r)^{3-\beta}}}.$$
Comparing Fig. 3 and Fig. 5 we observe a narrowing of the flux tube. This is to be expected as the action of $f(p)$ is to introduce additional gluonic correlations. That said, there is no major qualitative change in the field distribution.
![\[fig4\] $R^2 \langle \E^2(x) \rangle$ for | 1 | member_9 |
$Q{\bar Q}$ from [Eq. (\[el\])]{} with $\alpha = 1/2$ and $\beta=1$. The units and the same setting as in Fig. 1, except that the contribution from $f(p)$ has been included.](Fig4.eps){width="2.5in"}
Summary
=======
We have calculated the distribution of the longitudinal chromo-electric field in the presence of static $Q{\bar Q}$ and $QQQ$ sources using a variational model for the ground state wave functional. Despite this wave functional having no string-like correlations a flux tube like picture does emerge. In particular the on-axis energy density of the $Q{\bar Q}$ system behaves as $b_c/R^2$ for large inter-quark separation, $R$ and the field falls off like $b_c R^2/x^4$ at large distances from the center of mass of the $Q{\bar Q}$ system, $x$. This is weaker than in the Abelian case ($\sim R^2/x^6$) and implies that moments of the average transverse spread of the tube, defined as proportional to $\langle |x_\perp|^n \E^2(z,x_\perp) \rangle$, are finite for $n<2$ only. Thus there is no finite correlation length for the longitudinal component of the chromo-electric field, as expected for the state which does not take into account screening of the Coulomb line by the transverse gluons (flux tube). This also leads to large Van der Waals forces, which | 1 | member_9 |
is bothersome, but it is consistent with the scenario of “no confinement without Coulomb confinement” of Zwanziger. The Coulomb potential leads to a variational (stronger) upper bound to the true confining interaction.
Similar behavior at large distances is also true for the three quark sources, except that here we find the emergence of the “Y”-shape junction. This is consistent with lattice simulations, but is remarkable in our case as it arises from two-body forces. It will be interesting to examine field distributions which include transverse field excitations. In that case the only lattice results available are for the potential, not for the field distributions. Finally we note that, since the mean field calculation provides a variational upper bound, the long range behavior of the field distribution falls-off more slowly than expected for the Van der Waals force. Certainly as the complete string develops this is expected to disappear and it would be interesting to build a string-like model for the ansatz ground state to verify this assertion.
The authors wish to thank J. Greensite, H. Reinhardt, Y. Simonov, F. Steffen, H. Suganuma and D. Zwanziger for helpful feedback. This work was supported in part by the US Department of Energy | 1 | member_9 |
under contract DE-FG0287ER40365. The numerical computations were performed on the AVIDD Linux Clusters at Indiana University funded in part by the National Science Foundation under grant CDA-9601632.
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C. Alexandrou, P. De | 1 | member_9 |
---
abstract: |
This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given $n$, sum the previous two terms and divide them by the largest possible power of $n$. The behavior of such sequences depends on $n$. We analyze these sequences for small $n$: 2, 3, 4, and 5. Surprisingly these behaviors are very different. We also talk about any $n$. Many statements about these sequences are difficult or impossible to prove, but they can be supported by probabilistic arguments, we have plenty of those in this paper.
We also introduce ten new sequences. Most of the new sequences are also related to Fibonacci numbers proper, not just free Fibonacci numbers.
author:
- |
Brandon Avila\
MIT
- |
Tanya Khovanova\
MIT
title: Free Fibonacci Sequences
---
Introduction {#sec:intro}
============
John Horton Conway likes playing with the Fibonacci sequence. Instead of summing the two previous terms, he sums them up and then adds a twist: some additional operation. John Conway discussed these sequences with the second author and this is how we got interested in them. The second author already wrote about one class of such sequences called subprime Fibonacci sequences jointly with | 1 | member_10 |
Richard Guy and Julian Salazar [@GKS]. Here we discuss another variation called $n$-free Fibonacci sequences.
An $n$-free Fibonacci sequence starts with any two integers: $a_1$ and $a_2$. After that it is defined by the recurrence $a_k = (a_{k-1} +a_{k-2})/n^i$, where $n^i$ is the largest power of $n$ that is a factor of $a_{k-1} +a_{k-2}$.
It appears that many other people like twisting Fibonacci sequences. After we started working on this paper and made our calculations we checked, as everyone should, the On-Line Encyclopedia of Integer Sequences (OEIS [@OEIS]) and discovered that some $n$-free Fibonacci sequences were already submitted by three other people. Surprisingly, the first sequence submitted was the sequence of 7-free Fibonacci numbers (A078414) entered by Yasutoshi Kohmoto in Dec 2002. After that the sequence of 5-free Fibonacci numbers (A214684) was submitted by John W. Layman in Jul 2012. It followed by the sequence of 4-free Fibonacci numbers (A224382) submitted by Vladimir Shevelev in Apr 2013. As the reader will see very soon 2-free and 3-free Fibonacci numbers do not constitute new sequences. We filled the gap and submitted 6-free Fibonacci numbers (A232666) in Nov 2013.
In Section \[sec:definitions\] we introduce useful facts about Fibonacci numbers. In Section \[sec:2free\] | 1 | member_10 |
we show that all 2-free sequences end in a cycle of length 1. The 3-free Fibonacci sequences are much more complicated and we study them in Section \[sec:3free\]. All our computational experiments ended in a cycle of length 3. On the other hand, we show that 3-free sequences may contain arbitrary long increasing substrings. We prove this in Section \[sec:customized\]. Nevertheless, we give a probabilistic argument that a 3-free sequence should end in a cycle in Section \[sec:3free\].
The 4-free Fibonacci sequences are vastly different from 2-free and 3-free sequences (Section \[sec:4free\]). We did not find a sequence that ends in a cycle: all of them grow in our experiments. The proof that all of them grow seems intractable, but we supply a probabilistic argument that this is the case. Yet 5-free sequence bring something new (see Section \[sec:5free\]). They contain sequences that are never divided by 5 and provably grow indefinitely. At the same time 5-free sequences contain cycles too.
We continue with Section \[sec:division-free\] where we find other numbers $n$ that provide examples of sequences that never need to be divided by $n$. Now we wonder where the cycles disappeared to and discuss there potential properties in Section | 1 | member_10 |
\[sec:cycles\].
We finish with a discussion of our computational results. Section \[sec:growthdivfree\] explains why the average growth for some $n$-free sequences is close to the golden ratio and Section \[sec:growthomni\] explains the growth behavior for other values of $n$.
Fibonacci Numbers and $n$-free Fibonacci Sequences {#sec:definitions}
==================================================
Let us denote *Fibonacci numbers* by $F_k$. We assume that $F_0=0$ and $F_1=1$. The sequence is defined by the Fibonacci recurrence: $F_{n+1} =F_n + F_{n-1}$ (See A000045). We call an integer sequence $a_n$ *Fibonacci-like* if it satisfies the Fibonacci recurrence: $a_k = a_{k-1} +a_{k-2}$. A Fibonacci-like sequence is similar to the Fibonacci sequence, except it starts with any two integers. The second-famous Fibonacci-like sequence is the sequence of *Lucas numbers* $L_i$ that starts with $L_0=2$ and $L_1=1$: 2, 1, 3, 4, 7, 11, $\ldots$ (See A000032).
An *$n$-free Fibonacci* sequence starts with any two integers: $a_1$ and $a_2$ and is defined by the recurrence $a_k = (a_{k-1} +a_{k-2})/n^i$, where $n^i$ is the largest power of $n$ that is a factor of $a_{k-1} +a_{k-2}$. To continue the tradition we call numbers in the $n$-free Fibonacci sequence that starts with $a_0=0$ and $a_1=1$ *$n$-free Fibonacci numbers*.
In the future we will consider only sequences starting | 1 | member_10 |
with two non-negative integers. It is not that we do not care about other starting pairs, but positive sequences cover all essential cases. Indeed, if we start with two negative numbers we can multiply the sequence by $-1$ and get an all-positive sequence. If we start with numbers of different signs, the sequence eventually will become the same-sign sequence.
If we start with two zeros, we get an all-zero sequence. So we will consider only sequences that do not have two zeros at the beginning. Note, that a non-negative sequence can have a zero only in one of the two starting positions, never later.
The $n$-free Fibonacci sequence coincides with the Fibonacci-like sequence with the same beginning until the first occurrence of a multiple of $n$ in the Fibonacci-like sequence.
Given a positive integer $m > 1$, the smallest positive index $k$ for which $n$ divides the $k$-th Fibonacci number $F_k$ is called the *entry point* of $m$ and is denoted by $Z(m)$ (see sequence A001177 of Fibonacci entry points). For example, $Z(10) = 15$ and the 10-free Fibonacci numbers coincide with the Fibonacci numbers for indices $ < 15$.
Now that all the preparation is done, let us take | 1 | member_10 |
a closer look at the simplest $n$-free Fibonacci sequences: 2-free Fibonacci sequences.
2-free Fibonacci Sequences {#sec:2free}
==========================
Consider some examples. The sequence that starts with 0, 1 continues as 1, 1, 1, .... The only two 2-free Fibonacci numbers are 0 and 1. The sequence eventually stabilizes, or in other words, turns into a cycle of length 1. Let us look at other starting points. The sequence that starts as 1, 2 continues as 3, 5, 1, 3, 1, 1, and stabilizes at 1. This sequence turns into the same cycle. The sequence that starts as 100, 220, continues as, 5, 225, 115, 85, 25, 55, 5, 15, 5, 5, 5, and stabilizes at 5. It turns into a different cycle, but the length of the cycle is again equal to 1.
Every 2-free Fibonacci sequence eventually turns into a cycle of length 1: $x$, $x$, $x$, $\ldots$, for an odd $x$.
It is clear that after the second term all elements of the sequence are odd. Consider the maximum of the two consecutive terms of the sequence: $m_k = \max\{a_k,a_{k-1}\}$. If two consecutive terms $a_{k-1}$, $a_k$ of the sequence are odd and not equal to each other, then the | 1 | member_10 |
maximum decreases: $m_{k+1} < m_k$. Hence, the sequence has to stabilize.
It follows from the proof that for a sequence starting with $a_1$, $a_2$ the number of steps until the cycle is reached is not more than $\max\{a_1,a_2\}$. On the other hand, the subsequence before the cycle can be arbitrary long. It follows from the following lemma.
For any two odd numbers $a_1$, $a_2$, a preceding odd number $a_0$ can be found so that $a_0$, $a_1$, and $a_2$ form a 2-free Fibonacci sequence.
Pick a positive integer $k$ so that $2^k a_2 > a_1$ and set $a_0$ to be equal to $2^k a_2 - a_1$.
There are many ways to build predecessors to a given 2-free Fibonacci sequence. The minimal such sequence is built when we choose the smallest power of 2 that still allows us to have positive members in the sequence. We explicitly build such an example starting with $a_1=3$, and $a_2=1$. Reversing the indexing direction we get: 1, 3, 1, 5, 3, 7, 5, 9, 1, $\ldots$, which is now sequence A233526.
Next, we want to continue with 3-free Fibonacci sequences. Are they as simple as 2-free sequences?
3-free Fibonacci Sequences {#sec:3free}
==========================
Let us look at | 1 | member_10 |
3-free Fibonacci sequences. Consider an example of 3-free Fibonacci numbers: 0, 1, 1, 2, 1, 1, 2, and so on. The sequence turns into a cycle of length 3. There are only 3 different 3-free Fibonacci numbers.
We can multiply a 3-free sequence by a number not divisible by 3 to get another 3-free sequence. Thus, in general we can get cycles of the form $k$, $k$, $2k$, where $k$ is not divisible by 3.
Any cycle of length 3 in a 3-free Fibonacci sequence is of the form $k$, $k$, $2k$.
Consider the length 3 cycle $a$, $b$, $c$. From the definition of 3-free Fibonacci sequences, we know the following relations:
$$\begin{aligned}
a+b=3^xc \nonumber \\
b+c=3^ya \label{eq:b+c}\\
c+a=3^zb. \label{eq:c+a}\end{aligned}$$
Furthermore, no term in the sequence is divisible by 3. Then, by the pigeon-hole principle, at least two of the terms $a,b,c$ must be congruent modulo 3. Without loss of generality, take $a \equiv b$ (mod 3). Then $a+b \not\equiv 0$ (mod 3), so we have that $x=0$ and $a+b=c$. Now substitute for $c$ and add equations (\[eq:b+c\]) and (\[eq:c+a\]) to get that $a+b=3^{y-1}a+3^{z-1}b$. Since $3\nmid a+b$, either $y=1$ or $z=1$. If $y=1$, then $b=3^{z-1}b$, hence $z=1$. Similarly, $z=1$ implies | 1 | member_10 |
$y=1$. In either case, $y=z=1$. Then we may solve for our initial variables to show that $a=b$ and $c=a+b$. Restated, $a=k$, $b=k$, and $c=2k$.
The number $k$ in the cycle is the greatest common divisor of the sequence.
Because of the Fibonacci additive property, if any number divides two or more elements of the sequence (excluding the first two, which may be divisible by 3), it must divide all numbers in the sequence. Thus, $k$ must divide every element. The least of these elements, then, can only be $k$ itself, making it the greatest common divisor.
Will it be the case that all 3-free Fibonacci sequences end in cycles of length 3? We will build suspense by delaying this discussion, meanwhile we have a lemma about the length of any potential cycle:
\[thm:parity\] Any cycle in a 3-free Fibonacci sequence is of length $3n$ for some integer $n$.
Begin with any 3-free Fibonacci sequence, and divide out the highest power of 2 in the GCD of all its elements. The resulting sequence is a 3-free Fibonacci sequence with at least one odd element. It is clear that dividing or multiplying any number by 3 does not change its parity. Thus, | 1 | member_10 |
any sequence, regardless of how many factors of 3 are divided out from each term, will have the same underlying structure in its parity. Since we have reduced the sequence to the point where there exists at least one odd number, we know that the sequence reduced modulo 2 must be congruent to 1, 1, 0, 1, 1, 0. Only cycles of length $3n$ are permitted in this structure, and therefore permitted in 3-free sequences.
We checked all the starting pairs of numbers from 1 to 1000, and all these sequences end in a 3-cycle.
The 3-free Fibonacci sequences are in many ways similar to the notorious Collatz sequences [@La10], for which it is still not known if every sequence eventually cycles. We do not expect that it is easy to prove or disprove that ever 3-free Fibonacci sequence ends in a cycle. On the other hand, it is possible to make probabilistic arguments to support different claims about free Fibonacci sequences.
Here is the base of the argument. Suppose we encounter a number in a sequence that is divisible by 3, then after removing all powers of 3, let us assume that the resulting number has the remainder 1 | 1 | member_10 |
modulo 3 with probability $1/2$. If the sum of two consecutive terms in a sequence is large and divisible by 3, then we also assume that this number is divisible by $3^k$ with probability $1/3^{k-1}$.
How often do we divide by 3 in a 3-free Fibonacci sequence? The following lemma is obvious.
In a 3-free Fibonacci sequence the division happens for every term or for every other term.
In other words, we can not have a subsequence of length 3 such that each term is the sum of the previous two terms. We want to study two polar cases first: stretches where we divide every term and stretches were we divide every other term.
We will call a subsequence of a 3-free Fibonacci sequence where we divide at each step a *division-rich subsequence*. Conversely, we will call a subsequence of a 3-free Fibonacci sequence where we divide at every other step a *division-poor subsequence*.
\[thm:rich\] There exist arbitrary long division-rich subsequences.
The proof is done by explicit construction. Consider the definition of a division-rich subsequence. In this case, we divide by a power of 3 after every addition step, so that $3^{i_n} \cdot a_n = a_{n-1}+a_{n-2}$ for $i_n>0$. Equivalently, $a_{n-2}=3^{i_n} | 1 | member_10 |
\cdot a_n - a_{n-1}$. Thus, by choosing $a_n$ and $a_{n-1}$, and selecting a sequence $\{i_m\}$ that satisfies this relationship, the sequence can easily be constructed backwards. Our only requirements are that every term of the sequence is positive, and every step contains a division, so it will suffice to construct a sequence $\{i_m\}$ such that $3^{i_n} \cdot a_n - a_{n-1}>0$ and $i_n>0$ for all $n$.
As an example, let us begin with $a_n=1$, $a_{n-1}=1$. At each step let us choose the smallest possible power for $i_m$. Then $i_n = 1$ satisfies our inequality for the first step, and $a_{n-2}=2$. Continuing, $i_{n-1}=1$ satisfies the inequality for the next step, yielding $a_{n-3}=1$. Next, $i_{n-2}=1$ and $a_{n-3}=1$, followed by $i_{n-3}=2$ and $a_{n-4}=5$. Reading the sequence backwards we get: 1, 1, 2, 1, 5, 4, 11, 1, 32, 49, $\ldots$. This is now sequence A233525 in the OEIS [@OEIS]. when read forward, this sequence is a 3-free sequence containing eight divisions in a row. The process can be continued to arbitrarily many terms for arbitrarily many consecutive divisions.
The growth bound for division-rich subsequences is estimated by the following lemma.
For a division-rich subsequence: $\max\{a_{2k+1},a_{2k+2}\} \leq 2\max\{a_{2k-1},a_{2k}\}/3$.
We can estimate that $a_{2k+1} \leq (a_{2k-1}+a_{2k})/3 | 1 | member_10 |
\leq 2\max\{a_{2k-1},a_{2k}\}/3$ and $a_{2k+2} \leq (a_{2k}+a_{2k+1})/3 \leq 5\max\{a_{2k-1},a_{2k}\}/9 < 2\max\{a_{2k-1},a_{2k}\}/3$.
So we can expect that with probability $1/2^n$ there would be a subsequence of length $2n$, where the maximum of the next two terms does not exceed the maximum of the previous two terms by $2/3$. Clearly it cannot go down forever. We need to start with very large numbers to get a long stretch of a division-rich subsequence.
\[thm:poor\] There exist arbitrary long division-poor subsequences.
This proof is more complicated than the previous one, so we will do it together with a proof of a more powerful theorem in next Section \[sec:customized\].
If we index a division-poor subsequence in such a way that division happens on the odd term, then all the even terms form an increasing subsequence: $a_{2k} > a_{2k-2}$.
As every even term is the sum of the previous two terms we get: $a_{2k} = a_{2k-1} + a_{2k-2} > a_{2k-2}$.
That means that both division-rich and division-poor subsequences can not form a cycle. In particular, it means we can have sequences of arbitrary length without entering a cycle.
We showed that there exist 3-free Fibonacci sequences that have long increasing subsequences. Still, we want to present a | 1 | member_10 |
probabilistic argument that any 3-free Fibonacci sequence ends in a cycle.
According to our probabilistic assumptions we divide either every term or every other term with the same probability $1/2$. So on average we divide on every $1.5$ step. But for how much we divide on average?
\[thm:average3\] On average we divide by $3^{3/2}$.
Now we want to use the fact that when we divide by a power of 3 we on average divide by more than 3. If the number is large, we divide by 3 with probability $2/3$, by 9 with probability $2/9$ and so on. So the average division is by $$3^{2/3} \cdot 9^{2/9} \cdot 27^{2/27} \cdot \ldots.$$ We can say it differently. We can say that we divide by 3 with probability 1 and additionally we divide by 3 more with probability $1/3$, and by 3 more with probability $1/9$, so the result is 3 to the power $$1+1/3+1/9+1/27 + 1/81 + \ldots = 3/2.$$ Since the above sum is equal to $3/2$, every time we divide, we on average divide by $3^{3/2}$.
Notice that the average number we divide by is approximately 5.2 which is more than 5.
Let us build a probabilistic sequence that | 1 | member_10 |
capture some of the behavior of 3-free Fibonacci sequences. We start with two numbers $a_1$ and $a_2$ and flip a coin. If the coin turns heads we add the next number $a_3 = (a_1+a_2)/5$ to the sequence. If the coin turns tails we add two more terms: $a_3 = a_1+a_2$ and $a_4 = (a_2+a_3)/5$ to the sequence. Then repeat. We expect that this sequence on average grows faster than 3-free Fibonacci sequences, because we divide by a smaller number.
Now we want to bound the maximum of the last two terms of this probabilistic sequence after two coin flips. Let $M = \max\{a_1,a_2\}$. We have the following cases:
- After two heads, the sequence becomes: $a_1$, $a_2$, $(a_1+a_2)/5$, $(a_1+6a_2)/25$. The last two terms do not exceed $2M/5$.
- After head, tail, the sequence becomes: $a_1$, $a_2$, $(a_1+a_2)/5$, $(a_1+6a_2)/5$, $(2a_1+7a_2)/25$. The last two terms do not exceed $7M/5$.
- After tail, head, the sequence becomes: $a_1$, $a_2$, $a_1+a_2$, $(a_1+2a_2)/5$, $(6a_1+7a_2)/25$. The last two terms do not exceed $3M/5$.
- After two tails the sequence becomes: $a_1$, $a_2$, $a_1+a_2$, $(a_1+2a_2)/5$, $(6a_1+7a_2)/5$, $(7a_1+9a_2)/25$. The last two terms do not exceed $13M/5$.
As each event happens with the same probability $1/4$, the average growth | 1 | member_10 |
after $2n$ coin flips is $(2\cdot 3 \cdot 7 \cdot 13)^{1/4}/5$, which is below 0.97. So the overall trend for this sequence is to decrease.
Based on our computational experiments and probabilistic discussions above we conjecture:
Any 3-free Fibonacci sequence ends in a cycle.
So 2-free Fibonacci sequences provably end in a cycle, 3-free sequences conjecturally end in a cycle. Will 4-free Fibonacci sequences end in cycles too? Before discussing 4-free Fibonacci sequences, we want to make a detour and prove the promised result that an arbitrarily long division-poor sequence exists (See Lemma \[thm:poor\]) as a corollary to a much stronger and a more general theorem.
Customized-division subsequences {#sec:customized}
================================
We promised to give a proof that there exist arbitrary long division-poor sequences that are 3-free Fibonacci sequences. Now we want to prove a stronger statement. We want to allow any $n$ and show that we can build a customized $n$-free Fibonacci sequence that will have a division by a prescribed power of $n$ with the prescribed remainder.
Let us correspond to any $n$-free Fibonacci sequence the list of numbers by which we divide at every step. We call this list *a signature*. For example, a 3-free sequence 5, 4, | 1 | member_10 |
1, 5, 2, 7, 1, has signature \*, \*, 9, 1, 3, 1, 9. We placed stars at the first two places, because we do not know the preceding members of the sequence and, hence, do not know the powers.
Given an $n$-free Fibonacci sequence with a given signature and a given set of remainders, we can build many other sequences with the same signature and the set of remainders.
\[thm:adjustement\] Suppose an $n$-free Fibonacci sequence $s_1$ starts with $a_1$ and $a_2$ that are not divisible by $n$ and the product of the numbers that we divide by while calculating the first $k$ terms is strictly less than $n^m$. Consider an $n$-free Fibonacci sequence $s_2$ that starts with $b_1=a_1+ d_1 n^m$ and $b_2=a_2+ d_2 n^m$ for any integers $d_1$ and $d_2$. The first $k$ terms of both sequences have the same signature and the same set of remainders modulo $n$.
The initial terms of both sequences have the same remainders modulo $n^m$. Hence their sums have the same remainders. The first time we need to divide, we divide by the same power of $n$, say $m_1$, and the result will have the same remainders modulo $n^{m-m_1}$. The next time we | 1 | member_10 |
divide, the result will have the same remainders modulo $n^{m-m_1-m_2}$. And so on. When we complete the subsequence, all the remainders will be the same modulo $n$.
This lemma allows us to find a positive sequence with a given signature if we already found a sequence that might not be all positive.
If there exists some finite $n$-free Fibonacci sequence with a given signature and a set of remainders, then there exists a sequence with the same signature and remainders such that every term is positive.
Adjust the initial terms according to Lemma \[thm:adjustement\].
We say that a finite sequence of remainders $r_i$ modulo $n$ and a finite signature of the same length *match* each other, if the signature has a positive power of $n$ in place $k$ if and only $r_{k-2} + r_{k-1} | n$. We call a sequence of remainders modulo $n$ *legal* if $r_{k-2} + r_{k-1} = r_k$ unless $r_{k-2} + r_{k-1} | n$. Non-legal sequences can not be sequences of remainders of an $n$-free Fibonacci sequence.
Given a legal finite sequence of remainders and a matching signature, there exists an $n$-free Fibonacci sequence with the given sequence of remainders and signature.
By Lemma \[thm:adjustement\] it is | 1 | member_10 |
enough to find such a sequence that does not need to have positive terms. We can produce such a sequence by building it backwards, similar to what we did in Lemma \[thm:rich\].
There exist arbitrary long division-poor 3-free Fibonacci subsequences.
Let us build an example of a division-poor 3-free sequence, where each division is by exactly 3. Begin with $a_n=1$, $a_{n-1}=1$, and build a sequence that is not necessarily all-positive. That means we will have $a_{n-2k}=3a_{n-2k+2}-a_{n-2k+1}$ and $a_{n-2k-1}=3a_{n-2k+1}-a_{n-2k}$. Here are several terms: $-8$, 7, $-1$, 2, 1, 1. For a positive version we need to add $3^3$ to $-8$, as outlined in Lemma \[thm:adjustement\] as we had two divisions by 3. The adjusted all-positive division-poor sequence is 19, 7, 26, 11, 37, 16.
4-free Fibonacci sequences {#sec:4free}
==========================
Consider the 4-free Fibonacci sequence starting with 0, 1. This sequence is A224382: 0, 1, 1, 2, 3, 5, 2, 7, 9, 1, 10, 11, 21, 2, 23, 25, $\ldots$. It seems that this sequence grows and does not cycle.
In checking many other 4-free Fibonacci sequences, we still did not find any cycles. The behavior of 4-free sequences is completely different from the behavior of 3-free sequences.
For 3-free sequences we | 1 | member_10 |
expected that all of them cycle. Here it might be possible that none of them cycles.
Before making any claims, let us see how these sequences behave modulo 4.
A 4-free Fibonacci sequence contains an odd number.
Suppose there exists a 4-free Fibonacci sequence containing only even numbers. Then ignoring the initial terms, all the elements of the sequence equal 2 modulo 2. Therefore, we divide by a power of 4 every time. This cannot last forever.
After the first occurrence of an odd number, a 4-free Fibonacci sequence cannot have two even numbers in a row.
Start with the first odd number. The steps that do not include division generate a parity pattern: odd, odd, even, odd, odd, even and so on. So there are no two even numbers in a row. That means we can get a multiple of 4 only after summing two odd numbers. We might get an even number after the division, but the next number must be odd again.
Let us analyze the 4-free Fibonacci sequences probabilistically, similar to what we did for $n=3$.
\[thm:average4\] An average division is by a factor of $4^{4/3} \approx 6.35$.
When we divide, we divide by 4 with | 1 | member_10 |
probability 1, additionally with probability $1/4$ we divide by 4 more, and so on. So the result is 4 to the power $$1+1/4+1/16+\ldots = 4/3.$$
How often on average do we divide? Let us assume that we passed stretches of all even numbers. Each time we divide after that, the previous two numbers are odd. After the division, the remainder is 1, 2, or 3. So the following six cases describe what happens after the division:
If $a \equiv 1$ (mod 4) and $b \equiv 3$ (mod 4), then the following set of remainders might happen until the next division:
- 1;
- 2, 1, 3;
- 3, 2, 1, 3.
If $a \equiv 3$ (mod 4) and $b \equiv 1$ (mod 4), then the following set of remainders might happen until the next division:
- 3;
- 2, 3, 1;
- 1, 2, 3, 1.
\[thm:averagesteps\] The average number of steps between divisions is $8/3$.
We assume that during the division each remainder is generated with probability $1/3$, so the stretches of length 1, 3, and 4 between divisions are equally probable.
We want to build a probabilistic model that reflects some behavior of 4-free Fibonacci sequences. Let us | 1 | member_10 |
denote the average factor by which we divide by $x$. In this model, we simply divide by $x$ each time we need to divide. The sequence stops being an integer sequence, but we artificially assign a remainder mod 4 to every element of the sequence to see when we need to divide. We want to show that our model sequence grows with probability 1.
First, we want to estimate the ratio of two consecutive numbers in the model sequence.
\[thm:ratio\] If $a$ and $b$ are two consecutive numbers in the model sequence, starting from index 3, then $b > a\frac{2+x}{(1+x)x}$ and $a > b\frac{1}{1+x}$.
Let $v$ be the element before $a$. Then $b \geq (a+v)/x > a/x$. Analogously, $b \leq a+v$, and by the previous sentence, $xa > v$. Therefore, $b < a(1+x)$, which means that $a/(1+x) < v$. Plugging this back into $b \geq (a+v)/x$, we get $b > a(2+x)/(1+x)x$.
Now we are ready to prove the theorem:
In our probabilistic model, a sequence grows with probability 1.
Suppose we have two consecutive numbers in the sequence $a$ and $b$ whose sum is divisible by 4. By Lemma \[thm:averagesteps\] we have 1, 3, or 4 terms until the following | 1 | member_10 |
division with the same probability. That is, the following continuations until the next division are equally probable:
- $a$, $b$, $(a+b)/x$.
- $a$, $b$, $(a+b)/x$, $(a+(x+1)b)/x$, $(2a+(x+2)b)/x$.
- $a$, $b$, $(a+b)/x$, $(a+(x+1)b)/x$, $(2a+(x+2)b)/x$, $(3a+(2x+3)b)/x$.
If $b>a$, then the maximum of the last two terms is: $b$, $(2a+(x+2)b)/x$, and $(3a+(2x+3)b)/x$ correspondingly. Adjusting for the fact that $a > b\frac{1}{1+x}$ (see Lemma \[thm:ratio\]), the maximum of the last two terms is at least $b$, $b((x+2)+2/(x+1))/x$, and $b((2x+3)+3/(x+1))/x$ correspondingly. We want to estimate the ratio of the maximum of the last two terms to $\max\{a,b\}$. Counting probabilities we get the following lower bound for the ratio $$1^{1/3}(((x+2)+2/(x+1))/x)^{1/3}(((2x+3)+3/(x+1))/x)^{1/3}= 1.51023.$$
If $b \leq a$, then the maximum of the last two terms is at least $(a+b)/x$, $(2a+(x+2)b)/x$, or $(3a+(2x+3)b)/x$, respectively. Using $b > a\frac{2+x}{(1+x)x}$ from Lemma \[thm:ratio\], we get that the maximum of the last two terms is at least $a\frac{2+2x+x^2}{(1+x)x^2}$, $a\frac{2x^2+6x+4}{(1+x)x^2}$, or $a\frac{5x^2+10x+6}{(1+x)x^2}$, respectively. Counting probabilities, we get that the maximum is multiplied by at least $$\frac{2+2x+x^2}{(1+x)x^2}^{1/3}\frac{3x^2+6x+4}{(1+x)x^2}^{1/3}\frac{5x^2+10x+6}{(1+x)x^2}^{1/3}=0.453822.$$
Notice that if we have 3 or 4 terms until the next division, then the last two terms before the division are in the increasing order. That means the case when $b \leq a$ is at least | 1 | member_10 |
twice less probable, so the average growth is at least the cube root of $1.51023^2\cdot 0.453822 = 1.03507$, which is greater than 1.
The result is greater than 1, which means that our model sequence does not cycle all the time. Therefore, extending the argument to 4-free sequences, we can safely say that 4-free sequences do not cycle all the time. Taking our computational experiments and our intuition into account we are comfortable with the following conjecture:
With probability 1 a 4-free Fibonacci sequence does not cycle.
So 4-free Fibonacci sequences do not cycle. If $n$ grows will it mean that $n$-free Fibonacci sequences for $n > 4$ will not cycle either?
5-free Fibonacci sequences {#sec:5free}
==========================
Let us look at the Lucas sequence modulo 5: $2$, $1$, $3$, $4$, $2$, 1 $\ldots$ and see that no term is divisible by 5. Clearly, no term in the Lucas sequence will require that we factor out a power of 5, and the terms will grow indefinitely. Thus, the Lucas sequence is itself a 5-free Fibonacci sequence. This is something new. We do not need a probabilistic argument to show that there are 5-free Fibonacci sequences that do not cycle.
On the | 1 | member_10 |
other hand, it becomes quickly evident that the sequence of 5-free Fibonacci numbers: 0, 1. 1, 2, 3, 1, 4, 1, 1, 2, $\ldots$ (see A214684) cycles. Some sequences cycle, and some clearly do not!
But how often will we come upon a sequence that grows indefinitely? To answer this question, let us look at a couple of terms from a few sequences of Fibonacci numbers modulo 5. Begin with $1, 1, \ldots$, to obtain the sequence
1, 1, 2, 3, 0,
3, 3, 1, 4, 0,
4, 4, 3, 2, 0,
2, 2, 4, 1, 0,
and so on. We write it like this for clarity: at the end of each line, the last term is divisible by 5. In particular, the table above shows that $Z(5)$—the entry point of 5—is 5. Furthermore, since 5 is prime, we could know beforehand that each of these lines would be the same length. We simply had to start with the line beginning $1, 1, \ldots$, and multiply each term by 2, then 3, then 4. Clearly no term in the line could become 0 after the multiplication (except, of course, 0 itself), since there are no two non-zero numbers that multiply | 1 | member_10 |
---
abstract: 'If $G$ is a complex simply connected semisimple algebraic group and if ${\lambda}$ is a dominant weight, we consider the compactification $X_{\lambda}\subset {\mathbb P}\big(\operatorname{End}({V({{\lambda}})})\big)$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of ${\lambda}$ so that $X_{\lambda}$ is normal; as well, we give necessary and sufficient conditions on the support of ${\lambda}$ so that $X_{\lambda}$ is smooth.'
author:
- 'Paolo Bravi, Jacopo Gandini, Andrea Maffei, Alessandro Ruzzi'
title: 'Normality and non-normality of group compactifications in simple projective spaces'
---
Introduction {#introduction .unnumbered}
============
Consider a semisimple simply connected algebraic group $G$ over an algebraically closed field ${\Bbbk}$ of characteristic zero. If ${\lambda}$ is a dominant weight (with respect to a fixed maximal torus $T$ and a fixed Borel subgroup $B\supset T$) and if ${V({{\lambda}})}$ is the simple $G$-module of highest weight ${\lambda}$, then $\operatorname{End}\big({V({{\lambda}})}\big)$ is a simple $G\times G$-module. Let $I_{\lambda}\in \operatorname{End}\big({V({{\lambda}})}\big)$ be the identity map and consider the variety $X_{\lambda}\subset {\mathbb P}\big(\operatorname{End}({V({{\lambda}})})\big)$ given by the closure of the $G\times G$-orbit of $[I_{\lambda}]$. In [@Ka], S. Kannan studied for which ${\lambda}$ this variety is projectively normal, and this happens precisely when ${\lambda}$ is minuscule. In | 1 | member_11 |
[@Ti], D. Timashev studied the more general situation of a sum of irreducible representations, giving necessary and sufficient conditions for the normality and smoothness of these compactifications; however the conditions for normality are not completely explicit. In this paper we give an explicit characterization of the normality of $X_{\lambda}$, which allows to simplify the conditions for the smoothness as well.
To explain our results we need some notation. Let ${\Delta}$ be the set of simple roots (w.r.t. $T\subset B$) and identify ${\Delta}$ with the vertices of the Dynkin diagram. Define the support of ${\lambda}$ as the set $\operatorname{Supp}({\lambda})=\{{\alpha}\in {\Delta}{\, : \,}\langle {\lambda}, {\alpha}^\vee \rangle \neq 0
\}$.
\[see Theorem \[teo:normalita\]\] The variety $X_{\lambda}$ is normal if and only if ${\lambda}$ satisfies the following property:
- For every non-simply laced connected component $\Delta'$ of $\Delta$, if $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root, then it contains also the short root which is adjacent to a long simple root.
In particular, if the Dynkin diagram of $G$ is simply laced then $X_{\lambda}$ is normal, for all ${\lambda}$. In the paper we will prove the theorem in a more general form, for simple (i.e. with a unique closed orbit) linear projective compactifications of an | 1 | member_11 |
adjoint group (see section \[ssez:XSigma\]). We will make use of the wonderful compactification of $G_\mathrm{ad}$, the adjoint group of $G$, and of the results on projective normality of these compactifications proved by S. Kannan in [@Ka]. These results hold in the more general case of a symmetric variety; however our method does not apply to this more general situation (see section \[ssez:simmetrichenormalita\]).
\[see Theorem \[smooth Xsigma\]\] The variety $X_{\lambda}$ is smooth if and only if $\lambda$ satisfies property $(\star)$ of Theorem A together with the following properties:
- For every connected component $\Delta'$ of $\Delta$, $\operatorname{Supp}(\lambda)\cap \Delta'$ is connected and, in case it contains a unique element, then this element is an extreme of ${\Delta}'$;
- $\operatorname{Supp}(\lambda)$ contains every simple root which is adjacent to three other simple roots and at least two of the latter;
- Every connected component of ${\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ is of type ${\mathsf A}$.
Theorem B can be generalized to any simple and normal adjoint symmetric variety. Following a criterion of ${\mathbb Q}$-factoriality for spherical varieties given by M. Brion in [@Br2], properties i) and ii) characterize the $\mathbb{Q}$-factoriality of the normalization of $X_{\lambda}$ (see Proposition \[Q-fattorialita\]), while property iii) arises from a criterion of smoothness given | 1 | member_11 |
by D. Timashev in [@Ti] in the case of a linear projective compactification of a reductive group.
As a corollary of Theorem B, we get that $X_{\lambda}$ is smooth if and only if its normalization is smooth.
The paper is organized as follows. In the first section we introduce the wonderful compactification of $G_\mathrm{ad}$ and the normalization of the variety $X_{\lambda}$. In the second section we prove Theorem A, and in the third section Theorem B. In the last section we discuss some possible generalizations of our results.
Preliminaries {#sez:preliminari}
=============
Notation {#ssez:notazioni}
--------
Recall that $G$ is semisimple and simply connected. Fix a Borel subgroup $B\subset G$, a maximal torus $T\subset B$ and let $U$ denote the unipotent radical of $B$. Lie algebras of groups denoted by upper-case latin letters ($G,U,L,\ldots$) will be denoted by the corresponding lower-case german letter (${\mathfrak g}, \mathfrak u, \mathfrak l,\ldots$). Let $\Phi$ denote the set of roots of $G$ relatively to $T$ and ${\Delta}\subset \Phi$ the basis associated to the choice of $B$. For all ${\alpha}\in {\Delta}$ let $e_{\alpha}, {\alpha}^{{\vee}},f_{\alpha}$ be an $\mathfrak{sl}(2)$-triple of $T$-weights ${\alpha},0,-{\alpha}$. Let ${\Lambda}$ denote the weight lattice of $T$ and ${\Lambda}^+$ the subset of dominant weights. For all | 1 | member_11 |
${\alpha}\in{\Delta}$, denote by $\omega_{\alpha}$ the corresponding fundamental weight.
If ${\lambda}\in {\Lambda}$, recall the definition of its *support*: $$\operatorname{Supp}({\lambda}) = \{{\alpha}\in {\Delta}{\, : \,}\langle {\lambda}, {\alpha}^\vee \rangle \neq 0 \}.$$
If $I \subset {\Delta}$, define its *border* $\partial{I}$, its *interior* $I^\circ$ and its *closure* ${\overline}{I}$ as follows: $$\partial{I} =
\{ {\alpha}\in {\Delta}{\smallsetminus}I {\, : \,}{\exists\,}{\beta}\in I \mathrm{ \; such \, that \; }
\langle {\beta},{\alpha}^\vee \rangle \neq 0\};$$ $$I^\circ = I {\smallsetminus}\partial{({\Delta}{\smallsetminus}I)};$$ $$\overline{I} = I \cup \partial{I}.$$
For ${\lambda}\in \Lambda$, denote by ${\mathcal L}_{\lambda}$ the line bundle on $G/B$ whose $T$-weight in the point fixed by $B$ is $-{\lambda}$. For ${\lambda}$ dominant, ${V({{\lambda}})} = {\Gamma}(G/B,{\mathcal L}_{\lambda})^*$ is an irreducible $G$-module of highest weight ${\lambda}$; when we deal with different groups we will use the notation ${V_G({{\lambda}})}$.
Denote by ${\Pi({{\lambda}})}$ the set of weights occurring in ${V({{\lambda}})}$ and set ${\Pi^+({{\lambda}})}={\Pi({{\lambda}})}\cap {\Lambda}^+$. Let ${\lambda}\mapsto {\lambda}^*$ be the linear involution of ${\Lambda}$ defined by $({V({{\lambda}})})^*{\simeq}{V({{\lambda}^*})}$, for any dominant weight ${\lambda}$.
The weight lattice ${\Lambda}$ is endowed with the dominance order ${\leqslant}$ defined as follows: $\mu {\leqslant}{\lambda}$ if and only if ${\lambda}- \mu \in
{\mathbb N}{\Delta}$. If ${\beta}= \sum_{{\alpha}\in {\Delta}} n_{\alpha}{\alpha}\in {\mathbb Z}{\Delta}$, define its *support over* ${\Delta}$ (not to be confused with the previous | 1 | member_11 |
one) as follows: $$\operatorname{Supp}_{\Delta}({\beta}) = \{ {\alpha}\in {\Delta}{\, : \,}n_{\alpha}\neq 0 \}.$$
We introduce also some notations about the multiplication of sections. Notice that, for all ${\lambda},\mu \in\Lambda$, ${\mathcal L}_{\lambda}\otimes {\mathcal L}_\mu ={\mathcal L}_{{\lambda}+\mu}$. Therefore, if ${\lambda},\mu$ are dominant weights and $n\in {\mathbb N}$, the multiplication of sections defines maps as follows: $$m_{{\lambda},\mu}:{V({{\lambda}})}\times {V({\mu})} {\rightarrow}{V({{\lambda}+\mu})} \; \text{ and }
m_{\lambda}^n : {V({{\lambda}})} {\rightarrow}{V({n{\lambda}})}.$$ We will also write $uv$ for $m_{{\lambda},\mu}(u,v)$ and $u^n$ for $m^n_{\lambda}(u)$. Since $G/B$ is irreducible, $m_{{\lambda},\mu}$ and $m^n_{\lambda}$ induce the following maps at the level of projective spaces: $$\psi_{{\lambda},\mu}: {\mathbb P}({V({{\lambda}})}) \times {\mathbb P}({V({\mu})}) {\rightarrow}{\mathbb P}({V({{\lambda}+ \mu})})
\;{\text{ and }}\; \psi^n_{\lambda}: {\mathbb P}({V({{\lambda}})}) {\rightarrow}{\mathbb P}({V({n{\lambda}})}).$$
The following lemma is certainly well known; however we do not know any reference.
\[lem:immersioni\] Let ${\lambda},\mu$ be dominant weights.
1. If $\operatorname{Supp}({\lambda}) \cap \operatorname{Supp}(\mu) = {\varnothing}$, then the map $\psi_{{\lambda},\mu}\colon {\mathbb P}({V({{\lambda}})}) \times {\mathbb P}({V({\mu})}) \to {\mathbb P}({V({{\lambda}+
\mu})})$ is a closed embedding.
2. For any $n>0$, the map $\psi_{\lambda}^n: {\mathbb P}({V({{\lambda}})}) {\rightarrow}{\mathbb P}({V({n{\lambda}})})$ is a closed embedding.
$i)$. Fix highest weight vectors $v_{\lambda}\in {V({{\lambda}})}$, $v_\mu \in {V({\mu})}$ and $v_{{\lambda}+\mu}= v_{\lambda}v_\mu \in {V({{\lambda}+\mu})}$.
If $V$ is irreducible, then ${\mathbb P}(V)$ has a unique closed orbit, namely the orbit of the highest | 1 | member_11 |
weight vector. Consequently, since ${\mathbb P}(V({\lambda})) \times {\mathbb P}(V(\mu))$ has a unique closed orbit, in order to prove the claim it suffices to prove that $\psi_{{\lambda},\mu}$ is smooth in $x=([v_{\lambda}],[v_\mu])$ and that the inverse image of $[v_{{\lambda}+\mu}]$ is $x$. The second claim is clear for weight reasons.
In order to prove that $\psi_{{\lambda},\mu}$ is smooth in $x$, consider $T$-stable complements $U \subset {V({{\lambda}})}$, $V \subset {V({\mu})}$ and $W\subset
{V({{\lambda}+\mu})}$ of ${\Bbbk}\,v_{\lambda}$, ${\Bbbk}\,v_\mu$ and ${\Bbbk}\,v_{{\lambda}+\mu}$. So in a neighbourhood of $x$ the map $\psi_{{\lambda},\mu}$ can be described as $$\psi:U\times V {\longrightarrow}W \;\text{ where } \psi(u,v)= u v_\mu + v_{\lambda}v + u v.$$ The differential of $\psi_{{\lambda},\mu}$ in $x$ is then given by the differential of $\psi$ in $(0,0)$, thus it is described as follows: $$d\psi_x(u,v)=uv_\mu+v_{\lambda}v.$$ Suppose that $d\psi_x$ is not injective. Since it is $T$-equivariant, consider a maximal weight $\eta \in {\Pi({{\lambda}+\mu})}{\smallsetminus}\{{\lambda}+\mu\}$ such that there exists a couple of non-zero $T$-eigenvectors $(u,v)\in \ker d
\psi_x$ with weights respectively $\eta - \mu$ and $\eta - {\lambda}$. Suppose that $\eta - \mu \in {\Pi({{\lambda}})}{\smallsetminus}\{{\lambda}\}$ is not maximal and take ${\alpha}\in {\Delta}$ such that $\eta - \mu + {\alpha}\in
{\Pi({{\lambda}})}{\smallsetminus}\{{\lambda}\}$ and $e_{\alpha}u\neq 0$: then $$(e_{\alpha}u)v_\mu + v_{\lambda}(e_{\alpha}v) = e_{\alpha}(u v_\mu + v_{\lambda}v) = 0$$ | 1 | member_11 |
and $\eta + {\alpha}\in {\Pi({{\lambda}+ \mu})}{\smallsetminus}\{{\lambda}+\mu\}$, against the maximality of $\eta$. Thus $\eta - \mu$ is maximal in ${\Pi({{\lambda}})}
{\smallsetminus}\{{\lambda}\}$ and similarly $\eta - {\lambda}$ is maximal in ${\Pi({\mu})}
{\smallsetminus}\{\mu\}$. Therefore, on one hand it must be $$\eta - \mu = {\lambda}- {\alpha}$$ with ${\alpha}\in \operatorname{Supp}({\lambda})$, while on the other hand it must be $$\eta - {\lambda}= \mu - {\beta}$$ with ${\beta}\in \operatorname{Supp}(\mu)$. Since $\operatorname{Supp}({\lambda}) \cap \operatorname{Supp}(\mu) =
{\varnothing}$, this is impossible and shows that, if $(u, v) \in \ker d
\psi_x$, then it must be $u = 0$ or $v = 0$. Suppose now that $(u,0)
\in \ker d\psi_x$: then $u v_\mu = 0$ and by the irreducibility of $G/B$ also $u=0$. A similar argument applies if $v=0$.
$ii).$ Suppose that $v,w \in V({\lambda})$ are such that $v^{n} = w^{n}$: then $v=tw$ for some $t\in {\Bbbk}$. Thus $\psi_{\lambda}^n$ is injective. Let us show now that $\psi_{\lambda}^n$ is smooth; it is enough to show it in $x = [v_{\lambda}]$ where $v_{\lambda}\in V({\lambda})$ is a highest weight vector. Let $V \subset V({\lambda})$ be the $T$-stable complement of ${\Bbbk}v_{\lambda}$, identified with the tangent space $T_{x} {\mathbb P}(V({\lambda}))$. If $v
\in V$, the differential $d (\psi_{\lambda}^n)_{x}$ is described as follows $$d (\psi^n_{\lambda})_{x} (v) | 1 | member_11 |
= n v_{\lambda}^{n-1} v.$$ Thus $d (\psi^n_{\lambda})_{x}$ is injective and $\psi_{\lambda}^n$ is smooth.
The variety $X_{\lambda}$ {#ssez:Xlambda}
-------------------------
If ${\lambda}$ is a dominant weight, denote by ${E({{\lambda}})}$ the $G\times
G$-module $\operatorname{End}({V({{\lambda}})})$ and set $X_{\lambda}$ the closure of the $G\times G$-orbit of $[I_{\lambda}] \in {\mathbb P}({E({{\lambda}})})$. More generally if ${\lambda}_1,\dots,{\lambda}_m$ are dominant weights we define $$X_{{\lambda}_1, \ldots, {\lambda}_m}
= {\overline}{G\times G([I_{{\lambda}_1}], \ldots, [I_{{\lambda}_m}])} \subset
{\mathbb P}({E({{\lambda}_1})}) \times \cdots \times {\mathbb P}({E({{\lambda}_m})}).$$ Since ${E({{\lambda}})}$ is an irreducible $G\times G$-module of highest weight $({\lambda},{\lambda}^*)$, as a consequence of Lemma \[lem:immersioni\] we get that if ${\lambda}$ and $\mu$ have non-intersecting supports and if $n\in {\mathbb N}$ then $$X_{{\lambda}+\mu}\simeq X_{{\lambda},\mu} \qquad \mathrm{ and } \qquad X_{n{\lambda}}{\simeq}X_{\lambda}.$$ As a consequence we get the following proposition:
\[prp:supporto\]Let ${\lambda},\mu$ be dominant weights. Then $X_{\lambda}\simeq X_\mu$ as $G\times G$-varieties if and only if ${\lambda}$ and $\mu$ have the same support. Moreover, if $\operatorname{Supp}({\lambda})=\{{\alpha}_1,\dots,{\alpha}_m\}$ then $$X_{\lambda}{\simeq}X_{\omega_{{\alpha}_1},\dots,\omega_{{\alpha}_m}}.$$
By the discussion above we have to prove only that the condition is necessary. This follows by noticing that if $X_{\lambda}$ and $X_\mu$ are $G\times G$-isomorphic then also their closed $G\times G$-orbits are isomorphic, which is equivalent to the fact that ${\lambda}$ and $\mu$ have the same support.
The wonderful compactification of $G_{\mathrm{ad}}$ and | 1 | member_11 |
the normalization of $X_{\lambda}$ {#ssez:meravigliosa}
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When ${\lambda}$ is a regular weight (i.e. $\operatorname{Supp}({\lambda})={\Delta}$) the variety $X_{\lambda}$ is called the wonderful compactification of $G_\mathrm{ad}$ and it has been studied by C. De Concini and C. Procesi in [@CP]. We will denote this variety by $M$: it is smooth and the complement of its open orbit is the union of smooth prime divisors with normal crossings whose intersection is the closed orbit. The closed orbit of $M$ is isomorphic to $G/B \times G/B$ and the restriction of line bundles determines an embedding of $\operatorname{Pic}(M)$ into $\operatorname{Pic}(G/B\times G/B)$, that we identify with ${\Lambda}\times {\Lambda}$ as before; the image of this map is the set of weights of the form $({\lambda},{\lambda}^*)$. Therefore $\operatorname{Pic}(M)$ is identified with ${\Lambda}$ and we denote by ${\mathcal M}_{{\lambda}}$ a line bundle on $M$ whose restriction to $G/B \times G/B$ is isomorphic to ${\mathcal L}_{\lambda}\boxtimes {\mathcal L}_{{\lambda}^*}$. If $D\subset M$ is a $G\times G$-stable prime divisor then the line bundle defined by $D$ is of the form ${\mathcal M}_{{\alpha}_D}$, where ${\alpha}_D$ is a simple root. The map $D\mapsto {\alpha}_D$ defines a bijection between the set of $G\times G$-stable prime divisors and ${\Delta}$, and we denote by $M_{\alpha}$ the prime | 1 | member_11 |
divisor which corresponds to a simple root ${\alpha}$. We denote by $s_{\alpha}$ a section of ${\mathcal M}_{\alpha}$ whose associated divisor is $M_{\alpha}$; notice that such a section is $G\times G$-invariant. More generally if $\nu = \sum_{{\alpha}\in {\Delta}} n_{{\alpha}}{\alpha}\in {\mathbb N}{\Delta}$, set $s^\nu = \prod_{{\alpha}\in {\Delta}} s_{{\alpha}} ^{n_{{\alpha}}} \in
{\Gamma}(M,{\mathcal M}_\nu)$. Then, given any ${\lambda}\in {\Lambda}$, the multiplication by $s^\nu$ injects ${\Gamma}(M,{\mathcal M}_{{\lambda}- \nu})$ in ${\Gamma}(M,{\mathcal M}_{\lambda})$.
If ${\lambda}$ is a dominant weight, the map $G_\mathrm{ad}{\longrightarrow}{\mathbb P}({E({{\lambda}})})$ extends to a map $q_{\lambda}:M{\longrightarrow}{\mathbb P}({E({{\lambda}})})$ (see [@CP]) whose image is $X_{\lambda}$ and such that ${\mathcal M}_{\lambda}=q_{\lambda}^*({\mathcal O}_{{\mathbb P}({E({{\lambda}})})}(1))$. If we pull back the homogeneous coordinates of ${\mathbb P}({E({{\lambda}})})$ to $M$, we get then a submodule of ${\Gamma}(M,{\mathcal M}_{\lambda})$ which is isomorphic to ${E({{\lambda}})}^*$; by abuse of notation we will denote this submodule by ${E({{\lambda}})}^*$.
If ${\lambda}\in {\Lambda}$, in [@CP Theorem 8.3] the following decomposition of ${\Gamma}(M,{\mathcal M}_{\lambda})$ is given: $${\Gamma}(M,{\mathcal M}_{\lambda}) = \bigoplus_{\mu\in {\Lambda}^+ {\, : \,}\mu {\leqslant}{\lambda}} s^{{\lambda}-\mu}{E({\mu})}^*.$$
Consider the graded algebra $A({\lambda})=\bigoplus_{n=0}^\infty A_n({\lambda})$, where $A_n({\lambda})= {\Gamma}(M,{\mathcal M}_{n{\lambda}})$, and set $\widetilde{X}_{\lambda}=
\operatorname{Proj}A({\lambda})$. We have then a commutative diagram as follows: $$\xymatrix{M \ar@{->>}^{p_{\lambda}}[r] \ar@{->>}_{q_{\lambda}}[dr]
& \widetilde{X}_{\lambda}\ar@{->>}^{r_{\lambda}}[d]\\ & X_{\lambda}}$$
In [@Ka], it has been shown that $A({\lambda})$ is generated in degree $1$ and in | 1 | member_11 |
[@DC] that $r=r_{\lambda}$ is the normalization of $X_{\lambda}$. Notice that the projective coordinate ring of $X_{\lambda}\subset
{\mathbb P}({E({{\lambda}})})$ is given by the graded subalgebra $B({\lambda})=\bigoplus_{n=0}^\infty
B_n({\lambda})$ of $A({\lambda})$ generated by ${E({{\lambda}})}^* \subset {\Gamma}(M,{\mathcal M}_{\lambda})$.
The variety $X_\Sigma$ {#ssez:XSigma}
----------------------
We consider now a generalization of the variety $X_{\lambda}$. Let $\Sigma$ be a finite set of dominant weights and denote ${E({\Sigma})} =
\bigoplus_{\mu \in \Sigma} {E({\mu})}$; let $x_\Sigma=[(I_\mu)_{\mu\in\Sigma}]\in {\mathbb P}({E({\Sigma})})$ and define $X_\Sigma$ as the closure of the $G\times G$-orbit of $x_\Sigma$ in ${\mathbb P}({E({\Sigma})})$. If $\Sigma = \{{\lambda}\}$, then we get the variety $X_{\lambda}$, while if ${\Sigma}= {\Pi^+({{\lambda}})}$ we get its normalization $\widetilde X _{\lambda}$. Notice that the diagonal action of $G$ fixes the point $x_\Sigma$ so we have a $G\times G$ equivariant map $G{\longrightarrow}X_\Sigma$ given by $g \longmapsto (g,1)x_\Sigma$. This map induces a map from $G_{\mathrm{ad}}$ to $X_\Sigma$ if and only if the action of the center of $G\times G$ on $E({\lambda})$ is the same for all ${\lambda}\in
\Sigma$ or equivalently if ${\Sigma}$ is contained in a coset of ${\Lambda}$ modulo ${\mathbb Z}{\Delta}$. In this case we say that $X_\Sigma$ is a *semi-compactification* of $G_{ad}$. If $G_{\mathrm{ad}}$ is a simple group and and $\Sigma \neq \{ 0 \}$ then | 1 | member_11 |
$X_\Sigma$ is a compactification of $G_{\mathrm{ad}}$, while if $G_{\mathrm{ad}}$ is not simple we can only say that is a compactification of a group which is a quotient of $G_{\mathrm{ad}}$.
We say that $\Sigma$ is *simple* if there exists ${\lambda}\in\Sigma$ such that $\Sigma \subset {\Pi^+({{\lambda}})}$ or equivalently if ${\Sigma}$ contains a unique maximal element with respect to the dominance order ${\leqslant}$. Notice also that if ${\lambda}\in \Sigma$ is such that for all $\mu \in \Sigma$ different from ${\lambda}$ the vector $\mu - {\lambda}$ is not in ${\mathbb Q}_{{\geqslant}0}[{\Delta}]$ then is easy to construct a cocharacter $\chi
: {\Bbbk}^*{\longrightarrow}G\times G$ such that $\lim_{t\to 0}\chi(t)x_\Sigma $ is the highest weight line in ${\mathbb P}(E({\lambda}))$. In particular $X_\Sigma$ is a simple $G\times G$ semi-compactification of $G_{\mathrm{ad}}$ if and only if $\Sigma$ is simple.
By the description of the normalization of $X_{\lambda}$ is $\Sigma$ is simple and ${\lambda}\in {\Sigma}$ is the maximal element, then we get $$\xymatrix{\widetilde X_{\lambda}\ar[r]^{r} & X_\Sigma \ar[r] & X_{\lambda}}$$ In particular, it follows that $r=r_\Sigma:\widetilde X_{\lambda}{\longrightarrow}X_\Sigma$ is the normalization of $X_\Sigma$.
If ${\Sigma}$ is simple, denote $B(\Sigma)=\bigoplus_{n=0}^\infty
B_n(\Sigma)$ the projective coordinate ring of $X_\Sigma\subset
{\mathbb P}({E({\Sigma})})$: it is the subalgebra of $A({\lambda})$ generated by ${E({\Sigma})}^*\subset {\Gamma}(M,{\mathcal M}_{\lambda})$.
The discussion above and | 1 | member_11 |
the fact that in ${\mathbb P}(E({\lambda}))$ there is only one point fixed by the diagonal action of $G$ (the line of scalar matrices) proves that any $G\times G$ linear projective compactification of $G_{\mathrm{ad}}$ is of the form $X_\Sigma$. A projective $G\times G$-variety $X$ is said to be *linear* if there exists an equivariant embedding $X\subset {\mathbb P}(V)$ where $V$ is a finite dimensional rational $G\times G$-module. In particular as a consequence of Sumihiro’s Theorem (see for example [@KKLV Corollary 2.6]) all normal projective compactifications are linear. In this paper we study only linear compactifications.
Normality {#sez:normalita}
=========
In this section we determine for which simple $\Sigma$ the variety $X_\Sigma$ is normal, proving in particular Theorem A. In the following, by $\lambda$ we will always denote the maximal element of $\Sigma$.
Let ${\varphi}_{\lambda}\in {E({{\lambda}})}^*$ be a highest weight vector and set $X_\Sigma^\circ \subset X_\Sigma$ the open affine subset defined by the non-vanishing of ${\varphi}_{\lambda}$. In particular, we set ${\widetilde}X_{\lambda}=
X_{{\Pi^+({{\lambda}})}}$ and notice that ${\widetilde}X ^\circ_{\lambda}=
r^{-1}(X_\Sigma^\circ)$. Notice that $X^\circ_\Sigma$ is $B\times B$-stable and, since it intersects the closed orbit, it intersects every orbit: therefore $X_\Sigma$ is normal if and only if $X_\Sigma^\circ$ is normal if and only if the restriction $r{\bigr|}_{{\widetilde}X
| 1 | member_11 |
^\circ_{\lambda}} : {\widetilde}X^\circ_{\lambda}\to X^\circ_\Sigma$ is an isomorphism. Denote by $\bar B(\Sigma)$ the coordinate ring of $X^\circ_\Sigma$ and by $\bar A({\lambda})$ the coordinate ring of ${\widetilde}X^\circ_{\lambda}$; then we have $$\bar A({\lambda})= \{\frac{{\varphi}}{{\varphi}_{\lambda}^n}{\, : \,}{\varphi}\in A_n({\lambda})\}
\supset \{\frac{{\varphi}}{{\varphi}_{\lambda}^n}{\, : \,}{\varphi}\in B_n(\Sigma)\} = \bar B(\Sigma)$$ and $X_\Sigma$ is normal if and only if $\bar A({\lambda})=\bar B
(\Sigma)$. The rings $\bar A({\lambda})$ and $\bar B(\Sigma)$ are not $G\times G$-modules, however since $X^\circ_\Sigma$ is an open subset of $X_\Sigma$ we still have an action of the Lie algebra ${\mathfrak g}\oplus{\mathfrak g}$ on them.
By [@Ka], $\bar A({\lambda})$ is generated by the elements of the form ${\varphi}/{\varphi}_{\lambda}$ with ${\varphi}\in A_1({\lambda})$. In particular we have the following lemma.
\[lem:general-normality\] The variety $X_\Sigma$ is normal if and only if for all $\mu\in{\Lambda}^+$ such that $\mu{\leqslant}{\lambda}$ there exists $n>0$ such that $$s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*
\subset B_n(\Sigma).$$
Let ${\varphi}_\mu \in s^{{\lambda}-\mu}{E({\mu})}^*$ be a highest weight vector and suppose that $X_\Sigma$ is normal. Then, by the descriptions of $\bar A({\lambda})$ and $\bar B(\Sigma)$, for every dominant weight $\mu{\leqslant}{\lambda}$ there exist $n>0$ and $\varphi\in B_n(\Sigma)$ such that ${{\varphi}}/{{\varphi}_{\lambda}^n} = {{\varphi}_\mu}/{{\varphi}_{\lambda}}$ or equivalently ${\varphi}={\varphi}_\mu {\varphi}_{\lambda}^{n-1}\in B_n(\Sigma)$. Since ${\varphi}$ is a highest weight vector of the module $s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*$ the claim follows.
Conversely assume that for every | 1 | member_11 |
dominant weight $\mu{\leqslant}{\lambda}$ there exists $n$ such that $$s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*\subset B_n(\Sigma);$$ in particular ${\varphi}={\varphi}_\mu {\varphi}_{\lambda}^{n-1}\in
B_n(\Sigma)$. Let’s prove that ${\varphi}/{\varphi}_{\lambda}\in \bar B({\Sigma})$ for every ${\varphi}\in s^{{\lambda}-\mu}{E({\mu})}^*$; this implies the thesis since $\bar A({\lambda})$ is generated in degree one. If ${\varphi}={\varphi}_\mu$ this is clear. Using the action of the Lie algebra ${\mathfrak g}\oplus{\mathfrak g}$ on $\bar
B(\Sigma)$, let’s show that if ${\varphi}/{\varphi}_{\lambda}\in \bar B(\Sigma)$ then $f_{\alpha}({\varphi})/{\varphi}_{\lambda}\in \bar B(\Sigma)$: indeed we have $$\frac{f_{\alpha}({\varphi})}{{\varphi}_{\lambda}} = f_{\alpha}(\frac{{\varphi}}{{\varphi}_{\lambda}}) +
\frac{{\varphi}}{{\varphi}_{\lambda}} \cdot \frac{f_{\alpha}({\varphi}_{\lambda})}{{\varphi}_{\lambda}}$$ and the claim follows since $f_{\alpha}({\varphi}_{\lambda})\in {E({{\lambda}})}^*\subset B_1(\Sigma)$.
We can describe the set $B_n(\Sigma)$ more explicitly. Indeed, as in [@DC] or in [@Ka], it is possible to identify sections of a line bundle on $M$ with functions on $G$ and use the description of the multiplication of matrix coefficients. Recall that as a $G\times G$-module we have ${\Bbbk}[G]=\bigoplus_{{\lambda}\in {\Lambda}^+}{E({{\lambda}})}^* {\simeq}\bigoplus_{{\lambda}\in
{\Lambda}^+}{V({{\lambda}})}^*\otimes {V({{\lambda}})}$. More explicitly if $V$ is a representation of $G$, define $c_V:V^*\otimes V {\longrightarrow}{\Bbbk}[G]$ as usual by $c_V(\psi \otimes v)(g)= \langle \psi, gv\rangle$. If we multiply functions in ${\Bbbk}[G]$ of this type then we get $$c_{V}( \psi \otimes v) \cdot c_{W}(\chi \otimes w) =
c_{V\otimes W} \big((\psi\otimes\chi)\otimes(v\otimes w)\big):$$ in particular we get that the image of the multiplication ${E({{\lambda}})}^*
\otimes {E({\mu})}^* {\longrightarrow}{\Bbbk}[G]$ is | 1 | member_11 |
the sum of all ${E({\nu})}^*$ with ${V({\nu})}\subset {V({{\lambda}})}\otimes {V({\mu})}$.
As a consequence we obtain the following Lemma:
\[lem:coefficientimatriciali\] Let $\nu, \nu'$ be dominant weights, then the image of ${E({\nu})}^*
\otimes {E({\nu'})}^*$ in ${\Gamma}(M, {\mathcal M}_{\nu+\nu'})$ via the multiplication map is $$\bigoplus_{{V({\mu})}\subset {V({\nu})}\otimes {V({\nu'})}}\!\!\!\!s^{\nu+\nu'-\mu}{E({\mu})}^*.$$
Indeed let $ \pi : G {\rightarrow}M$ be the map induced by the inclusion $G_\mathrm{ad}\subset M$. Then any line bundle on $G$ can be trivialized so that the image of $\pi^*:{E({{\lambda}})}^*\subset{\Gamma}(M,{\mathcal M}_\nu){\longrightarrow}{\Bbbk}[G]$ is the image of $c_{{V({{\lambda}})}}$ and the claim follows from previous remarks.
Together with Lemma \[lem:general-normality\], this gives the following
\[prp:normalita\] The variety $X_\Sigma$ is normal if and only if, for every $\mu \in {\Lambda}^+$ such that $\mu {\leqslant}{\lambda}$, there exist $n>0$ and ${\lambda}_1,\dots,{\lambda}_n\in \Sigma$ such that $${V({\mu + (n-1){\lambda}})} \subset {V({{\lambda}_1})}\otimes \cdots \otimes {V({{\lambda}_n})} .$$
Remarks on tensor products {#ssez:prodottitensore}
--------------------------
By Proposition \[prp:normalita\], in order to establish the normality (or the non-normality) of $X_\Sigma$, we need some results on tensor product decomposition.
\[lem:riduzionelevi\] Let ${\lambda}, \mu, \nu$ be dominant weights and let ${\Delta}'\subset {\Delta}$ be such that $\operatorname{Supp}_{\Delta}({\lambda}+ \mu - \nu) \subset {\Delta}'$; let $L \subset G$ be the standard Levi subgroup associated to ${\Delta}'$. If $\pi \in {\Lambda}^+$, denote by ${V_L({\pi})}$ the | 1 | member_11 |
simple $L$-module of highest weight $\pi$. Then $${V({\nu})} \subset {V({{\lambda}})} \otimes {V({\mu})} \iff {V_L({\nu})} \subset {V_L({{\lambda}})} \otimes {V_L({\mu})}.$$
If $\mathfrak a$ is any Lie algebra, denote ${\mathfrak U}(\mathfrak a)$ the corresponding universal enveloping algebra.
Suppose that ${V_L({\nu})} \subset {V_L({{\lambda}})} \otimes {V_L({\mu})}$; fix maximal vectors $v_{\lambda}\in {V_L({{\lambda}})}$ and $v_\mu \in {V_L({\mu})}$ for the Borel subgroup $B\cap L \subset L$ and fix $p \in {\mathfrak U}(\mathfrak
l\cap\mathfrak u^-) \otimes {\mathfrak U}(\mathfrak l\cap\mathfrak u^-)$ such that $p\,(v_{\lambda}\otimes v_\mu) \in {V_L({{\lambda}})} \otimes {V_L({\mu})}$ is a maximal vector of weight $\nu$. Since ${V_L({{\lambda}})} \otimes {V_L({\mu})}
\subset {V({{\lambda}})} \otimes {V({\mu})}$, we only need to prove that $p\,(v_{\lambda}\otimes v_\mu)$ is a maximal vector for $B$ too. If ${\alpha}\in {\Delta}'$ then we have $e_{\alpha}p\,(v_{\lambda}\otimes v_\mu) = 0$ by hypothesis. On the other hand, if ${\alpha}\in {\Delta}{\smallsetminus}{\Delta}'$, notice that $e_{\alpha}$ commutes with $p$, since by its definition $p$ is supported only on the $f_{\alpha}$’s with ${\alpha}\in {\Delta}'$. Since $v_{\lambda}\otimes
v_\mu$ is a maximal vector for $B$, then we get $$e_{\alpha}p\,(v_{\lambda}\otimes v_\mu) = p\, e_{\alpha}(v_{\lambda}\otimes v_\mu) = 0;$$ thus $p\,(v_{\lambda}\otimes v_\mu)$ generates a simple $G$-module of highest weight $\nu$.
Assume conversely that ${V({\nu})} \subset {V({{\lambda}})} \otimes {V({\mu})}$ and fix $p
\in {\mathfrak U}(\mathfrak u^-) \otimes {\mathfrak U}(\mathfrak u^-)$ such that | 1 | member_11 |
$p\,(v_{\lambda}\otimes v_\mu) \in {V({{\lambda}})} \otimes {V({\mu})}$ is a maximal vector of weight $\nu$. Since $\operatorname{Supp}_{\Delta}({\lambda}+ \mu - \nu) \subset {\Delta}'$, we may assume that the only $f_{\alpha}$’s appearing in $p$ are those with ${\alpha}\in {\Delta}'$; therefore $p\,(v_{\lambda}\otimes v_\mu) \in {V_L({{\lambda}})}
\otimes {V_L({\mu})}$ and it generates a simple $L$-module of highest weight $\nu$.
\[lem:traslazione\] Fix ${\lambda}, \mu, \nu \in {\Lambda}^+$ such that ${V({\nu})} \subset {V({{\lambda}})}
\otimes {V({\mu})}$. Then, for any $\nu' \in {\Lambda}^+$, it also holds $${V({\nu + \nu'})} \subset {V({{\lambda}+ \nu'})} \otimes {V({\mu})}.$$
Fix a maximal vector $v_{\nu'} \in {V({\nu'})}$ and consider the $U$-equivariant map $$\begin{array}{cccc}
\phi: & {V({{\lambda}})} \otimes {V({\mu})} & {\longrightarrow}& {V({{\lambda}+ \nu'})}\otimes {V({\mu})}\\
& w_1 \otimes w_2 & \longmapsto & m_{{\lambda},\nu'}(w_1,v_{\nu'}) \otimes w_2
\end{array}$$ The claim follows since, if $v_\nu \in {V({{\lambda}})}\otimes {V({\mu})}$ is a $U$-invariant vector of weight $\nu$, then $\phi(v_\nu) \in {V({{\lambda}+
\nu'})} \otimes {V({\mu})}$ is a $U$-invariant vector of weight $\nu +
\nu'$.
We now describe some more explicit results. When we deal with explicit irreducible root systems, unless otherwise stated, we always use the numbering of simple roots and fundamental weights of Bourbaki [@Bo].
In order to describe the simple subsets ${\Sigma}\subset {\Lambda}^+$ which give rise to a non-normal variety $X_{\Sigma}$, we will | 1 | member_11 |
make use of following lemma.
\[lem:BGno\]
1. Let $G$ be of type ${\sf B}_r$. Then, for any $n$, ${V({(n-1){\omega}_1})} \not \subset {V({{\omega}_1})}^{\otimes n}$.
2. Let $G$ be of type ${\sf G}_2$. Then, for any $n$, ${V({{\omega}_1+(n-1){\omega}_2})} \not \subset {V({{\omega}_2})}^{\otimes n}$.
We consider only the first case, the second is similar. Fix a highest weight vector $v_1 \in {V({{\omega}_1})}$. If ${\alpha}$ is any simple root and if $1 {\leqslant}s {\leqslant}r$, notice that $f_{\alpha}$ acts non-trivially on $f_{{\alpha}_{s-1}}\cdots f_{{\alpha}_1}v_1$ if and only if ${\alpha}=
{\alpha}_s$. The $T$-eigenspace of weight 0 in ${V({{\omega}_1})}$ is spanned by $v_0=f_{{\alpha}_r}\cdots f_{{\alpha}_1}v_1$, and similarly the $T$-eigenspace of weight $(n-1){\omega}_1$ in ${V({{\omega}_1})}^{\otimes n}$ is spanned by $v_1^{\otimes i-1} \otimes v_0 \otimes v_1^{\otimes n-i}$, where $1 {\leqslant}i {\leqslant}n$. Since the vectors $$e_{{\alpha}_r} (v_1^{\otimes i-1} \otimes v_0 \otimes v_1^{\otimes
n-i}) = v_1^{\otimes i-1} \otimes (e_{{\alpha}_r} v_0) \otimes
v_1^{\otimes n-i}$$ are linearly independent, there exists no maximal vector of weight $(n-1){\omega}_1$ in ${V({{\omega}_1})}^{\otimes n}$.
Dual results will be needed to describe the subsets ${\Sigma}$ which give rise to a normal variety $X_{\Sigma}$, but before we need to introduce some further notation.
If $\Phi$ is an irreducible root system and ${\Delta}$ is a basis for $\Phi$ we will denote by $\eta$ the | 1 | member_11 |
highest root if $\Phi$ is simply laced or the highest short root if $\Phi$ is not simply laced. For the convenience of the reader we list the highest short root of every irreducible root system in Table \[tab:hsr\].
type of $\Phi$ highest short root
----------------- ------------------------------------------------------------------------------------------------------------
${\mathsf A}_r$ ${\alpha}_1+\cdots+{\alpha}_r=\omega_1+\omega_r$
${\mathsf B}_r$ ${\alpha}_1+\cdots+{\alpha}_r=\omega_1$
${\mathsf C}_r$ ${\alpha}_1+2({\alpha}_2+\cdots+{\alpha}_{r-1})+{\alpha}_r=\omega_2$
${\mathsf D}_r$ ${\alpha}_1+2({\alpha}_2+\cdots+{\alpha}_{r-2})+{\alpha}_{r-1}+{\alpha}_r=\omega_2$
${\mathsf E}_6$ ${\alpha}_1+2{\alpha}_2+2{\alpha}_3+3{\alpha}_4+2{\alpha}_5+{\alpha}_6=\omega_2$
${\mathsf E}_7$ $2{\alpha}_1+2{\alpha}_2+3{\alpha}_3+4{\alpha}_4+3{\alpha}_5+2{\alpha}_6+{\alpha}_7=\omega_1$
${\mathsf E}_8$ $2{\alpha}_1+3{\alpha}_2+4{\alpha}_3+6{\alpha}_4+5{\alpha}_5+4{\alpha}_6+3{\alpha}_7+2{\alpha}_8=\omega_8$
${\mathsf F}_4$ ${\alpha}_1+2{\alpha}_2+3{\alpha}_3+2{\alpha}_4=\omega_4$
${\mathsf G}_2$ $2{\alpha}_1+{\alpha}_2=\omega_1$
: []{data-label="tab:hsr"}
Recall the condition $(\star)$ defined in the introduction: a dominant weight ${\lambda}$ satisfies $(\star)$ if, for every non-simply laced connected component ${\Delta}'\subset {\Delta}$, if $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root then it contains also the short root which is adjacent to a long simple root.
\[twin\] If ${\Delta}' \subset {\Delta}$ is a non-simply laced connected component, order the simple roots in ${\Delta}'= \{ {\alpha}_1, \ldots,
{\alpha}_r\}$ starting from the extreme of the Dynkin diagram of ${\Delta}'$ which contains a long root and denote ${\alpha}_q$ the first short root in ${\Delta}'$. If ${\lambda}$ is a dominant weight such that ${\alpha}_q\not \in \operatorname{Supp}({\lambda})$ and such that $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root, denote ${\alpha}_p$ the last long root which occurs in $\operatorname{Supp}({\lambda})\cap {\Delta}'$; for instance, if ${\Delta}'$ | 1 | member_11 |
is not of type ${\mathsf G}_2$, then the numbering is as follows: $$\begin{picture}(9000,1800)(2000,-900)
\put(0,0){\multiput(0,0)(3600,0){2}{\circle*{150}}\thicklines\multiput(0,0)(2500,0){2}{\line(1,0){1100}}\multiput(1300,0)(400,0){3}{\line(1,0){200}}}
\put(3600,0){\multiput(0,0)(3600,0){2}{\circle*{150}}\thicklines\multiput(0,0)(2500,0){2}{\line(1,0){1100}}\multiput(1300,0)(400,0){3}{\line(1,0){200}}}
\put(7200,0){\multiput(0,0)(1800,0){2}{\circle*{150}}\thicklines\multiput(0,-60)(0,150){2}{\line(1,0){1800}}\multiput(1050,0)(-25,25){10}{\circle*{50}}\multiput(1050,0)(-25,-25){10}{\circle*{50}}}
\put(9000,0){\multiput(0,0)(3600,0){2}{\circle*{150}}\thicklines\multiput(0,0)(2500,0){2}{\line(1,0){1100}}\multiput(1300,0)(400,0){3}{\line(1,0){200}}}
\put(-150,-700){\tiny $\alpha_1$}
\put(3450,-700){\tiny $\alpha_p$}
\put(8850,-700){\tiny $\alpha_q$}
\put(12450,-700){\tiny $\alpha_r$}
\end{picture}$$ The *little brother* of ${\lambda}$ with respect to ${\Delta}'$ is the dominant weight $${\lambda}_{{\Delta}'}^{\mathrm{lb}}= {\lambda}- \sum_{i=p}^q {\alpha}_i =
\left\{ \begin{array}{ll}
{\lambda}-\omega_1+\omega_2 & \textrm{ if $G$ is of type $\sf{G}_2$} \\
{\lambda}+ {\omega}_{p-1} - {\omega}_{p} + {\omega}_{q+1} & \textrm{ otherwise}
\end{array} \right.$$ where ${\omega}_i$ is the fundamental weight associated to ${\alpha}_i$ if $1{\leqslant}i {\leqslant}r$, while ${\omega}_0 = {\omega}_{r+1} = 0$. The set of the little brothers of ${\lambda}$ will be denoted by ${\mathrm{LB}}({\lambda})$; notice that ${\mathrm{LB}}({\lambda})$ is empty if and only if ${\lambda}$ satisfies condition $(\star)$ of Theorem A. For convenience, define $\overline {\mathrm{LB}}({\lambda})={\mathrm{LB}}({\lambda})\cup\{{\lambda}\}$, while if ${\Delta}$ is connected and non-simply laced set ${\lambda}^{\mathrm{lb}}= {\lambda}_{\Delta}^{\mathrm{lb}}$.
\[lem:eta\] Assume $G$ to be simple and let ${\lambda}\in {\Lambda}^+{\smallsetminus}\{0\}$. Denote $\eta$ the highest root of $\Phi$ if the latter is simply laced or the highest short root otherwise.
1. If ${\lambda}$ satisfies the condition $(\star)$ then $${V({{\lambda}})} \subset {V({\eta})} \otimes {V({{\lambda}})}.$$
2. If ${\lambda}$ does not satisfy the condition $(\star)$ and if ${\lambda}^{\mathrm{lb}}$ is the little brother of ${\lambda}$ then $${V({{\lambda}})} \subset {V({\eta})} \otimes {V({{\lambda}^{\mathrm{lb}}})}.$$
If ${\Delta}$ is simply | 1 | member_11 |
laced, then ${V({\eta})}\simeq{\mathfrak g}$ is the adjoint representation: in this case the claim follows straightforward by considering the map ${\mathfrak g}\otimes {V({{\lambda}})} \to {V({{\lambda}})}$ induced by the ${\mathfrak g}$-module structure on ${V({{\lambda}})}$, which is non-zero since ${\lambda}$ is non-zero.
Suppose now that ${\Delta}$ is not simply laced. If ${\lambda}$ satisfies condition $(\star)$, then by Lemma \[lem:traslazione\] it is enough to study the case ${\lambda}= \omega_{\alpha}$ where ${\alpha}$ is a short simple root:
*Type ${\mathsf B}_r$*: ${V({\omega_r})}\subset{V({\omega_1})}\otimes {V({\omega_r})}$.
*Type ${\mathsf C}_r$*: ${V({\omega_i})}\subset{V({\omega_2})}\otimes {V({\omega_i})}$, with $i<r$.
*Type ${\mathsf F}_4$*: ${V({\omega_3})}\subset{V({\omega_4})}\otimes {V({\omega_3})}$ and ${V({\omega_4})}\subset{V({\omega_4})}\otimes {V({\omega_4})}$.
*Type ${\mathsf G}_2$*: ${V({\omega_1})}\subset{V({\omega_1})}\otimes {V({\omega_1})}$.
If ${\lambda}$ does not satisfy condition $(\star)$, by Lemma \[lem:traslazione\] we can assume that ${\lambda}={\omega}_{\alpha}$ with ${\alpha}$ a long root:
*Type ${\mathsf B}_r$*: ${V({\omega_i})}\subset{V({\omega_1})}\otimes {V({\omega_{i-1}})}$, if $1<i<r$, and ${V({\omega_1})}\subset{V({\omega_1})}\otimes {V({0})}$.
*Type ${\mathsf C}_r$*: ${V({\omega_r})}\subset{V({\omega_2})}\otimes {V({\omega_{r-2}})}$.
*Type ${\mathsf F}_4$*: ${V({\omega_1})}\subset{V({\omega_4})}\otimes {V({\omega_4})}$ and ${V({\omega_2})}\subset{V({\omega_4})}\otimes {V({\omega_1+\omega_4})}$.
*Type ${\mathsf G}_2$*: ${V({\omega_2})}\subset{V({\omega_1})}\otimes {V({\omega_1})}$.
The above mentioned inclusion relations for tensor products are essentially known: let us treat the case of type ${\mathsf C}_r$ with ${\lambda}=\omega_i$ and $i<r$, the other cases are easier or can be checked directly.
Let $v_0$ be a highest weight vector of ${V({\omega_2})}$ and $w_0$ be a highest weight vector of ${V({\omega_i})}$. Let $f$ be | 1 | member_11 |
the following product (in the universal enveloping algebra $\mathfrak U(\mathfrak u^-)$) $$f=f_{{\alpha}_i}\cdots f_{{\alpha}_1}\cdot f_{{\alpha}_{i+1}}\cdots f_{{\alpha}_{r-1}}\cdot f_{{\alpha}_r}\cdots f_{{\alpha}_2},$$ and consider all the factorizations $f = p\cdot q$ such that $p,q \in\mathfrak U(\mathfrak u^-)$. If ${\beta}_1,\ldots,{\beta}_j\in{\Delta}$, set $$\,^\mathrm r(f_{{\beta}_1}\cdots f_{{\beta}_j})=(-1)^j2^\delta f_{{\beta}_j}\cdots f_{{\beta}_1},$$ where $\delta$ equals 0 (resp. 1) if $\alpha_i$ occurs an even (resp. odd) number of times in $\{{\beta}_1,\ldots,{\beta}_j\}$. Then it is easy to check that the vector $$\sum_{p\cdot q=f} p.v_0 \otimes \,^\mathrm r\!q.w_0$$ is a $U$-invariant vector in ${V({\omega_2})}\otimes{V({\omega_i})}$ of $T$-weight $\omega_i$.
If the Dynkin diagram of $G$ is not simply laced we will need some further properties of tensor products.
If ${\Delta}$ is connected but not simply laced, we will denote by $\alpha_S$ the short simple root that is adjacent to a long simple root $\alpha_L$; moreover, we will denote the associated fundamental weights by $\omega_S$ and $\omega_L$. Finally, define $\zeta$ as the sum of all simple roots and notice that $\omega_S + \zeta$ is dominant.
\[lem:zeta\] Let ${\lambda}$ be a non-zero dominant weight.
1. If $G$ is of type ${\mathsf F}_4$ or ${\mathsf C}_r$ ($r{\geqslant}3$) and if $\operatorname{Supp}({\lambda})$ contains a long root then $${V({{\lambda}+\omega_S})} \subset {V({\zeta + \omega_S})} \otimes {V({{\lambda}})}.$$
2. If $G$ is of type ${\mathsf G}_2$ | 1 | member_11 |
and if ${\lambda}$ does not satisfy $(\star)$ then $${V({{\lambda}+\omega_1})} \subset {V({\omega_2})} \otimes {V({{\lambda}^{\mathrm{lb}}})}.$$
3. If $G$ is of type ${\mathsf G}_2$ and if ${\alpha}_S \in \operatorname{Supp}({\lambda})$ then $${V({{\lambda}+\omega_1})} \subset {V({\omega_2})} \otimes {V({{\lambda}})}.$$
By Lemma \[lem:traslazione\] it is enough to check the statements for ${\lambda}=\omega_{\alpha}$ with ${\alpha}$ a long root in the first two cases and ${\alpha}={\alpha}_S$ in the last case.
*Type ${\mathsf C}_r$*: by Lemma \[lem:traslazione\] it is enough to check that ${V({\omega_{r-1}})}\subset {V({\omega_1})} \otimes {V({\omega_r})}$.
*Type ${\mathsf F}_4$*: we have ${\lambda}=\omega_1$ or ${\lambda}=\omega_2$ and $\omega_S+\zeta=\omega_1+\omega_4$.
*Type ${\mathsf G}_2$*: we have ${\lambda}=\omega_2$ and ${\lambda}^{\mathrm{lb}}=\omega_1$ in point (2) and ${\lambda}=\omega_1$ in point (3).
Normality and non-normality of $X_\Sigma$ {#ssez:normalita}
-----------------------------------------
We are now able to state the main theorem.
\[teo:normalita\]Let $\Sigma$ be a simple set of dominant weights and let ${\lambda}$ be its maximal element. The variety $X_\Sigma$ is normal if and only if $\Sigma \supset {\mathrm{LB}}({\lambda})$.
Theorem A stated in the introduction follows immediately by considering the case $\Sigma=\{{\lambda}\}$. The remaining part of this section will be devoted to the proof of Theorem \[teo:normalita\]. The general strategy will be based on Proposition \[prp:normalita\] and will proceed by induction on the dominance order of weights. The ingredients of this induction will | 1 | member_11 |